Adaptive Momentum Estimation (Adam), which combines Adaptive Learning Rate and Momentum, is the most popular stochastic optimizer for accelerating training of deep neural networks. But Adam often generalizes significantly worse than Stochastic Gradient Descent (SGD). It is still mathematically unclear how Adaptive Learning Rate and Momentum affect saddle-point escaping and minima selection. Based on the diffusion theoretical framework, we separate the effects of Adaptive Learning Rate and Momentum on saddle-point escaping and minima selection. We find that SGD escapes saddle points very slowly along the directions of small-magnitude eigenvalues of the Hessian. We prove that Adaptive Learning Rate can make learning dynamics near saddle points approximately Hessian-independent, but cannot select flat minima as SGD does. In contrast, Momentum provides a momentum drift effect to help passing through saddle points, and almost does not affect flat minima selection. This mathematically explains why SGD (with Momentum) generalizes better, while Adam generalizes worse but converges faster. Motivated by the diffusion theoretical analysis, we design a novel adaptive optimizer named Adaptive Inertia Estimation (Adai), which uses parameter-wise adaptive inertia to accelerate training and provably favors flat minima as much as SGD. Our real-world experiments demonstrate that Adai can converge similarly fast to Adam, but generalize significantly better. Adai even generalizes better than SGD.

通过研究神经网络中的优化算法，提出了一个名为“自适应惯性”的新方法，能够更好地训练神经网络并提高泛化性能。

自适应惯性：解离自适应学习率和动量的影响