Source-linked AI summary
An improvement of the convergence proof of the ADAM-Optimizer
Sebastian Bock, Josef Goppold, Martin Weiß
TL;DR
ADAM is widely used for neural-network optimization, but its published convergence proof contains mistakes. This paper improves parts of that proof and derives a convergence estimate under boundedness assumptions, while leaving a key lemma as a conjecture.
Problem
The paper addresses mistakes in ADAM's existing convergence proof, whose correctness matters for understanding the optimizer's convergence behavior.
Method
The paper develops an improved convergence proof using convexity lemmas, an error-sum decomposition, and bounded-gradient and bounded-iterate assumptions.
Results
Under bounded-gradient and bounded-iterate assumptions, the paper states a convergence estimation for ADAM and identifies convergence speed O(...).
Takeaways & Limitations
The paper improves understanding of ADAM's convergence behavior but leaves a conjecture unresolved for future work.
Takeaways & Limitations
A lemma used in the convergence proof could not be proven and is treated as a conjecture.
Abstract
from arXiv · showhide
A common way to train neural networks is the Backpropagation. This algorithm includes a gradient descent method, which needs an adaptive step size. In the area of neural networks, the ADAM-Optimizer is one of the most popular adaptive step size methods. It was invented in \cite{Kingma.2015} by Kingma and Ba. The $5865$ citations in only three years shows additionally the importance of the given paper. We discovered that the given convergence proof of the optimizer contains some mistakes, so that the proof will be wrong. In this paper we give an improvement to the convergence proof of the ADAM-Optimizer.
1 INTRODUCTION
The paper presents ADAM as a widely used adaptive-step optimizer for neural networks, while identifying errors in its published convergence proof and proposing partial improvements plus a conjecture toward completion.
- Motivation: ADAM is a popular adaptive step-size method used in neural-network frameworks and associated with faster convergence in experiments.The paper discusses its use with backpropagation and reports Kingma and Ba's experimental comparison.
- Evidence context: Figure 1 compares different optimizers while training multilayer neural networks on MNIST images.
- Research gap: The convergence proof for ADAM contains mistakes that make the original proof incorrect.
- Contribution: The paper improves parts of the convergence proof and formulates a conjecture that would complete it.
2 NEURAL NETWORKS
Neural-network training represents neuron parameters as a weight vector and uses an error function with backpropagation to optimize those weights.
- Network representation: Neural-network weights are stored in a vector w ∈ R^n and modified during learning to obtain the required network behavior.
- Objective: The error function e(w) compares the input label with the network output.
- Optimization: Backpropagation uses gradient descent to minimize e(w), with ADAM available as the optimizer.
3 METHOD OF MOMENTS - ADAM
The paper introduces the method of moments and then specifies ADAM as an adaptive optimizer using moment estimates, decay parameters, and a weight-update rule.
- 3.1 Method of moments: The method of moments uses an adaptive step size and defines a rule for changing the network weight vector.
- 3.1 Method of moments: The weight-change rule updates w(t) using the prior weight vector, step size η, and the current training time t.
- 3.1 Method of moments: The method of moments adds a fraction of the previous weight change to the gradient-descent step.
- 3.1 Method of moments: Convergence of the method of moments requires the decay rate α to lie between 0 and 1.
- 3.2 ADAM-Optimizer: ADAM maintains first- and second-moment estimates using decay rates β1,t and β2, with initial moment vectors set to zero.
- 3.2 ADAM-Optimizer: The paper states that ADAM experimentally converges faster than other optimizers for multilayer and convolutional neural networks, while noting errors in its original convergence proof.
4 CONVERGENCE PROOF
The paper improves ADAM’s convergence proof under bounded-gradient and bounded-iterate-difference assumptions, while relying on an unproven conjecture. It derives a convergence estimate and concludes an O(·) convergence speed for the ADAM method.
- Proof setup: Lemma 4.1 characterizes convex differentiable functions through a gradient inequality on a convex set.The proof uses this lemma as its entry point for analyzing the error function.
- Error-sum analysis: The proof defines an error sum measuring the difference between the minimum and the current value of e(w(t)), then separates its bound into five terms.The terms are handled individually and combined after summing over gradient coordinates and time stamps.
- Convergence result: The proof is completed by showing that the error-sum term R(T) converges with respect to T and combining all terms.This step is presented as the route to finishing the convergence proof.
- Assumptions: The analysis assumes bounded gradients, bounded differences between iterates, β1, β2 ∈ (0, 1), and γ := β2.The theorem also uses time-varying β1,t := β1λ^(t−1) with λ ∈ (0, 1).
- Error-sum analysis: Terms 4 and 5 use conjecture 4.2 before being combined with the other terms to obtain the theorem’s assertion.The paper explicitly states that the proof of the referenced lemma could not be completed and treats it as a conjecture.
- Convergence result: Theorem 4.4 provides an ADAM convergence estimate under the stated boundedness assumptions, and Corollary 4.5 follows by dividing the inequality by T.The paper states that this establishes convergence speed O(·) for the ADAM method.
5 CONCLUSION AND OUTLOOK
The paper presents an improvement to ADAM’s convergence proof while acknowledging that at least one conjecture remains unresolved. It identifies future proof work and possible opportunities to improve ADAM’s speed and efficiency.
- Conclusion and outlook: The paper improves ADAM’s convergence proof but leaves at least one conjecture unresolved.The authors hope future work will prove the conjecture so ADAM can be used without concerns.
- Conclusion and outlook: Future developments may use the proof to identify opportunities for improving ADAM’s speed and efficiency.The paper connects this possibility to reducing learning time, especially for big data.