Early-Stopping Thresholds for ES-HyperNEAT: A Data-Driven Approach from Fitness Dynamics

Neuroevolution of Augmenting Topologies (NEAT) and its advanced version, Evolvable-Substrate HyperNEAT (ES-HyperNEAT), have shown great potential in developing neural networks. However, their effectiveness heavily depends on the selection of hyperparameters. This study investigates the optimization of ES-HyperNEAT hyperparameters using the Tree-structured Parzen Estimator (TPE) on the MNIST classification task, exploring a search space of over 3 billion potential combinations. TPE effectively navigates this vast space, significantly outperforming random search in terms of mean, median, and best accuracy. During the validation process, the best hyperparameter configuration found by TPE achieves an accuracy of 29.00% on MNIST, surpassing previous studies while using a smaller population size and fewer generations. The transferability of the optimized hyperparameters is explored in logic operations and Fashion-MNIST tasks, revealing successful transfer to the more complex Fashion-MNIST problem but limited to simpler logic operations. This study emphasizes a method to unlock the full potential of neuroevolutionary algorithms and provides insights into the hyperparameters' transferability across tasks of varying complexity.

In neuroevolution, indirect encoding generates neural network connectivity from a compact genome rather than specifying each connection. ES-HyperNEAT automatically discovers where to place hidden nodes by examining CPPN output patterns: it recursively subdivides space using a quadtree, expanding regions where CPPN outputs show high variance. This adaptive approach discovers network topology without manual substrate specification, extending the fixed-grid HyperNEAT framework built on NEAT. However, the quadtree resists tensorization. Each depth level depends on the parent's variance, forcing sequential evaluation. Different CPPNs produce different subdivision patterns, preventing batching. And variable leaf counts are incompatible with JAX's static shape requirement for JIT compilation. Our prior work confirmed these limits at depths exceeding 5, and a JAX reimplementation of the quadtree yielded only marginal speedup despite batched optimizations, motivating the eager reformulation presented here. We present EMR-HyperNEAT, which evaluates all positions at all resolutions up front, then filters using the same variance criterion: ES-HyperNEAT's subdivide_if(var > $θ$) becomes eval_all(); filter(var > $θ$). This performs more CPPN queries than necessary, but all queries become independent and parallelizable across both cores and population members, reducing complexity from \BigO($4^D$) to \BigO($4^D/P$) across $P$ parallel cores. Recurrent substrate configurations become feasible through a connection type taxonomy. The experiments section validates 12-34$\times$ on-device GPU speedup on XOR at depths 5-7, and empirically higher solve rates across benchmarks.

Adam and its variants dominate neural network training, but a single run only reveals whether a configuration works well after most of its budget is spent, a poor fit for hyperparameter or architecture search, where configurations must be ranked cheaply and pruned early. We introduce EvE (Evolutionary Explorer), a steady-state, population-of-four differential evolution (DE) optimizer with a targeted Adam fallback: each iteration proposes one candidate via DE, running a short burst of gradient descent only if the DE step fails to improve on the incumbent. Selection is greedy, so on a deterministic objective the best-so-far value is provably monotone non-increasing, and since gradients are used only as a targeted rescue, per-iteration cost stays within a constant factor of a single Adam step regardless of dimension. Under a fixed, evaluation-cost-matched budget, EvE wins or ties Adam on 76% of 70 (problem, dimension) cells across seven scalable benchmarks up to one million variables. On three real neural-network tasks (an MLP on MNIST, and LoRA fine-tuning of a 1.5B-parameter language model on two datasets) EvE finishes the same charged budget 1.7-3.9x faster, at a modest cost in final quality (about one accuracy point on MNIST, 9-11% higher relative test loss on the two fine-tuning tasks; on GSM8K, Adam is about 5 accuracy points more accurate, and fine-tuning lowers accuracy below the base model for both). Inside successive halving on UCI Adult, EvE completes hyperparameter and architecture searches 3.1-3.5x faster, ranking configurations about as consistently with Adam as Adam does with itself across seeds (Kendall's tau 0.66-0.69). EvE is not a total replacement for Adam as a final-stage trainer, but a fast, gradient-aware proxy for the search-heavy, budget-constrained regime one level up.