In this work, we develop linear bandit algorithms that automatically adapt to different environments. By plugging a novel loss estimator into the optimization problem that characterizes the instance-optimal strategy, our first algorithm not only achieves nearly instance-optimal regret in stochastic environments, but also works in corrupted environments with additional regret being the amount of corruption, while the state-of-the-art (Li et al., 2019) achieves neither instance-optimality nor the optimal dependence on the corruption amount. Moreover, by equipping this algorithm with an adversarial component and carefully-designed testings, our second algorithm additionally enjoys minimax-optimal regret in completely adversarial environments, which is the first of this kind to our knowledge. Finally, all our guarantees hold with high probability, while existing instance-optimal guarantees only hold in expectation.
We develop several new algorithms for learning Markov Decision Processes in an infinite-horizon average-reward setting with linear function approximation. Using the optimism principle and assuming that the MDP has a linear structure, we first propose a computationally inefficient algorithm with optimal O( √ T ) regret and another computationally efficient variant with O(T 3 4 ) regret, where T is the number of interactions. Next, taking inspiration from adversarial linear bandits, we develop yet another efficient algorithm with O( √ T ) regret under a different set of assumptions, improving the best existing result by Hao et al. [16] with O(T 2 3 ) regret. Moreover, we draw a connection between this algorithm and the Natural Policy Gradient algorithm proposed by Kakade [22], and show that our analysis improves the sample complexity bound recently given by Agarwal et al. [4].Preprint. Under review.
We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent's best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under self-play), agnostic (no need to know the actions played by the opponent), symmetric (players taking symmetric roles in the algorithm), and enjoying a finite-time last-iterate convergence guarantee, all of which are desirable properties of decentralized algorithms. * Equal contribution.
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