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A trust-region algorithm is presented for finding approximate minimizers ofsmooth unconstrained functions whose values and derivatives are subject torandom noise. It is shown that, under suitable probabilistic assumptions, thenew method finds (in expectation) an $\epsilon$-approximate minimizer ofarbitrary order $ q \geq 1$ in at most $\mathcal{O}(\epsilon^{-(q+1)})$ inexactevaluations of the function and its derivatives, providing the first suchresult for general optimality orders. The impact of intrinsic noise limitingthe validity of the assumptions is also discussed and it is shown thatdifficulties are unlikely to occur in the first-order version of the algorithmfor sufficiently large gradients. Conversely, should these assumptions fail forspecific realizations, then "degraded" optimality guarantees are shown to holdwhen failure occurs. These conclusions are then discussed and illustrated inthe context of subsampling methods for finite-sum optimization.

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