High-probability Convergence Guarantees of Decentralized SGD

Convergence in high-probability (HP) has attracted increasing interest, due to implying exponentially decaying tail bounds and strong guarantees for individual runs of an algorithm. While many works study HP guarantees in centralized settings, much less is understood in the decentralized setup, where existing works require strong assumptions, like uniformly bounded gradients, or asymptotically vanishing noise. This results in a significant gap between the assumptions used to establish convergence in the HP and the mean-squared error (MSE) sense, and is also contrary to centralized settings, where it is known that $\mathtt{SGD}$ converges in HP under the same conditions on the cost function as needed for MSE convergence. Motivated by these observations, we study the HP convergence of Decentralized $\mathtt{SGD}$ ($\mathtt{DSGD}$) in the presence of light-tailed noise, providing several strong results. First, we show that $\mathtt{DSGD}$ converges in HP under the same conditions on the cost as in the MSE sense, removing the restrictive assumptions used in prior works. Second, our sharp analysis yields order-optimal rates for both non-convex and strongly convex costs. Third, we establish a linear speed-up in the number of users, leading to matching, or strictly better transient times than those obtained from MSE results, further underlining the tightness of our analysis. To the best of our knowledge, this is the first work that shows $\mathtt{DSGD}$ achieves a linear speed-up in the HP sense. Our relaxed assumptions and sharp rates stem from several technical results of independent interest, including a result on the variance-reduction effect of decentralized methods in the HP sense, as well as a novel bound on the MGF of strongly convex costs, which is of interest even in centralized settings. Finally, we provide experiments that validate our theory.

💡 Research Summary

This paper addresses a notable gap in the theory of decentralized stochastic optimization: while high‑probability (HP) convergence guarantees are well‑understood for centralized stochastic gradient descent (SGD) under light‑tailed noise, analogous results for decentralized SGD (DSGD) have required restrictive assumptions such as uniformly bounded gradients or asymptotically vanishing noise. The authors show that these strong assumptions are unnecessary.

Problem setting.
A network of (n\ge 2) agents cooperatively minimizes the average loss
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