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Theory/Online Learning and Bandits

Nicolas Emmenegger, Mojmir Mutny, Andreas Krause

Certifiable, adaptive uncertainty estimates for unknown quantities are an essential ingredient of sequential decision-making algorithms. Standard approaches rely on problem-dependent concentration results and are limited to a specific combination of parameterization, noise family, and estimator. In this paper, we revisit the likelihood-based inference principle and propose to use \emph{likelihood ratios} to construct \emph{any-time valid} confidence sequences without requiring specialized treatment in each application scenario. Our method is especially suitable for problems with well-specified likelihoods, and the resulting sets always maintain the prescribed coverage in a model-agnostic manner. The size of the sets depends on a choice of estimator sequence in the likelihood ratio. We discuss how to provably choose the best sequence of estimators and shed light on connections to online convex optimization with algorithms such as Follow-the-Regularized-Leader. To counteract the initially large bias of the estimators, we propose a reweighting scheme that also opens up deployment in non-parametric settings such as RKHS function classes. We provide a \emph{non-asymptotic} analysis of the likelihood ratio confidence sets size for generalized linear models, using insights from convex duality and online learning. We showcase the practical strength of our method on generalized linear bandit problems, survival analysis, and bandits with various additive noise distributions.

Theory/Online Learning and Bandits

Hyun-jun Choi, Rajan Udwani, Min-hwan Oh

We present a new combinatorial bandit model, the \textit{cascading contextual assortment bandit}. This model serves as a generalization of both existing cascading bandits and assortment bandits, broadening their applicability in practice. For this model, we propose our first UCB bandit algorithm, UCB-CCA. We prove that this algorithm achieves a $T$-step regret upper-bound of $\tilde{\mathcal{O}}(\frac{1}{\kappa}d\sqrt{T})$, sharper than existing bounds for cascading contextual bandits by eliminating dependence on cascade length $K$. To improve the dependence on problem-dependent constant $\kappa$, we introduce our second algorithm, UCB-CCA+, which leverages a new Bernstein-type concentration result. This algorithm achieves $\tilde{\mathcal{O}}(d\sqrt{T})$ without dependence on $\kappa$ in the leading term. We substantiate our theoretical claims with numerical experiments, demonstrating the practical efficacy of our proposed methods.

Optimization/Stochastic

Lijun Zhang, Peng Zhao, Zhen-Hua Zhuang, Tianbao Yang, Zhi-Hua Zhou

This paper investigates group distributionally robust optimization (GDRO), with the purpose to learn a model that performs well over $m$ different distributions. First, we formulate GDRO as a stochastic convex-concave saddle-point problem, and demonstrate that stochastic mirror descent (SMD), using $m$ samples in each iteration, achieves an $O(m (\log m)/\epsilon^2)$ sample complexity for finding an $\epsilon$-optimal solution, which matches the $\Omega(m/\epsilon^2)$ lower bound up to a logarithmic factor. Then, we make use of techniques from online learning to reduce the number of samples required in each round from $m$ to $1$, keeping the same sample complexity. Specifically, we cast GDRO as a two-players game where one player simply performs SMD and the other executes an online algorithm for non-oblivious multi-armed bandits. Next, we consider a more practical scenario where the number of samples that can be drawn from each distribution is different, and propose a novel formulation of weighted GDRO, which allows us to derive distribution-dependent convergence rates. Denote by $n_i$ the sample budget for the $i$-th distribution, and assume $n_1 \geq n_2 \geq \cdots \geq n_m$. In the first approach, we incorporate non-uniform sampling into SMD such that the sample budget is satisfied in expectation, and prove that the excess risk of the $i$-th distribution decreases at an $O(\sqrt{n_1 \log m}/n_i)$ rate. In the second approach, we use mini-batches to meet the budget exactly and also reduce the variance in stochastic gradients, and then leverage stochastic mirror-prox algorithm, which can exploit small variances, to optimize a carefully designed weighted GDRO problem. Under appropriate conditions, it attains an $O((\log m)/\sqrt{n_i})$ convergence rate, which almost matches the optimal $O(\sqrt{1/n_i})$ rate of only learning from the $i$-th distribution with $n_i$ samples.

