Sub-sampled Trust-Region Methods with Deterministic Worst-Case Complexity Guarantees

In this paper, we develop and analyze sub-sampled trust-region methods for solving finite-sum optimization problems. These methods employ subsampling strategies to approximate the gradient and Hessian of the objective function, significantly reducing the overall computational cost. We propose a novel adaptive procedure for deterministically adjusting the sample size used for gradient (or gradient and Hessian) approximations. Furthermore, we establish worst-case iteration complexity bounds for obtaining approximate stationary points. More specifically, for a given εg, εH∈ (0,1), it is shown that an εg-approximate first-order stationary point is reached in at most O(εg⁻² ) iterations, whereas an (εg,εH)-approximate second-order stationary point is reached in at most O(max\εg⁻²εH⁻¹,εH⁻³\) iterations. Finally, numerical experiments illustrate the effectiveness of our new subsampling technique.