Implicit Bias of AdamW: $\ell_\infty$ Norm Constrained Optimization (2404.04454v1)

Abstract: Adam with decoupled weight decay, also known as AdamW, is widely acclaimed for its superior performance in LLMing tasks, surpassing Adam with $\ell_2$ regularization in terms of generalization and optimization. However, this advantage is not theoretically well-understood. One challenge here is that though intuitively Adam with $\ell_2$ regularization optimizes the $\ell_2$ regularized loss, it is not clear if AdamW optimizes a specific objective. In this work, we make progress toward understanding the benefit of AdamW by showing that it implicitly performs constrained optimization. More concretely, we show in the full-batch setting, if AdamW converges with any non-increasing learning rate schedule whose partial sum diverges, it must converge to a KKT point of the original loss under the constraint that the $\ell_\infty$ norm of the parameter is bounded by the inverse of the weight decay factor. This result is built on the observation that Adam can be viewed as a smoothed version of SignGD, which is the normalized steepest descent with respect to $\ell_\infty$ norm, and a surprising connection between normalized steepest descent with weight decay and Frank-Wolfe.

• The paper establishes that AdamW implicitly enforces an ℓ∞ norm constraint by converging to a KKT point under a non-increasing learning rate.
• It reveals that AdamW operates similarly to a smoothed SignGD, aligning its dynamics with normalized steepest descent and Frank-Wolfe principles.
• The study provides tight convergence bounds and practical insights for optimizing deep learning models with weight decay regularization.

Overview of AdamW's Implicit Bias in Constrained Optimization

The research paper "Implicit Bias of AdamW: $\ell_\infty$ Norm Constrained Optimization" by Shuo Xie and Zhiyuan Li provides a detailed theoretical exploration into the implicit bias associated with the AdamW optimizer, focusing on its dynamical behavior. AdamW is recognized for its exemplary performance over Adam with $\ell_2$ regularization, especially in the domain of LLMing. This paper endeavors to address the gap in theoretical understanding by establishing that AdamW implicitly enforces a $\ell_\infty$ norm constraint on optimization.

Main Contributions

1. Implicit Constrained Optimization: The authors establish that AdamW, when converging under a non-increasing learning rate whose partial sum diverges, reaches a KKT point of the original loss subject to the constraint that the $\ell_\infty$ norm of the parameters is bounded by the inverse of the weight decay factor. This assertion aligns AdamW with constrained optimization principles.
2. Relationship with SignGD: The paper uncovers the link between Adam and SignGD, demonstrating that Adam can be interpreted as a smoothed version of SignGD, which conducts normalized steepest descent with regard to the $\ell_\infty$ norm. This connects the working of Adam to known theoretical frameworks of steepest descent and Frank-Wolfe algorithms, elaborating on the geometric benefits of $\ell_\infty$ over other norm constraints.
3. Robust Theoretical Results: The paper delivers a robust theoretical framework, including a lemma providing a convergence bound for normalized steepest descent with weight decay, showcasing how convex problems are resolved within these constrained boundaries.
4. Tight Bound on Update Size: A novel and tight upper bound on Adam's average update size is introduced, applicable to non-deterministic settings as well, which contributes significantly to understanding the optimizer's dynamics, offering valuable insights for practical applications.
5. Experiments Supporting Theoretical Claims: Through empirical exploration, the paper underscores its theoretical insights, demonstrating the boundaries within which AdamW converges in practical scenarios, including LLMing tasks and synthetic experiments illustrating norm impacts.

Theoretical and Practical Implications

Theoretically, the implication of this work lies in its ability to cast light on the implicit bias of state-of-the-art optimization algorithms like AdamW. It links the bias to constrained optimization problems, providing a more comprehensive understanding of optimization process nuances in the deep learning landscape. By leveraging properties like normalized steepest descent with $\ell_\infty$ norm, this paper suggests latent geometric advantages that could reshape perspectives on model training strategies.

Practically, the work's conclusions offer guidance for hyperparameter tuning and algorithm selection based on underlying norm constraints applicable in extensive deep learning applications. The insights provided could refine model training approaches, particularly for architectures and tasks where parameter constraints inherently impact performance outcomes.

Speculation on Future Developments

Looking ahead, this paper's conclusions suggest further exploration in several directions. Firstly, it opens avenues for examining the implications of different norm constraints in varied deep learning architectures and tasks, potentially driving algorithmic innovations. Moreover, the distinct dynamics between stochastic and deterministic settings remain a fertile ground for future research, particularly in understanding optimizer performance amidst noisy gradients and large-scale models. Lastly, the potential for generalizing this approach to other adaptive methods (including those with higher-order moments) could yield significant advancements in the understanding and application of optimization in AI.

In summary, this paper constitutes a substantial theoretical advancement in comprehending AdamW's implicit bias, linking it to constrained optimization and offering a nuanced perspective on the underlying principles guiding modern machine learning optimizers.