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Theory/Online Learning and Bandits

Dimitrios Christou, Stratis Skoulakis, Volkan Cevher

In this work we consider an online learning problem, called Online $k$-Clustering with Moving Costs, at which a learner maintains a set of $k$ facilities over $T$ rounds so as to minimize the connection cost of an adversarially selected sequence of clients. The learner is informed on the positions of the clients at each round $t$ only after its facility-selection and can use this information to update its decision in the next round. However, updating the facility positions comes with an additional moving cost based on the moving distance of the facilities. We present the first $\mathcal{O}(\log n)$-regret polynomial-time online learning algorithm guaranteeing that the overall cost (connection $+$ moving) is at most $\mathcal{O}(\log n)$ times the time-averaged connection cost of the best fixed solution. Our work improves on the recent result of (Fotakis et al., 2021) establishing $\mathcal{O}(k)$-regret guarantees only on the connection cost.

Theory/Online Learning and Bandits

Marc Jourdan, Rémy Degenne

A Top Two sampling rule for bandit identification is a method which selects the next arm to sample from among two candidate arms, a *leader* and a *challenger*. Due to their simplicity and good empirical performance, they have received increased attention in recent years. However, for fixed-confidence best arm identification, theoretical guarantees for Top Two methods have only been obtained in the asymptotic regime, when the error level vanishes. In this paper, we derive the first non-asymptotic upper bound on the expected sample complexity of a Top Two algorithm, which holds for any error level. Our analysis highlights sufficient properties for a regret minimization algorithm to be used as leader. These properties are satisfied by the UCB algorithm, and our proposed UCB-based Top Two algorithm simultaneously enjoys non-asymptotic guarantees and competitive empirical performance.

Theory/Online Learning and Bandits

Xiao Zhang, Ninglu Shao, Zihua Si, Jun Xu, Wenhan Wang, Hanjing Su, Ji-Rong Wen

Contextual batched bandit (CBB) is a setting where a batch of rewards is observed from the environment at the end of each episode, but the rewards of the non-executed actions are unobserved, resulting in partial-information feedback. Existing approaches for CBB often ignore the rewards of the non-executed actions, leading to underutilization of feedback information. In this paper, we propose an efficient approach called Sketched Policy Updating with Imputed Rewards (SPUIR) that completes the unobserved rewards using sketching, which approximates the full-information feedbacks. We formulate reward imputation as an imputation regularized ridge regression problem that captures the feedback mechanisms of both executed and non-executed actions. To reduce time complexity, we solve the regression problem using randomized sketching. We prove that our approach achieves an instantaneous regret with controllable bias and smaller variance than approaches without reward imputation. Furthermore, our approach enjoys a sublinear regret bound against the optimal policy. We also present two extensions, a rate-scheduled version and a version for nonlinear rewards, making our approach more practical. Experimental results show that SPUIR outperforms state-of-the-art baselines on synthetic, public benchmark, and real-world datasets.

Theory/Online Learning and Bandits

Mathieu Molina, Nicolas Gast, Patrick Loiseau, Vianney Perchet

We consider the problem of online allocation subject to a long-term fairness penalty. Contrary to existing works, however, we do not assume that the decision-maker observes the protected attributes---which is often unrealistic in practice. Instead they can purchase data that help estimate them from sources of different quality; and hence reduce the fairness penalty at some cost. We model this problem as a multi-armed bandit problem where each arm corresponds to the choice of a data source, coupled with the fair online allocation problem. We propose an algorithm that jointly solves both problems and show that it has a regret bounded by $\mathcal{O}(\sqrt{T})$. A key difficulty is that the rewards received by selecting a source are correlated by the fairness penalty, which leads to a need for randomization (despite a stochastic setting). Our algorithm takes into account contextual information available before the source selection, and can adapt to many different fairness notions.

