Stochastic Gradient Descent in the Viewpoint of Graduated Optimization

Stochastic gradient descent (SGD) method is popular for solving non-convex optimization problems in machine learning. This work investigates SGD from a viewpoint of graduated optimization, which is a widely applied approach for non-convex optimization problems. Instead of the actual optimization problem, a series of smoothed optimization problems that can be achieved in various ways are solved in the graduated optimization approach. In this work, a formal formulation of the graduated optimization is provided based on the nonnegative approximate identity, which generalizes the idea of Gaussian smoothing. Also, an asymptotic convergence result is achieved with the techniques in variational analysis. Then, we show that the traditional SGD method can be applied to solve the smoothed optimization problem. The Monte Carlo integration is used to achieve the gradient in the smoothed problem, which may be consistent with distributed computing schemes in real-life applications. From the assumptions on the actual optimization problem, the convergence results of SGD for the smoothed problem can be derived straightforwardly. Numerical examples show evidence that the graduated optimization approach may provide more accurate training results in certain cases.