Compute Allocation in Evolutionary Search: From Depth-Breadth to Multi-Armed Bandits

Large language model (LLM)-driven evolution has shown promise for program search and algorithm discovery, but relying on strong models throughout long evolutionary runs is costly. A natural alternative is to combine cheap and strong models under a fixed inference budget. However, existing approaches typically allocate models at the level of individual queries or mutation steps, overlooking that evolutionary search is \textit{stateful}: each generated candidate changes the population from which subsequent mutations are produced. We empirically analyze LLM-driven evolutionary trajectories and find that search progress is strongly front-loaded, early trajectory performance is informative but noisy, and cheap models recover much of the early progress achieved by strong models at lower cost. Motivated by these findings, we propose \textbf{\model}, a training-free framework that shifts budget allocation from individual calls to evolving populations through adaptive \textit{population handoff}. A cheap model explores multiple trajectories in short blocks allocated by a bandit scheduler. Relay Gain, defined as the marginal improvement of a compact, quality-diverse candidate bank constructed for handoff, serves as the scheduler reward and determines when to hand off. The curated candidates initialize a shared strong model population for refinement. Across four benchmarks and three budgets, \model achieves the highest mean score in 11 of 12 settings, outperforming competitive baselines. Our results suggest that in stateful search, budget allocation should be organized around the population, not the individual call.

LLM-guided evolutionary methods such as AlphaEvolve have proven effective in domains like math, systems research, and algorithmic discovery, but their reliance on frontier models makes each run expensive. We argue this is largely an artifact of how existing frameworks allocate search: archives that fail to preserve solution diversity force compensation through stronger mutation models; blind model use spends frontier dollars on local edits a smaller model could handle; and full-set evaluation wastes rollouts on redundant examples. We introduce LEVI, a harness-first evolutionary framework built on the bet that stronger search architectures can substitute for or even outperform larger LLMs in evolutionary search. LEVI improves on three core components of evolutionary search: a solution database that establishes diversity from the beginning, and then maintains it throughout the run; a smarter mutation router that plays into the strengths of large and small LLMs; and a rank-preserving proxy benchmark for rollout-heavy settings. Across systems-research benchmarks LEVI attains the highest score on a budget 3.3-6.7x smaller than the published frontier-model runs of existing frameworks like ShinkaEvolve, GEPA, and AdaEvolve; on one problem, LEVI matches the existing best at a 35x lower cost. On prompt optimization, LEVI matches or exceeds GEPA at less than half of its rollout budget on four different benchmarks. LEVI is available as an open-source framework at https://github.com/ttanv/levi.

Large Language Model (LLM)-guided evolutionary search is increasingly used for automated algorithm discovery, yet most current methods track search progress primarily through executable programs and scalar fitness. Even when natural-language reasoning is used through heuristic descriptions or reflection, it typically remains transient mutation context or unstructured memory, rather than organized as persistent population-level state over strategic directions. As a result, evolutionary search can struggle to distinguish syntactically different implementations of the same idea, preserve lower-fitness but strategically promising directions, or detect when an entire family of strategies has saturated. We introduce \model, a modular strategy-space layer that turns language-level strategic reasoning into first-class population-level evolutionary state in LLM-driven program search. \model represents each candidate program with an explicit natural-language strategy, clusters the archive by strategy semantics, retrieves behaviorally complementary inspirations, and periodically navigates the strategy landscape to avoid saturated directions. Without modifying the underlying evolutionary algorithms, \model improves existing evolutionary backbones across algorithm discovery, systems optimization, and agent-scaffold design tasks in most settings. Across four systems benchmarks, \model achieves a 20.6% average relative improvement, with the best single run on Prism scoring 3$\times$ higher. These results suggest that persistent strategy representations provide a practical mechanism for improving the effectiveness and cost-efficiency of LLM-guided evolutionary search, pointing toward compound AI systems whose search capabilities benefit from the structured accumulation and reuse of algorithmic strategies.