Theory/Online Learning and Bandits

Haolin Liu, Chen-Yu Wei, Julian Zimmert

We consider the adversarial linear contextual bandit problem, where the loss vectors are selected fully adversarially and the per-round action set (i.e. the context) is drawn from a fixed distribution. Existing methods for this problem either require access to a simulator to generate free i.i.d. contexts, achieve a sub-optimal regret no better than $\tilde{\mathcal{O}}(T^{\frac{5}{6}})$, or are computationally inefficient. We greatly improve these results by achieving a regret of $\tilde{\mathcal{O}}(\sqrt{T})$ without a simulator, while maintaining computational efficiency when the action set in each round is small. In the special case of sleeping bandits with adversarial loss and stochastic arm availability, our result answers affirmatively the open question by [SGV20] on whether there exists a polynomial-time algorithm with $poly(d)\sqrt{T}$ regret. Our approach naturally handles the case where the loss is linear up to an additive misspecification error, and our regret shows near-optimal dependence on the magnitude of the error.

Theory/Online Learning and Bandits

Junpei Komiyama, Masaaki Imaizumi

In this research, we investigate the high-dimensional linear contextual bandit problem where the number of features $p$ is greater than the budget $T$, or it may even be infinite. Differing from the majority of previous works in this field, we do not impose sparsity on the regression coefficients. Instead, we rely on recent findings on overparameterized models, which enables us to analyze the performance of the minimum-norm interpolating estimator when data distributions have small effective ranks. We propose an explore-then-commit (EtC) algorithm to address this problem and examine its performance. Through our analysis, we derive the optimal rate of the ETC algorithm in terms of $T$ and show that this rate can be achieved by balancing exploration and exploitation. Moreover, we introduce an adaptive explore-then-commit (AEtC) algorithm that adaptively finds the optimal balance. We assess the performance of the proposed algorithms through a series of simulations.

Theory/Online Learning and Bandits

Mengfan Xu, Diego Klabjan

We study a decentralized multi-agent multi-armed bandit problem in which multiple clients are connected by time dependent random graphs provided by an environment. The reward distributions of each arm vary across clients and rewards are generated independently over time by an environment based on distributions that include both sub-exponential and sub-gaussian distributions. Each client pulls an arm and communicates with neighbors based on the graph provided by the environment. The goal is to minimize the overall regret of the entire system through collaborations. To this end, we introduce a novel algorithmic framework, which first provides robust simulation methods for generating random graphs using rapidly mixing markov chains or the random graph model, and then combines an averaging-based consensus approach with a newly proposed weighting technique and the upper confidence bound to deliver a UCB-type solution. Our algorithms account for the randomness in the graphs, removing the conventional doubly stochasticity assumption, and only require the knowledge of the number of clients at initialization. We derive optimal instance-dependent regret upper bounds of order $\log{T}$ in both sub-gaussian and sub-exponential environments, and a nearly optimal instance-free regret upper bound of order $\sqrt{T}\log T$ up to a $\log T$ factor. Importantly, our regret bounds hold with high probability and capture graph randomness, whereas prior works consider expected regret under assumptions and require more stringent reward distributions.

Theory/Online Learning and Bandits

Yue Kang, Cho-Jui Hsieh, Thomas Chun Man Lee

Lipschitz bandit is a variant of stochastic bandits that deals with a continuous arm set defined on a metric space, where the reward function is subject to a Lipschitz constraint. In this paper, we introduce a new problem of Lipschitz bandits in the presence of adversarial corruptions where an adaptive adversary corrupts the stochastic rewards up to a total budget $C$. The budget is measured by the sum of corruption levels across the time horizon $T$. We consider both weak and strong adversaries, where the weak adversary is unaware of the current action before the attack, while the strong one can observe it. Our work presents the first line of robust Lipschitz bandit algorithms that can achieve sub-linear regret under both types of adversary, even when the total budget of corruption $C$ is unrevealed to the agent. We provide a lower bound under each type of adversary, and show that our algorithm is optimal under the strong case. Finally, we conduct experiments to illustrate the effectiveness of our algorithms against two classic kinds of attacks.

