Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Methodology Establishing Linear Convergence of Adaptive Gradient Methods under PL Inequality

Kushal Chakrabarti, Mayank Baranwal

TL;DR

This work addresses proving linear convergence of adaptive gradient methods, specifically AdaGrad and Adam, for smooth objectives that satisfy the Polyak-Łojasiewicz inequality. It introduces a simple, unified discrete-time analysis that covers both batch and stochastic gradients, deriving explicit contraction factors and step-size restrictions. The main contributions are rigorous linear-convergence results for AdaGrad and Adam (with bias-corrected variants) and their stochastic-gradient extensions, which converge to a neighborhood of the optimum with rate $\rho\in(0,1)$ and neighborhood size tied to gradient-noise scale $M$. The results bridge the gap between theory and practice for adaptive methods, and the presented framework can be extended to analyze other variants like AdaBelief, AMSGrad, and RAdam.

Abstract

Adaptive gradient-descent optimizers are the standard choice for training neural network models. Despite their faster convergence than gradient-descent and remarkable performance in practice, the adaptive optimizers are not as well understood as vanilla gradient-descent. A reason is that the dynamic update of the learning rate that helps in faster convergence of these methods also makes their analysis intricate. Particularly, the simple gradient-descent method converges at a linear rate for a class of optimization problems, whereas the practically faster adaptive gradient methods lack such a theoretical guarantee. The Polyak-Łojasiewicz (PL) inequality is the weakest known class, for which linear convergence of gradient-descent and its momentum variants has been proved. Therefore, in this paper, we prove that AdaGrad and Adam, two well-known adaptive gradient methods, converge linearly when the cost function is smooth and satisfies the PL inequality. Our theoretical framework follows a simple and unified approach, applicable to both batch and stochastic gradients, which can potentially be utilized in analyzing linear convergence of other variants of Adam.

A Methodology Establishing Linear Convergence of Adaptive Gradient Methods under PL Inequality

TL;DR

This work addresses proving linear convergence of adaptive gradient methods, specifically AdaGrad and Adam, for smooth objectives that satisfy the Polyak-Łojasiewicz inequality. It introduces a simple, unified discrete-time analysis that covers both batch and stochastic gradients, deriving explicit contraction factors and step-size restrictions. The main contributions are rigorous linear-convergence results for AdaGrad and Adam (with bias-corrected variants) and their stochastic-gradient extensions, which converge to a neighborhood of the optimum with rate and neighborhood size tied to gradient-noise scale . The results bridge the gap between theory and practice for adaptive methods, and the presented framework can be extended to analyze other variants like AdaBelief, AMSGrad, and RAdam.

Abstract

Adaptive gradient-descent optimizers are the standard choice for training neural network models. Despite their faster convergence than gradient-descent and remarkable performance in practice, the adaptive optimizers are not as well understood as vanilla gradient-descent. A reason is that the dynamic update of the learning rate that helps in faster convergence of these methods also makes their analysis intricate. Particularly, the simple gradient-descent method converges at a linear rate for a class of optimization problems, whereas the practically faster adaptive gradient methods lack such a theoretical guarantee. The Polyak-Łojasiewicz (PL) inequality is the weakest known class, for which linear convergence of gradient-descent and its momentum variants has been proved. Therefore, in this paper, we prove that AdaGrad and Adam, two well-known adaptive gradient methods, converge linearly when the cost function is smooth and satisfies the PL inequality. Our theoretical framework follows a simple and unified approach, applicable to both batch and stochastic gradients, which can potentially be utilized in analyzing linear convergence of other variants of Adam.
Paper Structure (4 sections, 3 theorems, 55 equations)

This paper contains 4 sections, 3 theorems, 55 equations.

Table of Contents

  1. Linear convergence of AdaGrad
  2. Linear convergence of Adam
  3. AdaGrad with stochastic gradients
  4. Summary

Key Result

Theorem 1

Consider the AdaGrad algorithm in eqn:y_adagrad-eqn:x_adagrad with initialization $y_0 = 0_d$ and $x_0 \in \mathbb{R}^d$ and the parameter $\epsilon \in (0,1)$. If Assumptions assump_1-assump_3 hold, then there exists $\overline{h} > 0$ such that if the step size $0 < h < \overline{h}$, then $(f(x_{

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3
  • proof
  • Remark 4