Evolutionary Optimization

Population-based optimizers combine update rules such as mutation, selection, and recombination. When one rule changes, it is often unclear which convergence guarantees survive or how the new combination should be assessed. We develop an operator calculus: an operator is a population-update rule, and the calculus specifies how separately checked effects can be combined. Under explicit regularity and small-step conditions, the leading changes caused by the updates add, yielding reusable building blocks for convergence analysis. The framework distinguishes finding and retaining a good solution, reducing the population's mean objective, and concentrating candidates near an optimizer, and identifies the extra approximation conditions needed for finite evaluation-budget guarantees. Applications include distribution adaptation, recombinative evolution, and consensus dynamics, with verified nonconvex cases. Controlled experiments on a common nonconvex problem collection show how component effects change with population geometry and the performance measure: a rule can worsen the mean objective yet produce better candidates.

Large Language Models (LLMs) have advanced Automatic Heuristic Design (AHD) by enabling heuristic generation through reasoning and code synthesis. In LLM-based AHD, the LLM reasons about algorithm design and generates executable heuristic code. Existing architectures adopt two main paradigms: Natural Evolution applies crossover and mutation to this code to explore diverse strategies, but discards the reasoning traces behind the design decisions, weakening knowledge retention; Metacognitive Evolution retains these reasoning traces and refines them through reflection, but lacks population-level recombination, limiting exploration. These limitations reduce search efficiency, stability, and solution quality on complex problems. To address this gap, we propose MeEvo, an AHD framework that cyclically couples Natural Evolution and Metacognitive Evolution with operator balance that shifts from exploration to exploitation. Natural Evolution explores heuristic code while recording LLM-generated reasoning traces, fitness values, errors and best heuristic into a shared history; Metacognitive Evolution then reflects on this history to generate improved heuristics that feed into the next Natural Evolution cycle. This design enables population-driven exploration and reflection-driven refinement to reinforce each other. Experiments on five optimization problems show that MeEvo achieves stronger performance and lower variance than tested LLM-based AHD architectures, especially on complex constrained tasks.

Evolutionary optimization of spiking neural networks (SNNs) becomes increasingly difficult as task complexity grows because they must search a combined topology--parameter space that grows super-exponentially with network size. We address this scaling challenge through a co-evolutionary ensemble framework in which a population of candidate SNNs is evolved with fitness defined by each network's marginal contribution to group performance. Grounded in cooperative game theory and difference evaluation functions from multiagent systems, this credit assignment rewards networks that consistently improve ensemble performance and penalizes redundancy, encouraging complementary specialization during evolution rather than relying on post-hoc combination of independently trained networks. We evaluate the approach on classification, regression, and control tasks under $μ$Caspian neuromorphic hardware constraints. Co-evolved ensembles achieve statistically significant improvements over both single-network evolution and post-hoc ensembles across all tasks, with the most pronounced gains in control, where standard evolution fails to discover effective policies and co-evolution enables a qualitative transition to near-optimal performance.

We study the $(1 + 1)$-EA in dynamic linear environments, where in every generation selection is performed with respect to a freshly sampled linear function with positive weights. We consider the Dynamic Binary Value problem, where each generation uses a uniformly random permutation of $1,2,4,\dots,2^{n-1}$, and a Uniform weight variant, where the weights are drawn independently from $\mathrm{Unif}(0,1)$. Both of them have recently been integrated into the IOHprofiler platform and empirically studied. For both models we prove a sharp threshold in the mutation parameter $χ$ for mutation rate $χ/n$. Below the threshold, the expected optimisation time is $\mathcal{O}(n\log n)$, whereas above it the runtime becomes $2^{Ω(n)}$. For the Dynamic Binary Value problem in the exponential regime, we also quantify at what distance from the optimum the optimisation process stagnates. We show that there is a second threshold: a distance that is efficiently reached, but reaching any smaller distance takes exponential time. This quantifies and proves previous empirical findings.

