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In this paper, we study risk-sensitive Reinforcement Learning (RL), focusing on the objective of Conditional Value at Risk (CVaR) with risk tolerance $\tau$. Starting with multi-arm bandits (MABs), we show the minimax CVaR regret rate is $\Omega(\sqrt{\tau^{-1}AK})$, where $A$ is the number of actions and $K$ is the number of episodes, and that it is achieved by an Upper Confidence Bound algorithm with a novel Bernstein bonus. For online RL in tabular Markov Decision Processes (MDPs), we show a minimax regret lower bound of $\Omega(\sqrt{\tau^{-1}SAK})$ (with normalized cumulative rewards), where $S$ is the number of states, and we propose a novel bonus-driven Value Iteration procedure. We show that our algorithm achieves the optimal regret of $\widetilde O(\sqrt{\tau^{-1}SAK})$ under a continuity assumption and in general attains a near-optimal regret of $\widetilde O(\tau^{-1}\sqrt{SAK})$, which is minimax-optimal for constant $\tau$. This improves on the best available bounds. By discretizing rewards appropriately, our algorithms are computationally efficient.

A crucial problem in reinforcement learning is learning the optimal policy. We study this in tabular infinite-horizon discounted Markov decision processes under the online setting. The existing algorithms either fail to achieve regret optimality or have to incur a high memory and computational cost. In addition, existing optimal algorithms all require a long burn-in time in order to achieve optimal sample efficiency, i.e., their optimality is not guaranteed unless sample size surpasses a high threshold. We address both open problems by introducing a model-free algorithm that employs variance reduction and a novel technique that switches the execution policy in a slow-yet-adaptive manner. This is the first regret-optimal model-free algorithm in the discounted setting, with the additional benefit of a low burn-in time.

Actor-critic (AC) methods are widely used in reinforcement learning (RL) and benefit from the flexibility of using any policy gradient method as the actor and value-based method as the critic. The critic is usually trained by minimizing the TD error, an objective that is potentially decorrelated with the true goal of achieving a high reward with the actor. We address this mismatch by designing a joint objective for training the actor and critic in a decision-aware fashion. We use the proposed objective to design a generic, AC algorithm that can easily handle any function approximation. We explicitly characterize the conditions under which the resulting algorithm guarantees monotonic policy improvement, regardless of the choice of the policy and critic parameterization. Instantiating the generic algorithm results in an actor that involves maximizing a sequence of surrogate functions (similar to TRPO, PPO) and a critic that involves minimizing a closely connected objective. Using simple bandit examples, we provably establish the benefit of the proposed critic objective over the standard squared error. Finally, we empirically demonstrate the benefit of our decision-aware actor-critic framework on simple RL problems.

We consider the adversarial linear contextual bandit setting, which allows for the loss functions associated with each of $K$ arms to change over time without restriction. Assuming the $d$-dimensional contexts are drawn from a fixed known distribution, the worst-case expected regret over the course of $T$ rounds is known to scale as $\tilde O(\sqrt{Kd T})$. Under the additional assumption that the density of the contexts is log-concave, we obtain a second-order bound of order $\tilde O(K\sqrt{d V_T})$ in terms of the cumulative second moment of the learner's losses $V_T$, and a closely related first-order bound of order $\tilde O(K\sqrt{d L_T^*})$ in terms of the cumulative loss of the best policy $L_T^*$. Since $V_T$ or $L_T^*$ may be significantly smaller than $T$, these improve over the worst-case regret whenever the environment is relatively benign. Our results are obtained using a truncated version of the continuous exponential weights algorithm over the probability simplex, which we analyse by exploiting a novel connection to the linear bandit setting without contexts.

