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Doucet, Arnaud


Pruning untrained neural networks: Principles and Analysis

arXiv.org Machine Learning

Overparameterized neural networks display state-of-the art performance. However, there is a growing need for smaller, energy-efficient, neural networks to be able to use machine learning applications on devices with limited computational resources. A popular approach consists of using pruning techniques. While these techniques have traditionally focused on pruning pre-trained neural networks (e.g. LeCun et al. (1990) and Hassabi et al. (1993)), recent work by Lee et al. (2018) showed promising results where pruning is performed at initialization. However, such procedures remain unsatisfactory as the resulting pruned networks can be difficult to train and, for instance, these procedures do not prevent one layer being fully pruned. In this paper we provide a comprehensive theoretical analysis of pruning at initialization and training sparse architectures. This analysis allows us to propose novel principled approaches which we validate experimentally on a variety of network architectures. We particularly show that we can prune up to 99.9% of the weights while keeping the model trainable.


Bayesian Nonparametric Models on Decomposable Graphs

Neural Information Processing Systems

Over recent years Dirichlet processes and the associated Chinese restaurant process (CRP) have found many applications in clustering while the Indian buffet process (IBP) is increasingly used to describe latent feature models. In the clustering case, we associate to each data point a latent allocation variable. These latent variables can share the same value and this induces a partition of the data set. The CRP is a prior distribution on such partitions. In latent feature models, we associate to each data point a potentially infinite number of binary latent variables indicating the possession of some features and the IBP is a prior distribution on the associated infinite binary matrix.


Hamiltonian Variational Auto-Encoder

Neural Information Processing Systems

Variational Auto-Encoders (VAE) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBO). Com- bined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is neces- sary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick.


Filtering Variational Objectives

Neural Information Processing Systems

When used as a surrogate objective for maximum likelihood estimation in latent variable models, the evidence lower bound (ELBO) produces state-of-the-art results. Inspired by this, we consider the extension of the ELBO to a family of lower bounds defined by a particle filter's estimator of the marginal likelihood, the filtering variational objectives (FIVOs). FIVOs take the same arguments as the ELBO, but can exploit a model's sequential structure to form tighter bounds. We present results that relate the tightness of FIVO's bound to the variance of the particle filter's estimator by considering the generic case of bounds defined as log-transformed likelihood estimators. Experimentally, we show that training with FIVO results in substantial improvements over training the same model architecture with the ELBO on sequential data.


Clone MCMC: Parallel High-Dimensional Gaussian Gibbs Sampling

Neural Information Processing Systems

We propose a generalized Gibbs sampler algorithm for obtaining samples approximately distributed from a high-dimensional Gaussian distribution. Similarly to Hogwild methods, our approach does not target the original Gaussian distribution of interest, but an approximation to it. Contrary to Hogwild methods, a single parameter allows us to trade bias for variance. We show empirically that our method is very flexible and performs well compared to Hogwild-type algorithms. Papers published at the Neural Information Processing Systems Conference.


Expectation Particle Belief Propagation

Neural Information Processing Systems

We propose an original particle-based implementation of the Loopy Belief Propagation (LPB) algorithm for pairwise Markov Random Fields (MRF) on a continuous state space. This is achieved by considering proposal distributions in the exponential family whose parameters are updated iterately in an Expectation Propagation (EP) framework. The proposed particle scheme provides consistent estimation of the LBP marginals as the number of particles increases. We demonstrate that it provides more accurate results than the Particle Belief Propagation (PBP) algorithm of Ihler and McAllester (2009) at a fraction of the computational cost and is additionally more robust empirically. The computational complexity of our algorithm at each iteration is quadratic in the number of particles.