We study the $(μ+1)$ EA on the Binary Value function BinVal. We show that it needs at most $O(μ\log μ\cdot n \log n)$ function evaluations to find the optimum when $μ= o(n/\log n)$. This substantially improves upon the recent upper bound of $O(μ^5 n \log(n/μ^4))$ by Krejca, Neumann and Witt. Our results hold for several mutation operators including standard bit mutation. In particular, our bound implies that the $(μ+1)$ EA is at most a factor $O(\log μ\cdot \log n)$ slower on BinVal than on OneMax.

We introduce a new simple model to study the fitness progress of Evolution Strategies (ES) in generic problems. In this model, we bypass the underlying fitness landscape and assume that the mutation of any individual produces an offspring whose fitness relative to the parent is given by an invariant distribution $Z$, such as a mean-shifted Gaussian. This serves as a prototypical model for the optimisation landscape when an evolution algorithm operates far from the global optimum. This simple model can be used to approximate the optimisation process for problems where it is intractable to model the exact fitness function, including tasks such as hyperparameter tuning in machine learning models. We rigorously analyse the expected growth rate $\mathcal{R}_μ$ of the continuous steady-state $(μ+1)$-ES in this model. Unlike comma-selection strategies, the steady-state $(μ+1)$-ES maintains overlapping generations, introducing complex mathematical dependencies among surviving parents that make it harder to analyse. We give a general technique to analyse the the $(μ+ 1)$-ES by constructing modified processes whose growth rates provably sandwich that of the original process. These modified processes are then easier to analyse but still close enough to the true process to give a tight bound on the expected growth rate. When $Z = \mathcal{N}(-δ, 1)$ and $μ\le e^δ$, we show that $\mathcal{R}_μ = \frac{\log^{1 + o(1)} μ}μ \mathcal{R}_1$.

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) is a popular and prominent evolutionary algorithm for solving multi-objective optimisation problems. Despite its popularity, theoretical analyses of SPEA2 have only appeared recently. Moreover, these analyses focus exclusively on how SPEA2 handles non-dominated solutions and disregard the algorithmic components responsible for handling dominated solutions. We conduct a first runtime analysis of SPEA2 for which these components are analysed. We prove that, unlike other prominent algorithms, including NSGA-II, NSGA-III and SMS-EMOA under the same setting of constant population size and duplicate elimination, SPEA2 is unable to cover the Pareto front of the OneTrapZeroTrap benchmark efficiently. Our results indicate that using k-th nearest-neighbour distance in the fitness assignment provides an insufficient signal to maintain diversity among dominated individuals. To address this issue, we propose an improved variant, SPEA2$^+$, that considers all pairwise distances. The new algorithm achieves the same performance guarantees as the other prominent algorithms on OneTrapZeroTrap, while matching the performance of the original SPEA2 on simpler problems. Experimental results complement our theoretical findings.

Large Language Models are highly sensitive to prompt formulation, necessitating automatic prompt optimization to unlock their full potential. While evolutionary algorithms have emerged as the dominant paradigm, they suffer from a critical bottleneck: data efficiency. Current methods treat the development dataset as a static benchmark, wasting significant compute budget on uninformative data. In this work, we introduce APEX (Automatic Prompt Engineering eXpert), a novel framework that optimizes the data usage alongside the prompt search. APEX dynamically stratifies the dataset into Easy, Hard, and Mixed tiers based on the optimization lineage. By prioritizing the Mixed tier, which identifies the data where the LLM has mixed performance, we identify two high-leverage subsets: the addressable frontier for generating informative mutations and the rank-sensitive frontier for distinguishing candidate quality. We evaluate APEX across three diverse benchmarks: IFBench, SimpleQA Verified, and FACTS Grounding. Under a fixed budget of 5,000 evaluation calls, due to its data efficiency, APEX outperforms the initial prompt by an average of 11.2% on Gemini 2.5 Flash and 6.8% on Gemma 3 27B, demonstrating that a data-centric approach is key to efficient and effective prompt optimization.

