Gradient-variation online learning has drawn increasing attention due to its deep connections to game theory and optimization. It has been studied extensively in the full-information setting, but is underexplored with bandit feedback. In this work, we focus on gradient variation in Bandit Convex Optimization (BCO) with two-point feedback. By proposing a refined analysis of the non-consecutive gradient variation, a fundamental quantity in gradient variation with bandit feedback, we improve the dimension dependence for both convex and strongly convex functions compared with the best known results (Chiang et al., 2013). Our improved analysis of the non-consecutive gradient variation also implies other favorable problem-dependent guarantees, such as gradient-variance and small-loss regret bounds. Beyond the two-point setup, we demonstrate the versatility of our technique by achieving the first gradient-variation bound for one-point bandit linear optimization over hyper-rectangular domains. Finally, we validate the effectiveness of our results in more challenging tasks such as dynamic and universal regret minimization, establishing the first gradient-variation dynamic and universal regret bounds for two-point BCO.
Leiden University（莱顿大学）；Boston University（波士顿大学）；University of Southern California（南加州大学）
We study bandit convex optimization methods that adapt to the norm of the comparator, a topic that has only been studied before for its full-information counterpart. Specifically, we develop convex bandit algorithms with regret bounds that are small whenever the norm of the comparator is small. We first use techniques from the full-information setting to develop comparator-adaptive algorithms for linear bandits. Then, we extend the ideas to convex bandits with Lipschitz or smooth loss functions, using a new single-point gradient estimator and carefully designed surrogate losses.
Motivated by applications in clinical trials and finance, we study the problem of online convex optimization (with bandit feedback) where the decision maker is risk-averse. We provide two algorithms to solve this problem. The first one is a descent-type algorithm which is easy to implement. The second algorithm, which combines the ellipsoid method and a center point device, achieves (almost) optimal regret bounds with respect to the number of rounds. To the best of our knowledge this is the first attempt to address risk-aversion in the online convex bandit problem.
Center for Advanced Intelligence Project（高级智能研究中心）；RIKEN（理化学研究所）
The dueling bandit is a learning framework wherein the feedback information in the learning process is restricted to a noisy comparison between a pair of actions. In this research, we address a dueling bandit problem based on a cost function over a continuous space. We propose a stochastic mirror descent algorithm and show that the algorithm achieves an $O(\sqrt{T\log T})$-regret bound under strong convexity and smoothness assumptions for the cost function. Subsequently, we clarify the equivalence between regret minimization in dueling bandit and convex optimization for the cost function. More…
University of Alberta（阿尔伯塔大学）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Imperial College London（伦敦帝国理工学院）
Algorithms for bandit convex optimization and online learning often rely on constructing noisy gradient estimates, which are then used in appropriately adjusted first-order algorithms, replacing actual gradients. Depending on the properties of the function to be optimized and the nature of ``noise'' in the bandit feedback, the bias and variance of gradient estimates exhibit various tradeoffs. In this paper we propose a novel framework that replaces the specific gradient estimation methods with an abstract oracle. With the help of the new framework we unify previous works, reproducing their res…
We consider the closely related problems of bandit convex optimization with two-point feedback, and zero-order stochastic convex optimization with two function evaluations per round. We provide a simple algorithm and analysis which is optimal for convex Lipschitz functions. This improves on \cite{dujww13}, which only provides an optimal result for smooth functions; Moreover, the algorithm and analysis are simpler, and readily extend to non-Euclidean problems. The algorithm is based on a small but surprisingly powerful modification of the gradient estimator.
University of Southern California（南加利福尼亚大学）；National Key Laboratory for Novel Software Technology, Nanjing University（南京大学新型软件技术国家重点实验室）
We consider the problem of adversarial bandit convex optimization, that is, online learning over a sequence of arbitrary convex loss functions with only one function evaluation for each of them. While all previous works assume known and homogeneous curvature on these loss functions, we study a heterogeneous setting where each function has its own curvature that is only revealed after the learner makes a decision. We develop an efficient algorithm that is able to adapt to the curvature on the fly. Specifically, our algorithm not only recovers or \emph{even improves} existing results for several…
We consider the problem of online convex optimization against an arbitrary adversary with bandit feedback, known as bandit convex optimization. We give the first $\tilde{O}(\sqrt{T})$-regret algorithm for this setting based on a novel application of the ellipsoid method to online learning. This bound is known to be tight up to logarithmic factors. Our analysis introduces new tools in discrete convex geometry.
