通过自排斥随机游走加速分布式随机优化
Accelerating Distributed Stochastic Optimization via Self-Repellent Random Walks
AI总结：
本文提出由广义自排斥随机游走SRRW驱动的随机近似算法SA-SRRW，用于分布式随机优化，并证明其迭代误差几乎必然收敛及渐近协方差以O(1/α^2)降低。
摘要：
我们研究了一类分布式随机优化算法，其中梯度由一个以随机游走方式遍历智能体网络的令牌进行采样。通常，这些随机游走被选为马尔可夫链，能够渐近地从期望的目标分布中采样，并在优化迭代的收敛中发挥关键作用。本文采用一种新颖方法，将标准的线性马尔可夫令牌替换为遵循非线性马尔可夫链的令牌，即自排斥随机游走（Self-Repellent Radom Walk，SRRW）。SRRW针对任意给定的“基础”马尔可夫链定义，由正标量α参数化，它转移到过去被高频访问状态的可能性较低，因此得名。在图上MCMC采样的背景下，Doshi等人（2023）的一项最新突破表明，SRRW在采样的渐近方差方面实现了O(1/α)的降低。我们提出使用“广义”版本的SRRW，以随机近似的形式驱动用于分布式随机优化的令牌算法，称为SA-SRRW。我们证明了所得SA-SRRW的优化迭代误差几乎必然收敛于零，并证明了中心极限定理，导出了与迭代误差对应的所得渐近协方差矩阵的显式形式。该渐近协方差始终小于由基础马尔可夫链驱动的算法的协方差，并以O(1/α^2)的速率下降——因此使用SRRW的性能收益在随机优化背景下被放大。实证结果支持了我们的理论发现。
英文摘要：
We study a family of distributed stochastic optimization algorithms where gradients are sampled by a token traversing a network of agents in random-walk fashion. Typically, these random-walks are chosen to be Markov chains that asymptotically sample from a desired target distribution, and play a critical role in the convergence of the optimization iterates. In this paper, we take a novel approach by replacing the standard linear Markovian token by one which follows a nonlinear Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar α, is less likely to transition to states that were highly visited in the past, thus the name. In the context of MCMC sampling on a graph, a recent breakthrough in Doshi et al. (2023) shows that the SRRW achieves O(1/α) decrease in the asymptotic variance for sampling. We propose the use of a 'generalized' version of the SRRW to drive token algorithms for distributed stochastic optimization in the form of stochastic approximation, termed SA-SRRW. We prove that the optimization iterate errors of the resulting SA-SRRW converge to zero almost surely and prove a central limit theorem, deriving the explicit form of the resulting asymptotic covariance matrix corresponding to iterate errors. This asymptotic covariance is always smaller than that of an algorithm driven by the base Markov chain and decreases at rate O(1/α^2) - the performance benefit of using SRRW thereby amplified in the stochastic optimization context. Empirical results support our theoretical findings.
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