Overview
This research investigates the convergence of adaptive gradient methods, specifically AdaGrad, in the presence of heavy-tailed gradient noise during non-convex optimization. Adaptive gradient methods, which include optimizers like Adam and AdamW, are widely used in machine learning. The study focuses on understanding if these methods can converge under heavy-tailed noise without requiring additional algorithmic modifications such as gradient clipping or normalization.
Research Context
Modern machine learning tasks frequently encounter heavy-tailed gradient noise during optimization. To address this, current approaches often involve new mechanisms like gradient clipping and gradient normalization to ensure the convergence of first-order algorithms. However, adaptive gradient methods often demonstrate strong performance even without these extra operations. This observation prompts the question of whether adaptive gradient methods can intrinsically converge under heavy-tailed noise. This work specifically examines AdaGrad, which is considered the origin of adaptive gradient methods, as a case study to explore this question.
Approach
The research analyzes the theoretical convergence properties of AdaGrad in a non-convex optimization setting under heavy-tailed noise. It focuses on the tail index $p$ of the noise distribution. The study aims to derive a provable convergence rate for AdaGrad under these conditions. Additionally, it develops an algorithm-dependent lower bound to assess the attainability of existing minimax rates. The work also considers AdaGrad-Norm, a variant of AdaGrad used in theoretical studies, investigating its convergence properties under similar conditions.
Findings
- The research provides the first provable convergence rate for AdaGrad in non-convex optimization when the tail index $p$ satisfies $4/3 \leq p \leq 2$.
- This derived convergence rate is adaptive to the tail index $p$, meaning it does not require prior knowledge of $p$.
- An algorithm-dependent lower bound was developed, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by AdaGrad.
- For AdaGrad-Norm, a popular theoretical variant, an improved convergence rate is shown. This improved rate holds for any $1 \leq p \leq 2$ under an additional mild assumption.
Why This Matters
The findings contribute to a theoretical understanding of adaptive gradient methods, specifically AdaGrad, when operating in the challenging environment of heavy-tailed noise. The established convergence rates, particularly their adaptivity to the tail index, offer insights into the inherent robustness of these optimizers. The identification of a lower bound for AdaGrad's performance in this context also helps to delineate its capabilities relative to theoretical minimax rates.