Theory/Online Learning and Bandits

Evgenii Chzhen, Christophe Giraud, Zhen LI, Gilles Stoltz

We consider contextual bandit problems with knapsacks [CBwK], a problem where at each round, a scalar reward is obtained and vector-valued costs are suffered. The learner aims to maximize the cumulative rewards while ensuring that the cumulative costs are lower than some predetermined cost constraints. We assume that contexts come from a continuous set, that costs can be signed, and that the expected reward and cost functions, while unknown, may be uniformly estimated---a typical assumption in the literature. In this setting, total cost constraints had so far to be at least of order $T^{3/4}$, where $T$ is the number of rounds, and were even typically assumed to depend linearly on $T$. We are however motivated to use CBwK to impose a fairness constraint of equalized average costs between groups: the budget associated with the corresponding cost constraints should be as close as possible to the natural deviations, of order $\sqrt{T}$. To that end, we introduce a dual strategy based on projected-gradient-descent updates, that is able to deal with total-cost constraints of the order of $\sqrt{T}$ up to poly-logarithmic terms. This strategy is more direct and simpler than existing strategies in the literature. It relies on a careful, adaptive, tuning of the step size.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Binhui Xie, Shuang Li, Qingju Guo, Chi Liu, Xinjing Cheng

Active learning, a label-efficient paradigm, empowers models to interactively query an oracle for labeling new data. In the realm of LiDAR semantic segmentation, the challenges stem from the sheer volume of point clouds, rendering annotation labor-intensive and cost-prohibitive. This paper presents Annotator, a general and efficient active learning baseline, in which a voxel-centric online selection strategy is tailored to efficiently probe and annotate the salient and exemplar voxel girds within each LiDAR scan, even under distribution shift. Concretely, we first execute an in-depth analysis of several common selection strategies such as Random, Entropy, Margin, and then develop voxel confusion degree (VCD) to exploit the local topology relations and structures of point clouds. Annotator excels in diverse settings, with a particular focus on active learning (AL), active source-free domain adaptation (ASFDA), and active domain adaptation (ADA). It consistently delivers exceptional performance across LiDAR semantic segmentation benchmarks, spanning both simulation-to-real and real-to-real scenarios. Surprisingly, Annotator exhibits remarkable efficiency, requiring significantly fewer annotations, e.g., just labeling five voxels per scan in the SynLiDAR → SemanticKITTI task. This results in impressive performance, achieving 87.8% fully-supervised performance under AL, 88.5% under ASFDA, and 94.4% under ADA. We envision that Annotator will offer a simple, general, and efficient solution for label-efficient 3D applications.

Theory/Online Learning and Bandits

Eric Neyman, Tim Roughgarden

For each of $T$ time steps, $m$ experts report probability distributions over $n$ outcomes; we wish to learn to aggregate these forecasts in a way that attains a no-regret guarantee. We focus on the fundamental and practical aggregation method known as *logarithmic pooling* -- a weighted average of log odds -- which is in a certain sense the optimal choice of pooling method if one is interested in minimizing log loss (as we take to be our loss function). We consider the problem of learning the best set of parameters (i.e. expert weights) in an online adversarial setting. We assume (by necessity) that the adversarial choices of outcomes and forecasts are consistent, in the sense that experts report calibrated forecasts. Imposing this constraint creates a (to our knowledge) novel semi-adversarial setting in which the adversary retains a large amount of flexibility. In this setting, we present an algorithm based on online mirror descent that learns expert weights in a way that attains $O(\sqrt{T} \log T)$ expected regret as compared with the best weights in hindsight.

