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Generative Adversarial Networks
Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio
TL;DR
Deep generative models require alternatives to difficult probabilistic computations and Markov-chain-based training. This paper introduces adversarial nets, jointly training a generator and discriminator with backpropagation; theoretically, sufficient-capacity models recover the data distribution, and the paper demonstrates the framework’s viability.
Problem
Deep generative modeling is limited by difficult probabilistic computations and, in some approaches, the need for tractable densities or Markov chains.
Method
Adversarial nets jointly train a multilayer-perceptron generator and discriminator in a minimax game using backpropagation, with noise mapped into data space.
Results
The global minimum is achieved if and only if pg = pdata, with C(G) = −log 4, and the paper demonstrates the framework’s viability.
Takeaways & Limitations
The framework can generate samples without Markov chains or inference during learning, while supporting extensions including conditional modeling and semi-supervised learning.
Takeaways & Limitations
Convergence relies on sufficient model capacity and allowing the discriminator to reach its optimum at each algorithm step.
Abstract
from arXiv · showhide
We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G that captures the data distribution, and a discriminative model D that estimates the probability that a sample came from the training data rather than G. The training procedure for G is to maximize the probability of D making a mistake. This framework corresponds to a minimax two-player game. In the space of arbitrary functions G and D, a unique solution exists, with G recovering the training data distribution and D equal to 1/2 everywhere. In the case where G and D are defined by multilayer perceptrons, the entire system can be trained with backpropagation. There is no need for any Markov chains or unrolled approximate inference networks during either training or generation of samples. Experiments demonstrate the potential of the framework through qualitative and quantitative evaluation of the generated samples.
1 Introduction
The framework addresses the difficulty of training deep generative models by setting a generator against a discriminator in an adversarial game. For multilayer perceptrons, both models train with backpropagation and samples require only forward propagation.
- Deep generative models have had less impact because many probabilistic computations are difficult to approximate.
- The proposed framework pits a generative model against a discriminator that distinguishes generated samples from data samples.Competition drives the two models toward indistinguishable generated and genuine samples.
- For multilayer perceptrons, both models can be trained with backpropagation and dropout, while generation uses only forward propagation.The framework requires neither approximate inference nor Markov chains.
2 Related work
Prior generative-model approaches face difficulties from intractable normalization, density specification, or Markov-chain training. Sampling-oriented methods reduce some of these burdens but still use parameterized generative Markov chains.
- Undirected latent-variable models require a partition function and gradient that are intractable except in trivial cases.Restricted Boltzmann machines, deep Boltzmann machines, and variants estimate these quantities with Markov chain Monte Carlo.
- Deep belief networks combine directed and undirected layers, inheriting computational difficulties from both model types.
- Score matching and noise-contrastive estimation require densities specified analytically up to a normalization constant.For several-layer latent-variable models, even a tractable unnormalized density may be unavailable.
- Generative stochastic networks and generalized denoising auto-encoders can be trained by backpropagation but parameterize one step of a generative Markov chain.
3 Adversarial nets
Adversarial nets train a generator and discriminator as multilayer perceptrons in a two-player minimax game. The discriminator guides generated samples toward the data distribution, while an alternative generator objective supplies stronger early-learning gradients.
- The generator maps noise z through G to data space, while D outputs the probability that x came from data rather than the generator.
- D and G play a two-player minimax game whose value function combines log D(x) on data with log(1 − D(G(z))) on generated samples.
- In practice, training alternates discriminator and generator updates because optimizing D to completion is computationally prohibitive and risks overfitting finite data.
- Maximizing log D(G(z)) instead of minimizing log(1 − D(G(z))) preserves the fixed point while providing stronger gradients early in learning.
- At convergence with sufficient capacity, the generated distribution matches the data distribution and the discriminator cannot distinguish between them.The figure describes G moving samples toward regions more likely to be classified as data.
