Variational Gaussian Inference for Bilinear Models of Count Data

Bilinear models of count data with Poisson distribution are popular in applications such as matrix factorization for recommendation systems, modeling of receptive fields of sensory neurons, and modeling of neural-spike trains. Bayesian inference in such models remains challenging due to the product term of two Gaussian random vectors. In this paper, we propose new algorithms for such models based on variational Gaussian (VG) inference. We make two contributions. First, we show that the VG lower bound for these models, previously known to be intractable, is available in closed form under certain non-trivial constraints on the form of the posterior. Second, we show that the lower bound is biconcave and can be efficiently optimized for mean-field approximations. We also show that bi-concavity generalizes to the larger family of log-concave likelihoods, that subsume the Poisson distribution. We present new inference algorithms based on these results and demonstrate better performance on real-world problems at the cost of a modest increase in computation. Our contributions in this paper, therefore, provide more choices for Bayesian inference in terms of a speed-vs-accuracy tradeoff.

Scalable Collaborative Bayesian Preference Learning

Mohammad Emtiyaz Khan, Young Jun Ko, Matthias Seeger

Learning about users’ utilities from preference, discrete choice or implicit feedback data is of integral importance in e-commerce, targeted advertising and web search. Due to the sparsity and diffuse nature of data, Bayesian approaches hold much promise, yet most prior work does not scale up to realistic data sizes. We shed light on why inference for such settings is computationally difficult for standard machine learning methods, most of which focus on predicting explicit ratings only. To simplify the difficulty, we present a novel expectation maximization algorithm, driven by expectation propagation approximate inference, which scales to very large datasets without requiring strong factorization assumptions. Our utility model uses both latent bilinear collaborative filtering and non-parametric Gaussian process (GP) regression. In experiments on large real-world datasets, our method gives substantially better results than either matrix factorization or GPs in isolation, and converges significantly faster.

2014

Applications of Approximate Learning and Inference for Probabilistic Models

Young Jun Ko

We develop approximate inference and learning methods for facilitating the use of probabilistic modeling techniques motivated by applications in two different areas. First, we consider the ill-posed inverse problem of recovering an image from an underdetermined system of linear measurements corrupted by noise. Second, we consider the problem of inferring user preferences for items from counts, pairwise comparisons and user activity logs, instances of implicit feedback. Plausible models for images and the noise, incurred when recording them, render posterior inference intractable, while the scale of the inference problem makes sampling based approximations ineffective. Therefore, we develop deterministic approximate inference algorithms for two different augmentations of a typical sparse linear model: first, for the rectified-linear Poisson likelihood, and second, for tree-structured super-Gaussian mixture models. The rectified-linear Poisson likelihood is an alternative noise model, applicable in astronomical and biomedical imaging applications, that operate in intensity regimes in which quantum effects lead to observations that are best described by counts of particles arriving at a sensor, as well as in general Poisson regression problems arising in various fields. In this context we show, that the model-specific computations for Expectation Propagation can be robustly solved by a simple dynamic program. Next, we develop a scalable approximate inference algorithm for structured mixture models, that uses a discrete graphical model to represent dependencies between the latent mixture components of a collection of mixture models. Specifically, we use tree-structured mixtures of super-Gaussians to model the persistence across scales of large coefficients of the Wavelet transform of an image for improved reconstruction. In the second part on models of user preference, we consider two settings: the global static and the contextual dynamic setting. In the global static setting, we represent user-item preferences by a latent low-rank matrix. Instead of using numeric ratings we develop methods to infer this latent representation for two types of implicit feedback: aggregate counts of users interacting with a service and the binary outcomes of pairwise comparisons. We model count data using a latent Gaussian bilinear model with Poisson likelihoods. For this model, we show that the Variational Gaussian approximation can be further relaxed to be available in closed-form by adding additional constraints, leading to an efficient inference algorithm. In the second implicit feedback scenario, we infer the latent preference matrix from pairwise preference statements. We combine a low-rank bilinear model with non-parameteric item- feature regression and develop a novel approximate variational Expectation Maximization algorithm that mitigates the computational challenges due to latent couplings induced by the pairwise comparisons. Finally, in the contextual dynamic setting, we model sequences of user activity at the granularity of single interaction events instead of aggregate counts. Routinely gathered in the background at a large scale in many applications, such sequences can reveal temporal and contextual aspects of user behavior through recurrent patterns. To describe such data, we propose a generic collaborative sequence model based on recurrent neural networks, that combines ideas from collaborative filtering and language modeling.

EPFL2017

Mean-Field methods for Structured Deep-Learning in Computer Vision

Pierre Bruno Baqué

In recent years, Machine Learning based Computer Vision techniques made impressive progress. These algorithms proved particularly efficient for image classification or detection of isolated objects. From a probabilistic perspective, these methods can predict marginals, over single or multiple variables, independently, with high accuracy. However, in many tasks of practical interest, we need to predict jointly several correlated variables. Practical applications include people detection in crowded scenes, image segmentation, surface reconstruction, 3D pose estimation and others. A large part of the research effort in today's computer-vision community aims at finding task-specific solutions to these problems, while leveraging the power of Deep-Learning based classifiers. In this thesis, we present our journey towards a generic and practical solution based on mean-field (MF) inference. Mean-field is a Statistical Physics-inspired method which has long been used in Computer-Vision as a variational approximation to posterior distributions over complex Conditional Random Fields. Standard mean-field optimization is based on coordinate descent and in many situations can be impractical. We therefore propose a novel proximal gradient-based approach to optimizing the variational objective. It is naturally parallelizable and easy to implement. We prove its convergence, and then demonstrate that, in practice, it yields faster convergence and often finds better optima than more traditional mean-field optimization techniques. Then, we show that we can replace the fully factorized distribution of mean-field by a weighted mixture of such distributions, that similarly minimizes the KL-Divergence to the true posterior. Our extension of the clamping method proposed in previous works allows us to both produce a more descriptive approximation of the true posterior and, inspired by the diverse MAP paradigms, fit a mixture of mean-field approximations. We demonstrate that this positively impacts real-world algorithms that initially relied on mean-fields. One of the important properties of the mean-field inference algorithms is that the closed-form updates are fully differentiable operations. This naturally allows to do parameter learning by simply unrolling multiple iterations of the updates, the so-called back-mean-field algorithm. We derive a novel and efficient structured learning method for multi-modal posterior distribution based on the Multi-Modal Mean-Field approximation, which can be seamlessly combined to modern gradient-based learning methods such as CNNs. Finally, we explore in more details the specific problem of structured learning and prediction for multiple-people detection in crowded scenes. We then present a mean-field based structured deep-learning detection algorithm that provides state of the art results on this dataset.