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Flow-GAN: Combining Maximum Likelihood and Adversarial Learning in Generative Models
Aditya Grover, Manik Dhar, Stefano Ermon
TL;DR
GANs can generate strong samples but lack explicit densities, making quantitative evaluation difficult. Flow-GANs combine an invertible flow generator with adversarial and likelihood training; hybrid training achieves high held-out likelihood while retaining sample fidelity on MNIST and CIFAR-10.
Problem
GANs can generate better samples than maximum-likelihood models, but their lack of explicit densities makes quantitative evaluation challenging.
Source: §1 · p. 1Method
Flow-GANs use an invertible normalizing-flow generator to enable exact likelihood evaluation alongside adversarial and maximum-likelihood training.
Source: §1 · pp. 1–2Results
Hybrid training attains good log-likelihoods and high-quality samples on MNIST and CIFAR-10.
Source: §1 · p. 2Takeaways & Limitations
On CIFAR-10 the hybrid objective trades off sample quality and held-out likelihood, while on MNIST it outperforms MLE and adversarial learning on both.
Source: §1 · p. 2
Abstract
from arXiv · showhide
Adversarial learning of probabilistic models has recently emerged as a promising alternative to maximum likelihood. Implicit models such as generative adversarial networks (GAN) often generate better samples compared to explicit models trained by maximum likelihood. Yet, GANs sidestep the characterization of an explicit density which makes quantitative evaluations challenging. To bridge this gap, we propose Flow-GANs, a generative adversarial network for which we can perform exact likelihood evaluation, thus supporting both adversarial and maximum likelihood training. When trained adversarially, Flow-GANs generate high-quality samples but attain extremely poor log-likelihood scores, inferior even to a mixture model memorizing the training data; the opposite is true when trained by maximum likelihood. Results on MNIST and CIFAR-10 demonstrate that hybrid training can attain high held-out likelihoods while retaining visual fidelity in the generated samples.
1 Introduction
GANs can produce strong samples without explicit densities, complicating quantitative evaluation and density-based applications. Flow-GANs enable exact likelihood evaluation and support both adversarial and maximum-likelihood training; hybrid training improves the balance between likelihood and sample quality on MNIST and CIFAR-10.
1 Introduction
- GANs lack explicit densities, complicating quantitative evaluation; sample-quality metrics can miss memorization or missing modes, while AIS and KDE can be slow and rely on Gaussian observation assumptions.The paper also demonstrates limitations of AIS and KDE for evaluating and ranking implicit models.Source: §1 · pp. 1–2
- Flow-GANs pair a GAN with an invertible normalizing-flow generator, enabling exact likelihood evaluation, exact latent posterior inference, and efficient ancestral sampling.The generator transforms a prior noise density into a model density.Source: §1 · pp. 1–2
- Adversarial learning yields better sample quality than MLE but orders-of-magnitude-worse log-likelihoods; a Gaussian mixture baseline outperforms adversarial models on both sample quality and held-out likelihood.The analysis attributes the poor likelihoods to an ill-conditioned generator Jacobian, suggesting mode collapse rather than training-set overfitting.Source: §1 · pp. 1–2
- The hybrid objective combines adversarial training with observed-data log-likelihood, trading off sample quality and held-out likelihood on CIFAR-10 and improving both metrics over MLE and adversarial learning on MNIST.It is intended to address the opposing strengths of adversarial learning and MLE.Source: §1 · p. 2
2 Preliminaries
The preliminaries frame generative modeling as learning a probability density through a chosen divergence and optimization procedure. They contrast likelihood-based and adversarial objectives, then describe how GANs use latent-variable generators and discriminators.
2 Preliminaries
- Generative modeling is framed as learning a parameterized density from i.i.d. data by minimizing a divergence between data and model distributions.The choice of divergence and optimization procedure determines the learning objective.Source: §2 · p. 2
2.1 Maximum likelihood estimation
- MLE minimizes the KL divergence from the data distribution to the model and requires evaluating the model density, so it applies naturally to prescribed generative models.The text notes that maximum-likelihood estimators are asymptotically efficient under conditions.Source: §2.1 · p. 2, §2.3 · p. 2
2.2 Adversarial learning
- Adversarial learning extends beyond KL to divergences including Jensen–Shannon and Wasserstein; its Monte Carlo objective requires model samples rather than an explicit likelihood.This allows tractably sampled implicit models to be learned adversarially.Source: §2.2 · p. 2
2.3 Adversarial learning of latent variable models
- GANs transform samples from a tractable prior through a differentiable generator, while a discriminator supplies the adversarial objective; their implicit density makes likelihood evaluation challenging.The passage also situates adversarial learning among approaches to latent-variable models, whose likelihoods can require intractable marginalization.Source: §2.3 · p. 3
3 Flow Generative Adversarial Networks
Flow-GAN uses an invertible flow generator to support exact likelihood evaluation alongside adversarial training. Across MNIST and CIFAR-10, adversarial training improves sample-quality scores over MLE but yields much worse likelihoods; a training-point GMM can outperform it on both measures for some CIFAR-10 bandwidths.
3 Flow Generative Adversarial Networks
- An invertible flow generator lets Flow-GAN evaluate data likelihood by change of variables when the prior density and Jacobian determinant are tractable.Compositions of invertible transformations can produce complex densities while retaining tractable likelihood evaluation.Source: §3 · p. 3
3.1 Learning objectives
- Flow-GAN supports MLE training with a redundant discriminator and adversarial training with ancestral sampling from the prior.The adversarial objective can use a GAN loss such as WGAN.Source: §3.1 · pp. 3–4
3.2 Evaluation metrics and experimental setup
- The experiments compare held-out log-likelihood and image sample quality on MNIST and CIFAR-10, using NICE and Real-NVP flow generators, respectively.The adversarial objective uses Wasserstein distance, with a gradient penalty on the critic.Source: §3.2 · p. 4
3.3 Evaluation results
- After sufficient training, ADV log-likelihoods are orders of magnitude worse than MLE’s, although ADV achieves significantly better final MODE/Inception scores.The NLL gap appears in both training and validation curves, while visual inspection confirms the sample-quality comparison.Source: Figure 1 · p. 4, Figure 2 · p. 5, §3.3 · p. 5
3.4 Gaussian mixture models
- For CIFAR-10, some bandwidths let the training-point GMM outperform ADV on both held-out NLL and sample-quality metrics.The GMM’s shared bandwidth is selected using validation NLL; the observations are reported to hold for MNIST as well.Source: §3.4 · p. 5, Figure 3 · p. 6
4 Hybrid learning of Flow-GANs
The hybrid objective combines adversarial training with a data log-likelihood term, and its tunable weight interpolates between ADV and MLE. Results show a trade-off on CIFAR-10, while on MNIST the hybrid objective outperforms both objectives on test likelihood and sample quality.
4 Hybrid learning of Flow-GANs
- The hybrid objective augments the GAN minimax objective with an observed-data log-likelihood term, and λ interpolates between ADV and MLE.λ = 0 gives plain adversarial training; a very high λ gives MLE.Source: §4 · p. 6
- The reported comparisons use test log-likelihoods for the best validated MLE and ADV models and the highest MODE/Inception scores observed during training.The MNIST and CIFAR-10 results are summarized in Tables 1 and 2, respectively.Source: §4 · p. 6, Table 1 · p. 6, Table 2 · p. 6
- On MNIST, the hybrid objective outperforms MLE and ADV on both test log-likelihood and sample-quality metrics.On CIFAR-10, results follow the expected trade-off between the two goals.Source: §4 · p. 6, Table 1 · p. 6, Table 2 · p. 6
5 Interpreting the results
The paper finds that AIS and KDE can misestimate and misrank Flow-GAN likelihoods relative to exact evaluation. Its analysis links ADV’s poor likelihoods to a widely spread, ill-conditioned Jacobian singular-value distribution, while the hybrid objective shifts that distribution toward MLE’s.
5 Interpreting the results
- On MNIST, AIS and KDE estimates differ substantially from exact Flow-GAN likelihoods and each gives a ranking that disagrees with the exact MLE > Hybrid > ADV order.AIS ranks ADV > Hybrid > MLE, whereas KDE ranks Hybrid > MLE > ADV.Source: §5 · p. 6
5.1 Explaining log-likelihood trends
- ADV’s Jacobian singular values span a wide range, making the Jacobian ill-conditioned; its average log determinant is −15588.34, versus −4170.34 for MLE and −5184.40 for Hybrid.The analysis says ADV compresses latent-space volume into a smaller observed-space support, so tiny training or held-out data perturbations can manifest as poor likelihoods.Source: §5.1 · p. 7
- The hybrid objective corrects ADV’s singular-value behavior, with a distribution of magnitudes that closely matches MLE’s.The trend toward lower singular values also appears for standard DCGAN architectures trained with Wasserstein distance or Jensen-Shannon divergence.Source: §5.1 · p. 7
6 Discussion
Normalizing flows avoid the approximate inference required by some hybrid latent-variable approaches, but require an invertible generator; this enables exact likelihood evaluation. The discussion also relates adversarial learning to low-dimensional support and mode collapse.
