Generative Adversarial Networks are a popular method for learning distributions from data by modeling the target distribution as a function of a known distribution. The function, often referred to as the generator, is optimized to minimize a chosen distance measure between the generated and target distributions. One commonly used measure for this purpose is the Wasserstein distance. However, Wasserstein distance is hard to compute and optimize, and in practice entropic regularization techniques are used to improve numerical convergence. The influence of regularization on the learned solution, however, remains not well-understood. In this paper, we study how several popular entropic regularizations of Wasserstein distance impact the solution in a simple benchmark setting where the generator is linear and the target distribution is high-dimensional Gaussian. We show that entropy regularization promotes the solution sparsification, while replacing the Wasserstein distance with the Sinkhorn divergence recovers the unregularized solution. Both regularization techniques remove the curse of dimensionality suffered by Wasserstein distance. We show that the optimal generator can be learned to accuracy $\epsilon$ with $O(1/\epsilon^2)$ samples from the target distribution. We thus conclude that these regularization techniques can improve the quality of the generator learned from empirical data for a large class of distributions.
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This paper serves as a survey of recent advances in large margin training and its theoretical foundations, mostly for (nonlinear) deep neural networks (DNNs) that are probably the most prominent machine learning models for large-scale data in the community over the past decade. We generalize the formulation of classification margins from classical research to latest DNNs, summarize theoretical connections between the margin, network generalization, and robustness, and introduce recent efforts in enlarging the margins for DNNs comprehensively. Since the viewpoint of different methods is discrepant, we categorize them into groups for ease of comparison and discussion in the paper. Hopefully, our discussions and overview inspire new research work in the community that aim to improve the performance of DNNs, and we also point to directions where the large margin principle can be verified to provide theoretical evidence why certain regularizations for DNNs function well in practice. We managed to shorten the paper such that the crucial spirit of large margin learning and related methods are better emphasized.
Deep reinforcement learning (RL) algorithms have shown an impressive ability to learn complex control policies in high-dimensional environments. However, despite the ever-increasing performance on popular benchmarks such as the Arcade Learning Environment (ALE), policies learned by deep RL algorithms often struggle to generalize when evaluated in remarkably similar environments. In this paper, we assess the generalization capabilities of DQN, one of the most traditional deep RL algorithms in the field. We provide evidence suggesting that DQN overspecializes to the training environment. We comprehensively evaluate the impact of traditional regularization methods, $\ell_2$-regularization and dropout, and of reusing the learned representations to improve the generalization capabilities of DQN. We perform this study using different game modes of Atari 2600 games, a recently introduced modification for the ALE which supports slight variations of the Atari 2600 games traditionally used for benchmarking. Despite regularization being largely underutilized in deep RL, we show that it can, in fact, help DQN learn more general features. These features can then be reused and fine-tuned on similar tasks, considerably improving the sample efficiency of DQN.
Both generative adversarial network models and variational autoencoders have been widely used to approximate probability distributions of datasets. Although they both use parametrized distributions to approximate the underlying data distribution, whose exact inference is intractable, their behaviors are very different. In this report, we summarize our experiment results that compare these two categories of models in terms of fidelity and mode collapse. We provide a hypothesis to explain their different behaviors and propose a new model based on this hypothesis. We further tested our proposed model on MNIST dataset and CelebA dataset.
Batch Normalization (BN) improves both convergence and generalization in training neural networks. This work understands these phenomena theoretically. We analyze BN by using a basic block of neural networks, consisting of a kernel layer, a BN layer, and a nonlinear activation function. This basic network helps us understand the impacts of BN in three aspects. First, by viewing BN as an implicit regularizer, BN can be decomposed into population normalization (PN) and gamma decay as an explicit regularization. Second, learning dynamics of BN and the regularization show that training converged with large maximum and effective learning rate. Third, generalization of BN is explored by using statistical mechanics. Experiments demonstrate that BN in convolutional neural networks share the same traits of regularization as the above analyses.