Generative design of robots requires navigating a vast search-space, encompassing physical configurations and behavioural parameters. Evolutionary Algorithms (EAs) have shown promising results, but often converge prematurely to a small set of sub-optimal designs. Most EAs fail to maintain sufficient diversity in the population that would allow the discovery of distinct functional robots. To counter premature convergence, we introduce a superquadrics-based representation (SQs) for robot bodies. SQs are interpretable, compact and computationally efficient mathematical representations of 3D geometrical shapes that can be tuned to specific design-spaces. To encourage morphological diversity, we combine this representation with a quality-diversity (QD) algorithm (MAP-Elites). We compare SQs and Compositional Pattern Producing Networks representations as generators of morphologies, combining them with standard EAs and MAP-Elites. In two test environments, we find that using SQs to generate morphology in conjunction with the MAP-Elites algorithm reaches the highest QD-score across both environments, maximising diversity of design and functionality of generated robots. The findings highlight the benefits of using a compact and interpretable geometric representation for exploring a complex design-space and suggest that combining SQs with an explicit diversity mechanism increases the quality and number of designs generated.

Recent advances in LLM-driven code evolution have enabled automated discovery by iteratively generating and improving programs. However, applying these methods to adversarial multi-agent games introduces a fundamental challenge: the evaluation landscape shifts as strategies improve, causing fixed evaluators to become unreliable and evolution to stagnate. We propose three mechanisms to address this challenge: evaluator co-evolution, which incorporates discovered champions into the opponent pool; hierarchical deep evaluation, which replaces noisy few-game scores with statistically reliable assessments; and weakness pressure, which dynamically up-weights the most difficult opponents to break through plateaus. We implement these mechanisms within FAMOU, a framework built upon the same foundation-model code-evolution paradigm as OpenEvolve and ShinkaEvolve. On the MCTF 2026 3v3 maritime capture-the-flag task, FAMOU consistently outperforms both baselines under two backbone LLMs, achieving the highest combined score (0.526) and the best generalization to unseen opponents (61.7% win rate), while ablations confirm that each mechanism contributes to performance. Notably, the LLM mutation process generates tactical structures entirely absent from the seed strategies -- including lookahead search and adaptive interception -- demonstrating that code-level evolution can produce nontrivial algorithmic innovations in adversarial settings. The FAMOU-evolved strategy further achieved 1st place in the hardware round-robin and 3rd in simulation at the AAMAS 2026 MCTF Competition, validating its real-world transferability. The optimized implementation and corresponding evaluation codes developed through our evolutionary process are available at: https://github.com/1xiangliu1/FAMOU-CoEvo

While deep Reinforcement Learning (deep-RL) has been increasingly applied to parameter control in evolutionary algorithms, rigorous theoretical analysis of parameter control remains largely restricted to single-parameter settings, owing to the difficulty of deriving effective, interpretable multi-parameter policies amenable to formal study. We demonstrate how deep-RL can be leveraged to overcome this barrier, using the (1+($λ$,$λ$))-genetic algorithm optimizing OneMax, one of the few problems where a super-constant speedup of dynamic control has been formally proven, as a representative case study. We first show that standard approaches struggle to converge in this multi-parameter setting, and introduce algorithm-agnostic enhancements targeting action-space decomposition, reward shifting, and long-horizon discounting. With these in place, we compare common deep-RL methods and find that Double Deep Q-Networks uniquely avoid the policy collapse observed in Proximal Policy Optimization, yielding trajectories suitable for downstream analysis. Crucially, we move beyond the ``black-box'' nature of neural networks by distilling the learned behaviors into a transparent, symbolic control policy. This resulting policy does not only offer interpretability for future theoretical analysis but also yields exceptional performance, consistently outperforming existing baselines across a wide range of problem sizes.

