We investigate stochastic gradient methods and stochastic counterparts of the Barzilai-Borwein steplengths and their application to finite-sum minimization problems. Our proposal is based on the Trust-Region-ish (TRish) framework introduced in [F. E. Curtis, K. Scheinberg, R. Shi, A stochastic trust region algorithm based on careful step normalization, Informs Journal on Optimization, 1, 2019]. The new framework, named TRishBB, aims at enhancing the performance of TRish and at reducing the computational cost of the second-order TRish variant. We propose three different methods belonging to the TRishBB framework and present the convergence analysis for possibly nonconvex objective functions, considering biased and unbiased gradient approximations. Our analysis requires neither diminishing step-sizes nor full gradient evaluation. The numerical experiments in machine learning applications demonstrate the effectiveness of applying the Barzilai-Borwein steplength with stochastic gradients and show improved testing accuracy compared to the TRish method.

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