6 Discussion
- Flow models avoid the extra approximate-inference complexity of VAE-GAN approaches, but require an invertible generator and provide tractable exact likelihoods.The invertibility constraint is the trade-off relative to models such as VAEs.Source: §6 · p. 7
- Earlier flow models were used for maximum-likelihood estimation of fully observed and latent-variable models.The cited examples include Dinh, Krueger, and Bengio (2014); Rezende and Mohamed (2015); Kingma, Salimans, and Welling (2016); and Dinh, Sohl-Dickstein, and Bengio (2017).Source: §6 · p. 7
- Adversarial learning can produce low-dimensional-support distributions, a tendency associated with mode collapse.Prior approaches described here detect mode collapse visually or heuristically and focus on stabilizing GAN training.Source: §6 · p. 7
7 Conclusion
Flow-GAN enables quantitative comparison of adversarial learning and maximum likelihood using tractable likelihood evaluation. The paper reports that adversarial learning can yield superior samples despite very low likelihoods, while a hybrid objective combines both training goals.
7 Conclusion
- Adversarial learning yields superior-quality samples but very low likelihoods on training and validation data, while a naive Gaussian mixture can outperform it on both likelihood and sample metrics.AIS- and KDE-based estimates can fail to detect this pattern and can poorly approximate true likelihoods for the studied models.Source: §7 · p. 7
- Jacobian analysis indicates that adversarial learning tends toward low-support distributions, which can lead to low likelihoods.The paper proposes a hybrid objective combining MLE and adversarial loss terms to address this behavior.Source: §7 · p. 7
- The hybrid objective is proposed for applications that require both density estimation and sample generation.The paper identifies such applications as a future direction.Source: §7 · p. 7
A Experimental setup details
The experiments use MNIST and CIFAR-10 images, with a held-out validation set for CIFAR-10. Pixel values are dequantized and scaled before modeling.
A Experimental setup details
- The discrete pixel data are dequantized by adding uniform noise between 0 and 1 to every pixel, then scaled to [0, 1].This follows Uria, Murray, and Larochelle (2013).Source: A Experimental setup details · p. 9
- The MNIST and CIFAR-10 Flow-GANs use logistic and isotropic prior densities, respectively.The cited architecture precedents are Dinh, Krueger, and Bengio (2014) and Dinh, Sohl-Dickstein, and Bengio (2017).Source: A Experimental setup details · p. 9
B Sample quality
Adversarial training achieves stronger sample-quality scores than maximum likelihood on MNIST and CIFAR-10. On MNIST, Gaussian mixtures can outperform adversarial models on both held-out likelihood and sample-quality metrics.
B Sample quality
- Adversarial training significantly outperforms maximum likelihood on final MODE/Inception sample-quality scores for both MNIST and CIFAR-10.The curves track sample-quality metrics during training.Source: B Sample quality · p. 9
C Gaussian mixture models
- Gaussian mixtures outperform adversarially learned models on MNIST in both held-out log-likelihood and sampling metrics within Figure 6’s green shaded region.The comparison holds for any bandwidth parameter in that region.Source: Figure 6 · p. 10, C Gaussian mixture models · p. 10
- For relatively small bandwidths, Gaussian-mixture samples are high quality, and their held-out likelihoods exceed those of adversarial models with the best MODE/Inception scores.The GMM fits Gaussian densities around every training point.Source: C Gaussian mixture models · p. 10
D Explaining log-likelihood trends
On CIFAR-10, ADV generator Jacobian singular values are spread out, while Hybrid values are more concentrated, similar to MLE. The Jacobian log determinants follow ADV<Hybrid<MLE.
D Explaining log-likelihood trends
- Figure 8 suggests ADV generator Jacobians are ill-conditioned, with broadly spread singular values, while Hybrid values are more concentrated, similar to MLE.The figure plots the CDF of singular-value magnitudes for CIFAR-10.Source: D Explaining log-likelihood trends · p. 10, Figure 8 · p. 10
- The log determinants of the Jacobian are −12818.84 for MLE, −21848.09 for ADV, and −14729.51 for Hybrid, following ADV<Hybrid<MLE.The authors interpret this trend as empirical evidence that adversarial training favors distributions with smaller support.Source: D Explaining log-likelihood trends · p. 10
- Figure 7 reports that the Gaussian Mixture Model baseline has better MODE/Inception scores than ADV models on MNIST and CIFAR-10.Source: Figure 7 · p. 10