  Gradient dominance property is a condition weaker than strong convexity, yet
sufficiently ensures global convergence even in non-convex optimization. This
property finds wide applications in machine learning, reinforcement learning
(RL), and operations management. In this paper, we propose the stochastic
homogeneous second-order descent method (SHSODM) for stochastic functions
enjoying gradient dominance property based on a recently proposed
homogenization approach. Theoretically, we provide its sample complexity
analysis, and further present an enhanced result by incorporating variance
reduction techniques. Our findings show that SHSODM matches the best-known
sample complexity achieved by other second-order methods for gradient-dominated
stochastic optimization but without cubic regularization. Empirically, since
the homogenization approach only relies on solving extremal eigenvector problem
at each iteration instead of Newton-type system, our methods gain the advantage
of cheaper computational cost and robustness in ill-conditioned problems.
Numerical experiments on several RL tasks demonstrate the better performance of
SHSODM compared to other off-the-shelf methods.
