Frugality in second-order optimization: floating-point approximations for Newton's method

Minimizing loss functions is central to machine-learning training. Although first-order methods dominate practical applications, higher-order techniques such as Newton's method can deliver greater accuracy and faster convergence, yet are often avoided due to their computational cost. This work analyzes the impact of finite-precision arithmetic on Newton steps and establishes a convergence theorem for mixed-precision Newton optimizers, including "quasi" and "inexact" variants. The theorem provides not only convergence guarantees but also a priori estimates of the achievable solution accuracy. Empirical evaluations on standard regression benchmarks demonstrate that the proposed methods outperform Adam on the Australian and MUSH datasets. The second part of the manuscript introduces GN_k, a generalized Gauss-Newton method that enables partial computation of second-order derivatives. GN_k attains performance comparable to full Newton's method on regression tasks while requiring significantly fewer derivative evaluations.