Complexity of trust-region methods in the presence of unbounded Hessian approximations
Bibliographic record
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 \(O([(1-p)ε^{-2}]^{1/(1-p)})\) evaluation complexity to find an \(ε\)-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 \(O(\exp(c_1ε^{-2}))\) 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 \(O((1 - \log(ε))\exp(c_2ε^{-2}))\) 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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.007 | 0.040 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".