Posts in category: CMU

Figure 1: On LinkedIn, people are commonly connected with members from the same field who are likely to share skills and/or job preferences. Graph Convolutional Networks (GCNs) leverage this feature of the LinkedIn network and make better job recommendations by aggregating information from a member’s connections. For instance, to recommend a job to Member A, GCNs will aggregate information from Members B, C, and D who worked/are working in the same companies or have the same major.

TL;DR: Graph Convolutional Networks (GCNs) complement each node embedding with their neighboring node embeddings under a ‘homophily’ assumption, “connected nodes are relevant.” This leads to two critical problems when applying GCNs to real-world graphs: 1) scalability: numbers of neighboring nodes are sometimes too large to aggregate everything (e.g., Cristiano Ronaldo has 358 million connected accounts — his followers — on the Instagram’s member-to-member network),  2) low accuracy: nodes are sometimes connected with irrelevant nodes (e.g., people make connections with their personal friends who work in the totally different fields on LinkedIn). Here, we introduce a performance-adaptive sampling strategy for GCNs to solve both scalability and accuracy problems at once.

Graphs are ubiquitous. Any entities and interactions among them could be presented as graphs — nodes correspond to the individual entities and edges are generated between nodes when the corresponding entities have interactions between them. For instance, there are who-follows-whom graphs in social networks, who-pays-whom transaction networks in banking systems, and who-buys-which-products graphs in online malls. In addition to those originally graph-structured data, recently, few other computer science fields build new types of graphs by abstracting their concept (e.g., scene graphs in computer vision or knowledge graphs in NLP).

What are Graph Convolutional Networks?

As graphs contain rich contextual information — relationships among entities, various approaches have been proposed to include graph information in deep learning models. One of the most successful deep learning models combining graph information is Graph Convolutional Networks (GCNs) [1]. Intuitively, GCNs complement each node embeddings with their neighboring node embeddings, assuming neighboring nodes are relevant (we call this ‘homophily’), thus their information would help to improve a target node’s embedding. In Figure 1, on LinkedIn’s member-to-member networks, we refer to Member A‘s previous/current colleagues to make a job recommendation for Member A, assuming their jobs or skills are related to Member A‘s. GCNs aggregate neighboring node embeddings by borrowing the convolutional filter concept from Convolutional Neural Networks (CNNs) and replacing it with a first-order graph spectral filter.

When (h_i^{(l)}) denotes the hidden embedding of node (v_i) in the (l)-th layer, one-step convolution (we also call it one-step aggregation or one-step message-passing) in GCNs is described as follows:

[h^{(l+1)}_i = alpha left( frac{1}{N(i)}sum_{j=1}^{N}a(v_i, v_j)h^{(l)}_jW^{(l)} right), quad l = 0,dots,L-1 tag{1}label{1}]

where (a(v_i, v_j)) =1 when there is an edge from (v_i) to (v_j), otherwise 0; (N(i) = sum_{j=1}^{N} a(v_i, v_j)) is the degree of node (v_i); (alpha(cdot)) is a nonlinear function; (W^{(l)}) is the learnable transformation matrix. In short, GCNs average neighboring nodes (v_j)‘s embeddings (h_j^{(l)}), transform them with (W^{(l)}) and (alpha(cdot)), then update node (v_i)‘s embedding (h_i^{(l+1)}) using the aggregated and transformed neighboring embeddings. In practice, (h_i^{(0)}) is set with input node attributes and (h_i^{(L)}) is passed to an output layer specialized to a given downstream task. By stacking graph convolutional layers (L) times, (L)-layered GCNs complement each node embeddings with its neighboring nodes within (L) hops (Figure 2).

GCNs have garnered considerable attention as a powerful deep learning tool for representation learning of graph data. They demonstrate state-of-the-art performance on node classification, link prediction, and graph property prediction tasks. Currently, GCNs are one of most hot topics in graph mining and deep learning fields.

GCNs do not scale to large-scale real-world graphs.

However, when we adapt GCNs to million or billion-scaled real-world graphs (even trillion-scaled graphs for Google or Facebook), GCNs show a scalability issue. The main challenge comes from neighborhood expansion — GCNs expand neighbors recursively in the aggregation operations (i.e., convolution operations), leading to high computation and memory footprints. For instance, given a graph whose average degree is (d), (L)-layer GCNs access (d^L) neighbors per node on average (Figure 2). If the graph is dense or has many high degree nodes (e.g., Cristiano Ronaldo has 358 million followers on Instagram), GCNs need to aggregate a huge number of neighbors for most of the training/test examples.

The only way to alleviate this neighbor explosion problem is to sample a fixed number of neighbors in the aggregation operation, thereby regulating the computation time and memory usage. We first recast the original Equation (eqref{1}) as follows:

[h^{(l+1)}_i = alpha left( mathbb{E}_{jsim p(j|i)}[h^{(l)}_j]W^{(l)} right), quad l = 0,dots,L-1tag{2}label{2}]

where (p(j|i) = frac{a(v_i, v_j)}{N(i)}) defines the probability of sampling (v_j) given (v_i). Then we approximate the expectation by Monte-Carlo sampling as follows [2]:

[h^{(l+1)}_i = alpha left( frac{1}{k}sum_{jsim p(j|i)}^{k}h^{(l)}_jW^{(l)} right), quad l = 0,dots,L-1tag{3}label{3}]

where (k) is the number of sampled neighbors for each node. Now, we can regulate the GCNS’ computation costs using the sampling number (k).

GCN performance is affected by how neighbors are sampled, more specifically, how sampling policies — (q(j|i)), a probability of sampling a neighboring node (v_j) given a source node (v_i) — are defined. Various sampling policies [2-5] have been proposed to improve the GCN performance. Most of them target to minimize the variance caused by sampling (i.e., variance of the estimator (h^{(l+1)}_i) in Equation (eqref{3})). Variance minimization makes the aggregation of the sampled neighborhood to approximate the original aggregation of the full neighborhood. In other words, their sampling policies set the full neighborhood as the optimum they should approximate. But, is the full neighborhood the optimum?

Are all neighbors really helpful?

To answer this question, let’s go back to the motivation of the convolution operation in GCNs. When two nodes are connected with each other in graphs, we regard them as related to each other. Based on this ‘homophily’ assumption, GCNs aggregate neighboring nodes’ embeddings via the convolution operation to complement a target node’s embedding. So the convolution operation in GCNs will shine only when neighbors are informative for the task.

However, real-world graphs always contain unintended noisy neighbors. For example, in LinkedIn’s member-to-member networks, members might make connections not only with her colleagues working in the same field, but also with her family members or personal friends who may have totally different career paths (Figure 3). These family members or personal friends are uninformative for the job recommendation task. When their embeddings are aggregated into the target member’s embedding via the convolution operations, the recommendation quality becomes degraded. Thus, to fully enjoy benefits of the convolution operations, we need to filter out noisy neighbors.

How could we filter out noisy neighbors? We find the answer in the sampling policy: we sample neighbors only informative for a given task. How could we sample informative neighbors for the task? We train a sampler to maximize the target task’s performance (instead of minimizing sampling variance).

PASS: performance-adaptive sampling strategy for GCNs

We propose PASS, a performance-adaptive sampling strategy that optimizes a sampling policy directly for GCN performance. The key idea behind our approach is that we learn a sampling policy by propagating gradients of the GCN performance loss through the non-differentiable sampling operation. We first describe a learnable sampling policy function and how it operates in the GCN. Then we describe how to learn the parameters of the sampling policy by back-propagating gradients through the sampling operation.

Sampling policy: Figure 4 shows an overview of PASS. In the forward pass, PASS samples neighbors with its sampling policy (Figure 4(a)), then propagates their embeddings through the GCN (Figure 4(b)). Here, we introduce our parameterized sampling policy (q^{(l)}(j|i)) that estimates the probability of sampling node (v_j) given node (v_i) at the (l)-th layer. The policy (q^{(l)}(j|i)) is composed of two methodologies, importance (q^{(l)}_{imp}(j|i)) and random sampling (q^{(l)}_{rand}(j|i)) as follows:

[q^{(l)}_{imp}(j|i) = (W_scdot h^{(l)}_i)cdot(W_scdot h^{(l)}_j)\
q^{(l)}_{rand}(j|i) = frac{1}{N(i)}\
tilde{q}^{(l)}(j|i) = a_scdot[q^{(l)}_{imp}(j|i), quad q^{(l)}_{rand}(j|i)] \
q^{(l)}(j|i) = tilde{q}^{(l)}(j|i) / sum_{k=1}^{N(i)}tilde{q}^{(l)}(k|i)]

where (W_s) is a transformation matrix; (h^{(l)}_i) is the hidden embedding of node (v_i) at the (l)-th layer; (N(i)) is the degree of node (v_i); (a_s) is an attention vector; and (q^{(l)}(cdot|i)) is normalized to sum to 1. (W_s) and (a_s) are learnable parameters of our sampling policy, which will be updated toward performance improvement.

When a graph is well-clustered (i.e., less noisy neighbors), nodes are connected with all informative neighbors. Then random sampling becomes effective since its randomness helps aggregate diverse informative neighbors, thus preventing the GCN from overfitting. By capitalizing on both importance and random samplings, our sampling policy better generalizes across various graphs. Since we don’t know whether a given graph is well-clustered or not in advance, (a_s) learns which sampling methodology is more effective on a given task.

Training the Sampling Policy: after a forward pass with sampling, the GCN computes the performance loss (e.g., cross-entropy for node classification) then back-propagates gradients of the loss (Figure 4(c)). To learn a sampling policy maximizing the GCN performance, PASS trains the sampling policy based on gradients of the performance loss passed through the GCN. When (theta) denotes parameters ((W_s, a_s)) in our sampling policy (q^{(l)}_{theta}), we can write the sampling operation with (q^{(l)}_theta(j|i)) as follows:

[h^{(l+1)}_i = alpha_{W^{(l)}}(mathbb{E}_{jsim q^{(l)}_{theta}(j|i)}[h^{(l)}_j]), quad l = 0,dots,L-1]

Before being fed as input to the GCN transformation (alpha_{W^{(l)}})((cdot)), the hidden embeddings (h^{(l)}_j) go through an expectation operation (mathbb{E}_{jsim q^{(l)}_{theta}(j|i)})[(cdot)] under the sampling policy, which is non-differentiable. To pass gradients of the loss through the expectation, we apply the log derivative trick [6], widely used in reinforcement learning to compute gradients of stochastic policies. Then the gradient (nabla_theta mathcal{L}) of the loss (mathcal{L}) w.r.t. the sampling policy (q^{(l)}_{theta(j|i)}) is computed as follows:

Based on Theorem 4.1, we pass the gradients of the GCN performance loss to the sampling policy through the non-differentiable sampling operation and optimize the sampling policy for the GCN performance. You can find proof of the theorem in our original paper. PASS optimizes the sampling policy jointly with the GCN parameters to minimize the task performance loss, resulting in a considerable performance improvement.

Experimental Results

To examine the effectiveness of PASS, we run PASS on seven public benchmarks and two LinkedIn production datasets in comparison to four state-of-the-art sampling algorithms. GraphSage [2] samples neighbors randomly, while FastGCN [3], AS-GCN [4], and GCN-BS [5] do importance sampling with various sampling policy designs. Note that FastGCN, AS-GCN, and GCN-BS all target to minimize variance caused by neighborhood sampling. In Table 1, our proposed PASS method shows the highest accuracy among all baselines across all datasets on the node classification tasks. One interesting result is that GraphSage, which samples neighbors randomly, still shows good performance as compared to carefully designed importance sampling algorithms. The seven public datasets are well-clustered, which means most neighbors are relevant rather than noisy to a target node; thus there is not much room for improvement using importance sampling.

In the following experiment, we add noise to graphs. We investigate two different noise scenarios: 1) fake connections among existing nodes, and 2) fake neighbors with random feature vectors. These two scenarios are common in real-world graphs. The first “fake connection” scenario simulates connections made by mistake or unfit for the purpose (e.g., connections between family members in LinkedIn). The second “fake neighbor” scenario simulates fake accounts with random attributes used for fraudulent activities. For each node, we generate five true neighbors and five fake neighbors.

Table 2 shows that PASS consistently maintains high accuracy across all scenarios, while the performance of all other methods plummets. GraphSage, which gives the same sampling probability to true neighbors and fake neighbors, shows a sharp drop in accuracy. Other importance sampling-based methods, FastGCN, AS-GCN, and GCN-BS, also see a sharp drop in accuracy. They target to minimize sampling variance; thus they are likely to sample high-degree or dense-feature nodes, which help stabilize the variance, regardless of their relationship with the target node. Then, they all fail to distinguish fake neighbors from true neighbors. On the other hand, PASS learns which neighbors are informative or fake from gradients of the performance loss. These results show that the optimization of the sampling policy towards performance brings robustness to graph noise.

How does PASS learn which neighbors to sample?

PASS demonstrates superior performance in sampling informative neighbors for a given task. How could PASS learn whether a neighbor is informative for the task? How could PASS decide a certain sampling probability for each neighbor? To understand how PASS actually works, we dissect the back-propagation process of PASS. In Theorem 5.1., we find out that, during the back-propagation phase, PASS measures the alignment between (-dmathcal{L}/dh^{(l)}_i) and (h^{(l)}_j) and increases the sampling probability (q^{(l)}(j|i)) in proportion to this alignment. Proof of Theorem 5.1. can be found in the original paper.

This is an intuitively reasonable learning mechanism. GCNs train their parameters to move the node embeddings (h^{(l)}_i) in the direction that minimizes the performance loss (mathcal{L}), i.e., the gradient (-dmathcal{L} / dh^{(l)}_i). PASS promotes this process by sampling neighbors whose embeddings are aligned with the gradient (-dmathcal{L}/dh^{(l)}_i). When (h^{(l)}_i) is aggregated with the embedding (h^{(l)}_j) of a sampled neighbor aligned with the gradient, it moves in the direction that reduces the loss (mathcal{L}).

Let’s think about a simple example. In Figure 5, (h^{(l)}_3) is better aligned with (-dmathcal{L}/dh^{(l)}_2) than (h^{(l)}_5). Then PASS considers (v_3) more informative than (v_5) for (v_2) because node (v_3)’s embedding (h^{(l)}_3) helps (v_2)’s embedding (h^{(l)}_2) move in the direction (-dmathcal{L} / dh^{(l)}_2) that decreases the performance loss (mathcal{L}), while aggregating with node (v_5)’s embedding would move (h^{(l)}_2) in the opposite direction.

