Algorithm Selection as a Bandit Problem with Unbounded Losses
IDSIA / USI-SUPSI Istituto Dalle Molle di studi sull’intelligenza artificiale（IDSIA / USI-SUPSI 达莱莫莱人工智能研究所）；Università della Svizzera italiana (USI)（瑞士意大利语区大学）；University of Lugano（卢加诺大学）；TU Munich（慕尼黑工业大学）
Algorithm selection is typically based on models of algorithm performance, learned during a separate offline training sequence, which can be prohibitively expensive. In recent work, we adopted an online approach, in which a performance model is iteratively updated and used to guide selection on a sequence of problem instances. The resulting exploration-exploitation trade-off was represented as a bandit problem with expert advice, using an existing solver for this game, but this required the setting of an arbitrary bound on algorithm runtimes, thus invalidating the optimal regret of the solver. In this paper, we propose a simpler framework for representing algorithm selection as a bandit problem, with partial information, and an unknown bound on losses. We adapt an existing solver to this game, proving a bound on its expected regret, which holds also for the resulting algorithm selection technique. We present preliminary experiments with a set of SAT solvers on a mixed SAT-UNSAT benchmark.
We use surrogate losses to obtain several new regret bounds and new algorithms for contextual bandit learning. Using the ramp loss, we derive new margin-based regret bounds in terms of standard sequential complexity measures of a benchmark class of real-valued regression functions. Using the hinge loss, we derive an efficient algorithm with a $\sqrt{dT}$-type mistake bound against benchmark policies induced by $d$-dimensional regressors. Under realizability assumptions, our results also yield classical regret bounds.
Inria Lille（法国国家信息与自动化研究所里尔分部）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；MIT（麻省理工学院）
Multi-armed bandits are widely applied in scenarios like recommender systems, for which the goal is to maximize the click rate. However, more factors should be considered, e.g., user stickiness, user growth rate, user experience assessment, etc. In this paper, we model this situation as a problem of $K$-armed bandit with multiple losses. We define relative loss vector of an arm where the $i$-th entry compares the arm and the optimal arm with respect to the $i$-th loss. We study two goals: (a) finding the arm with the minimum $\ell^\infty$-norm of relative losses with a given confidence level (…
This paper tackles the risk averse multi-armed bandits problem when incurred losses are non-stationary. The conditional value-at-risk (CVaR) is used as the objective function. Two estimation methods are proposed for this objective function in the presence of non-stationary losses, one relying on a weighted empirical distribution of losses and another on the dual representation of the CVaR. Such estimates can then be embedded into classic arm selection methods such as epsilon-greedy policies. Simulation experiments assess the performance of the arm selection algorithms based on the two novel es…
We study nonstochastic bandits and experts in a delayed setting where delays depend on both time and arms. While the setting in which delays only depend on time has been extensively studied, the arm-dependent delay setting better captures real-world applications at the cost of introducing new technical challenges. In the full information (experts) setting, we design an algorithm with a first-order regret bound that reveals an interesting trade-off between delays and losses. We prove a similar first-order regret bound also for the bandit setting, when the learner is allowed to observe how many …
We study the adversarial multi-armed bandit problem and create a completely online algorithmic framework that is invariant under arbitrary translations and scales of the arm losses. We study the expected performance of our algorithm against a generic competition class, which makes it applicable for a wide variety of problem scenarios. Our algorithm works from a universal prediction perspective and the performance measure used is the expected regret against arbitrary arm selection sequences, which is the difference between our losses and a competing loss sequence. The competition class can be d…
Classical multi-armed bandit problems use the expected value of an arm as a metric to evaluate its goodness. However, the expected value is a risk-neutral metric. In many applications like finance, one is interested in balancing the expected return of an arm (or portfolio) with the risk associated with that return. In this paper, we consider the problem of selecting the arm that optimizes a linear combination of the expected reward and the associated Conditional Value at Risk (CVaR) in a fixed budget best-arm identification framework. We allow the reward distributions to be unbounded or even h…
Università degli Studi di Milano（米兰大学）；Weizmann Institute of Science（魏茨曼科学研究所）
We study how the regret guarantees of nonstochastic multi-armed bandits can be improved, if the effective range of the losses in each round is small (e.g. the maximal difference between two losses in a given round). Despite a recent impossibility result, we show how this can be made possible under certain mild additional assumptions, such as availability of rough estimates of the losses, or advance knowledge of the loss of a single, possibly unspecified arm. Along the way, we develop a novel technique which might be of independent interest, to convert any multi-armed bandit algorithm with regr…
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 study the bandit problem where the underlying expected reward is a Bounded Mean Oscillation (BMO) function. BMO functions are allowed to be discontinuous and unbounded, and are useful in modeling signals with infinities in the do-main. We develop a toolset for BMO bandits, and provide an algorithm that can achieve poly-log $δ$-regret -- a regret measured against an arm that is optimal after removing a $δ$-sized portion of the arm space.
