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We revisit the problem of offline reinforcement learning with value function realizability but without Bellman completeness. Previous work by Xie and Jiang (2021) and Foster et al. (2022) left open the question whether a bounded concentrability coefficient along with trajectory-based offline data admits a polynomial sample complexity. In this work, we provide a negative answer to this question for the task of offline policy evaluation. In addition to addressing this question, we provide a rather complete picture for offline policy evaluation with only value function realizability. Our primary findings are threefold: 1) The sample complexity of offline policy evaluation is governed by the concentrability coefficient in an aggregated Markov Transition Model jointly determined by the function class and the offline data distribution, rather than that in the original MDP. This unifies and generalizes the ideas of Xie and Jiang (2021) and Foster et al. (2022), 2) The concentrability coefficient in the aggregated Markov Transition Model may grow exponentially with the horizon length, even when the concentrability coefficient in the original MDP is small and the offline data is admissible (i.e., the data distribution equals the occupancy measure of some policy), 3) Under value function realizability, there is a generic reduction that can convert any hard instance with admissible data to a hard instance with trajectory data, implying that trajectory data offers no extra benefits over admissible data. These three pieces jointly resolve the open problem, though each of them could be of independent interest.

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While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when the utilities are non-concave, a situation that is common in machine learning applications where the agents' strategies are parameterized by deep neural networks, or the agents' utilities are computed by a neural network, or both. Indeed, non-concave games present a host of game-theoretic and optimization challenges: (i) Nash equilibria may fail to exist; (ii) local Nash equilibria exist but are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria have infinite support in general, and are intractable. To sidestep these challenges we propose a new solution concept, termed $(\varepsilon, \Phi(\delta))$-local equilibrium, which generalizes local Nash equilibrium in non-concave games, as well as (coarse) correlated equilibrium in concave games. Importantly, we show that two instantiations of this solution concept capture the convergence guarantees of Online Gradient Descent and no-regret learning, which we show efficiently converge to this type of equilibrium in non-concave games with smooth utilities.

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We study policy optimization algorithms for computing correlated equilibria in multi-player general-sum Markov Games. Previous results achieve $O(T^{-1/2})$ convergence rate to a correlated equilibrium and an accelerated $O(T^{-3/4})$ convergence rate to the weaker notion of coarse correlated equilibrium. In this paper, we improve both results significantly by providing an uncoupled policy optimization algorithm that attains a near-optimal $\tilde{O}(T^{-1})$ convergence rate for computing a correlated equilibrium. Our algorithm is constructed by combining two main elements (i) smooth value updates and (ii) the optimistic-follow-the-regularized-leader algorithm with the log barrier regularizer.

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We study online reinforcement learning in linear Markov decision processes with adversarial losses and bandit feedback, without prior knowledge on transitions or access to simulators. We introduce two algorithms that achieve improved regret performance compared to existing approaches. The first algorithm, although computationally inefficient, ensures a regret of $\widetilde{\mathcal{O}}\left(\sqrt{K}\right)$, where $K$ is the number of episodes. This is the first result with the optimal $K$ dependence in the considered setting. The second algorithm, which is based on the policy optimization framework, guarantees a regret of $\widetilde{\mathcal{O}}\left(K^{\frac{3}{4}} \right)$ and is computationally efficient. Both our results significantly improve over the state-of-the-art: a computationally inefficient algorithm by Kong et al. [2023] with $\widetilde{\mathcal{O}}\left(K^{\frac{4}{5}}+poly\left(\frac{1}{\lambda_{\min}}\right) \right)$ regret, for some problem-dependent constant $\lambda_{\min}$ that can be arbitrarily close to zero, and a computationally efficient algorithm by Sherman et al. [2023b] with $\widetilde{\mathcal{O}}\left(K^{\frac{6}{7}} \right)$ regret.

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We consider the adversarial linear contextual bandit problem, where the loss vectors are selected fully adversarially and the per-round action set (i.e. the context) is drawn from a fixed distribution. Existing methods for this problem either require access to a simulator to generate free i.i.d. contexts, achieve a sub-optimal regret no better than $\widetilde{O}(T^{\frac{5}{6}})$, or are computationally inefficient. We greatly improve these results by achieving a regret of $\widetilde{O}(\sqrt{T})$ without a simulator, while maintaining computational efficiency when the action set in each round is small. In the special case of sleeping bandits with adversarial loss and stochastic arm availability, our result answers affirmatively the open question by Saha et al. [2020] on whether there exists a polynomial-time algorithm with $poly(d)\sqrt{T}$ regret. Our approach naturally handles the case where the loss is linear up to an additive misspecification error, and our regret shows near-optimal dependence on the magnitude of the error.

