Stochastic model-based minimization of weakly convex functions
This work addresses optimization challenges in machine learning and related fields by offering theoretical guarantees for stochastic algorithms, though it appears incremental as it builds on existing principles like the Moreau envelope.
The paper tackles the problem of minimizing weakly convex functions using stochastic model-based algorithms, establishing a convergence rate of O(k^{-1/4}) for stationarity measures and providing first complexity guarantees for methods like stochastic proximal point and proximal subgradient.
We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate $O(k^{-1/4})$. As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.