Abstract
We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed scheme is a stochastic zeroth-order optimization algorithm which demands only the capability to evaluate small subsets of components of the cost function. It can be depicted as a bank of samplers that generate particle approximations of several sequences of probability measures. These measures are constructed in such a way that they have associated probability density functions whose global maxima coincide with the global minima of the original cost function. The algorithm selects the best performing sampler and uses it to approximate a global minimum of the cost function. We prove analytically that the resulting estimator converges to a global minimum of the cost function almost surely and provide explicit convergence rates in terms of the number of generated Monte Carlo samples and the dimension of the search space. We show, by way of numerical examples, that the algorithm can tackle cost functions with multiple minima or with broad “flat” regions which are hard to minimize using gradient-based techniques.
Highlights
In signal processing and machine learning, optimization problems of the form n min θ ∈Θ f (θ ) = i =1 fi (θ ), (1.1)where Θ ⊂ Rd is the d-dimensional compact search space, have attracted significant attention in recent years for problems where n is very large
We have proposed a parallel sequential Monte Carlo optimization algorithm which does not require the computation of gradients and, can be applied to the minimization of challenging cost functions, e.g., with multiple global minima or with broad “flat” regions
We have provided a detailed analysis of the proposed scheme
Summary
In signal processing and machine learning, optimization problems of the form n min θ ∈Θ f (θ ). We analytically prove that the estimate provided by each sampler converges almost surely to a global minimum of the cost function and provide explicit convergence rates in terms of the number of Monte Carlo samples generated by the algorithm This type of analysis goes beyond standard results for particle filters: It tackles the problem of stochastic optimization directly and it yields stronger theoretical guarantees compared to other stochastic optimization methods in the literature. The main contribution of this paper includes the theoretical analysis of the proposed PSMCO scheme and its numerical demonstration on three problems where classical stochastic optimization methods (especially gradient-based algorithms) struggle to perform. The notation x indicates the floor function for a real number x, which returns the biggest integer k ≤ x
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