Faster doubly stochastic functional gradient by gradient preconditioning for scalable kernel methods

Abstract

The doubly stochastic functional gradient descent algorithm (DSG) that is memory friendly and computationally efficient can effectively scale up kernel methods. However, in solving the highly ill-conditioned large-scale nonlinear machine learning problem, the convergence speed of DSG is quite slow. This is because the condition number of the Hessian matrix of this problem is quite large, which will make stochastic gradient methods converge very slowly. Fortunately, gradient preconditioning is a well-established technique in optimization aiming to reduce the condition number. Therefore, we propose a preconditioned doubly stochastic functional gradient descent algorithm (P-DSG) by combining DSG with gradient preconditioning. P-DSG first uses the gradient preconditioning to adaptively scale the individual components of the estimated functional gradient obtained by DSG, and then utilizes the preconditioned functional gradient as the descent direction in each iteration. Theoretically, an appropriate preconditioner is always the inverse of the Hessian matrix at the optimum, which is not easy to get due to its high computation cost. Therefore, we first choose an empirical covariance matrix of random Fourier features to approximate the Hessian matrix, and then perform a low-rank approximation to the empirical covariance matrix. P-DSG has a fast convergence rate $\mathcal {O}(1/t)$ and produces a smaller constant factor in the boundary than that of DSG while remains $\mathcal {O}(t)$ memory friendly and $\mathcal {O}(td)$ computationally efficient. Finally, we test the performance of P-DSG on the kernel ridge regression, kernel support vector machines, and kernel logistic regression, respectively. The experimental results show that P-DSG speeds up convergence and achieves better performance.