Adversarial Bandits

We show upper and lower bounds on the regret of $m$-set adversarial bandits for different utilities (winner reward or sum of rewards) and feedback models (winner index, winner reward, sum of rewards, and their combinations). By comparing to standard bounds for combinatorial and MNL bandits, our results reveal how subtle changes in the setting can have a dramatic impact on the learning rates. Our main technical contributions are the information-theoretic lower bounds on the regret. Experiments on synthetic data confirm our theoretical analyses.

We study adversarial multiplayer bandits with $K$ arms and $2\le m<K$ labeled players, without collision information, shared randomness, or an external communication channel. We design a constructive communication and synchronization protocol with a Monte Carlo public constructor. With probability at least $1-CN^{-32}$ over preprocessing, where $N=2Km(T+1)$, its fixed published output satisfies $$R_T\le C K^{5/2}\sqrt T\log^2(2Km(T+1))$$ simultaneously for every oblivious reward sequence chosen after preprocessing. Here $R_T$ is expected regret over the players' private execution randomness. Positive reward observations establish a common learning schedule and synchronize players before learning begins. The cost of delayed communication is charged to the support of positive rewards, ensuring that periods with little useful feedback incur only limited regret. A slow--fast learning procedure then maintains valid reward estimates while assignments and scores are exchanged.

We study Nash regret minimization in unknown finite matrix games with bandit payoff feedback and observed opponent actions. We develop Optimistic Payoff Balancing (OPB), which achieves instance-dependent $\mathcal{O}(\log^2 T)$ Nash regret against arbitrary adaptive opponents, including games with nonunique equilibria. This resolves the open problem posed by Maiti et al. (2025), extending their polylogarithmic guarantee under bandit feedback from $2\times2$ games to arbitrary finite dimensions. To handle nonunique equilibria, we construct a reference strategy that leaves room for local adjustments. We order independent payoff differences by estimation accuracy and scale these adjustments by uncertainty, allowing the learner to exploit the opponent's imbalance to offset estimation costs. Our result thus shows that observing opponent actions suffices for polylogarithmic Nash regret in general finite matrix games.

We study last-iterate convergence in unknown two-player zero-sum matrix games with bandit payoff feedback and observed opponent actions. For games with $d$ actions per player, we develop an algorithm achieving a duality gap of $\widetilde{\mathcal{O}}(\sqrt{d/t})$ with high probability, simultaneously at every round $t$. This improves the dimension dependence of the best previously known guarantee by a factor of $d^{3/2}$. The rate matches a standard bandit lower bound, establishing minimax optimality in both the number of actions and the number of rounds, up to logarithmic factors. The algorithm is computationally efficient, requiring only $\mathcal{O}(d)$ time and memory per round. Our technical contribution is a joint design of adaptive averaging and corrected exponential weights that absorbs estimation variance, together with a potential argument that bounds phase durations.

A learner probes at most $k$ of $n$ arms each round, receives the maximum of their rewards in $[0,1]$, and competes with the best fixed arm. When does the probing advantage pay for learning? We determine two minimax laws. Under independent stochastic rewards with winner feedback (the maximum and a winning label), or on arbitrary fixed sequences given a single signed contrast between block maxima, the minimax regret has order $Φ_{n,k}(T)=\min\{\frac{n-k}{n}T,\frac{n-k}{k}\}$, $2\le k<n$. Under winner feedback, both arbitrary joint i.i.d. rewards and fixed sequences have minimax regret of order $R_{n,k}(T)=\frac{n-k}{n}\min\{T,\frac{n+T}{k},\sqrt{\frac{nT}{k}}\}$. Both laws have universal constants and anytime upper bounds. The first reduces regret to a pure coverage cost: same-round contrasts absorb the stability cost, and independence permits exact resampling whose gains fund sample advancement. The second adds a learning cost that becomes comparable to coverage at horizon $n$; beyond $nk$, numerical maxima improve over labels alone. The lower bound allows every adaptive action size.

