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In this paper, we study a generic direct-search algorithm in which the polling directions are defined using random subspaces. Complexity guarantees for such an approach are derived thanks to probabilistic properties related to both the subspaces and the directions used within these subspaces. Our analysis crucially extends previous deterministic and probabilistic arguments by relaxing the need for directions to be deterministically bounded in norm. As a result, our approach encompasses a wide range of new optimal polling strategies that can be characterized using our subspace and direction properties. By leveraging results on random subspace embeddings and sketching matrices, we show that better complexity bounds are obtained for randomized instances of our framework. A numerical investigation confirms the benefit of randomization, particularly when done in subspaces, when solving problems of moderately large dimension.

Many machine learning solutions are framed as optimization problems which rely on good hyperparameters. Algorithms for tuning these hyperparameters usually assume access to exact solutions to the underlying learning problem, which is typically not practical. Here, we apply a recent dynamic accuracy derivative-free optimization method to hyperparameter tuning, which allows inexact evaluations of the learning problem while retaining convergence guarantees. We test the method on the problem of learning elastic net weights for a logistic classifier, and demonstrate its robustness and efficiency compared to a fixed accuracy approach. This demonstrates a promising approach for hyperparameter tuning, with both convergence guarantees and practical performance., Comment: Accepted to the 12th OPT Workshop on Optimization for Machine Learning at NeurIPS 2020

We present DFO-GN, a derivative-free version of the Gauss-Newton method for solving nonlinear least-squares problems. As is common in derivative-free optimization, DFO-GN uses interpolation of function values to build a model of the objective, which is then used within a trust-region framework to give a globally-convergent algorithm requiring $O(\epsilon^{-2})$ iterations to reach approximate first-order criticality within tolerance $\epsilon$. This algorithm is a simplification of the method from [H. Zhang, A. R. Conn, and K. Scheinberg, A Derivative-Free Algorithm for Least-Squares Minimization, SIAM J. Optim., 20 (2010), pp. 3555-3576], where we replace quadratic models for each residual with linear models. We demonstrate that DFO-GN performs comparably to the method of Zhang et al. in terms of objective evaluations, as well as having a substantially faster runtime and improved scalability.