We consider the classical problem of sequential probability assignment under logarithmic loss while competing against an arbitrary, potentially nonparametric class of experts. We obtain tight bounds on the minimax regret via a new approach that exploits the self-concordance property of the logarithmic loss. We show that for any expert class with (sequential) metric entropy $\mathcal{O}(γ^{-p})$ at scale $γ$, the minimax regret is $\mathcal{O}(n^{p/(p+1)})$, and that this rate cannot be improved without additional assumptions on the expert class under consideration. As an application of our techniques, we resolve the minimax regret for nonparametric Lipschitz classes of experts.
We develop a new theoretical framework, the \emph{envelope complexity}, to analyze the minimax regret with logarithmic loss functions and derive a Bayesian predictor that adaptively achieves the minimax regret over high-dimensional $\ell_1$-balls within a factor of two. The prior is newly derived for achieving the minimax regret and called the \emph{spike-and-tails~(ST) prior} as it looks like. The resulting regret bound is so simple that it is completely determined with the smoothness of the loss function and the radius of the balls except with logarithmic factors, and it has a generalized fo…
We prove non-asymptotic lower bounds on the expectation of the maximum of $d$ independent Gaussian variables and the expectation of the maximum of $d$ independent symmetric random walks. Both lower bounds recover the optimal leading constant in the limit. A simple application of the lower bound for random walks is an (asymptotically optimal) non-asymptotic lower bound on the minimax regret of online learning with expert advice.
Göteborgs Universitet（哥德堡大学）；Chalmers University of Technology（查尔姆斯理工大学）
Bayesian methods suffer from the problem of how to specify prior beliefs. One interesting idea is to consider worst-case priors. This requires solving a stochastic zero-sum game. In this paper, we extend well-known results from bandit theory in order to discover minimax-Bayes policies and discuss when they are practical.
We prove a new minimax theorem connecting the worst-case Bayesian regret and minimax regret under partial monitoring with no assumptions on the space of signals or decisions of the adversary. We then generalise the information-theoretic tools of Russo and Van Roy (2016) for proving Bayesian regret bounds and combine them with the minimax theorem to derive minimax regret bounds for various partial monitoring settings. The highlight is a clean analysis of `non-degenerate easy' and `hard' finite partial monitoring, with new regret bounds that are independent of arbitrarily large game-dependent co…
Northwestern University（西北大学）；University of Florida（佛罗里达大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $Ω(\sqrt{dT (\log T) (\log n)})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results …
University of Toronto（多伦多大学）；Spanflug Technologies GmbH（Spanflug技术有限公司）；Boston University（波士顿大学）
Quantile (and, more generally, KL) regret bounds, such as those achieved by NormalHedge (Chaudhuri, Freund, and Hsu 2009) and its variants, relax the goal of competing against the best individual expert to only competing against a majority of experts on adversarial data. More recently, the semi-adversarial paradigm (Bilodeau, Negrea, and Roy 2020) provides an alternative relaxation of adversarial online learning by considering data that may be neither fully adversarial nor stochastic (i.i.d.). We achieve the minimax optimal regret in both paradigms using FTRL with separate, novel, root-logarit…
UC Berkeley（加州大学伯克利分校）；University of Pennsylvania（宾夕法尼亚大学）
We study the regret of optimal strategies for online convex optimization games. Using von Neumann's minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary's action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen's inequality for a …
We study the stochastic Multi-Armed Bandit (MAB) problem under worst-case regret and heavy-tailed reward distribution. We modify the minimax policy MOSS for the sub-Gaussian reward distribution by using saturated empirical mean to design a new algorithm called Robust MOSS. We show that if the moment of order $1+ε$ for the reward distribution exists, then the refined strategy has a worst-case regret matching the lower bound while maintaining a distribution-dependent logarithm regret.
In this paper, we consider learning scenarios where the learned model is evaluated under an unknown test distribution which potentially differs from the training distribution (i.e. distribution shift). The learner has access to a family of weight functions such that the test distribution is a reweighting of the training distribution under one of these functions, a setting typically studied under the name of Distributionally Robust Optimization (DRO). We consider the problem of deriving regret bounds in the classical learning theory setting, and require that the resulting regret bounds hold uni…
As noted in the works of \cite{lattimore2020bandit}, it has been mentioned that it is an open problem to characterize the minimax regret of linear bandits in a wide variety of action spaces. In this article we present an optimal regret lower bound for a wide class of convex action spaces.