Tel Aviv University（特拉维夫大学）；Institute for Advanced Study（高等研究院）；Microsoft Research（微软研究院）；Università degli Studi di Milano（米兰大学）；University of Insubria（因苏布里亚大学）；Technion（以色列理工学院）；Weizmann Institute of Science（魏茨曼科学研究所）
We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). We provide se…
Google Research（谷歌研究院）；John Hopcroft Center（约翰·霍普克罗夫特中心）；Shanghai Jiao Tong University（上海交通大学）
Motivated by problems of learning to rank long item sequences, we introduce a variant of the cascading bandit model that considers flexible length sequences with varying rewards and losses. We formulate two generative models for this problem within the generalized linear setting, and design and analyze upper confidence algorithms for it. Our analysis delivers tight regret bounds which, when specialized to vanilla cascading bandits, results in sharper guarantees than previously available in the literature. We evaluate our algorithms on a number of real-world datasets, and show significantly imp…
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；IBM Research（IBM研究院）；University of Pennsylvania（宾夕法尼亚大学）
We address the problem of \emph{`Internal Regret'} in \emph{Sleeping Bandits} in the fully adversarial setup, as well as draw connections between different existing notions of sleeping regrets in the multiarmed bandits (MAB) literature and consequently analyze the implications: Our first contribution is to propose the new notion of \emph{Internal Regret} for sleeping MAB. We then proposed an algorithm that yields sublinear regret in that measure, even for a completely adversarial sequence of losses and availabilities. We further show that a low sleeping internal regret always implies a low ext…
The Multi-armed bandit offer the advantage to learn and exploit the already learnt knowledge at the same time. This capability allows this approach to be applied in different domains, going from clinical trials where the goal is investigating the effects of different experimental treatments while minimizing patient losses, to adaptive routing where the goal is to minimize the delays in a network. This article provides a review of the recent results on applying bandit to real-life scenario and summarize the state of the art for each of these fields. Different techniques has been proposed to sol…
We study the adversarial online learning problem and create a completely online algorithmic framework that has data dependent regret guarantees in both full expert feedback and bandit feedback settings. We study the expected performance of our algorithm against general comparators, which makes it applicable for a wide variety of problem scenarios. Our algorithm works from a universal prediction perspective and the performance measure used is the expected regret against arbitrary comparator sequences, which is the difference between our losses and a competing loss sequence. The competition clas…
Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université Toulouse 3 Paul Sabatier（图卢兹第三大学）；University of Alberta（阿尔伯塔大学）
We provide new lower bounds on the regret that must be suffered by adversarial bandit algorithms. The new results show that recent upper bounds that either (a) hold with high-probability or (b) depend on the total lossof the best arm or (c) depend on the quadratic variation of the losses, are close to tight. Besides this we prove two impossibility results. First, the existence of a single arm that is optimal in every round cannot improve the regret in the worst case. Second, the regret cannot scale with the effective range of the losses. In contrast, both results are possible in the full-infor…
Italian Institute of Technology（意大利理工学院）；Google Research（谷歌研究院）；University College London（伦敦大学学院）
Motivated by a natural problem in online model selection with bandit information, we introduce and analyze a best arm identification problem in the rested bandit setting, wherein arm expected losses decrease with the number of times the arm has been played. The shape of the expected loss functions is similar across arms, and is assumed to be available up to unknown parameters that have to be learned on the fly. We define a novel notion of regret for this problem, where we compare to the policy that always plays the arm having the smallest expected loss at the end of the game. We analyze an arm…
We consider the Scale-Free Adversarial Multi Armed Bandits(MAB) problem. At the beginning of the game, the player only knows the number of arms $n$. It does not know the scale and magnitude of the losses chosen by the adversary or the number of rounds $T$. In each round, it sees bandit feedback about the loss vectors $l_1,\dots, l_T \in \mathbb{R}^n$. The goal is to bound its regret as a function of $n$ and norms of $l_1,\dots, l_T$. We design a bandit Follow The Regularized Leader (FTRL) algorithm, that uses an adaptive learning rate and give two different regret bounds, based on the explorat…