This reasoning process leads to two important considerations. First, it crystallizes our understanding of the aggregation operation in GCNs. The aggregation operation enables a node’s embedding to move towards its informative neighbors’ embeddings to reduce the performance loss. Second, this reasoning process shows the benefits of joint optimization of the GCN and sampling policy. Without optimizing the sampling policy jointly, the GCN depends solely on its parameters to move node embeddings towards the minimum performance loss. Joint optimization with the sampling policy helps the GCN to move the node embeddings more efficiently by aggregating with informative neighbors’ embeddings, leading to the minimum loss more efficiently.

PASS catches two birds, “accuracy” and “scalability”, with one stone. 

Today, we introduced a novel sampling algorithm PASS for graph convolutional networks. By sampling neighbors informative for task performance, PASS improves both the accuracy and scalability of CGNs. In nine different real-world graphs, PASS consistently outperforms state-of-the-art samplers, being up to 10.4% more accurate. In the presence of graph noises, PASS shows up to 53.1% higher accuracy than the baselines, proving its ability to read the context and distinguish the noises. By dissecting the back-propagation process, PASS explains why a neighbor is considered informative and assigned a high sampling probability.

In this era of big data, new graphs and tasks are generated every day. Graphs become bigger and bigger, and different tasks require different relational information within the graphs. By sampling informative neighbors adaptively for a given task, PASS allows GCNs to be applied on larger-scale graphs and a more diverse range of tasks. We believe that PASS can bring even more impact on a wider range of users across academia and industry in the future.

Links: paper, video, slide, code will be released at the end of 2021.

If you would like to reference this article in an academic publication, please use this BibTeX:

@inproceedings{yoon2021performance,
  title={Performance-Adaptive Sampling Strategy Towards Fast and Accurate Graph Neural Networks},
  author={Yoon, Minji and Gervet, Th{'e}ophile and Shi, Baoxu and Niu, Sufeng and He, Qi and Yang, Jaewon},
  booktitle={Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining},
  pages={2046--2056},
  year={2021}
}

References

1. Thomas N Kipf and Max Welling. 2016. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907 (2016).
2. Will Hamilton, Zhitao Ying, and Jure Leskovec. 2017. Inductive representation learning on large graphs. In Advances in neural information processing systems.
3. Jie Chen, Tengfei Ma, and Cao Xiao. 2018. Fastgcn: fast learning with graph convolutional networks via importance sampling. arXiv preprint arXiv:1801.10247 (2018).
4. Wenbing Huang, Tong Zhang, Yu Rong, and Junzhou Huang. 2018. Adaptive sampling towards fast graph representation learning. In Advances in neural information processing systems. 4558–4567
5. Ziqi Liu, Zhengwei Wu, Zhiqiang Zhang, Jun Zhou, Shuang Yang, Le Song, and Yuan Qi. 2020. Bandit Samplers for Training Graph Neural Networks. arXiv preprint arXiv:2006.05806 (2020).
6. Ronald J Williams. 1992. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning 8, 3-4 (1992), 229–256.

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Fig. 1: The safety envelope (in green) is an imaginary surface that separates the robot from all obstacles in its environment. As long as the robot never intersects the safety envelope, it is guaranteed to not collide with any obstacle. Our task is to estimate this envelope.

Safe navigation and obstacle detection

Consider the scene in Fig. 1 that a mobile robot wishes to navigate safely. The scene contains many obstacles such as walls, poles, and walking people. Obstacles could be arbitrarily distributed, their motion might be haphazard, and they may enter and leave the environment in an undetermined manner. This situation is commonly encountered in a variety of robotics tasks such as indoor and outdoor robot navigation, autonomous driving, and robot delivery. The robot must accurately and reliably detect all obstacles (static and dynamic) in the scene to avoid colliding with them and navigate safely. Therefore, it must estimate the safety envelope of the scene.

What is a safety envelope?

We define the safety envelope as an imaginary surface that separates the robot from all obstacles in its environment. As long as the robot never intersects the safety envelope, it is guaranteed to not collide with any obstacles! How can the robot accurately estimate the location of the safety envelope? Can it provide any guarantees about its ability to discover obstacles? In our recent paper published at RSS 2021, we answer these questions in the affirmative using a novel sensor called programmable light curtains.

What are light curtains?

A programmable light curtain is a 3D sensor recently invented at CMU. It can measure the depth of any user-specified 2D vertical surface (“curtain”) in the environment. A common strategy for 3D sensing is to use LiDARs. LiDARs have become ubiquitous in robotics and autonomous driving. However, they can be low resolution, expensive and slow. In contrast, light curtains are relatively inexpensive, faster, and of much higher resolution!

LiDAR (Velodyne)	Light curtain
Low resolution (128 rows)	High resolution (1280 rows)
Expensive (~$100,000)	Inexpensive (~$1,000)
Slow (5-20 Hz)	Fast (60 Hz)
No control required	User control required

Most importantly, light curtains are a controllable sensor: the user selects a vertical 2D surface, and the light curtain detects objects intersecting that surface. This is a fundamentally different sensing paradigm from LiDARs. LiDARs passively sense the entire scene without any user input. However, light curtains can be actively controlled by the user to focus their sensing capacity on specific regions of interest. While controllability is clearly a desirable feature, it also presents a unique challenge: light curtains require the user to select which locations to sense. How should we place light curtains to accurately estimate the safety envelope of a scene?

Random curtains can reliably discover obstacles

Suppose we have a scene with no prior knowledge of where the obstacles are. How should we place light curtains to discover them? Surprisingly, we answered this question by taking inspiration from heist films such as Ocean’s Twelve. In a scene from this movie shown in Fig. 3, the robber attempts to evade a museum’s security system consisting of randomly moving laser detectors. The robber needs to try extremely hard, literally bending over backward, to avoid intersecting the lasers. Although the robber managed to pull it off in the movie, it is clear that this would be virtually impossible in the real world.

Light curtains are nothing but moving laser detectors! Therefore, our insight is to place curtains at random locations in the scene. We refer to them as “random curtains”, as shown in Fig. 4. It turns out to be incredibly hard for (sufficiently large) obstacles to avoid getting detected by random curtains. We place random curtains to quickly discover unknown objects and estimate the safety envelope of the scene.

In the section on theoretical analysis of random curtains near the end of this blog post, we will present a novel analytical technique that actually computes the probability of random curtains intersecting and detecting obstacles. The analytical probabilities act as safety guarantees for our perception system towards detecting and avoiding obstacles.

Forecasting safety envelopes

Assume that we have already estimated the safety envelope in the current timestep. As objects move and the scene changes with time, we wish to estimate the safety envelope for the next timestep. In this case, it may be inefficient to explore the scene from scratch by randomly placing curtains. Instead, we use machine learning to train a neural network to forecast how the safety envelope will evolve in the next timestep. The inputs to the network are all light curtain measurements from previous timesteps. The output is the predicted change in the envelope’s position in the next timestep. We use DAgger [Ross et. al. 2011], a standard imitation learning algorithm, to train such a forecasting model from data. By predicting how the safety envelope will move, we can directly sense the locations where obstacles are anticipated to be and efficiently track the safety envelope.

Active light curtain placement pipeline

Our overall pipeline for placing light curtains to estimate and track the safety envelope is as follows. Given previous light curtain measurements, we train a neural network to forecast how the safety envelope of the scene will evolve in the next timestep. We then place light curtains to sense the predicted locations. At the same time, we place random light curtains to discover obstacles and update our predictions. Finally, the light curtain measurements are input back to the forecasting method, closing the loop.

Real-world results

Here is our method in action in the real world! The scene consists of multiple people walking in front of the light curtain device at various speeds and in different patterns. Our method, which combines learning-based forecasting and random curtain placement, tries to estimate the safety envelope of this dynamic scene at each timestep.

The middle video shows the light curtain placed at the estimated location of the safety envelope in black. It also shows a LiDAR point cloud in red, used only for visualization purposes (our method only uses light curtain measurements). The video on the right shows intersection points, i.e. the points detected by the light curtain when it intersects object surfaces in green. These are aggregated across multiple frames to visualize the motion of obstacles.

Brisk Walking

Relaxed Walking

Many people (structured walking)

Many people (haphazard, occluded walking)

Fast motion

In all of the above videos, the light curtain is able to accurately estimate the safety envelope and produce a large number of intersection points. Due to the guarantees of high detection probability, our method generalizes to a varying number of obstacles (one vs. two vs. five people), a large range of motion (relaxed vs. brisk vs. extremely fast and sudden motion), and different patterns of motion (structured vs. complicated and haphazard).

Theoretical analysis of random curtain detection

Previously, we mentioned that random light curtains can detect obstacles with a high probability. Can we perform any theoretical analysis to actually compute this probability? Can we compute the probability of a random curtain detecting an obstacle of a given shape, size, and location? If so, these probabilities can act as safety guarantees that help certify the ability of our perception system to detect and avoid obstacles. Before we begin analyzing random curtains, we must first understand how they are generated.

Constraint Graph and generating random curtains

In order to generate any light curtain, we need to account for the physical constraints of the light curtain device. These are encoded into a constraint graph (see Fig. 8). The nodes of the graph represent locations where the light curtain might be placed. The nodes are organized into “camera rays” indexed by (t in {1, dots, T}) from left to right. A light curtain is created by imaging one node per ray, from left to right. An edge exists between two nodes if they can be imaged consecutively.

What decides whether an edge exists between two nodes? The light curtain device contains a rotating mirror that redirects and shoots light into the scene. By specifying the angle of rotation, light can be beamed at the desired locations to be imaged. However, the mirror, being a physical device, has velocity and acceleration limits. Therefore, we add an edge between two nodes only if the mirror can rotate fast enough to image those two locations one after the other.

This means that any path in the constraint graph connecting the leftmost ray (t=1) to the rightmost ray (t=T) represents a feasible light curtain! Furthermore, random curtains can be generated by performing a random walk through the graph. The random walk is performed by starting from a node on the leftmost ray. In each iteration, a node on the next ray is randomly sampled among the neighbors of the current node from some probability distribution (e.g. uniform distribution). This process is repeated till the rightmost ray is reached. Fig. 9 shows examples of actual random curtains generated this way.

Computing detection probability using Dynamic Programming

Assume that we are given the shape, size, and location of an obstacle (blue–red shape in Fig. 10). Some random curtains will intersect and detect the obstacle (detections are shown in yellow), but other random curtains will miss the obstacle. Can we compute the probability of detection?

A naive approach would be to enumerate the set of all feasible light curtains and sum the probabilities of sampling those curtains that detect the object. Unfortunately, this is impractical because the number of feasible curtains is exponentially large! Another approach is to use Monte Carlo sampling for estimating the detection probability. In this method, we sample a large number of random curtains and output the fraction of the sampled curtains that detect the obstacle. While this approach is simple, we will show later that it requires a large number of samples to be drawn, and only produces stochastic estimates of the true probability.

Instead, we have developed an analytical and efficient approach to compute the detection probability, using dynamic programming. We first divide the overall problem into multiple sub-problems. Let us represent (mathbf{x}_t) to be a node on the (t)-th ray. Let us define (P_mathrm{det}(mathbf{x}_t)) to be the the probability that “sub-curtains” i.e. partial random curtains starting from node (mathbf{x}_t) and ending at a rightmost node will detect the obstacle. We wish to compute (P_mathrm{det}(cdot)) for every node in the constraint graph. Conveniently, these sub-problems satisfy a recursive relationship! If the obstacle is detected at node (mathbf{x}_t), (P_mathrm{det}(mathbf{x}_t)) is trivially equal to (1). If not, it is equal to the sum of detection probabilities of sub-curtains starting at (mathbf{x}_t)’s child nodes (mathbf{x}_{t+1}), weighted by the probability (P(mathbf{x}_t rightarrow mathbf{x}_{t+1})) of transitioning to (mathbf{x}_{t+1}). This is expressed by the equation below.

$$
P_mathrm{det}(mathbf{x}_{t}) =
begin{cases}
1 & text{if obstacle is detected at node } mathbf{x}_{t}\
sum_{mathbf{x}_{t+1}} P(mathbf{x}_{t} rightarrow mathbf{x}_{t+1})~P_mathrm{det}(mathbf{x}_{t+1}) & text{otherwise}
end{cases} tag{1}
$$

An example of random curtain analysis

Conclusion

We presented a method to estimate the safety envelope of a scene: a hypothetical surface that separates the robot from all obstacles in the environment. We used programmable light curtains, an actively controllable, resource-efficient sensor to directly estimate the safety envelope. Using a dynamic programming-based approach, we showed that random light curtains can discover obstacles with high-probability guarantees. We combined this with a machine learning-based forecasting method to efficiently track the safety envelope. This enables our robot perception system to accurately estimate safety envelopes, while our probabilistic guarantees help certify its accuracy and safety. This work is a step towards safe robot navigation using inexpensive controllable sensors.

Further reading

If you’re interested in more details, please check out the links to the full paper, the project website, talk, demo, and more!

Citation

This blog post is based on the following paper :

Siddharth Ancha, Gaurav Pathak, Srinivasa Narasimhan, and David Held.
Active Safety Envelopes using Light Curtains with Probabilistic Guarantees.
In Proceedings of Robotics: Science and Systems (RSS), July 2021.

Acknowledgements

Thanks to David Held, Srinivasa Narasimhan and Paul Liang for feedback on this post!

This material is based upon work supported by the National Science Foundation under Grants No. IIS-1849154, IIS-1900821 and by the United States Air Force and DARPA under Contract No. FA8750-18-C-0092. All opinions, findings, and conclusions or recommendations expressed in this post are those of the author(s) and do not necessarily reflect the views of Carnegie Mellon University, National Science Foundation, United States Air Force and DARPA.

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Imitation learning (IL) is the problem of finding a policy, (pi), that is as close as possible to an expert’s policy, (pi_E). IL algorithms can be grouped broadly into (a) online, (b) offline, and (c) interactive methods. We provide, for each setting, performance bounds for learned policies that apply for all algorithms, provably efficient algorithmic templates for achieving said bounds, and practical realizations that out-perform recent work.

From beating the world champion at Go (Silver et al.) to getting cars to drive themselves (Bojarski et al.), we’ve seen unprecedented successes in learning to make sequential decisions over the last few years. When viewed from an algorithmic viewpoint, many of these accomplishments share a common paradigm: imitation learning (IL). In imitation learning, one is given access to samples of expert behavior (e.g. moves chosen by Monte-Carlo Tree Search or steering angles recorded from an expert driver) and tries to learn a policy that mimics this behavior. Unlike reinforcement learning, imitation learning does not require careful tuning of a reward function, making it easier to scale to real-world tasks where one is able to gather expert behavior (like Go or driving). As we continue to apply imitation learning algorithms to safety-critical problems, it becomes increasingly important for us to have strong guarantees on their performance: while wrong steps in Go lead to a lost game at worst, mistakes of self-driving cars could result in far worse. In our ICML’21 Paper Of Moments and Matching: A Game Theoretic Framework for Closing the Imitation Gap, we provide bounds on how well any imitation algorithm can do, as well as provably efficient algorithms for achieving these bounds.

