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.

翻译：我们提出AdaSub算法，这是一种在低维子空间中基于二阶信息计算搜索方向的随机优化方法，其子空间由当前与历史信息自适应定义。与一阶方法相比，二阶方法具有更优的收敛特性，但每次迭代需计算海森矩阵导致计算成本过高，使其难以实际应用。为解决此问题，我们的方法可通过选择搜索子空间维度来管理计算开销与算法效率。代码已在GitHub上开源，初步数值实验表明，在达到指定精度所需的时间与迭代次数方面，AdaSub优于主流随机优化器。