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Probabilistic Methods/Variational Inference

Dieterich Lawson, Michael Li, Scott Linderman

Sequential latent variable models (SLVMs) are essential tools in statistics and machine learning, with applications ranging from healthcare to neuroscience. As their flexibility increases, analytic inference and model learning can become challenging, necessitating approximate methods. Here we introduce neural adaptive smoothing via twisting (NAS-X), a method that extends reweighted wake-sleep (RWS) to the sequential setting by using smoothing sequential Monte Carlo (SMC) to estimate intractable posterior expectations. Combining RWS and smoothing SMC allows NAS-X to provide low-bias and low-variance gradient estimates, and fit both discrete and continuous latent variable models. We illustrate the theoretical advantages of NAS-X over previous methods and explore these advantages empirically in a variety of tasks, including a challenging application to mechanistic models of neuronal dynamics. These experiments show that NAS-X substantially outperforms previous VI- and RWS-based methods in inference and model learning, achieving lower parameter error and tighter likelihood bounds.

Probabilistic Methods/Variational Inference

Anshuk Uppal, Kristoffer Stensbo-Smidt, Wouter Boomsma, Jes Frellsen

In variational inference, the benefits of Bayesian models rely on accurately capturing the true posterior distribution. We propose using neural samplers that specify implicit distributions, which are well-suited for approximating complex multimodal and correlated posteriors in high-dimensional spaces. Our approach introduces novel bounds for approximate inference using implicit distributions by locally linearising the neural sampler. This is distinct from existing methods that rely on additional discriminator networks and unstable adversarial objectives. Furthermore, we present a new sampler architecture that, for the first time, enables implicit distributions over tens of millions of latent variables, addressing computational concerns by using differentiable numerical approximations. We empirically show that our method is capable of recovering correlations across layers in large Bayesian neural networks, a property that is crucial for a network's performance but notoriously challenging to achieve. To the best of our knowledge, no other method has been shown to accomplish this task for such large models. Through experiments in downstream tasks, we demonstrate that our expressive posteriors outperform state-of-the-art uncertainty quantification methods, validating the effectiveness of our training algorithm and the quality of the learned implicit approximation.

Probabilistic Methods/Variational Inference

Veit David Wild, Sahra Ghalebikesabi, Dino Sejdinovic, Jeremias Knoblauch

We establish the first mathematically rigorous link between Bayesian, variational Bayesian, and ensemble methods. A key step towards this it to reformulate the non-convex optimisation problem typically encountered in deep learning as a convex optimisation in the space of probability measures. On a technical level, our contribution amounts to studying generalised variational inference through the lense of Wasserstein gradient flows. The result is a unified theory of various seemingly disconnected approaches that are commonly used for uncertainty quantification in deep learning---including deep ensembles and (variational) Bayesian methods. This offers a fresh perspective on the reasons behind the success of deep ensembles over procedures based on parameterised variational inference, and allows the derivation of new ensembling schemes with convergence guarantees. We showcase this by proposing a family of interacting deep ensembles with direct parallels to the interactions of particle systems in thermodynamics, and use our theory to prove the convergence of these algorithms to a well-defined global minimiser on the space of probability measures.

Probabilistic Methods/Variational Inference

Kyurae Kim, Jisu Oh, Kaiwen Wu, Yian Ma, Jacob Gardner

We provide the first convergence guarantee for black-box variational inference (BBVI) with the reparameterization gradient. While preliminary investigations worked on simplified versions of BBVI (e.g., bounded domain, bounded support, only optimizing for the scale, and such), our setup does not need any such algorithmic modifications. Our results hold for log-smooth posterior densities with and without strong log-concavity and the location-scale variational family. Notably, our analysis reveals that certain algorithm design choices commonly employed in practice, such as nonlinear parameterizations of the scale matrix, can result in suboptimal convergence rates. Fortunately, running BBVI with proximal stochastic gradient descent fixes these limitations and thus achieves the strongest known convergence guarantees. We evaluate this theoretical insight by comparing proximal SGD against other standard implementations of BBVI on large-scale Bayesian inference problems.

