Median Clipping for Zeroth-order Non-Smooth Convex Optimization and Multi Arm Bandit Problem with Heavy-tailed Symmetric Noise
arxiv(2024)
摘要
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 κ-th moment
with κ∈ (1,2], our results allow even heavier noise with any κ
> 0, e.g., the noise distribution can have unbounded 1-st moment. Moreover,
our results match the best-known ones for the case of the bounded variance. To
achieve this, we use the mini-batched median estimate of the sampled gradient
differences, apply gradient clipping to the result, and plug in the final
estimate into the accelerated method. We apply this technique to the stochastic
multi-armed bandit problem with heavy-tailed distribution of rewards and
achieve O(√(Td)) regret by incorporating noise symmetry.
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