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$\ell_0$ constrained optimization is prevalent in machine learning,particularly for high-dimensional problems, because it is a fundamentalapproach to achieve sparse learning. Hard-thresholding gradient descent is adominant technique to solve this problem. However, first-order gradients of theobjective function may be either unavailable or expensive to calculate in a lotof real-world problems, where zeroth-order (ZO) gradients could be a goodsurrogate. Unfortunately, whether ZO gradients can work with thehard-thresholding operator is still an unsolved problem. To solve this puzzle,in this paper, we focus on the $\ell_0$ constrained black-box stochasticoptimization problems, and propose a new stochastic zeroth-order gradienthard-thresholding (SZOHT) algorithm with a general ZO gradient estimatorpowered by a novel random support sampling. We provide the convergence analysisof SZOHT under standard assumptions. Importantly, we reveal a conflict betweenthe deviation of ZO estimators and the expansivity of the hard-thresholdingoperator, and provide a theoretical minimal value of the number of randomdirections in ZO gradients. In addition, we find that the query complexity ofSZOHT is independent or weakly dependent on the dimensionality under differentsettings. Finally, we illustrate the utility of our method on a portfoliooptimization problem as well as black-box adversarial attacks.

Zeroth-Order Hard-Thresholding: Gradient Error vs. Expansivity