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Generative Multi-Adversarial Networks

Ishan Durugkar, Ian Gemp, Sridhar Mahadevan

arXiv:1611.01673v3cs.LGcs.MAcs.NE

TL;DR

GANs are difficult to train and often require modifying the original minimax objective for useful early gradients. GMAN extends GANs to multiple discriminators, enabling alternative feedback schemes and untampered-objective training; its variants generally converge faster to higher-quality steady states by GMAM.

  • Problem

    GANs are difficult to train, and practical methods modify the original minimax objective to strengthen early gradient signals.

  • Method

    GMAN extends GANs with multiple independently trained discriminators whose objectives are aggregated through flexible generator-side functions.

  • Results

    GMAN variants generally converged faster to higher-quality steady states across image-generation tasks, as measured by GMAM.

  • Takeaways & Limitations

    GMAN makes training with the original GAN objective possible by increasing the odds that the generator receives constructive feedback.

  • Takeaways & Limitations

    The formidable-adversary analysis is oversimplified because practical discriminator optimization does not reach convergence or global optimality, and finite datasets induce point-mass distributions.

Abstract

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Generative adversarial networks (GANs) are a framework for producing a generative model by way of a two-player minimax game. In this paper, we propose the \emph{Generative Multi-Adversarial Network} (GMAN), a framework that extends GANs to multiple discriminators. In previous work, the successful training of GANs requires modifying the minimax objective to accelerate training early on. In contrast, GMAN can be reliably trained with the original, untampered objective. We explore a number of design perspectives with the discriminator role ranging from formidable adversary to forgiving teacher. Image generation tasks comparing the proposed framework to standard GANs demonstrate GMAN produces higher quality samples in a fraction of the iterations when measured by a pairwise GAM-type metric.

1 INTRODUCTION

GMAN extends GANs to multiple discriminators to address difficult training while retaining the original minimax objective. Its variants and GMAM evaluation target faster convergence and higher-quality image generation.

  • GANs generate realistic samples through a two-player minimax game between a generator and discriminator.
  • GANs have succeeded across image generation and other application domains, but they remain reputably difficult to train.
  • GMAN generalizes GANs to multiple discriminators and explores roles from formidable adversary to forgiving teacher.
  • GMAN introduces GMAM, a pairwise metric for evaluating separately trained generative frameworks on image-generation tasks.
  • GMAN* automatically regulates training, reaches higher GMAM performance, and requires a fraction of standard GAN training time.

2 GENERATIVE ADVERSARIAL NETWORKS TO GMAN

GMAN reformulates the generator’s objective by aggregating feedback from multiple independently trained discriminators. This creates a flexible extension designed to accelerate generator training while retaining the broader GAN framework.

  • The original GAN is a minimax game between generator Gθ(z): z → x and discriminator Dω(x): x → [0, 1].
  • pdata(x) denotes the true data distribution, pz(z) a simple sampling distribution, and pG(x) the distribution induced by the generator.
  • The original convergence result assumes sufficient capacities and an optimal discriminator, while practical training replaces log(1 − D(G(z))) with −log(D(G(z))) for stronger initial gradients.
  • Because the optimal discriminator is rarely available, minimizing the practical GAN objective provides only a lower bound rather than direct access to the ideal divergence objective.
  • Prior work reformulates GAN objectives using alternative distribution divergences, moment matching, or real-valued energies.
  • GMAN instead reformulates generator feedback to accelerate training toward a more robust state irrespective of the choice of V.
  • GMAN uses minG max F(V(D1,G), …, V(DN,G)) while each discriminator independently maximizes its own objective.

3 A FORMIDABLE ADVERSARY

GMAN’s formidable-adversary variants aggregate multiple discriminator objectives to challenge the generator more strongly. Max aggregation, boosting, and their practical limitations define this design space.