This study addresses the challenges composers and sound designers face in creating and refining tools to achieve their musical goals. Using evolutionary processes to promote diversity and foster serendipitous discoveries, we automate the search through uncharted sonic spaces for sound discovery, arguing that diversity-promoting algorithms can bridge the gap between the theoretical realisation and practical accessibility of sounds. We describe a system for generative sound synthesis combining Quality Diversity (QD) algorithms with a supervised discriminative model, inspired by the Innovation Engine algorithm, and explore different configurations and the interplay between the chosen synthesis approach and the discriminative model. We examine the interaction between Compositional Pattern Producing Networks (CPPNs) and Digital Signal Processing (DSP) graphs, introducing a novel approach that uses multiple specialised CPPNs for different frequency ranges; this yields simpler networks while maintaining performance comparable to single-CPPN setups. We also investigate evolutionary stepping stones by analysing goal switches between musical and non-musical contexts, revealing how lineages traverse unlikely paths to current elites. Expanding the behaviour space of a previous study to include various sound durations, we uncover specialisation within temporal niches. Results indicate that CPPN and DSP graphs coupled with a Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) and a deep learning classifier can generate a substantial variety of synthetic sounds, diverse and innovative across temporal and contextual dimensions. We present the generated sound objects through an online explorer and as rendered sound files, and, in the context of music composition, an experimental application that showcases their creative potential across various durations and contexts.

Covariance matrix adaptation evolution strategy (CMA-ES) is a state-of-the-art black-box optimization algorithm. In general, CMA-ES uses a portfolio of multiple stopping criteria to automatically determine when to stop the search. This mechanism aims to avoid unnecessary consumption of the function evaluation budget during stagnation. Stopping criteria play an important role in CMA-ES, particularly when restart strategies are employed. However, the effectiveness of stopping criteria in CMA-ES remains poorly understood. To address this issue, this paper investigates how the 11 stopping criteria in CMA-ES behave on the noiseless BBOB function set. The performance of the stopping criteria is quantitatively evaluated based on the optimal stopping point in terms of the number of function evaluations in a single run of CMA-ES. Our results show that, although which stopping criterion is triggered first depends significantly on the sample size $λ$ and the dimension $n$, \texttt{tolflatfitness} and \texttt{tolfun} are frequently the first criteria to be triggered among the portfolio of 11 stopping criteria. We also demonstrate that \texttt{tolfunhist} and the portfolio achieve the highest stopping accuracy in most cases. In addition, our results show that the \texttt{tolfun} and \texttt{tolfunhist} criteria are frequently triggered before CMA-ES reaches complete stagnation.

Gray-box optimization is an approach for making some problem-specific information available to the algorithm while still relying on fitness information as the main guide to an optimum. This approach was shown to be beneficial in various combinatorial optimization tasks and neatly captures the continuum between fully black-box algorithms and tailored algorithms. In this work, we discuss different flavors of gray-box algorithms. We show that RLS can find a proper $2$-coloring in a bipartite graph starting from a random $2$-coloring, in an expected time of $\mathcal{O}(n \log n)$. In contrast, when starting from a proper $n$-coloring, the (1+1) EA cannot find such a coloring except when offered additional guiding on plateaus of the search space. Finally, we show the run time for this setting can be much improved by using gray-box operators.

Designing high-performance legged robots requires jointly optimizing morphology and control. Model-free Reinforcement Learning (RL) offers an alternative to model-predictive control for developing robust controllers without explicitly specifying robot dynamics. Thus, we have seen theuse of RL to train controllers and evaluate designs for robot morphology optimization. While RL has shown success inlocomotion, using it in the co-design inner loop is expensive due to repeated policy training. Universal policies conditioned on morphology offer a promising alternative, but suffer from behavioral diversity collapse, converging to a single strategy that performs sub-optimally across designs. On the other hand, end-to-end Mixture-of-Experts (MoE) architectures fail due to a collapse in its representation. We propose Gaussian Evolutionary Specialists (GES), a framework that decouples design-space partitioning from policy learning to capture diverse behaviors explicitly. GES assigns specialist policies to evolving Gaussian regions and iteratively refines them via training, probing, and territory expansion. The resulting specialists are integrated into a design sampling loop, replacing costly re-training with direct evaluation. When tested on the Buoyancy-Assisted Light Legged Unit (BALLU), GES discovers designs with 5 - 25% higher performance than naive universal policies. On hardware, a GES optimized design overcomes a 24 cm tall obstacle - 3x improvement over the baseline BALLU design. Moreover, GES curtails design optimization time by 37%.

