Trust-region algorithms: probabilistic complexity and intrinsic noise with applications to subsampling techniques

Bellavia S., Gurioli G., Morini B., Toint Ph. L.

Modeling and Simulation  65K05  Management Science and Operations Research  Computational Mathematics  Control and Optimization  FOS: Mathematics  90C26  G.1.6  65C50  Trust-region methods  Evaluation complexity  Probabilistic analysis  Subsampling methods  Inexact functions and derivatives  Finite-sum optimization  Optimization and Control (math.OC)  Mathematics - Optimization and Control  F.2.1 

A trust-region algorithm is presented for finding approximate minimizers of smooth unconstrained functions whose values and derivatives are subject to random noise. It is shown that, under suitable probabilistic assumptions, the new method finds (in expectation) an epsilon-approximate minimizer of arbitrary order q >= 1 in at most O(epsilon(-(q+1))) inexact evaluations of the function and its derivatives, providing the first such result for general optimality orders. The impact of intrinsic noise limiting the validity of the assumptions is also discussed and it is shown that difficulties are unlikely to occur in the first-order version of the algorithm for sufficiently large gradients. Conversely, should these assumptions fail for specific realizations, then "degraded" optimality guarantees are shown to hold when failure occurs. These conclusions are then discussed and illustrated in the context of subsampling methods for finite-sum optimization. (C) 2022 The Author(s). Published by Elsevier Ltd on behalf of Association of European Operational Research Societies (EURO).

Source: EURO journal on computational optimization (Print) 10 (2022). doi:10.1016/j.ejco.2022.100043

Publisher: Springer, Heidelberg, Germania

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