Median Clipping for Zeroth-order Non-Smooth Convex Optimization and Multi Arm Bandit Problem with Heavy-tailed Symmetric Noise

Nikita Maksimovich Kornilov, Yuriy Dorn, Aleksandr Lobanov, Nikolay Kutuzov, Innokentiy Andreevich Shibaev, Eduard Gorbunov, Alexander Gasnikov, Alexander V. Nazin

Keywords: optimization, median clipping, heavy tails, multi arm bandit

TL;DR: In case of symmetric stochastic noise corrupting function values, median clipping allows building zeroth-order and MAB methods which beat SOTA results both in practice and theory.

Abstract: In this paper, we consider non-smooth convex optimization with a zeroth-order oracle corrupted by symmetric stochastic noise. Unlike the existing high-probability results requiring the noise to have bounded $\kappa$-th moment with $\kappa \in (1,2]$, our results allow even heavier noise with any $\kappa > 0$, e.g., the noise distribution can have unbounded expectation. Our convergence rates match the best-known ones for the case of the bounded variance, namely, to achieve function accuracy $\varepsilon$ our methods with Lipschitz oracle require $\tilde{O}(d^2\varepsilon^{-2})$ iterations for any $\kappa > 0$. We build the median gradient estimate with bounded second moment as the mini-batched median of the sampled gradient differences. We apply this technique to the stochastic multi-armed bandit problem with heavy-tailed distribution of rewards and achieve $\tilde{O}(\sqrt{dT})$ regret. We demonstrate the performance of our zeroth-order and MAB algorithms for different $\kappa$ on synthetic and real-world data. Our methods do not lose to SOTA approaches and dramatically outperform them for $\kappa \leq 1$.

Primary Area: optimization

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Submission Number: 11809