Accelerate Stochastic Subgradient Method by Leveraging Local Growth Condition
This work provides a theoretical and practical advancement for researchers and practitioners in machine learning by accelerating stochastic optimization methods, though it is incremental as it builds on existing subgradient methods with new theoretical insights and practical variants.
The paper tackles the problem of improving convergence rates in first-order stochastic convex optimization by showing that the global convergence rate can be quantified by a local growth condition of the objective function, achieving an optimal iteration complexity of $\widetilde O(1/ε^{2(1-θ)})$ for an $ε$-optimal solution, where $θ$ is a growth rate parameter.
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 $F(\mathbf w)$ in the $ε$-sublevel set grows as fast as $\|\mathbf w - \mathbf w_*\|_2^{1/θ}$, where $\mathbf w_*$ represents the closest optimal solution to $\mathbf w$ and $θ\in(0,1]$ quantifies the local growth rate, the iteration complexity of first-order stochastic optimization for achieving an $ε$-optimal solution can be $\widetilde O(1/ε^{2(1-θ)})$, which is optimal at most up to a logarithmic factor. To achieve the faster global convergence, we develop two different 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 includes 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 $θ$; (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. We also characterize the complexity of the proposed algorithms for ensuring the gradient is small without the smoothness assumption.