Convergence Rate Analysis of LION

TL;DR

This paper provides a rigorous convergence rate analysis of the LION optimizer, showing it converges at an optimal rate to critical points in nonconvex stochastic optimization, supported by extensive experiments.

Contribution

It establishes the first comprehensive convergence rate for LION, matching the theoretical lower bounds and confirming empirical performance advantages over SGD.

Findings

LION converges to KKT points at rate O(√d K^{-1/4})

LION outperforms SGD in loss and performance

Empirical gradient norm ratios match theoretical predictions

Abstract

The LION (evoLved sIgn mOmeNtum) optimizer for deep neural network training was found by Google via program search, with the simple sign update yet showing impressive performance in training large scale networks. Although previous studies have investigated its convergence properties, a comprehensive analysis, especially the convergence rate, is still desirable. Recognizing that LION can be regarded as solving a specific constrained problem, this paper focuses on demonstrating its convergence to the Karush-Kuhn-Tucker (KKT) point at the rate of $\cal O(\sqrt{d}K^{-1/4})$ measured by gradient $\ell_1$ norm, where $d$ is the problem dimension and $K$ is the number of iteration steps. Step further, we remove the constraint and establish that LION converges to the critical point of the general unconstrained problem at the same rate. This rate not only delivers the currently optimal dependence on the problem dimension $d$ but also tightly matches the theoretical lower bound for nonconvex stochastic optimization algorithms, which is typically measured using the gradient $\ell_2$ norm, with respect to the number of iterations $K$. Through extensive experiments, we not only demonstrate that LION achieves lower loss and higher performance compared to standard SGD, but also empirically confirm that the gradient $\ell_1/\ell_2$ norm ratio aligns with $\Theta(\sqrt{d})$, thus proving that our convergence rate matches the theoretical lower bound with respect to $d$ in the empirical sense.