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Theory · Reinforcement Learning and Planning

Tal Lancewicki, Yishay Mansour

We study online finite-horizon Markov Decision Processes with adversarially changing loss and aggregate bandit feedback (a.k.a full-bandit). Under this type of feedback, the agent observes only the total loss incurred over the entire trajectory, rather than the individual losses at each intermediate step within the trajectory. We introduce the first Policy Optimization algorithms for this setting. In the known-dynamics case, we achieve the first *optimal* regret bound of $\tilde \Theta(H^2\sqrt{SAK})$, where $K$ is the number of episodes, $H$ is the episode horizon, $S$ is the number of states, and $A$ is the number of actions. In the unknown dynamics case we establish regret bound of $\tilde O(H^3 S \sqrt{AK})$, significantly improving the best known result by a factor of $H^2 S^5 A^2$.

Theory · Online Learning and Bandits

Joongkyu Lee, Min-hwan Oh

In this paper, we propose an improved online confidence bound for multinomial logistic (MNL) models and apply this result to MNL bandits, achieving variance-dependent optimal regret. Recently, Lee & Oh (2024) established an online confidence bound for MNL models and achieved nearly minimax-optimal regret in MNL bandits. However, their results still depend on the norm-boundedness of the unknown parameter $B$ and the maximum size of possible outcomes $K$. To address this, we first derive an online confidence bound of $\mathcal{O} (\sqrt{d \log t} + B )$, which is a significant improvement over the previous bound of $\mathcal{O} (B \sqrt{d} \log t \log K )$ (Lee & Oh, 2024). This is mainly achieved by establishing tighter self-concordant properties of the MNL loss and introducing a novel intermediary term to bound the estimation error. Using this new online confidence bound, we propose a constant-time algorithm, **OFU-MNL++**, which achieves a variance-dependent regret bound of $\mathcal{O} \Big( d \log T \sqrt{ \sum_{t=1}^T \sigma_t^2 } \Big) $ for sufficiently large $T$, where $\sigma_t^2$ denotes the variance of the rewards at round $t$, $d$ is the dimension of the contexts, and $T$ is the total number of rounds. Furthermore, we introduce an Maximum Likelihood Estimation (MLE)-based algorithm that achieves an anytime, **OFU-M$^2$NL**, $\operatorname{poly}(B)$-free regret of $\mathcal{O} \Big( d \log (BT) \sqrt{ \sum_{t=1}^T \sigma_t^2 } \Big) $.

Theory · Online Learning and Bandits

Bingshan Hu, Zhiming Huang, Tianyue Zhang, Mathias Lécuyer, Nidhi Hegde

We address differentially private stochastic bandit problems by leveraging Thompson Sampling with Gaussian priors and Gaussian differential privacy (GDP). We propose DP-TS-UCB, a novel parametrized private algorithm that enables trading off privacy and regret. DP-TS-UCB satisfies $ \tilde{O} \left(T^{0.25(1-\alpha)}\right)$-GDP and achieves $O \left(K\ln^{\alpha+1}(T)/\Delta \right)$ regret bounds, where $K$ is the number of arms, $ \Delta$ is the sub-optimality gap, $T$ is the learning horizon, and $\alpha \in [0,1]$ controls the trade-off between privacy and regret. Theoretically, DP-TS-UCB relies on anti-concentration bounds for the Gaussian distributions, linking the exploration mechanisms of Thompson Sampling and Upper Confidence Bound, which may be of independent research interest.

Theory · Online Learning and Bandits

Kapilan Balagopalan, Tuan Nguyen, Yao Zhao, Kwang-Sung Jun

The best arm identification problem requires identifying the best alternative (i.e., arm) in active experimentation using the smallest number of experiments (i.e., arm pulls), which is crucial for cost-efficient and timely decision-making processes. In the fixed confidence setting, an algorithm must stop data-dependently and return the estimated best arm with a correctness guarantee. Since this stopping time is random, we desire its distribution to have light tails. Unfortunately, many existing studies focus on high probability or in expectation bounds on the stopping time, which allow heavy tails and, for high probability bounds, even not stopping at all. We first prove that this never-stopping event can indeed happen for some popular algorithms. Motivated by this, we propose algorithms that provably enjoy an exponential-tailed stopping time, which improves upon the polynomial tail bound reported by Kalyanakrishnan et al. (2012). The first algorithm is based on a fixed budget algorithm called Sequential Halving along with a doubling trick. The second algorithm is a meta algorithm that takes in any fixed confidence algorithm with a high probability stopping guarantee and turns it into one that enjoys an exponential-tailed stopping time. Our results imply that there is much more to be desired for contemporary fixed confidence algorithms.