Meta-learning has emerged as an effective paradigm for transferring knowledge across sequential bandit tasks. While substantial progress has been made for stochastic bandits and non-contextual adversarial bandits, meta-learning for adversarial linear contextual bandits (ALCBs) with random action sets remains largely unexplored. To address this problem, we propose Meta-LinEXP3, an online-within-online algorithm that constructs a predictable task-level prior from completed tasks to guide the inner LinEXP3 learner. For known context distributions, we develop a policy-centered estimator that achieves an intrinsic-dimension $\mathcal{O}(\sqrt{n})$ per-task regret bound. For unknown distributions, we introduce a past-only regularized moment estimator with an $\mathcal{O}(n^{2/3})$ leading regret term and explicit finite-sample error. We further establish a direct connection between prior accuracy and transfer regret, showing that increasingly accurate priors yield sublinear transfer-dependent regret across tasks. Experiments demonstrate the effectiveness of Meta-LinEXP3, including its application to structured hyperspectral tensor sampling.

We study adversarial combinatorial bandits with $m$-set actions, where at each round the learner selects $m$ out of $d$ items and observes only the aggregate loss of the selected items. The resulting action set contains $K=\binom{d}{m}$ elements and can therefore be exponentially large. Nevertheless, the loss of every action is determined by the same $d$-dimensional vector of item losses. We propose a computationally efficient algorithm that exploits this structure without explicitly enumerating the action set. Against adaptive non-anticipating adversaries, it guarantees, with probability at least $1-δ$, regret against the best fixed action of $$R_T = O\left(\sqrt{dT\log(K/δ)}\right).$$ This matches the high-probability regret bound of the finite-action EXP3-KW algorithm of Zimmert and Lattimore, whose direct implementation may require exponential space. Our algorithm instead represents each sampling distribution with $d$ parameters and runs in polynomial time without enumerating the action set. Thus, it resolves the open problem posed by Maiti et al.

We consider the extensive-form bandit problem where on each trial the learner plays an extensive-form game against an oblivious adversary. We focus on the notion of switching regret, which measures the expected performance of the learner against that of any switching sequence of mixed strategies in retrospect. Our algorithm takes a parameter $ρ>0$ and achieves a switching regret of $\tilde{\mathcal{O}}((1/ρ+ρK)\sqrt{H A T})$ where $K$ is the number of switches in the comparator sequence, $H$ is the maximum number of the learner's information sets that can be traversed during a play of the game and $A$ is the number of actions that the learner can possibly take. Our algorithm is extremely efficient, taking a per trial time of only $\mathcal{O}(H B)$ where $B$ is the maximum number of actions available to the learner at any of its information sets.

This note aims to serve as an entry point to the literature on learning in games, a topic with significant theoretical appeal and a wide range of applications -- from machine learning and data science to economics and beyond. Our presentation is structured around two complementary viewpoints: We first consider a single agent -- the learner -- engaged in a sequential decision process in an unknown, non-stationary, and possibly adversarial environment. We then examine what happens when the environment is shaped by the decisions of several interacting agents, not necessarily aware of each other's actions or goals, and all seeking to improve their individual rewards. In this general context, we examine a family of regularized learning policies based on best-responding to the past history of play, up to a regularization penalty intended to encourage exploration and prevent over-commitment to suboptimal choices. In the single-agent setting, we present some basic regret bounds for regularized learning in adversarial multi-armed bandits; in the multi-agent setting, we describe an ergodic equilibrium convergence result for zero-sum games in the spirit of classical results on fictitious play, as well as a "folk theorem" linking strategic and dynamic notions of stability -- Nash equilibria and attracting points of regularized learning, respectively. We pay special attention to the information available to the players and, through a unified analysis framework, we study both oracle- and payoff-based (bandit) methods. Our goal is to provide a coherent and comprehensible -- albeit, by necessity, not comprehensive -- account of some recent ideas in the field, and to discuss their implications for the study of rationality.

