A Pontryagin Perspective on Reinforcement Learning

Reinforcement learning has traditionally focused on learning state-dependent policies to solve optimal control problems in a closed-loop fashion. In this work, we introduce the paradigm of open-loop reinforcement learning where a fixed action sequence is learned instead. We present three new algorithms: one robust model-based method and two sample-efficient model-free methods. Rather than basing our algorithms on Bellman's equation from dynamic programming, our work builds on Pontryagin's principle from the theory of open-loop optimal control. We provide convergence guarantees and evaluate all methods empirically on a pendulum swing-up task, as well as on two high-dimensional MuJoCo tasks, demonstrating remarkable performance compared to existing baselines.

Prior-Dependent Allocations for Bayesian Fixed-Budget Best-Arm Identification in Structured Bandits

We study the problem of Bayesian fixed-budget best-arm identification (BAI) in structured bandits. We propose an algorithm that uses fixed allocations based on the prior information and the structure of the environment. We provide theoretical bounds on its performance across diverse models, including the first prior-dependent upper bounds for linear and hierarchical BAI. Our key contribution is introducing new proof methods that result in tighter bounds for multi-armed BAI compared to existing methods. We extensively compare our approach to other fixed-budget BAI methods, demonstrating its consistent and robust performance in various settings. Our work improves our understanding of Bayesian fixed-budget BAI in structured bandits and highlights the effectiveness of our approach in practical scenarios.

Beyond Average Return in Markov Decision Processes

What are the functionals of the reward that can be computed and optimized exactly in Markov Decision Processes? In the finite-horizon, undiscounted setting, Dynamic Programming (DP) can only handle these operations efficiently for certain classes of statistics. We summarize the characterization of these classes for policy evaluation, and give a new answer for the planning problem. Interestingly, we prove that only generalized means can be optimized exactly, even in the more general framework of Distributional Reinforcement Learning (DistRL).DistRL permits, however, to evaluate other functionals approximately. We provide error bounds on the resulting estimators, and discuss the potential of this approach as well as its limitations.These results contribute to advancing the theory of Markov Decision Processes by examining overall characteristics of the return, and particularly risk-conscious strategies.

POMRL: No-Regret Learning-to-Plan with Increasing Horizons

We study the problem of planning under model uncertainty in an online meta-reinforcement learning (RL) setting where an agent is presented with a sequence of related tasks with limited interactions per task. The agent can use its experience in each task and across tasks to estimate both the transition model and the distribution over tasks. We propose an algorithm to meta-learn the underlying structure across tasks, utilize it to plan in each task, and upper-bound the regret of the planning loss. Our bound suggests that the average regret over tasks decreases as the number of tasks increases and as the tasks are more similar. In the classical single-task setting, it is known that the planning horizon should depend on the estimated model's accuracy, that is, on the number of samples within task. We generalize this finding to meta-RL and study this dependence of planning horizons on the number of tasks. Based on our theoretical findings, we derive heuristics for selecting slowly increasing discount factors, and we validate its significance empirically.

Non-stationary Bandits and Meta-Learning with a Small Set of Optimal Arms

We study a sequential decision problem where the learner faces a sequence of $K$-armed stochastic bandit tasks. The tasks may be designed by an adversary, but the adversary is constrained to choose the optimal arm of each task in a smaller (but unknown) subset of $M$ arms. The task boundaries might be known (the bandit meta-learning setting), or unknown (the non-stationary bandit setting), and the number of tasks $N$ as well as the total number of rounds $T$ are known ($N$ could be unknown in the meta-learning setting). We design an algorithm based on a reduction to bandit submodular maximization, and show that its regret in both settings is smaller than the simple baseline of $\tilde{O}(\sqrt{KNT})$ that can be obtained by using standard algorithms designed for non-stationary bandit problems. For the bandit meta-learning problem with fixed task length $\tau$, we show that the regret of the algorithm is bounded as $\tilde{O}(N\sqrt{M \tau}+N^{2/3})$. Under additional assumptions on the identifiability of the optimal arms in each task, we show a bandit meta-learning algorithm with an improved $\tilde{O}(N\sqrt{M \tau}+N^{1/2})$ regret.

EigenGame Unloaded: When playing games is better than optimizing

We build on the recently proposed EigenGame that views eigendecomposition as a competitive game. EigenGame's updates are biased if computed using minibatches of data, which hinders convergence and more sophisticated parallelism in the stochastic setting. In this work, we propose an unbiased stochastic update that is asymptotically equivalent to EigenGame, enjoys greater parallelism allowing computation on datasets of larger sample sizes, and outperforms EigenGame in experiments. We present applications to finding the principal components of massive datasets and performing spectral clustering of graphs. We analyze and discuss our proposed update in the context of EigenGame and the shift in perspective from optimization to games.

Asymptotically Optimal Information-Directed Sampling

We introduce a computationally efficient algorithm for finite stochastic linear bandits. The approach is based on the frequentist information-directed sampling (IDS) framework, with an information gain potential that is derived directly from the asymptotic regret lower bound. We establish frequentist regret bounds, which show that the proposed algorithm is both asymptotically optimal and worst-case rate optimal in finite time. Our analysis sheds light on how IDS trades off regret and information to incrementally solve the semi-infinite concave program that defines the optimal asymptotic regret. Along the way, we uncover interesting connections towards a recently proposed two-player game approach and the Bayesian IDS algorithm.

The Elliptical Potential Lemma Revisited

This note proposes a new proof and new perspectives on the so-called Elliptical Potential Lemma. This result is important in online learning, especially for linear stochastic bandits. The original proof of the result, however short and elegant, does not give much flexibility on the type of potentials considered and we believe that this new interpretation can be of interest for future research in this field.

EigenGame: PCA as a Nash Equilibrium

We present a novel view on principal component analysis (PCA) as a competitive game in which each approximate eigenvector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this PCA game and the behavior of its gradient based updates. The resulting algorithm which combines elements from Oja's rule with a generalized Gram-Schmidt orthogonalization is naturally decentralized and hence parallelizable through message passing. We demonstrate the scalability of the algorithm with experiments on large image datasets and neural network activations. We discuss how this new view of PCA as a differentiable game can lead to further algorithmic developments and insights.

Confident Off-Policy Evaluation and Selection through Self-Normalized Importance Weighting

We consider off-policy evaluation in the contextual bandit setting for the purpose of obtaining a robust off-policy selection strategy, where the selection strategy is evaluated based on the value of the chosen policy in a set of proposal (target) policies. We propose a new method to compute a lower bound on the value of an arbitrary target policy given some logged data in contextual bandits for a desired coverage. The lower bound is built around the so-called Self-normalized Importance Weighting (SN) estimator. It combines the use of a semi-empirical Efron-Stein tail inequality to control the concentration and Harris' inequality to control the bias. The new approach is evaluated on a number of synthetic and real datasets and is found to be superior to its main competitors, both in terms of tightness of the confidence intervals and the quality of the policies chosen.