A Deterministic Gradient-Based Approach to Avoid Saddle Points
This addresses a key bottleneck in training machine learning models efficiently, though it is incremental as it builds on an existing method.
The paper tackles the problem of gradient descent converging to saddle points in loss functions by proposing a modified Laplacian smoothing gradient descent (mLSGD) method, showing that its attraction region dimension is floor((n-1)/2) compared to n-1 for gradient descent, which reduces the risk of getting stuck.
Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning models efficiently. First-order methods such as gradient descent are usually the methods of choice for training machine learning models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent [Osher et al., arXiv:1806.06317], called modified Laplacian smoothing gradient descent (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region's dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is floor((n-1)/2), and hence it is significantly smaller than that of the gradient descent whose dimension is n-1.