We study the bandit-feedback version of online principal component analysis (Bandit PCA): in each round $t = 1,\dots,T$, the adversary selects a $d \times d$ symmetric gain matrix $G_t$ with spectrum in $[0,1]$ and rank at most $r$; the learner simultaneously selects a unit vector $w_t \in S^{d-1}$ and receives the reward $w_t^\top G_t w_t$. The learner receives no other feedback, and aims to minimize the regret against the best unit vector in hindsight. This problem was introduced by Kotlowski and Neu (2019), who gave an algorithm with regret $O(d\sqrt{rT \log T})$ and showed the lower bound of $Ω(r\sqrt{T/\log T})$. We improve upon both of these bounds and essentially bridge the gap between them, establishing the minimax regret of order $r\sqrt{dT}$ up to polylogarithmic factors in $d$ and $T$. The upper bound is attained by a novel algorithm, which combines online mirror descent on the spectrahedron of (real) density matrices with a multiscale exploration scheme in which the eigenspaces with different spectral magnitudes are updated at different rates. For the lower bound, we construct an adaptive adversary that refines a hidden large-reward subspace based on the learner's actions, in such a way that low regret is impossible without estimating the subspace; as a result, lower-bounding the regret reduces to studying the arising subspace estimation problem. Finally, we discuss connections of Bandit PCA with adaptive-measurement quantum tomography.

We study an adversarial bandit problem for entanglement-based quantum-network routing over a modest graph corpus. Alice selects an end-to-end repeater route for an Ekert-91 protocol (E91) representing her move, while Eve selects an attack surface, either edge intercept--resend or repeater memory degradation. Payoffs are drawn from cached SeQUeNCe-simulated E91 transcripts, and Alice accepts a turn when the finite-sample statistic violates the Clauser-Horne-Shimony-Holt (CHSH) bound. Performing adversarial co-learning across 50 structured topologies, we find that learned retention tracks a full-matrix minimax reference closely (Pearson $r=0.99$): under a one-surface Eve action model, bottleneck families have zero retention, while non-bottleneck families follow a $1-1/N$ coverage principle. We then fit decision-tree explanation models to graph-, attack-, and route-level topology-corpus targets and report their faithfulness. Finally, we construct prompt records for local language models to summarize the tree evidence, resulting in an open-source explanation workflow for quantum-repeater network games.

We study distributed online submodular maximization under partition matroid constraints, in which multiple agents select a limited number of actions from their own subsets sequentially to maximize the cumulative value of a sequence of objective functions. We develop a unified algorithmic framework that accommodates full-information and bandit feedback models. For both feedback models, we prove that the proposed algorithms achieve sublinear $(1-1/e)$-regret guarantees, which are comparable to those achieved by existing centralized counterparts. Furthermore, to tackle the sampling violation issue caused by continuous relaxation and rounding, we develop a bounded stochastic pipage rounding scheme and show that the probability of sampling violation vanishes asymptotically. As a result, the cumulative sampling violation remains sublinear in $T$, which is further shown to be not improvable under certain conditions. Numerical results validate the theoretical findings in this paper.

In many online learning and bandit problems, the actions we consider possess inherent similarities--for instance because they share latent traits, tags, or hierarchical structure. We study online learning with a similarity-structured action set, encoded by a rooted tree whose leaves are the actions and whose levels quantify how closely two actions are related. The loss sequence is assumed tree-compatible: losses of similar actions are constrained to be close. We establish an impossibility result showing that usual one-point bandit feedback cannot, in general, leverage range or tree-induced similarity, even under very strong similarity constraints. We then provide a unified set of algorithms which adapt to a wide range of richer feedback models, from semi-bandit feedback down to multi-point bandit protocols, including the minimal two-point feedback setting. We show these algorithms exhibit best-of-both-worlds guarantees and provably exploit action similarities by replacing the number of actions $K$ by a similarity-aware effective number of actions $K_{\mathrm{eff}}$ in the regret bounds. As an application, we show that under two-point feedback, it is possible to achieve $\sqrt{T}$ regret in Lipschitz bandits when $d \leq 2$.

We study adversarial bandit optimization in which the loss functions may be non-convex and non-smooth. In each round, the learner selects an action and observes only the loss incurred at that action. The loss consists of an underlying convex and $β$-smooth component and an adversarial perturbation that may be chosen after observing the learner's action. The perturbations are subject to a global budget controlling their cumulative magnitude over time. This framework extends the globally budgeted, post-action perturbation model from underlying linear losses to general convex and $β$-smooth losses. For this broader class, we establish expected regret guarantees that explicitly characterize the effect of the perturbation budget. To establish these guarantees, we modify a standard bandit optimization algorithm and develop an analysis that controls the additional regret caused by the perturbations. In the absence of perturbations, our results reduce to regret guarantees for the standard bandit convex optimization setting with $β$-smooth losses.

