Mixed Simultaneous Perturbation Stochastic Approximation for Gradient-Free Optimization with Noisy Measurements

This paper considers the minimization of a loss function, whose variables contain both continuous and discrete components. When only noisy measurements of the loss function are available, we propose the mixed simultaneous perturbation stochastic approximation (MSPSA) algorithm that uses the pseudo-gradient information to iteratively update the estimates while maintaining the mixed-variables-type constraints. Our work unifies two practical algorithms: the simultaneous perturbation stochastic approximation (SPSA) and discrete simultaneous perturbation stochastic approximation (DSPSA). MSPSA algorithm can be applied to hyper-parameter estimation, optimal experimental design, and many real-world applications. We establish the almost sure convergence for MSPSA under some mild conditions. Numerical results illustrate this algorithm's practicality and advantages.