General Machine Learning · Online Learning, Active Learning and Bandits

Jingruo Sun, Wenzhi Gao, Ellen Vitercik, Yinyu Ye

Online linear programming (OLP) has found broad applications in revenue management and resource allocation. State-of-the-art OLP algorithms achieve low regret by repeatedly solving linear programming (LP) subproblems that incorporate updated resource information. However, LP-based methods are computationally expensive and often inefficient for large-scale applications. By contrast, recent first-order OLP algorithms are more computationally efficient but typically suffer from weaker regret guarantees. To address these shortcomings, we propose a new algorithm that combines the strengths of LP-based and first-order OLP algorithms. Our algorithm re-solves the LP subproblems periodically at a predefined frequency $f$ and uses the latest dual prices to guide online decision-making. In parallel, a first-order method runs during each interval between LP re-solves and smooths resource consumption. Our algorithm achieves $\mathcal{O}(\log (T/f) + \sqrt{f})$ regret and delivers a "wait-less" online decision-making process that balances computational efficiency and regret guarantees. Extensive experiments demonstrate at least $10$-fold improvements in regret over first-order methods and $100$-fold improvements in runtime over LP-based methods.

Theory · Online Learning and Bandits

Mohammadreza Daneshvaramoli, Helia Karisani, Adam Lechowicz, Bo Sun, Cameron Musco, Mohammad Hajiesmaili

This paper introduces a family of learning-augmented algorithms for online knapsack problems that achieve near Pareto-optimal consistency-robustness trade-offs through a simple combination of trusted learning-augmented and worst-case algorithms. Our approach relies on succinct, practical predictions—single values or intervals estimating the minimum value of any item in an offline solution. Additionally, we propose a novel fractional-to-integral conversion procedure, offering new insights for online algorithm design.

Theory · Online Learning and Bandits

Jing Wang, Yu-Jie Zhang, Peng Zhao, Zhi-Hua Zhou

We study the stochastic linear bandits with heavy-tailed noise. Two principled strategies for handling heavy-tailed noise, truncation and median-of-means, have been introduced to heavy-tailed bandits. Nonetheless, these methods rely on specific noise assumptions or bandit structures, limiting their applicability to general settings. The recent work [Huang et al.2024] develop a soft truncation method via the adaptive Huber regression to address these limitations. However, their method suffers undesired computational cost: it requires storing all historical data and performing a full pass over these data at each round. In this paper, we propose a \emph{one-pass} algorithm based on the online mirror descent framework. Our method updates using only current data at each round, reducing the per-round computational cost from $\mathcal{O}(t \log T)$ to $\mathcal{O}(1)$ with respect to current round $t$ and the time horizon $T$, and achieves a near-optimal and variance-aware regret of order $\widetilde{\mathcal{O}}\big(d T^{\frac{1-\varepsilon}{2(1+\varepsilon)}} \sqrt{\sum_{t=1}^T \nu_t^2} + d T^{\frac{1-\varepsilon}{2(1+\varepsilon)}}\big)$ where $d$ is the dimension and $\nu_t^{1+\varepsilon}$ is the $(1+\varepsilon)$-th central moment of reward at round $t$.

General Machine Learning · Online Learning, Active Learning and Bandits

Osman Berke Guney, Ketan Saichandran, Karim Elzokm, Ziming Zhang, Vijaya Kolachalama

In many practical applications, including medicine, acquiring all relevant data for machine learning models is often infeasible due to constraints on time, cost, and resources. This makes it important to selectively acquire only the most informative features, yet traditional static feature selection methods fall short in scenarios where feature importance varies across instances. Here, we propose an active feature acquisition (AFA) framework, which dynamically selects features based on their importance to each individual case. Our method leverages local explanation techniques to generate instance-specific feature importance rankings. We then reframe the AFA problem as a feature prediction task, introducing a policy network grounded in a decision transformer architecture. This policy network is trained to select the next most informative feature by learning from the feature importance rankings. As a result, features are acquired sequentially, ordered by their predictive significance, leading to more efficient feature selection and acquisition. Extensive experiments on multiple datasets demonstrate that our approach outperforms current state-of-the-art AFA methods in both predictive accuracy and feature acquisition efficiency. These findings highlight the promise of an explainability-driven AFA strategy in scenarios where the cost of feature acquisition is a key concern.

