David Kristjanson Duvenaud

We perform scalable approximate inference in a recently-proposed family of continuous-depth Bayesian neural networks. In this model class, uncertainty about separate weights in each layer produces dynamics that follow a stochastic differential equation (SDE). We demonstrate gradient-based stochastic variational inference in this infinite-parameter setting, producing arbitrarily-flexible approximate posteriors. We also derive a novel gradient estimator that approaches zero variance as the approximate posterior approaches the true posterior. This approach inherits the memory-efficient training and tunable precision of neural ODEs.

We generalize gradient descent with momentum for learning in differentiable games to have complex-valued momentum. We give theoretical motivation for our method by proving convergence on bilinear zero-sum games for simultaneous and alternating updates. Our method gives real-valued parameter updates, making it a drop-in replacement for standard optimizers. We empirically demonstrate that complex-valued momentum can improve convergence in adversarial games-like generative adversarial networks-by showing we can find better solutions with an almost identical computational cost. We also show a practical generalization to a complex-valued Adam variant, which we use to train BigGAN to better inception scores on CIFAR-10.

Pre-training large models is useful, but adds many hyperparameters, such as task weights or augmentations in SimCLR. We give a scalable, gradient-based way to tune these hyperparamters. Because exact pre-training gradients are intractable, we approximate them. Specifically, we compose implicit differentiation for the long, almost-converged pre-training stage, with backprop through training for the short fine-tuning stage. We applied approximate pre-training gradients to tune thousands of task weights for graph-based protein function prediction, and to learn an entire data augmentation neural net for contrastive learning on electrocardiograms.

We attempt to combine the clarity and safety of high-level functional languages with the efficiency and parallelism of low-level numerical languages. We treat arrays as eagerly-memoized functions on typed index sets, allowing abstract function manipulations, such as currying, to work on arrays. In contrast to composing primitive bulk-array operations, we argue for an explicit nested indexing style that mirrors application of functions to arguments. We also introduce a fine-grained typed effects system which affords concise and automatically-parallelized in-place updates.

We propose a general and scalable approximate sampling strategy for probabilistic models with discrete variables. Our approach uses gradients of the likelihood function with respect to its discrete inputs to propose updates in a Metropolis-Hastings sampler. We show empirically that this approach outperforms generic samplers in a number of difficult settings including Ising models, Potts models, restricted Boltzmann machines, and factorial hidden Markov models. We also demonstrate the use of our improved sampler for training deep energy-based models on high dimensional discrete data. This approach outperforms variational auto-encoders and existing energy-based models. Finally, we give bounds showing that our approach is near-optimal in the class of samplers which propose local updates.

We meta-learn information helpful for training on a particular task or dataset, leveraging recent work on implicit differentiation. We explore applications such as learning weights for individual training examples, parameterizing label-dependent data augmentation policies, and representing attention masks that highlight salient image regions.

Energy-Based Models (EBMs) present a flexible and appealing way to represent uncertainty. In this work, we present a simple method for training EBMs at scale which uses an entropy-regularized generator to amortize the MCMC sampling typically used in EBM training. We apply our estimator to the recently proposed Joint Energy Model (JEM), where we match the original performance with faster and stable training. This allows us to extend JEM models to semi-supervised classification on tabular data from a variety of continuous domains.

We explore the use of exact per-sample Hessian-vector products and gradients to construct optimizers that are self-tuning and hyperparameter-free. Based on a dynamical model, we derive a curvature-corrected, noise-adaptive online gradient estimate. We prove that our model-based procedure converges in the noisy quadratic setting. Though we do not see similar gains in deep learning tasks, we match the performance of well-tuned optimizers. Our initial experiments indicate that when training deep nets our optimizer works too well, in a sense - it descends into regions of high variance and high curvature early on in the optimization, and gets stuck there.

Neural ODEs become expensive to solve numerically as training progresses. We introduce a differentiable surrogate for the time cost of standard numerical solvers using higher-order derivatives of solution trajectories. These derivatives are efficient to compute with Taylor-mode automatic differentiation. Optimizing this additional objective trades model performance against the time cost of solving the learned dynamics.

