Martin Jankowiak

Martin Jankowiak

Bayesian variable selection is a powerful tool for data analysis, as it offers a principled method for variable selection that accounts for prior information and uncertainty. However, wider adoption of Bayesian variable selection has been hampered by computational challenges, especially in difficult regimes with a large number of covariates P or non-conjugate likelihoods. To scale to the large P regime we introduce an efficient MCMC scheme whose cost per iteration is sublinear in P. In addition we show how this scheme can be extended to generalized linear models for count data, which are prevalent in biology, ecology, economics, and beyond. In particular we design efficient algorithms for variable selection in binomial and negative binomial regression, which includes logistic regression as a special case. In experiments we demonstrate the effectiveness of our methods, including on cancer and maize genomic data.

Martin Jankowiak, Du Phan

Traditional approaches to variational inference rely on parametric families of variational distributions, with the choice of family playing a critical role in determining the accuracy of the resulting posterior approximation. Simple mean-field families often lead to poor approximations, while rich families of distributions like normalizing flows can be difficult to optimize and usually do not incorporate the known structure of the target distribution due to their black-box nature. To expand the space of flexible variational families, we revisit Variational Rejection Sampling (VRS) [Grover et al., 2018], which combines a parametric proposal distribution with rejection sampling to define a rich non-parametric family of distributions that explicitly utilizes the known target distribution. By introducing a low-variance reparameterized gradient estimator for the parameters of the proposal distribution, we make VRS an attractive inference strategy for models with continuous latent variables. We argue theoretically and demonstrate empirically that the resulting method--Reparameterized Variational Rejection Sampling (RVRS)--offers an attractive trade-off between computational cost and inference fidelity. In experiments we show that our method performs well in practice and that it is well-suited for black-box inference, especially for models with local latent variables.

Martin Jankowiak, Du Phan

Variational inference is a powerful paradigm for approximate Bayesian inference with a number of appealing properties, including support for model learning and data subsampling. By contrast MCMC methods like Hamiltonian Monte Carlo do not share these properties but remain attractive since, contrary to parametric methods, MCMC is asymptotically unbiased. For these reasons researchers have sought to combine the strengths of both classes of algorithms, with recent approaches coming closer to realizing this vision in practice. However, supporting data subsampling in these hybrid methods can be a challenge, a shortcoming that we address by introducing a surrogate likelihood that can be learned jointly with other variational parameters. We argue theoretically that the resulting algorithm permits the user to make an intuitive trade-off between inference fidelity and computational cost. In an extensive empirical comparison we show that our method performs well in practice and that it is well-suited for black-box inference in probabilistic programming frameworks.

Martin Jankowiak

Bayesian variable selection is a powerful tool for data analysis, as it offers a principled method for variable selection that accounts for prior information and uncertainty. However, wider adoption of Bayesian variable selection has been hampered by computational challenges, especially in difficult regimes with a large number of covariates or non-conjugate likelihoods. Generalized linear models for count data, which are prevalent in biology, ecology, economics, and beyond, represent an important special case. Here we introduce an efficient MCMC scheme for variable selection in binomial and negative binomial regression that exploits Tempered Gibbs Sampling (Zanella and Roberts, 2019) and that includes logistic regression as a special case. In experiments we demonstrate the effectiveness of our approach, including on cancer data with seventeen thousand covariates.

Martin Jankowiak, Geoff Pleiss

We introduce a simple and scalable method for training Gaussian process (GP) models that exploits cross-validation and nearest neighbor truncation. To accommodate binary and multi-class classification we leverage Pòlya-Gamma auxiliary variables and variational inference. In an extensive empirical comparison with a number of alternative methods for scalable GP regression and classification, we find that our method offers fast training and excellent predictive performance. We argue that the good predictive performance can be traced to the non-parametric nature of the resulting predictive distributions as well as to the cross-validation loss, which provides robustness against model mis-specification.

David Eriksson, Martin Jankowiak

Bayesian optimization (BO) is a powerful paradigm for efficient optimization of black-box objective functions. High-dimensional BO presents a particular challenge, in part because the curse of dimensionality makes it difficult to define -- as well as do inference over -- a suitable class of surrogate models. We argue that Gaussian process surrogate models defined on sparse axis-aligned subspaces offer an attractive compromise between flexibility and parsimony. We demonstrate that our approach, which relies on Hamiltonian Monte Carlo for inference, can rapidly identify sparse subspaces relevant to modeling the unknown objective function, enabling sample-efficient high-dimensional BO. In an extensive suite of experiments comparing to existing methods for high-dimensional BO we demonstrate that our algorithm, Sparse Axis-Aligned Subspace BO (SAASBO), achieves excellent performance on several synthetic and real-world problems without the need to set problem-specific hyperparameters.

Geoff Pleiss, Martin Jankowiak, David Eriksson et al.

Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians $\mathcal{N}(\mathbf 0, \mathbf K)$ or whitening a vector $\mathbf b$ against covariance matrix $\mathbf K$. While existing methods typically require $O(N^3)$ computation, we introduce a highly-efficient quadratic-time algorithm for computing $\mathbf K^{1/2} \mathbf b$, $\mathbf K^{-1/2} \mathbf b$, and their derivatives through matrix-vector multiplication (MVMs). Our method combines Krylov subspace methods with a rational approximation and typically achieves $4$ decimal places of accuracy with fewer than $100$ MVMs. Moreover, the backward pass requires little additional computation. We demonstrate our method's applicability on matrices as large as $50,\!000 \times 50,\!000$ - well beyond traditional methods - with little approximation error. Applying this increased scalability to variational Gaussian processes, Bayesian optimization, and Gibbs sampling results in more powerful models with higher accuracy.

Martin Jankowiak, Geoff Pleiss, Jacob R. Gardner

We introduce Deep Sigma Point Processes, a class of parametric models inspired by the compositional structure of Deep Gaussian Processes (DGPs). Deep Sigma Point Processes (DSPPs) retain many of the attractive features of (variational) DGPs, including mini-batch training and predictive uncertainty that is controlled by kernel basis functions. Importantly, since DSPPs admit a simple maximum likelihood inference procedure, the resulting predictive distributions are not degraded by any posterior approximations. In an extensive empirical comparison on univariate and multivariate regression tasks we find that the resulting predictive distributions are significantly better calibrated than those obtained with other probabilistic methods for scalable regression, including variational DGPs--often by as much as a nat per datapoint.

Du Phan, Neeraj Pradhan, Martin Jankowiak

NumPyro is a lightweight library that provides an alternate NumPy backend to the Pyro probabilistic programming language with the same modeling interface, language primitives and effect handling abstractions. Effect handlers allow Pyro's modeling API to be extended to NumPyro despite its being built atop a fundamentally different JAX-based functional backend. In this work, we demonstrate the power of composing Pyro's effect handlers with the program transformations that enable hardware acceleration, automatic differentiation, and vectorization in JAX. In particular, NumPyro provides an iterative formulation of the No-U-Turn Sampler (NUTS) that can be end-to-end JIT compiled, yielding an implementation that is much faster than existing alternatives in both the small and large dataset regimes.

Adam Foster, Martin Jankowiak, Matthew O'Meara et al.

We introduce a fully stochastic gradient based approach to Bayesian optimal experimental design (BOED). Our approach utilizes variational lower bounds on the expected information gain (EIG) of an experiment that can be simultaneously optimized with respect to both the variational and design parameters. This allows the design process to be carried out through a single unified stochastic gradient ascent procedure, in contrast to existing approaches that typically construct a pointwise EIG estimator, before passing this estimator to a separate optimizer. We provide a number of different variational objectives including the novel adaptive contrastive estimation (ACE) bound. Finally, we show that our gradient-based approaches are able to provide effective design optimization in substantially higher dimensional settings than existing approaches.

Fritz Obermeyer, Eli Bingham, Martin Jankowiak et al.

It is a significant challenge to design probabilistic programming systems that can accommodate a wide variety of inference strategies within a unified framework. Noting that the versatility of modern automatic differentiation frameworks is based in large part on the unifying concept of tensors, we describe a software abstraction for integration --functional tensors-- that captures many of the benefits of tensors, while also being able to describe continuous probability distributions. Moreover, functional tensors are a natural candidate for generalized variable elimination and parallel-scan filtering algorithms that enable parallel exact inference for a large family of tractable modeling motifs. We demonstrate the versatility of functional tensors by integrating them into the modeling frontend and inference backend of the Pyro programming language. In experiments we show that the resulting framework enables a large variety of inference strategies, including those that mix exact and approximate inference.

Martin Jankowiak, Geoff Pleiss, Jacob R. Gardner

The combination of inducing point methods with stochastic variational inference has enabled approximate Gaussian Process (GP) inference on large datasets. Unfortunately, the resulting predictive distributions often exhibit substantially underestimated uncertainties. Notably, in the regression case the predictive variance is typically dominated by observation noise, yielding uncertainty estimates that make little use of the input-dependent function uncertainty that makes GP priors attractive. In this work we propose two simple methods for scalable GP regression that address this issue and thus yield substantially improved predictive uncertainties. The first applies variational inference to FITC (Fully Independent Training Conditional; Snelson et.~al.~2006). The second bypasses posterior approximations and instead directly targets the posterior predictive distribution. In an extensive empirical comparison with a number of alternative methods for scalable GP regression, we find that the resulting predictive distributions exhibit significantly better calibrated uncertainties and higher log likelihoods--often by as much as half a nat per datapoint.

Martin Jankowiak, Jacob Gardner

We construct flexible likelihoods for multi-output Gaussian process models that leverage neural networks as components. We make use of sparse variational inference methods to enable scalable approximate inference for the resulting class of models. An attractive feature of these models is that they can admit analytic predictive means even when the likelihood is non-linear and the predictive distributions are non-Gaussian. We validate the modeling potential of these models in a variety of experiments in both the supervised and unsupervised setting. We demonstrate that the flexibility of these `neural' likelihoods can improve prediction quality as compared to simpler Gaussian process models and that neural likelihoods can be readily combined with a variety of underlying Gaussian process models, including deep Gaussian processes.

Adam Foster, Martin Jankowiak, Eli Bingham et al.

Bayesian optimal experimental design (BOED) is a principled framework for making efficient use of limited experimental resources. Unfortunately, its applicability is hampered by the difficulty of obtaining accurate estimates of the expected information gain (EIG) of an experiment. To address this, we introduce several classes of fast EIG estimators by building on ideas from amortized variational inference. We show theoretically and empirically that these estimators can provide significant gains in speed and accuracy over previous approaches. We further demonstrate the practicality of our approach on a number of end-to-end experiments.

Fritz Obermeyer, Eli Bingham, Martin Jankowiak et al.

A wide class of machine learning algorithms can be reduced to variable elimination on factor graphs. While factor graphs provide a unifying notation for these algorithms, they do not provide a compact way to express repeated structure when compared to plate diagrams for directed graphical models. To exploit efficient tensor algebra in graphs with plates of variables, we generalize undirected factor graphs to plated factor graphs and variable elimination to a tensor variable elimination algorithm that operates directly on plated factor graphs. Moreover, we generalize complexity bounds based on treewidth and characterize the class of plated factor graphs for which inference is tractable. As an application, we integrate tensor variable elimination into the Pyro probabilistic programming language to enable exact inference in discrete latent variable models with repeated structure. We validate our methods with experiments on both directed and undirected graphical models, including applications to polyphonic music modeling, animal movement modeling, and latent sentiment analysis.

Martin Jankowiak

In this note we consider setups in which variational objectives for Bayesian neural networks can be computed in closed form. In particular we focus on single-layer networks in which the activation function is piecewise polynomial (e.g. ReLU). In this case we show that for a Normal likelihood and structured Normal variational distributions one can compute a variational lower bound in closed form. In addition we compute the predictive mean and variance in closed form. Finally, we also show how to compute approximate lower bounds for other likelihoods (e.g. softmax classification). In experiments we show how the resulting variational objectives can help improve training and provide fast test time predictions.

Eli Bingham, Jonathan P. Chen, Martin Jankowiak et al.

Pyro is a probabilistic programming language built on Python as a platform for developing advanced probabilistic models in AI research. To scale to large datasets and high-dimensional models, Pyro uses stochastic variational inference algorithms and probability distributions built on top of PyTorch, a modern GPU-accelerated deep learning framework. To accommodate complex or model-specific algorithmic behavior, Pyro leverages Poutine, a library of composable building blocks for modifying the behavior of probabilistic programs.

Martin Jankowiak, Theofanis Karaletsos

We exploit the link between the transport equation and derivatives of expectations to construct efficient pathwise gradient estimators for multivariate distributions. We focus on two main threads. First, we use null solutions of the transport equation to construct adaptive control variates that can be used to construct gradient estimators with reduced variance. Second, we consider the case of multivariate mixture distributions. In particular we show how to compute pathwise derivatives for mixtures of multivariate Normal distributions with arbitrary means and diagonal covariances. We demonstrate in a variety of experiments in the context of variational inference that our gradient estimators can outperform other methods, especially in high dimensions.

Martin Jankowiak, Fritz Obermeyer

We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterization trick: Gamma, Beta, and Dirichlet. We further observe that when the reparameterization trick is applied to the Cholesky-factorized multivariate Normal distribution, the resulting gradients are suboptimal in the sense of optimal transport. We derive the optimal gradients and show that they have reduced variance in a Gaussian Process regression task. We demonstrate with a variety of synthetic experiments and stochastic variational inference tasks that our pathwise gradients are competitive with other methods.