Theory · Online Learning and Bandits

Xiaoyu Wang, Wenhao Yang, Chang Yao, Mingli Song, Yuanyu Wan

Although the differential privacy (DP) of decentralized online learning has garnered considerable attention recently, existing algorithms are unsatisfactory due to their inability to achieve $(\epsilon, 0)$-DP over all $T$ rounds, recover the optimal regret in the non-private case, and maintain the lightweight computation under complex constraints. To address these issues, we first propose a new decentralized online learning algorithm satisfying $(\epsilon, 0)$-DP over $T$ rounds, and show that it can achieve $\widetilde{O}(n(\rho^{-1/4}+\epsilon^{-1}\rho^{1/4})\sqrt{T})$ and $\widetilde{O}(n (\rho^{-1/2}+\epsilon^{-1}))$ regret bounds for convex and strongly convex functions respectively, where $n$ is the number of local learners and $\rho$ is the spectral gap of the communication matrix. As long as $\epsilon=\Omega(\sqrt{\rho})$, these bounds nearly match existing lower bounds in the non-private case, which implies that $(\epsilon, 0)$-DP of decentralized online learning may be ensured nearly for free. Our key idea is to design a block-decoupled accelerated gossip strategy that can be incorporated with the classical tree-based private aggregation, and also enjoys a faster average consensus among local learners. Furthermore, we develop a projection-free variant of our algorithm to keep the efficiency under complex constraints. As a trade-off, the above regret bounds degrade to $\widetilde{O}(n(T^{3/4}+\epsilon^{-1}T^{1/4}))$ and $\widetilde{O}(n(T^{2/3}+\epsilon^{-1}))$ respectively, which however are even better than the existing private centralized projection-free online algorithm.

Theory · Online Learning and Bandits

Cyrille Kone, Emilie Kaufmann, Laura Richert

In this paper, we address the problem of identifying the Pareto Set under feasibility constraints in a multivariate bandit setting. Specifically, given a $K$-armed bandit with unknown means $\mu_1, \dots, \mu_K \in \mathbb{R}^d$, the goal is to identify the set of arms whose mean is not uniformly worse than that of another arm (i.e., not smaller for all objectives), while satisfying some known set of linear constraints, expressing, for example, some minimal performance on each objective. Our focus lies in fixed-confidence identification, for which we introduce an algorithm that significantly outperforms racing-like algorithms and the intuitive two-stage approach that first identifies feasible arms and then their Pareto Set. We further prove an information-theoretic lower bound on the sample complexity of any algorithm for constrained Pareto Set identification, showing that the sample complexity of our approach is near-optimal. Our theoretical results are supported by an extensive empirical evaluation on a series of benchmarks.

Reinforcement Learning · Online

Canzhe Zhao, Yutian Cheng, Jing Dong, Baoxiang Wang, Shuai Li

We investigate learning approximate Nash equilibrium (NE) policy profiles in two-player zero-sum imperfect information extensive-form games (IIEFGs) with last-iterate convergence guarantees. Existing algorithms either rely on full-information feedback or provide only asymptotic convergence rates. In contrast, we focus on the bandit feedback setting, where players receive feedback solely from the rewards associated with the experienced information set and action pairs in each episode. Our proposed algorithm employs a negentropy regularizer weighted by a "virtual transition" over the information set-action space to facilitate an efficient approximate policy update. Through a carefully designed virtual transition and leveraging the entropy regularization technique, we demonstrate finite-time last-iterate convergence to the NE with a rate of $\widetilde{\mathcal{O}}(k^{-\frac{1}{8}})$ under bandit feedback in each episode $k$. Empirical evaluations across various IIEFG instances show its competitive performance compared to baseline methods.

Theory · Online Learning and Bandits

Steve Hanneke, Amirreza Shaeiri

List learning is an important topic in both theoretical and empirical machine learning research, playing a key role in the recent breakthrough result of (Brukhim et al., 2022) on the characterization of multiclass PAC learnability, as well as addressing label ambiguity in computer vision classification tasks, among others. In this paper, we study the problem of list transductive online learning. In this framework, the learner outputs a list of multiple labels for each instance rather than just one, as in traditional multiclass classification. In the realizable setting, we demonstrate a trichotomy of possible rates of the minimax number of mistakes. In particular, if the learner plays for $\text{T} \in \mathbb{N}$ rounds, its minimax number of mistakes can only be of the orders $\Theta(\text{T})$, $\Theta(\log \text{T})$, or $\Theta(1)$. This resolves an open question raised by (Hanneke et al., 2024). On the other hand, in the agnostic setting, we characterize the learnability by constructively proving the $\widetilde{\mathcal{O}}(\sqrt{\text{T}})$ upper bound on the minimax expected regret. Along the way, we also answer another open question asked by (Moran et al., 2023). To establish these results, we introduce two new combinatorial complexity dimensions, called the Level-constrained $(\mathrm{L+1})$-Littlestone dimension and Level-constrained $(\mathrm{L+1})$-Branching dimension, if the list size is $\mathrm{L} \in \mathbb{N}$. Eventually, we conclude our work by raising an open question regarding eliminating the factor list size, which seems to be a crucial step, as it has consistently appeared in previous works on this subject.