The relationship between the number of training data points, the number of parameters in a statistical model, and the generalization capabilities of the model has been widely studied. Previous work has shown that double descent can occur in the over-parameterized regime, and believe that the standard bias-variance trade-off holds in the under-parameterized regime. In this paper, we present a simple example that provably exhibits double descent in the under-parameterized regime. For simplicity, we look at the ridge regularized least squares denoising problem with data on a line embedded in high-dimension space. By deriving an asymptotically accurate formula for the generalization error, we observe sample-wise and parameter-wise double descent with the peak in the under-parameterized regime rather than at the interpolation point or in the over-parameterized regime. Further, the peak of the sample-wise double descent curve corresponds to a peak in the curve for the norm of the estimator, and adjusting $\mu$, the strength of the ridge regularization, shifts the location of the peak. We observe that parameter-wise double descent occurs for this model for small $\mu$. For larger values of $\mu$, we observe that the curve for the norm of the estimator has a peak but that this no longer translates to a peak in the generalization error. Moreover, we study the training error for this problem. The considered problem setup allows for studying the interaction between two regularizers. We provide empirical evidence that the model implicitly favors using the ridge regularizer over the input data noise regularizer. Thus, we show that even though both regularizers regularize the same quantity, i.e., the norm of the estimator, they are not equivalent.

The problem of constrained reinforcement learning (CRL) holds significant importance as it provides a framework for addressing critical safety satisfaction concerns in the field of reinforcement learning (RL). However, with the introduction of constraint satisfaction, the current CRL methods necessitate the utilization of second-order optimization or primal-dual frameworks with additional Lagrangian multipliers, resulting in increased complexity and inefficiency during implementation. To address these issues, we propose a novel first-order feasible method named Constrained Proximal Policy Optimization (CPPO). By treating the CRL problem as a probabilistic inference problem, our approach integrates the Expectation-Maximization framework to solve it through two steps: 1) calculating the optimal policy distribution within the feasible region (E-step), and 2) conducting a first-order update to adjust the current policy towards the optimal policy obtained in the E-step (M-step). We establish the relationship between the probability ratios and KL divergence to convert the E-step into a convex optimization problem. Furthermore, we develop an iterative heuristic algorithm from a geometric perspective to solve this problem. Additionally, we introduce a conservative update mechanism to overcome the constraint violation issue that occurs in the existing feasible region method. Empirical evaluations conducted in complex and uncertain environments validate the effectiveness of our proposed method, as it performs at least as well as other baselines.

Electroencephalogram (EEG) provides noninvasive measures of brain activity and is found to be valuable for diagnosis of some chronic disorders. Specifically, pre-treatment EEG signals in alpha and theta frequency bands have demonstrated some association with anti-depressant response, which is well-known to have low response rate. We aim to design an integrated pipeline that improves the response rate of major depressive disorder patients by developing an individualized treatment policy guided by the resting state pre-treatment EEG recordings and other treatment effects modifiers. We first design an innovative automatic site-specific EEG preprocessing pipeline to extract features that possess stronger signals compared with raw data. We then estimate the conditional average treatment effect using causal forests, and use a doubly robust technique to improve the efficiency in the estimation of the average treatment effect. We present evidence of heterogeneity in the treatment effect and the modifying power of EEG features as well as a significant average treatment effect, a result that cannot be obtained by conventional methods. Finally, we employ an efficient policy learning algorithm to learn an optimal depth-2 treatment assignment decision tree and compare its performance with Q-Learning and outcome-weighted learning via simulation studies and an application to a large multi-site, double-blind randomized controlled clinical trial, EMBARC.

We introduce DeepNash, an autonomous agent capable of learning to play the imperfect information game Stratego from scratch, up to a human expert level. Stratego is one of the few iconic board games that Artificial Intelligence (AI) has not yet mastered. This popular game has an enormous game tree on the order of $10^{535}$ nodes, i.e., $10^{175}$ times larger than that of Go. It has the additional complexity of requiring decision-making under imperfect information, similar to Texas hold'em poker, which has a significantly smaller game tree (on the order of $10^{164}$ nodes). Decisions in Stratego are made over a large number of discrete actions with no obvious link between action and outcome. Episodes are long, with often hundreds of moves before a player wins, and situations in Stratego can not easily be broken down into manageably-sized sub-problems as in poker. For these reasons, Stratego has been a grand challenge for the field of AI for decades, and existing AI methods barely reach an amateur level of play. DeepNash uses a game-theoretic, model-free deep reinforcement learning method, without search, that learns to master Stratego via self-play. The Regularised Nash Dynamics (R-NaD) algorithm, a key component of DeepNash, converges to an approximate Nash equilibrium, instead of 'cycling' around it, by directly modifying the underlying multi-agent learning dynamics. DeepNash beats existing state-of-the-art AI methods in Stratego and achieved a yearly (2022) and all-time top-3 rank on the Gravon games platform, competing with human expert players.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.