We present MotionDisco, a framework that discovers contact-rich, long-horizon humanoid loco-manipulation motions from scratch, without relying on teleoperation or motion retargeting from human demonstrations. This is challenging because the space of possible contact interactions grows combinatorially with the task horizon and the number of objects in the scene. MotionDisco enables rapid discovery of novel motions by coupling a large language model (LLM) guided evolutionary search over sequences of interactions with an efficient sequential kinodynamic trajectory optimizer and pruning strategy, enabling the rapid discovery of novel skills. Through extensive ablation studies, we show that our LLM-guided search discovers successful whole-body trajectories across several challenging long-horizon tasks. Finally, by training reinforcement learning tracking policies on the discovered trajectories, we transfer the motions to a real humanoid robot. This is the first work to discover and deploy long-horizon humanoid loco-manipulation skills entirely through automated evolutionary search. Supplementary videos of the experiments are available at: https://youtu.be/DHiVz34QYlw.

Noisy evolution strategies under fixed evaluation budgets face a depth-fidelity trade-off: spending evaluations to denoise intra-generation rankings reduces the number of distribution updates the optimizer can execute. We argue for depth over fidelity and propose probabilistic elite membership (PEM), which replaces hard rank-based weights in evolution strategies with conditional expected rank weights that integrate over ranking uncertainty. PEM preserves the conditional mean update while reducing conditional update dispersion, a Rao-Blackwellization of the noisy rank-based step. We instantiate PEM via residual bootstrapping (RB-PEM) with capped per-generation overhead, complemented by an adaptive probe-and-switch mechanism for low-noise regimes. Across the COCO bbob-noisy suite and external tasks including RL policy search and hyperparameter optimization, RB-PEM achieves consistent gains in high-misranking, budget-constrained settings.

Neuroevolution is a representative neural architecture search paradigm that evolves both network topology and weights through evolutionary algorithms. In this paper, we propose Seq103, a unified NEAT-style neuroevolution framework for compact sequence architecture discovery. Seq103 consists of a shared evolutionary backbone and an optional recurrent extension. The shared backbone includes an elementary node-and-connection representation, per-class RMSE-based evaluation, mutation-based evolution with class-wise recombination, and elitism. The optional hidden-state mechanism extends the search space with hidden-state nodes and hidden connections, enabling temporal memory when step-wise recurrent inference is required. With this design, Seq103 applies the same core search pipeline to both step-wise recurrent and sample-wise feedforward sequence classification. In recurrent tasks, the hidden-state extension is enabled to provide temporal memory; in feedforward tasks, it is disabled while the shared evolutionary backbone remains unchanged. We evaluate Seq103 on 8 text classification datasets and the full UCRArchive2018 benchmark with 128 univariate time-series datasets. On step-wise tasks, Seq103 retains 86.96% of the best-baseline accuracy on average while using 34.6x to 3218.0x fewer parameters. On sample-wise tasks over the full UCRArchive2018 benchmark, Seq103 retains 81.95% of the best-baseline accuracy on average while using 11.8x to 160,601.0x fewer parameters.

While Large Language Models (LLMs) have recently shown promise in Automated Heuristic Design (AHD), existing methods typically generate and evolve heuristics as a single operator or search strategy, limiting their ability to model strong coupling among multiple decision substructures in problems such as the Traveling Thief Problem (TTP) and the Traveling Purchaser Problem (TPP). In this work, we propose CoEvo-AHD, an LLM-driven dual-population co-evolutionary framework for automated heuristic design in coupled combinatorial optimization. Unlike prior methods that evolve individual heuristics in isolation, CoEvo-AHD leverages LLMs to co-evolve two closely related operator populations. A cooperative evaluation mechanism explicitly captures interactions between route and selection operators, while pairwise scoring and synergistic joint crossover help discover complementary operator logic for joint improvement across coupled decision subspaces. We further design a tool-invocation environment library that encapsulates frequently used core operations, such as local-search delta computation, into callable functions, enabling LLM-generated operators to use standardized interfaces instead of reimplementing inefficient and error-prone problem-specific loops. Experiments on TTP and TPP show that CoEvo-AHD automatically discovers cooperative heuristic combinations and achieves competitive solution quality against traditional heuristics.