We study a multi-agent multi-armed bandit problem in the competitive setup with two-sided matching markets under a human centric decision making model. To capture human preferences, we use cumulative prospect theory (CPT) that weighs the actions of the agent in a nonlinear fashion using a ($α$-Hölder continuous) weight function. CPT has been widely used in behavioral economics and risk sensitive machine learning to emulate human preferences. We analyze the state-of-the-art learning algorithm with CPT weight distorted rewards and obtain a player optimal regret of $\mathcal{O}(K\log T \left(\frac{1}Δ\right)^{2/α})$, where $K$ denotes the number of arms, $T$ is the learning horizon, and $Δ$ represents (suitably defined) players' minimum preference gap. Noticing the dependence on $Δ$ to be sub-optimal, we further improve this regret by judiciously selecting the active set of arms during exploration, which removes the dependence on $K$ in the dominant term and achieves an improved (optimal) regret guarantees in the setting where the number of arms $K$ is significantly larger than the number of players $N$. In addition, we consider adversarial markets where the observed rewards of the agents may be corrupted. We propose and analyze algorithms for robust markets with CPT as risk sensitive measure in both settings where the total corruption budget is known and where it is unknown, and establish logarithmic player-optimal regret guarantees in both cases.

Modern recommendation systems increasingly rely on dynamically routing diverse queries to multiple embedding models. Despite its practical significance, this problem remains poorly understood under realistic conditions like adversarial queries, bandit feedback, and limited observability of models. We formalize embedding model routing as an adversarial contextual linear bandit with low-rank experts, where contexts are queries, actions are items, and experts are the embedding models working on low-rank latent representation spaces. We first establish that standard regret notions suffer from structural misspecification or statistical intractability, and we identify a log-quadratic policy class that is expressive enough to capture query-dependent model routing, yet structured enough to allow efficient online learning. Second, we propose a policy gradient algorithm called Hypentropy Policy Gradient (HPG). It provably adapts to the unknown low-rank structure under incomplete information and attains $\tilde{\mathcal O}(s\sqrt{M T})$ linearized policy regret -- where $s, M$, and $T$ are the intrinsic rank of the experts, the number of models, and the number of rounds -- thus avoiding a curse of dimensionality. Finally, we also provide an computationally efficient and parameter-free implementation of HPG.

Follow-the-regularized-leader framework has shown effectiveness and flexibility in online learning problems, where the choice of learning rates are known to be crucial. Recently, adaptive learning rates defined in terms of the arm-selection probabilities, obtained by solving convex optimization, have achieved improved best-of-both-worlds (BOBW) guarantees in various bandit problems. In contrast, BOBW guarantees for its computationally efficient alternative, follow-the-perturbed-leader (FTPL), remain relatively limited since its optimization-free nature ironically makes the design of adaptive, probability-dependent learning rates non-trivial. To address this challenge, we propose an adaptive learning rate for FTPL by introducing surrogate probability functions that can be computed only from the available quantities, without requiring the exact probabilities. Based on these learning rates with surrogate functions, we provide the BOBW guarantee for FTPL with Pareto perturbations for any shape parameter $α>1$, generalizing prior results restricted to specific choices of $α=2$. We further show the BOBW guarantees for FTPL with adaptive learning rates in the bandit problem with expert advices. Our approach preserves the computational simplicity of FTPL while enabling probability-dependent adaptivity, and the surrogate-based methodology may be of independent interest in other algorithmic frameworks beyond FTPL and learning rate designs.

We study the two-action apple-tasting problem with switching costs against an oblivious adversary. In an equivalent normalized formulation, at each round the learner chooses between a revealing action and a blind action: the revealing action gives reward $0$ and reveals the hidden value $x_t\in[-1,1]$ of the blind action; the blind action gives reward $x_t$ but reveals nothing. The learner pays one unit whenever they switches actions, and regret is measured against the best fixed action in hindsight. General feedback-graph algorithms with switching costs give $\widetilde O(T^{2/3})$ regret guarantees for this problem. The two-action apple-tasting graph was the natural candidate for the missing $Ω(T^{2/3})$ obstruction in the switching-cost classification: such a lower bound would have transferred to a large family of still-unclassified feedback graphs. We prove that this obstruction is not there: the oblivious minimax expected regret for this problem satisfies $$\frac{1}{2\sqrt3}\cdot\sqrt T \le R_T^\star \le 2\sqrt{3}\cdot \sqrt{T}.$$