Theory/Online Learning and Bandits

Ruo-Chun Tzeng, Po-An Wang, Alexandre Proutiere, Chi-Jen Lu

We study the best arm identification problem in combinatorial semi-bandits in the fixed confidence setting. We present Perturbed Frank-Wolfe Sampling (P-FWS), an algorithm that (i) runs in polynomial time, (ii) achieves the instance-specific minimal sample complexity in the high confidence regime, and (iii) enjoys polynomial sample complexity guarantees in the moderate confidence regime. To our best knowledge, existing algorithms cannot achieve (ii) and (iii) simultaneously in vanilla bandits. With P-FWS, we close the computational-statistical gap in best arm identification in combinatorial semi-bandits. The design of P-FWS starts from the optimization problem that defines the information-theoretical and instance-specific sample complexity lower bound. P-FWS solves this problem in an online manner using, in each round, a single iteration of the Frank-Wolfe algorithm. Structural properties of the problem are leveraged to make the P-FWS successive updates computationally efficient. In turn, P-FWS only relies on a simple linear maximization oracle.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Pier Giuseppe Sessa, Pierre Laforgue, Nicolò Cesa-Bianchi, Andreas Krause

Multitask learning is a powerful framework that enables one to simultaneously learn multiple related tasks by sharing information between them. Quantifying uncertainty in the estimated tasks is of pivotal importance for many downstream applications, such as online or active learning. In this work, we provide novel confidence intervals for multitask regression in the challenging agnostic setting, i.e., when neither the similarity between tasks nor the tasks' features are available to the learner. The obtained intervals do not require i.i.d. data and can be directly applied to bound the regret in online learning. Through a refined analysis of the multitask information gain, we obtain new regret guarantees that, depending on a task similarity parameter, can significantly improve over treating tasks independently. We further propose a novel online learning algorithm that achieves such improved regret without knowing this parameter in advance, i.e., automatically adapting to task similarity. As a second key application of our results, we introduce a novel multitask active learning setup where several tasks must be simultaneously optimized, but only one of them can be queried for feedback by the learner at each round. For this problem, we design a no-regret algorithm that uses our confidence intervals to decide which task should be queried. Finally, we empirically validate our bounds and algorithms on synthetic and real-world (drug discovery) data.

Theory/Online Learning and Bandits

Licong Lin, Mufang Ying, Suvrojit Ghosh, Koulik Khamaru, Cun-Hui Zhang

Estimation and inference in statistics pose significant challenges when data are collected adaptively. Even in linear models, the Ordinary Least Squares (OLS) estimator may fail to exhibit asymptotic normality for single coordinate estimation and have inflated error. This issue is highlighted by a recent minimax lower bound, which shows that the error of estimating a single coordinate can be enlarged by a multiple of $\sqrt{d}$ when data are allowed to be arbitrarily adaptive, compared with the case when they are i.i.d. Our work explores this striking difference in estimation performance between utilizing i.i.d. and adaptive data. We investigate how the degree of adaptivity in data collection impacts the performance of estimating a low-dimensional parameter component in high-dimensional linear models. We identify conditions on the data collection mechanism under which the estimation error for a low-dimensional parameter component matches its counterpart in the i.i.d. setting, up to a factor that depends on the degree of adaptivity. We show that OLS or OLS on centered data can achieve this matching error. In addition, we propose a novel estimator for single coordinate inference via solving a Two-stage Adaptive Linear Estimating equation (TALE). Under a weaker form of adaptivity in data collection, we establish an asymptotic normality property of the proposed estimator.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Yu-Jie Zhang, Masashi Sugiyama

This paper investigates the logistic bandit problem, a variant of the generalized linear bandit model that utilizes a logistic model to depict the feedback from an action. While most existing research focuses on the binary logistic bandit problem, the multinomial case, which considers more than two possible feedback values, offers increased practical relevance and adaptability for use in complex decision-making problems such as reinforcement learning. In this paper, we provide an algorithm that enjoys both statistical and computational efficiency for the logistic bandit problem. In the binary case, our method improves the state-of-the-art binary logistic bandit method by reducing the per-round computation cost from $\mathcal{O}(\log T)$ to $\mathcal{O}(1)$ with respect to the time horizon $T$, while still preserving the minimax optimal guarantee up to logarithmic factors. In the multinomial case, with $K+1$ potential feedback values, our algorithm achieves an $\tilde{\mathcal{O}}(K\sqrt{T})$ regret bound with $\mathcal{O}(1)$ computational cost per round. The result not only improves the $\tilde{\mathcal{O}}(K\sqrt{\kappa T})$ bound for the best-known tractable algorithm—where the large constant $\kappa$ increases exponentially with the diameter of the parameter domain—but also reduces the $\mathcal{O}(T)$ computational complexity demanded by the previous method.

