Automated Algorithm Discovery through Mathematical Reformulation: A Study in Visual-Token Reduction

Automated algorithm discovery typically searches within a fixed understanding of a task, mutating programs or recombining familiar heuristics. For algorithm-design tasks that admit useful mathematical abstraction, the mathematical formulation shapes algorithm design before implementation. Such a task may be expressed as different mathematical problems, each revealing different properties to preserve and different algorithmic structures. Each formulation admits multiple solution procedures with different states, updates, and outcomes. Our central insight is therefore to make both the mathematical formulation and its solution procedure explicit objects of search. We introduce a formulation-centric framework in which each proposal records a formulation–solution pair. An AI–Math–AI loop generates diverse formulations and a growing toolbox of mathematical mechanisms, retrieves relevant external knowledge, derives solution procedures, and implements them as executable programs. Its feedback mechanism balances exploitation and diversity: it allocates more search effort to mathematical areas associated with stronger proposals, while discouraging repeated exploration and redirecting attempts toward underexplored areas and tools. We study visual-token reduction in frozen multimodal large language models, where the framework discovers effective selectors grounded in reconstruction, information gain, and coverage. Their performance across tasks, token budgets, and model backbones demonstrates the practical value of exploring alternative mathematical views of the same problem. These results support searching over mathematical formulations and solution procedures as a productive direction for automated algorithm discovery.