We consider bandit optimization of a smooth reward function, where the goal is cumulative regret minimization. This problem has been studied for $α$-Hölder continuous (including Lipschitz) functions with $0<α\leq 1$. Our main result is in generalization of the reward function to Hölder space with exponent $α>1$ to bridge the gap between Lipschitz bandits and infinitely-differentiable models such as linear bandits. For Hölder continuous functions, approaches based on random sampling in bins of a discretized domain suffices as optimal. In contrast, we propose a class of two-layer algorithms that deploy misspecified linear/polynomial bandit algorithms in bins. We demonstrate that the proposed algorithm can exploit higher-order smoothness of the function by deriving a regret upper bound of $\tilde{O}(T^\frac{d+α}{d+2α})$ for when $α>1$, which matches existing lower bound. We also study adaptation to unknown function smoothness over a continuous scale of Hölder spaces indexed by $α$, with a bandit model selection approach applied with our proposed two-layer algorithms. We show that it achieves regret rate that matches the existing lower bound for adaptation within the $α\leq 1$ subset.
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Microsoft Research（微软研究院）
Designing efficient general-purpose contextual bandit algorithms that work with large -- or even continuous -- action spaces would facilitate application to important scenarios such as information retrieval, recommendation systems, and continuous control. While obtaining standard regret guarantees can be hopeless, alternative regret notions have been proposed to tackle the large action setting. We propose a smooth regret notion for contextual bandits, which dominates previously proposed alternatives. We design a statistically and computationally efficient algorithm -- for the proposed smooth r…
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Université Paris-Sud（巴黎南大学）
In the context of stochastic continuum-armed bandits, we present an algorithm that adapts to the unknown smoothness of the objective function. We exhibit and compute a polynomial cost of adaptation to the H{ö}lder regularity for regret minimization. To do this, we first reconsider the recent lower bound of Locatelli and Carpentier [20], and define and characterize admissible rate functions. Our new algorithm matches any of these minimal rate functions. We provide a finite-time analysis and a thorough discussion about asymptotic optimality.
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.
STOR-i Centre for Doctoral Training, Lancaster University（兰卡斯特大学STOR-i博士训练中心）；Department of Mathematics and Statistics, Lancaster University（兰卡斯特大学数学与统计系）
Thompson Sampling is a well established approach to bandit and reinforcement learning problems. However its use in continuum armed bandit problems has received relatively little attention. We provide the first bounds on the regret of Thompson Sampling for continuum armed bandits under weak conditions on the function class containing the true function and sub-exponential observation noise. Our bounds are realised by analysis of the eluder dimension, a recently proposed measure of the complexity of a function class, which has been demonstrated to be useful in bounding the Bayesian regret of Thom…
We introduce a new stochastic smoothing perspective to study adversarial contextual bandit problems. We propose a general algorithm template that represents random perturbation based algorithms and identify several perturbation distributions that lead to strong regret bounds. Using the idea of smoothness, we provide an $O(\sqrt{T})$ zero-order bound for the vanilla algorithm and an $O(L^{*2/3}_{T})$ first-order bound for the clipped version. These bounds hold when the algorithms use with a variety of distributions that have a bounded hazard rate. Our algorithm template includes EXP4 as a speci…
We consider the neural contextual bandit problem. In contrast to the existing work which primarily focuses on ReLU neural nets, we consider a general set of smooth activation functions. Under this more general setting, (i) we derive non-asymptotic error bounds on the difference between an overparameterized neural net and its corresponding neural tangent kernel, (ii) we propose an algorithm with a provably sublinear regret bound that is also efficient in the finite regime as demonstrated by empirical studies. The non-asymptotic error bounds may be of broader interest as a tool to establish the …
We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. In practice, however, the smoothness of payoff functions is typically not known in advance, and misspecification of smoothness may severely deteriorate the performance of existing methods. In this work, we consider a framework where the smoothness of payoff functions is not known,…
University of Colorado, Boulder（科罗拉多大学博尔德分校）；University of Virginia（弗吉尼亚大学）
We investigate the sparse linear contextual bandit problem where the parameter $θ$ is sparse. To relieve the sampling inefficiency, we utilize the "perturbed adversary" where the context is generated adversarilly but with small random non-adaptive perturbations. We prove that the simple online Lasso supports sparse linear contextual bandit with regret bound $\mathcal{O}(\sqrt{kT\log d})$ even when $d \gg T$ where $k$ and $d$ are the number of effective and ambient dimension, respectively. Compared to the recent work from Sivakumar et al. (2020), our analysis does not rely on the precondition p…
We consider the problem of controlling a known linear dynamical system under stochastic noise, adversarially chosen costs, and bandit feedback. Unlike the full feedback setting where the entire cost function is revealed after each decision, here only the cost incurred by the learner is observed. We present a new and efficient algorithm that, for strongly convex and smooth costs, obtains regret that grows with the square root of the time horizon $T$. We also give extensions of this result to general convex, possibly non-smooth costs, and to non-stochastic system noise. A key component of our al…