Theory/Online Learning and Bandits

Julia Olkhovskaya, Jack Mayo, Tim van Erven, Gergely Neu, Chen-Yu Wei

We consider the adversarial linear contextual bandit setting, whichallows for the loss functions associated with each of $K$ arms to changeover time without restriction. Assuming the $d$-dimensional contexts aredrawn from a fixed known distribution, the worst-case expected regretover the course of $T$ rounds is known to scale as $\tilde O(\sqrt{KdT})$. Under the additional assumption that the density of the contextsis log-concave, we obtain a second-order bound of order $\tildeO(K\sqrt{d V_T})$ in terms of the cumulative second moment of thelearner's losses $V_T$, and a closely related first-order bound of order$\tilde O(K\sqrt{d L_T^*})$ in terms of the cumulative loss of the bestpolicy $L_T^*$. Since $V_T$ or $L_T^*$ may be significantly smaller than$T$, these improve over the worst-case regret whenever the environmentis relatively benign. Our results are obtained using a truncated versionof the continuous exponential weights algorithm over the probabilitysimplex, which we analyse by exploiting a novel connection to the linearbandit setting without contexts.

Theory/Online Learning and Bandits

Evelyn Xiao-Yue Gong, Mark Sellke

We study pure exploration with infinitely many bandit arms generated \iid from an unknown distribution. Our goal is to efficiently select a single high quality arm whose average reward is, with probability $1-\delta$, within $\varepsilon$ of being with the top $\eta$-fraction of arms; this is a natural adaptation of the classical PAC guarantee for infinite action sets. We consider both the fixed confidence and fixed budget settings, aiming respectively for optimal \emph{expected} and \emph{fixed} sample complexity.For fixed confidence, we give an algorithm with expected sample complexity $O\left(\frac{\log (1/\eta)\log (1/\delta)}{\eta\varepsilon^2}\right)$. This is optimal except for the $\log (1/\eta)$ factor, and the $\delta$-dependence closes a quadratic gap in the literature. For fixed budget, we show the asymptotically optimal sample complexity as $\delta\to 0$ is $c^{-1}\log(1/\delta)\big(\log\log(1/\delta)\big)^2$ to leading order; equivalently, the optimal failure probability with exactly $N$ samples decays as $\exp\big(-(1\pm o(1))\frac{cN}{\log^2 N}\big)$.The value of $c$ depends explicitly on the problem parameters (including the unknown arm distribution) through a certain Fisher information distance. Even the strictly super-linear dependence on $\log(1/\delta)$ was not known and resolves a question of Grossman-Moshkovitz (FOCS 2015).

Theory/Online Learning and Bandits

Hamish Flynn, David Reeb, Melih Kandemir, Jan Peters

We present improved algorithms with worst-case regret guarantees for the stochastic linear bandit problem. The widely used "optimism in the face of uncertainty" principle reduces a stochastic bandit problem to the construction of a confidence sequence for the unknown reward function. The performance of the resulting bandit algorithm depends on the size of the confidence sequence, with smaller confidence sets yielding better empirical performance and stronger regret guarantees. In this work, we use a novel tail bound for adaptive martingale mixtures to construct confidence sequences which are suitable for stochastic bandits. These confidence sequences allow for efficient action selection via convex programming. We prove that a linear bandit algorithm based on our confidence sequences is guaranteed to achieve competitive worst-case regret. We show that our confidence sequences are tighter than competitors, both empirically and theoretically. Finally, we demonstrate that our tighter confidence sequences give improved performance in several hyperparameter tuning tasks.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Haoting Zhang, Jinghai He, Rhonda Righter, Zuo-Jun Shen, Zeyu Zheng