We consider the Scale-Free Adversarial Multi-Armed Bandit (MAB) problem with unrestricted feedback delays. In contrast to the standard assumption that all losses are $[0,1]$-bounded, in our setting, losses can fall in a general bounded interval $[-L, L]$, unknown to the agent beforehand. Furthermore, the feedback of each arm pull can experience arbitrary delays. We propose a novel approach named Scale-Free Delayed INF (SFD-INF) for this novel setting, which combines a recent "convex combination trick" together with a novel doubling and skipping technique. We then present two instances of SFD-I…
Delayed Bandits: When Do Intermediate Observations Help?
Università degli Studi di Milano（米兰大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；University of Copenhagen（哥本哈根大学）；University of Amsterdam（阿姆斯特丹大学）；Politecnico di Milano（米兰理工大学）
Recent work on follow the perturbed leader (FTPL) algorithms for the adversarial multi-armed bandit problem has highlighted the role of the hazard rate of the distribution generating the perturbations. Assuming that the hazard rate is bounded, it is possible to provide regret analyses for a variety of FTPL algorithms for the multi-armed bandit problem. This paper pushes the inquiry into regret bounds for FTPL algorithms beyond the bounded hazard rate condition. There are good reasons to do so: natural distributions such as the uniform and Gaussian violate the condition. We give regret bounds f…
Wichita State University（威奇托州立大学）；Department of Electrical Engineering and Computer Science（电气工程与计算机科学系（不单独作为机构提取，此处主要机构为大学））
This paper addresses the problem of unknown delays in adversarial multi-armed bandit (MAB) with multiple play. Existing work on similar game setting focused on only the case where the learner selects an arm in each round. However, there are lots of applications in robotics where a learner needs to select more than one arm per round. It is therefore worthwhile to investigate the effect of delay when multiple arms are chosen. The multiple arms chosen per round in this setting are such that they experience the same amount of delay. There can be an aggregation of feedback losses from different com…
We consider the adversarial multi-armed bandit problem under delayed feedback. We analyze variants of the Exp3 algorithm that tune their step-size using only information (about the losses and delays) available at the time of the decisions, and obtain regret guarantees that adapt to the observed (rather than the worst-case) sequences of delays and/or losses. First, through a remarkably simple proof technique, we show that with proper tuning of the step size, the algorithm achieves an optimal (up to logarithmic factors) regret of order $\sqrt{\log(K)(TK + D)}$ both in expectation and in high pro…
University of Torontoland（多伦多兰大学）；University of Edenborrow（伊登伯勒大学）
We study the stochastic Multi-Armed Bandit (MAB) problem with random delays in the feedback received by the algorithm. We consider two settings: the reward-dependent delay setting, where realized delays may depend on the stochastic rewards, and the reward-independent delay setting. Our main contribution is algorithms that achieve near-optimal regret in each of the settings, with an additional additive dependence on the quantiles of the delay distribution. Our results do not make any assumptions on the delay distributions: in particular, we do not assume they come from any parametric family of …
Department of Computer Science, ETH Zurich（苏黎世联邦理工学院计算机科学系）
We consider a setting where multiple players sequentially choose among a common set of actions (arms). Motivated by a cognitive radio networks application, we assume that players incur a loss upon colliding, and that communication between players is not possible. Existing approaches assume that the system is stationary. Yet this assumption is often violated in practice, e.g., due to signal strength fluctuations. In this work, we design the first Multi-player Bandit algorithm that provably works in arbitrarily changing environments, where the losses of the arms may even be chosen by an adversar…
Max Planck Institut für Informatik, Saarland University（马克斯·普朗克信息学研究所，萨尔大学）；RWTH Aachen University（亚琛工业大学）
We study a variation of the classical multi-armed bandits problem. In this problem, the learner has to make a sequence of decisions, picking from a fixed set of choices. In each round, she receives as feedback only the loss incurred from the chosen action. Conventionally, this problem has been studied when losses of the actions are drawn from an unknown distribution or when they are adversarial. In this paper, we study this problem when the losses of the actions also satisfy certain structural properties, and especially, do show a trend structure. When this is true, we show that using \textit{…
This paper studies a multi-armed bandit problem where the decision-maker is loss averse, in particular she is risk averse in the domain of gains and risk loving in the domain of losses. The focus is on large horizons. Consequences of loss aversion for asymptotic (large horizon) properties are derived in a number of analytical results. The analysis is based on a new central limit theorem for a set of measures under which conditional variances can vary in a largely unstructured history-dependent way subject only to the restriction that they lie in a fixed interval.
Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
We give an oracle-based algorithm for the adversarial contextual bandit problem, where either contexts are drawn i.i.d. or the sequence of contexts is known a priori, but where the losses are picked adversarially. Our algorithm is computationally efficient, assuming access to an offline optimization oracle, and enjoys a regret of order $O((KT)^{\frac{2}{3}}(\log N)^{\frac{1}{3}})$, where $K$ is the number of actions, $T$ is the number of iterations and $N$ is the number of baseline policies. Our result is the first to break the $O(T^{\frac{3}{4}})$ barrier that is achieved by recently introduc…
Motivated by applications of bandit algorithms in education, we consider a stochastic multi-armed bandit problem with $\varepsilon$-contaminated rewards. We allow an adversary to give arbitrary unbounded contaminated rewards with full knowledge of the past and future. We impose the constraint that for each time $t$ the proportion of contaminated rewards for any action is less than or equal to $\varepsilon$. We derive concentration inequalities for two robust mean estimators for sub-Gaussian distributions in the $\varepsilon$-contamination context. We define the $\varepsilon$-contaminated stoch…
University of Southern California（南加州大学）；Massachusetts Institute of Technology（麻省理工学院）
When an AI system interacts with multiple users, it frequently needs to make allocation decisions. For instance, a virtual agent decides whom to pay attention to in a group setting, or a factory robot selects a worker to deliver a part. Demonstrating fairness in decision making is essential for such systems to be broadly accepted. We introduce a Multi-Armed Bandit algorithm with fairness constraints, where fairness is defined as a minimum rate that a task or a resource is assigned to a user. The proposed algorithm uses contextual information about the users and the task and makes no assumption…
Criteo AI Lab（Criteo人工智能实验室）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Grenoble INP（格勒诺布尔国立理工学院）
In many online decision processes, the optimizing agent is called to choose between large numbers of alternatives with many inherent similarities; in turn, these similarities imply closely correlated losses that may confound standard discrete choice models and bandit algorithms. We study this question in the context of nested bandits, a class of adversarial multi-armed bandit problems where the learner seeks to minimize their regret in the presence of a large number of distinct alternatives with a hierarchy of embedded (non-combinatorial) similarities. In this setting, optimal algorithms based…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；Microsoft Research（微软研究院）
Online learning algorithms are designed to learn even when their input is generated by an adversary. The widely-accepted formal definition of an online algorithm's ability to learn is the game-theoretic notion of regret. We argue that the standard definition of regret becomes inadequate if the adversary is allowed to adapt to the online algorithm's actions. We define the alternative notion of policy regret, which attempts to provide a more meaningful way to measure an online algorithm's performance against adaptive adversaries. Focusing on the online bandit setting, we show that no bandit algo…
CentraleSupélec（中央高等电力学院）；Université Paris-Saclay（巴黎-萨克雷大学）；Institut de recherche technologique Saint Exupéry（圣埃克苏佩里技术研究院）；Institut de mathématiques de Toulouse（图卢兹数学研究所）；Université Paul Sabatier（保罗·萨巴蒂埃大学）；CNRS（法国国家科学研究中心）；HEC Paris（巴黎高等商学院）
Optimal Regret for Policy Optimization in Contextual Bandits
School of Computer Science and AI, Tel-Aviv University, Tel-Aviv, Israel（计算机科学与人工智能学院，特拉维夫大学，特拉维夫，以色列）；Google Research, Tel-Aviv, Israel（谷歌研究，特拉维夫，以色列）
Google Research（谷歌研究院）；Tel-Aviv University（特拉维夫大学）
We study a novel variant of online finite-horizon Markov Decision Processes with adversarially changing loss functions and initially unknown dynamics. In each episode, the learner suffers the loss accumulated along the trajectory realized by the policy chosen for the episode, and observes aggregate bandit feedback: the trajectory is revealed along with the cumulative loss suffered, rather than the individual losses encountered along the trajectory. Our main result is a computationally efficient algorithm with $O(\sqrt{K})$ regret for this setting, where $K$ is the number of episodes. We establ…
We study the Bayesian regret of the renowned Thompson Sampling algorithm in contextual bandits with binary losses and adversarially-selected contexts. We adapt the information-theoretic perspective of \cite{RvR16} to the contextual setting by considering a lifted version of the information ratio defined in terms of the unknown model parameter instead of the optimal action or optimal policy as done in previous works on the same setting. This allows us to bound the regret in terms of the entropy of the prior distribution through a remarkably simple proof, and with no structural assumptions on th…
Université de Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）
We study the corrupted bandit problem, i.e. a stochastic multi-armed bandit problem with $k$ unknown reward distributions, which are heavy-tailed and corrupted by a history-independent adversary or Nature. To be specific, the reward obtained by playing an arm comes from corresponding heavy-tailed reward distribution with probability $1-\varepsilon \in (0.5,1]$ and an arbitrary corruption distribution of unbounded support with probability $\varepsilon \in [0,0.5)$. First, we provide $\textit{a problem-dependent lower bound on the regret}$ of any corrupted bandit algorithm. The lower bounds indi…
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…
The stochastic multi-arm bandit problem has been extensively studied under standard assumptions on the arm's distribution (e.g bounded with known support, exponential family, etc). These assumptions are suitable for many real-world problems but sometimes they require knowledge (on tails for instance) that may not be precisely accessible to the practitioner, raising the question of the robustness of bandit algorithms to model misspecification. In this paper we study a generic Dirichlet Sampling (DS) algorithm, based on pairwise comparisons of empirical indices computed with re-sampling of the a…
This work studies the problem of learning episodic Markov Decision Processes with known transition and bandit feedback. We develop the first algorithm with a ``best-of-both-worlds'' guarantee: it achieves $\mathcal{O}(log T)$ regret when the losses are stochastic, and simultaneously enjoys worst-case robustness with $\tilde{\mathcal{O}}(\sqrt{T})$ regret even when the losses are adversarial, where $T$ is the number of episodes. More generally, it achieves $\tilde{\mathcal{O}}(\sqrt{C})$ regret in an intermediate setting where the losses are corrupted by a total amount of $C$. Our algorithm is …
Tsinghua University（清华大学）；University of Southern California（南加利福尼亚大学）
We consider regret minimization for Adversarial Markov Decision Processes (AMDPs), where the loss functions are changing over time and adversarially chosen, and the learner only observes the losses for the visited state-action pairs (i.e., bandit feedback). While there has been a surge of studies on this problem using Online-Mirror-Descent (OMD) methods, very little is known about the Follow-the-Perturbed-Leader (FTPL) methods, which are usually computationally more efficient and also easier to implement since it only requires solving an offline planning problem. Motivated by this, we take a c…
Lancaster University（兰卡斯特大学）；Columbia University（哥伦比亚大学）；DeepMind；University of Alberta（阿尔伯塔大学）
We study a variant of the stochastic $K$-armed bandit problem, which we call "bandits with delayed, aggregated anonymous feedback". In this problem, when the player pulls an arm, a reward is generated, however it is not immediately observed. Instead, at the end of each round the player observes only the sum of a number of previously generated rewards which happen to arrive in the given round. The rewards are stochastically delayed and due to the aggregated nature of the observations, the information of which arm led to a particular reward is lost. The question is what is the cost of the inform…