We study best-arm identification with fixed confidence in bandit models with graph smoothness constraint. We provide and analyze an efficient gradient ascent algorithm to compute the sample complexity of this problem as a solution of a non-smooth max-min problem (providing in passing a simplified analysis for the unconstrained case). Building on this algorithm, we propose an asymptotically optimal strategy. We furthermore illustrate by numerical experiments both the strategy's efficiency and the impact of the smoothness constraint on the sample complexity. Best Arm Identification (BAI) is an i…
Harvard University（哈佛大学）；Microsoft Research NYC（微软纽约研究院）；Microsoft Research, New York（微软纽约研究院）；Carnegie Mellon University（卡内基梅隆大学）
Online learning algorithms, widely used to power search and content optimization on the web, must balance exploration and exploitation, potentially sacrificing the experience of current users in order to gain information that will lead to better decisions in the future. While necessary in the worst case, explicit exploration has a number of disadvantages compared to the greedy algorithm that always "exploits" by choosing an action that currently looks optimal. We ask under what conditions inherent diversity in the data makes explicit exploration unnecessary. We build on a recent line of work o…
Technion, Israel Institute of Technology（以色列理工学院）；Nvidia Research（英伟达研究院）
The Combinatorial Multi-Armed Bandit problem is a sequential decision-making problem in which an agent selects a set of arms on each round, observes feedback for each of these arms and aims to maximize a known reward function of the arms it chose. While previous work proved regret upper bounds in this setting for general reward functions, only a few works provided matching lower bounds, all for specific reward functions. In this work, we prove regret lower bounds for combinatorial bandits that hold under mild assumptions for all smooth reward functions. We derive both problem-dependent and pro…
We study a nonparametric contextual bandit problem where the expected reward functions belong to a Hölder class with smoothness parameter $β$. We show how this interpolates between two extremes that were previously studied in isolation: non-differentiable bandits ($β\leq1$), where rate-optimal regret is achieved by running separate non-contextual bandits in different context regions, and parametric-response bandits (satisfying $β=\infty$), where rate-optimal regret can be achieved with minimal or no exploration due to infinite extrapolatability. We develop a novel algorithm that carefully adju…
In this paper, we analyze the continuous armed bandit problems for nonconvex cost functions under certain smoothness and sublevel set assumptions. We first derive an upper bound on the expected cumulative regret of a simple bin splitting method. We then propose an adaptive bin splitting method, which can significantly improve the performance. Furthermore, a minimax lower bound is derived, which shows that our new adaptive method achieves locally minimax optimal expected cumulative regret.
Bandit learning algorithms typically involve the balance of exploration and exploitation. However, in many practical applications, worst-case scenarios needing systematic exploration are seldom encountered. In this work, we consider a smoothed setting for structured linear contextual bandits where the adversarial contexts are perturbed by Gaussian noise and the unknown parameter $θ^*$ has structure, e.g., sparsity, group sparsity, low rank, etc. We propose simple greedy algorithms for both the single- and multi-parameter (i.e., different parameter for each context) settings and provide a unifi…
Bridging Adversarial and Nonstationary Multi-armed Bandit
Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）；The Hong Kong University of Science and Technology（香港科技大学）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
We consider the combinatorial multi-armed bandit (CMAB) problem, where the reward function is nonlinear. In this setting, the agent chooses a batch of arms on each round and receives feedback from each arm of the batch. The reward that the agent aims to maximize is a function of the selected arms and their expectations. In many applications, the reward function is highly nonlinear, and the performance of existing algorithms relies on a global Lipschitz constant to encapsulate the function's nonlinearity. This may lead to loose regret bounds, since by itself, a large gradient does not necessari…
University College London（伦敦大学学院）；University of Oxford（牛津大学）
We consider a stochastic linear bandit problem with multiple users, where the relationship between users is captured by an underlying graph and user preferences are represented as smooth signals on the graph. We introduce a novel bandit algorithm where the smoothness prior is imposed via the random-walk graph Laplacian, which leads to a single-user cumulative regret scaling as $\tilde{\mathcal{O}}(Ψd \sqrt{T})$ with time horizon $T$, feature dimensionality $d$, and the scalar parameter $Ψ\in (0,1)$ that depends on the graph connectivity. This is an improvement over $\tilde{\mathcal{O}}(d \sqrt…
We consider the problem of optimising functions in the reproducing kernel Hilbert space (RKHS) of a Matérn kernel with smoothness parameter $ν$ over the domain $[0,1]^d$ under noisy bandit feedback. Our contribution, the $π$-GP-UCB algorithm, is the first practical approach with guaranteed sublinear regret for all $ν>1$ and $d \geq 1$. Empirical validation suggests better performance and drastically improved computational scalablity compared with its predecessor, Improved GP-UCB.
Bandit learning is characterized by the tension between long-term exploration and short-term exploitation. However, as has recently been noted, in settings in which the choices of the learning algorithm correspond to important decisions about individual people (such as criminal recidivism prediction, lending, and sequential drug trials), exploration corresponds to explicitly sacrificing the well-being of one individual for the potential future benefit of others. This raises a fairness concern. In such settings, one might like to run a "greedy" algorithm, which always makes the (myopically) opt…
National University of Singapore（新加坡国立大学）；Cornell University（康奈尔大学）
Consider a nonparametric contextual multi-arm bandit problem where each arm $a \in [K]$ is associated to a nonparametric reward function $f_a: [0,1] \to \mathbb{R}$ mapping from contexts to the expected reward. Suppose that there is a large set of arms, yet there is a simple but unknown structure amongst the arm reward functions, e.g. finite types or smooth with respect to an unknown metric space. We present a novel algorithm which learns data-driven similarities amongst the arms, in order to implement adaptive partitioning of the context-arm space for more efficient learning. We provide regre…
University of Southern California（南加利福尼亚大学）；Nanjing University（南京大学）
We consider the problem of combining and learning over a set of adversarial bandit algorithms with the goal of adaptively tracking the best one on the fly. The CORRAL algorithm of Agarwal et al. (2017) and its variants (Foster et al., 2020a) achieve this goal with a regret overhead of order $\widetilde{O}(\sqrt{MT})$ where $M$ is the number of base algorithms and $T$ is the time horizon. The polynomial dependence on $M$, however, prevents one from applying these algorithms to many applications where $M$ is poly$(T)$ or even larger. Motivated by this issue, we propose a new recipe to corral a l…
Columbia University（哥伦比亚大学）；University of Michigan（密歇根大学）
The $K$-armed dueling bandit problem, where the feedback is in the form of noisy pairwise comparisons, has been widely studied. Previous works have only focused on the sequential setting where the policy adapts after every comparison. However, in many applications such as search ranking and recommendation systems, it is preferable to perform comparisons in a limited number of parallel batches. We study the batched $K$-armed dueling bandit problem under two standard settings: (i) existence of a Condorcet winner, and (ii) strong stochastic transitivity and stochastic triangle inequality. For bot…
University of Washington（华盛顿大学）；Harvard University（哈佛大学）
We study a multi-armed bandit problem in a dynamic environment where arm rewards evolve in a correlated fashion according to a Markov chain. Different than much of the work on related problems, in our formulation a learning algorithm does not have access to either a priori information or observations of the state of the Markov chain and only observes smoothed reward feedback following time intervals we refer to as epochs. We demonstrate that existing methods such as UCB and $\varepsilon$-greedy can suffer linear regret in such an environment. Employing mixing-time bounds on Markov chains, we d…
University of California Berkeley（加州大学伯克利分校）；University of Massachusetts（马萨诸塞大学）；DeepMind（深度思维）；Adobe（奥多比）；Google Research（谷歌研究院）；University of Alberta（阿尔伯塔大学）
We study bandit model selection in stochastic environments. Our approach relies on a meta-algorithm that selects between candidate base algorithms. We develop a meta-algorithm-base algorithm abstraction that can work with general classes of base algorithms and different type of adversarial meta-algorithms. Our methods rely on a novel and generic smoothing transformation for bandit algorithms that permits us to obtain optimal $O(\sqrt{T})$ model selection guarantees for stochastic contextual bandit problems as long as the optimal base algorithm satisfies a high probability regret guarantee. We …
A conversion theorem and minimax optimality for continuum contextual bandits
University of Oxford CMAP, Ecole Polytechnique, IP Paris（牛津大学计算机应用数学中心，巴黎高等技术学院，IP巴黎）；CMAP, Ecole Polytechnique, IP Paris（计算机应用数学中心，巴黎高等技术学院，IP巴黎）；CSML, Istituto Italiano di Tecnologia University College London（计算机科学与机器学习，意大利技术研究所，伦敦大学学院）；CREST, ENSAE, IP Paris（CREST，ENSAE，IP巴黎）
