Complexity of trust-region methods in the presence of unbounded Hessian approximations

TL;DR

This paper extends complexity analysis of trust-region methods to cases with unbounded Hessian approximations, providing new bounds depending on the growth rate of the Hessians, which are relevant for practical quasi-Newton methods.

Contribution

It introduces novel complexity bounds for trust-region methods with unbounded Hessians, covering different growth regimes and improving understanding of convergence behavior.

Findings

Established sharp evaluation complexity bounds for Hessians bounded by a power of successful iterations.

Derived exponential complexity bounds for linearly growing Hessians, improving previous double exponential estimates.

Provided new complexity results for convex objectives under unbounded Hessian assumptions.

Abstract

We extend traditional complexity analyses of trust-region methods for unconstrained, possibly nonconvex, optimization. Whereas most complexity analyses assume uniform boundedness of the model Hessians, we work with potentially unbounded model Hessians. Boundedness is not guaranteed in practical implementations, in particular ones based on quasi-Newton updates such as PSB, BFGS and SR1. We examine two regimes of Hessian growth: one bounded by a power of the number of successful iterations, and one bounded by a power of the number of iterations. This allows us to formalize and address the intuition of Powell [IMA J. Numer. Ana. 30(1):289-301,2010], who studied convergence under a special case of our assumptions, but whose proof contained complexity arguments. Specifically, for \(0 \leq p < 1\), we establish sharp \(O([(1-p)\epsilon^{-2}]^{1/(1-p)})\) evaluation complexity to find an \(\epsilon\)-stationary point when model Hessians are \(O(|\mathcal{S}_{k-1}|^p)\), where \(|\mathcal{S}_{k-1}|\) is the number of iterations where the step was accepted, up to iteration \(k-1\). For \(p = 1\), which is the case studied by Powell, we establish a sharp \(O(\exp(c_1\epsilon^{-2}))\) evaluation complexity for a certain constant \(c_1 > 0\). This is far better than the double exponential bound that \citet{powell-2010} suspected, and is far worse than other bounds surmised elsewhere in the literature. We establish similar sharp bounds when model Hessians are \(O(k^p)\), where \(k\) is the iteration counter, for \(0 \leq p < 1\). When \(p = 1\), the complexity bound depends on the parameters of the family, but reduces to \(O((1 - \log(\epsilon))\exp(c_2\epsilon^{-2}))\) for a certain constant \(c_2 > 0\) for the special case of the standard trust-region method. As special cases, we derive novel complexity bounds for (strongly) convex objectives under the same growth assumptions.