Generative Modeling by Estimating Gradients of the Data Distribution

https://arxiv.org/pdf/1907.05600

https://github.com/ermongroup/ncsn

Oct 2020 (NeurIPS 2019)

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Abstract

We introduce a new generative model where samples are produced via Langevin dynamics using gradients of the data distribution estimated with score matching. Because gradients can be ill-defined and hard to estimate when the data resides on low-dimensional manifolds, we perturb the data with different levels of Gaussian noise, and jointly estimate the corresponding scores, i.e., the vector fields of gradients of the perturbed data distribution for all noise levels. For sampling, we propose an annealed Langevin dynamics where we use gradients corresponding to gradually decreasing noise levels as the sampling process gets closer to the data manifold. Our framework allows flexible model architectures, requires no sampling during training or the use of adversarial methods, and provides a learning objective that can be used for principled model comparisons. Our models produce samples comparable to GANs on MNIST, CelebA and CIFAR-10 datasets, achieving a new state-of-the-art inception score of 8.87 on CIFAR-10. Additionally, we demonstrate that our models learn effective representations via image inpainting experiments.

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1. Introduction

Generative models have many applications in machine learning. To list a few, they have been used to generate high-fidelity images [26, 6], synthesize realistic speech and music fragments [58], improve the performance of semi-supervised learning [28, 10], detect adversarial examples and other anomalous data [54], imitation learning [22], and explore promising states in reinforcement learning [41]. Recent progress is mainly driven by two approaches: likelihood-based methods [17, 29, 11, 60] and generative adversarial networks (GAN [15]). The former uses log-likelihood (or a suitable surrogate) as the training objective, while the latter uses adversarial training to minimize f-divergences [40] or integral probability metrics [2, 55] between model and data distributions.

 

Although likelihood-based models and GANs have achieved great success, they have some intrinsic limitations. For example, likelihood-based models either have to use specialized architectures to build a normalized probability model (e.g., autoregressive models, flow models), or use surrogate losses (e.g., the evidence lower bound used in variational auto-encoders [29], contrastive divergence in energy-based models [21]) for training. GANs avoid some of the limitations of likelihood-based models, but their training can be unstable due to the adversarial training procedure. In addition, the GAN objective is not suitable for evaluating and comparing different GAN models. While other objectives exist for generative modeling, such as noise contrastive estimation [19] and minimum probability flow [50], these methods typically only work well for low-dimensional data.

 

In this paper, we explore a new principle for generative modeling based on estimating and sampling from the (Stein) score [33] of the logarithmic data density, which is the gradient of the log-density function at the input data point. This is a vector field pointing in the direction where the log data density grows the most. We use a neural network trained with score matching [24] to learn this vector field from data. We then produce samples using Langevin dynamics, which approximately works by gradually moving a random initial sample to high density regions along the (estimated) vector field of scores. However, there are two main challenges with this approach. First, if the data distribution is supported on a low dimensional manifold—as it is often assumed for many real world datasets—the score will be undefined in the ambient space, and score matching will fail to provide a consistent score estimator. Second, the scarcity of training data in low data density regions, e.g., far from the manifold, hinders the accuracy of score estimation and slows down the mixing of Langevin dynamics sampling. Since Langevin dynamics will often be initialized in low-density regions of the data distribution, inaccurate score estimation in these regions will negatively affect the sampling process. Moreover, mixing can be difficult because of the need of traversing low density regions to transition between modes of the distribution.

 

To tackle these two challenges, we propose to perturb the data with random Gaussian noise of various magnitudes. Adding random noise ensures the resulting distribution does not collapse to a low dimensional manifold. Large noise levels will produce samples in low density regions of the original (unperturbed) data distribution, thus improving score estimation. Crucially, we train a single score network conditioned on the noise level and estimate the scores at all noise magnitudes. We then propose an annealed version of Langevin dynamics, where we initially use scores corresponding to the highest noise level, and gradually anneal down the noise level until it is small enough to be indistinguishable from the original data distribution. Our sampling strategy is inspired by simulated annealing [30, 37] which heuristically improves optimization for multimodal landscapes.

 

Our approach has several desirable properties. First, our objective is tractable for almost all parameterizations of the score networks without the need of special constraints or architectures, and can be optimized without adversarial training, MCMC sampling, or other approximations during training. The objective can also be used to quantitatively compare different models on the same dataset. Experimentally, we demonstrate the efficacy of our approach on MNIST, CelebA [34], and CIFAR-10 [31]. We show that the samples look comparable to those generated from modern likelihood-based models and GANs. On CIFAR-10, our model sets the new state-of-the-art inception score of 8.87 for unconditional generative models, and achieves a competitive FID score of 25.32. We show that the model learns meaningful representations of the data by image inpainting experiments.

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2. Score-based generative modeling

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2.1. Score matching for score estimation

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2.2. Sampling with Langevin dynamics

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3. Challenges of score-based generative modeling

In this section, we analyze more closely the idea of score-based generative modeling. We argue that there are two major obstacles that prevent a naïve application of this idea.

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3.1. The manifold hypothesis

The manifold hypothesis states that data in the real world tend to concentrate on low dimensional manifolds embedded in a high dimensional space (a.k.a., the ambient space). This hypothesis empirically holds for many datasets, and has become the foundation of manifold learning [3, 47]. Under the manifold hypothesis, score-based generative models will face two key difficulties. First, since the score ∇x log pdata(x) is a gradient taken in the ambient space, it is undefined when x is confined to a low dimensional manifold. Second, the score matching objective Eq. (1) provides a consistent score estimator only when the support of the data distribution is the whole space (cf ., Theorem 2 in [24]), and will be inconsistent when the data reside on a low-dimensional manifold.

 

The negative effect of the manifold hypothesis on score estimation can be seen clearly from Fig. 1, where we train a ResNet (details in Appendix B.1) to estimate the data score on CIFAR-10. For fast training and faithful estimation of the data scores, we use the sliced score matching objective (Eq. (3)). As Fig. 1 (left) shows, when trained on the original CIFAR-10 images, the sliced score matching loss first decreases and then fluctuates irregularly. In contrast, if we perturb the data with a small Gaussian noise (such that the perturbed data distribution has full support over R^D), the loss curve will converge (right panel). Note that the Gaussian noise N (0, 0.0001) we impose is very small for images with pixel values in the range [0, 1], and is almost indistinguishable to human eyes.

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3.2. Low data density regions

The scarcity of data in low density regions can cause difficulties for both score estimation with score matching and MCMC sampling with Langevin dynamics.

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4. Noise Conditional Score Networks: learning and inference

We observe that perturbing data with random Gaussian noise makes the data distribution more amenable to score-based generative modeling. First, since the support of our Gaussian noise distribution is the whole space, the perturbed data will not be confined to a low dimensional manifold, which obviates difficulties from the manifold hypothesis and makes score estimation well-defined. Second, large Gaussian noise has the effect of filling low density regions in the original unperturbed data distribution; therefore score matching may get more training signal to improve score estimation. Furthermore, by using multiple noise levels we can obtain a sequence of noise-perturbed distributions that converge to the true data distribution. We can improve the mixing rate of Langevin dynamics on multimodal distributions by leveraging these intermediate distributions in the spirit of simulated annealing [30] and annealed importance sampling [37].

 

Built upon this intuition, we propose to improve score-based generative modeling by 1) perturbing the data using various levels of noise; and 2) simultaneously estimating scores corresponding to all noise levels by training a single conditional score network. After training, when using Langevin dynamics to generate samples, we initially use scores corresponding to large noise, and gradually anneal down the noise level. This helps smoothly transfer the benefits of large noise levels to low noise levels where the perturbed data are almost indistinguishable from the original ones. In what follows, we will elaborate more on the details of our method, including the architecture of our score networks, the training objective, and the annealing schedule for Langevin dynamics.

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4.1. Noise Conditional Score Networks

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4.2. Learning NCSNs via score matching

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4.3. NCSN inference via annealed Langevin dynamics

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5. Experiments

In this section, we demonstrate that our NCSNs are able to produce high quality image samples on several commonly used image datasets. In addition, we show that our models learn reasonable image representations by image inpainting experiments.

Setup

We use MNIST, CelebA [34], and CIFAR-10 [31] datasets in our experiments. For CelebA, the images are first center-cropped to 140 × 140 and then resized to 32 × 32. All images are rescaled so that pixel values are in [0, 1]. We choose L = 10 different standard deviations such that {σi} L i=1 is a geometric sequence with σ1 = 1 and σ10 = 0.01. Note that Gaussian noise of σ = 0.01 is almost indistinguishable to human eyes for image data. When using annealed Langevin dynamics for image generation, we choose T = 100 and  = 2 × 10−5 , and use uniform noise as our initial samples. We found the results are robust w.r.t. the choice of T, and  between 5 × 10−6 and 5 × 10−5 generally works fine. We provide additional details on model architecture and settings in Appendix A and B.

Image generation

In Fig. 5, we show uncurated samples from annealed Langevin dynamics for MNIST, CelebA and CIFAR-10. As shown by the samples, our generated images have higher or comparable quality to those from modern likelihood-based models and GANs. To intuit the procedure of annealed Langevin dynamics, we provide intermediate samples in Fig. 4, where each row shows how samples evolve from pure random noise to high quality images. More samples from our approach can be found in Appendix C. We also show the nearest neighbors of generated images in the training dataset in Appendix C.2, in order to demonstrate that our model is not simply memorizing training images. To show it is important to learn a conditional score network jointly for many noise levels and use annealed Langevin dynamics, we compare against a baseline approach where we only consider one noise level {σ1 = 0.01} and use the vanilla Langevin dynamics sampling method. Although this small added noise helps circumvent the difficulty of the manifold hypothesis (as shown by Fig. 1, things will completely fail if no noise is added), it is not large enough to provide information on scores in regions of low data density. As a result, this baseline fails to generate reasonable images, as shown by samples in Appendix C.1.

