Zeroth-Order Regularized Optimization (ZORO): Approximately Sparse Gradients and Adaptive Sampling

We consider the problem of minimizing a high-dimensional objective function, which may include a regularization term, using only noisy evaluations of the function. Such optimization is also called derivative-free, zeroth-order, or black-box optimization. We propose a new Zeroth-Order Regularized Optimization method, dubbed ZORO. When the underlying gradient is approximately sparse at an iterate, ZORO needs very few objective function evaluations to obtain a new iterate that decreases the objective function. We achieve this with an adaptive, randomized gradient estimator, followed by an inexact proximal-gradient scheme. Under a novel approximately sparse gradient assumption and various different convex settings, we show the (theoretical and empirical) convergence rate of ZORO is only logarithmically dependent on the problem dimension. Numerical experiments show ZORO outperforms existing methods on both synthetic and real datasets.