  • 3 A FORMIDABLE ADVERSARY: GMAN explores discriminator variants that better approximate maxD V(D,G), producing a harsher critic for the generator.
  • 3.1 MAXIMIZING V(D,G): With F := max, the generator receives the largest objective among N discriminators, effectively using parallel random restarts.
  • 3.1 MAXIMIZING V(D,G): Max aggregation forces generated samples to withstand scrutiny from all N discriminators, each potentially representing a distinct maximum.
  • 3.1 MAXIMIZING V(D,G): The max-discriminator interpretation is oversimplified because practical optimization does not reach convergence or global optimality.
  • 3.1 MAXIMIZING V(D,G): Multiple discriminators alter game dynamics and discriminator trajectories, preventing a general claim that max performance is uniformly higher over time.
  • 3.2 BOOSTING: Online boosting combines predictions from weaker discriminators to classify whether samples come from the generator or dataset.
  • 3.2 BOOSTING: Unlike max selection, boosting can use linear combinations and must predict before receiving its loss function.
  • 3.2 BOOSTING: Boosting failed to produce promising results on the image-generation tasks, possibly because it creates too strong an adversary.

4 A FORGIVING TEACHER

GMAN softens the discriminator’s criticism by combining multiple discriminators, allowing constructive feedback while preserving the original minimax objective. The ensemble also maintains diverse hypotheses that can diffuse probability mass across the data space.

  • 4 A FORGIVING TEACHER: A superior discriminator can give uniformly negative feedback, indicating where to reduce generator mass without identifying where to increase it.This creates a whack-a-mole dilemma because reducing probability in one region necessarily increases it elsewhere.
  • 4 A FORGIVING TEACHER: GMAN softens the max operator with parameterized Pythagorean means, ranging from the mean at λ = 0 to the max as λ →∞.The softmax variant is differentiable and can correspond to a convex combination of discriminators.
  • 4 A FORGIVING TEACHER: The generator receives a vanishing gradient only when all discriminators classify a sample as fake, so fooling one discriminator can provide constructive feedback.This permits minimizing the original generator objective, log(1 − D), rather than replacing it with −log(D).
  • 4 A FORGIVING TEACHER: Setting λ closer to zero early in training increases the odds of constructive feedback and reduces feedback variance while discriminators learn decision boundaries.The ensemble is especially useful when discriminators are far from optimal.
  • 4.3 MAINTAINING MULTIPLE HYPOTHESES: Training multiple limited-capacity discriminators can produce diverse hypotheses whose averaged distribution diffuses probability mass across the data space.Under reasonable assumptions about the true distribution, the averaged distribution is expected to have higher likelihood.

5 EVALUATION

The evaluation compares GMAN variants with standard and modified GAN baselines across MNIST, CelebA, and CIFAR-10, emphasizing convergence and GMAM-based image quality. Increasing discriminator multiplicity accelerates convergence, while GMAN∗ achieves strong overall performance and GMAN is reported as robust to mode collapse.

  • The evaluation covers MNIST, CIFAR-10, and CelebA, comparing convergence rates and steady-state generator quality using GMAM.
  • MNIST: 2x fewer iterations reach steady state on MNIST as the discriminator ensemble grows, with lower minimax-objective variance across runs.Figure 4 separately reports GMAN∗ with N = 5 reaching steady state at approximately twice the speed of GAN with N = 1.
  • MNIST: GMAN∗ achieves the best overall performance by GMAM, agreeing with the relative image quality observed for GMAN-0 across MNIST epochs.
  • MNIST: GMAN∗ adjusts λ by easing the game initially and increasing difficulty as training progresses; Figure 7 compares this schedule with fixed λ values using GMAM.
  • CelebA & CIFAR-10: Accelerated convergence also appears on CelebA, while GMAN-0 generates images on CIFAR-10 and improves image quality as the number of generators increases at fixed iterations.
  • CelebA & CIFAR-10: GMAN is reported as robust to mode collapse, with linear rather than quadratic batch-size complexity compared with minibatch discrimination.The paper attributes robustness to requiring generated samples to satisfy a diverse set of discriminators.

6 CONCLUSION

GMAN extends GANs with multiple discriminators spanning adversarial roles, enabling the original GAN objective and faster convergence to higher-quality steady states. GMAN* also outperformed single-discriminator GANs on MNIST, while future work targets generator control and discriminator diversity.