We generalize the adjoint sensitivity method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients with high-order adaptive solvers. Specifically, we derive a stochastic differential equation whose solution is the gradient, a memory-efficient algorithm for caching noise, and conditions under which numerical solutions converge. In addition, we combine our method with gradient-based stochastic variational inference for latent stochastic differential equations. We use our method to fit stochastic dynamics defined by neural networks, achieving competitive performance on a 50-dimensional motion capture dataset.

We use the implicit function theorem to scalably approximate gradients of the validation loss with respect to hyperparameters. This lets us train networks with millions of weights and millions of hyperparameters. For instance, we learn a data-augmentation network - where every weight is a hyperparameter tuned for validation performance - that outputs augmented training examples, from scratch. We also learn a distilled dataset where each feature in each datapoint is a hyperparameter, and tune millions of regularization hyperparameters.

We show that you can reinterpret standard classification architectures as energy-based generative models and train them as such. Doing this allows us to achieve state-of-the-art performance at both generative and discriminative modeling in a single model. Adding this energy-based training also improves calibration, out-of-distribution detection, and adversarial robustness.

We introduce an unbiased estimator of the log marginal likelihood and its gradients for latent variable models. In an encoder-decoder architecture, the parameters of the encoder can be optimized to minimize its variance of this estimator. We show that models trained using our estimator give better test-set likelihoods than a standard importance-sampling based approach for the same average computational cost.

We introduce a family of restricted neural network architectures that allow efficient computation of a family of differential operators involving dimension-wise derivatives, such as the divergence. Our proposed architecture has a Jacobian matrix composed of diagonal and hollow (zero-diagonal) components. We demonstrate these cheap differential operators on root-finding problems, exact density evaluation for continuous normalizing flows, and evaluating the Fokker-Planck equation.

We propose a new family of efficient and expressive deep generative models of graphs. We use graph neural networks to generate new edges conditioned on the already-sampled parts of the graph, reducing dependence on node ordering and bypasses the bottleneck caused by the sequential nature of RNNs. We achieve state-of-the-art time efficiency and sample quality compared to previous models, and generate graphs of up to 5000 nodes.

Time series with non-uniform intervals occur in many applications, and are difficult to model using standard recurrent neural networks. We generalize RNNs to have continuous-time hidden dynamics defined by ordinary differential equations. These models can naturally handle arbitrary time gaps between observations, and can explicitly model the probability of observation times using Poisson processes.

Invertible residual networks provide transformations where only Lipschitz conditions rather than architectural constraints are needed for enforcing invertibility. We give a tractable unbiased estimate of the log density, and improve these models in other ways. The resulting approach, called Residual Flows, achieves state-of-the-art performance on density estimation amongst flow-based models.

We show that standard ResNet architectures can be made invertible, allowing the same model to be used for classification, density estimation, and generation. Our approach only requires adding a simple normalization step during training. Invertible ResNets define a generative model which can be trained by maximum likelihood on unlabeled data. To compute likelihoods, we introduce a tractable approximation to the Jacobian log-determinant of a residual block. Our empirical evaluation shows that invertible ResNets perform competitively with both stateof-the-art image classifiers and flow-based generative models, something that has not been previously achieved with a single architecture.

Hyperparameter optimization can be formulated as a bilevel optimization problem, where the optimal parameters on the training set depend on the hyperparameters. We adapt regularization hyperparameters for neural networks by fitting compact approximations to the best-response function, which maps hyperparameters to optimal weights and biases. We show how to construct scalable best-response approximations for neural networks by modeling the best-response as a single network whose hidden units are gated conditionally on the regularizer.

Training normalized generative models such as Real NVP or Glow requires restricting their architectures to allow cheap computation of Jacobian determinants. Alternatively, if the transformation is specified by an ordinary differential equation, then the Jacobian's trace can be used. We use Hutchinson's trace estimator to give a scalable unbiased estimate of the log-density. The result is a continuous-time invertible generative model with unbiased density estimation and one-pass sampling, while allowing unrestricted neural network architectures. We demonstrate our approach on high-dimensional density estimation, image generation, and variational inference, improving the state-of-the-art among exact likelihood methods with efficient sampling.

