Catalyst Acceleration for Gradient-Based Non-Convex Optimization
This provides a method for machine learning and signal processing practitioners to use efficient convex algorithms on non-convex problems without prior convexity knowledge, though it is incremental as it adapts existing methods.
The paper tackles the problem of applying gradient-based convex optimization methods to non-convex functions by introducing a generic scheme that works for weakly convex objectives, achieving stationary points with first-order efficiency and automatic acceleration to near-optimal rates when convexity holds, as demonstrated in experiments on sparse matrix factorization and neural network learning.
We introduce a generic scheme to solve nonconvex optimization problems using gradient-based algorithms originally designed for minimizing convex functions. Even though these methods may originally require convexity to operate, the proposed approach allows one to use them on weakly convex objectives, which covers a large class of non-convex functions typically appearing in machine learning and signal processing. In general, the scheme is guaranteed to produce a stationary point with a worst-case efficiency typical of first-order methods, and when the objective turns out to be convex, it automatically accelerates in the sense of Nesterov and achieves near-optimal convergence rate in function values. These properties are achieved without assuming any knowledge about the convexity of the objective, by automatically adapting to the unknown weak convexity constant. We conclude the paper by showing promising experimental results obtained by applying our approach to incremental algorithms such as SVRG and SAGA for sparse matrix factorization and for learning neural networks.