A Taxonomy of Imitation Learning Algorithms

Let’s focus on the problem of trying to teach a car to drive around a track from expert demonstrations. We instrument the car with cameras and sensors that measure the angle of the wheel and how hard the pedals are being pushed. Then, in terms of increasing requirements, the approaches we could take are:

• Offline: Have the expert drive laps, recording their states (camera images) and actions (pedals/wheel). Use your favorite supervised learning algorithm to regress from states to actions. This approach is called Behavioral Cloning.
• Online: Record expert states and actions. Then, have the car try to drive around the track and measure the delta between learner and expert trajectories. Train the policy to minimize this delta. GAIL is an algorithm that uses a discriminator network to measure this delta.
• Interactive: (0) Start with an empty dataset D. (1) Record the car driving a sample lap. (2) Ask the expert driver what they would have done for each recorded image. Append this data to D. (3) Regress over data in D. (4) Go back to 1. This approach is known as DAgger.

One of our key insights is that all three of these approaches can be seen as minimizing a sort of divergence from expert behavior. Concretely,

• Offline: We measure a divergence between learner and expert actions on states from expert demonstrations.
• Online: We measure a divergence between learner and expert trajectories.
• Interactive: We measure a divergence between learner and expert actions but on states from learner rollouts.

Also notice that as we transition from Offline to Online IL, we add a requirement of access to the environment or an accurate simulator. As we move from Online to Interactive IL, we also need access to a queryable expert. Let (pi) denote the policy, (pi_E) denote the expert’s policy, and (f) denote the divergence. We can visualize our thoughts thus far as:

With this divergence-minimizing perspective in mind, we’re able to introduce a unifying, game-theoretic perspective.

A Game-Theoretic Perspective on IL

A natural question at this point might be: what divergence should one use to measure the difference between learner and expert behavior? Examples abound in the literature: Kullback-Liebler? Wasserstein? Jensen-Shannon? Total Variation? Maximum Mean Discrepancy? Without prior knowledge about the problem, it’s really hard to say. For example, KL Divergence has a mode-covering effect — this means that if half the data was the expert swerving left to avoid a tree and half the data was them swerving right, the learner would learn to pick a point in the middle and drive straight into the tree!

If we’re not sure what divergence is the right choice, we can just minimize all of them, which is equivalent to minimizing a worst-case or adversarially-chosen one. Using (pi) and (pi_E) to denote the learner and expert policies, we can write out the optimization problem for each setting:

• Offline: $$ min_{pi} max_f mathbb{E}_{s, a sim pi_E}[f(s, pi(s)) – f(s, a)] $$
• Online: $$ min_{pi} max_f mathbb{E}_{s, a sim pi}[f(s, a)] – mathbb{E}_{s, a sim pi_E}[f(s, a)]$$
• Interactive: $$ min_{pi} max_f mathbb{E}_{s, a sim pi}[f(s, a) – f(s, pi_E(s))] $$

Each of these equations is in the form of a two-player zero-sum game between a learner (pi) and a discriminator (f). Two-player zero-sum games have been extensively studied in game theory, allowing us to use standard tools to analyze and solve them. Notice the similarity of the forms of these games — the only real difference is which state-action distributions the divergence is calculated between. Thus, we can view all three classes of imitation learning as solving a games with different classes of discriminators. This game-theoretic perspective is extremely powerful for a few reasons:

1. As we have access to more information (e.g. a simulator or a queryable expert), we’re able to evaluate more powerful discriminators. Minimizing these more powerful discriminators leads to tighter performance bounds. Specifically, we show that the difference between learner and expert performance for offline IL scales quadratically with the horizon of the problem, and linearly for online / interactive IL. Quadratically compounding errors translate to poor real-world performance. Thus, one perspective on our bounds is that they show that access to a simulator or a queryable expert is both necessary and sufficient for learning performant policies. We recommend checking out the full paper for the precise upper and lower bounds.
2. These performance bounds apply for all algorithms in each class — after all, you can’t do better by considering a more restrictive class of divergences. This means our bounds apply for a lot of prior work (e.g. Behavioral Cloning, GAIL, DAgger, MaxEnt IRL, …). Importantly, these bounds also apply for all non-adversarial algorithms: they’re just optimizing over a singleton discriminator class.
3. Our game-theoretic perspective also tells us that finding a policy that minimizes the worst-case divergence is equivalent to finding a Nash Equilibrium of the corresponding game, a problem we know how to solve provably efficiently for two-player zero-sum games. By solving a particular game, we inherit the performance bounds that come with the class of divergences considered.

Together, these three points tell us that a game-theoretic perspective allows us to unify imitation learning as well as efficiently find strong policies!

A Practical Prescription for each IL Setting

Let’s dig into how we can compute Nash equilibria efficiently in theory and in practice for all three games. Intuitively, a Nash equilibrium is a strategy for each player such that no player wants to unilaterally deviate. This means that each player is playing a best-response to every other player. We can find such an equilibrium by competing two types of algorithms:

• No-Regret: slow, stable, choosing best option over history.
• Best-Response: fast, choosing best option to last iterate of other player.

Classic analysis shows that having one player follow a no-regret algorithm and the other player follow a best-response algorithm will, within a polynomial number of iterations, converge to an approximate Nash equilibrium of the game. The intuition of the proof is that if player 1 is steadily converging to a strategy that performs well even when player 2 choses their strategy adversarially, player 1 can’t have much of an incentive to deviate, meaning their strategy must be half of a Nash equilibrium.

We’d like to emphasize the generality of this approach to imitation learning: you can plug in any no-regret algorithm and both our policy performance and efficiency results still hold. There’s a plethora of algorithms that can be developed from this no-regret reduction perspective!

We instantiate this general template into an implementable procedure for each setting. We compare our approaches against similar recent work. We plot the performance of our methods in orange. (J(pi)) refers to learner’s expected cumulative reward while (pi_E) in green is the expert’s performance. As stated above, our goal is for the learner to match expert performance.

Offline: We adopt a model similar to a Wasserstein GAN where the learner acts as the generator and the discriminator tries to distinguish between learner and expert actions on expert states. We set the learner’s learning rate to be much lower than that of the discriminator, simulating no-regret on policy vs. best response on divergence. We term this approach Adversarial Value-moment IL, or AdVIL. We find it to be competitive with recent work:

Online: We repurpose the replay buffer of an off-policy RL algorithm as the discriminator by assigning negative rewards to actions that don’t directly match the expert. We impute a reward of +1 for expert behavior and -1/k for learner behavior from a past round, where k is the round number. The slow-moving append-only replay buffer implements a no-regret algorithm against a policy that best-responds via RL at each round. We term this approach Adversarial Reward-moment IL, or AdRIL, and find that it can significantly outperform other online IL algorithms at some tasks:

Interactive: We modify DAgger to use adversarially chosen losses at each round instead of a fixed function. At each round, a discriminator network is trained between the last policy and the expert. Then, for all samples for that round, this discriminator network is used as the loss function. Then, just like DAgger, the learner minimizes loss over the history of samples and loss functions for all rounds. Thus, the learner is following a no-regret algorithm against a best-response by the discriminator. We call this algorithm DAgger-esque Qu-moment IL, or DAeQuIL.

To demonstrate the potential advantages of DAeQuIL over DAgger, we test out both algorithms on a simulated UAV forest navigation task, where the expert demonstrates a wide variety of tree avoidance behaviors (left). DAgger attempts to match the mean of these interactively queried action labels, leading to it learning to crash directly into the first tree it sees (center). DAeQuIL, on the other hand, is able to learn to swerve out of the way of trees and navigate successfully through the forest (right).

Parting Thoughts

We provide, for all three settings of imitation learning, performance bounds for learned policies, a provably efficient reduction to no-regret online learning, and practical algorithms. If you’re interested in learning more, I recommend you check out:

• Our paper.
• Our video explanations.
• Our code.

There are lots of interesting areas left to explore in imitation learning, including imitation from observation alone that would allow one to leverage the large corpus of instructional videos online to train robots. Another direction that we’re particularly excited about is mimicking expert behavior, even in the presence of unobserved confounders. Stay tuned!

DISCLAIMER: All opinions expressed in this post are those of the author and do not represent the views of CMU.

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Figure 1: Overview: To avoid out-of-distribution actions in the Offline Reinforcement Learning problem, we propose to implicitly constrain the policy by modifying the action space instead of enforcing explicit constraints. The policy is trained to output a latent action which will be passed into a decoder pretrained with the dataset. We demonstrate that our method provides competitive performance in both simulation and real-robot experiments.

Offline RL: Learning a policy from a static dataset

The goal of Reinforcement Learning (RL) is to learn to perform a task by interacting with the environment. It has achieved significant success in a lot of applications such as games and robotics. One major challenge in RL is that it requires a huge amount of interactive data collected in the environment to learn a policy. However, data collection is expensive and potentially dangerous in many real-world applications, such as robotics in safety-critical situations (e.g., around humans) or healthcare problems. Worse, RL algorithms also usually assume that the dataset used to update the policy comes from the current policy or its own training process.

To use data more wisely, we may consider Offline Reinforcement Learning. The goal of offline RL is to learn a policy from a static dataset of transitions without further data collection. Although we may still need a large amount of data, the assumption of static datasets allows more flexibility in data collection. For example, in robotics, we can include human demonstrations, reuse rollouts from previous experiments, and share data within the community. In this way, the dataset is more likely to be scaled up in size even when data collection is expensive.

One important feature of offline RL is that it requires no assumption about the performance of the policy that is used to collect the dataset. This is in contrast to behavior cloning, where we assume that the dataset is collected by an expert, so that we can directly “copy” the actions given states without reasoning about the future reward. In offline RL, the dataset could be collected by a policy (or several policies) with arbitrary performance.

At first glance, off-policy algorithms seem to be able to meet the above requirements. Off-policy algorithms save the agent’s interactions during training in a replay buffer and train the policy by sampling transitions from the replay buffer (Lillicrap 2015, Haarnoja 2018). However, as shown in previous work (Fujimoto 2018b), when we apply off-policy algorithms to a static dataset, the performance can be very poor due to out-of-distribution actions. In the off-policy algorithm, the Q-function is updated by the Bellman operator:

$$ mathcal{T} hat{Q}^pi(s_t, a_t) = mathbb{E}_{r_t, s_{t+1}}[r_t + gamma hat{Q}^pi(s_{t+1}, pi(s_{t+1}))] $$

As explained in Fujimoto (2018b), if the policy selects an action (pi(s_{t+1})) that is not included in this static dataset, then the term (hat{Q}^pi(s_{t+1},pi(s_{t+1}))) may have a large extrapolation error. The extrapolation error will be accumulated by the Bellman operator and exacerbated by the policy updates. These errors eventually lead to significant overestimation bias that can hurt the performance. This has always been an issue for Q-learning-based methods (Thrun 1993, Fujimoto 2018a), but it is especially problematic when applied on a static dataset because the policy is not able to try out and correct the overly-optimistic actions. The problem is more significant when the action space is large, such as continuous action space with high dimensions.

Objectives for Offline RL algorithms to avoid out-of-distribution actions

To fix the issue discussed above and to fully utilize the dataset, we need two objectives in offline RL. First, the policy should be constrained to select actions within the support of the dataset. The policy that represents the probability distribution (p(a|s)) of the dataset is usually called the behavior policy, denoted as (pi_B). We aim to maximize the return of the policy (G_t) subject to the constraint that (pi_B(a|s)) is larger than a threshold (epsilon):

$$ max_{asim pi(cdot|s)} mathbb{E}[G_t]$$
$$ s.t. pi_B(a|s) > epsilon$$

An illustration of this constraint is shown in Figure 3 below. Given a behavior policy ( pi_B(a|s) ) on the top figure, the agent policy ( pi) should only choose actions within the green region where ( pi_B(a|s) > epsilon). On the other hand, the constraint cannot be overly restrictive.  Specifically, the policy should not be affected by the density of (pi_B). In the example in Figure 3, the policy should have the flexibility to choose any action within the green region even if it deviates from the most probable action of (pi_B) and the “shape” of the distribution (pi) is very different from the shape of (pi_B).

Figure 4 below shows a more intuitive example. Consider an agent in a grid world. Suppose that the agent has a dataset of transitions marked as blue dots and arrows. The agent aims to find a path to get to the goal without the information of the other parts of the map. As shown on the left figure, it cannot select out-of-distribution actions because it might be dangerous. As shown on the right figure, if action (a_1) appears 10 times in the dataset, and action (a_0) appears 5 times in the dataset, it should not choose action (a_1) just because it appears more often in the dataset; as shown, this might be a suboptimal action for the task.

Previous methods struggled with achieving both of these objectives. For example, BCQ (Fujimoto 2018b) proposes to sample from the behavior policy, perturb around it and then take the action that maximizes the Q-value. This method will be restricted by the density of the behavior policy distribution if the sample size 𝑁 is not large enough. Another line of work uses explicit policy constraints in the optimization process (Jaques 2019, Kumar 2019, Wu 2019). They try to force the agent policy to be close to the behavior policy in terms of different measures of distance, such as KL or MMD (Figure 1). The explicit constraints create difficulties in the optimization and distance metrics such as KL will be affected by the density (see Appendix E in our paper). 

Proposed Method: Policy in Latent Action Space (PLAS)

In our paper, PLAS: Latent Action Space for Offline Reinforcement Learning (CoRL 2020), we propose a method that can satisfy both the objectives discussed above by simply modifying the action space of the policy – i.e., the policy will only select actions when ( pi_B(a|s) > epsilon), but will not be restricted by the density of the distribution ( pi_B(a|s)). In our method, we first model the behavior policy using a Conditional Variational Autoencoder (CVAE) as in previous work (Fujimoto 2018b, Kumar 2019). The CVAE is trained to reconstruct actions conditioned on the states. The decoder of the CVAE creates a mapping from the latent space to the action space. Instead of training a policy in the action space of the environment, we propose to learn a Policy in the Latent Action Space (PLAS) of the CVAE and then use the pretrained decoder to output an action in the original action space.

Using the above approach, we can naturally constrain the policy to select actions within the dataset because the action is chosen from the latent space. The prior of the latent variable of CVAE is set to be a normal distribution for simplicity, following the common practice. To constrain the latent policy from selecting actions that are too “far away” from this prior, we use a tanH activation at the output of the policy; this implicitly constrains the policy to select within a fixed number of standard deviations of the mean of the latent prior. It is important to note that the action output from the decoder should be conditioned on the state because we care about (pi_B(a|s) > epsilon) instead of (pi_B(a)>epsilon). This approach also satisfies the second objective because the policy can select any action within the latent space and will not be affected by the density of the behavior policy. 

Experiments: Cloth Sliding and D4RL benchmark

This modification over the action space can be built on top of any off-policy algorithm with either a stochastic or deterministic policy. In our experiment, we use TD3 (Fujimoto 2018a) with a deterministic latent policy. We evaluate our algorithm on a wide range of continuous control tasks, including a real robot experiment on cloth sliding and the D4RL benchmark. 

The task for the real-robot experiment is to slide along the edge of the cloth without dropping it. The dataset we use consists of a replay buffer from a previous experiment (around 7000 timesteps) and 5 episodes of expert trajectories (around 300 timesteps). Our method outperforms all the baselines and achieves similar performance as the expert.

Figure 6: Results from the real robot experiment. Left: Performance of PLAS on the cloth-sliding task. More videos can be found here. Right: Training curves of PLAS and the baselines. PLAS outperforms the other baselines and achieves similar performance as the expert. 

On the D4RL benchmark, our method also achieves consistent performance across a wide range of datasets with different environments and qualities despite its simplicity. We provide some of the qualitative and quantitative results below in Figures 7 and 8. Check out the full results on the D4RL benchmark in the paper. More videos can be found on our website.

To further analyze the result, we plot the estimation error of the learned Q-functions in Figure 9. During the evaluation, we compare the estimated Q-value of the state-action pairs with their true return from the rollouts. Our method has the lowest mean squared error (MSE) while the baselines have either more significant overestimation or underestimation.

As mentioned earlier in the objectives, our method focuses on avoiding out-of-distribution actions. In our experiment, we analyze the effect of out-of-distribution actions by introducing an additional component: we add a perturbation layer that is trained together with the latent policy, inspired by Fujimoto 2018b. The perturbation layer outputs a residual over the action output of the decoder. This allows the final policy output to deviate from the support of the dataset in a controlled way. More precisely, restricting the range of the output of the perturbation layer is essentially constraining the action output to be close to the dataset in terms of the L-infinity norm. In Figure 10, we plot the performance of our method with different ranges of allowed perturbation. We found that out-of-distribution actions introduced by the perturbation layer are usually harmful to datasets with high-quality rollouts such as the medium-expert datasets. However, it could be helpful for some of the random or medium datasets depending on the environment. The full analysis of the perturbation layer can be found in Appendix D. The results shed light on the disentangled contributions of in-distribution generalization and out-of-distribution generalization in offline reinforcement learning.

Conclusion and Discussion

We propose a simple and straightforward approach to offline RL: Policy in the Latent Action Space (PLAS). To summarize:

• Our approach naturally avoids out-of-distribution actions while allowing the flexibility for improvement over the performance of the dataset through implicit constraints.
• It achieves competitive performance in both simulated environments and a real robot experiment on cloth manipulation.
• We provided the analyses on Q-function estimation error and the separation of in-distribution vs. out-of-distribution generalization in Offline RL.

Please visit our website for the paper, code, and more videos.

Our method can be extended in different ways. First, it will benefit from a better generative model. For example, using normalizing flow to replace VAE could potentially lead to theoretical guarantees and better evaluation performance. Second, it can also be extended to allow better “out-of-distribution” generalization. We hope that our method will pave the way for future possibilities of applying reinforcement learning algorithms to real-world applications by using the static datasets more efficiently.

Acknowledgements

This material is based upon work supported by the United States Air Force and DARPA under Contract No. FA8750-18-C-0092, LG Electronics and the National Science Foundation under Grant No. IIS-1849154. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of United States Air Force and DARPA and the National Science Foundation.

DISCLAIMER: All opinions expressed in this post are those of the author and do not represent the views of Carnegie Mellon University.

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Fig 1: A formal relationship between interpretability and complexity. Going from left to right, we consider increasingly complex functions. As the complexity increases, local linear explanations can approximate the function only in smaller and smaller neighborhoods. These neighborhoods, in other words, need to become more and more disjoint as the function becomes more complex. Indeed, we quantify “disjointedness” of the neighborhoods via a term denoted by (rho_S) and relate it to the complexity of the function class, and subsequently, its generalization properties.

Introduction

There has been a growing interest in interpretable machine learning (IML), towards helping users better understand how their ML models behave. IML has become a particularly relevant concern especially as practitioners aim to apply ML in important domains such as healthcare [Caruana et al., ’15], financial services [Chen et al., ’18], and scientific discovery [Karpatne et al., ’17].

While much of the work in IML has been qualitative and empirical, in our recent ICLR21 paper, we study how concepts in interpretability can be formally related to learning theory. At a high level, the connection between these two fields seems quite natural. Broadly, one can consider there to be a trade-off between notions of a model’s interpretability and its complexity. That is, as a model’s decision boundary gets more complicated (e.g., compare a sparse linear model vs. a neural network) it is harder in a general sense for a human to understand how it makes decisions. At the same time, learning theory commonly analyzes relationships between notions of complexity for a class of functions (e.g., the number of parameters required to represent those functions) and the functions’ generalization properties (i.e., their predictive accuracy on unseen test data). Therefore, it is natural to suspect that, through some appropriate notion of complexity, one can establish connections between interpretability and generalization.

How do we establish this connection? First, IML encompasses a wide range of definitions and problems, spanning both the design of inherently interpretable models as well as post-hoc explanations for black-boxes (e.g. including but not limited to approximation [Riberio et al., ’16], counterfactual [Dhurandhar et al., ’18], and feature importance-based explanations [Lundberg & Lee, ’17]). In our work, we focus on a notion of interpretability that is based on the quality of local approximation explanations. Such explanations are a common and flexible post-hoc approach for IML, used by popular methods such as LIME [Riberio et al., ’16] and MAPLE [Plumb et al., ‘18] which we’ll briefly outline later in the blog. We then answer two questions that relate this notion of local explainability to important statistical properties of the model:

1. Performance Generalization: How can a model’s predictive accuracy on unseen test data be related to the interpretability of the learned model? 
2. Explanation Generalization: We look at a novel statistical problem that arises in a growing subclass of local approximation algorithms (such as MAPLE and RL-LIM [Yoon et al., ‘19]). Since these algorithms learn explanations by fitting them on finite data, the explanations may not necessarily fit unseen data well. Hence, we ask, what is the quality of those explanations on unseen data?

In what follows, we’ll first provide a quick introduction to local explanations. Then, we’ll motivate and answer our two main questions by discussing a pair of corresponding generalization guarantees that are in terms of how “accurate” and “complex” the explanations are. Here, the “complexity” of the local explanations corresponds to how large of a local neighborhood the explanations span (the larger the neighborhood, the lower the complexity — see Fig 1 for a visualization). For question (1), this results in a bound that roughly captures the idea that an easier-to-locally-approximate (f) enjoys better performance generalization. For question (2), our bound tells us that, when the explanations can accurately fit all the training data that fall within large neighborhoods, the explanations are likely to fit unseen data better. Finally, we’ll examine our insights in practice by verifying that these guarantees capture non-trivial relationships in a few real-world datasets. 

Local Explainability

Local approximation explanations operate on a basic idea: use a model from a simple family (like a linear model) to locally mimic a model from a more complex family (like a neural network model). Then, one can directly inspect the approximation (e.g. by looking at the weights of the linear model). More formally, for a given black-box model (f : mathcal{X} rightarrow mathcal{Y}), the explanation system produces at any input (x in mathcal{X}) , a ”simple” function (g_x(cdot) : mathcal{X} rightarrow mathcal{Y}) that approximates (f) in a local neighborhood around (x) . Here, we assume (f in mathcal{F}) (the complex model class) and (g_x in mathcal{G}_{text{local}} ) (the simple model class). 

As an example, here’s what LIME (Local Interpretable Model-agnostic Explanations) does. At any point (x), in order to produce the corresponding explanation (g_x), LIME would sample a bunch of perturbed points in the neighborhood around (x) and label them using the complex function (f(x)). It would then learn a simple function that fits the resulting dataset. One can then use this simple function to better understand how (f) behaves in the locality around (x).

Performance Generalization

Our first result is a generalization bound on the squared error loss of the function (f). Now, a typical generalization bound would look something like

$$text{TestLoss}(f) ≤ text{TrainLoss}(f) + sqrt{ frac{text{Complexity}(mathcal{F})}{text{# of training examples}}  }$$

where the bound is in terms of how well (f) fits the training data, and also how “complex” the function class (mathcal{F}) is. In practice though, (mathcal{F}) can often be a very complex class, rendering these bounds too large to be meaningful.

Yet while (mathcal{F}) is complex, what if the function (f) might itself have been picked from a subset of (mathcal{F}) that is in some way much simpler? For example, this is the case in neural networks trained by gradient descent [Zhang et al., ‘17, Neyshabur et al., ‘15]). Capturing this sort of simplicity could lead to more interesting bounds that (a) aren’t as loose and/or (b) shed insight into what meaningful  properties of (f) can influence how well it generalizes. While there are many different notions of simplicity that different learning theory results have studied, here we are interested in quantifying simplicity in terms of how “interpretable” (f) is, and relate that to generalization. 

To state our  result, imagine that we have a training set (S = {(x_i,y_i)}_{i=1}^{m})sampled from the data distribution (D). Then, we show the following bound on the test-time squared error loss:

$$underbrace{mathbb{E}_{D}[(f(x)-y)^2]}_{text{Test Loss}} leq underbrace{hat{mathbb{E}}_{S}[(f(x)-y)^2]}_{text{Train Loss}} + underbrace{mathbb{E}_{x sim D} mathbb{E}_{x’ sim N_{x}}[(g_{x’}(x) – f(x))^2]}_{text{Explanation Quality (MNF)}} + underbrace{rho_S cdot mathcal{R}(mathcal{G}_{local})}_{substack{text{Complexity of} \ text{Explanation System}}}.$$

Let’s examine these terms one by one.

Train Loss: The first term, as is typical of many generalization bounds, is simply the training error of (f) on (S). 

Explanation Quality (MNF): The second term captures a notion of how interpretable (f) is, measuring how accurate the set of local explanations (g) is with a quantity that we call the “mirrored neighborhood fidelity” (MNF). This metric is actually a slight modification of a standard notion of local explanation quality used in IML literature, called neighborhood fidelity (NF) [Riberio et al., ’16; Plumb et al., ‘18]. More concretely, we explain how MNF and NF are calculated below in Fig 2.

While notationally the differences between MNF and the standard notion of NF are slight, there are some noteworthy differences and potential (dis)advantages of using MNF over NF from an interpretability point of view. At a high level, we argue that MNF offers more robustness (when compared to NF) to any potential irregular behavior of (f) off the manifold of the data distribution. We discuss this in greater detail in the full paper.

Complexity of Explanation System: Finally, the third and perhaps the most interesting term measures how complex the infinite set of explanation functions ({g_x}_{x in mathcal{X}}) is. As it turns out, this system of explanations ({g_x}_{x in mathcal{X}}), which we will call (g), has a complexity that can be nicely decomposed into two factors. One factor, namely (mathcal{R}(mathcal{G}_{text{local}})), corresponds to the (Rademacher) complexity of the simple local class (mathcal{G}_{text{local}}), which is going to be a very small quantity, much smaller than the complexity of (mathcal{F}). Think of this factor as typically scaling linearly with the number of parameters for (mathcal{G}_{text{local}}) and also with the dataset size (m) as (1/sqrt{m}). The second factor is (rho_S), and is what we call the “neighborhood disjointedness” factor. This factor lies between ([1, sqrt{m}]) and is defined by how little overlap there is between the different local neighborhoods specified for each of the training datapoints in (S). When there is absolutely no overlap, (rho_S) can be as large as (sqrt{m}), but when all these neighborhoods are exactly the same, (rho_S) equals (1). 

Implications of the overall bound: Having unpacked all the terms, let us take a step back and ask: assuming that (f) has fit the training data well (i.e., the first term is small), when are the other two terms large or small? We visualize this in Fig 1. Consider the case where MNF can be made small by approximating (f) by (g) on very large neighborhoods (Fig 1 left). In such a case, the neighborhoods would overlap heavily, thus keeping (rho_S) small as well. Intuitively, this suggests good generalization when (f) is “globally simple”. On the other hand, when (f) is too complex, then we need to either shrink the neighborhoods or increase the complexity of (mathcal{G}_{text{local}}) to keep MNF small. Thus, one would either suffer from MNF or (rho_S) exploding, suggesting bad generalization. In fact, when (rho_S) is as large as (sqrt{m}), the bound is “vacuous” as the complexity term no longer decreases with the dataset size (m), suggesting no generalization!

Explanation Generalization

We’ll now turn to a different, novel statistical question which arises when considering a number of recent IML algorithms. Here we are concerned with how well explanations learned from finite data generalize to unseen data. 

To motivate this question more clearly, we need to understand a key difference between canonical and finite-sample-based IML approaches. In canonical approaches (e.g. LIME), at different values of (x’), the explanations (g_{x’}) are learned by fitting on a fresh bunch of points (S_{x’}) from a (user-defined) neighborhood distribution (N_{x’}) (see Fig 3, top). But a growing number of approaches such as MAPLE and RL-LIM learn their explanations by fitting (g_{x’}) on a “realistic” dataset (S) drawn from (D) (rather than from an arbitrary distribution) and then re-weighting the datapoints in (S) depending on a notion of their closeness to (x’) (see Fig 3, bottom).

Now, while the canonical approaches effectively train (g) on an infinite dataset (cup_{x’ in mathcal{X}} S_{x’}), recent approaches train (g) on only that finite dataset (S) (reused for every (x’)). 

Using a realistic (S) has certain advantages (as motivated in this blog post), but on the flip side, since (S) is finite, it can potentially result in a severe chance of overfitting (we visualize this in Fig 3 right). This makes it valuable to seek a guarantee on the approximation error of (g) on test data (“Test MNF”) in terms of its fit on the training data (S) (“Train MNF”). In our paper, we derive such a result below: 

$$underbrace{mathbb{E}_{x sim D} mathbb{E}_{x’ sim N_{x}}[(g_{x’}(x) – f(x))^2]}_{text{Test MNF}} leq underbrace{hat{mathbb{E}}_{x sim S} mathbb{E}_{x’ sim N_{x}}[(g_{x’}(x) – f(x))^2]}_{text{Train MNF}} + rho_S cdot mathcal{R}(mathcal{G}_{local}). $$

As before, what this bound implies is that when the neighborhoods have very little overlap, there is poor generalization. This indeed makes sense: if the neighborhoods are too tiny, any explanation (g_{x’}) would have been trained on a very small subset of (S) that falls within its neighborhood. Thus the fit of (g_{x’}) won’t generalize to other neighboring points.

Experiments

While our theoretical results offer insights that make sense qualitatively, we also want to make a case empirically that they indeed capture meaningful relationships between the quantities involved. Particularly, we explore this for neural networks trained on various regression tasks from the UCI suite and in the context of the “explanation generalization” bound. That is we learn explanations to fit a finite dataset (S) by minimizing Train MNF, and then evaluate what Test MNF is like. Here, there are two important relationships we empirically establish:

1. Dependence on (rho_S): Given that our bounds may be vacuous for large (rho_S=sqrt{m}), does this quantity actually scale well in practice (i.e. less than (O(sqrt{m})))? Indeed, we observe that we can find reasonably large choices of the neighborhood size ( sigma ) without causing Train MNF to become too high (somewhere around (sigma = 1) in Fig 4 bottom) and for which we can also achieve a reasonably small (rho_S approx O(m^{0.2})) (Fig 4 top).
2. Dependence on neighborhood size: Do wider neighborhoods actually lead to improved generalization gaps? From Fig 4 bottom, we do observe that as the neighborhood width increases, TrainMNF and TestMNF overall get closer to each other, indicating that the generalization gap decreases (Fig 4 bottom).

Conclusion

We have shown how a model’s local explainability can be formally connected to some of its various important statistical properties. One direction for future work is to consider extending these ideas to high-dimensional datasets, a challenging setting where our current bounds become prohibitively large. Another direction would be to more thoroughly explore these bounds in the context of neural networks, for which researchers are in search of novel types of bounds [Zhang et al., ‘17; Nagarajan and Kolter ‘19].

Separately, when it comes to the interpretability community, it would be interesting to explore the advantages/disadvantages of evaluating and learning explanations via MNF rather than NF. As discussed here, MNF appears to have reasonable connections to generalization, and as we show in the paper, it may also promise more robustness to off-manifold behavior.

To learn more about our work, check out our upcoming ICLR paper. Moreover, for a broader discussion about IML and some of the most pressing challenges in the field, here is a link to a recent white paper we wrote.

References

Jeffrey Li, Vaishnavh Nagarajan, Gregory Plumb, and Ameet Talwalkar, 2021, “A Learning Theoretic Perspective on Local Explainability“, ICLR 2021.

Rich Caruana, Yin Lou, Johannes Gehrke, Paul Koch, Marc Sturm, and Noemie Elhadad, 2015, “Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission.” ACM SIGKDD, 2015.

Chaofan Chen, Kancheng Lin, Cynthia Rudin, Yaron Shaposhnik, Sijia Wang, and Tong Wang, 2018, “An interpretable model with globally consistent explanations for credit risk.” NeurIPS 2018 Workshop on Challenges and Opportunities for AI in Financial Services: the Impact of Fairness, Explainability, Accuracy, and Privacy, 2018.

Anuj Karpatne, Gowtham Atluri, James H. Faghmous, Michael Steinbach, Arindam Banerjee, Auroop Ganguly, Shashi Shekhar, Nagiza Samatova, and Vipin Kumar, 2017, “Theory-guided data science: A new paradigm for scientific discovery from data.” IEEE Transactions on Knowledge and Data Engineering, 2017.

Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin, 2016, “Why should I trust you?: Explaining the predictions of any classifier.” ACM SIGKDD, 2016.

Scott M. Lundberg, and Su-In Lee, 2017, “A unified approach to interpreting model predictions.” NeurIPS, 2017.

Amit Dhurandhar, Pin-Yu Chen, Ronny Luss, Chun-Chen Tu, Paishun Ting, Karthikeyan Shanmugam, and Payel Das, 2018, “Explanations based on the Missing: Towards Contrastive Explanations with Pertinent Negatives” NeurIPS, 2018.

Jinsung Yoon, Sercan O. Arik, and Tomas Pfister, 2019, “RL-LIM: Reinforcement learning-based locally interpretable modeling”, arXiv 2019 1909.12367.

Gregory Plumb, Denali Molitor and Ameet S. Talwalkar, 2018, “Model Agnostic Supervised Local Explanations“, NeurIPS 2018.

Vaishnavh Nagarajan and J. Zico Kolter, 2019, “Uniform convergence may be unable to explain generalization in deep learning”, NeurIPS 2019

Behnam Neyshabur, Ryota Tomioka, Nathan Srebro, 2015, “In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning”, ICLR 2015 Workshop.

Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, Oriol Vinyals, 2017, “Understanding deep learning requires rethinking generalization”, ICLR’ 17.

Valerie Chen, Jeffrey Li, Joon Sik Kim, Gregory Plumb, and Ameet Talwalkar, 2021, “Towards Connecting Use Cases and Methods in Interpretable Machine Learning“, arXiv 2021 2103.06254.

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Figure 1. Due to feasibility or ethical requirements, a prediction model may only access a subset of the confounding factors that affect both the decision and outcome. We propose a procedure for learning valid counterfactual predictions in this setting.

In machine learning, we often want to predict the likelihood of an outcome if we take a proposed decision or action. A healthcare setting, for instance, may require predicting whether a patient will be re-admitted to the hospital if the patient receives a particular treatment. In the child welfare setting, a social worker needs to assess the likelihood of adverse outcomes if the agency offers family services. In such settings, algorithmic predictions can be used to help decision-makers. Since the prediction target depends on a particular decision (e.g., the particular medical treatment, or offering family services), we refer to these predictions as counterfactual.

In general, for valid counterfactual inference, we need to measure all factors that affect both the decision and the outcome of interest. However, we may not want to use all such factors in our prediction model. Some factors such as race or gender may be too sensitive to use for prediction. Some factors may be too complex to use when model interpretability is desired, or some factors may be difficult to measure at prediction time.

Child welfare example: The child welfare screening task requires a social worker to decide which calls to the child welfare hotline should be investigated. In jurisdictions such as Allegheny County, the social worker makes their decision based on allegations in the call and historical information about individuals associated with the call, such as their prior child welfare interaction and criminal justice history. Both the call allegations and historical information may contain factors that affect both the decision and future child outcomes, but the child welfare agency may be unable to parse and preprocess call information in real-time for use in a prediction system. The social worker would still benefit from a prediction that summarizes the risk based on historical information. Therefore, the goal is a prediction based on a subset of the confounding factors.

Healthcare example: Healthcare providers may make decisions based on the patient’s history as well as lab results and diagnostic tests, but the patient’s health record may not be in a form that can be easily input to a prediction algorithm.

How can we make counterfactual predictions using only a subset of confounding factors?

We propose a method for using offline data to build a prediction model that only requires access to the available subset of confounders at prediction time. Offline data is an important part of the solution because if we know nothing about the unmeasured confounders, then in general we cannot make progress. Fortunately, in our settings of interest, it is often possible to obtain an offline dataset that contains measurements of the full set of confounders as well as the outcome of interest and historical decision.

What is “runtime confounding?”

Runtime confounding occurs when all confounding factors are recorded in the training data, but the prediction model cannot use all confounding factors as features due to sensitivity, interpretability, or feasibility requirements. As examples,

• It may not be possible to measure factors efficiently enough for use in the prediction model but it is possible to measure factors offline with sufficient processing time. Child welfare agencies typically do record call allegations for offline processing.
• It may be undesirable to use some factors that are too sensitive or too complex for use in a prediction model.

Formally, let (V in mathbb{R}^{d_v}) denote the vector of factors available for prediction and (Z in mathbb{R}^{d_z}) denote the vector of confounding factors unavailable for prediction (but available in the training data). Given (V), our goal is to predict an outcome under a proposed decision; we wish to predict the potential outcome (Y^{A=a}) that we would observe under decision (a).

Prediction target: $$nu(v) := mathbb{E}[Y^{A=a} mid V = v] .$$ In order to estimate this hypothetical counterfactual quantity, we need assumptions that enable us to identify this quantity with observable data. We require three assumptions that are standard in causal inference:

Assumption 1: The decision assigned to one unit does not affect the potential outcomes of another unit.
Assumption 2: All units have some non-zero probability of receiving decision (a) (the decision of interest for prediction).
Assumption 3: (V,Z) describe all factors that jointly affect the decision and outcome.

These assumptions enable us to identify our target estimand as $$nu(v) = mathbb{E}[ mathbb{E}[Y mid A = a, V = v, Z =z] mid V =v].$$

This suggests that we can estimate an outcome model (mu(v,z) := mathbb{E}[Y mid A = a, V = v, Z =z]) and then regress the outcome model estimates on (V).

The simple plug-in (PL) approach:

1. Estimate the outcome model (mu(v,z)) by regressing (Y sim V, Zmid A = a). Use this model to construct pseudo-outcomes (hat{mu}(V,Z)) for each case in our training data.
2. Regress (hat{mu}(V,Z) sim V) to yield a prediction model that only requires knowledge of (V).

How does the PL approach perform?

• Yields valid counterfactual predictions under our three causal assumptions.
• Not optimal: Consider the setting in which (d_z >> d_v), for instance, in the child welfare setting where (Z) corresponds to the natural language in the hotline call. The PL approach requires us to efficiently estimate a more challenging high-dimensional target (mathbb{E}[Y mid A = a, V = v, Z =z]) when our target is a lower-dimensional quantity (nu(V)).

We can better take advantage of the lower-dimensional structure of our target estimand using doubly-robust techniques, which are popular in causal inference because they give us two chances to get our estimation right.

Our proposed doubly-robust (DR) approach

In addition to estimating the outcome model like the PL approach, a doubly-robust approach also estimates a decision model (pi(v,z) := mathbb{E}[mathbb{I}{A=a} mid V = v, Z =z]), which is known as the propensity model in causal inference. This is particularly helpful in settings where it is easier to estimate the decision model than the outcome model.

We propose a doubly-robust (DR) approach that also involves two stages:

1. Regress (Y sim V, Zmid A = a) to yield outcome model (hat{mu}(v,z)). Regress (mathbb{I}{A=a} sim V, Z) to yield decision model (hat{pi}(v,z)).
2. Regress $$frac{mathbb{I}{A=a}}{hat{pi}(V,Z)}(Y – hat{mu}(V,Z)) + hat{mu}(V,Z) sim V.$$

When does the DR approach perform well?

• When we can build either a very good outcome model or a very good decision model
• If both the decision model and outcome model are somewhat good

The DR approach can achieve oracle optimality–that is, it achieves the same regression error (up to constants) as an oracle with access to the true potential outcomes (Y^a).

We can see this by bounding the error of our method (hat{nu}) with the sum of the oracle error and a product of error terms on the outcome and decision models:

begin{align}
mathbb{E}[(hat{nu}(v) – nu(v))^2] ≲
& mathbb{E}[(tilde{nu}(v) – nu(v))^2] + \
& mathbb{E}[(hat{pi}(V,Z) -pi(V,Z))^2 mid V = v]mathbb{E}[(hat{mu}(V,Z) -mu(V,Z))^2 mid V = v].
end{align}

where (tilde{nu}(v)) denotes the function we would get in our second-stage estimation if we had oracle access to (Y^a).

So as long as we can estimate the outcome and decision models such that their product of errors is smaller than the oracle error, then the DR approach is oracle-efficient. This result holds for any regression method, assuming that we have used sample-splitting to learn (hat{nu}), (hat{mu}), and (hat{pi}).

While the DR approach has this desirable theoretical guarantee, in practice is it possible that the PL approach may perform better depending on the dimensionality of the problem.

How do I know which method I should use?

To determine which method will work best in a given setting, we provide an evaluation procedure that can be applied to any prediction method to estimate its mean-squared error. Under our three causal assumptions, the prediction error of a model (hat{nu}) is identified as

$$mathbb{E}[(Y^a – hat{nu}(V))^2] = mathbb{E}[mathbb{E}[(Y-hat{nu}(V)^2 mid V, Z, A = a]].$$

Defining the error regression (eta(v,z) = mathbb{E}[(Y-hat{nu}(V))^2 mid V = v, Z =a, A = a] ), we propose the following doubly-robust estimator for the MSE on a validation sample of (n) cases:

$$frac{1}{n} sum_{i=1}^n left[ frac{mathbb{I}{A_i = a }}{hat{pi}(V_i, Z_i)} left( (Y_i -hat{nu}(V_i))^2 – hat{eta}(V_i, Z_i) right) + hat{eta}(V_i, Z_i) right] .$$

Under mild assumptions, this estimator is (sqrt{n}) consistent, enabling us to get error estimates with confidence intervals.

DR achieves lowest MSE in synthetic experiments

We perform simulations on synthetic data to show how the level of confounding and dimensionalities of (V) and (Z) determine which method performs best. Synthetic experiments enable us to evaluate the methods on the ground-truth counterfactual outcomes. We compare the PL and DR approaches to a biased single-stage approach that estimates (mathbb{E}[Y mid V, A =a]), which we refer to as the treatment-conditional regression (TCR) approach.

In the left-hand panel above, we compare the method as we vary the amount of confounding. When there is no confounding ((k_z = 0)), the TCR approach performs best as expected. Under no confounding, the TCR approach is no longer biased and efficiently estimates the target of interest in one stage. However, as we increase the level of confounding, the TCR performance degrades faster than the PL and DR methods. The DR method performs best under any non-zero level of confounding.

The right-hand panel compares the methods as we vary the dimensionality of our predictors. We hold the total dimensionality of ((V, Z)) fixed at (500) (so (d_z = 500 – d_v)). The DR approach performs best across the board, and the TCR approach performs well when the dimensionality is low because TCR avoids the high-dimensional second stage regression. However, this advantage disappears as (d_v) increases. The gap between the PL and DR methods is largest for low (d_v) because the DR method is able to take advantage of the lower dimensional target. At high (d_v) the PL error approaches the DR error.

DR is comparable to PL in a real-world task

We compare the methods on a real-world child welfare screening task where the goal is to predict the likelihood that a case will require services under the decision “screened in for investigation” using historical information as predictors and controlling for confounders that are sensitive (race) and hard to process (the allegations in the call). Our dataset consists of over 30,000 calls to the child welfare hotline in Allegheny County, PA. We evaluate the methods using our proposed real-world evaluation procedure since we do not have access to the ground-truth outcomes for cases that were not screened in for investigation.

We find that the DR and PL approach perform comparably on this task, both outperforming the TCR method.

Recap

• Runtime confounding arises when it is undesirable or impermissible to use some confounding factors in the prediction model.
• We propose a generic procedure to build counterfactual predictions when the factors are available in offline training data.
• In theory, our approach is provably efficient in the oracle sense
• In practice, we recommend building the DR, PL, and TCR approaches and using our proposed evaluation scheme to choose the best performing model.
• Our full paper is available in the Proceedings of NeurIPS 2020.

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Figure 1. A toy linear regression example illustrating Tilted Empirical Risk Minimization (TERM) as a function of the tilt hyperparameter (t). Classical ERM ((t=0)) minimizes the average loss and is shown in pink. As (t to -infty ) (blue), TERM finds a line of best fit while ignoring outliers. In some applications, these ‘outliers’ may correspond to minority samples that should not be ignored. As (t to + infty) (red), TERM recovers the min-max solution, which minimizes the worst loss. This can ensure the model is a reasonable fit for all samples, reducing unfairness related to representation disparity.

In machine learning, models are commonly estimated via empirical risk minimization (ERM), a principle that considers minimizing the average empirical loss on observed data. Unfortunately, minimizing the average loss alone in ERM has known drawbacks—potentially resulting in models that are susceptible to outliers, unfair to subgroups in the data, or brittle to shifts in distribution. Previous works have thus proposed numerous bespoke solutions for these specific problems.

In contrast, in this post, we describe our work in tilted empirical risk minimization (TERM), which provides a unified view on the deficiencies of ERM (Figure 1). TERM considers a modification to ERM that can be used for diverse applications such as enforcing fairness between subgroups, mitigating the effect of outliers, and addressing class imbalance—all in one unified framework.

What is Tilted ERM (TERM)?

Empirical risk minimization is a popular technique for statistical estimation where the model, (theta in R^d), is estimated by minimizing the average empirical loss over data, ({x_1, dots, x_N}):

$$overline{R} (theta) := frac{1}{N} sum_{i in [N]} f(x_i; theta).$$

Despite its popularity, ERM is known to perform poorly in situations where average performance is not an appropriate surrogate for the problem of interest. In our work (ICLR 2021), we aim to address deficiencies of ERM through a simple, unified framework—tilted empirical risk minimization (TERM). TERM encompasses a family of objectives, parameterized by the hyperparameter (t):

$$widetilde{R} (t; theta) := frac{1}{t} logleft(frac{1}{N} sum_{i in [N]} e^{t f(x_i; theta)}right). $$

TERM recovers ERM when (t to 0). It also recovers other popular alternatives such as the max-loss ((t to +infty)) and min-loss ((t to -infty)). While the tilted objective used in TERM is not new and is commonly used in other domains,¹ it has not seen widespread use in machine learning.

In our work, we investigate tilting by: (i) rigorously studying properties of the TERM objective, and (ii) exploring its utility for a number of ML applications. Surprisingly, we find that this simple and general extension to ERM is competitive with state-of-the-art, problem-specific solutions for a wide range of problems in ML. 

TERM: Properties and Interpretations

Given the modifications that TERM makes to ERM, the first question we ask is: What happens to the TERM objective when we vary (t)? Below we explore properties of TERM with varying (t) to better understand the potential benefits of (t)-tilted losses. In particular, we find:

• TERM with varying (t)’s reweights samples to magnify/suppress outliers (as in Figure 1).
• TERM smoothly moves between traditional ERM (pink line), the max-loss (red line), and min-loss (blue line), and can be used to trade-off between these problems. 
• TERM approximates a popular family of quantile losses (such as median loss, shown in the orange line) with different tilting parameters. Quantile losses have nice properties but can be hard to directly optimize.

Next, we discuss these properties and interpretations in more detail.

Varying (t) reweights the importance of outlier samples

First, we take a closer look at the gradient of the t-tilted loss, and observe that the gradients of the t-tilted objective (widetilde{R}(t; theta)) are of the form:

$$nabla_{theta} widetilde{R}(t; theta) = sum_{i in [N]} w_i(t; theta) nabla_{theta} f(x_i; theta), text{where } w_i propto e^{tf(x_i; theta)}.$$

This indicates that the tilted gradient is a weighted average of the gradients of the original individual losses, and the weights are exponentially proportional to the loss values. As illustrated in Figure 1, for positive values of (t), TERM will thus magnify outliers (samples with large losses), and for negative t’s, it will suppress outliers by downweighting them.

TERM offers a trade-off between the average loss and min-/max-loss

Another perspective on TERM is that it offers a continuum of solutions between the min and max losses. As (t) goes from 0 to (+infty), the average loss will increase, and the max-loss will decrease (going from the pink star to the red star in Figure 2), smoothly trading average-loss for max-loss. Similarly, for (t<0), the solutions achieve a smooth tradeoff between average-loss and min-loss. Additionally, as (t >0) increases, the empirical variance of the losses across all samples also decreases. In the applications below we will see how these properties can be used in practice.

TERM solutions approximate superquantile methods

Finally, we note a connection to another popular variant on ERM: superquantile methods. The (k)-th quantile loss is defined as the (k)-th largest individual loss among all samples, which may be useful for many applications. For example, optimizing for the median loss instead of mean may be desirable for applications in robustness, and the max-loss is an extreme of the quantile loss which can be used to enforce fairness. However, minimizing such objectives can be challenging especially in large-scale settings, as they are non-smooth (and generally non-convex). The TERM objective offers an upper bound on the given quantile of the losses, and the solutions of TERM can provide close approximations to the solutions of the quantile loss optimization problem.

[Note] All discussions above assume that the loss functions belong to generalized linear models. However, we empirically observe competitive performance when applying TERM to broader classes of objectives, including deep neural networks. Please see our paper for full statements and proofs.

TERM Applications

TERM is a general framework applicable to a variety of real-world machine learning problems. Using our understanding of TERM from the previous section, we can consider ‘tilting’ at different hierarchies of the data to adjust to the problem of interest. For instance, one can tilt at the sample level, as in the linear regression toy example. It is also natural to perform tilting at the group level to upweight underrepresented groups. Further, we can tilt at multiple levels to address practical applications requiring multiple objectives. In our work, we consider applying TERM to the following problems:

• [negative (t)’s]: robust regression, robust classification, mitigating noisy annotators
• [positive (t)’s]: handling class imbalance, fair PCA, variance reduction for generalization
• [hierarchical tilting with (t_1<0, t_2>0)]: jointly addressing robustness and fairness

For all applications considered, we find that TERM is competitive with or outperforms state-of-the-art, problem-specific tailored baselines. In this post, we discuss three examples on robust classification with (t<0), fair PCA with (t>0), and hierarchical tilting. 

Robust classification 

Crowdsourcing is a popular technique for obtaining data labels from a large crowd of annotators. However, the quality of annotators varies significantly as annotators may be unskilled or even malicious. Thus, handling a large amount of noise is essential for the crowdsourcing setting. Here we consider applying TERM ((t<0)) to the application of mitigating noisy annotators.

Specifically, we explore a common benchmark—taking the CIFAR10 dataset and simulating 100 annotators where 20 of them are always correct and 80 of them assign labels uniformly at random. We use negative (t)’s for annotator-level tilting, which is equivalent to assigning annotator-level weights based on the aggregate value of their loss.

Figure 3 demonstrates that our approach performs on par with the oracle method that knows the qualities of annotators in advance. Additionally, we find that the accuracy of TERM alone is 5% higher than that reported by previous approaches which are specifically designed for this problem.

Fair principal component analysis (PCA)

While the previous application explored TERM with (t<0), here we consider an application of TERM with positive (t)’s to fair PCA. PCA is commonly used for dimension reduction while preserving useful information of the original data for downstream tasks. The goal of fair PCA  is to learn a projection that achieves similar (or the same) reconstruction errors across subgroups. 

Applying standard PCA can be unfair to underrepresented groups. Figure 4 demonstrates that the classical PCA algorithm results in a large gap in the representation quality between two groups (G1 and G2).

To promote fairness, previous methods have proposed to solve a min-max problem via semidefinite programming, which scales poorly with the problem dimension. We apply TERM to this problem, reweighting the gradients based on the loss on each group. We see that TERM with a large (t) can recover the min-max results where the resulting losses on two groups are almost identical. In addition, with moderate values of (t), TERM offers more flexible tradeoffs between performance and fairness by reducing the performance gap less aggressively. 

Solving compound issues: multi-objective tilting

Finally, we note that in practice, multiple issues can co-exist in the data, e.g., we may have issues with both class imbalance and label noise. In these settings we can adopt hierarchical TERM as described previously to address compound issues. Depending on the application, one can choose whether to apply tilting at each level (e.g., possibly more than two levels of hierarchies exist in the data), and at either direction ((t>0) or (t<0)). For example, we can perform negative tilting at the sample level within each group to mitigate outlier samples, and perform positive tilting across all groups to promote fairness.

We test this protocol on the HIV-1 data with logistic regression. In Table 1 below, we find that TERM is superior to all baselines which perform well in their respective problem settings (only showing a subset here) when considering noisy samples and class imbalance simultaneously. 

More broadly, the idea of tilting can be applied to other learning problems, like GAN training, meta-learning, and improving calibration and generalization for deep learning. We encourage interested readers to view our paper, which explores a more comprehensive set of applications.

[Solving TERM] Wondering how to solve TERM? In our work, we discuss the properties of the objective in terms of its smoothness and convexity behavior. Based on that, we develop both batch and (scalable) stochastic solvers for TERM, where the computation cost is within 2(times) of standard ERM solvers. We describe these algorithms as well as their convergence guarantees in our paper.

Discussion

Our work explores tilted empirical risk minimization (TERM), a simple and general alternative to ERM, which is ubiquitous throughout machine learning. Our hope is that the TERM framework will allow machine learning practitioners to easily modify the ERM objective to handle practical concerns such as enforcing fairness amongst subgroups, mitigating the effect of outliers, and ensuring robust performance on new, unseen data. Critical to the success of such a framework is understanding the implications of the modified objective (i.e., the impact of varying (t)), both theoretically and empirically. Our work rigorously explores these effects—demonstrating the potential benefits of tilted objectives across a wide range of applications in machine learning.

Interested in learning more?

• Full paper here
• Code here

ACKNOWLEDGEMENT

Thanks to Maruan Al-Shedivat, Ahmad Beirami, Virginia Smith, and Ivan Stelmakh for feedback on this blog post.

Footnotes

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This week researchers from all across computer science, social sciences and humanities are gathering for the flagship conference of the emerging field of Fairness, Accountability and Transparency in algorithmic systems: FAccT. FAccT (previously FAT*) is dedicated to the inherent risks that come with the increasing adoption of data-driven algorithmic decision making systems in socially consequential domains such as policing, criminal justice, health care and education. The conference was formed as a venue for the increasing volume of work in this area in 2018 and has since become one of the top venues in the study of societal impacts of machine learning – submissions have more than quadrupled since the inaugural conference!

Now in its 4th year, the fully-virtual event spans 82 paper presentations from 15 different countries across 14 time zones as well as 13 tutorials, a doctoral consortium and 10 CRAFT sessions aimed at Critiquing and Rethinking Accountability, Fairness and Transparency. Complementing paper presentations and tutorials, the CRAFT sessions aim for interaction between participants with different backgrounds including academics, journalists, advocates, activists, educators and artists with the idea of reflection and discussion of the field from a more holistic perspective.

Many influential papers have been published at FAccT even within these first few years of the conference. Examples include Joy Buolamwini and Timnit Gebru’s 2018 study on Gender Shades in which the authors uncover significantly higher error rates in commercial gender classification for darker-skinned females which led companies to adjust their algorithms and sparked a wider discussion of similar problems in computer vision. Leading up to this year’s conference, the paper ‘On the Dangers of Stochastic Parrots: Can Language Models Be Too Big?’ coming out of Google and the University of Washington has gotten much attention in the wider field as it led to the firing of Timnit Gebru as co-lead of the Ethical AI team from Google leaving both room for speculations and sparking a discussion on the future of AI ethics research in private companies.

As one of the main contributing institutions, Carnegie Mellon University is proud to present 10 papers and one tutorial at this year’s conference. Contributions are made from all across campus with authors from the Machine Learning Department, the Department of Statistics and Data Science, the Institute for Software Research, the Computer Science Department, Heinz College of Public Policy, and the Philosophy Department. Several of the studies focus on auditing existing systems in the context of predictive policing [4], image representations learned in an unsupervised manner [5], or the use of mobility data for Covid-19 policy [6]. Others propose new algorithmic solutions to analyze the allocation of opportunities for intergenerational mobility [1], post-process predictions in risk assessment [2], examine the equity of cash bail decisions [3], or understand the fairness implications of leave-one-out training data [8]. The authors of [7] focus on disparity amplification avoidance under different world views and fairness notions, while [9] introduce Value Cards, an educational toolkit for teaching the societal impacts of machine learning. Finally, the authors of [10] provide counternarratives on data sharing in Africa using a storytelling approach based on a series of interviews. We give a short description of each of the papers along with the session times at the conference and links to the preprints below.

Papers

[1] Allocating Opportunities in a Dynamic Model of Intergenerational Mobility
Hoda Heidari (Carnegie Mellon University), Jon Kleinberg (Cornell University)
Session: March 8, 22:00 – 23:45 UTC 
Tags: Algorithm Development, Fairness
Summary: The authors develop a model for analyzing the allocation of opportunities for intergenerational mobility such as higher education and find that purely payoff-maximizing objectives can still lead to a form of socioeconomic affirmative action in the optimal allocation.

[2] Fairness in Risk Assessment Instruments: Post-Processing to Achieve Counterfactual Equalized Odds
Alan Mishler (Carnegie Mellon University), Edward Kennedy (Carnegie Mellon University), Alexandra Chouldechova (Carnegie Mellon University)
Session: March 10, 20:00 – 21:30 UTC
Tags: Algorithm Development, Causality, Evaluation, Fairness
Summary: The authors develop a method to post-process existing binary predictors used in risk assessment, e.g. for recidivism prediction, to satisfy approximate counterfactual equalized odds. They discuss the convergence rate to an optimal fair predictor and propose doubly robust estimation of the risk and fairness properties of a fixed post-processed predictor.

[3] A Bayesian Model of Cash Bail Decisions
Joshua Williams (Carnegie Mellon University), Zico Kolter (Carnegie Mellon University)
Session: March 8, 20 – 21:45 UTC
Tags: Algorithm Development, Data, Fairness, Law & Policy
Summary: The authors create a hierarchical Bayesian model of cash bail assignments to analyze fairness between racial groups while overcoming the problem of infra-marginality. Results on 50 judges uniformly show that they are more likely to assign cash bail to black defendants than to white defendants given the same likelihood of skipping a court appearance.

[4] The effect of differential victim crime reporting on predictive policing systems
Nil-Jana Akpinar (Carnegie Mellon University), Maria De-Arteaga (University of Texas at Austin), Alexandra Chouldechova (Carnegie Mellon University)
Session: March 8, 20:00 – 21:45 UTC
Tags: Auditing, Data, Evaluation
Summary: The authors audit place-based predictive policing algorithms trained on victim crime reporting data and find that geographical bias arises when victim crime reporting rates vary within a city. This result requires no use of arrest data or data from police initiated contact.

[5] Image Representations Learned With Unsupervised Pre-Training Contain Human-like Biases
Ryan Steed (Carnegie Mellon University), Aylin Caliskan (George Washington University)
Session: March 8, 12:00 – 13:45 UTC
Tags: Computer Vision, Data, Evaluation, Fairness, Humanistic Theory & Critique
Summary: The authors develop a method for quantifying biased associations between representations of social concepts and attributes in images using image representations learned in an unsupervised manner. The results closely match hypotheses about intersectional bias from social psychology and suggest that machine learning models can automatically learn bias from the way people are stereotypically portrayed on the web.

[6] Leveraging Administrative Data for Bias Audits: Assessing Disparate Coverage with Mobility Data for COVID-19 Policy
Amanda Coston (Carnegie Mellon University), Neel Guha (Stanford University), Derek Ouyang (Stanford University), Lisa Lu (Stanford University), Alexandra Chouldechova (Carnegie Mellon University), Daniel E. Ho (Stanford University)
Session: March 9, 14:00 – 15:45 UTC
Tags: Auditing, Data, Evaluation
Summary: The authors audit the use of smartphone-based mobility data for COVID-19 policy by leveraging administrative voter roll data in the absence of demographic information. Their results suggest that older and non-white voters are less liekely to be captured by mobility data which can disproportionally harm these groups if allocation of public health resources is based on such data sets.

[7] Avoiding Disparity Amplification under Different Worldviews
Samuel Yeom (Carnegie Mellon University), Michael Carl Tschantz (International Computer Science Institute)
Session: Match 10, 14:00 – 15:45 UTC
Tags: Data, Evaluation, Metrics
Summary: The authors mathematically compare competing definitions of group-level fairness and their properties under various worldviews which are assumptions about how, if at all, the observed data is biased. They discuss the criterion of disparity amplification and introduce a new world view with a corresponding notion of fairness as a more realistic perspective.

[8] Leave-one-out Unfairness
Emily Black (Carnegie Mellon University), Matt Fredrikson (Carnegie Mellon University)
Session: March 9, 22:00 – 23:45 UTC
Tags: Algorithm Development, Data, Evaluation, Fairness, Metrics 
Summary: The authors introduce leave-one-out unfairness which focuses on the change of prediction for an individual due to inclusion or exclusion of a single other individual from the training data. They discuss the relation of this concept to robustness, memorization and individual fairness in deep models.

[9] Value Cards: An Educational Toolkit for Teaching Social Impacts of Machine Learning through Deliberation
Hong Shen (Carnegie Mellon University), Wesley Deng (UC Berkeley), Aditi Chattopadhyay (Carnegie Mellon University), Steven Wu (Carnegie Mellon University), Xu Wang (University of Michigan), Haiyi Zhu (Carnegie Mellon University)
Session: March 8, 22:00 – 23:45 UTC
Tags: Accountability, Education, Human Factors
Summary: The authors introduce Value Cards, an educational toolkit with topics related to Fairness, Accountability, and Ethics, and present an early use of the approach in a college-level computer science course. Results suggest that the use of the toolkit can improve students’ understanding of both technical definitions and trade-offs of performance metrics and apply them in real-world contexts.

[10] Narratives and Counternarratives on Data Sharing in Africa
Rediet Abebe (UC Berkeley), Kehinde Aruleba (University of Witwatersrand), Abeba Birhane (University College Dublin), Sara Kingsley (Carnegie Mellon University), George Obaido (University of Witwatersrand), Sekou L. Remy (IBM Research Africa), Swathi Sadagopan (Deloitte)
Session: March 9, 12:00 – 13:50 UTC
Tags: Data, Ethics, Humanistic Theory & Critique 
Summary: The authors use storytelling via fictional personas built from a series of interviews with African data experts to complicate dominant narratives and provide counternarratives on data sharing in Africa. They discuss issues arising from power imbalances and Western-centric policies in the context of open data initiatives centered around data extracted from African communities and discuss avenues for addressing these issues.

Tutorials

Sociocultural diversity in machine learning: Lessons from philosophy, psychology, and organizational science
Sina Fazelpour (Carnegie Mellon University) and Maria De-Arteaga (University of Texas at Austin)
Session: March 4, 14:00 – 15:30 UTC
Summary: The current discussion of sociocultural diversity in machine learning research leaves a gap between the conversation about measures and benefits and the philosophical, psychological and organizational research on the underlying concepts. This tutorial addresses the concepts and consequences of sociocultural diversity and situates this understanding and its implications for the discussion of sociocultural diversity in machine learning.

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TL;DR: motivated to better understand the fundamental tradeoffs in federated learning, we present a probabilistic perspective that generalizes and improves upon federated optimization and enables a new class of efficient federated learning algorithms.

Thanks to deep learning, today we can train better machine learning models when given access to massive data. However, the standard, centralized training is impossible in many interesting use-cases—due to the associated data transfer and maintenance costs (most notably in video analytics), privacy concerns (e.g., in healthcare settings), or sensitivity of the proprietary data (e.g., in drug discovery). And yet, different parties that own even a small amount of data want to benefit from access to accurate models. This is where federated learning comes to the rescue!

Broadly, federated learning (FL) allows multiple data owners (or clients) to train shared models collaboratively under the orchestration of a central server without having to share any data. Typically, FL proceeds in multiple rounds of communication between the server and the clients: the clients compute model updates on their local data and send them to the server which aggregates and applies these updates to the shared model. While gaining popularity very quickly, FL is a relatively new subfield with many open questions and unresolved challenges.

Here is one interesting conundrum driving our work:

Client-server communication is often too slow and expensive. To speed up training (often x10-100) we can make clients spend more time at each round on local training (e.g., do more local SGD steps), thereby reducing the total number of communication rounds. However, because of client data heterogeneity (natural in practice), it turns out that increasing the amount of local computation per round results in convergence to inferior models!

This phenomenon is illustrated below in Figure 1 on a toy convex problem, where we see that more local steps lead the classical federated averaging (FedAvg) algorithm to converge to points that are much further away from the global optimum. But why does this happen?

In this post, we will present a probabilistic perspective on federated learning that will help us better understand this phenomenon and design new FL algorithms that can utilize local computation much more efficiently, converging faster, to better optima.

The classical approach: FL as a distributed optimization problem

Federated learning was originally introduced as a new setting for distributed optimization with a few distinctive properties such as a massive number of distributed nodes (or clients), slow and expensive communication, and unbalanced and non-IID data scattered across the nodes. The main goal of FL is to approximate centralized training (the gold-standard) and converge to the same optimum as the centralized optimization would have, at the fastest rate possible.

Mathematically, FL is formulated as minimization of a linear combination of local objectives, (f_i): $$min_{theta in mathbb{R}^d} left{F(theta) := sum_{i=1}^N q_i f_i(theta) right}$$ where the weights (q_i) are usually set proportional to the sizes (n_i) of the local datasets to make (F(theta)) match the centralized training objective. So, how can we solve this optimization problem within a minimal number of communication rounds?

The trick is simple: at each round (t), instead of asking clients to estimate and send gradients of their local objective functions (as done in conventional distributed optimization), let them optimize their objectives for multiple steps (or even epochs!) to obtain (theta^t_{i}) and send differences (or “deltas”) between the initial (theta^t) and updated states (theta^t_{i}) to the server as pseudo-gradients, which the server then averages, scales by a learning rate (alpha_t), and uses to update the model state: $$theta^{t+1} = theta^t + alpha_t sum_{i=1}^N q_i Delta_i^t, quad text{where} Delta_i^t := theta^t – theta_i^t$$ This approach, known as FedAvg or local SGD, allows clients to make more progress at each round. And since taking additional SGD steps locally is orders of magnitude faster than communicating with the server, the method converges much faster both in the number of rounds and in wall-clock time.

The problem (a.k.a. “client drift”): as we mentioned in the beginning, allowing multiple local SGD steps between client-server synchronization makes the algorithm converge to an inferior optimum in the non-IID setting (i.e., when clients have different data distributions) since the resulting pseudo-gradients turn out to be somehow biased compared to centralized training.

There are ways to overcome client drift using local regularization, carefully setting learning rate schedules, or using different control variate methods, but most of these mitigation strategies intentionally have to limit the optimization progress clients can make at each round.

Fundamentally, viewing FL as a distributed optimization problem runs into a tradeoff between the amount of local progress allowed and the quality of the final solution.

So, is there a way around this fundamental limitation?

An alternative approach: FL via posterior inference

Typically, client objectives (f_i(theta)) correspond to log-likelihoods of their local data. Therefore, statistically speaking, FL is solving a maximum likelihood estimation (MLE) problem. Instead of solving it using distributed optimization techniques, however, we can take a Bayesian approach: first, infer the posterior distribution, (P(theta mid D)), then identify its mode which will be the solution.

Why is posterior inference better than optimization? Because any posterior can be exactly decomposed into a product of sub-posteriors: $$P(theta mid D) propto prod_{i=1}^N P(theta mid D_i)$$

Thus, we are guaranteed to find the correct solution in three simple steps:

1. Infer local sub-posteriors on each client and send their sufficient statistics to the server.
2. Multiplicatively aggregate sub-posteriors on the server into the global posterior.
3. Find and return the mode of the global posterior.

Wait, isn’t posterior inference intractable!?

Indeed, there is a reason why posterior inference is not as popular as optimization: it is either intractable or often significantly more complex and computationally expensive. Moreover, posterior distributions rarely have closed form expressions and require various approximations.

For example, consider federated least squares regression, with quadratic local objectives: (f_i(theta) = frac{1}{2} |X_i^toptheta – y_i|^2.) In this case, the global posterior mode has a closed form expression: $$theta^star = left( sum_{i=1}^N q_i Sigma_i^{-1} right)^{-1} left( sum_{i=1}^N q_i Sigma_i^{-1} mu_i right)$$ where (mu_i) and (Sigma_i) are the means and covariances of the local posteriors. Even though in this simple case the posterior is Gaussian and inference is technically tractable, computing (theta^star) requires inverting multiple matrices and communicating local means and covariances from the clients to the server. In comparison to FedAvg, which requires only (O(d)) computation and (O(d)) communication per round, posterior inference seems like a very bad idea…

Approximate inference FTW!

Turns out that we can compute approximately using an elegant distributed inference algorithm which we call federated posterior averaging (or FedPA):

1. On the server, we can compute iteratively over multiple rounds: $$theta^{t+1} = theta^t – alpha_t sum_{i=1}^N q_i underbrace{Sigma_i^{-1}left( theta^t – mu_i right)}_{:= Delta_i^t}$$ where (alpha_t) is the server learning rate. This procedure avoids the outer matrix inverse and requires clients to send to the server only some delta vectors instead of full covariance matrices. Also, the summation can be substituted with a stochastic approximation, i.e., only a subset of clients must participate in each round. Note how similar it is to FedAvg!
2. On the clients, we can compute (Delta_i^t := Sigma_i^{-1}left( theta^t – mu_i right)) very efficiently in two steps:
   1. Use stochastic gradient Markov chain Monte Carlo (SG-MCMC) to produce multiple approximate samples from the local posterior.
   2. Use an efficient dynamic programming procedure to compute the inverse covariance matrix multiplied by a vector in (O(d)) time and memory.

Note: in the case of arbitrary non-Gaussian likelihoods (which is the case for deep neural nets), FedPA essentially approximates the local and global posteriors with the best fitting Gaussians (a.k.a. the Laplace approximation).

What is the difference between FedAvg and FedPA?

FedPA has the same computation and communication complexity as FedAvg. In fact, the algorithms differ only in how the client updates (Delta_i^t) are computed. Since FedAvg computes (Delta_i^t approx theta^t – mu_i), we can also view it as an approximate posterior inference algorithm that estimates local covariances (Sigma_i) with identity matrices, which results in biased updates!

Figure 2 illustrates the difference between FedAvg and FedPA in terms of the bias and variance of updates they compute at each round as functions of the number of SGD steps:

• More local SGD steps increase the bias of FedAvg updates, leading the algorithm to converge to a point further away from the optimum.
• FedPA uses local SGD steps to produce more posterior samples, which improves the estimates of the local means and covariances and reduces the bias of model updates.

Does FedPA actually work in practice?

The bias-variance tradeoff argument seems great in theory, but does it actually work in practice? First, let’s revisit our toy 2D example with 2 clients and quadratic objectives:

We see that not only is FedPA as fast as FedAvg initially but it also converges to a point that is significantly closer to the global optimum. At the end of convergence, FedPA exhibits some oscillations that could be further eliminated by increasing the number of local posterior samples.

Next, let’s compare FedPA with FedAvg head-to-head on realistic and challenging benchmarks, such as the federated CIFAR100 and StackOverflow datasets:

For clients to be able to sample from local posteriors using SG-MCMC, their models have to be close enough to local optima in the parameter space. Therefore, we first “burn-in” FedPA for a few rounds by running it in the FedAvg regime (i.e., compute the deltas the same way as FedAvg). At some point, we switch to local SG-MCMC sampling. Figures 4 and 5 show the evaluation metrics over the course of training. We clearly see a significant jump in performance right at the point when the algorithm was essentially switched from FedAvg to FedPA.

Concluding thoughts & what’s next?

Viewing federated learning through the lens of probabilistic inference turned out to be fruitful. Not only were we able to reinterpret FedAvg as a biased approximate inference algorithm and explain the strange effect of multiple local SGD steps on its convergence, but this new perspective allowed us to design a new FL algorithm that blends together optimization with local MCMC-based posterior sampling and utilizes local computation efficiently.

We believe that FedPA is just the beginning of a new class of approaches to federated learning. One of the biggest advantages of the distributed optimization over posterior inference so far is a strong theoretical understanding of FedAvg’s convergence and its variations in different IID and non-IID settings, which was developed over the past few years by the optimization community. Convergence analysis of posterior inference in different federated settings is an important research avenue to pursue next.

While FedPA relies on a number of specific design choices we had to make (the Laplace approximation, MCMC-based local inference, the shrinkage covariance estimation, etc.), our inferential perspective connects FL to a rich toolbox of techniques from Bayesian machine learning literature: variational inference, expectation propagation, ensembling and Bayesian deep learning, privacy guarantees for posterior sampling, among others. Exploring application of these techniques in different FL settings may lead us to even more interesting discoveries!

Want to learn more?

• Check out our ICLR 2021 paper.
• Play with all our FedPA demos using this code.

ACKNOWLEDGEMENTS: Thanks to Jenny Gillenwater, Misha Khodak, Peter Kairouz, and Afshin Rostamizadeh for feedback on this blog post.

DISCLAIMER: All opinions expressed in this post are those of the author and do not represent the views of CMU.

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Top and Bottom Right: RealNVP [3] uses checkerboard and channel-wise partitioning schemes in order to factor out parameters and ensure that there aren’t redundant partitions from previous layers. GLOW [4] uses an invertible 1×1 convolution which allows the partitioned to be ‘learned’ by a linear layer. We show that arbitrary partitions can be simulated in a constant number of layers with a fixed partition, showing that these ideas increase representational power by at most a constant factor. Bottom Left: Random points are well-separated with high probability on a high-dimensional sphere, which allows us to construct a distribution that is challenging for flows.

The promise of unsupervised learning lies in its potential to take advantage of cheap and plentiful unlabeled data to learn useful representations or generate high-quality samples. For the latter task, neural network-based generative models have recently enjoyed a lot of success in producing realistic images and text. Two major paradigms in deep generative modeling are generative adversarial networks (GANs) and normalizing flows. When successfully scaled up and trained, both can generate high-quality and diverse samples from high-dimensional distributions. The training procedure for GANs involves min-max (saddle-point) optimization, which is considerably more difficult than standard loss minimization, leading to problems like mode dropping.

Normalizing flows [1] have been proposed as an alternative type of generative model which allows not only efficient sampling but also training via maximum likelihood through a closed-form computation of the likelihood function. They are written as pushforwards of a simple distribution (typically a Gaussian) through an invertible transformation (f), typically parametrized as a composition of simple invertible transformations. The main reason for this parametrization is the change-of-variables formula: if (z) is a random variable sampled from a known base distribution (P(z)) (typically a standard multivariate normal), (f: mathbb{R}^dto mathbb{R}^d) is invertible and differentiable, and (x = f^{-1}(z)) then $$p(x) = p(f(x))left|detleft(frac{partial f(x)}{partial x^T}right)right|.$$ Here, (frac{partial f(x)}{partial x^T}) is the Jacobian of (f).

Normalizing flows are trained by maximizing the likelihood using gradient descent. However, in practice, training normalizing flows runs into difficulties as well: models which produce good samples typically need to be extremely deep — which comes with accompanying vanishing/exploding gradient problems. A very related problem is that they are often poorly conditioned. Data like images are often inherently lower-dimensional than the ambient space, the map from low dimensional data to high dimensional latent variables can be difficult to invert and therefore train.

In our recent work [2], we tackle representational questions around depth and conditioning of normalizing flows—first for general invertible architectures, then for a particular common architecture—where the normalizing flow is a composition of so-called affine couplings.

Depth Bound on General Invertible Architectures

The most fundamental restriction of the normalizing flow paradigm is that each layer needs to be invertible. We ask whether this restriction has any ‘cost’ in terms of the size, and in particular the depth, of the model. Here we’re counting depth in terms of the number of the invertible transformations that make up the flow. A requirement for large depth would explain training difficulties due to exploding (or vanishing) gradients.

A natural way of formalizing this question is by exhibiting a distribution which is easy to model for an unconstrained generator network but hard for a shallow normalizing flow. Precisely, we ask: is there a probability distribution that can be represented by a shallow generator with a small number of parameters that could not be approximately represented by a shallow composition of invertible transformations?

We demonstrate that such a distribution exists. Specifically, we show that

Theorem: For every k, s.t. (k = o(exp(d))) and any parameterized family of compositions of Lipschitz invertible transformations with (p) parameters per transformation and at most (O(k / p)) transformations, there exists a generator (g:mathbb{R}^{d+1} to mathbb{R}) with depth (O(1)) and (O(k)) parameters s.t the pushforward of a Gaussian through (g) cannot be approximated in either KL or Wasserstein-1 distance by a network in this family.

The result above is extremely general: it only requires a bound on the number of parameters per transformation in the parametrization of the normalizing flow and Lipschitzness of these maps. As such it easily includes common choices used in practice like affine couplings with at most (p) parameters per layer or invertible feedforward networks, where each intermediate layer is of dimension (d) and the nonlinearity is invertible (e.g. leaky ReLU). On the flip side, for possible architectures with a large number of parameters per transformation, this theorem gives a (possibly loose) lower bound of a small number of transformations.

Proof Sketch: The generator for our construction approximates a mixture of (k) Gaussians with means placed uniformly randomly on a (d)-dimensional sphere in the ambient space. We will use the probabilistic method to show there is a family of such mixtures, s.t. each pair of members in this family are far apart (say, in Wasserstein distance). Furthermore, by an epsilon net discretization argument we can count how many “essentially” distinct invertible Lipschitz networks there are. If the number of mixtures in the family is much larger than the size of the epsilon net, at least one mixture must be far from all invertible networks.

The family of mixtures is constructed by choosing the (k) means for the components uniformly at random on a sphere. It’s well known that (exp(o(d))) randomly chosen points on a unit sphere will, with high probability, have constant pairwise distance. Similarly, coding-theoretic arguments (used to prove the so-called Gilbert-Varshamov bound) can be used to show that selecting (exp(o(d))) (k)-tuples of those means will, with high probability, ensure that each pair of (k)-tuples is such that the average pair of means is at constant distance. This suffices to ensure the Wasserstein distance between pairs of mixtures is large. ∎

Results for Affine Couplings

Affine Couplings [3] are one of the most common transformations in scalable architectures for normalizing flows. An affine coupling is a map (f: mathbb{R}^dto mathbb{R}^d) such that for some partition into a set containing approximately half of the coordinates (S) and it’s complement, $$f(x_S, x_{[d]setminus S}) = (x_S, x_{[d]setminus S} odot s(x_s) + t(x_s))$$ for some scaling and translation functions (typically parameterized by neural networks) (s) and (t). Clearly, an affine coupling block only transforms one partition of the coordinates at a time by an affine function while leaving the other partition intact. It’s easy to see that an affine coupling is invertible if each coordinate of (s) is invertible. Moreover, the Jacobian of this function is $$begin{bmatrix}I & 0\frac{partial t}{partial x_S^T} & text{diag}(s(x_S))end{bmatrix}$$ In particular it’s lower triangular, so we can calculate the determinant in linear time by multiplying the (d) diagonal elements (in general determinants take (O(d^3)) time to compute). This allows us to efficiently compute likelihoods and their gradients for SGD on large models via the change of variables formula.
These affine coupling blocks are stacked, often while changing the part of the partition that is updated or more generally, permuting the elements in between the application of the coupling.

Effect of the Choice of Partition on Representational Power

The choice of partition is often somewhat ad-hoc and involves domain knowledge (e.g. for image datasets, a typical choice is a checkerboard pattern or a channel-wise partition). In fact, some recent approaches like GLOW [4] try to “learn” permutations to apply between each pair of affine couplings. (Technically, since a permutation is a discrete object, in [4] the authors learn a 1×1 convolutions instead.)

While ablation experiments provide definite evidence that including learned 1×1 convolutions is beneficial for modeling image data in practice, it’s unclear whether this effect is from increased modeling power or algorithmic effects — and even less so how to formally quantify it. In this section, we come to a clear understanding of the representational value of adding this flexibility in partitioning. We knew from GLOW that adding these partitions helped. Now we know why!

We formalize the representational question as follows: how many affine couplings with a fixed partition are needed to simulate an arbitrary linear map? Since a linear map is more general than a 1×1 convolution, if it’s possible to do so with a small (say constant) number of affine couplings, we can simulate any affine coupling-based normalizing flow including 1×1 convolutions by one that does not include them which is merely a constant factor larger.

Concretely, we consider linear functions of the form $$T = prod_{i=1}^Kbegin{bmatrix}I & 0\A_i & B_iend{bmatrix} begin{bmatrix}C_i & D_i\0 & Iend{bmatrix},$$ for matrices (A_i, D_i in mathbb{R}^{dtimes d}) and diagonal matrices (B_i, C_i in mathbb{R}^{dtimes d}). The right hand side is precisely a composition of affine coupling blocks with linear maps (s, t) with a fixed partition (the parts of the input that are being updated alternate). We show the following result:

Theorem: To represent an arbitrary invertible (T), (K) must be at most 24. Additionally, there exist invertible matrices (T) such that (K geq 3).

Proof sketch: The statement hopefully reminds the reader of the standard LU decomposition — the twist of course being that the matrices on the right-hand side have a more constrained structure than merely being triangular. Our proof starts with the existence of a (LUP) decomposition for every matrix.

We first show that we can construct an arbitrary permutation (up to sign) using at most 21 alternating matrices of the desired form. The argument is group theoretic: we use the fact that a permutation decomposes into a composition of two permutations of order 2, which must be disjoint products of swaps and show that swapping elements can be implemented “in parallel” using several partitioned matrices of the type we’re considering.

Next, we show that we can produce an arbitrary triangular matrix with our partitioned matrices. We use similar techniques as above to reduce the matrix to a regular system of block linear equations which we can then solve. Our upper bound comes from just counting the total number of matrices required for these operations: the 21 for the permutation and 13 for each triangular matrix (upper and lower), giving a total of 47 required matrices. ∎

To reiterate the takeaway: a GLOW-style linear layer in between affine couplings could in theory make your network between 5 and 47 times smaller while representing the same function. We now have a precise understanding of the value of that architectural choice!

We also verified empirically in the figure below how well these linear models would fit randomly chosen (i.e. with iid Gaussian entries) linear functions. It seems empirically that at least for this ensemble our upper bound is loose and we can fit the functions well without using the full 47 layers. Closing this gap is an interesting problem for future work.

Universal Approximation with Poorly Conditioned Networks

In our earlier result on the depth of invertible networks, we assumed that our network was Lipschitz and therefore well-conditioned. A natural question is then, if we remove this requirement, how powerful is the resulting class of models? In particular, we ask: are poorly conditioned affine coupling-based normalizing flows universal approximators as they are used in practice?

Curiously, this question has in fact not been answered in prior work. In a very recent work [5], it was shown that if we allow for padding of our data with extra dimensions that take a constant value 0, affine couplings are universal approximators. (Note, this kind of padding clearly results in a singular Jacobian — as the value in the added dimensions is constant.) The idea for why padding helps is that these extra dimensions are used as a “scratch pad” for the computation the network is performing. Another recent work [6] gives a proof of universal approximation for affine couplings assuming arbitrary permutations in between the layers are allowed (ala Glow) and a partition separating (d -1) dimensions from the other. However, in practice, these models are trained using a roughly half-half split and often without linear layers in between couplings (which already works quite well). We prove that none of these architectural modifications to affine couplings are necessary for universal approximation and additionally suggest a trade-off between the conditioning of the model and the quality of its approximation. Concretely, we show:

Theorem: For any bounded and absolutely continuous distribution (Q) over (mathbb{R}^n) and any (epsilon > 0), there exists a 3-layer affine coupling (g) with maps (s, t) represented by feedforward ReLU networks such that (W_2(g_# P, Q) leq epsilon), where (g_# P) is the pushforward of a standard Gaussian through (g).

We note that the construction for the theorem trades off quality of approximation ((epsilon)) with conditioning: the smallest singular value of the Jacobian in the construction for our theorem above will scale like (1/epsilon) — thus suggesting that if we want to use affine couplings as universal approximators, conditioning may be an issue even if we don’t pad with a constant value for the added dimensions like prior works — which obviously results in a singular Jacobian.

Proof sketch: The proof is based on two main ideas.

The first is a deep result from optimal transport, Brenier’s theorem, which for sufficiently “regular” distributions (p) over (mathbb{R}^d) guarantees an invertible map (phi), s.t. the pushforward of the Gaussian through (phi) equals (p). This reduces our problem to approximating (phi) using a sequence of affine couplings.

The difficulty in approximating (phi) is the fact that affine couplings are only allowed to change one part of the input, and in a constrained way. The trick we use to do this without a “scratchpad” to store intermediate computation as in prior works is to instead hide information in the “low order bits” of the other partition. For details, refer to our paper. ∎

Finally, on the experimental front, we wanted to experiment with how padding affects the conditioning of a learned model. We considered synthetic 2d datasets (see figure below) and found that padding with zeros resulted in a very poorly conditioned model which produced poor samples, as might be expected. We also considered a type of padding which is reasonable but for which we have no theory — namely, to use iid Gaussian samples as values for the added dimensions (in this case, the resulting Jacobians are not prima facie singular, and the model can still use them as a “scratch pad”). While we have no result that this in any formal sense can result in better-conditioned networks, we found that in practice it frequently does and it also results in better samples. This seems like a very fruitful direction for future research. Finally, without padding, the model produces samples of middling quality and has a condition number in between that of zero and Gaussian padding.

Conclusions

Normalizing flows are one of the most popular generative models across various domains, though we still have a relatively narrow understanding of their relative pros and cons compared to other models. We show in this work that there are fundamental tradeoffs between depth and conditioning and representational power of this type of function. Though we have cleared up considerably the representational aspects of these models, the algorithmic and statistical questions are still wide open. We hope that this work guides both users of flows and theoreticians as to the fine-grained properties of flows as compared to other generative models.

Bibliography

[1] Rezende and Mohamed, 2015, Variational Inference with Normalizing Flows, ICML 2015

[2] Koehler, Mehta, and Risteski, 2020, Representational aspects of depth and conditioning in normalizing flows, Under Submission.

[3] Dinh, Sohl-Dickstein, and S. Bengio, 2016, Density estimation using Real NVP, ICLR 2016

[4] Kingma and Dhariwal, 2018, GLOW: Generative flow with 1×1 convolutions, NeurIPS 2018

[5] Huang, Dinh, and Courville, 2020, Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models

[6] Teshima, Ishikawa, Tojo, Oono, Ikeda, and Sugiyama, 2020, Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators, NeurIPS 2020

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