Liyu Chen

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.

Liyu Chen, Haipeng Luo, Aviv Rosenberg

Policy optimization is among the most popular and successful reinforcement learning algorithms, and there is increasing interest in understanding its theoretical guarantees. In this work, we initiate the study of policy optimization for the stochastic shortest path (SSP) problem, a goal-oriented reinforcement learning model that strictly generalizes the finite-horizon model and better captures many applications. We consider a wide range of settings, including stochastic and adversarial environments under full information or bandit feedback, and propose a policy optimization algorithm for each setting that makes use of novel correction terms and/or variants of dilated bonuses (Luo et al., 2021). For most settings, our algorithm is shown to achieve a near-optimal regret bound. One key technical contribution of this work is a new approximation scheme to tackle SSP problems that we call \textit{stacked discounted approximation} and use in all our proposed algorithms. Unlike the finite-horizon approximation that is heavily used in recent SSP algorithms, our new approximation enables us to learn a near-stationary policy with only logarithmic changes during an episode and could lead to an exponential improvement in space complexity.

Liyu Chen, Rahul Jain, Haipeng Luo

We introduce two new no-regret algorithms for the stochastic shortest path (SSP) problem with a linear MDP that significantly improve over the only existing results of (Vial et al., 2021). Our first algorithm is computationally efficient and achieves a regret bound $\widetilde{O}\left(\sqrt{d^3B_{\star}^2T_{\star} K}\right)$, where $d$ is the dimension of the feature space, $B_{\star}$ and $T_{\star}$ are upper bounds of the expected costs and hitting time of the optimal policy respectively, and $K$ is the number of episodes. The same algorithm with a slight modification also achieves logarithmic regret of order $O\left(\frac{d^3B_{\star}^4}{c_{\min}^2\text{gap}_{\min}}\ln^5\frac{dB_{\star} K}{c_{\min}} \right)$, where $\text{gap}_{\min}$ is the minimum sub-optimality gap and $c_{\min}$ is the minimum cost over all state-action pairs. Our result is obtained by developing a simpler and improved analysis for the finite-horizon approximation of (Cohen et al., 2021) with a smaller approximation error, which might be of independent interest. On the other hand, using variance-aware confidence sets in a global optimization problem, our second algorithm is computationally inefficient but achieves the first "horizon-free" regret bound $\widetilde{O}(d^{3.5}B_{\star}\sqrt{K})$ with no polynomial dependency on $T_{\star}$ or $1/c_{\min}$, almost matching the $Ω(dB_{\star}\sqrt{K})$ lower bound from (Min et al., 2021).

Liyu Chen, Mehdi Jafarnia-Jahromi, Rahul Jain et al.

We introduce a generic template for developing regret minimization algorithms in the Stochastic Shortest Path (SSP) model, which achieves minimax optimal regret as long as certain properties are ensured. The key of our analysis is a new technique called implicit finite-horizon approximation, which approximates the SSP model by a finite-horizon counterpart only in the analysis without explicit implementation. Using this template, we develop two new algorithms: the first one is model-free (the first in the literature to our knowledge) and minimax optimal under strictly positive costs; the second one is model-based and minimax optimal even with zero-cost state-action pairs, matching the best existing result from [Tarbouriech et al., 2021b]. Importantly, both algorithms admit highly sparse updates, making them computationally more efficient than all existing algorithms. Moreover, both can be made completely parameter-free.

Mehdi Jafarnia-Jahromi, Liyu Chen, Rahul Jain et al.

We consider the problem of online reinforcement learning for the Stochastic Shortest Path (SSP) problem modeled as an unknown MDP with an absorbing state. We propose PSRL-SSP, a simple posterior sampling-based reinforcement learning algorithm for the SSP problem. The algorithm operates in epochs. At the beginning of each epoch, a sample is drawn from the posterior distribution on the unknown model dynamics, and the optimal policy with respect to the drawn sample is followed during that epoch. An epoch completes if either the number of visits to the goal state in the current epoch exceeds that of the previous epoch, or the number of visits to any of the state-action pairs is doubled. We establish a Bayesian regret bound of $O(B_\star S\sqrt{AK})$, where $B_\star$ is an upper bound on the expected cost of the optimal policy, $S$ is the size of the state space, $A$ is the size of the action space, and $K$ is the number of episodes. The algorithm only requires the knowledge of the prior distribution, and has no hyper-parameters to tune. It is the first such posterior sampling algorithm and outperforms numerically previously proposed optimism-based algorithms.

Liyu Chen, Haipeng Luo, Chen-Yu Wei

We study the stochastic shortest path problem with adversarial costs and known transition, and show that the minimax regret is $\widetilde{O}(\sqrt{DT^\star K})$ and $\widetilde{O}(\sqrt{DT^\star SA K})$ for the full-information setting and the bandit feedback setting respectively, where $D$ is the diameter, $T^\star$ is the expected hitting time of the optimal policy, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. Our results significantly improve upon the existing work of (Rosenberg and Mansour, 2020) which only considers the full-information setting and achieves suboptimal regret. Our work is also the first to consider bandit feedback with adversarial costs. Our algorithms are built on top of the Online Mirror Descent framework with a variety of new techniques that might be of independent interest, including an improved multi-scale expert algorithm, a reduction from general stochastic shortest path to a special loop-free case, a skewed occupancy measure space, and a novel correction term added to the cost estimators. Interestingly, the last two elements reduce the variance of the learner via positive bias and the variance of the optimal policy via negative bias respectively, and having them simultaneously is critical for obtaining the optimal high-probability bound in the bandit feedback setting.