Recent literature has demonstrated promising results on the training of Generative Adversarial Networks by employing a set of discriminators, as opposed to the traditional game involving one generator against a single adversary. Those methods perform single-objective optimization on some simple consolidation of the losses, e.g. an average. In this work, we revisit the multiple-discriminator approach by framing the simultaneous minimization of losses provided by different models as a multi-objective optimization problem. Specifically, we evaluate the performance of multiple gradient descent and the hypervolume maximization algorithm on a number of different datasets. Moreover, we argue that the previously proposed methods and hypervolume maximization can all be seen as variations of multiple gradient descent in which the update direction computation can be done efficiently. Our results indicate that hypervolume maximization presents a better compromise between sample quality and diversity, and computational cost than previous methods. Generative Adversarial Networks (GANs) BID13 offer a new approach to generative modeling, using game-theoretic training schemes to implicitly learn a given probability density. Prior to the emergence of GAN architectures, realistic generative modeling remained elusive. When offering unparalleled realism, GAN training remains fraught with stability issues. Commonly reported shortcomings involved in the GAN game are the lack of useful gradients provided by the discriminator, and mode collapse, i.e. lack of diversity in the generator's samples.Considerable research effort has been devoted in recent literature in order to overcome training instability 1 within the GAN framework. Some architectures such as BEGAN BID4 ) have applied auto-encoders as discriminators and proposed a new loss to help stabilize training. Methods such as TTUR BID16 , in turn, have attempted to define schedules for updating the generator and discriminator differently. The PacGAN algorithm (Lin et al., 2017) proposes to modify the discriminator's architecture which will receive m concatenated samples as input, while modifications to alternate updates in SGD were introduced in (Yadav et al., 2017) . These samples are jointly classified as either real or generated, and authors show that this enforces sample diversity. In SNGAN (Miyato et al., 2018) , authors introduce spectral normalization on the discriminator aiming to ensure Lipschitz continuity, which is empirically shown to consistently yield high quality samples when different sets of hyperparameters are used.Recent works have proposed to tackle GANs instability issues using multiple discriminators. Neyshabur et al. (2017) propose a GAN variation in which one generator is trained against a set of discriminators, where each discriminator sees a fixed random projection of the inputs. Prior work, including GMAN BID9 has also explored training against multiple discriminators.In this paper, we build upon Neyshabur et al.'s introduced framework and propose reformulating the average loss minimization aiming to further stabilize GAN training. Specifically, we propose treating the loss signal provided by each discriminator as an independent objective function. To achieve this, we simultaneously minimize the losses using multi-objective optimization techniques. Namely, we exploit previously introduced methods in literature such as the multiple gradient descent algorithm (MGD) BID7 . However, due to MGD's prohibitively high cost in the case of large neural networks, we propose the use of more efficient alternatives such as maximization of the hypervolume of the region defined between a fixed, shared upper bound on those losses, which we will refer to as the nadir point η * , and each of the component losses.In contrast to Neyshabur et al. (2017) 's approach, where the average loss is minimized when training the generator, hypervolume maximization (HV) optimizes a weighted loss, and the generator's training will adaptively assign greater importance to feedback from discriminators against which it performs poorly.Experiments performed on MNIST show that HV presents a good compromise in the computational cost-samples quality trade-off, when compared to average loss minimization or GMAN's approach (low quality and cost), and MGD (high quality and cost). Also, the sensitivity to introduced hyperparameters is studied and results indicate that increasing the number of discriminators consequently increases the generator's robustness along with sample quality and diversity. Experiments on CIFAR-10 indicate the method described produces higher quality generator samples in terms of quantitative evaluation. Moreover, image quality and sample diversity are once more shown to consistently improve as we increase the number of discriminators.In summary, our main contributions are the following:1. We offer a new perspective on multiple-discriminator GAN training by framing it in the context of multi-objective optimization, and draw similarities between previous research in GANs variations and MGD, commonly employed as a general solver for multi-objective optimization. 2. We propose a new method for training multiple-discriminator GANs: Hypervolume maximization, which weighs the gradient contributions of each discriminator by its loss.The remainder of this document is organized as follows: Section 2 introduces definitions on multiobjective optimization and MGD. In Section 3 we describe prior relevant literature. Hypervolume maximization is detailed in Section 4, with experiments and results presented in Section 5. Conclusions and directions for future work are drawn in Section 6. In this work we have shown that employing multiple discriminators is a practical approach allowing us to trade extra capacity, and thereby extra computational cost, for higher quality and diversity of generated samples. Such an approach is complimentary to other advances in GANs training and can be easily used together with other methods. We introduced a multi-objective optimization framework for studying multiple discriminator GANs, and showed strong similarities between previous work and the multiple gradient descent algorithm. The proposed approach was observed to consistently yield higher quality samples in terms of FID. Furthermore, increasing the number of discriminators was shown to increase sample diversity and generator robustness.Deeper analysis of the quantity || K k=1 α k ∇l k || is the subject of future investigation. We hypothesize that using it as a penalty term might reduce the necessity of a high number of discriminators. In BID16 , authors proposed to use as a quality metric the squared Fréchet distance BID11 between Gaussians defined by estimates of the first and second order moments of the outputs obtained through a forward pass in a pretrained classifier of both real and generated data. They proposed the use of Inception V3 (Szegedy et al., 2016) for computation of the data representation and called the metric Fréchet Inception Distance (FID), which is defined as: DISPLAYFORM0 where m d , Σ d and m g , Σ g are estimates of the first and second order moments from the representations of real data distributions and generated data, respectively.We employ FID throughout our experiments for comparison of different approaches. However, for each dataset in which FID was computed, the output layer of a pretrained classifier on that particular dataset was used instead of Inception. m d and Σ d were estimated on the complete test partitions, which are not used during training.
We introduce hypervolume maximization for training GANs with multiple discriminators, showing performance improvements in terms of sample quality and diversity. 
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