Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence

In this paper, a new theory is developed for first-order stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal solutions. In particular, if the objective function $F(\w)$ in the $\epsilon$-sublevel set grows as fast as $\|\w - \w_*\|_2^{1/\theta}$, where $\w_*$ represents the closest optimal solution to $\w$ and $\theta\in(0,1]$ quantifies the local growth rate, the iteration complexity of first-order stochastic optimization for achieving an $\epsilon$-optimal solution can be $\widetilde O(1/\epsilon^{2(1-\theta)})$, which is {\it optimal at most} up to a logarithmic factor. This result is fundamentally better in contrast with the previous works that either assume a global growth condition in the entire domain or achieve a local faster convergence under the local faster growth condition. To achieve the faster global convergence, we develop two different {\bf accelerated stochastic subgradient} methods by iteratively solving the original problem approximately in a local region around a historical solution with the size of the local region gradually decreasing as the solution approaches the optimal set. Besides the theoretical improvements, this work also include new contributions towards making the proposed algorithms practical: (i) we present practical variants of accelerated stochastic subgradient methods that can run without the knowledge of multiplicative growth constant and even the growth rate $\theta$; (ii) we consider a broad family of problems in machine learning to demonstrate that the proposed algorithms enjoy faster convergence than traditional stochastic subgradient method. For example, when applied to the $\ell_1$ regularized empirical polyhedral loss minimization (e.g., hinge loss, absolute loss), the proposed stochastic methods have a logarithmic iteration complexity.