Motivated by practical needs such as large-scale learning, we study the impact of adaptivity constraints to linear contextual bandits, a central problem in online active learning. We consider two popular limited adaptivity models in literature: batch learning and rare policy switches. We show that, when the context vectors are adversarially chosen in $d$-dimensional linear contextual bandits, the learner needs $O(d \log d \log T)$ policy switches to achieve the minimax-optimal regret, and this is optimal up to $\mathrm{poly}(\log d, \log \log T)$ factors; for stochastic context vectors, even in the more restricted batch learning model, only $O(\log \log T)$ batches are needed to achieve the optimal regret. Together with the known results in literature, our results present a complete picture about the adaptivity constraints in linear contextual bandits. Along the way, we propose the distributional optimal design, a natural extension of the optimal experiment design, and provide a both statistically and computationally efficient learning algorithm for the problem, which may be of independent interest.
Naveen Jindal School of Management, University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）
In this paper we study the non-stationary stochastic optimization question with bandit feedback and dynamic regret measures. The seminal work of Besbes et al. (2015) shows that, when aggregated function changes is known a priori, a simple re-starting algorithm attains the optimal dynamic regret. In this work, we designed a stochastic optimization algorithm with fixed step sizes, which combined together with the multi-scale sampling framework of Wei and Luo (2021) achieves the optimal dynamic regret in non-stationary stochastic optimization without requiring prior knowledge of function change b…
Tsinghua University（清华大学）；Northwestern University（西北大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study multinomial logit bandit with limited adaptivity, where the algorithms change their exploration actions as infrequently as possible when achieving almost optimal minimax regret. We propose two measures of adaptivity: the assortment switching cost and the more fine-grained item switching cost. We present an anytime algorithm (AT-DUCB) with $O(N \log T)$ assortment switches, almost matching the lower bound $Ω(\frac{N \log T}{ \log \log T})$. In the fixed-horizon setting, our algorithm FH-DUCB incurs $O(N \log \log T)$ assortment switches, matching the asymptotic lower bound. We also pre…
This paper extends the Distributionally Robust Optimization (DRO) approach for offline contextual bandits. Specifically, we leverage this framework to introduce a convex reformulation of the Counterfactual Risk Minimization principle. Besides relying on convex programs, our approach is compatible with stochastic optimization, and can therefore be readily adapted tothe large data regime. Our approach relies on the construction of asymptotic confidence intervals for offline contextual bandits through the DRO framework. By leveraging known asymptotic results of robust estimators, we also show how…
Multi-armed bandit problems provide a framework to identify the optimal intervention over a sequence of repeated experiments. Without additional assumptions, minimax optimal performance (measured by cumulative regret) is well-understood. With access to additional observed variables that d-separate the intervention from the outcome (i.e., they are a d-separator), recent "causal bandit" algorithms provably incur less regret. However, in practice it is desirable to be agnostic to whether observed variables are a d-separator. Ideally, an algorithm should be adaptive; that is, perform nearly as wel…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Google Research（谷歌研究院）；The Hong Kong University of Science and Technology（香港科技大学）
Recently, several studies (Zhou et al., 2021a; Zhang et al., 2021b; Kim et al., 2021; Zhou and Gu, 2022) have provided variance-dependent regret bounds for linear contextual bandits, which interpolates the regret for the worst-case regime and the deterministic reward regime. However, these algorithms are either computationally intractable or unable to handle unknown variance of the noise. In this paper, we present a novel solution to this open problem by proposing the first computationally efficient algorithm for linear bandits with heteroscedastic noise. Our algorithm is adaptive to the unkno…
Microsoft Research（微软研究院）；University of Arizona（亚利桑那大学）
We study contextual bandit learning with an abstract policy class and continuous action space. We obtain two qualitatively different regret bounds: one competes with a smoothed version of the policy class under no continuity assumptions, while the other requires standard Lipschitz assumptions. Both bounds exhibit data-dependent "zooming" behavior and, with no tuning, yield improved guarantees for benign problems. We also study adapting to unknown smoothness parameters, establishing a price-of-adaptivity and deriving optimal adaptive algorithms that require no additional information.
