ICLR 2023 Papers

Learning mappings between infinite-dimensional function spaces have achieved empirical success in many disciplines of machine learning, including generative modeling, functional data analysis, causal inference, and multi-agent reinforcement learning. In this paper, we study the statistical limit of learning a Hilbert-Schmidt operator between two infinite-dimensional Sobolev reproducing kernel Hilbert spaces. We establish the information-theoretic lower bound in terms of the Sobolev Hilbert-Schmidt norm and show that a regularization that learns the spectral components below the bias contour and ignores the ones above the variance contour can achieve the optimal learning rate. At the same time, the spectral components between the bias and variance contours give us flexibility in designing computationally feasible machine learning algorithms. Based on this observation, we develop a multilevel kernel operator learning algorithm that is optimal when learning linear operators between infinite-dimensional function spaces.

Neural networks can be trained to solve partial differential equations (PDEs) by using the PDE residual as the loss function. This strategy is called "physics-informed neural networks" (PINNs), but it currently cannot produce high-accuracy solutions, typically attaining about $0.1\%$ relative error. We present an adversarial approach that overcomes this limitation, which we call competitive PINNs (CPINNs). CPINNs train a discriminator that is rewarded for predicting mistakes the PINN makes. The discriminator and PINN participate in a zero-sum game with the exact PDE solution as an optimal strategy. This approach avoids squaring the large condition numbers of PDE discretizations, which is the likely reason for failures of previous attempts to decrease PINN errors even on benign problems. Numerical experiments on a Poisson problem show that CPINNs achieve errors four orders of magnitude smaller than the best-performing PINN. We observe relative errors on the order of single-precision accuracy, consistently decreasing with each epoch. To the authors' knowledge, this is the first time this level of accuracy and convergence behavior has been achieved. Additional experiments on the nonlinear Schr{\"o}dinger, Burgers', and Allen--Cahn equation show that the benefits of CPINNs are not limited to linear problems.

We study the type of solutions to which stochastic gradient descent converges when used to train a single hidden-layer multivariate ReLU network with the quadratic loss. Our results are based on a dynamical stability analysis. In the univariate case, it was shown that linearly stable minima correspond to network functions (predictors), whose second derivative has a bounded weighted $L^1$ norm. Notably, the bound gets smaller as the step size increases, implying that training with a large step size leads to `smoother' predictors. Here we generalize this result to the multivariate case, showing that a similar result applies to the Laplacian of the predictor. We demonstrate the tightness of our bound on the MNIST dataset, and show that it accurately captures the behavior of the solutions as a function of the step size. Additionally, we prove a depth separation result on the approximation power of ReLU networks corresponding to stable minima of the loss. Specifically, although shallow ReLU networks are universal approximators, we prove that stable shallow networks are not. Namely, there is a function that cannot be well-approximated by stable single hidden-layer ReLU networks trained with a non-vanishing step size. This is while the same function can be realized as a stable two hidden-layer ReLU network. Finally, we prove that if a function is sufficiently smooth (in a Sobolev sense) then it can be approximated arbitrarily well using shallow ReLU networks that correspond to stable solutions of gradient descent.

Finding equilibria via gradient play in competitive multi-agent games has been attracting a growing amount of attention in recent years, with emphasis on designing efficient strategies where the agents operate in a decentralized and symmetric manner with guaranteed convergence. While significant efforts have been made in understanding zero-sum two-player matrix games, the performance in zero-sum multi-agent games remains inadequately explored, especially in the presence of delayed feedbacks, leaving the scalability and resiliency of gradient play open to questions. In this paper, we make progress by studying asynchronous gradient plays in zero-sum polymatrix games under delayed feedbacks. We first establish that the last iterate of entropy-regularized optimistic multiplicative weight updates (OMWU) method converges linearly to the quantal response equilibrium (QRE), the solution concept under bounded rationality, in the absence of delays. The linear convergence continues to hold even when the feedbacks are randomly delayed under mild statistical assumptions, albeit at a slower rate. Moving beyond random delays, we further demonstrate entropy-regularized OMWU with two-timescale learning rates enjoys faster last-iterate convergence under fixed delays, and continues to converge provably even when the delays are arbitrarily bounded. Our methods also lead to finite-time guarantees to approximate the Nash equilibrium (NE) by moderating the amount of regularization. To the best of our knowledge, this work is the first that aims to understand asynchronous gradient play in zero-sum polymatrix games under a wide range of delay assumptions.

