Dual stochastic natural gradient descent

Abstract The multinomial logistic regression (MLR) model is widely used in statistics and machine learning. On the one hand, stochastic gradient descent (SGD) is the most common approach for determining the parameters of a such model in big data scenarios, due to its simplicity and low computational complexity property. Furthermore, SGD has proven convergence under reasonable conditions. However, SGD has slow sub-linear rates of convergence and it often reduces convergence speed due to the plateau phenomenon. On the other hand, stochastic natural gradient descent (SNGD), proposed by Amari, is a manifold optimization method shown to be Fisher efficient when it converges, but its convergence properties remain unproven and it is often computationally prohibitive for models with a large number of parameters. Here, we propose dual stochastic natural gradient descent (DSNGD), a stochastic optimization method for MLR based on manifold optimization concepts. In the discrete scenario, DSNGD (i) has linear per-iteration computational complexity in the number of parameters, and (ii) is proven to converge. To achieve (i) we leverage the dual flatness of the family of joint distributions for MLR to simplify computations. To ensure (ii) DSNGD builds on the foundational ideas of convergent stochastic natural gradient descent (CSNGD), a variant of SNGD with guaranteed convergence, using an independent sequence to construct a bounded approximation of the natural gradient. By generalizing a result from Sunehag et al., we prove that DSNGD converges in the discrete case and maintains linear computational complexity per iteration. Beyond its convergence property and linear computational complexity, DSNGD empirically demonstrates fast convergence comparable to SNGD, improves upon SGD performance, and exhibits stability where SNGD does not.