In this paper, we present refined probabilistic bounds on empirical reward estimates for off-policy learning in bandit problems. We build on the PAC-Bayesian bounds from Seldin et al. (2010) and improve on their results using a new parameter optimization approach introduced by Rodríguez et al. (2024). This technique is based on a discretization of the space of possible events to optimize the "in probability" parameter. We provide two parameter-free PAC-Bayes bounds, one based on Hoeffding-Azuma's inequality and the other based on Bernstein's inequality. We prove that our bounds are almost optimal as they recover the same rate as would be obtained by setting the "in probability" parameter after the realization of the data.
We consider Bayesian optimization in settings where observations can be adversarially biased, for example by an uncontrolled hidden confounder. Our first contribution is a reduction of the confounded setting to the dueling bandit model. Then we propose a novel approach for dueling bandits based on information-directed sampling (IDS). Thereby, we obtain the first efficient kernelized algorithm for dueling bandits that comes with cumulative regret guarantees. Our analysis further generalizes a previously proposed semi-parametric linear bandit model to non-linear reward functions, and uncovers in…
Cambridge University（剑桥大学）；University of British Columbia（不列颠哥伦比亚大学）；Oxford University（牛津大学）
We address the problem of finding the maximizer of a nonlinear smooth function, that can only be evaluated point-wise, subject to constraints on the number of permitted function evaluations. This problem is also known as fixed-budget best arm identification in the multi-armed bandit literature. We introduce a Bayesian approach for this problem and show that it empirically outperforms both the existing frequentist counterpart and other Bayesian optimization methods. The Bayesian approach places emphasis on detailed modelling, including the modelling of correlations among the arms. As a result, …
Bosch Center for Artificial Intelligence（博世人工智能中心）；University of Southern Denmark（南丹麦大学）；Technische Universität Darmstadt（达姆施塔特工业大学）
We present a PAC-Bayesian analysis of lifelong learning. In the lifelong learning problem, a sequence of learning tasks is observed one-at-a-time, and the goal is to transfer information acquired from previous tasks to new learning tasks. We consider the case when each learning task is a multi-armed bandit problem. We derive lower bounds on the expected average reward that would be obtained if a given multi-armed bandit algorithm was run in a new task with a particular prior and for a set number of steps. We propose lifelong learning algorithms that use our new bounds as learning objectives. O…
BayesOpt is a library with state-of-the-art Bayesian optimization methods to solve nonlinear optimization, stochastic bandits or sequential experimental design problems. Bayesian optimization is sample efficient by building a posterior distribution to capture the evidence and prior knowledge for the target function. Built in standard C++, the library is extremely efficient while being portable and flexible. It includes a common interface for C, C++, Python, Matlab and Octave.
Uppsala University（乌普萨拉大学）；Department of Information Technology（信息技术系）；Division of Scientific Computing（科学计算部）；SE-751 05, Uppsala（乌普萨拉，SE-751 05）；Sweden（瑞典）
Approximate Bayesian computation is an established and popular method for likelihood-free inference with applications in many disciplines. The effectiveness of the method depends critically on the availability of well performing summary statistics. Summary statistic selection relies heavily on domain knowledge and carefully engineered features, and can be a laborious time consuming process. Since the method is sensitive to data dimensionality, the process of selecting summary statistics must balance the need to include informative statistics and the dimensionality of the feature vector. This p…
We consider the problem of identifying any $k$ out of the best $m$ arms in an $n$-armed stochastic multi-armed bandit. Framed in the PAC setting, this particular problem generalises both the problem of `best subset selection' and that of selecting `one out of the best m' arms [arcsk 2017]. In applications such as crowd-sourcing and drug-designing, identifying a single good solution is often not sufficient. Moreover, finding the best subset might be hard due to the presence of many indistinguishably close solutions. Our generalisation of identifying exactly $k$ arms out of the best $m$, where $…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Chair for Information Technology, University of Leoben（莱奥本大学信息技术教席）；Université Laval（拉瓦尔大学）；University College London（伦敦大学学院）
We develop a new tool for data-dependent analysis of the exploration-exploitation trade-off in learning under limited feedback. Our tool is based on two main ingredients. The first ingredient is a new concentration inequality that makes it possible to control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. The second ingredient is an application of this inequality to the exploration-exploitation trade-off via importance weighted sampling. We apply the new tool to the stochastic multiarmed bandit problem, how…
This habilitation thesis presents several contributions to (1) the PAC-Bayesian analysis of statistical learning, (2) the three aggregation problems: given d functions, how to predict as well as (i) the best of these d functions (model selection type aggregation), (ii) the best convex combination of these d functions, (iii) the best linear combination of these d functions, (3) the multi-armed bandit problems.