Large Language Models (LLMs) remain highly vulnerable to diverse attacks, particularly in black-box settings where the internals of target models are inaccessible. Existing black-box defenses typically rely on pre-defined filtering heuristics, which often fail to generalize to unseen attack types and target model architectures. We introduce EvoDefense, an experience-guided co-evolving black-box defense paradigm. EvoDefense employs a guard LLM to detect malicious queries and an experience memory module to accumulate defense knowledge from previous interactions. At the core of EvoDefense is a continuous attack-defense evolution loop, where an attack generator and the guard model iteratively refine their attack strategies and defense policies through experience-guided optimization. This design enables EvoDefense to generalize across unseen attacks and target models without retraining. Experiments on HarmBench, AdvBench, and AlpacaEval show that EvoDefense achieves consistently strong defense performance across seven popular models and five representative LLM attacks, while preserving competitive general capabilities. On HarmBench, EvoDefense reduces the attack success rate (ASR) of AutoDAN-turbo on Gemini-3-flash and LLaMA-3-8B-Instruct from 29.4% and 43.4% to 8.4% and 6.2%, respectively.

Combinatorial and optimization problems are fundamental to many industrial AI applications. Solving large-scale real-world instances of such problems typically requires careful problem formalization, specialized solvers, and expert-designed heuristics. Thus, experts need to specify not only what solutions are, but also how they are derived. By introducing the tool CHECKMATE, we show that algorithm generation via code evolution represents a paradigm shift by eliminating the need to formulate the how. CHECKMATE solely relies on the what. Specifically, a formal specification ensures solutions' correctness and enables systematic performance evaluation of the generated programs, while a natural language description guides the evolutionary process. The effectiveness of our method is demonstrated on selected problems from two industrial domains: configuration and scheduling. In all cases, the evolved algorithms consistently outperform state-of-the-art solvers. This underscores the potential of formal methods in guiding code evolution for automatically solving complex real-world problems.

GP-GOMEA is a state-of-the-art evolutionary algorithm for symbolic regression, known for discovering small and interpretable models. However, its computational cost remains substantial, limiting its applicability to larger datasets and more complex target expressions. In contrast, the rise of modern subsymbolic approaches, particularly deep learning, has been driven largely by the massive parallelism offered by GPUs. In this work, we take the first major step toward a fully GPU-accelerated GP-GOMEA by introducing a GPU-based fitness evaluation scheme. We design a GPU-friendly representation of GP-GOMEA's template-based individuals and a corresponding evaluation strategy that exploits the inherent parallelism of population-based search. This substantially increases evaluation throughput, enabling orders of magnitude more evaluations within the same time budget. Across four standard symbolic regression benchmarks, this increased evaluation capacity yields performance improvements, particularly for larger datasets and larger population sizes. Moreover, the ability to efficiently evaluate much larger datasets and more complex templates enables analyses that were previously infeasible, allowing us to systematically analyze what makes expressions increasingly difficult for GP-GOMEA, providing new insights into how expression structure affects search difficulty. Finally, for the first time, this expanded capability allows a problem-agnostic evolutionary algorithm to reliably regress one of the largest Feynman equations within four hours.

Evolution Strategies (ES) has recently emerged as a competitive alternative to reinforcement learning (RL) for large language model (LLM) fine-tuning, offering advantages through simplicity, scalability, and inference-only training. However, recent work suggests that ES fine-tuning on new tasks may induce forgetting of prior tasks. First, this paper shows that prior task forgetting (1) is better characterized as performance drift rather than irreversible forgetting, with prior-task performance often recovering during ES training; and (2) is not a specific failure mode of ES, but can also arise for fine-tuning with RL methods. Second, it analyzes when and why such drift arises, highlighting its dependence on ES training dynamics, particularly random walk behavior in weakly constrained directions of the weight space. Third, based on these insights, it introduces Anchored Weight Decay (AWD) as a parameter-space regularization technique that constrains optimization toward the initial model parameters. AWD effectively stabilizes prior-task performance while preserving target-task performance, achieving benefits comparable to large ES population sizes at much lower computational cost. Thus, contrary to previous beliefs, the paper shows that prior-task forgetting under ES is largely avoidable, positioning ES as a promising approach for continual learning in LLMs.