Probabilistic Methods/Variational Inference

Peter Nickl, Lu Xu, Dharmesh Tailor, Thomas Möllenhoff, Mohammad Emtiyaz Khan

Understanding model’s sensitivity to its training data is crucial but can also be challenging and costly, especially during training. To simplify such issues, we present the Memory-Perturbation Equation (MPE) which relates model's sensitivity to perturbation in its training data. Derived using Bayesian principles, the MPE unifies existing sensitivity measures, generalizes them to a wide-variety of models and algorithms, and unravels useful properties regarding sensitivities. Our empirical results show that sensitivity estimates obtained during training can be used to faithfully predict generalization on unseen test data. The proposed equation is expected to be useful for future research on robust and adaptive learning.

Probabilistic Methods/Variational Inference

Chirag Modi, Robert Gower, Charles Margossian, Yuling Yao, David Blei, Lawrence Saul

Variational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics. Typically, VI fits a simple parametric distribution to be close to the target posterior, optimizing an appropriate objective such as the evidence lower bound (ELBO). In this work, we present a new approach to VI. Our method is based on the principle of score matching---namely, that if two distributions are equal then their score functions (i.e., gradients of the log density) are equal at every point on their support. With this principle, we develop score-matching VI, an iterative algorithm that seeks to match the scores between the variational approximation and the exact posterior. At each iteration, score-matching VI solves an inner optimization, one that minimally adjusts the current variational estimate to match the scores at a newly sampled value of the latent variables. We show that when the variational family is a Gaussian, this inner optimization enjoys a closed-form solution, which we call Gaussian score matching VI (GSM-VI). GSM-VI is a ``black box'' variational algorithm in that it only requires a differentiable joint distribution, and as such it can be applied to a wide class of models. We compare GSM-VI to black box variational inference (BBVI), which has similar requirements but instead optimizes the ELBO. We first study how GSM-VI behaves as a function of the problem dimensionality, the condition number of the target covariance matrix (when the target is Gaussian), and the degree of mismatch between the approximating and exact posterior distribution. We then study GSM-VI on a collection of real-world Bayesian inference problems from the posteriorDB database of datasets and models. We find that GSM-VI is faster than BBVI and equally or more accurate. Specifically, over a wide range of target posteriors, GSM-VI requires 10-100x fewer gradient evaluations than BBVI to obtain a comparable quality of approximation.

Probabilistic Methods/Variational Inference

Feynman Liang, Liam Hodgkinson, Michael Mahoney

Despite the successes of probabilistic models based on passing noise through neural networks, recent work has identified that such methods often fail to capture tail behavior accurately---unless the tails of the base distribution are appropriately calibrated. To overcome this deficiency, we propose a systematic approach for analyzing the tails of random variables, and we illustrate how this approach can be used during the static analysis (before drawing samples) pass of a probabilistic programming language (PPL) compiler. To characterize how the tails change under various operations, we develop an algebra which acts on a three-parameter family of tail asymptotics and which is based on the generalized Gamma distribution. Our algebraic operations are closed under addition and multiplication; they are capable of distinguishing sub-Gaussians with differing scales; and they handle ratios sufficiently well to reproduce the tails of most important statistical distributions directly from their definitions. Our empirical results confirm that inference algorithms that leverage our heavy-tailed algebra attain superior performance across a number of density modeling and variational inference (VI) tasks.

Probabilistic Methods/Variational Inference

Zongyu Guo, Gergely Flamich, Jiajun He, Zhibo Chen, José Miguel Hernández-Lobato

Many common types of data can be represented as functions that map coordinates to signal values, such as pixel locations to RGB values in the case of an image. Based on this view, data can be compressed by overfitting a compact neural network to its functional representation and then encoding the network weights. However, most current solutions for this are inefficient, as quantization to low-bit precision substantially degrades the reconstruction quality. To address this issue, we propose overfitting variational Bayesian neural networks to the data and compressing an approximate posterior weight sample using relative entropy coding instead of quantizing and entropy coding it. This strategy enables direct optimization of the rate-distortion performance by minimizing the $\beta$-ELBO, and target different rate-distortion trade-offs for a given network architecture by adjusting $\beta$. Moreover, we introduce an iterative algorithm for learning prior weight distributions and employ a progressive refinement process for the variational posterior that significantly enhances performance. Experiments show that our method achieves strong performance on image and audio compression while retaining simplicity.

Probabilistic Methods/Variational Inference

Veit David Wild, Sahra Ghalebikesabi, Dino Sejdinovic, Jeremias Knoblauch

We establish the first mathematically rigorous link between Bayesian, variational Bayesian, and ensemble methods. A key step towards this it to reformulate the non-convex optimisation problem typically encountered in deep learning as a convex optimisation in the space of probability measures. On a technical level, our contribution amounts to studying generalised variational inference through the lense of Wasserstein gradient flows. The result is a unified theory of various seemingly disconnected approaches that are commonly used for uncertainty quantification in deep learning---including deep ensembles and (variational) Bayesian methods. This offers a fresh perspective on the reasons behind the success of deep ensembles over procedures based on parameterised variational inference, and allows the derivation of new ensembling schemes with convergence guarantees. We showcase this by proposing a family of interacting deep ensembles with direct parallels to the interactions of particle systems in thermodynamics, and use our theory to prove the convergence of these algorithms to a well-defined global minimiser on the space of probability measures.

Probabilistic Methods/Variational Inference

Justin Domke, Robert Gower, Guillaume Garrigos

Black-box variational inference is widely used in situations where there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs—namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides rigorous guarantees that methods similar to those used in practice converge on realistic inference problems.

Probabilistic Methods/Variational Inference

Kevin Course, Prasanth Nair

We consider the problem of inferring latent stochastic differential equations (SDEs) with a time and memory cost that scales independently with the amount of data, the total length of the time series, and the stiffness of the approximate differential equations. This is in stark contrast to typical methods for inferring latent differential equations which, despite their constant memory cost, have a time complexity that is heavily dependent on the stiffness of the approximate differential equation. We achieve this computational advancement by removing the need to solve differential equations when approximating gradients using a novel amortization strategy coupled with a recently derived reparametrization of expectations under linear SDEs. We show that, in practice, this allows us to achieve similar performance to methods based on adjoint sensitivities with more than an order of magnitude fewer evaluations of the model in training.

Probabilistic Methods/Variational Inference

Tianle Liu, Promit Ghosal, Krishnakumar Balasubramanian, Natesh Pillai

Stein Variational Gradient Descent (SVGD) is a nonparametric particle-based deterministic sampling algorithm. Despite its wide usage, understanding the theoretical properties of SVGD has remained a challenging problem. For sampling from a Gaussian target, the SVGD dynamics with a bilinear kernel will remain Gaussian as long as the initializer is Gaussian. Inspired by this fact, we undertake a detailed theoretical study of the Gaussian-SVGD, i.e., SVGD projected to the family of Gaussian distributions via the bilinear kernel, or equivalently Gaussian variational inference (GVI) with SVGD. We present a complete picture by considering both the mean-field PDE and discrete particle systems. When the target is strongly log-concave, the mean-field Gaussian-SVGD dynamics is proven to converge linearly to the Gaussian distribution closest to the target in KL divergence. In the finite-particle setting, there is both uniform in time convergence to the mean-field limit and linear convergence in time to the equilibrium if the target is Gaussian. In the general case, we propose a density-based and a particle-based implementation of the Gaussian-SVGD, and show that several recent algorithms for GVI, proposed from different perspectives, emerge as special cases of our unified framework. Interestingly, one of the new particle-based instance from this framework empirically outperforms existing approaches. Our results make concrete contributions towards obtaining a deeper understanding of both SVGD and GVI.

Probabilistic Methods/Variational Inference

Prateek Jaiswal, Harsha Honnappa, Vinayak Rao

We study data-driven decision-making problems in the Bayesian framework, where the expectation in the Bayes risk is replaced by a risk-sensitive entropic risk measure with respect to the posterior distribution. We focus on problems where calculating the posterior distribution is intractable, a typical situation in modern applications with large datasets and complex data generating models. We leverage a dual representation of the entropic risk measure to introduce a novel risk-sensitive variational Bayesian (RSVB) framework for jointly computing a risk-sensitive posterior approximation and the corresponding decision rule. Our general framework includes \textit{loss-calibrated} VB (Lacoste-Julien et al. [2011] ) as a special case. We also study the impact of these computational approximations on the predictive performance of the inferred decision rules. We compute the convergence rates of the RSVB approximate posterior and the corresponding optimal value. We illustrate our theoretical findings in parametric and nonparametric settings with the help of three examples.