Theory · Reinforcement Learning and Planning

Heyang Zhao, Chenlu Ye, Wei Xiong, Quanquan Gu, Tong Zhang

Recent advances in Reinforcement Learning from Human Feedback (RLHF) have shown that KL-regularization plays a pivotal role in improving the efficiency of RL fine-tuning for large language models (LLMs). Despite its empirical advantage, the theoretical difference between KL-regularized RL and standard RL remains largely under-explored. While there is a recent line of work on the theoretical analysis of KL-regularized objective in decision making (Xiong et al., 2024a; Xie et al., 2024; Zhao et al., 2024), these analyses either reduce to the traditional RL setting or rely on strong coverage assumptions. In this paper, we propose an optimism-based KL-regularized online contextual bandit algorithm, and provide a novel analysis of its regret. By carefully leveraging the benign optimization landscape induced by the KL-regularization and the optimistic reward estimation, our algorithm achieves an $\mathcal{O}\big(\eta\log (N_{\mathcal R} T)\cdot d_{\mathcal R}\big)$ logarithmic regret bound, where $\eta, N_{\mathcal R},T,d_{\mathcal R}$ denote the KL-regularization parameter, the cardinality of the reward function class, number of rounds, and the complexity of the reward function class. Furthermore, we extend our algorithm and analysis to reinforcement learning by developing a novel decomposition over transition steps and also obtain a similar logarithmic regret bound.

Theory · Online Learning and Bandits

Zilong Wang, Zhiyao Zhang, Shuai Li

Multi-player multi-armed bandit (MP-MAB) has been widely studied owing to its diverse applications across numerous domains. We consider an MP-MAB problem where $N$ players compete for $K$ arms in $T$ rounds. The reward distributions are heterogeneous where each player has a different expected reward for the same arm. When multiple players select the same arm, they collide and obtain zero rewards. In this paper, our target is to find the max-min fairness matching that maximizes the reward of the player who receives the lowest reward. This paper improves the existing max-min regret upper bound of $O(\exp(1/\Delta) + K^3 \log T\log \log T)$. More specifically, our decentralized fair elimination algorithm (DFE) deals with heterogeneity and collision carefully and attains a regret upper bound of $O((N^2+K)\log T / \Delta)$, where $\Delta$ is the minimum reward gap between max-min value and sub-optimal arms.In addition, this paper also provides an $\Omega(\max\\{ N^2, K \\} \log T / \Delta)$ regret lower bound for this problem, which indicates that our algorithm is optimal with respect to key parameters $T, N, K$, and $\Delta$. Additional numerical experiments also show the efficiency and improvement of our algorithms.

General Machine Learning · Online Learning, Active Learning and Bandits

Zhiyong Wang, Jiahang Sun, Mingze Kong, Jize Xie, Qinghua Hu, John C. S. Lui, Zhongxiang Dai

The contextual multi-armed bandit (MAB) is a widely used framework for problems requiring sequential decision-making under uncertainty, such as recommendation systems. In applications involving a large number of users, the performance of contextual MAB can be significantly improved by facilitating collaboration among multiple users. This has been achieved by the clustering of bandits (CB) methods, which adaptively group the users into different clusters and achieve collaboration by allowing the users in the same cluster to share data. However, classical CB algorithms typically rely on numerical reward feedback, which may not be practical in certain real-world applications. For instance, in recommendation systems, it is more realistic and reliable to solicit preference feedback between pairs of recommended items rather than absolute rewards. To address this limitation, we introduce the first "clustering of dueling bandit algorithms" to enable collaborative decision-making based on preference feedback. We propose two novel algorithms: (1) Clustering of Linear Dueling Bandits (COLDB) which models the user reward functions as linear functions of the context vectors, and (2) Clustering of Neural Dueling Bandits (CONDB) which uses a neural network to model complex, non-linear user reward functions. Both algorithms are supported by rigorous theoretical analyses, demonstrating that user collaboration leads to improved regret bounds. Extensive empirical evaluations on synthetic and real-world datasets further validate the effectiveness of our methods, establishing their potential in real-world applications involving multiple users with preference-based feedback.

