Mutation Without Variation: Convergence Dynamics in LLM-Driven Program Evolution

LLM-driven program evolution has emerged as a powerful tool for automated scientific discovery, yet existing frameworks offer no principled guide for designing their individual components and provide no guarantee that the search converges. We introduce SMCEvolve, which recasts program search as sampling from a reward-tilted target distribution and approximates it with a Sequential Monte Carlo (SMC) sampler. From this view, three core mechanisms emerge as principled components: adaptive parent resampling, mixture of mutation with acceptance, and automatic convergence control. We further provide a finite-sample complexity analysis that bounds the LLM-call budget required to reach a target approximation error. Across math, algorithm efficiency, symbolic regression, and end-to-end ML research benchmarks, SMCEvolve surpasses state-of-the-art evolving systems while using fewer LLM calls under self-determined termination. The code is available at https://github.com/kongwanbianjinyu/SMCEvolve.

Recent LLM-guided evolutionary search methods have shown that iterative program mutation can discover strong algorithms, but they typically optimize each task independently, even when related tasks share reusable structure. We introduce Evolutionary Multi-Task Optimization (EMO) for LLM-guided program discovery, and propose EMO-STA (Shared-Then-Adapt), a two-stage framework that first evolves a shared archive of executable programs across a task family and then adapts selected shared candidates to each target task. Within EMO-STA, we explore multiple adaptation strategies, including warm-starting from the shared archive, adapting the best average shared program, and adapting the shared program that performs best on each target task. Across eight task families spanning continuous optimization, geometric construction, modeling, and algorithmic optimization, EMO-STA improves over matched-compute single-task evolution in most settings, with STA Best-Local providing the strongest in-distribution adaptation and STA Best-Shared yielding robust transfer to unseen tasks. Compute-allocation experiments show that allocating a substantial fraction of the family-level budget to shared evolution is consistently beneficial, with roughly balanced shared and adaptation budgets often being optimal. Beyond compute efficiency, we show that shared evolution can mitigate overfitting in low-evidence settings (e.g. few training data), including ARC tasks and time-series feature engineering, by favoring programs that generalize across all tasks rather than exploiting task-specific brittle artifacts.

LLM-based program evolution systems such as FunSearch and AlphaEvolve have shown strong ability to discover novel algorithms, but typically optimize each task in isolation, discarding search experience after completion. We introduce $\varepsilon$-MemEvo, a framework for cross-task knowledge transfer in LLM program evolution. $\varepsilon$-MemEvo stores prior experience as task-agnostic tactic memories: compact natural-language summaries of successful algorithmic strategies rather than raw code, enabling transfer across tasks with different APIs and evaluators. To avoid negative transfer from semantically mismatched memories, $\varepsilon$-MemEvo uses an adaptive injection gate that decides whether retrieved memories should be injected, and at what intensity. We evaluate $\varepsilon$-MemEvo on 8 diverse optimization benchmarks spanning mathematical optimization and systems engineering, using a content-level Leave-One-Out protocol that excludes target-task memory entries. On the primary GPT-5 backbone, $\varepsilon$-MemEvo improves AUCC over AdaEvolve on all 8 tasks, with a mean relative gain of +8.7%, and improves early-stage convergence by +9.4% on average. Ablations show that naive memory injection can fail catastrophically, while adaptive gating remains safe across all five ablation tasks. The data-updated posterior is interpretable in observed states: it favors skip during improving search and shifts from skip to hint across early and late plateaus. These gains incur less than 1% computational overhead.