Contextual decision-making problems have witnessed extensive applications in various fields such as online content recommendation, personalized healthcare, and autonomous vehicles, where a core practical challenge is to select a suitable surrogate model for capturing unknown complicated reward functions. It is often the case that both high approximation accuracy and explicit uncertainty quantification are desired. In this work, we propose a neural network-accompanied Gaussian process (NN-AGP) model, which leverages neural networks to approximate the unknown and potentially complicated reward function regarding the contextual variable, and maintains a Gaussian process surrogate model with respect to the decision variable. Our model is shown to outperform existing approaches by offering better approximation accuracy thanks to the use of neural networks and possessing explicit uncertainty quantification from the Gaussian process. We also analyze the maximum information gain of the NN-AGP model and prove regret bounds for the corresponding algorithms. Moreover, we conduct experiments on both synthetic and practical problems, illustrating the effectiveness of our approach.

Theory/Online Learning and Bandits

Johannes Kirschner, Alireza Bakhtiari, Kushagra Chandak, Volodymyr Tkachuk, Csaba Szepesvari

A long line of works characterizes the sample complexity of regret minimization in sequential decision-making by min-max programs. In the corresponding saddle-point game, the min-player optimizes the sampling distribution against an adversarial max-player that chooses confusing models leading to large regret. The most recent instantiation of this idea is the decision-estimation coefficient (DEC), which was shown to provide nearly tight lower and upper bounds on the worst-case expected regret in structured bandits and reinforcement learning. By re-parametrizing the offset DEC with the confidence radius and solving the corresponding min-max program, we derive an anytime variant of the Estimation-To-Decisions algorithm (Anytime-E2D). Importantly, the algorithm optimizes the exploration-exploitation trade-off online instead of via the analysis. Our formulation leads to a practical algorithm for finite model classes and linear feedback models. We further point out connections to the information ratio, decoupling coefficient and PAC-DEC, and numerically evaluate the performance of E2D on simple examples.

Theory/Online Learning and Bandits

Arun Verma, Zhongxiang Dai, YAO SHU, Bryan Kian Hsiang Low

We study a novel variant of the parameterized bandits problem in which the learner can observe additional auxiliary feedback that is correlated with the observed reward. The auxiliary feedback is readily available in many real-life applications, e.g., an online platform that wants to recommend the best-rated services to its users can observe the user's rating of service (rewards) and collect additional information like service delivery time (auxiliary feedback). In this paper, we first develop a method that exploits auxiliary feedback to build a reward estimator with tight confidence bounds, leading to a smaller regret. We then characterize the regret reduction in terms of the correlation coefficient between reward and its auxiliary feedback. Experimental results in different settings also verify the performance gain achieved by our proposed method.

Theory/Online Learning and Bandits

Zhiyu Zhang, Ashok Cutkosky, Yannis Paschalidis

Motivated by the challenge of nonstationarity in sequential decision making, we study Online Convex Optimization (OCO) under the coupling of two problem structures: the domain is unbounded, and the comparator sequence $u_1,\ldots,u_T$ is arbitrarily time-varying. As no algorithm can guarantee low regret simultaneously against all comparator sequences, handling this setting requires moving from minimax optimality to comparator adaptivity. That is, sensible regret bounds should depend on certain complexity measures of the comparator relative to one's prior knowledge. This paper achieves a new type of such adaptive regret bounds leveraging a sparse coding framework. The complexity of the comparator is measured by its energy and its sparsity on a user-specified dictionary, which offers considerable versatility. For example, equipped with a wavelet dictionary, our framework improves the state-of-the-art bound (Jacobsen & Cutkosky, 2022) by adapting to both ($i$) the magnitude of the comparator average $||\bar u||=||\sum_{t=1}^Tu_t/T||$, rather than the maximum $\max_t||u_t||$; and ($ii$) the comparator variability $\sum_{t=1}^T||u_t-\bar u||$, rather than the uncentered sum $\sum_{t=1}^T||u_t||$. Furthermore, our proof is simpler due to decoupling function approximation from regret minimization.