Theory · Online Learning and Bandits

Aymeric Capitaine, Etienne Boursier, Eric Moulines, Michael Jordan, Alain Oliviero Durmus

The framework of uncoupled online learning in multiplayer games has made significant progress in recent years. In particular, the development of time-varying games has considerably expanded its modeling capabilities. However, current regret bounds quickly become vacuous when the game undergoes significant variations over time, even when these variations are easy to predict. Intuitively, the ability of players to forecast future payoffs should lead to tighter guarantees, yet existing approaches fail to incorporate this aspect. This work aims to fill this gap by introducing a novel prediction-aware framework for time-varying games, where agents can forecast future payoffs and adapt their strategies accordingly. In this framework, payoffs depend on an underlying state of nature that agents predict in an online manner. To leverage these predictions, we propose the POMWU algorithm, a contextual extension of the optimistic Multiplicative Weight Update algorithm, for which we establish theoretical guarantees on social welfare and convergence to equilibrium. Our results demonstrate that, under bounded prediction errors, the proposed framework achieves performance comparable to the static setting. Finally, we empirically demonstrate the effectiveness of POMWU in a traffic routing experiment.

Theory · Online Learning and Bandits

Aditya Gangrade, Aldo Pacchiano, Clay Scott, Venkatesh Saligrama

We study the 'feasible action search' (FAS) problem for linear bandits, wherein a learner attempts to discover a feasible point for a set of linear constraints $\Phi_* a \ge 0,$ without knowledge of the matrix $\Phi_* \in \mathbb{R}^{m \times d}$. A FAS learner selects a sequence of actions $a_t,$ and uses observations of the form $\Phi_* a_t + \mathrm{noise}$ to either find a point with nearly optimal 'safety margin', or detect that the constraints are infeasible, where the safety margin of an action measures its (signed) distance from the constraint boundary. While of interest in its own right, the FAS problem also directly addresses a key deficiency in the extant theory of 'safe linear bandits' (SLBs), by discovering a safe initialisation for low-regret SLB methods.We propose and analyse a novel efficient FAS-learner. Our method, FAST, is based on Thompson Sampling. It applies a _coupled_ random perturbation to an estimate of $\Phi_*,$ and plays a maximin point of a game induced by this perturbed matrix. We prove that FAST stops in $\tilde{O}(d^3/\varepsilon^2 M_*^2)$ steps, and incurs $\tilde{O}(d^3/|M_*|)$ safety costs, to either correctly detect infeasibility, or output a point that is at least $(1-\varepsilon) M_*$-safe, where $M_*$ is the _optimal safety margin_ of $\Phi_*$. Further, instantiating prior SLB methods with the output of FAS yields the first SLB methods that incur $\tilde{O}(\sqrt{d^3 T/M_*^2})$ regret and $O(1)$ risk without a priori knowledge of a safe action. The main technical novelty lies in the extension of Thompson Sampling to this multiobjective setting, for which we both propose a coupled noise design, and provide an analysis that avoids convexity considerations.

General Machine Learning · Online Learning, Active Learning and Bandits

Ramya Ramalingam, Shayan Kiyani, Aaron Roth

Existing algorithms for online conformal prediction---guaranteeing marginal coverage in adversarial settings---are variants of online gradient descent (OGD), but their analyses of worst-case coverage do not follow from the regret guarantee of OGD. What is the relationship between no-regret learning and online conformal prediction? We observe that although standard regret guarantees imply marginal coverage in i.i.d. settings, this connection fails as soon as we either move to adversarial environments or ask for group conditional coverage. On the other hand, we show a tight connection between *threshold calibrated* coverage and swap-regret in adversarial settings, which extends to group-conditional (multi-valid) coverage. We also show that algorithms in the *follow the regularized leader* family of no regret learning algorithms (which includes online gradient descent) can be used to give group-conditional coverage guarantees in adversarial settings for arbitrary grouping functions. Via this connection we analyze and conduct experiments using a multi-group generalization of the ACI algorithm of Gibbs & Candes (2021).

Theory · Online Learning and Bandits

Christophe Roux, David Martinez-Rubio, Sebastian Pokutta

We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.

General Machine Learning · Online Learning, Active Learning and Bandits

Côme Fiegel, Pierre Menard, Tadashi Kozuno, Michal Valko, Vianney Perchet

We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performances, with the best attainable rate being $\mathcal{O}(T^{-1/4})$ in contrast to the usual $\mathcal{O}(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate. The first algorithm leverages a straightforward tradeoff between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.