Towards Understanding Adam Convergence on Highly Degenerate Polynomials
This work provides theoretical insights into Adam's auto-convergence properties for a specific class of functions, which is incremental but clarifies its advantages in optimization for machine learning practitioners.
The paper tackled the problem of understanding Adam's convergence on highly degenerate polynomials, showing that Adam achieves local linear convergence without external schedulers, significantly outperforming Gradient Descent and Momentum which have sub-linear convergence.
Adam is a widely used optimization algorithm in deep learning, yet the specific class of objective functions where it exhibits inherent advantages remains underexplored. Unlike prior studies requiring external schedulers and $β_2$ near 1 for convergence, this work investigates the "natural" auto-convergence properties of Adam. We identify a class of highly degenerate polynomials where Adam converges automatically without additional schedulers. Specifically, we derive theoretical conditions for local asymptotic stability on degenerate polynomials and demonstrate strong alignment between theoretical bounds and experimental results. We prove that Adam achieves local linear convergence on these degenerate functions, significantly outperforming the sub-linear convergence of Gradient Descent and Momentum. This acceleration stems from a decoupling mechanism between the second moment $v_t$ and squared gradient $g_t^2$, which exponentially amplifies the effective learning rate. Finally, we characterize Adam's hyperparameter phase diagram, identifying three distinct behavioral regimes: stable convergence, spikes, and SignGD-like oscillation.