Theory/Online Learning and Bandits

Rui Castro, Fredrik Hellström, Tim van Erven

We consider online prediction of a binary sequence with expert advice. For this setting, we devise label-efficient forecasting algorithms, which use a selective sampling scheme that enables collecting much fewer labels than standard procedures. For the general case without a perfect expert, we prove best-of-both-worlds guarantees, demonstrating that the proposed forecasting algorithm always queries sufficiently many labels in the worst case to obtain optimal regret guarantees, while simultaneously querying much fewer labels in more benign settings. Specifically, for a scenario where one expert is strictly better than the others in expectation, we show that the label complexity of the label-efficient forecaster is roughly upper-bounded by the square root of the number of rounds. Finally, we present numerical experiments empirically showing that the normalized regret of the label-efficient forecaster can asymptotically match known minimax rates for pool-based active learning, suggesting it can optimally adapt to benign settings.

Theory/Online Learning and Bandits

Raunak Kumar, Sarah Dean, Robert Kleinberg

Online convex optimization (OCO) is a widely used framework in online learning. In each round, the learner chooses a decision in a convex set and an adversary chooses a convex loss function, and then the learner suffers the loss associated with their current decision. However, in many applications the learner's loss depends not only on the current decision but on the entire history of decisions until that point. The OCO framework and its existing generalizations do not capture this, and they can only be applied to many settings of interest after a long series of approximation arguments. They also leave open the question of whether the dependence on memory is tight because there are no non-trivial lower bounds. In this work we introduce a generalization of the OCO framework, ``Online Convex Optimization with Unbounded Memory'', that captures long-term dependence on past decisions. We introduce the notion of $p$-effective memory capacity, $H_p$, that quantifies the maximum influence of past decisions on present losses. We prove an $O(\sqrt{H_p T})$ upper bound on the policy regret and a matching (worst-case) lower bound. As a special case, we prove the first non-trivial lower bound for OCO with finite memory~\citep{anavaHM2015online}, which could be of independent interest, and also improve existing upper bounds. We demonstrate the broad applicability of our framework by using it to derive regret bounds, and to improve and simplify existing regret bound derivations, for a variety of online learning problems including online linear control and an online variant of performative prediction.

Theory/Online Learning and Bandits

Shipra Agrawal, Yiding Feng, Wei Tang

We consider a novel dynamic pricing and learning setting where in addition to setting prices of products in sequential rounds, the seller also ex-ante commits to ‘advertising schemes’. That is, in the beginning of each round the seller can decide what kind of signal they will provide to the buyer about the product’s quality upon realization. Using the popular Bayesian persuasion framework to model the effect of these signals on the buyers’ valuation and purchase responses, we formulate the problem of finding an optimal design of the advertising scheme along with a pricing scheme that maximizes the seller’s expected revenue. Without any apriori knowledge of the buyers’ demand function, our goal is to design an online algorithm that can use past purchase responses to adaptively learn the optimal pricing and advertising strategy. We study the regret of the algorithm when compared to the optimal clairvoyant price and advertisingscheme. Our main result is a computationally efficient online algorithm that achieves an $O(T^{2/3}(m \log T )^{1/3})$ regret bound when the valuation function is linear in the product quality. Here $m$ is the cardinality of the discrete product quality domain and $T$ is the time horizon. This result requires some natural monotonicity and Lipschitz assumptions on the valuation function, but no Lipschitz or smoothness assumption on the buyers’ demand function. For constant $m$, our result matches the regret lower bound for dynamic pricing within logarithmic factors, which is a special case of our problem. We also obtain several improved results for the widely considered special case of additive valuations, including an $\tilde{O}(T^{2/3})$ regret bound independent of $m$ when $m\le T^{1/3}$.