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
We consider prediction with expert advice when data are generated from distributions varying arbitrarily within an unknown constraint set. This semi-adversarial setting includes (at the extremes) the classical i.i.d. setting, when the unknown constraint set is restricted to be a singleton, and the unconstrained adversarial setting, when the constraint set is the set of all distributions. The Hedge algorithm -- long known to be minimax (rate) optimal in the adversarial regime -- was recently shown to be simultaneously minimax optimal for i.i.d. data. In this work, we propose to relax the i.i.d.…
We show that a version of the generalised information ratio of Lattimore and Gyorgy (2020) determines the asymptotic minimax regret for all finite-action partial monitoring games provided that (a) the standard definition of regret is used but the latent space where the adversary plays is potentially infinite; or (b) the regret introduced by Rustichini (1999) is used and the latent space is finite. Our results are complemented by a number of examples. For any $p \in [1/2,1]$ there exists an infinite partial monitoring game for which the minimax regret over $n$ rounds is $n^p$ up to subpolynomia…
Yale University（耶鲁大学）；Simons Institute for the Theory of Computing（西蒙斯理论计算研究所）；Massachusetts Institute of Technology（麻省理工学院）；University of Southern California（南加利福尼亚大学）
We study the problem of switching-constrained online convex optimization (OCO), where the player has a limited number of opportunities to change her action. While the discrete analog of this online learning task has been studied extensively, previous work in the continuous setting has neither established the minimax rate nor algorithmically achieved it. In this paper, we show that $ T $-round switching-constrained OCO with fewer than $ K $ switches has a minimax regret of $ Θ(\frac{T}{\sqrt{K}}) $. In particular, it is at least $ \frac{T}{\sqrt{2K}} $ for one dimension and at least $ \frac{T}{…
Princeton University（普林斯顿大学）；Università degli Studi di Milano（米兰大学）；Microsoft Research New England（微软新英格兰研究院）；Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）
We address the online linear optimization problem with bandit feedback. Our contribution is twofold. First, we provide an algorithm (based on exponential weights) with a regret of order $\sqrt{d n \log N}$ for any finite action set with $N$ actions, under the assumption that the instantaneous loss is bounded by 1. This shaves off an extraneous $\sqrt{d}$ factor compared to previous works, and gives a regret bound of order $d \sqrt{n \log n}$ for any compact set of actions. Without further assumptions on the action set, this last bound is minimax optimal up to a logarithmic factor. Interestingl…
UT Austin（德克萨斯大学奥斯汀分校）；UIUC（伊利诺伊大学厄巴纳-香槟分校）；Amazon（亚马逊）
Cascading bandits is a natural and popular model that frames the task of learning to rank from Bernoulli click feedback in a bandit setting. For the case of unstructured rewards, we prove matching upper and lower bounds for the problem-independent (i.e., gap-free) regret, both of which strictly improve the best known. A key observation is that the hard instances of this problem are those with small mean rewards, i.e., the small click-through rates that are most relevant in practice. Based on this, and the fact that small mean implies small variance for Bernoullis, our key technical result show…
We study reinforcement learning in an infinite-horizon average-reward setting with linear function approximation, where the transition probability function of the underlying Markov Decision Process (MDP) admits a linear form over a feature mapping of the current state, action, and next state. We propose a new algorithm UCRL2-VTR, which can be seen as an extension of the UCRL2 algorithm with linear function approximation. We show that UCRL2-VTR with Bernstein-type bonus can achieve a regret of $\tilde{O}(d\sqrt{DT})$, where $d$ is the dimension of the feature mapping, $T$ is the horizon, and $\…
We study minimax optimal reinforcement learning in episodic factored Markov decision processes (FMDPs), which are MDPs with conditionally independent transition components. Assuming the factorization is known, we propose two model-based algorithms. The first one achieves minimax optimal regret guarantees for a rich class of factored structures, while the second one enjoys better computational complexity with a slightly worse regret. A key new ingredient of our algorithms is the design of a bonus term to guide exploration. We complement our algorithms by presenting several structure-dependent l…
We study the nonstationary stochastic Multi-Armed Bandit (MAB) problem in which the distribution of rewards associated with each arm are assumed to be time-varying and the total variation in the expected rewards is subject to a variation budget. The regret of a policy is defined by the difference in the expected cumulative rewards obtained using the policy and using an oracle that selects the arm with the maximum mean reward at each time. We characterize the performance of the proposed policies in terms of the worst-case regret, which is the supremum of the regret over the set of reward distri…
In this work, we propose an efficient minimax optimal global optimization algorithm for multivariate Lipschitz continuous functions. To evaluate the performance of our approach, we utilize the average regret instead of the traditional simple regret, which, as we show, is not suitable for use in the multivariate non-convex optimization because of the inherent hardness of the problem itself. Since we study the average regret of the algorithm, our results directly imply a bound for the simple regret as well. Instead of constructing lower bounding proxy functions, our method utilizes a predetermin…
We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form $f_t(x) = g_t(\langle x, θ\rangle)$ for convex $g_t : \mathbb R \to \mathbb R$ and unknown $θ\in \mathbb R^d$ that is homogeneous over time. We provide a short information-theoretic proof that the minimax regret is at most $O(d \sqrt{n} \log(n \operatorname{diam}(\mathcal K)))$ where $n$ is the number of interactions, $d$ the dimension and $\operatorname{diam}(\mathcal K)$ is the diameter of the constraint set.