Unmanned Aerial Vehicles (UAVs) equipped with radar sensors are deployed for target search missions in diverse environments, where targets exhibit characteristic signatures (e.g., respiration micro-motion in human search) detectable through occlusions. A fundamental challenge arises from shifts in radar statistics as the UAV moves through a dynamic and potentially non-stationary environment, rendering any fixed signal-processing strategy suboptimal; yet perception and adaptation must run onboard a resource-constrained aerial node in real time. Since no single detector performs well across all conditions, we adopt a multi-policy paradigm and formulate UAV target search as an online policy selection problem over a library of specialized detectors, with performance measured by regret, the cumulative loss gap relative to the best policy in each scene. The setting couples in-scene stochastic noise with inter-scene shifts. Whereas prior methods capture only one regime, we account for both through the Stochastically Extended Adversary (SEA) framework, without requiring oracle knowledge of scene dynamics. Because adaptation must run at the UAV, we instantiate SEA through \textsc{SEArch}, a lightweight optimistic Follow the Regularized Leader (OFTRL) selector with an adaptive learning rate, achieving regret $O(\barσ_T \sqrt{T} + \sqrt{J})$, where $\barσ_T$ captures radar measurement noise and $J$ is the number of scene transitions over the mission horizon $T$. To enable rapid adaptation under frequent scene changes, we further introduce \textsc{W-SEArch}, a windowed variant that restarts every $w$ rounds and achieves regret $O(\barσ_I \sqrt{w})$ under at most one transition per window. Experiments show up to 30% regret reduction compared to non-adaptive baselines across a range of non-stationary settings.

We study two-player zero-sum games (TPZSGs) with bandit feedback under fairness constraints requiring every action to be played with probability at least $α/m$. Existing instance-dependent results target $\textit{pure}$ Nash equilibria, while fairness generically produces $\textit{mixed}$ equilibria, a harder learning target. Our key technical tool is a reparametrization: every fair strategy decomposes as $p = (α/m)\mathbf{1} + (1-α)\widetilde{p}$ with $\widetilde{p} \in Δ_m$, and substituting into the payoff form yields $p^{\top}Aq = \widetilde{p}^{\top}\widetilde{A} q$ for a fair payoff matrix $\widetilde{A} := (1-α)A + α\mathbf{1} c^{\top}$, where $c_j = \tfrac{1}{m}\sum_i A(i,j)$ is the column-mean vector. The fair game on $A$ is then equivalent to a standard zero-sum game on $\widetilde{A}$, so equilibrium existence, KKT structure, and LP basis stability reduce to classical results applied to $\widetilde{A}$. We derive the fair minimax value, fair Nash equilibrium, fair regret, and a clean dual representation showing the price of fairness is at most $α(1-1/m)$ and vanishes whenever the unconstrained equilibrium already has full support. Our main result is an $\widetilde{O}(T^{2/3})$ regret bound for an Explore-Then-Commit algorithm, $\texttt{Fair-ETC-TPZSG}$, applicable to general mixed fair equilibria, together with a discussion of why naive action elimination does not readily improve it. When the fair equilibrium has a single dominant action, equivalently when $\widetilde{p}^{\star}$ is a vertex of $Δ_m$, the bound sharpens to instance-dependent $\widetilde{O}(1/\widetildeΔ(α)^{2})$, where $\widetildeΔ(α)$ is the LP-margin gap.

We study bandit learning in matching markets, where players and arms constitute the two market sides, and the players' utilities are linear in the arm contexts. In each round, new arms arrive with observable contexts. Then, the algorithm matches them to players, aiming to minimize each player's regret against a stable matching benchmark. This contextual structure creates significant complexity: subtle context shifts can slightly alter one player's utility while completely reconfiguring the underlying benchmark, causing large regret spikes for others. We address this in two settings: stochastic contexts, drawn from a latent distribution, and adversarial contexts, which may be arbitrary. For the stochastic case, we introduce a novel minimum preference gap to capture learning difficulty and provide a fully adaptive algorithm with an instance-dependent poly-logarithmic regret upper bound. We also establish matching instance-independent regret upper and lower bounds under a mild distributional assumption. For the adversarial setting, we propose a tractable regret notion that remains valid under arbitrary contexts and achieves an instance-independent sublinear regret bound via an adaptive algorithm.

