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 *non-linear* Markov chain - namely the Self-Repellent Radom Walk (SRRW). Defined for any given 'base' Markov chain, the SRRW, parameterized by a positive scalar $\\alpha$, 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 $O(1/\\alpha)$ 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 $O(1/\\alpha^2)$ - the performance benefit of using SRRW thereby *amplified* in the stochastic optimization context. Empirical results support our theoretical findings.

Data pruning aims to obtain lossless performances with less overall cost. A common approach is to filter out samples that make less contribution to the training. This could lead to gradient expectation bias compared to the original data. To solve this problem, we propose InfoBatch, a novel framework aiming to achieve lossless training acceleration by unbiased dynamic data pruning. Specifically, InfoBatchrandomly prunes a portion of less informative samples based on the loss distribution and rescales the gradients of the remaining samples to approximate the original gradient. As a plug-and-play and architecture-agnostic framework, InfoBatch consistently obtains lossless training results on classification, semantic segmentation, vision pertaining, and instruction fine-tuning tasks. On CIFAR10/100, ImageNet-1K, and ADE20K, InfoBatch losslessly saves 40% overall cost. For pertaining MAE and diffusion model, InfoBatch can respectively save 24.8% and 27% cost. For LLaMA instruction fine-tuning, InfoBatch is also able to save 20% cost and is compatible with coreset selection methods. The code will be made public.

We show how to obtain improved active learning methods in the agnostic (adversarial noise) setting by combining marginal leverage score sampling with non-independent sampling strategies that promote spatial coverage. In particular, we propose an easily implemented method based on the \emph{pivotal sampling algorithm}, which we test on problems motivated by learning-based methods for parametric PDEs and uncertainty quantification. In comparison to independent sampling, our method reduces the number of samples needed to reach a given target accuracy by up to $50\%$.We support our findings with two theoretical results. First, we show that any non-independent leverage score sampling method that obeys a weak \emph{one-sided $\ell_{\infty}$ independence condition} (which includes pivotal sampling) can actively learn $d$ dimensional linear functions with $O(d\log d)$ samples, matching independent sampling. This result extends recent work on matrix Chernoff bounds under $\ell_{\infty}$ independence, and may be of interest for analyzing other sampling strategies beyond pivotal sampling. Second, we show that, for the important case of polynomial regression, our pivotal method obtains an improved bound of $O(d)$ samples.