Novelty Search with Local Competition (NSLC)

• NSLC is a quality-diversity evolutionary algorithm that integrates behavioral novelty with local competition to evolve diverse, high-quality solutions.
• It employs NSGA-II for multi-objective selection by evaluating both novelty and local performance via neighborhood comparisons.
• NSLC has been successfully applied to GAN evolution, showing enhanced diversity and niche protection compared to global competition strategies.

Novelty Search with Local Competition (NSLC) is a quality-diversity (QD) evolutionary algorithm that integrates the exploration of behavioral diversity with exploitation through locally bounded quality comparison. NSLC advances standard novelty search by introducing a criterion for “local competition”: individuals are ranked not only by novelty but also by their ability to outperform their nearest behavioral neighbors. This allows populations to evolve both diverse and high-performing solutions. NSLC has been applied in the evolution of generative adversarial networks (GANs) and forms a cornerstone for recent refinements within QD, such as Dominated Novelty Search, which further aims to simplify and generalize the local competition concept (Costa et al., 2020, Bahlous-Boldi et al., 1 Feb 2025).

1. Theoretical Foundations of NSLC

NSLC combines two objectives:

• Novelty: Maximizing behavioral diversity by encouraging solutions that differ from their nearest neighbors in a defined behavior space.
• Local Competition: Selective pressure for quality based on outperforming behavioral neighbors, as opposed to the entire population.

For an individual $x$ with behavior characterization $bc(x)$ and fitness $f(x)$, and an archive $A$ of past solutions, the key quantities are:

• Neighborhood: $N_k(x)$, the $k$ closest individuals to $x$ in behavior space, drawn from the union of the population $P$ and archive $A$.
• Novelty score:

$\rho(x) = \frac{1}{k}\sum_{y\in N_k(x)} D(x, y)$

where $bc(x)$0 is a domain-specific distance metric over $bc(x)$1.

• Local competition score:

$bc(x)$2

Normalization (optional): $bc(x)$3.

The framework permits contrasting with pure novelty (optimize $bc(x)$4 only) and global competition (compare $bc(x)$5 against all $bc(x)$6 in $bc(x)$7).

2. NSLC Algorithmic Structure

NSLC typically employs multi-objective selection—in practice, NSGA-II—with two axes: novelty ($bc(x)$8) and local competition ($bc(x)$9). The high-level pseudocode is as follows (Costa et al., 2020):

• Initialize a population $f(x)$0 and empty archive $f(x)$1.
• For each generation:
  • Evaluate fitness $f(x)$2 and behavior characterization $f(x)$3 for all $f(x)$4.
  • Update archive $f(x)$5 by inserting a fraction $f(x)$6 of the population.
  • For each $f(x)$7 compute $f(x)$8, $f(x)$9, and $A$0.
  • Form a selection pool from $A$1 and offspring.
  • Apply NSGA-II non-dominated sorting on ($A$2, $A$3) to select the next generation $A$4.
  • Apply variation operators (typically mutation) to generate offspring.

The Pareto front is constructed by jointly maximizing $A$5 and $A$6. No explicit crowding metric is necessary on the novelty axis since $A$7 itself ensures behavioral spread.

3. Adaptation of NSLC to Generative Adversarial Network (GAN) Evolution

In GAN evolution, NSLC operates on populations of generator and discriminator neural network architectures, each encoded as sequential lists of layer-genes. Behavior characterization is defined over the architecture space (not the data-generating output):

• Genome encoding: Layers (“genes”) specify architectural choices (e.g., Dense, Conv, Transpose-Conv, activation, channel count).
• Distance metric $A$8: NEAT-style genome distance based on matching/non-matching genes.
• Quality metric:
  • Generator: $A$9 (minimizing Fréchet Inception Distance as defined in equation (3) of (Costa et al., 2020)).
  • Discriminator: $N_k(x)$0 (negated “discriminator loss”).
• Variation operators: add/remove layer, mutate activation or units, no explicit crossover.
• Weight inheritance: overlapping weights are copied if child and parent genome structures match.

Typical hyperparameters include $N_k(x)$1 (MNIST), archive insertion $N_k(x)$2, genome max-length 4–5, and specific mutation rates.

4. Empirical Results and Comparative Analysis

When applied to GAN evolution on MNIST and CelebA, NSLC (COEGAN+NSLC) demonstrates:

• Increased Diversity: NSLC leads to greater architectural diversity. t-SNE visualization shows broader coverage of genotype space relative to baselines.
• Solution Quality: While global competition (COEGAN+NSGC) achieves best FID (MNIST: FID $N_k(x)$3), NSLC attains better average diversity (MNIST: FID $N_k(x)$4 vs. standard COEGAN $N_k(x)$5).
• Exploration-Exploitation: Local competition encourages niche protection and exploration of new architectures, though it may slow convergence to optimal FID compared to global competition.
• Reselection Patterns: NSLC preferentially reselects more recent and diverse genotypes, indicating ongoing exploration.

A plausible implication is that NSLC can protect new behavioral niches against premature convergence, at a slight cost to exploitation efficiency.

5. Practical Implementation Guidelines

Key parameters and operational pointers include:

• Neighborhood size ($N_k(x)$6):
  • Small $N_k(x)$7 (2–5): tight niche protection, slow fitness gain.
  • Larger $N_k(x)$8: local competition approaches global, faster convergence, less exploration.
• Archive management:
  • Archive insertion $N_k(x)$9; higher $k$0 increases diversity reference, but with greater computational load.
  • Periodic pruning may be required to limit memory.
• Exploration/Exploitation Tuning:
  • If fitness stagnates: reduce $k$1 or mutation strength.
  • If fitness is unstable: increase $k$2 or emphasize local competition in NSGA-II.
• Scaling:
  • Higher genome-length for high-resolution data (e.g., 8–10 layers for $k$3 images) increases computational demands.
• Genetic operators:
  • Mutation-only schemes suffice for simple architectures; more complex modeling may require crossover.

6. Relation to Dominated Novelty Search and Alternative QD Approaches

Dominated Novelty Search (DNS) (Bahlous-Boldi et al., 1 Feb 2025) reformulates NSLC by dispensing with explicit archives or fixed grid structures, instead expressing local competition through a dynamic fitness transformation:

• Core concept: Each individual’s new fitness ($k$4) is its mean distance to its $k$5-nearest fitter neighbors.
• Advantages over NSLC:
  • Removes the need for archive management and manually tuned distance thresholds.
  • Handles high-dimensional and irregular descriptor spaces automatically.
  • Single, interpretable hyperparameter ($k$6).
• Empirical findings: DNS outperforms NSLC (Threshold-Elites) and MAP-Elites across continuous-control, maze, and unsupervised descriptor benchmarks, especially under high-dimensionality.

The emergence of DNS suggests the local competition principle can be preserved without the overhead of population containers, providing a scalable and adaptable QD algorithmic core for a wider range of evolutionary domains.

7. Comparative Overview

Variant	Diversity Mechanism	Quality Pressure	Key Parameters
Pure novelty	Neighborhood diversity	None	$k$7 (neighbor count)
NSLC	Local novelty + comp.	Local competition	$k$8, archive size $k$9
NSGC	Local novelty + comp.	Global competition	$x$0, archive size $x$1
DNS	Fit-dist. to better $x$2	Dynamic, archive-free	$x$3

NSLC remains a foundational technique within the QD paradigm, particularly in settings where explicit niche protection and structural diversity are prioritized. The progression towards DNS and related archive-free approaches reflects a broader trend towards scalable, less parameter-sensitive quality-diversity optimization frameworks.