Recently, the runtime analysis of multi-valued estimation-of-distribution algorithms in the framework of Ben Jedidia et al. (TCS 2024) has made significant advancements. However, almost all existing analyses are limited to multi-valued objective functions that in each dimension only distinguish between two types, also called categories, of values and hence can be treated with similar methods as pseudo-Boolean problems. Only recently, Adak and Witt (GECCO 2025) have presented a first runtime analysis of a multi-valued compact genetic algorithm (cGA) on the multi-valued OneMax function G-OneMax$\colon \{0,\dots,r-1\}^n \to \mathbf{N}$ defined by G-OneMax$(x_1,\dots,x_n)=\sum_{i=1}^n {x}_i$ and truly depending on all $r$ categories. We improve their runtime result from $\textrm{O}\bigl(n r^3 \log^2( n)\log (r)\bigr)$ to $\textrm{O}\bigl(n r \log^3(n)\log^3(r)\bigr)$, both for an optimal choice of the update strength $K$. Our result matches, up to polylogarithmic factors, the existing bound for the simpler $r$-valued OneMax function depending essentially only on two values and analyzed in several previous works. To show the new bound, we use improved drift theorems for processes with high self-loop probabilities and specifically derived concentration inequalities to analyze how probability mass in the multi-valued cGA moves into successively smaller and smaller intervals of the $r$-valued frequency matrix.

Evolutionary model merging provides a powerful framework for the automated, training-free composition of LLMs through parameter-space search. However, existing methods predominantly rely on stochastic, hand-crafted operators that overlook the underlying performance landscape of the coefficient space. We propose Evolutionary Generative Merging (EvoGM), a framework that transcends manual heuristics by employing learnable generative modeling to optimize merging coefficients. Specifically, EvoGM features a dual-generator architecture with cycle-consistent learning to adaptively sample and refine promising merging candidates. By constructing winner-loser pairs from historical search trajectories, our framework effectively captures high-performance parameter distributions and maximizes data efficiency. This generative process is seamlessly integrated into a multi-round evolutionary pipeline, where elite merged models iteratively serve as new expert foundations. Extensive experiments across diverse benchmarks demonstrate that EvoGM significantly outperforms state-of-the-art baselines, exhibiting robust performance on both seen and unseen tasks. Code and data are available at https://github.com/JiangTao97/evogm.

LLM-guided evolutionary search (Evolve systems) has reached state-of-the-art results on mathematical and combinatorial tasks, yet most existing systems report only the best of many runs and leave the run-to-run distribution undocumented. We ask how a fixed budget of LLM calls should be allocated, and how reliably a single run reaches the reported numbers. Sweeping the depth-breadth grid over five models and three tasks, we identify two empirical regularities: a fitness-compute envelope along which capability ordering largely collapses on effective FLOPs, and a bilinear depth-breadth fit with task-specific interaction; both are gated by model-task capability. Motivated by these regularities, we propose BaSE (Bandit-based Self-Evolving), a multi-armed bandit that allocates LLM calls across parallel trajectories. Without changing the model, prompt, or evaluator, BaSE improves mean fitness by 12.3% over the strongest island-protocol baseline across 8 (model, task) cells, with the largest gains on high-variance settings: a reliability gain from allocation alone.