We study the adversarial kernel bandit problem, in which the loss at each round is induced by an arbitrary bounded element of a reproducing kernel Hilbert space (RKHS). We propose an exponential-weights algorithm built on a regularized importance-weighted loss estimator, together with an explicit correction term that cancels the bias introduced by the regularization. Our main result bounds the regret by $\widetilde{O}\big(\sqrt{T\, d_*(λ)\,\log|{X}|}\big)$, where $d_*(λ)$ is a widely-adopted notion of effective dimension that captures the complexity of the kernel. Up to logarithmic factors, this matches the known rate achieved in the related stochastic kernel bandit problem. A notable application is the Matérn$(ν,d)$ kernel with smoothness parameter $ν$ on $\mathbb{R}^d$, for which our bound specializes to $\widetilde{O}\big(T^{(ν+d)/(2ν+d)}\big)$, improving over the best-known prior rate of Chatterji et al. [2019] while simultaneously removing the rank-one adversary assumption required by their analysis. Moreover, this rate is the same as the known optimal rate for stochastic kernel bandits, and also matches a lower bound from concurrent work up to a $\log T$ factor.

We study adversarial multi-armed bandits with and without delayed feedback under a safety-aware goal: achieving minimax-optimal worst-case regret while keeping nearly constant regret relative to a designated "safe" baseline policy. Existing approaches can balance this trade-off with immediate feedback for smooth comparators, but arbitrary delays can mistime transitions between conservatism and exploration, endangering the safety guarantee. To bridge this gap, we propose Prudent-Banker, a novel algorithm that combines a delay-adapted variant of Online Mirror Descent with a modified phased-aggression mechanism. Its key technical contribution is a delay-calibrated restart threshold that rigorously accounts for the worst-case distortion induced by unobserved feedback and reliably detects comparator suboptimality. We also establish new lower bounds for safety-constrained adversarial delayed bandits, showing that the regret guarantees of Prudent-Banker are unimprovable, up to logarithmic factors, under the baseline-safety requirement. To the best of our knowledge, Prudent-Banker is the first algorithm to achieve the optimal safety--robustness trade-off: pseudo-regret $\widetilde{O}(\sqrt{T}+\sqrt{D})$ together with $\widetilde{O}(1)$ regret against the safe comparator, both with and without delays. Experiments across diverse delay distributions show that, unlike standard delay-robust baselines, Prudent-Banker effectively balances safety and learning.

We study an online market-making problem in which a learner sequentially posts bid and ask prices for a single asset while interacting with traders holding private valuations. Unlike existing online learning formulations that assume fully censored feedback, we introduce an action-dependent feedback model inspired by real limit order books: when a trade occurs, the trader's valuation remains hidden, whereas when no trade occurs, informative feedback about supply and demand is revealed. We show that this additional information fundamentally changes the learnability of the problem. In the stochastic setting with i.i.d. market prices, we propose an elimination-based algorithm that achieves $O(\sqrt T)$ regret with high probability, without requiring any smoothness assumptions on the distribution of trader valuations. We then extend this result to a broad class of mean-reverting price processes by considering both local, autoregressive dynamics and a weaker global drift condition based on cumulative deviations from the mean. Under either assumption, we establish high-probability $O(\sqrt T)$ regret bounds, relying on a new concentration inequality of independent interest. Finally, in the adversarial setting with oblivious prices, we design an explore-then-perturb algorithm that guarantees $O(T^{2/3})$ regret in expectation. Our results quantify the value of observing the order book in online market making and demonstrate that even limited, action-dependent feedback can substantially improve regret guarantees compared to standard bandit feedback models.

This paper studies kernelized bandits (also known as Gaussian process bandits) in an adversarial environment, where the reward functions in a known reproducing kernel Hilbert space (RKHS) may be adversarially chosen at each round. We show that the exponential-weight algorithm achieves $\tilde{O}(\sqrt{T γ_T})$ adversarial regret, where $T$ and $γ_T$ denote the number of total rounds and the maximum information gain, respectively. For squared exponential (SE) and $ν$-Matérn kernels, we also show algorithm-independent lower bounds that guarantee the optimality of our algorithm up to polylogarithmic factors. Furthermore, we present a computationally efficient variant of our algorithm using Nyström approximation while maintaining nearly optimal regret guarantees.

