arXiv subjects

Oscar Smee

Publications and source records attributed to Oscar Smee.

3 recordsLinked to original sources

arXiv2026

Primal-Dual Inexact Newton-MR for Nonconvex Optimization with Equality Constraints

Optimization problems with nonlinear equality constraints arise throughout science, engineering, and increasingly in machine learning. Prominent methods for solving such problems include sequential quadratic programming and, more broadly, primal-dual Newton methods. Classical analyses of these methods typically rely on strong assumptions, perhaps most notably positive definiteness of the Lagrangian Hessian on the null space of the constraint Jacobian. In practice, this assumption often necessitates strong regularization or the use of a positive definite Hessian surrogate. Moreover, in large-scale settings, solving the primal-dual Newton subproblem exactly is often computationally infeasible. To address these issues, we propose an inexact primal-dual Newton method with an inner solver based on the conjugate residual (CR) method. Exploiting recently established properties of CR, including negative-curvature detection, iterate monotonicity, and descent guarantees, our method handles indefiniteness in the subproblem directly as it arises. Our method thereby avoids detrimental regularization of the Lagrangian Hessian while naturally accommodating inexact solves. We establish worst-case global convergence guarantees and demonstrate strong empirical performance on large-scale, nonconvex problems.

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arXiv2025

First-ish Order Methods: Hessian-aware Scalings of Gradient Descent

Gradient descent is the primary workhorse for optimizing large-scale problems in machine learning. However, its performance is highly sensitive to the choice of the learning rate. A key limitation of gradient descent is its lack of natural scaling, which often necessitates expensive line searches or heuristic tuning to determine an appropriate step size. In this paper, we address this limitation by incorporating Hessian information to scale the gradient direction. By accounting for the curvature of the function along the gradient, our adaptive, Hessian-aware scaling method ensures a local unit step size guarantee, even in nonconvex settings. Near a local minimum that satisfies the second-order sufficient conditions, our approach achieves linear convergence with a unit step size. We show that our method converges globally under a significantly weaker version of the standard Lipschitz gradient smoothness assumption. Even when Hessian information is inexact, the local unit step size guarantee and global convergence properties remain valid under mild conditions. Finally, we validate our theoretical results empirically on a range of convex and nonconvex machine learning tasks, showcasing the effectiveness of the approach.

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arXiv2024

Inexact Newton-type Methods for Optimisation with Nonnegativity Constraints

We consider solving large scale nonconvex optimisation problems with nonnegativity constraints. Such problems arise frequently in machine learning, such as nonnegative least-squares, nonnegative matrix factorisation, as well as problems with sparsity-inducing regularisation. In such settings, first-order methods, despite their simplicity, can be prohibitively slow on ill-conditioned problems or become trapped near saddle regions, while most second-order alternatives involve non-trivially challenging subproblems. The two-metric projection framework, initially proposed by Bertsekas (1982), alleviates these issues and achieves the best of both worlds by combining projected gradient steps at the boundary of the feasible region with Newton steps in the interior in such a way that feasibility can be maintained by simple projection onto the nonnegative orthant. We develop extensions of the two-metric projection framework, which by inexactly solving the subproblems as well as employing non-positive curvature directions, are suitable for large scale and nonconvex settings. We obtain state-of-the-art convergence rates for various classes of non-convex problems and demonstrate competitive practical performance on a variety of problems.

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