  • GMAN introduces multiple discriminators whose roles range from formidable adversary to forgiving teacher.
  • GMAN makes the original GAN objective possible by increasing the odds that the generator receives constructive feedback.
  • GMAN variants achieved faster convergence to a higher-quality steady state across varied tasks according to the GMAM metric.
  • GMAN* outperformed single-discriminator GANs on MNIST and reached higher GMAM performance in a fraction of standard GAN training time.
  • Future work will investigate more sophisticated generator game control and mechanisms for maintaining discriminator diversity.

A.1 ACCELERATED CONVERGENCE & REDUCED VARIANCE

Increasing the discriminator ensemble size accelerates convergence of the generator objective to steady state and reduces its variance on CelebA and CIFAR-10.

  • Increasing N, the number of discriminators, accelerates convergence of the generator objective F to steady state on CelebA.Figure 10 averages F over five training runs; Figure 11 provides alternative convergence evidence.
  • Increasing N reduces the variance σ2 of the generator objective over training runs on both CelebA and CIFAR-10.The figures represent variance with filled shadows of ±1σ.
  • Increasing N, the number of discriminators, accelerates convergence of the generator objective F to steady state on CIFAR-10.Figure 12 averages F over five training runs; Figure 13 provides alternative convergence evidence.

A.2 ADDITIONAL GMAM TABLES

GMAN-0 with five discriminators reaches steady state about twice as fast as GAN, while increasing the discriminator count improves CIFAR-10 evaluation scores.

  • ≈2x: GMAN-0 with N = 5 reaches steady state at approximately twice the speed of GAN with N = 1.The standard deviation is measured over a sliding window of 500 iterations; lower values indicate a more steady-state.
  • ≈2x: GMAN-0 with N = 5 reaches steady state at approximately twice the speed of GAN with N = 1 on CIFAR-10.The standard deviation is measured over a sliding window of 500 iterations; lower values indicate a more steady-state.
  • Increasing the discriminator count from 2 to 5 significantly improves CIFAR-10 scores over standard GAN on both GMAM and Inception scores.

A.3 GENERATED IMAGES

The appendix provides generated-image samples and additional CIFAR-10 evaluation tables, including pairwise GMAM and Inception-score summaries for GMAN variants using two or five discriminators.

  • Tables 3 and 5 report pairwise GMAM means for CIFAR-10 models trained with two and five discriminators, respectively.Positive GMAM indicates better performance relative to the row opponent, while negative values indicate worse performance.
  • Tables 4 and 6 report CIFAR-10 Inception-score means with standard deviations for models trained with two and five discriminators, respectively.Higher Inception scores are better.
  • Generated samples are shown for the CelebA cropped dataset and for GMAN-0 on CIFAR-10.

A.4 SOMEWHAT RELATED WORK

GMAN differs from earlier multi-discriminator and adversarial-training approaches by targeting unsupervised settings and by formalizing continuous discriminator aggregation. The experiments use DCGAN-like architectures with varied discriminator designs and dataset-specific training schedules.

  • Related approaches: Earlier two-discriminator approaches applied only to semi-supervised settings with known dataset partitions, whereas GMAN targets unsupervised scenarios.The related-work passage also notes a possible extension to semi-supervised domain adaptation with multiple discriminators per domain.
  • Theoretical interpretation: The intermediate value theorem implies that softmax aggregation can be interpreted as evaluating the minimax objective at some discriminator in the discriminator space.This remains true even when that discriminator is not representable by the selected neural-network architecture.
  • Optimization: GMAN* minimax optimization is converted to an unconstrained formulation by introducing Λ and defining λ(Λ) = log(1 + e^Λ).The generator minimizes over Λ ∈ R through this auxiliary-variable parameterization.
  • Boosting perspective: AdaBoost.OL is used because it does not require knowing a weak learner’s edge γ and permits γ < 0, matching deep nets with unknown edges.The passage defines P(correct label) = 0.5 + γ with γ ∈ (0, 0.5] for positive edges.
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