We study adversarial noisy bandits given a known function class $\mathcal{F}$. In each round, the adversary selects a function $f \in \mathcal{F}$, the learner chooses an arm, and then observes a noisy reward determined by the chosen arm and the function $f$. The goal is to minimize the cumulative regret $R(T)$, defined as the difference between the learner's performance and that of the best fixed arm in hindsight over $T$ rounds. We say that a function class $\mathcal{F}$ is learnable if there exists an algorithm achieving sublinear regret. Our main result is a complete characterization of learnability for adversarial noisy bandits. The characterization is given in terms of a convexified variant of the generalized maximin volume introduced by Hanneke and Wang (2025): namely, the generalized maximin volume evaluated on the convex hull $\operatorname{co}(\mathcal F)$. We prove that $\mathcal F$ is learnable if and only if this convexified generalized maximin volume is positive at every scale. This condition characterizes learnability against both oblivious and adaptive adversaries, showing in particular that these two notions of learnability are equivalent in the noisy bandit setting. Our analysis reveals that the key complexity measure is closely connected to two new combinatorial notions, hitting set and distribution covering number, which may be of independent interest. These results establish the first complete characterization of learnability for adversarial noisy bandits.

We study \emph{multi-armed bandits} (MABs) augmented with \emph{best-action queries}, in which the learner may additionally query an oracle that reveals the best arm in the current round. This setting was recently characterized by Russo et al. [2024] in the \emph{full-feedback} model, where the learner observes the rewards of all arms after each round. They show that, in both \emph{stochastic} and \emph{adversarial} environments, $k$ best-action queries reduce the optimal $\widetilde{\mathcal{O}}(\sqrt{T})$ regret to $\widetilde{\mathcal{O}}(\min\{T/k,\sqrt{T}\})$. Whether this improvement extends to the more realistic \emph{bandit-feedback} model -- where the learner observes only the reward of the played arm -- was left as an open problem. We fully resolve this question. When rewards are stochastic but correlated among arms, we show that the full-feedback result does not carry over: any algorithm must incur regret at least $Ω(\sqrt{T-k})$. This lower bound directly extends to adversarial environments. On the positive side, we show that $\widetilde{\mathcal{O}}(\min\{T/k,\sqrt{T-k}\})$ regret is still achievable when rewards are stochastic and i.i.d., and establish a matching lower bound, up to logarithmic factors. Together, these results provide a complete characterization of the benefits of \emph{best-action queries} in the \emph{bandit-feedback} model.

We study budget-constrained contextual bandits with adversarial contexts, where each action yields a random reward and incurs a random cost. We adopt the standard realizability assumption: conditioned on the observed context, rewards and costs are drawn independently from fixed distributions whose expectations belong to known function classes. We focus on the continuing setting, in which the algorithm operates over the entire horizon even after the budget for cumulative cost is exhausted. In this setting, the objective is to simultaneously control regret and the violation of the budget constraint. Building on the seminal $\mathsf{SquareCB}$ framework of Foster et al. [2018], we propose a simple and modular framework that leverages online regression oracles to reduce the constrained problem to a standard unconstrained contextual bandit problem with adaptively defined surrogate reward functions. In contrast to prior works, which focus on stochastic contexts, our reduction yields improved guarantees for more general adversarial contexts, together with an efficient algorithm with a compact and transparent analysis.

We consider adversarial multi-armed bandit problems where the learner is allowed to observe losses of a number of arms beside the arm that it actually chose. We study the case where all non-chosen arms reveal their loss with a fixed but unknown probability $r$, independently of each other and the action of the learner. We propose two algorithms that work for different ranges of $r$. We show that after $T$ rounds in a bandit problem with $N$ arms, the expected regret of our first algorithm is $O(\sqrt{(T /r) \log N })$ whenever $r\ge(\log T)/(2N)$, while our second algorithm achieves a regret of $O(\sqrt{(T/r) \log (N+T)})$ for smaller values of $r$. We also give a quick estimation procedure that decides the range of~$r$. All our bounds are within logarithmic factors of the best achievable performance of any algorithm that is even allowed to know~$r$.

We consider online learning problems under a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem.