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A Comprehensive Framework for Analyzing the Convergence of Adam: Bridging the Gap with SGD

Ruinan Jin, Xiao Li, Yaoliang Yu, Baoxiang Wang

TL;DR

This work tackles the theoretical convergence gap for Adam by introducing a comprehensive framework that operates under SGD-like assumptions, specifically $L$-smoothness and the ABC inequality, rather than requiring almost-sure gradient boundedness. It unifies non-asymptotic sample complexity, almost-sure convergence, and asymptotic $L_1$ convergence, showing that Adam can achieve last-iterate convergence and SGD-like rates without dependence on $ rac{1}{\mu}$ under appropriate hyperparameter settings. A key innovation is the approximate descent inequality for Adam, supported by a Lyapunov product $\Pi_{\Delta,t}$ and a novel treatment of the extra error term via $\overline{\Delta}_t$ and Burkholder-based moment bounds. The results bridge Adam’s theoretical guarantees with those of SGD, enhancing reliability across a broad class of nonconvex problems and informing hyperparameter strategies for practical use. Overall, the framework broadens the applicability of Adam by aligning its convergence properties with SGD under realistic assumptions, with implications for variants and large-scale optimization in deep learning.

Abstract

Adaptive Moment Estimation (Adam) is a cornerstone optimization algorithm in deep learning, widely recognized for its flexibility with adaptive learning rates and efficiency in handling large-scale data. However, despite its practical success, the theoretical understanding of Adam's convergence has been constrained by stringent assumptions, such as almost surely bounded stochastic gradients or uniformly bounded gradients, which are more restrictive than those typically required for analyzing stochastic gradient descent (SGD). In this paper, we introduce a novel and comprehensive framework for analyzing the convergence properties of Adam. This framework offers a versatile approach to establishing Adam's convergence. Specifically, we prove that Adam achieves asymptotic (last iterate sense) convergence in both the almost sure sense and the \(L_1\) sense under the relaxed assumptions typically used for SGD, namely \(L\)-smoothness and the ABC inequality. Meanwhile, under the same assumptions, we show that Adam attains non-asymptotic sample complexity bounds similar to those of SGD.

A Comprehensive Framework for Analyzing the Convergence of Adam: Bridging the Gap with SGD

TL;DR

This work tackles the theoretical convergence gap for Adam by introducing a comprehensive framework that operates under SGD-like assumptions, specifically -smoothness and the ABC inequality, rather than requiring almost-sure gradient boundedness. It unifies non-asymptotic sample complexity, almost-sure convergence, and asymptotic convergence, showing that Adam can achieve last-iterate convergence and SGD-like rates without dependence on under appropriate hyperparameter settings. A key innovation is the approximate descent inequality for Adam, supported by a Lyapunov product and a novel treatment of the extra error term via and Burkholder-based moment bounds. The results bridge Adam’s theoretical guarantees with those of SGD, enhancing reliability across a broad class of nonconvex problems and informing hyperparameter strategies for practical use. Overall, the framework broadens the applicability of Adam by aligning its convergence properties with SGD under realistic assumptions, with implications for variants and large-scale optimization in deep learning.

Abstract

Adaptive Moment Estimation (Adam) is a cornerstone optimization algorithm in deep learning, widely recognized for its flexibility with adaptive learning rates and efficiency in handling large-scale data. However, despite its practical success, the theoretical understanding of Adam's convergence has been constrained by stringent assumptions, such as almost surely bounded stochastic gradients or uniformly bounded gradients, which are more restrictive than those typically required for analyzing stochastic gradient descent (SGD). In this paper, we introduce a novel and comprehensive framework for analyzing the convergence properties of Adam. This framework offers a versatile approach to establishing Adam's convergence. Specifically, we prove that Adam achieves asymptotic (last iterate sense) convergence in both the almost sure sense and the sense under the relaxed assumptions typically used for SGD, namely -smoothness and the ABC inequality. Meanwhile, under the same assumptions, we show that Adam attains non-asymptotic sample complexity bounds similar to those of SGD.
Paper Structure (57 sections, 22 theorems, 254 equations, 1 table, 1 algorithm)

This paper contains 57 sections, 22 theorems, 254 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Bounded Variance and Coordinate Affine Noise Variance:
  4. Almost Surely Bounded Stochastic Gradients:
  5. $L_2$ Bounded Stochastic Gradients:
  6. Randomly Reshuffled Stochastic Gradients:
  7. Preliminaries
  8. Adam
  9. Assumptions
  10. Comparison with Prior Works on Stochastic Gradient Assumptions
  11. Traditional Affine Noise Variance Condition
  12. Coordinate Affine Noise Variance Assumption
  13. Almost Surely Bounded Stochastic Gradient Assumption
  14. Hyperparameter Settings
  15. Theoretical Results
  16. ...and 42 more sections

Key Result

Theorem 3.1

Consider the Adam algorithm as specified in Algorithm Adam, and assume that Assumptions ass_1--ass_3 are satisfied. Then, for any initial point, any $T \geq 1$, and any $s \in (0,1)$, the following bound holds with probability at least $1 - s$: The constant implied by the $\mathcal{O}$ notation depends on the initial point, the constants in Assumptions ass_1--ass_3 (excluding $1/\mu$), and the pa

Theorems & Definitions (50)

  • Theorem 3.1: Non-Asymptotic Sample Complexity
  • Theorem 3.2: Asymptotic Almost Sure Convergence
  • Remark 1
  • Theorem 3.3: Asymptotic $L_1$-Convergence
  • Remark 4.1
  • Lemma 4.1: Approximate Descent Inequality
  • proof
  • proof
  • proof
  • proof
  • ...and 40 more