While Generative Adversarial Networks (GANs) have empirically produced impressive results on learning complex real-world distributions, recent work has shown that they suffer from lack of diversity or mode collapse. The theoretical work of Arora et al.~\cite{AroraGeLiMaZh17} suggests a dilemma about GANs' statistical properties: powerful discriminators cause overfitting, whereas weak discriminators cannot detect mode collapse. In contrast, we show in this paper that GANs can in principle learn distributions in Wasserstein distance (or KL-divergence in many cases) with polynomial sample complexity, if the discriminator class has strong distinguishing power against the particular generator class (instead of against all possible generators). For various generator classes such as mixture of Gaussians, exponential families, and invertible neural networks generators, we design corresponding discriminators (which are often neural nets of specific architectures) such that the Integral Probability Metric (IPM) induced by the discriminators can provably approximate the Wasserstein distance and/or KL-divergence. This implies that if the training is successful, then the learned distribution is close to the true distribution in Wasserstein distance or KL divergence, and thus cannot drop modes. Our preliminary experiments show that on synthetic datasets the test IPM is well correlated with KL divergence, indicating that the lack of diversity may be caused by the sub-optimality in optimization instead of statistical inefficiency.
We propose a novel regularizer to improve the training of Generative Adversarial Networks (GANs). The motivation is that when the discriminator D spreads out its model capacity in the right way, the learning signals given to the generator G are more informative and diverse. These in turn help G to explore better and discover the real data manifold while avoiding large unstable jumps due to the erroneous extrapolation made by D. Our regularizer guides the rectifier discriminator D to better allocate its model capacity, by encouraging the binary activation patterns on selected internal layers of D to have a high joint entropy. Experimental results on both synthetic data and real datasets demonstrate improvements in stability and convergence speed of the GAN training, as well as higher sample quality. The approach also leads to higher classification accuracies in semi-supervised learning.
Recently introduced generative adversarial network (GAN) has been shown numerous promising results to generate realistic samples. The essential task of GAN is to control the features of samples generated from a random distribution. While the current GAN structures, such as conditional GAN, successfully generate samples with desired major features, they often fail to produce detailed features that bring specific differences among samples. To overcome this limitation, here we propose a controllable GAN (ControlGAN) structure. By separating a feature classifier from a discriminator, the generator of ControlGAN is designed to learn generating synthetic samples with the specific detailed features. Evaluated with multiple image datasets, ControlGAN shows a power to generate improved samples with well-controlled features. Furthermore, we demonstrate that ControlGAN can generate intermediate features and opposite features for interpolated and extrapolated input labels that are not used in the training process. It implies that ControlGAN can significantly contribute to the variety of generated samples.
We present new intuitions and theoretical assessments of the emergence of disentangled representation in variational autoencoders. Taking a rate-distortion theory perspective, we show the circumstances under which representations aligned with the underlying generative factors of variation of data emerge when optimising the modified ELBO bound in $\beta$-VAE, as training progresses. From these insights, we propose a modification to the training regime of $\beta$-VAE, that progressively increases the information capacity of the latent code during training. This modification facilitates the robust learning of disentangled representations in $\beta$-VAE, without the previous trade-off in reconstruction accuracy.
Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time.
We investigate the training and performance of generative adversarial networks using the Maximum Mean Discrepancy (MMD) as critic, termed MMD GANs. As our main theoretical contribution, we clarify the situation with bias in GAN loss functions raised by recent work: we show that gradient estimators used in the optimization process for both MMD GANs and Wasserstein GANs are unbiased, but learning a discriminator based on samples leads to biased gradients for the generator parameters. We also discuss the issue of kernel choice for the MMD critic, and characterize the kernel corresponding to the energy distance used for the Cramer GAN critic. Being an integral probability metric, the MMD benefits from training strategies recently developed for Wasserstein GANs. In experiments, the MMD GAN is able to employ a smaller critic network than the Wasserstein GAN, resulting in a simpler and faster-training algorithm with matching performance. We also propose an improved measure of GAN convergence, the Kernel Inception Distance, and show how to use it to dynamically adapt learning rates during GAN training.
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