In this work we revisit two classic high-dimensional online learning problems, namely linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for b…
Georgia Institute of Technology（佐治亚理工学院）；Microsoft Research（微软研究院）
Model selection in contextual bandits is an important complementary problem to regret minimization with respect to a fixed model class. We consider the simplest non-trivial instance of model-selection: distinguishing a simple multi-armed bandit problem from a linear contextual bandit problem. Even in this instance, current state-of-the-art methods explore in a suboptimal manner and require strong "feature-diversity" conditions. In this paper, we introduce new algorithms that a) explore in a data-adaptive manner, and b) provide model selection guarantees of the form $\mathcal{O}(d^α T^{1- α})$ …
Stochastic sparse linear bandits offer a practical model for high-dimensional online decision-making problems and have a rich information-regret structure. In this work we explore the use of information-directed sampling (IDS), which naturally balances the information-regret trade-off. We develop a class of information-theoretic Bayesian regret bounds that nearly match existing lower bounds on a variety of problem instances, demonstrating the adaptivity of IDS. To efficiently implement sparse IDS, we propose an empirical Bayesian approach for sparse posterior sampling using a spike-and-slab Ga…
Chennai Mathematical Institute（金奈数学研究所）；Indian Institute of Science（印度科学学院）
We study how to adapt to smoothly-varying ('easy') environments in well-known online learning problems where acquiring information is expensive. For the problem of label efficient prediction, which is a budgeted version of prediction with expert advice, we present an online algorithm whose regret depends optimally on the number of labels allowed and $Q^*$ (the quadratic variation of the losses of the best action in hindsight), along with a parameter-free counterpart whose regret depends optimally on $Q$ (the quadratic variation of the losses of all the actions). These quantities can be signifi…
University of California San Diego（加州大学圣地亚哥分校）；University of Arizona（亚利桑那大学）
In many real-world applications, multiple agents seek to learn how to perform highly related yet slightly different tasks in an online bandit learning protocol. We formulate this problem as the $ε$-multi-player multi-armed bandit problem, in which a set of players concurrently interact with a set of arms, and for each arm, the reward distributions for all players are similar but not necessarily identical. We develop an upper confidence bound-based algorithm, RobustAgg$(ε)$, that adaptively aggregates rewards collected by different players. In the setting where an upper bound on the pairwise si…
A characterization of sample adaptivity in UCB data
School of Data Science, the Chinese University of Hong Kong, Shenzhen (CUHK Shenzhen)（香港中文大学（深圳）数据科学学院）；School of Management and Economics, the Chinese University of Hong Kong, Shenzhen (CUHK Shenzhen)（香港中文大学（深圳）经管学院）
We develop a novel and generic algorithm for the adversarial multi-armed bandit problem (or more generally the combinatorial semi-bandit problem). When instantiated differently, our algorithm achieves various new data-dependent regret bounds improving previous work. Examples include: 1) a regret bound depending on the variance of only the best arm; 2) a regret bound depending on the first-order path-length of only the best arm; 3) a regret bound depending on the sum of first-order path-lengths of all arms as well as an important negative term, which together lead to faster convergence rates fo…
We study the problem of estimating the value of a known smooth function $f$ at an unknown point $\boldsymbolμ \in \mathbb{R}^n$, where each component $μ_i$ can be sampled via a noisy oracle. Sampling more frequently components of $\boldsymbolμ$ corresponding to directions of the function with larger directional derivatives is more sample-efficient. However, as $\boldsymbolμ$ is unknown, the optimal sampling frequencies are also unknown. We design an instance-adaptive algorithm that learns to sample according to the importance of each coordinate, and with probability at least $1-δ$ returns an $…
Applied AI Institute（应用人工智能研究所）；Deakin University（迪肯大学）；University of Oxford（牛津大学）
Offline policy learning (OPL) leverages existing data collected a priori for policy optimization without any active exploration. Despite the prevalence and recent interest in this problem, its theoretical and algorithmic foundations in function approximation settings remain under-developed. In this paper, we consider this problem on the axes of distributional shift, optimization, and generalization in offline contextual bandits with neural networks. In particular, we propose a provably efficient offline contextual bandit with neural network function approximation that does not require any func…
Amazon（亚马逊）；North Carolina State University（北卡罗来纳州立大学）
Despite the great interest in the bandit problem, designing efficient algorithms for complex models remains challenging, as there is typically no analytical way to quantify uncertainty. In this paper, we propose Multiplier Bootstrap-based Exploration (MBE), a novel exploration strategy that is applicable to any reward model amenable to weighted loss minimization. We prove both instance-dependent and instance-independent rate-optimal regret bounds for MBE in sub-Gaussian multi-armed bandits. With extensive simulation and real data experiments, we show the generality and adaptivity of MBE.
