Diffusion-based generative models have demonstrated a capacity for perceptually impressive synthesis, but can they also be great likelihood-based models? We answer this in the affirmative, and introduce a family of diffusion-based generative models that obtain state-of-the-art likelihoods on standard image density estimation benchmarks. Unlike other diffusion-based models, our method allows for efficient optimization of the noise schedule jointly with the rest of the model. We show that the variational lower bound (VLB) simplifies to a remarkably short expression in terms of the signal-to-noise ratio of the diffused data, thereby improving our theoretical understanding of this model class. Using this insight, we prove an equivalence between several models proposed in the literature. In addition, we show that the continuous-time VLB is invariant to the noise schedule, except for the signal-to-noise ratio at its endpoints. This enables us to learn a noise schedule that minimizes the variance of the resulting VLB estimator, leading to faster optimization. Combining these advances with architectural improvements, we obtain state-of-the-art likelihoods on image density estimation benchmarks, outperforming autoregressive models that have dominated these benchmarks for many years, with often significantly faster optimization. In addition, we show how to turn the model into a bits-back compression scheme, and demonstrate lossless compression rates close to the theoretical optimum.

While energy-based models (EBMs) exhibit a number of desirable properties, training and sampling on high-dimensional datasets remains challenging. Inspired by recent progress on diffusion probabilistic models, we present a diffusion recovery likelihood method to tractably learn and sample from a sequence of EBMs trained on increasingly noisy versions of a dataset. Each EBM is trained by maximizing the recovery likelihood: the conditional probability of the data at a certain noise level given their noisy versions at a higher noise level. The recovery likelihood objective is more tractable than the marginal likelihood objective, since it only requires MCMC sampling from a relatively concentrated conditional distribution. Moreover, we show that this estimation method is theoretically consistent: it learns the correct conditional and marginal distributions at each noise level, given sufficient data. After training, synthesized images can be generated efficiently by a sampling process that initializes from a spherical Gaussian distribution and progressively samples the conditional distributions at decreasingly lower noise levels. Our method generates high fidelity samples on various image datasets. On unconditional CIFAR-10 our method achieves FID 9.60 and inception score 8.58, superior to the majority of GANs. Moreover, we demonstrate that unlike previous work on EBMs, our long-run MCMC samples from the conditional distributions do not diverge and still represent realistic images, allowing us to accurately estimate the normalized density of data even for high-dimensional datasets.

Creating noise from data is easy; creating data from noise is generative modeling. We present a stochastic differential equation (SDE) that smoothly transforms a complex data distribution to a known prior distribution by slowly injecting noise, and a corresponding reverse-time SDE that transforms the prior distribution back into the data distribution by slowly removing the noise. Crucially, the reverse-time SDE depends only on the time-dependent gradient field (a.k.a., score) of the perturbed data distribution. By leveraging advances in score-based generative modeling, we can accurately estimate these scores with neural networks, and use numerical SDE solvers to generate samples. We show that this framework encapsulates previous approaches in diffusion probabilistic modeling and score-based generative modeling, and allows for new sampling procedures. In particular, we introduce a predictor-corrector framework to correct errors in the evolution of the discretized reverse-time SDE. We also derive an equivalent neural ODE that samples from the same distribution as the SDE, which enables exact likelihood computation, and improved sampling efficiency. In addition, our framework enables conditional generation with an unconditional model, as we demonstrate with experiments on class-conditional generation, image inpainting, and colorization. Combined with multiple architectural improvements, we achieve record-breaking performance for unconditional image generation on CIFAR-10 with an Inception score of 9.89 and FID of 2.20, a competitive likelihood of 3.10 bits/dim, and demonstrate high fidelity generation of $1024 \times 1024$ images for the first time from a score-based generative model.

In discriminative settings such as regression and classification there are two random variables at play, the inputs X and the targets Y. Here, we demonstrate that the Variational Information Bottleneck can be viewed as a compromise between fully empirical and fully Bayesian objectives, attempting to minimize the risks due to finite sampling of Y only. We argue that this approach provides some of the benefits of Bayes while requiring only some of the work.

Non-saturating generative adversarial network (GAN) training is widely used and has continued to obtain groundbreaking results. However so far this approach has lacked strong theoretical justification, in contrast to alternatives such as f-GANs and Wasserstein GANs which are motivated in terms of approximate divergence minimization. In this paper we show that non-saturating GAN training does in fact approximately minimize a particular f-divergence. We develop general theoretical tools to compare and classify f-divergences and use these to show that the new f-divergence is qualitatively similar to reverse KL. These results help to explain the high sample quality but poor diversity often observed empirically when using this scheme.