Theory/Online Learning and Bandits

Ayush Sawarni, Soumyabrata Pal, Siddharth Barman

We obtain essentially tight upper bounds for a strengthened notion of regret in the stochastic linear bandits framework. The strengthening---referred to as Nash regret---is defined as the difference between the (a priori unknown) optimum and the geometric mean of expected rewards accumulated by the linear bandit algorithm. Since the geometric mean corresponds to the well-studied Nash social welfare (NSW) function, this formulation quantifies the performance of a bandit algorithm as the collective welfare it generates across rounds. NSW is known to satisfy fairness axioms and, hence, an upper bound on Nash regret provides a principled fairness guarantee. We consider the stochastic linear bandits problem over a horizon of $\mathsf{T}$ rounds and with a set of arms ${\cal X}$ in ambient dimension $d$. Furthermore, we focus on settings in which the stochastic reward---associated with each arm in ${\cal X}$---is a non-negative, sub-Poisson random variable. For this setting, we develop an algorithm that achieves a Nash regret of $O\left( \sqrt{\frac{d}{\mathsf{T}}} \log(\mathsf{T} |{\cal X}|)\right)$. In addition, addressing linear bandit instances in which the set of arms ${\cal X}$ is not necessarily finite, we obtain a Nash regret upper bound of $O\left( \frac{d^\frac{5}{4}}{\sqrt{\mathsf{T}}} \log(\mathsf{T})\right)$. Since bounded random variables are sub-Poisson, these results hold for bounded, non-negative rewards. Our linear bandit algorithm is built upon the successive elimination method with novel technical insights, including tailored concentration bounds and the use of sampling via John ellipsoid in conjunction with the Kiefer–Wolfowitz optimal design.

Theory/Online Learning and Bandits

Joe Suk, Arpit Agarwal

The $K$-armed dueling bandits problem, where the feedback is in the form of noisy pairwise preferences, has been widely studied due its applications in information retrieval, recommendation systems, etc. Motivated by concerns that user preferences/tastes can evolve over time, we consider the problem of _dueling bandits with distribution shifts_. Specifically, we study the recent notion of _significant shifts_ (Suk and Kpotufe, 2022), and ask whether one can design an _adaptive_ algorithm for the dueling problem with $O(\sqrt{K\tilde{L}T})$ dynamic regret,where $\tilde{L}$ is the (unknown) number of significant shifts in preferences. We show that the answer to this question depends on the properties of underlying preference distributions. Firstly, we give an impossibility result that rules out any algorithm with $O(\sqrt{K\tilde{L}T})$ dynamic regret under the well-studied Condorcet and SST classes of preference distributions. Secondly, we show that $\text{SST}\cap \text{STI}$ is the largest amongst popular classes of preference distributions where it is possible to design such an algorithm. Overall, our results provides an almost complete resolution of the above question for the hierarchy of distribution classes.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Volkan Cevher, Ashok Cutkosky, Ali Kavis, Georgios Piliouras, Stratis Skoulakis, Luca Viano

Motivated by alternating game-play in two-player games, we study an altenating variant of the \textit{Online Linear Optimization} (OLO). In alternating OLO, a \textit{learner} at each round $t \in [n]$ selects a vector $x^t$ and then an \textit{adversary} selects a cost-vector $c^t \in [-1,1]^n$. The learner then experiences cost $(c^t + c^{t-1})^\top x^t$ instead of $(c^t)^\top x^t$ as in standard OLO. We establish that under this small twist, the $\Omega(\sqrt{T})$ lower bound on the regret is no longer valid. More precisely, we present two online learning algorithms for alternating OLO that respectively admit $\mathcal{O}((\log n)^{4/3} T^{1/3})$ regret for the $n$-dimensional simplex and $\mathcal{O}(\rho \log T)$ regret for the ball of radius $\rho>0$. Our results imply that in alternating game-play, an agent can always guarantee $\mathcal{\tilde{O}}((\log n)^{4/3} T^{1/3})$ regardless the strategies of the other agent while the regret bound improves to $\mathcal{O}(\log T)$ in case the agent admits only two actions.