When an image classifier makes a prediction, which parts of the image are relevant and why? We can rephrase this question to ask: which parts of the image, if they were not seen by the classifier, would most change its decision? Producing an answer requires marginalizing over images that could have been seen but weren't. We can sample plausible image in-fills by conditioning a generative model on the rest of the image. We then optimize to find the image regions that most change the classifier's decision after in-fill. Our approach contrasts with ad-hoc in-filling approaches, such as blurring or injecting noise, which generate inputs far from the data distribution, and ignore informative relationships between different parts of the image. Our method produces more compact and relevant saliency maps, with fewer artifacts compared to previous methods.

Models are usually tuned by nesting optimization of model weights inside the optimization of hyperparameters. We collapse this nested optimization into joint stochastic optimization of weights and hyperparameters. Our method trains a neural net to output approximately optimal weights as a function of hyperparameters. This method converges to locally optimal weights and hyperparameters for sufficiently large hypernetworks. We compare this method to standard hyperparameter optimization strategies and demonstrate its effectiveness for tuning thousands of hyperparameters.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

Variational autoencoders can be regularized to produce disentangled representations, in which each latent dimension has a distinct meaning. However, existing regularization schemes also hurt the model's ability to model the data. We show a simple method to regularize only the part that causes disentanglement. We also give a principled, classifier-free measure of disentanglement called the mutual information gap.

Bayesian neural nets combine the flexibility of deep learning with uncertainty estimation, but are usually approximated using a fully-factorized Guassian. We show that natural gradient ascent with adaptive weight noise implicitly fits a variational Gassuain posterior. This insight allows us to train full-covariance, fully factorized, or matrix-variate Gaussian variational posteriors using noisy versions of natural gradient, Adam, and K-FAC, respectively, allowing us to scale to modern-size convnets. Our noisy K-FAC algorithm makes better predictions and has better-calibrated uncertainty than existing methods. This leads to more efficient exploration in active learning and reinforcement learning.

Amortized inference allows latent-variable models to scale to large datasets. The quality of approximate inference is determined by two factors: a) the capacity of the variational distribution to match the true posterior and b) the ability of the recognition net to produce good variational parameters for each datapoint. We show that the recognition net giving bad variational parameters is often a bigger problem than using a Gaussian approximate posterior, because the generator can adapt to it.
Chris Cremer, Xuechen Li, David Duvenaud
International Conference on Machine Learning, 2018
paper | bibtex | slides

We learn low-variance, unbiased gradient estimators for any function of random variables. We backprop through a neural net surrogate of the original function, which is optimized to minimize gradient variance during the optimization of the original objective. We train discrete latent-variable models, and do continuous and discrete reinforcement learning with an adaptive, action-conditional baseline.

We develop a molecular autoencoder, which converts discrete representations of molecules to and from a continuous representation. This allows gradient-based optimization through the space of chemical compounds. Continuous representations also let us generate novel chemicals by interpolating between molecules.

We give a simple recipe for reducing the variance of the gradient of the variational evidence lower bound. The entire trick is just removing one term from the gradient. Removing this term leaves an unbiased gradient estimator whose variance approaches zero as the approximate posterior approaches the exact posterior. We also generalize this trick to mixtures and importance-weighted posteriors.

The standard interpretation of importance-weighted autoencoders is that they maximize a tighter, multi-sample lower bound than the standard evidence lower bound. We give an alternate interpretation: it optimizes the standard lower bound, but using a more complex distribution, which we show how to visualize.

We propose a general modeling and inference framework that combines the complementary strengths of probabilistic graphical models and deep learning methods. Our model family composes latent graphical models with neural network observation likelihoods. All components are trained simultaneously. We use this framework to automatically segment and categorize mouse behavior from raw depth video.

How do people learn about complex functional structure? We propose that humans use compositionality: complex structure is decomposed into simpler building blocks. We formalize this idea using a grammar over Gaussian process kernels. We show that people prefer compositional extrapolations, and argue that this is consistent with broad principles of human cognition.

Autograd automatically differentiates native Python and Numpy code. It can handle loops, ifs, recursion and closures, and it can even take derivatives of its own derivatives. It uses reverse-mode differentiation (a.k.a. backpropagation), which means it's efficient for gradient-based optimization. Check out the tutorial and the examples directory.

Stochastic gradient descent samples from a nonparametric distribution, implicitly defined by the transformation of the initial distribution by an optimizer. We track the loss of entropy during optimization to get a scalable estimate of the marginal likelihood. This Bayesian interpretation of SGD gives a theoretical foundation for popular tricks such as early stopping and ensembling. We evaluate our marginal likelihood estimator on neural network models.

We introduce a convolutional neural network that operates directly on graphs, allowing end-to-end learning of the entire feature pipeline. This architecture generalizes standard molecular fingerprints. These data-driven features are more interpretable, and have better predictive performance on a variety of tasks. Related work led to our Nature Materials paper.

Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of the validation loss with respect to all hyperparameters by differentiating through the entire training procedure. This lets us optimize thousands of hyperparameters, including step-size and momentum schedules, weight initialization distributions, richly parameterized regularization schemes, and neural net architectures.

We show that some standard differential equation solvers are equivalent to Gaussian process predictive means, giving them a natural way to handle uncertainty. This work is part of the larger probabilistic numerics research agenda, which interprets numerical algorithms as inference procedures so they can be better understood and extended.

• Introduction - GPs are hard to scale, but easy to train, and don't overfit (mostly).
• Expressing structure with kernels - How to build various kinds of models, with examples.
• Automatic model building - Machines can build custom models for a dataset by combining kernels.
• Automatic model description - Machines can also describe these models visually and with text.
• Deep Gaussian processes - A Bayesian version of deep neural nets, easy to analyze.
• Additive Gaussian processes - A simple way to decompose high-dimensional functions.
• Warped mixture models - Let you learn arbitrarily complicated cluster shapes.

We wrote a program which automatically writes reports summarizing automatically constructed models. A prototype for the automatic statistician project.

To suggest better neural network architectures, we analyze the properties of different priors on compositions of functions. We study deep Gaussian processes, a type of infinitely-wide, deep neural net. We also examine infinitely deep covariance functions. Finally, we show that you get additive covariance if you do dropout on Gaussian processes.

If you fit a mixture of Gaussians to a single cluster that is curved or heavy-tailed, your model will report that the data contains many clusters! To fix this problem, we warp a latent mixture of Gaussians into nonparametric cluster shapes. The low-dimensional latent mixture model summarizes the properties of the high-dimensional density manifolds describing the data.

How could an AI do statistics? To search through an open-ended class of structured, nonparametric regression models, we introduce a simple grammar which specifies composite kernels. These structured models often allow an interpretable decomposition of the function being modeled, as well as long-range extrapolation. Many common regression methods are special cases of this large family of models.

Two short animations illustrate the differences between a Metropolis-Hastings (MH) sampler and a Hamiltonian Monte Carlo (HMC) sampler, to the tune of the Harlem shake. This inspired several followup videos - benchmark your MCMC algorithm on these distributions!

Instead of the usual Monte-Carlo based methods for computing integrals of likelihood functions, we instead construct a surrogate model of the likelihood function, and infer its integral conditioned on a set of evaluations. This allows us to evaluate the likelihood wherever is most informative, instead of running a Markov chain. This means fewer evaluations to estimate integrals.

We prove several connections between a numerical integration method that minimizes a worst-case bound (herding), and a model-based way of estimating integrals (Bayesian quadrature). It turns out that both optimize the same criterion, and that Bayesian Quadrature does this optimally.

When functions have additive structure, we can extrapolate further than with standard Gaussian process models. We show how to efficiently integrate over exponentially-many ways of modeling a function as a sum of low-dimensional functions.

How can we take advantage of images labeled only by what objects they contain? By combining information across different scales, we use image-level labels (such as this image contains a cat) to infer what different classes of objects look like at the pixel-level, and where they occur in images. This work formed my M.Sc. thesis at UBC.