中位数截断用于零阶非光滑凸优化及具有重尾对称噪声的多臂老虎机问题
Median Clipping for Zeroth-order Non-Smooth Convex Optimization and Multi-Armed Bandit Problem with Heavy-tailed Symmetric Noise
AI总结：
本文提出了一种中位数截断方法，用于解决受重尾对称噪声影响的零阶非光滑凸优化和多臂老虎机问题，实现了在不同噪声条件下优于现有方法的收敛速率和性能。
摘要：
在本文中，我们考虑受到对称随机噪声污染的零阶凸优化问题。与现有高概率结果要求噪声具有有界κ-阶矩（κ∈(1,2]）不同，我们的结果允许更重的噪声，任何κ>0，例如噪声分布可以有无界期望。我们的收敛速率与有界方差情况下的最佳已知结果一致，即为了达到函数精度ε，我们的方法在Lipschitz oracle下需要~O(d²ε⁻²)次迭代，对于任何κ>0。我们构建了具有有界二阶矩的中位数梯度估计，作为采样梯度差的迷你批次中位数。我们将此技术应用于具有重尾分布奖励的随机多臂老虎机问题，并实现了~O(√dT)的 regrets。我们在各种κ∈(0,2]上展示了我们的零阶和MAB算法在合成和现实数据中的性能。我们的方法不逊于最先进的方法，并且对于κ≤1显著优于它们。
英文摘要：
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 $κ\in (1,2]$, our results allow even heavier noise with any $κ> 0$, 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 $κ> 0$. 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 various $κ\in (0,2]$ on synthetic and real-world data. Our methods do not lose to SOTA approaches and dramatically outperform them for $κ\leq 1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