Volvo Car Corporation（沃尔沃汽车公司）；Chalmers University of Technology（查尔姆斯理工大学）
In this work, we address the problem of long-distance navigation for battery electric vehicles (BEVs), where one or more charging sessions are required to reach the intended destination. We consider the availability and performance of the charging stations to be unknown and stochastic, and develop a combinatorial semi-bandit framework for exploring the road network to learn the parameters of the queue time and charging power distributions. Within this framework, we first outline a pre-processing for the road network graph to handle the constrained combinatorial optimization problem in an effic…
Restless bandit problems are instances of non-stationary multi-armed bandits. These problems have been studied well from the optimization perspective, where the goal is to efficiently find a near-optimal policy when system parameters are known. However, very few papers adopt a learning perspective, where the parameters are unknown. In this paper, we analyze the performance of Thompson sampling in episodic restless bandits with unknown parameters. We consider a general policy map to define our competitor and prove an $\tilde{\mathcal{O}}(\sqrt{T})$ Bayesian regret bound. Our competitor is flexi…
Combinatorial Bandit Bayesian Optimization for Tensor Outputs
Department of Industrial Engineering, Tsinghua University, Beijing 100084, China（清华大学工业工程系）；College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China（南京航空航天大学经济管理学院）
We study PAC-Bayesian generalization bounds for Multilayer Perceptrons (MLPs) with the cross entropy loss. Above all, we introduce probabilistic explanations for MLPs in two aspects: (i) MLPs formulate a family of Gibbs distributions, and (ii) minimizing the cross-entropy loss for MLPs is equivalent to Bayesian variational inference, which establish a solid probabilistic foundation for studying PAC-Bayesian bounds on MLPs. Furthermore, based on the Evidence Lower Bound (ELBO), we prove that MLPs with the cross entropy loss inherently guarantee PAC- Bayesian generalization bounds, and minimizin…
Contextual multi-armed bandit problems arise frequently in important industrial applications. Existing solutions model the context either linearly, which enables uncertainty driven (principled) exploration, or non-linearly, by using epsilon-greedy exploration policies. Here we present a deep learning framework for contextual multi-armed bandits that is both non-linear and enables principled exploration at the same time. We tackle the exploration vs. exploitation trade-off through Thompson sampling by exploiting the connection between inference time dropout and sampling from the posterior over …
Inria Lille Nord-Europe（法国国家信息与自动化研究所里尔北欧中心）；Orange Labs（Orange实验室）；CNRS（法国国家科学研究中心）；Univ. Lille（里尔大学）
We study a variant of the stochastic multi-armed bandit (MAB) problem in which the rewards are corrupted. In this framework, motivated by privacy preservation in online recommender systems, the goal is to maximize the sum of the (unobserved) rewards, based on the observation of transformation of these rewards through a stochastic corruption process with known parameters. We provide a lower bound on the expected regret of any bandit algorithm in this corrupted setting. We devise a frequentist algorithm, KLUCB-CF, and a Bayesian algorithm, TS-CF and give upper bounds on their regret. We also pro…
By leveraging experience from previous tasks, meta-learning algorithms can achieve effective fast adaptation ability when encountering new tasks. However it is unclear how the generalization property applies to new tasks. Probably approximately correct (PAC) Bayes bound theory provides a theoretical framework to analyze the generalization performance for meta-learning. We derive three novel generalisation error bounds for meta-learning based on PAC-Bayes relative entropy bound. Furthermore, using the empirical risk minimization (ERM) method, a PAC-Bayes bound for meta-learning with data-depend…
Princeton University（普林斯顿大学）；Università degli Studi di Milano（米兰大学）
Multi-armed bandit problems are the most basic examples of sequential decision problems with an exploration-exploitation trade-off. This is the balance between staying with the option that gave highest payoffs in the past and exploring new options that might give higher payoffs in the future. Although the study of bandit problems dates back to the Thirties, exploration-exploitation trade-offs arise in several modern applications, such as ad placement, website optimization, and packet routing. Mathematically, a multi-armed bandit is defined by the payoff process associated with each option. In …
This paper studies regret minimization in a multi-armed bandit. It is well known that side information, such as the prior distribution of arm means in Thompson sampling, can improve the statistical efficiency of the bandit algorithm. While the prior is a blessing when correctly specified, it is a curse when misspecified. To address this issue, we introduce the assumption of a random-effect model to bandits. In this model, the mean arm rewards are drawn independently from an unknown distribution, which we estimate. We derive a random-effect estimator of the arm means, analyze its uncertainty, a…
Model misspecification is a major consideration in applications of statistical methods and machine learning. However, it is often neglected in contextual bandits. This paper studies a common form of misspecification, an inter-arm heterogeneity that is not captured by context. To address this issue, we assume that the heterogeneity arises due to arm-specific random variables, which can be learned. We call this setting a robust contextual bandit. The arm-specific variables explain the unknown inter-arm heterogeneity, and we incorporate them in the robust contextual estimator of the mean reward a…
University of Cambridge（剑桥大学）；Alan Turing Institute（艾伦·图灵研究所）
Drawing a sample from a discrete distribution is one of the building components for Monte Carlo methods. Like other sampling algorithms, discrete sampling suffers from the high computational burden in large-scale inference problems. We study the problem of sampling a discrete random variable with a high degree of dependency that is typical in large-scale Bayesian inference and graphical models, and propose an efficient approximate solution with a subsampling approach. We make a novel connection between the discrete sampling and Multi-Armed Bandits problems with a finite reward population and p…
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…
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…
Inria Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）；University College London（伦敦大学学院）
Generalised Bayesian learning algorithms are increasingly popular in machine learning, due to their PAC generalisation properties and flexibility. The present paper aims at providing a self-contained survey on the resulting PAC-Bayes framework and some of its main theoretical and algorithmic developments.
Tel Aviv University（特拉维夫大学）；Google AI（谷歌人工智能）；Princeton（普林斯顿大学）
PAC-Bayes is a useful framework for deriving generalization bounds which was introduced by McAllester ('98). This framework has the flexibility of deriving distribution- and algorithm-dependent bounds, which are often tighter than VC-related uniform convergence bounds. In this manuscript we present a limitation for the PAC-Bayes framework. We demonstrate an easy learning task that is not amenable to a PAC-Bayes analysis. Specifically, we consider the task of linear classification in 1D; it is well-known that this task is learnable using just $O(\log(1/δ)/ε)$ examples. On the other hand, we sho…
University College London（伦敦大学学院）；Inria（法国国家信息与自动化研究所）
When utilising PAC-Bayes theory for risk certification, it is usually necessary to estimate and bound the Gibbs risk of the PAC-Bayes posterior. Many works in the literature employ a method for this which requires a large number of passes of the dataset, incurring high computational cost. This manuscript presents a very general alternative which makes computational savings on the order of the dataset size.
Recent studies have empirically investigated different methods to train stochastic neural networks on a classification task by optimising a PAC-Bayesian bound via stochastic gradient descent. Most of these procedures need to replace the misclassification error with a surrogate loss, leading to a mismatch between the optimisation objective and the actual generalisation bound. The present paper proposes a novel training algorithm that optimises the PAC-Bayesian bound, without relying on any surrogate loss. Empirical results show that this approach outperforms currently available PAC-Bayesian tra…
Columbia University（哥伦比亚大学）；Moloco（Moloco公司）；University of New Hampshire（新罕布什尔大学）
Many efficient algorithms with strong theoretical guarantees have been proposed for the contextual multi-armed bandit problem. However, applying these algorithms in practice can be difficult because they require domain expertise to build appropriate features and to tune their parameters. We propose a new method for the contextual bandit problem that is simple, practical, and can be applied with little or no domain expertise. Our algorithm relies on decision trees to model the context-reward relationship. Decision trees are non-parametric, interpretable, and work well without hand-crafted featu…
LTS2；EPFL（洛桑联邦理工学院）；IDCOM；University of Edinburgh（爱丁堡大学）
Recently the generalization error of deep neural networks has been analyzed through the PAC-Bayesian framework, for the case of fully connected layers. We adapt this approach to the convolutional setting.
Meta learning automatically infers an inductive bias, that includes the hyperparameter of the base-learning algorithm, by observing data from a finite number of related tasks. This paper studies PAC-Bayes bounds on meta generalization gap. The meta-generalization gap comprises two sources of generalization gaps: the environment-level and task-level gaps resulting from observation of a finite number of tasks and data samples per task, respectively. In this paper, by upper bounding arbitrary convex functions, which link the expected and empirical losses at the environment and also per-task level…
Empirically, the PAC-Bayesian analysis is known to produce tight risk bounds for practical machine learning algorithms. However, in its naive form, it can only deal with stochastic predictors while such predictors are rarely used and deterministic predictors often performs well in practice. To fill this gap, we develop a new generalization error bound, the PAC-Bayesian transportation bound, unifying the PAC-Bayesian analysis and the chaining method in view of the optimal transportation. It is the first PAC-Bayesian bound that relates the risks of any two predictors according to their distance,…
We introduce vector optimization problems with stochastic bandit feedback, in which preferences among designs are encoded by a polyhedral ordering cone $C$. Our setup generalizes the best arm identification problem to vector-valued rewards by extending the concept of Pareto set beyond multi-objective optimization. We characterize the sample complexity of ($ε,δ$)-PAC Pareto set identification by defining a new cone-dependent notion of complexity, called the ordering complexity. In particular, we provide gap-dependent and worst-case lower bounds on the sample complexity and show that, in the wor…
The Ohio State University（俄亥俄州立大学）；AT&T Labs Research（AT&T实验室研究中心）
We consider stochastic multi-armed bandit problems with graph feedback, where the decision maker is allowed to observe the neighboring actions of the chosen action. We allow the graph structure to vary with time and consider both deterministic and Erdős-Rényi random graph models. For such a graph feedback model, we first present a novel analysis of Thompson sampling that leads to tighter performance bound than existing work. Next, we propose new Information Directed Sampling based policies that are graph-aware in their decision making. Under the deterministic graph case, we establish a Bayesia…
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…
Over the past decade, contextual bandit algorithms have been gaining in popularity due to their effectiveness and flexibility in solving sequential decision problems---from online advertising and finance to clinical trial design and personalized medicine. At the same time, there are, as of yet, surprisingly few options that enable researchers and practitioners to simulate and compare the wealth of new and existing bandit algorithms in a standardized way. To help close this gap between analytical research and empirical evaluation the current paper introduces the object-oriented R package "conte…
Multi-armed bandits are a quintessential machine learning problem requiring the balancing of exploration and exploitation. While there has been progress in developing algorithms with strong theoretical guarantees, there has been less focus on practical near-optimal finite-time performance. In this paper, we propose an algorithm for Bayesian multi-armed bandits that utilizes value-function-driven online planning techniques. Building on previous work on UCB and Gittins index, we introduce linearly-separable value functions that take both the expected return and the benefit of exploration into co…
We present a formal model of human decision-making in explore-exploit tasks using the context of multi-armed bandit problems, where the decision-maker must choose among multiple options with uncertain rewards. We address the standard multi-armed bandit problem, the multi-armed bandit problem with transition costs, and the multi-armed bandit problem on graphs. We focus on the case of Gaussian rewards in a setting where the decision-maker uses Bayesian inference to estimate the reward values. We model the decision-maker's prior knowledge with the Bayesian prior on the mean reward. We develop the…
As cellular networks become denser, a scalable and dynamic tuning of wireless base station parameters can only be achieved through automated optimization. Although the contextual bandit framework arises as a natural candidate for such a task, its extension to a parallel setting is not straightforward: one needs to carefully adapt existing methods to fully leverage the multi-agent structure of this problem. We propose two approaches: one derived from a deterministic UCB-like method and the other relying on Thompson sampling. Thanks to its bayesian nature, the latter is intuited to better preser…
We give tight concentration bounds for mixtures of martingales that are simultaneously uniform over (a) mixture distributions, in a PAC-Bayes sense; and (b) all finite times. These bounds are proved in terms of the martingale variance, extending classical Bernstein inequalities, and sharpening and simplifying prior work.
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecué and Mitchell, 2012; Zhivotovskiy and Hanneke…
CWI Amsterdam（阿姆斯特丹数学与计算机科学国家研究所）；Leiden University（莱顿大学）；Google Research（谷歌研究院）；Khoury College of Computer Sciences, Northeastern University（东北大学库里计算机科学学院）
We give a novel, unified derivation of conditional PAC-Bayesian and mutual information (MI) generalization bounds. We derive conditional MI bounds as an instance, with special choice of prior, of conditional MAC-Bayesian (Mean Approximately Correct) bounds, itself derived from conditional PAC-Bayesian bounds, where `conditional' means that one can use priors conditioned on a joint training and ghost sample. This allows us to get nontrivial PAC-Bayes and MI-style bounds for general VC classes, something recently shown to be impossible with standard PAC-Bayesian/MI bounds. Second, it allows us t…
When I first encountered PAC-Bayesian concentration inequalities they seemed to me to be rather disconnected from good old-fashioned results like Hoeffding's and Bernstein's inequalities. But, at least for one flavour of the PAC-Bayesian bounds, there is actually a very close relation, and the main innovation is a continuous version of the union bound, along with some ingenious applications. Here's the gist of what's going on, presented from a machine learning perspective.
