The theoretical understanding of AdamW's optimization dynamics can be leveraged to design even more effective optimization algorithms for deep learning by incorporating similar principles of implicit bias and regularization. By studying how AdamW optimizes the ℓ∞ norm-constrained problem and converges to KKT points, researchers can develop new optimization algorithms that explicitly target specific properties of the solution space.
One approach could be to design adaptive algorithms that incorporate constraints or regularization terms based on different norms, similar to how AdamW implicitly regularizes towards solutions with bounded ℓ∞ norms. By tailoring the optimization process to encourage desirable properties in the learned models, such as sparsity, robustness, or stability, new algorithms can be developed to address specific challenges in deep learning tasks.
Additionally, the insights from the analysis of AdamW's optimization dynamics can guide the development of hybrid optimization algorithms that combine the strengths of different approaches, such as adaptive methods and normalized steepest descent with weight decay. By leveraging the theoretical understanding of optimization biases and regularization effects, researchers can innovate and create more efficient and effective optimization algorithms for training deep learning models.