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）
In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.
We consider a novel setting of zeroth order non-convex optimization, where in addition to querying the function value at a given point, we can also duel two points and get the point with the larger function value. We refer to this setting as optimization with dueling-choice bandits since both direct queries and duels are available for optimization. We give the COMP-GP-UCB algorithm based on GP-UCB (Srinivas et al., 2009), where instead of directly querying the point with the maximum Upper Confidence Bound (UCB), we perform a constrained optimization and use comparisons to filter out suboptimal…
We construct a new map from a convex function to a distribution on its domain, with the property that this distribution is a multi-scale exploration of the function. We use this map to solve a decade-old open problem in adversarial bandit convex optimization by showing that the minimax regret for this problem is $\tilde{O}(\mathrm{poly}(n) \sqrt{T})$, where $n$ is the dimension and $T$ the number of rounds. This bound is obtained by studying the dual Bayesian maximin regret via the information ratio analysis of Russo and Van Roy, and then using the multi-scale exploration to solve the Bayesian…
Bandit Convex Optimization (BCO) is a fundamental framework for modeling sequential decision-making with partial information, where the only feedback available to the player is the one-point or two-point function values. In this paper, we investigate BCO in non-stationary environments and choose the \emph{dynamic regret} as the performance measure, which is defined as the difference between the cumulative loss incurred by the algorithm and that of any feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the path-length of the comparator sequence that reflects the non-stationa…
Technion - Israel Institute of Technology（以色列理工学院）
We revisit the challenge of designing online algorithms for the bandit convex optimization problem (BCO) which are also scalable to high dimensional problems. Hence, we consider algorithms that are \textit{projection-free}, i.e., based on the conditional gradient method whose only access to the feasible decision set, is through a linear optimization oracle (as opposed to other methods which require potentially much more computationally-expensive subprocedures, such as computing Euclidean projections). We present the first such algorithm that attains $O(T^{3/4})$ expected regret using only $O(T…
Microsoft Research（微软研究院）；Weizmann Institute of Science（魏茨曼科学研究所）
The problem of stochastic convex optimization with bandit feedback (in the learning community) or without knowledge of gradients (in the optimization community) has received much attention in recent years, in the form of algorithms and performance upper bounds. However, much less is known about the inherent complexity of these problems, and there are few lower bounds in the literature, especially for nonlinear functions. In this paper, we investigate the attainable error/regret in the bandit and derivative-free settings, as a function of the dimension d and the available number of queries T. W…
Microsoft Research（微软研究院）；Stanford University（斯坦福大学）；University of Southern California（南加州大学）
We study adaptive regret bounds in terms of the variation of the losses (the so-called path-length bounds) for both multi-armed bandit and more generally linear bandit. We first show that the seemingly suboptimal path-length bound of (Wei and Luo, 2018) is in fact not improvable for adaptive adversary. Despite this negative result, we then develop two new algorithms, one that strictly improves over (Wei and Luo, 2018) with a smaller path-length measure, and the other which improves over (Wei and Luo, 2018) for oblivious adversary when the path-length is large. Our algorithms are based on the w…
We analyze the minimax regret of the adversarial bandit convex optimization problem. Focusing on the one-dimensional case, we prove that the minimax regret is $\widetildeΘ(\sqrt{T})$ and partially resolve a decade-old open problem. Our analysis is non-constructive, as we do not present a concrete algorithm that attains this regret rate. Instead, we use minimax duality to reduce the problem to a Bayesian setting, where the convex loss functions are drawn from a worst-case distribution, and then we solve the Bayesian version of the problem with a variant of Thompson Sampling. Our analysis featur…