Theory/Online Learning and Bandits

David Woodruff, Fred Zhang, Samson Zhou

In the online learning with experts problem, an algorithm makes predictions about an outcome on each of $T$ days, given a set of $n$ experts who make predictions on each day. The algorithm is given feedback on the outcomes of each day, including the cost of its prediction and the cost of the expert predictions, and the goal is to make a prediction with the minimum cost, compared to the best expert in hindsight. However, often the predictions made by experts or algorithms at some time influence future outcomes, so that the input is adaptively generated. In this paper, we study robust algorithms for the experts problem under memory constraints. We first give a randomized algorithm that is robust to adaptive inputs that uses $\widetilde{O}\left(\frac{n}{R\sqrt{T}}\right)$ space for $M=O\left(\frac{R^2 T}{\log^2 n}\right)$, thereby showing a smooth space-regret trade-off. We then show a space lower bound of $\widetilde{\Omega}\left(\frac{nM}{RT}\right)$ for any randomized algorithm that achieves regret $R$ with probability $1-2^{-\Omega(T)}$, when the best expert makes $M$ mistakes. Our result implies that the natural deterministic algorithm, which iterates through pools of experts until each expert in the pool has erred, is optimal up to polylogarithmic factors. Finally, we empirically demonstrate the benefit of using robust procedures against a white-box adversary that has access to the internal state of the algorithm.

Theory/Online Learning and Bandits

Jing Dong, Yuichi Yoshida

We introduce a transformation framework that can be utilized to develop online algorithms with low $\epsilon$-approximate regret in the random-order model from offline approximation algorithms. We first give a general reduction theorem that transforms an offline approximation algorithm with low average sensitivity to an online algorithm with low $\epsilon$-approximate regret. We then demonstrate that offline approximation algorithms can be transformed into a low-sensitivity version using a coreset construction method. To showcase the versatility of our approach, we apply it to various problems, including online $(k,z)$-clustering, online matrix approximation, and online regression, and successfully achieve polylogarithmic $\epsilon$-approximate regret for each problem. Moreover, we show that in all three cases, our algorithm also enjoys low inconsistency, which may be desired in some online applications.

Theory/Online Learning and Bandits

Qinyi Chen, Negin Golrezaei, Djallel Bouneffouf

Traditional multi-armed bandit (MAB) frameworks, predominantly examined under stochastic or adversarial settings, often overlook the temporal dynamics inherent in many real-world applications such as recommendation systems and online advertising. This paper introduces a novel non-stationary MAB framework that captures the temporal structure of these real-world dynamics through an auto-regressive (AR) reward structure. We propose an algorithm that integrates two key mechanisms: (i) an alternation mechanism adept at leveraging temporal dependencies to dynamically balance exploration and exploitation, and (ii) a restarting mechanism designed to discard out-of-date information. Our algorithm achieves a regret upper bound that nearly matches the lower bound, with regret measured against a robust dynamic benchmark. Finally, via a real-world case study on tourism demand prediction, we demonstrate both the efficacy of our algorithm and the broader applicability of our techniques to more complex, rapidly evolving time series.

Theory/Online Learning and Bandits

Chung-Wei Lee, Qinghua Liu, Yasin Abbasi Yadkori, Chi Jin, Tor Lattimore, Csaba Szepesvari

We consider a contextual bandit problem with $S $ contexts and $K $ actions. In each round $t=1,2,\dots$ the learnerobserves a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into $r\le \min(S ,K)$ groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an $\epsilon$-optimal policy after using at most $\widetilde O(r (S +K )/\epsilon^2)$ samples with high probability and provide a matching $\widetilde\Omega(r (S +K )/\epsilon^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r ^3(S +K )T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O{\sqrt{\text{poly}(r)(S+K)T}}$ minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.