Search has been proposed as an effective method for self-improving language models and agentic systems, both for post-training sample generation and for inference. However, widely used methods such as best-of-N sampling and tree search face two fundamental limitations: they are guided by sparse verification signals, and they construct candidates primarily through autoregressive expansion, restricting exploration to regions with substantial model probability mass. To address these, we propose Bidirectional Evolutionary Search (BES), a search framework that couples forward candidate evolution with backward goal decomposition. In the forward search, BES augments standard expansion with evolution operators that recombine partial trajectories to generate candidates that are difficult to obtain from a single model rollout. In the backward search, BES recursively decomposes the original task into checkable subgoals, producing dense intermediate feedback that guides forward search. We provide theoretical motivation showing that candidates generated by expansion-only search are confined to a narrow entropy shell while evolutionary operators can escape it, and that backward search can exponentially reduce the number of required samples to find a correct answer. Experiments show that on challenging post-training tasks where mainstream post-training algorithms fail to improve, BES enables consistent gains, and on three open problem solving benchmarks at inference time, BES outperforms existing open-source frameworks in both average and best-case performance. Code and trained models are available at https://github.com/Embodied-Minds-Lab/BES.

Baldwinian and Lamarckian evolution have existed for a long time in evolutionary algorithms (EAs) without ever dominating the academic literature or practical applications. In this work, we use modern empirical and theoretical methods to revisit Lamarckian and Baldwinian evolution and rigorously compare them with the generic Darwinian evolution. On the empirical side, we run a comprehensive suite of experiments on graphs from six different datasets from the recent GraphBench benchmark on Maximum Independent Set and Maximum Cut problems. Our results show that Baldwinian and Lamarckian evolution consistently outperform Darwinian evolution, confirming the great potential of local search augmented evolutionary algorithms. Notably, in the great majority of cases, all EAs outperform recent deep learning baselines and approach the performance of highly specialised heuristic and exact solvers. We furthermore report a high-performing set of generalist parameters for all studied evolution types that we hope will be of use to practitioners in future. On the theoretical side, we extend the existing DeceptiveLeadingBlocks benchmark to arbitrary block length $k$. For all constant $k$, we then prove asymptotically tight runtime bounds for the $(1+1)$ EA in the three evolution types on this benchmark. For Baldwinian evolution, these are independent of $k$, whereas for the other two evolution types, the runtimes steeply increase with growing value of $k$.

Evolutionary computation offers a variety of tools to solve complex real-world optimization problems. However, research often focuses on smaller, simplified problems and optimization algorithms that sometimes miss expectations in real-world scenarios. Additionally, trust in the applied algorithm and the solutions it provides is often essential in such settings, but requires an understanding of the search process itself. This leads to evolutionary computation often not being seriously considered by practitioners in many application contexts, among them physics-based modeling. In this article, techniques from evolutionary computation are detailed that can alleviate these problems. First, five real-world physics-based optimization problems are introduced and described by domain experts. For each of these, the requirements for the evolutionary algorithm regarding performance and explainability to increase trust and usability are presented. We found that all domain experts expect fast convergence to a good solution and want some explanations for how the results were formed, while other requirements strongly depend on the respective problem. Finally, we present existing approaches that can be leveraged to improve those aspects of evolutionary algorithms but have to our knowledge never been employed in complex real-world scenarios. This implies a gap between both domains that needs to be closed to exploit the full potential of evolutionary computation.

Effective multi-agent systems cannot be designed by selecting prompts or communication graphs in isolation. Agent behavior depends on the information an agent receives, while the usefulness of a communication edge depends on how the receiving agent interprets and uses that information. We propose \textbf{TCP-MCP} (Topology-Coupled Prompting for Multi-Agent Collaborative Problem-Solving), a co-evolution framework that searches agent prompts and communication topologies as a unified genome. TCP-MCP uses an initialization-time landscape probe to calibrate early search behavior, and then relies on Pareto-front diagnostics to adapt exploration under three objectives: task performance, token cost, and structural complexity. Using the same DeepSeek-V3.2 backbone across all methods, TCP-MCP achieves 82.66%, 89.96%, and 96.61% accuracy on MMLU-Pro, MMLU, and GSM8K, respectively. Across the three benchmarks, it consistently outperforms automated graph-generation baselines and achieves competitive accuracy relative to debate-style systems, while using up to 5.69$\times$ fewer tokens than those systems at the reported operating points. These results show that jointly evolving prompts and communication structure provides a practical route to cost-aware and task-adaptive multi-agent system design in controlled evaluations.