---
title: Novelty Search with Local Competition (NSLC)
url: https://www.emergentmind.com/topics/novelty-search-with-local-competition-nslc
type: topic
---

# Novelty Search with Local Competition (NSLC)

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 [2007.06251, 2502.00593].

## 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 $D$ is a domain-specific distance metric over $bc(x)$.
- **Local competition score**:
  $$
  \varphi(x) = |\{y \in N_k(x): f(x) > f(y)\}|
  $$
  Normalization (optional): $\varphi_\text{norm}(x) = \varphi(x)/k$.

The framework permits contrasting with pure novelty (optimize $\rho$ only) and global competition (compare $f(x)$ against all $y$ in $P \cup A$).

## 2. NSLC Algorithmic Structure

NSLC typically employs multi-objective selection—in practice, NSGA-II—with two axes: novelty ($\rho$) and local competition ($\varphi$). The high-level pseudocode is as follows [2007.06251]:

- Initialize a population $P$ and empty archive $A$.
- For each generation:
  - Evaluate fitness $f(x)$ and behavior characterization $bc(x)$ for all $x \in P$.
  - Update archive $A$ by inserting a fraction $p$ of the population.
  - For each $x\in P$ compute $N_k(x)$, $\rho(x)$, and $\varphi(x)$.
  - Form a selection pool from $P$ and offspring.
  - Apply NSGA-II non-dominated sorting on ($\rho$, $\varphi$) to select the next generation $P$.
  - Apply variation operators (typically mutation) to generate offspring.

The Pareto front is constructed by jointly maximizing $\rho$ and $\varphi$. No explicit crowding metric is necessary on the novelty axis since $\rho$ 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 $D(x,y)$**: NEAT-style genome distance based on matching/non-matching genes.
- **Quality metric**:
  - Generator: $f_G(x) = -\mathrm{FID}(x)$ (minimizing Fréchet Inception Distance as defined in equation (3) of [2007.06251]).
  - Discriminator: $f_D(x) = -J^{(D)}(D,G)$ (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 $k=3$ (MNIST), archive insertion $p=0.10$, 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 $24.3 \pm 3.3$), NSLC attains better average diversity (MNIST: FID $35.2 \pm 12.5$ vs. standard COEGAN $36.8 \pm 18.6$).
- **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 ($k$)**:
  - Small $k$ (2–5): tight niche protection, slow fitness gain.
  - Larger $k$: local competition approaches global, faster convergence, less exploration.
- **Archive management**:
  - Archive insertion $p \in [0.05,0.20]$; higher $p$ increases diversity reference, but with greater computational load.
  - Periodic pruning may be required to limit memory.
- **Exploration/Exploitation Tuning**:
  - If fitness stagnates: reduce $k$ or mutation strength.
  - If fitness is unstable: increase $k$ or emphasize local competition in NSGA-II.
- **Scaling**:
  - Higher genome-length for high-resolution data (e.g., 8–10 layers for $128\times128$ 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) [2502.00593] 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 ($\tilde f_i$) is its mean distance to its $k$-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 ($k$).
- **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               | $k$ (neighbor count)      |
| NSLC         | Local novelty + comp.    | Local competition  | $k$, archive size $p$     |
| NSGC         | Local novelty + comp.    | Global competition | $k$, archive size $p$     |
| DNS          | Fit-dist. to better $k$  | Dynamic, archive-free | $k$                       |

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.

Source: https://www.emergentmind.com/topics/novelty-search-with-local-competition-nslc