  We introduce AdaSub, a stochastic optimization algorithm that computes a
search direction based on second-order information in a low-dimensional
subspace that is defined adaptively based on available current and past
information. Compared to first-order methods, second-order methods exhibit
better convergence characteristics, but the need to compute the Hessian matrix
at each iteration results in excessive computational expenses, making them
impractical. To address this issue, our approach enables the management of
computational expenses and algorithm efficiency by enabling the selection of
the subspace dimension for the search. Our code is freely available on GitHub,
and our preliminary numerical results demonstrate that AdaSub surpasses popular
stochastic optimizers in terms of time and number of iterations required to
reach a given accuracy.