Much as replacing hand-designed features with learned functions has revolutionized how we solve perceptual tasks, we believe learned algorithms will transform how we train models. In this work we focus on general-purpose learned optimizers capable of training a wide variety of problems with no user-specified hyperparameters. We introduce a new, neural network parameterized, hierarchical optimizer with access to additional features such as validation loss to enable automatic regularization. Most learned optimizers have been trained on only a single task, or a small number of tasks. We train our optimizers on thousands of tasks, making use of orders of magnitude more compute, resulting in optimizers that generalize better to unseen tasks. The learned optimizers not only perform well, but learn behaviors that are distinct from existing first order optimizers. For instance, they generate update steps that have implicit regularization and adapt as the problem hyperparameters (e.g. batch size) or architecture (e.g. neural network width) change. Finally, these learned optimizers show evidence of being useful for out of distribution tasks such as training themselves from scratch.

Invertible flow-based generative models are an effective method for learning to generate samples, while allowing for tractable likelihood computation and inference. However, the invertibility requirement restricts models to have the same latent dimensionality as the inputs. This imposes significant architectural, memory, and computational costs, making them more challenging to scale than other classes of generative models such as Variational Autoencoders (VAEs). We propose a generative model based on probability flows that does away with the bijectivity requirement on the model and only assumes injectivity. This also provides another perspective on regularized autoencoders (RAEs), with our final objectives resembling RAEs with specific regularizers that are derived by lower bounding the probability flow objective. We empirically demonstrate the promise of the proposed model, improving over VAEs and AEs in terms of sample quality.

Certain biological neurons demonstrate a remarkable capability to optimally compress the history of sensory inputs while being maximally informative about the future. In this work, we investigate if the same can be said of artificial neurons in recurrent neural networks (RNNs) trained with maximum likelihood. In experiments on two datasets, restorative Brownian motion and a hand-drawn sketch dataset, we find that RNNs are sub-optimal in the information plane. Instead of optimally compressing past information, they extract additional information that is not relevant for predicting the future. Overcoming this limitation may require alternative training procedures and architectures, or objectives beyond maximum likelihood estimation. Remembering past events is a critical component of predicting the future and acting in the world. An information-theoretic quantification of how much observing the past can help in predicting the future is given by the predictive information (Bialek et al., 2001). The predictive information is the mutual information (MI) between a finite set of observations (the past of a sequence) and an infinite number of additional draws from the same process (the future of a sequence).

Deploying machine learning systems in the real world requires both high accuracy on clean data and robustness to naturally occurring corruptions. While architectural advances have led to improved accuracy, building robust models remains challenging. Prior work has argued that there is an inherent trade-off between robustness and accuracy, which is exemplified by standard data augment techniques such as Cutout, which improves clean accuracy but not robustness, and additive Gaussian noise, which improves robustness but hurts accuracy. To overcome this trade-off, we introduce Patch Gaussian, a simple augmentation scheme that adds noise to randomly selected patches in an input image. Models trained with Patch Gaussian achieve state of the art on the CIFAR-10 and ImageNetCommon Corruptions benchmarks while also improving accuracy on clean data. We find that this augmentation leads to reduced sensitivity to high frequency noise(similar to Gaussian) while retaining the ability to take advantage of relevant high frequency information in the image (similar to Cutout). Finally, we show that Patch Gaussian can be used in conjunction with other regularization methods and data augmentation policies such as AutoAugment, and improves performance on the COCO object detection benchmark.

While normalizing flows have led to significant advances in modeling high-dimensional continuous distributions, their applicability to discrete distributions remains unknown. In this paper, we show that flows can in fact be extended to discrete events---and under a simple change-of-variables formula not requiring log-determinant-Jacobian computations. Discrete flows have numerous applications. We consider two flow architectures: discrete autoregressive flows that enable bidirectionality, allowing, for example, tokens in text to depend on both left-to-right and right-to-left contexts in an exact language model; and discrete bipartite flows that enable efficient non-autoregressive generation as in RealNVP. Empirically, we find that discrete autoregressive flows outperform autoregressive baselines on synthetic discrete distributions, an addition task, and Potts models; and bipartite flows can obtain competitive performance with autoregressive baselines on character-level language modeling for Penn Tree Bank and text8.