Population-Based Multi-Objective Training of Discriminators for Semi-Supervised GANs

Technical Summary: Population-Based Multi-Objective Training of Discriminators for Semi-Supervised GANs

Problem Statement
Semi-supervised Generative Adversarial Networks (SSL-GANs) aim to leverage large unlabeled datasets while training a discriminator that functions simultaneously as a classifier for labeled data and a detector for real versus generated samples. However, training these models is often unstable due to the non-stationary interaction between the generator and discriminator, leading to issues such as vanishing gradients, mode collapse, and oscillatory dynamics.

Existing approaches, including recent co-evolutionary methods like CE-SSL-GAN, attempt to stabilize training by maintaining populations of models. However, these methods typically aggregate the supervised (classification) and unsupervised (real/fake discrimination) objectives into a single scalar loss function. This aggregation forces a specific trade-off between the two distinct learning tasks, potentially limiting the model's ability to explore optimal solutions where classification accuracy and adversarial discrimination are balanced differently.

Methodology: COMOD-SSLGAN
The paper proposes COMOD-SSLGAN (CO-evolutionary Multi-Objective Discriminator SSL-GAN), a population-based training strategy that reformulates discriminator learning as an explicit bi-objective optimization problem.

• Multi-Objective Formulation: Instead of summing the supervised loss ($L_{D,s}$) and unsupervised loss ($L_{D,u}$), the method treats them as separate objectives to be minimized simultaneously: $\min_v (L_{D,s}(v), L_{D,u}(v; u))$.
• Population-Based Co-evolution: The algorithm maintains two populations of size $\mu$: one for generators ($G$) and one for discriminators ($D$).
• Competitive Evaluation: Individuals are evaluated through a competitive matchup policy (specifically "all-vs-all" in the experiments). The performance of a discriminator is assessed against multiple generators, and objectives are aggregated across these matchups to compute fitness.
• Pareto-Based Selection: Discriminators are ranked using Pareto dominance based on their bi-objective vector. This allows the population to maintain a diverse set of discriminators representing different trade-offs between classification accuracy and real/fake discrimination. Diversity is preserved using criteria such as crowding distance, following the NSGA-II selection mechanism.
• Variation and Elitism: The variation operator applies Stochastic Gradient Descent (SGD) as a mutation mechanism, training coupled SSL-GAN pairs for a set number of epochs. The authors also introduce an elitist variant, where the best individuals in each population are preserved unchanged for the next generation to mitigate the loss of high-quality solutions due to stochastic noise.
• Generator Selection: Unlike the multi-objective discriminator, generators are selected based on minimizing a single aggregated loss ($\hat{L}_G$).

Key Contributions

1. Bi-Objective Formulation: A novel training framework where discriminator learning is separated into supervised and unsupervised objectives, avoiding scalar aggregation.
2. Pareto-Based Selection: A mechanism that preserves diverse trade-offs between classification and discrimination tasks during the co-evolutionary process.
3. Empirical Analysis: A comprehensive evaluation of evolutionary variants, including an analysis of the impact of population size and the introduction of an elitist replacement strategy, compared against standard SSL-GAN and the mono-objective CE-SSL-GAN.

Experimental Results
Experiments were conducted on the MNIST dataset using a limited-label setting (100 labeled samples per class). The study compared the proposed method against standard SSL-GAN and CE-SSL-GAN across 30 independent runs.

• Classification Accuracy: The proposed method significantly improved discriminator classification accuracy. The elitist variant consistently achieved the highest median accuracy (0.90 for $\mu=5$ and $\mu=7$), representing a relative improvement of approximately 20% over the standard SSL-GAN baseline (0.75). Even the base configuration with $\mu=1$ (isolating the effect of Pareto selection) showed a significant improvement (+9.33%), suggesting the multi-objective formulation itself enhances stability.
• Sample Quality (SSIM): The Structural Similarity Index Measure (SSIM) for generated samples also improved. Population-based configurations achieved median SSIM values between 0.38 and 0.41, compared to 0.32 for the baseline. The best results were observed with $\mu=9$ using the CE-SSL-GAN variant, though visual inspection indicated that all co-evolutionary variants produced samples of similar perceptual quality.
• Training Dynamics: The study observed that discriminator accuracy tends to peak early (epochs 10–20) and then decline as the generator produces more realistic samples, making the discrimination task harder. The elitist strategy effectively mitigated this degradation, maintaining higher accuracy throughout training compared to non-elitist variants.
• Population Size: Increasing the population size beyond a certain point (e.g., from $\mu=5$ to $\mu=9$) did not consistently yield further performance gains, suggesting that the primary benefit of population-based training stems from maintaining diversity rather than simply increasing the number of individuals.

Significance and Claims
The paper claims that formulating discriminator training as a multi-objective optimization problem, combined with competitive co-evolution, provides a robust framework for stabilizing SSL-GAN training. The authors assert that:

• The Pareto-based selection mechanism allows the training process to explore different trade-offs between supervised and unsupervised objectives, leading to more accurate and stable classifiers.
• The elitist strategy is particularly effective in preserving high-quality solutions, resulting in the most consistent classification performance and training stability.
• While population-based training improves the consistency of generated samples, the most significant gains are observed in the classification task and the robustness of the training process.

The authors conclude that this approach offers a promising direction for stabilizing SSL-GANs, though they note that future work is required to validate these findings on more complex datasets (e.g., CIFAR-10, SVHN) and to explore multi-objective formulations for generator training as well.