A Complete Characterization of Learnability for Adversarial Noisy Bandits
Authors: Steve Hanneke, Kun Wang
Organizations: Department of Computer Science, Purdue University
Abstract
We study adversarial noisy bandits given a known function class F. In each round, the adversary selects a function f∈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 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 co(F). We prove that 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 bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.
Yasin Abbasi-Yadkori, Peter L. Bartlett, Victor Gabillon +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.
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 Φn,k(T)=min{nn−kT,kn−k}, 2≤k<n. Under winner feedback, both arbitrary joint i.i.d. rewards and fixed sequences have minimax regret of order Rn,k(T)=nn−kmin{T,kn+T,knT}. 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.