Pompeu Fabra University（庞培法布拉大学）；INRIA Lille – Nord Europe（法国国家信息与自动化研究所里尔-北欧分所）
This work addresses the problem of regret minimization in non-stochastic multi-armed bandit problems, focusing on performance guarantees that hold with high probability. Such results are rather scarce in the literature since proving them requires a large deal of technical effort and significant modifications to the standard, more intuitive algorithms that come only with guarantees that hold on expectation. One of these modifications is forcing the learner to sample arms from the uniform distribution at least $Ω(\sqrt{T})$ times over $T$ rounds, which can adversely affect performance if many of…
Improved Regret Bounds for Bandits with Expert Advice
Università degli Studi di Milano（米兰大学）；Politecnico di Milano（米兰理工大学）；Politecnico di Torino（都灵理工大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；TU Delft（代尔夫特理工大学）
Michigan State University（密歇根州立大学）；NEC Laboratories America（NEC美国实验室）
In citep{Hazan-2008-extract}, the authors showed that the regret of online linear optimization can be bounded by the total variation of the cost vectors. In this paper, we extend this result to general online convex optimization. We first analyze the limitations of the algorithm in \citep{Hazan-2008-extract} when applied it to online convex optimization. We then present two algorithms for online convex optimization whose regrets are bounded by the variation of cost functions. We finally consider the bandit setting, and present a randomized algorithm for online bandit convex optimization with a…
Microsoft Research（微软研究院）；Google Research India（谷歌印度研究院）
We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost/reward functions $\f_t$ admit a "pseudo-1d" structure, i.e. $\f_t(\w) = \loss_t(\pred_t(\w))$ where the output of $\pred_t$ is one-dimensional. At each round, the learner observes context $\x_t$, plays prediction $\pred_t(\w_t; \x_t)$ (e.g. $\pred_t(\cdot)=\langle \x_t, \cdot\rangle$) for some $\w_t \in \mathbb{R}^d$ and observes loss $\loss_t(\pred_t(\w_t))$ where $\loss_t$ is a convex Lipschitz-continuous function. The goal is to minimize the standard regret metric. This pseudo-1d b…
We consider the decision-making framework of online convex optimization with a very large number of experts. This setting is ubiquitous in contextual and reinforcement learning problems, where the size of the policy class renders enumeration and search within the policy class infeasible. Instead, we consider generalizing the methodology of online boosting. We define a weak learning algorithm as a mechanism that guarantees multiplicatively approximate regret against a base class of experts. In this access model, we give an efficient boosting algorithm that guarantees near-optimal regret against…
Online Learning on Hidden-Convex Losses via Algorithmic Equivalence: Optimal Regret, Geometric Barrier, and Bandit Feedback
Singapore University of Technology and Design（新加坡科技设计大学）；National Technical University of Athens（雅典国家技术大学）；National and Kapodistrian University of Athens（雅典国家与卡多斯大学）；University of California, Irvine（加州大学 Irvine 分校）；Archimedes, Athena Research Center, Greece（希腊阿提卡研究中心 Archimedes）；National University of Singapore, Centre for Quantum Technologies（新加坡国立大学 量子技术中心）
We consider a the general online convex optimization framework introduced by Zinkevich. In this setting, there is a sequence of convex functions. Each period, we must choose a signle point (from some feasible set) and pay a cost equal to the value of the next function on our chosen point. Zinkevich shows that, if the each function is revealed after the choice is made, then one can achieve vanishingly small regret relative the best single decision chosen in hindsight. We extend this to the bandit setting where we do not find out the entire functions but rather just their value at our chosen poi…
We consider the adversarial convex bandit problem and we build the first $\mathrm{poly}(T)$-time algorithm with $\mathrm{poly}(n) \sqrt{T}$-regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves $\tilde{O}(n^{9.5} \sqrt{T})$-regret, and we show that a simple variant of this algorithm can be run in $\mathrm{poly}(n \log(T))$-time per step…
The present paper deals with online convex optimization involving both time-varying loss functions, and time-varying constraints. The loss functions are not fully accessible to the learner, and instead only the function values (a.k.a. bandit feedback) are revealed at queried points. The constraints are revealed after making decisions, and can be instantaneously violated, yet they must be satisfied in the long term. This setting fits nicely the emerging online network tasks such as fog computing in the Internet-of-Things (IoT), where online decisions must flexibly adapt to the changing user pre…
This paper deals with bandit online learning problems involving feedback of unknown delay that can emerge in multi-armed bandit (MAB) and bandit convex optimization (BCO) settings. MAB and BCO require only values of the objective function involved that become available through feedback, and are used to estimate the gradient appearing in the corresponding iterative algorithms. Since the challenging case of feedback with \emph{unknown} delays prevents one from constructing the sought gradient estimates, existing MAB and BCO algorithms become intractable. For such challenging setups, delayed expl…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
We study the adversarial bandit problem with composite anonymous delayed feedback. In this setting, losses of an action are split into $d$ components, spreading over consecutive rounds after the action is chosen. And in each round, the algorithm observes the aggregation of losses that come from the latest $d$ rounds. Previous works focus on oblivious adversarial setting, while we investigate the harder non-oblivious setting. We show non-oblivious setting incurs $Ω(T)$ pseudo regret even when the loss sequence is bounded memory. However, we propose a wrapper algorithm which enjoys $o(T)$ policy…
The Hong Kong University of Science and Technology（香港科技大学）
Differentially private (DP) stochastic convex optimization (SCO) is ubiquitous in trustworthy machine learning algorithm design. This paper studies the DP-SCO problem with streaming data sampled from a distribution and arrives sequentially. We also consider the continual release model where parameters related to private information are updated and released upon each new data, often known as the online algorithms. Despite that numerous algorithms have been developed to achieve the optimal excess risks in different $\ell_p$ norm geometries, yet none of the existing ones can be adapted to the str…
Oxford University（牛津大学）；INRIA Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）
Online learning algorithms require to often recompute least squares regression estimates of parameters. We study improving the computational complexity of such algorithms by using stochastic gradient descent (SGD) type schemes in place of classic regression solvers. We show that SGD schemes efficiently track the true solutions of the regression problems, even in the presence of a drift. This finding coupled with an $O(d)$ improvement in complexity, where $d$ is the dimension of the data, make them attractive for implementation in the big data settings. In the case when strong convexity in the …
Non-stationary Delayed Online Convex Optimization: From Full-information to Bandit Setting
School of Software Technology, Zhejiang University（浙江大学软件学院）；State Key Laboratory of Blockchain and Data Security, Zhejiang University（浙江大学区块链与数据安全国家重点实验室）；National Key Laboratory for Novel Software Technology, Nanjing University（南京大学新型软件技术国家实验室）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）
In this paper, we improve the regret bound for online kernel selection under bandit feedback. Previous algorithm enjoys a $O((\Vert f\Vert^2_{\mathcal{H}_i}+1)K^{\frac{1}{3}}T^{\frac{2}{3}})$ expected bound for Lipschitz loss functions. We prove two types of regret bounds improving the previous bound. For smooth loss functions, we propose an algorithm with a $O(U^{\frac{2}{3}}K^{-\frac{1}{3}}(\sum^K_{i=1}L_T(f^\ast_i))^{\frac{2}{3}})$ expected bound where $L_T(f^\ast_i)$ is the cumulative losses of optimal hypothesis in $\mathbb{H}_{i}=\{f\in\mathcal{H}_i:\Vert f\Vert_{\mathcal{H}_i}\leq U\}$.…
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research（微软研究院）
Motivated by applications to online learning in sparse estimation and Bayesian optimization, we consider the problem of online unconstrained nonsubmodular minimization with delayed costs in both full information and bandit feedback settings. In contrast to previous works on online unconstrained submodular minimization, we focus on a class of nonsubmodular functions with special structure, and prove regret guarantees for several variants of the online and approximate online bandit gradient descent algorithms in static and delayed scenarios. We derive bounds for the agent's regret in the full in…