Continuum-armed bandits (a.k.a., black-box or $0^{th}$-order optimization) involves optimizing an unknown objective function given an oracle that evaluates the function at a query point, with the goal of using as few query points as possible. In the most well-studied case, the objective function is assumed to be Lipschitz continuous and minimax rates of simple and cumulative regrets are known in both noiseless and noisy settings. This paper studies continuum-armed bandits under more general smoothness conditions, namely Besov smoothness conditions, on the objective function. In both noiseless …
Zalando Research（Zalando研究）；OLX Berlin Hub（OLX柏林中心）
Deep Reinforcement Learning has been shown to be very successful in complex games, e.g. Atari or Go. These games have clearly defined rules, and hence allow simulation. In many practical applications, however, interactions with the environment are costly and a good simulator of the environment is not available. Further, as environments differ by application, the optimal inductive bias (architecture, hyperparameters, etc.) of a reinforcement agent depends on the application. In this work, we propose a multi-arm bandit framework that selects from a set of different reinforcement learning agents …
Pure exploration in multi-armed bandits has emerged as an important framework for modeling decision-making and search under uncertainty. In modern applications, however, one is often faced with a tremendously large number of options. Even obtaining one observation per option may be too costly rendering traditional pure exploration algorithms ineffective. Fortunately, one often has access to similar relationships amongst the options that can be leveraged. In this paper, we consider the pure exploration problem in stochastic multi-armed bandits where the similarities between the arms are capture…
Graph Dimensionality Reduction for Contextual Bandits: Structure-Specific Regret Bounds under Approximate Smoothness and Noisy Eigenspaces
Department of Computer and Information Sciences, University of Delaware, USA（德克萨斯大学达勒姆分校计算机与信息科学系）；Department of Computer Science, Iowa State University, USA（爱荷华州立大学计算机科学系）；Department of Civil, Construction & Environmental Engineering, Iowa State University, USA（爱荷华州立大学土木、建设与环境工程系）
Heidelberg University（海德堡大学）；Amazon Development Center（亚马逊开发中心）
The goal of counterfactual learning for statistical machine translation (SMT) is to optimize a target SMT system from logged data that consist of user feedback to translations that were predicted by another, historic SMT system. A challenge arises by the fact that risk-averse commercial SMT systems deterministically log the most probable translation. The lack of sufficient exploration of the SMT output space seemingly contradicts the theoretical requirements for counterfactual learning. We show that counterfactual learning from deterministic bandit logs is possible nevertheless by smoothing ou…
ENS de Lyon（里昂大学）；Inria Lille – Nord Europe, SequeL team（里尔-北欧研究所，SequeL团队）；DeepMind Paris（巴黎DeepMind）；Google Research（谷歌研究）；Columbia University（哥伦比亚大学）
We consider a stochastic linear bandit model in which the available actions correspond to arbitrary context vectors whose associated rewards follow a non-stationary linear regression model. In this setting, the unknown regression parameter is allowed to vary in time. To address this problem, we propose D-LinUCB, a novel optimistic algorithm based on discounted linear regression, where exponential weights are used to smoothly forget the past. This involves studying the deviations of the sequential weighted least-squares estimator under generic assumptions. As a by-product, we obtain novel devia…
Much of modern learning theory has been split between two regimes: the classical offline setting, where data arrive independently, and the online setting, where data arrive adversarially. While the former model is often both computationally and statistically tractable, the latter requires no distributional assumptions. In an attempt to achieve the best of both worlds, previous work proposed the smooth online setting where each sample is drawn from an adversarially chosen distribution, which is smooth, i.e., it has a bounded density with respect to a fixed dominating measure. We provide tight b…