4 Theoretical Results
The analysis shows that the adversarial criterion has a unique optimum when the generator distribution matches the data distribution, and establishes convergence under sufficient capacity and ideal discriminator updates. In practical multilayer-perceptron parameterizations, these guarantees do not directly extend because optimization occurs over generator parameters.
- Setting: In the non-parametric setting, the generator distribution is analyzed as an estimator of the data distribution given sufficient capacity and training time.The analysis studies convergence in the space of probability density functions.
- Optimal discriminator: For a fixed generator, the discriminator is trained by maximizing the log-likelihood of distinguishing data samples from generated samples.The discriminator estimates whether an input came from pdata or pg.
- Global optimum: The virtual training criterion reaches its global minimum, −log(4), if and only if pg = pdata.The criterion can be expressed using the Jensen–Shannon divergence, which is non-negative and zero only for equal distributions.
- Convergence: With enough capacity, an optimal discriminator at each iteration, and generator updates that improve the criterion, Algorithm 1 makes pg converge to pdata.The convergence claim assumes the discriminator reaches its optimum before each generator update.
- Practical parameterization: Multilayer perceptrons restrict the family of attainable generator distributions and introduce multiple critical points in parameter space, so the non-parametric guarantees do not directly apply.The paper nevertheless motivates this parameterization by the practical performance of multilayer perceptrons.
5 Experiments
The experiments evaluate adversarial nets on MNIST, TFD, and CIFAR-10 using generated samples and Parzen-window likelihood estimates. The reported visualizations emphasize fair, uncorrelated samples and non-memorization, while the likelihood estimate has recognized limitations.
- Adversarial nets were trained on MNIST, the Toronto Face Database, and CIFAR-10 with rectifier/sigmoid generators, maxout discriminators, and discriminator dropout.
- The evaluation estimated test-set log-likelihood by fitting a Gaussian Parzen window to generated samples.The Gaussian width σ was selected by cross-validation.
- The Parzen-window likelihood method has high variance and performs poorly in high-dimensional spaces, limiting its reliability as an evaluation measure.The paper describes it as the best available method to the authors' knowledge for these models.
- Figure 2 presents fair random draws rather than cherry-picked samples, with nearest training examples shown to demonstrate that the model did not memorize the training set.The figure covers MNIST, TFD, and two CIFAR-10 architectures.
- The generated samples are uncorrelated because sampling does not depend on Markov-chain mixing, and Figure 3 visualizes linear interpolation in the generator's z space.
6 Advantages and disadvantages
Adversarial models avoid several computational burdens of earlier generative approaches, but require explicit coordination between generator and discriminator. They also lack an explicit density representation and can suffer diversity collapse when synchronization fails.
- Adversarial models require no Markov chains or inference during learning, use backpropagation for gradients, and accommodate a wide variety of functions.
- The framework has no explicit representation of p_g(x), so it directly models sampling rather than an explicit probability density.
- Training requires keeping D synchronized with G; otherwise, G may collapse many z values to one x and lose sample diversity.The paper calls this failure mode the “Helvetica scenario.”
- Adversarial networks can represent sharp or degenerate distributions, whereas Markov-chain methods require distributions sufficiently blurry for chains to mix between modes.
7 Conclusions and future work
The paper identifies several extensions of adversarial nets, including conditional generation, learned inference, conditional modeling, semi-supervised learning, and efficiency improvements. It concludes that the framework is viable and that these directions may be useful.
- The framework supports conditional generation by supplying c as input to both the generator and discriminator.
- An auxiliary network can learn approximate inference by predicting z from x, including after the generator has finished training.
- Families of parameter-sharing conditional models can approximately represent conditionals p(x_S | x̸_S), extending adversarial nets toward stochastic structured modeling.
- Discriminator or inference-net features could support semi-supervised classification when labeled data are limited.
- Better coordination of G and D and improved noise-distribution choices could substantially accelerate training.
- The paper demonstrates the viability of adversarial modeling and suggests that its proposed research directions could prove useful.