 

For quantitative evaluation, we report inception [48] and FID [20] scores on CIFAR-10 in Tab. 1. As an unconditional model, we achieve the state-of-the-art inception score of 8.87, which is even better than most reported values for class-conditional generative models. Our FID score 25.32 on CIFAR-10 is also comparable to top existing models, such as SNGAN [36]. We omit scores on MNIST and CelebA as the scores on these two datasets are not widely reported, and different preprocessing (such as the center crop size of CelebA) can lead to numbers not directly comparable.

Image inpainting

 

In Fig. 6, we demonstrate that our score networks learn generalizable and semantically meaningful image representations that allow it to produce diverse image inpaintings. Note that some previous models such as PixelCNN can only impute images in the raster scan order. In contrast, our method can naturally handle images with occlusions of arbitrary shapes by a simple modification of the annealed Langevin dynamics procedure (details in Appendix B.3). We provide more image inpainting results in Appendix C.5.

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6. Related work

Our approach has some similarities with methods that learn the transition operator of a Markov chain for sample generation [4, 51, 5, 16, 52]. For example, generative stochastic networks (GSN [4, 1]) use denoising autoencoders to train a Markov chain whose equilibrium distribution matches the data distribution. Similarly, our method trains the score function used in Langevin dynamics to sample from the data distribution. However, GSN often starts the chain very close to a training data point, and therefore requires the chain to transition quickly between different modes. In contrast, our annealed Langevin dynamics are initialized from unstructured noise. Nonequilibrium Thermodynamics (NET [51]) used a prescribed diffusion process to slowly transform data into random noise, and then learned to reverse this procedure by training an inverse diffusion. However, NET is not very scalable because it requires the diffusion process to have very small steps, and needs to simulate chains with thousands of steps at training time.

 

Previous approaches such as Infusion Training (IT [5]) and Variational Walkback (VW [16]) also employed different noise levels/temperatures for training transition operators of a Markov chain. Both IT and VW (as well as NET) train their models by maximizing the evidence lower bound of a suitable marginal likelihood. In practice, they tend to produce blurry image samples, similar to variational autoencoders. In contrast, our objective is based on score matching instead of likelihood, and we can produce images comparable to GANs.

 

There are several structural differences that further distinguish our approach from previous methods discussed above. First, we do not need to sample from a Markov chain during training. In contrast, the walkback procedure of GSNs needs multiple runs of the chain to generate “negative samples”. Other methods including NET, IT, and VW also need to simulate a Markov chain for every input to compute the training loss. This difference makes our approach more efficient and scalable for training deep models. Secondly, our training and sampling methods are decoupled from each other. For score estimation, both sliced and denoising score matching can be used. For sampling, any method based on scores is applicable, including Langevin dynamics and (potentially) Hamiltonian Monte Carlo [38]. Our framework allows arbitrary combinations of score estimators and (gradient-based) sampling approaches, whereas most previous methods tie the model to a specific Markov chain. Finally, our approach can be used to train energy-based models (EBM) by using the gradient of an energy-based model as the score model. In contrast, it is unclear how previous methods that learn transition operators of Markov chains can be directly used for training EBMs.

 

Score matching was originally proposed for learning EBMs. However, many existing methods based on score matching are either not scalable [24] or fail to produce samples of comparable quality to VAEs or GANs [27, 49]. To obtain better performance on training deep energy-based models, some recent works have resorted to contrastive divergence [21], and propose to sample with Langevin dynamics for both training and testing [12, 39]. However, unlike our approach, contrastive divergence uses the computationally expensive procedure of Langevin dynamics as an inner loop during training. The idea of combining annealing with denoising score matching has also been investigated in previous work under different contexts. In [14, 7, 66], different annealing schedules on the noise for training denoising autoencoders are proposed. However, their work is on learning representations for improving the performance of classification, instead of generative modeling. The method of denoising score matching can also be derived from the perspective of Bayes least squares [43, 44], using techniques of Stein’s Unbiased Risk Estimator [35, 56].

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7. Conclusion

We propose the framework of score-based generative modeling where we first estimate gradients of data densities via score matching, and then generate samples via Langevin dynamics. We analyze several challenges faced by a naïve application of this approach, and propose to tackle them by training Noise Conditional Score Networks (NCSN) and sampling with annealed Langevin dynamics. Our approach requires no adversarial training, no MCMC sampling during training, and no special model architectures. Experimentally, we show that our approach can generate high quality images that were previously only produced by the best likelihood-based models and GANs. We achieve the new state-of-the-art inception score on CIFAR-10, and an FID score comparable to SNGANs.

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A. Architectures

The architecture of our NCSNs used in the experiments has three important components: instance normalization, dilated convolutions and U-Net-type architectures. Below we give more background on them and discuss how we modified them to suit our purpose. For more comprehensive details and a reference implementation, we recommend the readers to check our publicly available code base. Our score networks are implemented in PyTorch. Code and checkpoints are available at https://github.com/ermongroup/ncsn.

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