Theory/Online Learning and Bandits

Soumya Basu, Abishek Sankararaman

Double Auction enables decentralized transfer of goods between multiple buyers and sellers, thus underpinning functioning of many online marketplaces. Buyers and sellers compete in these markets through bidding, but do not often know their own valuation a-priori. As the allocation and pricing happens through bids, the profitability of participants, hence sustainability of such markets, depends crucially on learning respective valuations through repeated interactions. We initiate the study of Double Auction markets under bandit feedback on both buyers' and sellers' side. We show with confidence bound based bidding, and `Average Pricing' there is an efficient price discovery among the participants. In particular, the regret on combined valuation of the buyers and the sellers -- a.k.a. the social regret -- is $O(\log(T)/\Delta)$ in $T$ rounds, where $\Delta$ is the minimum price gap. Moreover, the buyers and sellers exchanging goods attain $O(\sqrt{T})$ regret, individually. The buyers and sellers who do not benefit from exchange in turn only experience $O(\log{T}/ \Delta)$ regret individually in $T$ rounds. We augment our upper bound by showing that $\omega(\sqrt{T})$ individual regret, and $\omega(\log{T})$ social regret is unattainable in certain Double Auction markets. Our paper is the first to provide decentralized learning algorithms in a two-sided market where \emph{both sides have uncertain preference} that need to be learned.

Theory/Online Learning and Bandits

Zihao Hu, Guanghui Wang, Jacob Abernethy

The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), for enforcing constraints and achieving optimal regret bounds. However, it suffers from computational complexity limitations in high-dimensional settings or when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue by replacing the projection oracle with more efficient optimization subroutines. But to date, these methods have been developed primarily in the Euclidean setting, and while there has been growing interest in optimization on Riemannian manifolds, there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions are generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization on curved spaces for two scenarios: when we have access to (a) a separation oracle or (b) a linear optimization oracle. For geodesically convex losses, and when a separation oracle is available, our algorithms achieve $O(T^{\frac{1}{2}})$, $O(T^{\frac{3}{4}})$ and $O(T^{\frac{1}{2}})$ adaptive regret guarantees in the full information setting, the bandit setting with one-point feedback and the bandit setting with two-point feedback, respectively. When a linear optimization oracle is available, we obtain regret rates of $O(T^{\frac{3}{4}})$ for geodesically convex losses and $O(T^{\frac{2}{3}}\log T)$ for strongly geodesically convex losses.

Theory/Online Learning and Bandits

Daniel Freund, Thodoris Lykouris, Wentao Weng

Queueing systems are widely applicable stochastic models with use cases in communication networks, healthcare, service systems, etc. Although their optimal control has been extensively studied, most existing approaches assume perfect knowledge of the system parameters. Of course, this assumption rarely holds in practice where there is parameter uncertainty, thus motivating a recent line of work on bandit learning for queueing systems. This nascent stream of research focuses on the asymptotic performance of the proposed algorithms. In this paper, we argue that an asymptotic metric, which focuses on late-stage performance, is insufficient to capture the intrinsic statistical complexity of learning in queueing systems which typically occurs in the early stage. Instead, we propose the *Cost of Learning in Queueing (CLQ)*, a new metric that quantifies the maximum increase in time-averaged queue length caused by parameter uncertainty.We characterize the CLQ of a single-queue multi-server system, and then extend these results to multi-queue multi-server systems and networks of queues. In establishing our results, we propose a unified analysis framework for CLQ that bridges Lyapunov and bandit analysis, provides guarantees for a wide range of algorithms, and could be of independent interest.

Theory/Online Learning and Bandits

El Mehdi Saad, Gilles Blanchard, Nicolas Verzelen

We consider the problem of best arm identification in the multi-armed bandit model, under fixed confidence. Given a confidence input $\delta$, the goal is to identify the arm with the highest mean reward with a probability of at least $1 - \delta$, while minimizing the number of arm pulls. While the literature provides solutions to this problem under the assumption of independent arms distributions, we propose a more flexible scenario where arms can be dependent and rewards can be sampled simultaneously. This framework allows the learner to estimate the covariance among the arms distributions, enabling a more efficient identification of the best arm. The relaxed setting we propose is relevant in various applications, such as clinical trials, where similarities between patients or drugs suggest underlying correlations in the outcomes. We introduce new algorithms that adapt to the unknown covariance of the arms and demonstrate through theoretical guarantees that substantial improvement can be achieved over the standard setting. Additionally, we provide new lower bounds for the relaxed setting and present numerical simulations that support their theoretical findings.

Theory/Online Learning and Bandits

Alankrita Bhatt, Nika Haghtalab, Abhishek Shetty

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret.