Pan Xu

Pan Xu, Hongkai Zheng, Eric Mazumdar et al.

We study the efficiency of Thompson sampling for contextual bandits. Existing Thompson sampling-based algorithms need to construct a Laplace approximation (i.e., a Gaussian distribution) of the posterior distribution, which is inefficient to sample in high dimensional applications for general covariance matrices. Moreover, the Gaussian approximation may not be a good surrogate for the posterior distribution for general reward generating functions. We propose an efficient posterior sampling algorithm, viz., Langevin Monte Carlo Thompson Sampling (LMC-TS), that uses Markov Chain Monte Carlo (MCMC) methods to directly sample from the posterior distribution in contextual bandits. Our method is computationally efficient since it only needs to perform noisy gradient descent updates without constructing the Laplace approximation of the posterior distribution. We prove that the proposed algorithm achieves the same sublinear regret bound as the best Thompson sampling algorithms for a special case of contextual bandits, viz., linear contextual bandits. We conduct experiments on both synthetic data and real-world datasets on different contextual bandit models, which demonstrates that directly sampling from the posterior is both computationally efficient and competitive in performance.

Tianyuan Jin, Pan Xu, Xiaokui Xiao et al.

We study the regret of Thompson sampling (TS) algorithms for exponential family bandits, where the reward distribution is from a one-dimensional exponential family, which covers many common reward distributions including Bernoulli, Gaussian, Gamma, Exponential, etc. We propose a Thompson sampling algorithm, termed ExpTS, which uses a novel sampling distribution to avoid the under-estimation of the optimal arm. We provide a tight regret analysis for ExpTS, which simultaneously yields both the finite-time regret bound as well as the asymptotic regret bound. In particular, for a $K$-armed bandit with exponential family rewards, ExpTS over a horizon $T$ is sub-UCB (a strong criterion for the finite-time regret that is problem-dependent), minimax optimal up to a factor $\sqrt{\log K}$, and asymptotically optimal, for exponential family rewards. Moreover, we propose ExpTS$^+$, by adding a greedy exploitation step in addition to the sampling distribution used in ExpTS, to avoid the over-estimation of sub-optimal arms. ExpTS$^+$ is an anytime bandit algorithm and achieves the minimax optimality and asymptotic optimality simultaneously for exponential family reward distributions. Our proof techniques are general and conceptually simple and can be easily applied to analyze standard Thompson sampling with specific reward distributions.

Yizhou Zhang, Guannan Qu, Pan Xu et al.

We study a multi-agent reinforcement learning (MARL) problem where the agents interact over a given network. The goal of the agents is to cooperatively maximize the average of their entropy-regularized long-term rewards. To overcome the curse of dimensionality and to reduce communication, we propose a Localized Policy Iteration (LPI) algorithm that provably learns a near-globally-optimal policy using only local information. In particular, we show that, despite restricting each agent's attention to only its $κ$-hop neighborhood, the agents are able to learn a policy with an optimality gap that decays polynomially in $κ$. In addition, we show the finite-sample convergence of LPI to the global optimal policy, which explicitly captures the trade-off between optimality and computational complexity in choosing $κ$. Numerical simulations demonstrate the effectiveness of LPI.

Zhen Qin, Zhishuai Liu, Pan Xu

signSGD is popular in nonconvex optimization due to its communication efficiency. Yet, existing analyses typically assume data are sampled with replacement in each iteration, contradicting a common practical implementation where data are randomly reshuffled and sequentially fed into the algorithm. This gap leaves the theoretical understanding of the more practical algorithm, signSGD with random reshuffling (SignRR), largely unexplored. We develop the first analysis of SignRR to identify the core technical challenge that prevents a thorough convergence analysis of this method. In particular, given a dataset of size $n$ and $T$ epochs, we show that the expected gradient norm of SignRR is upper bounded by $O(\log(nT)/\sqrt{nT} + σ)$, where $σ$ is the averaged conditional mean square error that may not vanish. To tackle this limitation, we develop two new sign-based algorithms under random reshuffling: SignRVR, which incorporates variance-reduced gradients, and SignRVM, which integrates momentum-based updates. Both algorithms achieve a faster convergence rate of ${O}(\log(nT)/\sqrt{nT} +\log(nT)\sqrt{n}/\sqrt{T})$. We further extend our algorithms to a distributed setting, with a convergence rate of ${O}(\log(n_0T)/\sqrt{n_0T} +\log (n_0T)\sqrt{n_0}/\sqrt{T})$, where $n_0$ is the size of the dataset of a single machine. These results mark the first step towards the theoretical understanding of practical implementation of sign-based optimization algorithms. Finally, we back up our theoretical findings through experiments on simulated and real-world problems, verifying that randomly reshuffled sign methods match or surpass existing baselines.

Weixin Wang, Haoyang Zheng, Guang Lin et al.

Most existing approximate Thompson Sampling (TS) algorithms for multi-armed bandits use Stochastic Gradient Langevin Dynamics (SGLD) or its variants in each round to sample from the posterior, relaxing the need for conjugacy assumptions between priors and reward distributions in vanilla TS. However, they often require approximating a different posterior distribution in different round of the bandit problem. This requires tricky, round-specific tuning of hyperparameters such as dynamic learning rates, causing challenges in both theoretical analysis and practical implementation. To alleviate this non-stationarity, we introduce TS-SA, which incorporates stochastic approximation (SA) within the TS framework. In each round, TS-SA constructs a posterior approximation only using the most recent reward(s), performs a Langevin Monte Carlo (LMC) update, and applies an SA step to average noisy proposals over time. This can be interpreted as approximating a stationary posterior target throughout the entire algorithm, which further yields a fixed step-size, a unified convergence analysis framework, and improved posterior estimates through temporal averaging. We establish near-optimal regret bounds for TS-SA, with a simplified and more intuitive theoretical analysis enabled by interpreting the entire algorithm as a simulation of a stationary SGLD process. Our empirical results demonstrate that even a single-step Langevin update with certain warm-up outperforms existing methods substantially on bandit tasks.

Pan Xu, Yifan Xu

We consider online resource allocation under a typical non-profit setting, where limited or even scarce resources are administered by a not-for-profit organization like a government. We focus on the internal-equity by assuming that arriving requesters are homogeneous in terms of their external factors like demands but heterogeneous for their internal attributes like demographics. Specifically, we associate each arriving requester with one or several groups based on their demographics (i.e., race, gender, and age), and we aim to design an equitable distributing strategy such that every group of requesters can receive a fair share of resources proportional to a preset target ratio. We present two LP-based sampling algorithms and investigate them both theoretically (in terms of competitive-ratio analysis) and experimentally based on real COVID-19 vaccination data maintained by the Minnesota Department of Health. Both theoretical and numerical results show that our LP-based sampling strategies can effectively promote equity, especially when the arrival population is disproportionately represented, as observed in the early stage of the COVID-19 vaccine rollout.