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We study the problem of computing an optimal policy of an infinite-horizon discounted constrained Markov decision process (constrained MDP). Despite the popularity of Lagrangian-based policy search methods used in practice, the oscillation of policy iterates in these methods has not been fully understood, bringing out issues such as violation of constraints and sensitivity to hyper-parameters. To fill this gap, we employ the Lagrangian method to cast a constrained MDP into a constrained saddle-point problem in which max/min players correspond to primal/dual variables, respectively, and develop two single-time-scale policy-based primal-dual algorithms with non-asymptotic convergence of their policy iterates to an optimal constrained policy. Specifically, we first propose a regularized policy gradient primal-dual (RPG-PD) method that updates the policy using an entropy-regularized policy gradient, and the dual via a quadratic-regularized gradient ascent, simultaneously. We prove that the policy primal-dual iterates of RPG-PD converge to a regularized saddle point with a sublinear rate, while the policy iterates converge sublinearly to an optimal constrained policy. We further instantiate RPG-PD in large state or action spaces by including function approximation in policy parametrization, and establish similar sublinear last-iterate policy convergence. Second, we propose an optimistic policy gradient primal-dual (OPG-PD) method that employs the optimistic gradient method to update primal/dual variables, simultaneously. We prove that the policy primal-dual iterates of OPG-PD converge to a saddle point that contains an optimal constrained policy, with a linear rate. To the best of our knowledge, this work appears to be the first non-asymptotic policy last-iterate convergence result for single-time-scale algorithms in constrained MDPs.

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Existing online learning algorithms for adversarial Markov Decision Processes achieve ${O}(\sqrt{T})$ regret after $T$ rounds of interactions even if the loss functions are chosen arbitrarily by an adversary, with the caveat that the transition function has to be fixed. This is because it has been shown that adversarial transition functions make no-regret learning impossible. Despite such impossibility results, in this work, we develop algorithms that can handle both adversarial losses and adversarial transitions, with regret increasing smoothly in the degree of maliciousness of the adversary. More concretely, we first propose an algorithm that enjoys $\widetilde{{O}}(\sqrt{T} + C^{\textsf{P}})$ regret where $C^{\textsf{P}}$ measures how adversarial the transition functions are and can be at most ${O}(T)$. While this algorithm itself requires knowledge of $C^{\textsf{P}}$, we further develop a black-box reduction approach that removes this requirement. Moreover, we also show that further refinements of the algorithm not only maintains the same regret bound, but also simultaneously adapts to easier environments (where losses are generated in a certain stochastically constrained manner as in Jin et al. [2021]) and achieves $\widetilde{{O}}(U + \sqrt{UC^{\textsf{L}}} + C^{\textsf{P}})$ regret, where $U$ is some standard gap-dependent coefficient and $C^{\textsf{L}}$ is the amount of corruption on losses.

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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.

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We revisit the problem of learning in two-player zero-sum Markov games, focusing on developing an algorithm that is $uncoupled$, $convergent$, and $rational$, with non-asymptotic convergence rates. We start from the case of stateless matrix game with bandit feedback as a warm-up, showing an $\mathcal{O}(t^{-\frac{1}{8}})$ last-iterate convergence rate. To the best of our knowledge, this is the first result that obtains finite last-iterate convergence rate given access to only bandit feedback. We extend our result to the case of irreducible Markov games, providing a last-iterate convergence rate of $\mathcal{O}(t^{-\frac{1}{9+\varepsilon}})$ for any $\varepsilon>0$. Finally, we study Markov games without any assumptions on the dynamics, and show a $path convergence$ rate, which is a new notion of convergence we defined, of $\mathcal{O}(t^{-\frac{1}{10}})$. Our algorithm removes the synchronization and prior knowledge requirement of [Wei et al., 2021], which pursued the same goals as us for irreducible Markov games. Our algorithm is related to [Chen et al., 2021, Cen et al., 2021] and also builds on the entropy regularization technique. However, we remove their requirement of communications on the entropy values, making our algorithm entirely uncoupled.

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Best-of-both-worlds algorithms for online learning which achieve near-optimal regret in both the adversarial and the stochastic regimes have received growing attention recently. Existing techniques often require careful adaptation to every new problem setup, including specialised potentials and careful tuning of algorithm parameters. Yet, in domains such as linear bandits, it is still unknown if there exists an algorithm that can simultaneously obtain $O(\log(T))$ regret in the stochastic regime and $\tilde{O}(\sqrt{T})$ regret in the adversarial regime. In this work, we resolve this question positively and present a general reduction from best of both worlds to a wide family of follow-the-regularized-leader (FTRL) and online-mirror-descent (OMD) algorithms. We showcase the capability of this reduction by transforming existing algorithms that are only known to achieve worst-case guarantees into new algorithms with best-of-both-worlds guarantees in contextual bandits, graph bandits and tabular Markov decision processes.

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