Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions

TL;DR

This paper presents a simplified convergence proof for AdaGrad on non-convex functions, achieving tighter iteration bounds and extending analysis to local smoothness conditions, thereby broadening understanding of AdaGrad's effectiveness.

Contribution

The authors introduce a novel auxiliary function to simplify convergence proofs for AdaGrad, improving bounds and extending analysis to local smoothness assumptions.

Findings

AdaGrad converges in (rac{1}{\u03b5^2}) iterations in over-parameterized regimes.

Tighter convergence rates than previous (rac{1}{^4}) bounds are established.

Convergence under (L_0,L_1) smoothness depends on learning rate, which is shown to be necessary.

Abstract

We provide a simple convergence proof for AdaGrad optimizing non-convex objectives under only affine noise variance and bounded smoothness assumptions. The proof is essentially based on a novel auxiliary function $\xi$ that helps eliminate the complexity of handling the correlation between the numerator and denominator of AdaGrad's update. Leveraging simple proofs, we are able to obtain tighter results than existing results \citep{faw2022power} and extend the analysis to several new and important cases. Specifically, for the over-parameterized regime, we show that AdaGrad needs only $\mathcal{O}(\frac{1}{\varepsilon^2})$ iterations to ensure the gradient norm smaller than $\varepsilon$, which matches the rate of SGD and significantly tighter than existing rates $\mathcal{O}(\frac{1}{\varepsilon^4})$ for AdaGrad. We then discard the bounded smoothness assumption and consider a realistic assumption on smoothness called $(L_0,L_1)$-smooth condition, which allows local smoothness to grow with the gradient norm. Again based on the auxiliary function $\xi$, we prove that AdaGrad succeeds in converging under $(L_0,L_1)$-smooth condition as long as the learning rate is lower than a threshold. Interestingly, we further show that the requirement on learning rate under the $(L_0,L_1)$-smooth condition is necessary via proof by contradiction, in contrast with the case of uniform smoothness conditions where convergence is guaranteed regardless of learning rate choices. Together, our analyses broaden the understanding of AdaGrad and demonstrate the power of the new auxiliary function in the investigations of AdaGrad.