# Making the Last Iterate of SGD Information Theoretically Optimal

## TL;DR

This paper introduces new step size sequences for SGD and GD that achieve information-theoretic optimal bounds on the suboptimality of the last iterate, addressing a longstanding gap in theoretical understanding and practical performance.

## Contribution

It proposes a novel modification scheme for step size sequences that ensures the last iterate's suboptimality matches the optimal bounds, improving theoretical guarantees and practical results.

## Key findings

- New step size sequences achieve optimal last iterate bounds
- Modified sequences match average-case guarantees for last point
- Simulations confirm significant improvement over standard step sizes

## Abstract

Stochastic gradient descent (SGD) is one of the most widely used algorithms for large scale optimization problems. While classical theoretical analysis of SGD for convex problems studies (suffix) \emph{averages} of iterates and obtains information theoretically optimal bounds on suboptimality, the \emph{last point} of SGD is, by far, the most preferred choice in practice. The best known results for last point of SGD \cite{shamir2013stochastic} however, are suboptimal compared to information theoretic lower bounds by a $\log T$ factor, where $T$ is the number of iterations. \cite{harvey2018tight} shows that in fact, this additional $\log T$ factor is tight for standard step size sequences of $\OTheta{\frac{1}{\sqrt{t}}}$ and $\OTheta{\frac{1}{t}}$ for non-strongly convex and strongly convex settings, respectively. Similarly, even for subgradient descent (GD) when applied to non-smooth, convex functions, the best known step-size sequences still lead to $O(\log T)$-suboptimal convergence rates (on the final iterate). The main contribution of this work is to design new step size sequences that enjoy information theoretically optimal bounds on the suboptimality of \emph{last point} of SGD as well as GD. We achieve this by designing a modification scheme, that converts one sequence of step sizes to another so that the last point of SGD/GD with modified sequence has the same suboptimality guarantees as the average of SGD/GD with original sequence. We also show that our result holds with high-probability. We validate our results through simulations which demonstrate that the new step size sequence indeed improves the final iterate significantly compared to the standard step size sequences.