Asynchronous Anytime Sequential Monte Carlo

Neural Information Processing Systems

We introduce a new sequential Monte Carlo algorithm we call the particle cascade. The particle cascade is an asynchronous, anytime alternative to traditional sequential Monte Carlo algorithms that is amenable to parallel and distributed implementations. It uses no barrier synchronizations which leads to improved particle throughput and memory efficiency. It is an anytime algorithm in the sense that it can be run forever to emit an unbounded number of particles while keeping within a fixed memory budget. We prove that the particle cascade provides an unbiased marginal likelihood estimator which can be straightforwardly plugged into existing pseudo-marginal methods.


Schr\"odinger Bridge Samplers

arXiv.org Machine Learning

Consider a reference Markov process with initial distribution $\pi_{0}$ and transition kernels $\{M_{t}\}_{t\in[1:T]}$, for some $T\in\mathbb{N}$. Assume that you are given distribution $\pi_{T}$, which is not equal to the marginal distribution of the reference process at time $T$. In this scenario, Schr\"odinger addressed the problem of identifying the Markov process with initial distribution $\pi_{0}$ and terminal distribution equal to $\pi_{T}$ which is the closest to the reference process in terms of Kullback--Leibler divergence. This special case of the so-called Schr\"odinger bridge problem can be solved using iterative proportional fitting, also known as the Sinkhorn algorithm. We leverage these ideas to develop novel Monte Carlo schemes, termed Schr\"odinger bridge samplers, to approximate a target distribution $\pi$ on $\mathbb{R}^{d}$ and to estimate its normalizing constant. This is achieved by iteratively modifying the transition kernels of the reference Markov chain to obtain a process whose marginal distribution at time $T$ becomes closer to $\pi_T = \pi$, via regression-based approximations of the corresponding iterative proportional fitting recursion. We report preliminary experiments and make connections with other problems arising in the optimal transport, optimal control and physics literatures.


Localised Generative Flows

arXiv.org Machine Learning

A BSTRACT We argue that flow-based density models based on continuous bijections are limited in their ability to learn target distributions with complicated topologies, and propose localised generative flows (LGFs) to address this problem. LGFs are composed of stacked continuous mixtures of bijections, which enables each bijection to learn a local region of the target rather than its entirety. Our method is a generalisation of existing flow-based methods, which can be used without modification as the basis for an LGF model. Unlike normalising flows, LGFs do not permit exact computation of log likelihoods, but we propose a simple variational scheme that performs well in practice. We show empirically that LGFs yield improved performance across a variety of density estimation tasks. 1 I NTRODUCTION Flow-based generative models, often referred to as normalising flows, have become popular methods for density estimation because of their flexibility, expressiveness, and tractable likelihoods. Given the problem of learning an unknown target density p null X on a data space X, normalising flows model p null X as the marginal of X obtained by the generative process Z p Z, X: g 1 ( Z), (1) where p Z is a prior density on a space Z, and g: X Z is a bijection. The parameters of g can be learned via maximum likelihood given i.i.d. To be effective, a normalising flow model must specify an expressive family of bijections with tractable Jacobians. Affine coupling layers (Dinh et al., 2014; 2016), autoregressive transformations (Germain et al., 2015; Papamakarios et al., 2017), ODEbased transformations (Grathwohl et al., 2018), and invertible ResNet blocks (Behrmann et al., 2019) are all examples of such bijections that can be composed to produce complicated flows. These models have demonstrated significant promise in their ability to model complex datasets (Papamakarios et al., 2017) and to synthesise novel data points (Kingma & Dhariwal, 2018). However, in all these cases, g is continuous in x .


Modular Meta-Learning with Shrinkage

arXiv.org Artificial Intelligence

Most gradient-based approaches to meta-learning do not explicitly account for the fact that different parts of the underlying model adapt by different amounts when applied to a new task. For example, the input layers of an image classification convnet typically adapt very little, while the output layers can change significantly. This can cause parts of the model to begin to overfit while others underfit. To address this, we introduce a hierarchical Bayesian model with per-module shrinkage parameters, which we propose to learn by maximizing an approximation of the predictive likelihood using implicit differentiation. Our algorithm subsumes Reptile and outperforms variants of MAML on two synthetic few-shot meta-learning problems.