Probabilistic Methods/Variational Inference

Saurav Jha, Dong Gong, He Zhao, Lina Yao

Continual learning (CL) aims to train deep neural networks efficiently on streaming data while limiting the forgetting caused by new tasks. However, learning transferable knowledge with less interference between tasks is difficult, and real-world deployment of CL models is limited by their inability to measure predictive uncertainties. To address these issues, we propose handling CL tasks with neural processes (NPs), a class of meta-learners that encode different tasks into probabilistic distributions over functions all while providing reliable uncertainty estimates. Specifically, we propose an NP-based CL approach (NPCL) with task-specific modules arranged in a hierarchical latent variable model. We tailor regularizers on the learned latent distributions to alleviate forgetting. The uncertainty estimation capabilities of the NPCL can also be used to handle the task head/module inference challenge in CL. Our experiments show that the NPCL outperforms previous CL approaches. We validate the effectiveness of uncertainty estimation in the NPCL for identifying novel data and evaluating instance-level model confidence. Code is available at https://github.com/srvCodes/NPCL.

Probabilistic Methods/Variational Inference

Ziheng Cheng, Shiyue Zhang, Longlin Yu, Cheng Zhang

Particle-based variational inference methods (ParVIs) such as Stein variational gradient descent (SVGD) update the particles based on the kernelized Wasserstein gradient flow for the Kullback-Leibler (KL) divergence. However, the design of kernels is often non-trivial and can be restrictive for the flexibility of the method. Recent works show that functional gradient flow approximations with quadratic form regularization terms can improve performance. In this paper, we propose a ParVI framework, called generalized Wasserstein gradient descent (GWG), based on a generalized Wasserstein gradient flow of the KL divergence, which can be viewed as a functional gradient method with a broader class of regularizers induced by convex functions. We show that GWG exhibits strong convergence guarantees. We also provide an adaptive version that automatically chooses Wasserstein metric to accelerate convergence. In experiments, we demonstrate the effectiveness and efficiency of the proposed framework on both simulated and real data problems.

Probabilistic Methods/Variational Inference

Sebastian Zeng, Florian Graf, Roland Kwitt

We consider the problem of variational Bayesian inference in a latent variable model where a (possibly complex) observed stochastic process is governed by the unobserved solution of a latent stochastic differential equation (SDE). Motivated by the challenges that arise when trying to learn a latent SDE in $\mathbb{R}^n$ from large-scale data, such as efficient gradient computation, we take a step back and study a specific subclass instead. In our case, the SDE evolves inside a homogeneous latent space and is induced by stochastic dynamics of the corresponding (matrix) Lie group. In the context of learning problems, SDEs on the $n$-dimensional unit sphere are arguably the most relevant incarnation of this setup. For variational inference, the sphere not only facilitates using a uniform prior on the initial state of the SDE, but we also obtain a particularly simple and intuitive expression for the KL divergence between the approximate posterior and prior process in the evidence lower bound. We provide empirical evidence that a latent SDE of the proposed type can be learned efficiently by means of an existing one-step geometric Euler-Maruyama scheme. Despite restricting ourselves to a less diverse class of SDEs, we achieve competitive or even state-of-the-art performance on a collection of time series interpolation and classification benchmarks.

Probabilistic Methods/Variational Inference

Thibault RANDRIANARISOA, Botond Szabo

By now Bayesian methods are routinely used in practice for solving inverse problems. In inverse problems the parameter or signal of interest is observed only indirectly, as an image of a given map, and the observations are typically further corrupted with noise. Bayes offers a natural way to regularize these problems via the prior distribution and provides a probabilistic solution, quantifying the remaining uncertainty in the problem. However, the computational costs of standard, sampling based Bayesian approaches can be overly large in such complex models. Therefore, in practice variational Bayes is becoming increasingly popular. Nevertheless, the theoretical understanding of these methods is still relatively limited, especially in context of inverse problems.In our analysis we investigate variational Bayesian methods for Gaussian process priors to solve linear inverse problems. We consider both mildly and severely ill-posed inverse problems and work with the popular inducing variable variational Bayes approach proposed by Titsias [Titsias, 2009]. We derive posterior contraction rates for the variational posterior in general settings and show that the minimax estimation rate can be attained by correctly tunned procedures. As specific examples we consider a collection of inverse problems including the heat equation, Volterra operator and Radon transform and inducing variable methods based on population and empirical spectral features.