Ilgam Latypov

Yuriy Dorn, Aleksandr Katrutsa, Ilgam Latypov et al.

In this study, we propose a new method for constructing UCB-type algorithms for stochastic multi-armed bandits based on general convex optimization methods with an inexact oracle. We derive the regret bounds corresponding to the convergence rates of the optimization methods. We propose a new algorithm Clipped-SGD-UCB and show, both theoretically and empirically, that in the case of symmetric noise in the reward, we can achieve an $O(\log T\sqrt{KT\log T})$ regret bound instead of $O\left (T^{\frac{1}{1+α}} K^{\fracα{1+α}} \right)$ for the case when the reward distribution satisfies $\mathbb{E}_{X \in D}[|X|^{1+α}] \leq σ^{1+α}$ ($α\in (0, 1])$, i.e. perform better than it is assumed by the general lower bound for bandits with heavy-tails. Moreover, the same bound holds even when the reward distribution does not have the expectation, that is, when $α<0$.

Yuriy Dorn, Aleksandr Katrutsa, Ilgam Latypov et al.

Bandit optimization usually refers to the class of online optimization problems with limited feedback, namely, a decision maker uses only the objective value at the current point to make a new decision and does not have access to the gradient of the objective function. While this name accurately captures the limitation in feedback, it is somehow misleading since it does not have any connection with the multi-armed bandits (MAB) problem class. We propose two new classes of problems: the functional multi-armed bandit problem (FMAB) and the best function identification problem. They are modifications of a multi-armed bandit problem and the best arm identification problem, respectively, where each arm represents an unknown black-box function. These problem classes are a surprisingly good fit for modeling real-world problems such as competitive LLM training. To solve the problems from these classes, we propose a new reduction scheme to construct UCB-type algorithms, namely, the F-LCB algorithm, based on algorithms for nonlinear optimization with known convergence rates. We provide the regret upper bounds for this reduction scheme based on the base algorithms' convergence rates. We add numerical experiments that demonstrate the performance of the proposed scheme.