Many Dec-POMDPs admit a qualitatively diverse set of ''reasonable'' joint policies, where reasonableness is indicated by symmetry equivariance, non-sabotaging behaviour and the graceful degradation of performance when paired with ad-hoc partners. Some of the work in diversity literature is concerned with generating these policies. Unfortunately, existing methods fail to produce teams of agents that are simultaneously diverse, high performing, and reasonable. In this work, we propose a novel approach, adversarial diversity (ADVERSITY), which is designed for turn-based Dec-POMDPs with public actions. ADVERSITY relies on off-belief learning to encourage reasonableness and skill, and on ''repulsive'' fictitious transitions to encourage diversity. We use this approach to generate new agents with distinct but reasonable play styles for the card game Hanabi and open-source our agents to be used for future research on (ad-hoc) coordination.

We present a smoothly broken power law functional form (referred to by us as a broken neural scaling law (BNSL)) that accurately models and extrapolates the scaling behaviors of deep neural networks (i.e. how the evaluation metric of interest varies as the amount of compute used for training, number of model parameters, training dataset size, or upstream performance varies) for various architectures and for each of various tasks within a large and diverse set of upstream and downstream tasks, in zero-shot, prompted, and fine-tuned settings. This set includes large-scale vision, language, audio, video, diffusion, generative modeling, multimodal learning, contrastive learning, AI alignment, robotics, out-of-distribution (OOD) generalization, continual learning, uncertainty estimation / calibration, out-of-distribution detection, adversarial robustness, molecules, computer programming/coding, math word problems, arithmetic, unsupervised/self-supervised learning, and reinforcement learning (single agent and multi-agent). When compared to other functional forms for neural scaling behavior, this functional form yields extrapolations of scaling behavior that are considerably more accurate on this set. Moreover, this functional form accurately models and extrapolates scaling behavior that other functional forms are incapable of expressing such as the non-monotonic transitions present in the scaling behavior of phenomena such as double descent and the delayed, sharp inflection points present in the scaling behavior of tasks such as arithmetic. Lastly, we use this functional form to glean insights about the limit of the predictability of scaling behavior. See arXiv for longer version of this paper. Code is available at https://github.com/ethancaballero/broken_neural_scaling_laws

Open-ended learning methods that automatically generate a curriculum of increasingly challenging tasks serve as a promising avenue toward generally capable reinforcement learning agents. Existing methods adapt curricula independently over either environment parameters (in single-agent settings) or co-player policies (in multi-agent settings). However, the strengths and weaknesses of co-players can manifest themselves differently depending on environmental features. It is thus crucial to consider the dependency between the environment and co-player when shaping a curriculum in multi-agent domains. In this work, we use this insight and extend Unsupervised Environment Design (UED) to multi-agent environments. We then introduce Multi-Agent Environment Design Strategist for Open-Ended Learning (MAESTRO), the first multi-agent UED approach for two-player zero-sum settings. MAESTRO efficiently produces adversarial, joint curricula over both environments and co-players and attains minimax-regret guarantees at Nash equilibrium. Our experiments show that MAESTRO outperforms a number of strong baselines on competitive two-player games, spanning discrete and continuous control settings.

MOBA games, e.g., Dota2 and Honor of Kings, have been actively used as the testbed for the recent AI research on games, and various AI systems have been developed at the human level so far. However, these AI systems mainly focus on how to compete with humans, less on exploring how to collaborate with humans. To this end, this paper makes the first attempt to investigate human-agent collaboration in MOBA games. In this paper, we propose to enable humans and agents to collaborate through explicit communication by designing an efficient and interpretable Meta-Command Communication-based framework, dubbed MCC, for accomplishing effective human-agent collaboration in MOBA games. The MCC framework consists of two pivotal modules: 1) an interpretable communication protocol, i.e., the Meta-Command, to bridge the communication gap between humans and agents; 2) a meta-command value estimator, i.e., the Meta-Command Selector, to select a valuable meta-command for each agent to achieve effective human-agent collaboration. Experimental results in Honor of Kings demonstrate that MCC agents can collaborate reasonably well with human teammates and even generalize to collaborate with different levels and numbers of human teammates. Videos are available at https://sites.google.com/view/mcc-demo.

Deep neural network (DNN) classifiers are often overconfident, producing miscalibrated class probabilities. In high-risk applications like healthcare, practitioners require fully calibrated probability predictions for decision-making. That is, conditioned on the prediction vector, every class’ probability should be close to the predicted value. Most existing calibration methods either lack theoretical guarantees for producing calibrated outputs, reduce classification accuracy in the process, or only calibrate the predicted class. This paper proposes a new Kernel-based calibration method called KCal. Unlike existing calibration procedures, KCal does not operate directly on the logits or softmax outputs of the DNN. Instead, KCal learns a metric space on the penultimate-layer latent embedding and generates predictions using kernel density estimates on a calibration set. We first analyze KCal theoretically, showing that it enjoys a provable full calibration guarantee. Then, through extensive experiments across a variety of datasets, we show that KCal consistently outperforms baselines as measured by the calibration error and by proper scoring rules like the Brier Score.

Despite evidence from cognitive sciences that larger groups of speakers tend to develop more structured languages in human communication, scaling up to populations has failed to yield significant benefits in emergent multi-agent communication. In this paper we advocate for an alternate population-level training paradigm for referential games based on the idea of "partitioning" the agents into sender-receiver pairs and limiting co-adaptation across pairs. We show that this results in optimizing a different objective at the population level, where agents maximize (1) their respective "internal" communication accuracy and (2) some measure of alignment between agents. In experiments, we find that this leads to the emergence of languages that are significantly more compositional. Moreover, when agents are trained in populations that are not fully connected (ie. not all agent pairs interact at training time), this approach reduces multi-linguality and improves zero-shot communication with new agents (ie. agents are able to communicate successfully with other agents outside their training partners).

Computing Nash equilibrium policies is a central problem in multi-agent reinforcement learning that has received extensive attention both in theory and in practice. However, in light of computational intractability barriers in general-sum games, provable guarantees have been thus far either limited to fully competitive or cooperative scenarios or impose strong assumptions that are difficult to meet in most practical applications. In this work, we depart from those prior results by investigating infinite-horizon \emph{adversarial team Markov games}, a natural and well-motivated class of games in which a team of identically-interested players---in the absence of any explicit coordination or communication---is competing against an adversarial player. This setting allows for a unifying treatment of zero-sum Markov games and Markov potential games, and serves as a step to model more realistic strategic interactions that feature both competing and cooperative interests. Our main contribution is the first algorithm for computing stationary $\epsilon$-approximate Nash equilibria in adversarial team Markov games with computational complexity that is polynomial in all the natural parameters of the game, as well as $1/\epsilon$. The proposed algorithm is based on performing independent policy gradient steps for each player in the team, in tandem with best responses from the side of the adversary; in turn, the policy for the adversary is then obtained by solving a carefully constructed linear program. Our analysis leverages non-standard techniques to establish the KKT optimality conditions for a nonlinear program with nonconvex constraints, thereby leading to a natural interpretation of the induced Lagrange multipliers.

Machine learning systems, especially with overparameterized deep neural networks, can generalize to novel test instances drawn from the same distribution as the training data. However, they fare poorly when evaluated on out-of-support test points. In this work, we tackle the problem of developing machine learning systems that retain the power of overparameterized function approximators while enabling extrapolation to out-of-support test points when possible. This is accomplished by noting that under certain conditions, a "transductive" reparameterization can convert an out-of-support extrapolation problem into a problem of within-support combinatorial generalization. We propose a simple strategy based on bilinear embeddings to enable this type of combinatorial generalization, thereby addressing the out-of-support extrapolation problem under certain conditions. We instantiate a simple, practical algorithm applicable to various supervised learning and imitation learning tasks.

In this paper, we propose a novel Collaborative Pure Exploration in Kernel Bandit model (CoPE-KB), where multiple agents collaborate to complete different but related tasks with limited communication. Our model generalizes prior CoPE formulation with the single-task and classic MAB setting to allow multiple tasks and general reward structures. We propose a novel communication scheme with an efficient kernelized estimator, and design optimal algorithms CoKernelFC and CoKernelFB for CoPE-KB with fixed-confidence and fixed-budget objectives, respectively. Nearly matching upper and lower bounds in both sampling and communication complexity are established to demonstrate the optimality of our algorithms. Our theoretical results explicitly quantify how task similarities influence learning speedup, and only depend on the effective dimension of feature space. Our novel techniques including an efficient kernelized estimator and linear structured instance transformation, which overcome the communication difficulty in high-dimensional feature space and derive communication round lower bounds, can be of independent interests.

We study decentralized learning in two-player zero-sum discounted Markov games where the goal is to design a policy optimization algorithm for either agent satisfying two properties. First, the player does not need to know the policy of the opponent to update its policy. Second, when both players adopt the algorithm, their joint policy converges to a Nash equilibrium of the game. To this end, we construct a meta-algorithm, dubbed as $\texttt{Homotopy-PO}$, which provably finds a Nash equilibrium at a global linear rate. In particular, $\texttt{Homotopy-PO}$ interweaves two base algorithms $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ via homotopy continuation. $\texttt{Local-Fast}$ is an algorithm that enjoys local linear convergence while $\texttt{Global-Slow}$ is an algorithm that converges globally but at a slower sublinear rate. By switching between these two base algorithms, $\texttt{Global-Slow}$ essentially serves as a ``guide'' which identifies a benign neighborhood where $\texttt{Local-Fast}$ enjoys fast convergence. However, since the exact size of such a neighborhood is unknown, we apply a doubling trick to switch between these two base algorithms. The switching scheme is delicately designed so that the aggregated performance of the algorithm is driven by $\texttt{Local-Fast}$. Furthermore, we prove that $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ can both be instantiated by variants of optimistic gradient descent/ascent (OGDA) method, which is of independent interest.

Communication is important in many multi-agent reinforcement learning (MARL) problems for agents to share information and make good decisions. However, when deploying trained communicative agents in a real-world application where noise and potential attackers exist, the safety of communication-based policies becomes a severe issue that is underexplored. Specifically, if communication messages are manipulated by malicious attackers, agents relying on untrustworthy communication may take unsafe actions that lead to catastrophic consequences. Therefore, it is crucial to ensure that agents will not be misled by corrupted communication, while still benefiting from benign communication. In this work, we consider an environment with $N$ agents, where the attacker may arbitrarily change the communication from any $C<\frac{N-1}{2}$ agents to a victim agent. For this strong threat model, we propose a certifiable defense by constructing a message-ensemble policy that aggregates multiple randomly ablated message sets. Theoretical analysis shows that this message-ensemble policy can utilize benign communication while being certifiably robust to adversarial communication, regardless of the attacking algorithm. Experiments in multiple environments verify that our defense significantly improves the robustness of trained policies against various types of attacks.

By enabling agents to communicate, recent cooperative multi-agent reinforcement learning (MARL) methods have demonstrated better task performance and more coordinated behavior. Most existing approaches facilitate inter-agent communication by allowing agents to send messages to each other through free communication channels, i.e., \emph{cheap talk channels}. Current methods require these channels to be constantly accessible and known to the agents a priori. In this work, we lift these requirements such that the agents must discover the cheap talk channels and learn how to use them. Hence, the problem has two main parts: \emph{cheap talk discovery} (CTD) and \emph{cheap talk utilization} (CTU). We introduce a novel conceptual framework for both parts and develop a new algorithm based on mutual information maximization that outperforms existing algorithms in CTD/CTU settings. We also release a novel benchmark suite to stimulate future research in CTD/CTU.

Multi-Agent Reinforcement Learning (MARL)---where multiple agents learn to interact in a shared dynamic environment---permeates across a wide range of critical applications. While there has been substantial progress on understanding the global convergence of policy optimization methods in single-agent RL, designing and analysis of efficient policy optimization algorithms in the MARL setting present significant challenges and new desiderata, which unfortunately, remain highly inadequately addressed by existing theory. In this paper, we focus on the most basic setting of competitive multi-agent RL, namely two-player zero-sum Markov games, and study equilibrium finding algorithms in both the infinite-horizon discounted setting and the finite-horizon episodic setting. We propose a single-loop policy optimization method with symmetric updates from both agents, where the policy is updated via the entropy-regularized optimistic multiplicative weights update (OMWU) method and the value is updated on a slower timescale. We show that, in the full-information tabular setting, the proposed method achieves a finite-time last-iterate linear convergence to the quantal response equilibrium of the regularized problem, which translates to a sublinear convergence to the Nash equilibrium by controlling the amount of regularization. Our convergence results improve upon the best known iteration complexities, and lead to a better understanding of policy optimization in competitive Markov games.

It is well known that the finite step-size ($h$) in Gradient descent (GD) implicitly regularizes solutions to flatter minimas. A natural question to ask is \textit{Does the momentum parameter $\beta$ (say) play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?}. To answer this question, first, we show that the trajectory traced by discrete H.B momentum update (GD+M) is $O(h^2)$ close to a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. This implicit regularizer for (GD+M) is indeed stronger than that of (GD) by factor of $(\frac{1+\beta}{1-\beta})$, thus explaining why (GD+M) shows better generalization performance and higher test accuracy than (GD). Furthermore, we extend our analysis to stochastic version of gradient descent with momentum (SGD+M) and propose a deterministic continuous trajectory that is $O(h^2)$ close to the discrete update of (SGD+M) in a strong approximation sense. We explore the implicit regularization in (SGD+M) and (GD+M) through a series of experiments validating our theory.

The recent observation of neural power-law scaling relations has made a significant impact in the field of deep learning. A substantial amount of attention has been dedicated as a consequence to the description of scaling laws, although mostly for supervised learning and only to a reduced extent for reinforcement learning frameworks. In this paper we present an extensive study of performance scaling for a cornerstone reinforcement learning algorithm, AlphaZero. On the basis of a relationship between Elo rating, playing strength and power-law scaling, we train AlphaZero agents on the games Connect Four and Pentago and analyze their performance. We find that player strength scales as a power law in neural network parameter count when not bottlenecked by available compute, and as a power of compute when training optimally sized agents. We observe nearly identical scaling exponents for both games. Combining the two observed scaling laws we obtain a power law relating optimal size to compute similar to the ones observed for language models. We find that the predicted scaling of optimal neural network size fits our data for both games. This scaling law implies that previously published state-of-the-art game-playing models are significantly smaller than their optimal size, given the respective compute budgets. We also show that large AlphaZero models are more sample efficient, performing better than smaller models with the same amount of training data.