Cornell University（康奈尔大学）；Microsoft Research（微软研究院）；University of Southern California（南加州大学）
A major challenge in contextual bandits is to design general-purpose algorithms that are both practically useful and theoretically well-founded. We present a new technique that has the empirical and computational advantages of realizability-based approaches combined with the flexibility of agnostic methods. Our algorithms leverage the availability of a regression oracle for the value-function class, a more realistic and reasonable oracle than the classification oracles over policies typically assumed by agnostic methods. Our approach generalizes both UCB and LinUCB to far more expressive possi…
Department of Statistics, Purdue University（普渡大学统计系）；Department of Electrical Engineering, Princeton University（普林斯顿大学电气工程系）；Purdue University（普渡大学）；Princeton University（普林斯顿大学）
In this paper, we propose a novel perturbation-based exploration method in bandit algorithms with bounded or unbounded rewards, called residual bootstrap exploration (\texttt{ReBoot}). The \texttt{ReBoot} enforces exploration by injecting data-driven randomness through a residual-based perturbation mechanism. This novel mechanism captures the underlying distributional properties of fitting errors, and more importantly boosts exploration to escape from suboptimal solutions (for small sample sizes) by inflating variance level in an \textit{unconventional} way. In theory, with appropriate varianc…
We propose two linear bandits algorithms with per-step complexity sublinear in the number of arms $K$. The algorithms are designed for applications where the arm set is extremely large and slowly changing. Our key realization is that choosing an arm reduces to a maximum inner product search (MIPS) problem, which can be solved approximately without breaking regret guarantees. Existing approximate MIPS solvers run in sublinear time. We extend those solvers and present theoretical guarantees for online learning problems, where adaptivity (i.e., a later step depends on the feedback in previous ste…
Statistical Inference for Misspecified Contextual Bandits
Department of Statistics, University of Wisconsin–Madison（威斯康星大学麦迪逊分校统计系）；School of Data Science and Society, University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校数据科学与社会学院）
A fundamental challenge in contextual bandits is to develop flexible, general-purpose algorithms with computational requirements no worse than classical supervised learning tasks such as classification and regression. Algorithms based on regression have shown promising empirical success, but theoretical guarantees have remained elusive except in special cases. We provide the first universal and optimal reduction from contextual bandits to online regression. We show how to transform any oracle for online regression with a given value function class into an algorithm for contextual bandits with …
The bias of the sample means of the arms in multi-armed bandits is an important issue in adaptive data analysis that has recently received considerable attention in the literature. Existing results relate in precise ways the sign and magnitude of the bias to various sources of data adaptivity, but do not apply to the conditional inference setting in which the sample means are computed only if some specific conditions are satisfied. In this paper, we characterize the sign of the conditional bias of monotone functions of the rewards, including the sample mean. Our results hold for arbitrary cond…
Several learning problems involve solving min-max problems, e.g., empirical distributional robust learning or learning with non-standard aggregated losses. More specifically, these problems are convex-linear problems where the minimization is carried out over the model parameters $w\in\mathcal{W}$ and the maximization over the empirical distribution $p\in\mathcal{K}$ of the training set indexes, where $\mathcal{K}$ is the simplex or a subset of it. To design efficient methods, we let an online learning algorithm play against a (combinatorial) bandit algorithm. We argue that the efficiency of s…