We introduce an online convex optimization algorithm which utilizes projected subgradient descent with optimal adaptive learning rates. Our method provides second-order minimax-optimal dynamic regret guarantee (i.e. dependent on the sum of squared subgradient norms) for a sequence of general convex functions, which may not have strong convexity, smoothness, exp-concavity or even Lipschitz-continuity. The regret guarantee is against any comparator decision sequence with bounded path variation (i.e. sum of the distances between successive decisions). We generate the lower bound of the worst-case…
Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret
Facebook AI Research（Facebook人工智能研究院）；Inria Lille（法国国家信息与自动化研究所里尔中心）；Tsinghua University（清华大学）；University of Washington（华盛顿大学）；Facebook AI Research Paris（Facebook人工智能研究院巴黎）；DeepMind Paris（DeepMind巴黎）
We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal $O(T^{1/2})$ worst-case regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is predictable, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst-case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret…
We study the kernelized bandit problem, that involves designing an adaptive strategy for querying a noisy zeroth-order-oracle to efficiently learn about the optimizer of an unknown function $f$ with a norm bounded by $M<\infty$ in a Reproducing Kernel Hilbert Space~(RKHS) associated with a positive definite kernel $K$. Prior results, working in a \emph{minimax framework}, have characterized the worst-case~(over all functions in the problem class) limits on regret achievable by \emph{any} algorithm, and have constructed algorithms with matching~(modulo polylogarithmic factors) worst-case perfor…
We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of $\tilde{O}( \sqrt{HSAT} + H^2S^2A+H\sqrt{T})$ where $H$ is the time horizon, $S$ the number of states, $A$ the number of actions and $T$ the number of time-steps. This result improves over the best previous known bound $\tilde{O}(HS \sqrt{AT})$ achieved by the UCRL2 algorithm of Jaksch et al., 2010. The key significance of our new results is that when $T\geq H^3S^3A$ and $SA\geq H$, it leads to a regret …
We study the Stochastic Shortest Path (SSP) problem in which an agent has to reach a goal state in minimum total expected cost. In the learning formulation of the problem, the agent has no prior knowledge about the costs and dynamics of the model. She repeatedly interacts with the model for $K$ episodes, and has to minimize her regret. In this work we show that the minimax regret for this setting is $\widetilde O(\sqrt{ (B_\star^2 + B_\star) |S| |A| K})$ where $B_\star$ is a bound on the expected cost of the optimal policy from any state, $S$ is the state space, and $A$ is the action space. Th…
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.
Multi-Armed Bernoulli Bandits via Minimax Single-Arm Stopping
Shanghai Jiao Tong University（上海交通大学）；The University of Sydney（悉尼大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Imperial College Business School（帝国理工学院商学院）
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）
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…
Asymptotically and Minimax Optimal Regret Bounds for Multi-Armed Bandits with Abstention
Institute of Operations Research and Analytics（运营研究与分析研究所）；National University of Singapore（新加坡国立大学）；Department of Electrical and Computer Engineering（电子与计算机工程系）；Department of Mathematics（数学系）
University of Pennsylvania（宾夕法尼亚大学）；Cornell University（康奈尔大学）；CREST-ENSAE（法国国家统计与经济管理学院（CREST-ENSAE））
We consider the random design regression model with square loss. We propose a method that aggregates empirical minimizers (ERM) over appropriately chosen random subsets and reduces to ERM in the extreme case, and we establish sharp oracle inequalities for its risk. We show that, under the $\varepsilon^{-p}$ growth of the empirical $\varepsilon$-entropy, the excess risk of the proposed method attains the rate $n^{-2/(2+p)}$ for $p\in(0,2)$ and $n^{-1/p}$ for $p>2$ where $n$ is the sample size. Furthermore, for $p\in(0,2)$, the excess risk rate matches the behavior of the minimax risk of functio…
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 se…
In this paper, we propose new problem-independent lower bounds on the sample complexity and regret in episodic MDPs, with a particular focus on the non-stationary case in which the transition kernel is allowed to change in each stage of the episode. Our main contribution is a novel lower bound of $Ω((H^3SA/ε^2)\log(1/δ))$ on the sample complexity of an $(\varepsilon,δ)$-PAC algorithm for best policy identification in a non-stationary MDP. This lower bound relies on a construction of "hard MDPs" which is different from the ones previously used in the literature. Using this same class of MDPs, w…
Green security domains feature defenders who plan patrols in the face of uncertainty about the adversarial behavior of poachers, illegal loggers, and illegal fishers. Importantly, the deterrence effect of patrols on adversaries' future behavior makes patrol planning a sequential decision-making problem. Therefore, we focus on robust sequential patrol planning for green security following the minimax regret criterion, which has not been considered in the literature. We formulate the problem as a game between the defender and nature who controls the parameter values of the adversarial behavior a…
We study the problem of online learning with dynamics, where a learner interacts with a stateful environment over multiple rounds. In each round of the interaction, the learner selects a policy to deploy and incurs a cost that depends on both the chosen policy and current state of the world. The state-evolution dynamics and the costs are allowed to be time-varying, in a possibly adversarial way. In this setting, we study the problem of minimizing policy regret and provide non-constructive upper bounds on the minimax rate for the problem. Our main results provide sufficient conditions for onlin…
