Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise

We consider a first order stochastic optimization framework where, at each iteration, $K$ independent identically distributed (i.i.d.) data point samples are drawn, based on which stochastic gradients can be queried. We allow gradient noise to be heavy-tailed, with possibly infinite variances. For the considered heavy-tailed setting, many algorithmic variants have recently been proposed based on gradient clipping or other nonlinear operators (e.g., normalization) applied over noisy gradients. In this paper, we take an alternative approach and propose a novel stochastic first order method dubbed Robust Stochastic Gradient Descent with medoid mini-batch gradient sampling, R-SGD-Mini for short. The core idea of R-SGD-Mini is to split the $K$-sized data batch into $M$ distinct data chunks, form for each chunk the stochastic gradient, and update the solution estimate with respect to the stochastic gradient direction of the chunk that is medoid of gradients of all data-chunks. Under a general class of symmetric heavy-tailed gradient noises and a standard non-convex setting, we establish explicit bounds on the expected time-averaged squared gradient norm. More precisely, we show that the latter quantity converges at rate $\mathcal{O}(T^{-1})$ to a small neighborhood of zero; we explicitly characterize this neighborhood in terms of noise and algorithm's parameters. Moreover, if the time horizon is known in advance, we establish the rate of $\mathcal{O}(T^{-\frac{1}{2}}).$ Furthermore, when clipping is incorporated, we obtain convergence guaranties in the high-probability sense and recover the same rate. Experimental results indicate that R-SGD-Mini and its clipped variant consistently perform favorably compared to SGD, clipped SGD and Median-of-Means based methods.

Low-precision pretraining (FP8, MXFP4, NVFP4) is now standard for frontier language models, yet the literature is almost entirely achievability -- algorithms and empirical scaling laws -- with no matching characterization of what is information-theoretically possible. We study a B-bit quantized stochastic first-order oracle: an optimizer interacts for T rounds and receives, each round, a B-bit adaptive public-coin description of its stochastic gradient. Our main contribution is an exact reduction from optimizing a strongly convex quadratic family to interactively compressed Gaussian mean estimation -- under the B-bit oracle the query carries no information, so optimization collapses exactly onto a sequential distributed-estimation problem. This yields two unconditional lower bounds, a communication bound TB = Omega(d) and a statistical bound T = Omega(sigma^2 d / eps^2), and the sharp product-form bound T = Omega((sigma^2 d / eps^2) max{1, d/B}). The product form is also unconditional: a B-bit transcript carries at most O(TB / sigma^2) of Fisher trace about the mean, so bits rather than dimension limit the recoverable information, and combined with the multivariate van Trees inequality this gives the bound directly, without bounded-likelihood-ratio truncation. We give a near-matching achievability result with exact per-round bit accounting under a bounded-dynamic-range oracle, tight up to a logarithmic factor; the lower bound is for truly Gaussian (unbounded) gradients, and closing this oracle gap is left open. A sequential rate-distortion perspective extends the reduction to correlated and drifting oracles and corrects an earlier conjecture: positive noise correlation raises the bound by (1+rho)/(1-rho) rather than relaxing it. The bounds give an information-theoretic baseline for any low-bit gradient path, not an optimality claim about deployed FP4 systems.

Many stochastic gradient methods are believed not to converge when the noise in stochastic gradients has only a finite $p$-th moment for $p\in\left(1,2\right)$, a setting known as the heavy-tailed noise assumption. However, some recent studies have found that Stochastic Gradient Descent ($\textsf{SGD}$), without any modification to its update rule, can surprisingly converge in expectation for convex problems with bounded domains, highlighting the potential of classical stochastic gradient methods. Inspired by this recent progress, we provide a comprehensive study of stochastic optimization under heavy-tailed noise and establish new in-expectation convergence results for Stochastic Mirror Descent ($\textsf{SMD}$) and Accelerated Stochastic Mirror Descent ($\textsf{ASMD}$) in convex optimization, and for $\textsf{SGD}$ and Stochastic Gradient Descent with Momentum ($\textsf{SGDM}$) in nonconvex optimization. Notably, our results not only hold without algorithmic changes but also avoid restrictive assumptions, such as bounded domains, imposed in prior work. More importantly, our analysis provides a new, elegant, and powerful framework for studying heavy-tailed stochastic optimization, opening a new route to understanding first-order stochastic gradient methods.