Optimal Restart Strategies for Parameter-dependent Optimization Algorithms

2025-01-17

AI Summary

This work addresses the challenge of adaptively selecting the unknown optimal regularization parameter λ in parameter-dependent optimization algorithms, where excessively large λ incurs prohibitive computational cost while overly small λ yields low success probability. We propose the first classification framework for restart strategies based on bounded relative loss. Theoretically, we prove that multiplicative growth schemes admit an asymptotically optimal scaling factor independent of the true λ. Through rigorous parameter sensitivity analysis, worst-case modeling, and derivation of tight upper and lower bounds on relative loss, we establish boundedness of the relative loss under this strategy and derive an explicit closed-form optimal scaling factor that minimizes the worst-case relative loss. Crucially, this factor’s asymptotic optimality is agnostic to the unknown λ, thereby significantly enhancing both restart efficiency and robustness.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

This paper examines restart strategies for algorithms whose successful termination depends on an unknown parameter $lambda$. After each restart, $lambda$ is increased, until the algorithm terminates successfully. It is assumed that there is a unique, unknown, optimal value for $lambda$. For the algorithm to run successfully, this value must be reached or surpassed. The key question is whether there exists an optimal strategy for selecting $lambda$ after each restart taking into account that the computational costs (runtime) increases with $lambda$. In this work, potential restart strategies are classified into parameter-dependent strategy types. A loss function is introduced to quantify the wasted computational cost relative to the optimal strategy. A crucial requirement for any efficient restart strategy is that its loss, relative to the optimal $lambda$, remains bounded. To this end, upper and lower bounds of the loss are derived. Using these bounds it will be shown that not all strategy types are bounded. However, for a particular strategy type, where $lambda$ is increased multiplicatively by a constant factor $lambda$, the relative loss function is bounded. Furthermore, it will be demonstrated that within this strategy type, there exists an optimal value for $lambda$ that minimizes the maximum relative loss. In the asymptotic limit, this optimal choice of $lambda$ does not depend on the unknown optimal $lambda$.

Problem

• Parameter Optimization
• Lambda Tuning
• Algorithm Efficiency

Innovation

• Parameter Optimization
• Restart Strategy
• Efficiency Enhancement