For the linear bandit problem, we extend the analysis of algorithm CombEXP from [R. Combes, M. S. Talebi Mazraeh Shahi, A. Proutiere, and M. Lelarge. Combinatorial bandits revisited. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, editors, Advances in Neural Information Processing Systems 28, pages 2116--2124. Curran Associates, Inc., 2015. URL http://papers.nips.cc/paper/5831-combinatorial-bandits-revisited.pdf] to the high-probability case against adaptive adversaries, allowing actions to come from an arbitrary polytope. We prove a high-probability regret of \(O(T^{2/3}…
Bandit Convex Optimization with Gradient Prediction Adaptivity
Department of Mathematics, National University of Singapore, Singapore（新加坡国立大学数学系）；Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India（印度理工学院德里分校计算机科学与工程系）；Department of Electrical and Computer Engineering, National University of Singapore, Singapore（新加坡国立大学电子与计算机工程系）
We introduce a simple and efficient algorithm for unconstrained zeroth-order stochastic convex bandits and prove its regret is at most $(1 + r/d)[d^{1.5} \sqrt{n} + d^3] polylog(n, d, r)$ where $n$ is the horizon, $d$ the dimension and $r$ is the radius of a known ball containing the minimiser of the loss.
In decision-making problems such as the multi-armed bandit, an agent learns sequentially by optimizing a certain feedback. While the mean reward criterion has been extensively studied, other measures that reflect an aversion to adverse outcomes, such as mean-variance or conditional value-at-risk (CVaR), can be of interest for critical applications (healthcare, agriculture). Algorithms have been proposed for such risk-aware measures under bandit feedback without contextual information. In this work, we study contextual bandits where such risk measures can be elicited as linear functions of the …
We prove that the information-theoretic upper bound on the minimax regret for zeroth-order adversarial bandit convex optimisation is at most $O(d^{2.5} \sqrt{n} \log(n))$, where $d$ is the dimension and $n$ is the number of interactions. This improves on $O(d^{9.5} \sqrt{n} \log(n)^{7.5}$ by Bubeck et al. (2017). The proof is based on identifying an improved exploratory distribution for convex functions.
We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(Δ,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{Δ_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $Δ_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the co…
The Chinese University of Hong Kong（香港中文大学）；Microsoft（微软）；University of Cambridge（剑桥大学）
We generalize the setting of online clustering of bandits by allowing non-uniform distribution over user frequencies. A more efficient algorithm is proposed with simple set structures to represent clusters. We prove a regret bound for the new algorithm which is free of the minimal frequency over users. The experiments on both synthetic and real datasets consistently show the advantage of the new algorithm over existing methods.
University of Copenhagen（哥本哈根大学）；ICREA（加泰罗尼亚高等研究院）；Pompeu Fabra University（庞培法布拉大学）
We present a new strategy for gap estimation in randomized algorithms for multiarmed bandits and combine it with the EXP3++ algorithm of Seldin and Slivkins (2014). In the stochastic regime the strategy reduces dependence of regret on a time horizon from $(\ln t)^3$ to $(\ln t)^2$ and eliminates an additive factor of order $Δe^{1/Δ^2}$, where $Δ$ is the minimal gap of a problem instance. In the adversarial regime regret guarantee remains unchanged.
Lewis-Sigler Institute（刘易斯-西格勒研究所）；Princeton University（普林斯顿大学）；Department of Computer Science（计算机科学系）；Center for Statistics and Machine Learning（统计与机器学习中心）
We address the problem of regret minimization in logistic contextual bandits, where a learner decides among sequential actions or arms given their respective contexts to maximize binary rewards. Using a fast inference procedure with Polya-Gamma distributed augmentation variables, we propose an improved version of Thompson Sampling, a Bayesian formulation of contextual bandits with near-optimal performance. Our approach, Polya-Gamma augmented Thompson Sampling (PG-TS), achieves state-of-the-art performance on simulated and real data. PG-TS explores the action space efficiently and exploits high…
