When Descent Is Too Stable: Event-Triggered Hamiltonian Learning to Optimize

2026-05-07

AI Summary

This work addresses the challenge of fixed-budget non-convex optimization, where conventional optimizers often become trapped in uninformative local minima due to excessive stability, thereby wasting computational resources. To overcome this limitation, the authors propose the SHAPE framework, which uniquely integrates port-Hamiltonian systems with event-triggered control by formulating optimization as a control problem. By constructing an augmented phase space $(q, p)$ and introducing a learned Hamiltonian vector field together with an event-triggered controller, SHAPE dynamically switches among descent, exploitation, or escape strategies while preserving system structure. This approach effectively balances local exploration and global exploitation, accommodates stochastic or estimated gradients, and consistently outperforms existing optimizers under fixed computational budgets, yielding significantly improved solution quality.

Abstract

Fixed-budget nonconvex optimization can fail not because local descent is unstable, but because it is too stable: after reaching a nearby stationary point, an optimizer may spend the remaining evaluations refining an uninformative local minimum. We formulate this failure mode as a control problem over optimizer dynamics, where the learner must decide when to descend, when to exploit a promising basin, and when stagnation should trigger movement elsewhere. We introduce SHAPE, a structured adaptive port-Hamiltonian task-family optimizer for event-triggered minima hunting under local information. Starting from gradient-descent dynamics, SHAPE lifts optimization to an augmented phase space $(q, p)$, where the primal state $q$ represents the candidate solution, the cotangent variable $p$ carries directional sensitivity, and a controller $u$ provides processed information from current gradient oracle. Within each stage, a learned Hamiltonian vector field induces structured local descent; across stages, a fixed event clock in the implementation updates ports and memory when local equilibria are detected, with stage-dependent horizons treated in the analysis as a direct generalization. This design preserves a passivity-compatible structure while allowing the same trained policy to use clean, stochastic, or estimated gradient inputs. Experiments on fixed-budget nonconvex optimization tasks show that SHAPE improves best-so-far performance compared with fixed-policy optimizers. These results suggest that adaptive Hamiltonian energy shaping provides a principled mechanism for balancing descent, exploration, and budget allocation in difficult optimization landscapes.

Problem

• fixed-budget optimization
• nonconvex optimization
• local minima
• optimizer stagnation
• event-triggered control

Innovation

• event-triggered optimization
• Hamiltonian learning
• port-Hamiltonian systems
• fixed-budget nonconvex optimization
• adaptive exploration