Yan Dai

Yan Dai, Haipeng Luo, Liyu Chen · tsinghua

We consider regret minimization for Adversarial Markov Decision Processes (AMDPs), where the loss functions are changing over time and adversarially chosen, and the learner only observes the losses for the visited state-action pairs (i.e., bandit feedback). While there has been a surge of studies on this problem using Online-Mirror-Descent (OMD) methods, very little is known about the Follow-the-Perturbed-Leader (FTPL) methods, which are usually computationally more efficient and also easier to implement since it only requires solving an offline planning problem. Motivated by this, we take a closer look at FTPL for learning AMDPs, starting from the standard episodic finite-horizon setting. We find some unique and intriguing difficulties in the analysis and propose a workaround to eventually show that FTPL is also able to achieve near-optimal regret bounds in this case. More importantly, we then find two significant applications: First, the analysis of FTPL turns out to be readily generalizable to delayed bandit feedback with order-optimal regret, while OMD methods exhibit extra difficulties (Jin et al., 2022). Second, using FTPL, we also develop the first no-regret algorithm for learning communicating AMDPs in the infinite-horizon setting with bandit feedback and stochastic transitions. Our algorithm is efficient assuming access to an offline planning oracle, while even for the easier full-information setting, the only existing algorithm (Chandrasekaran and Tewari, 2021) is computationally inefficient.

Yan Dai, Haipeng Luo, Chen-Yu Wei et al. · tsinghua

We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change arbitrarily over $K$ episodes and the state space can be arbitrarily large. We assume that the Q-function of any policy is linear in some known features, that is, a linear function approximation exists. The best existing regret upper bound for this setting (Luo et al., 2021) is of order $\tilde{\mathcal O}(K^{2/3})$ (omitting all other dependencies), given access to a simulator. This paper provides two algorithms that improve the regret to $\tilde{\mathcal O}(\sqrt K)$ in the same setting. Our first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free linear MDPs, which achieves $\tilde{\mathcal O}(K^{8/9})$ regret and greatly improves over the best existing bound $\tilde{\mathcal O}(K^{14/15})$. This algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu & Olkhovskaya (2020), which could again be of independent interest.

Yan Dai, Qiwen Cui, Simon S. Du · tsinghua

Markov Games (MG) is an important model for Multi-Agent Reinforcement Learning (MARL). It was long believed that the "curse of multi-agents" (i.e., the algorithmic performance drops exponentially with the number of agents) is unavoidable until several recent works (Daskalakis et al., 2023; Cui et al., 2023; Wang et al., 2023). While these works resolved the curse of multi-agents, when the state spaces are prohibitively large and (linear) function approximations are deployed, they either had a slower convergence rate of $O(T^{-1/4})$ or brought a polynomial dependency on the number of actions $A_{\max}$ -- which is avoidable in single-agent cases even when the loss functions can arbitrarily vary with time. This paper first refines the AVLPR framework by Wang et al. (2023), with an insight of designing *data-dependent* (i.e., stochastic) pessimistic estimation of the sub-optimality gap, allowing a broader choice of plug-in algorithms. When specialized to MGs with independent linear function approximations, we propose novel *action-dependent bonuses* to cover occasionally extreme estimation errors. With the help of state-of-the-art techniques from the single-agent RL literature, we give the first algorithm that tackles the curse of multi-agents, attains the optimal $O(T^{-1/2})$ convergence rate, and avoids $\text{poly}(A_{\max})$ dependency simultaneously.

Jiatai Huang, Yan Dai, Longbo Huang

In this paper, we generalize the concept of heavy-tailed multi-armed bandits to adversarial environments, and develop robust best-of-both-worlds algorithms for heavy-tailed multi-armed bandits (MAB), where losses have $α$-th ($1<α\le 2$) moments bounded by $σ^α$, while the variances may not exist. Specifically, we design an algorithm \texttt{HTINF}, when the heavy-tail parameters $α$ and $σ$ are known to the agent, \texttt{HTINF} simultaneously achieves the optimal regret for both stochastic and adversarial environments, without knowing the actual environment type a-priori. When $α,σ$ are unknown, \texttt{HTINF} achieves a $\log T$-style instance-dependent regret in stochastic cases and $o(T)$ no-regret guarantee in adversarial cases. We further develop an algorithm \texttt{AdaTINF}, achieving $\mathcal O(σK^{1-\nicefrac 1α}T^{\nicefrac{1}α})$ minimax optimal regret even in adversarial settings, without prior knowledge on $α$ and $σ$. This result matches the known regret lower-bound (Bubeck et al., 2013), which assumed a stochastic environment and $α$ and $σ$ are both known. To our knowledge, the proposed \texttt{HTINF} algorithm is the first to enjoy a best-of-both-worlds regret guarantee, and \texttt{AdaTINF} is the first algorithm that can adapt to both $α$ and $σ$ to achieve optimal gap-indepedent regret bound in classical heavy-tailed stochastic MAB setting and our novel adversarial formulation.