We introduce two complementary techniques for efficient optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) through step-size sharing. Subset-Norm (SN) reduces AdaGrad’s memory footprint from $O(d)$ to $O(\sqrt{d})$, where $d$ is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian noise, we show a noise-adapted high-probability convergence guarantee with improved dimensional dependence of SN over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state’s memory footprint by restricting momentum to a low-dimensional subspace while performing SGD in the orthogonal complement. We prove high-probability convergence rates for Subspace-Momentum under standard assumptions. Empirical evaluation on pre-training and fine-tuning LLMs demonstrates the effectiveness of our methods. For instance, combining Subset-Norm with Subspace-Momentum achieves Adam’s validation perplexity for LLaMA 1B in approximately half the training tokens (6.8B vs 13.1B) while reducing Adam’s optimizer-states memory footprint by more than 80% with minimal additional hyperparameter tuning.

In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. The method can be applied to the non-convex case. We demonstrate an $O((1+\sigma^{2}\log(1/\delta))/T+\sigma/\sqrt{T})$ convergence rate when the number of iterations $T$ is known and an $O((1+\sigma^{2}\log(T/\delta))/\sqrt{T})$ convergence rate when $T$ is unknown for SGD, where $1-\delta$ is the desired success probability. These bounds improve over existing bounds in the literature. We also revisit AdaGrad-Norm (Ward et al., 2019) and show a new analysis to obtain a high probability bound that does not require the bounded gradient assumption made in previous works. The full version of our paper contains results for the standard per-coordinate AdaGrad.

In this work, we study the convergence in high probability of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $p$th moments, for some $1<p\le2$. Prior works in this setting follow the same recipe of using concentration inequalities and an inductive argument with union bound to bound the iterates across all iterations. This method results in an increase in the failure probability by a factor of $T$, where $T$ is the number of iterations. We instead propose a new analysis approach based on bounding the moment generating function of a well chosen supermartingale sequence. We improve the dependency on $T$ in the convergence guarantee for a wide range of algorithms with clipped gradients, including stochastic (accelerated) mirror descent for convex objectives and stochastic gradient descent for nonconvex objectives. Our high probability bounds achieve the optimal convergence rates and match the best currently known in-expectation bounds. Our approach naturally allows the algorithms to use time-varying step sizes and clipping parameters when the time horizon is unknown, which appears difficult or even impossible using the techniques from prior works. Furthermore, we show that in the case of clipped stochastic mirror descent, several problem constants, including the initial distance to the optimum, are not required when setting step sizes and clipping parameters.