arXiv subjects

Chris J. Maddison

Publications and source records attributed to Chris J. Maddison.

53 records · Page 3Linked to original sources

arXiv2020

Direct Policy Gradients: Direct Optimization of Policies in Discrete Action Spaces

Direct optimization is an appealing framework that replaces integration with optimization of a random objective for approximating gradients in models with discrete random variables. A$^\star$ sampling is a framework for optimizing such random objectives over large spaces. We show how to combine these techniques to yield a reinforcement learning algorithm that approximates a policy gradient by finding trajectories that optimize a random objective. We call the resulting algorithms "direct policy gradient" (DirPG) algorithms. A main benefit of DirPG algorithms is that they allow the insertion of domain knowledge in the form of upper bounds on return-to-go at training time, like is used in heuristic search, while still directly computing a policy gradient. We further analyze their properties, showing there are cases where DirPG has an exponentially larger probability of sampling informative gradients compared to REINFORCE. We also show that there is a built-in variance reduction technique and that a parameter that was previously viewed as a numerical approximation can be interpreted as controlling risk sensitivity. Empirically, we evaluate the effect of key degrees of freedom and show that the algorithm performs well in illustrative domains compared to baselines.

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arXiv2020

Rao-Blackwellizing the Straight-Through Gumbel-Softmax Gradient Estimator

Gradient estimation in models with discrete latent variables is a challenging problem, because the simplest unbiased estimators tend to have high variance. To counteract this, modern estimators either introduce bias, rely on multiple function evaluations, or use learned, input-dependent baselines. Thus, there is a need for estimators that require minimal tuning, are computationally cheap, and have low mean squared error. In this paper, we show that the variance of the straight-through variant of the popular Gumbel-Softmax estimator can be reduced through Rao-Blackwellization without increasing the number of function evaluations. This provably reduces the mean squared error. We empirically demonstrate that this leads to variance reduction, faster convergence, and generally improved performance in two unsupervised latent variable models.

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arXiv2020

On Empirical Comparisons of Optimizers for Deep Learning

Selecting an optimizer is a central step in the contemporary deep learning pipeline. In this paper, we demonstrate the sensitivity of optimizer comparisons to the hyperparameter tuning protocol. Our findings suggest that the hyperparameter search space may be the single most important factor explaining the rankings obtained by recent empirical comparisons in the literature. In fact, we show that these results can be contradicted when hyperparameter search spaces are changed. As tuning effort grows without bound, more general optimizers should never underperform the ones they can approximate (i.e., Adam should never perform worse than momentum), but recent attempts to compare optimizers either assume these inclusion relationships are not practically relevant or restrict the hyperparameters in ways that break the inclusions. In our experiments, we find that inclusion relationships between optimizers matter in practice and always predict optimizer comparisons. In particular, we find that the popular adaptive gradient methods never underperform momentum or gradient descent. We also report practical tips around tuning often ignored hyperparameters of adaptive gradient methods and raise concerns about fairly benchmarking optimizers for neural network training.

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arXiv2019

Continuous Hierarchical Representations with Poincaré Variational Auto-Encoders

The variational auto-encoder (VAE) is a popular method for learning a generative model and embeddings of the data. Many real datasets are hierarchically structured. However, traditional VAEs map data in a Euclidean latent space which cannot efficiently embed tree-like structures. Hyperbolic spaces with negative curvature can. We therefore endow VAEs with a Poincaré ball model of hyperbolic geometry as a latent space and rigorously derive the necessary methods to work with two main Gaussian generalisations on that space. We empirically show better generalisation to unseen data than the Euclidean counterpart, and can qualitatively and quantitatively better recover hierarchical structures.

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arXiv2019

Hamiltonian descent for composite objectives

In optimization the duality gap between the primal and the dual problems is a measure of the suboptimality of any primal-dual point. In classical mechanics the equations of motion of a system can be derived from the Hamiltonian function, which is a quantity that describes the total energy of the system. In this paper we consider a convex optimization problem consisting of the sum of two convex functions, sometimes referred to as a composite objective, and we identify the duality gap to be the 'energy' of the system. In the Hamiltonian formalism the energy is conserved, so we add a contractive term to the standard equations of motion so that this energy decreases linearly (i.e., geometrically) with time. This yields a continuous-time ordinary differential equation (ODE) in the primal and dual variables which converges to zero duality gap, i.e., optimality. This ODE has several useful properties: it induces a natural operator splitting; at convergence it yields both the primal and dual solutions; and it is invariant to affine transformation despite only using first order information. We provide several discretizations of this ODE, some of which are new algorithms and others correspond to known techniques, such as the alternating direction method of multipliers (ADMM). We conclude with some numerical examples that show the promise of our approach. We give an example where our technique can solve a convex quadratic minimization problem orders of magnitude faster than several commonly-used gradient methods, including conjugate gradient, when the conditioning of the problem is poor. Our framework provides new insights into previously known algorithms in the literature as well as providing a technique to generate new primal-dual algorithms.

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arXiv2019

Tighter Variational Bounds are Not Necessarily Better

We provide theoretical and empirical evidence that using tighter evidence lower bounds (ELBOs) can be detrimental to the process of learning an inference network by reducing the signal-to-noise ratio of the gradient estimator. Our results call into question common implicit assumptions that tighter ELBOs are better variational objectives for simultaneous model learning and inference amortization schemes. Based on our insights, we introduce three new algorithms: the partially importance weighted auto-encoder (PIWAE), the multiply importance weighted auto-encoder (MIWAE), and the combination importance weighted auto-encoder (CIWAE), each of which includes the standard importance weighted auto-encoder (IWAE) as a special case. We show that each can deliver improvements over IWAE, even when performance is measured by the IWAE target itself. Furthermore, our results suggest that PIWAE may be able to deliver simultaneous improvements in the training of both the inference and generative networks.

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arXiv2018

Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives

Deep latent variable models have become a popular model choice due to the scalable learning algorithms introduced by (Kingma & Welling, 2013; Rezende et al., 2014). These approaches maximize a variational lower bound on the intractable log likelihood of the observed data. Burda et al. (2015) introduced a multi-sample variational bound, IWAE, that is at least as tight as the standard variational lower bound and becomes increasingly tight as the number of samples increases. Counterintuitively, the typical inference network gradient estimator for the IWAE bound performs poorly as the number of samples increases (Rainforth et al., 2018; Le et al., 2018). Roeder et al. (2017) propose an improved gradient estimator, however, are unable to show it is unbiased. We show that it is in fact biased and that the bias can be estimated efficiently with a second application of the reparameterization trick. The doubly reparameterized gradient (DReG) estimator does not suffer as the number of samples increases, resolving the previously raised issues. The same idea can be used to improve many recently introduced training techniques for latent variable models. In particular, we show that this estimator reduces the variance of the IWAE gradient, the reweighted wake-sleep update (RWS) (Bornschein & Bengio, 2014), and the jackknife variational inference (JVI) gradient (Nowozin, 2018). Finally, we show that this computationally efficient, unbiased drop-in gradient estimator translates to improved performance for all three objectives on several modeling tasks.

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arXiv2018

Hamiltonian Descent Methods

We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger class includes functions whose second derivatives may be singular or unbounded at their minima. Our methods are discretizations of conformal Hamiltonian dynamics, which generalize the classical momentum method to model the motion of a particle with non-standard kinetic energy exposed to a dissipative force and the gradient field of the function of interest. They are first-order in the sense that they require only gradient computation. Yet, crucially the kinetic gradient map can be designed to incorporate information about the convex conjugate in a fashion that allows for linear convergence on convex functions that may be non-smooth or non-strongly convex. We study in detail one implicit and two explicit methods. For one explicit method, we provide conditions under which it converges to stationary points of non-convex functions. For all, we provide conditions on the convex function and kinetic energy pair that guarantee linear convergence, and show that these conditions can be satisfied by functions with power growth. In sum, these methods expand the class of convex functions on which linear convergence is possible with first-order computation.

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arXiv2018

Conditional Neural Processes

Deep neural networks excel at function approximation, yet they are typically trained from scratch for each new function. On the other hand, Bayesian methods, such as Gaussian Processes (GPs), exploit prior knowledge to quickly infer the shape of a new function at test time. Yet GPs are computationally expensive, and it can be hard to design appropriate priors. In this paper we propose a family of neural models, Conditional Neural Processes (CNPs), that combine the benefits of both. CNPs are inspired by the flexibility of stochastic processes such as GPs, but are structured as neural networks and trained via gradient descent. CNPs make accurate predictions after observing only a handful of training data points, yet scale to complex functions and large datasets. We demonstrate the performance and versatility of the approach on a range of canonical machine learning tasks, including regression, classification and image completion.

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arXiv2017

Filtering Variational Objectives

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.

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arXiv2017

REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models

Learning in models with discrete latent variables is challenging due to high variance gradient estimators. Generally, approaches have relied on control variates to reduce the variance of the REINFORCE estimator. Recent work (Jang et al. 2016, Maddison et al. 2016) has taken a different approach, introducing a continuous relaxation of discrete variables to produce low-variance, but biased, gradient estimates. In this work, we combine the two approaches through a novel control variate that produces low-variance, \emph{unbiased} gradient estimates. Then, we introduce a modification to the continuous relaxation and show that the tightness of the relaxation can be adapted online, removing it as a hyperparameter. We show state-of-the-art variance reduction on several benchmark generative modeling tasks, generally leading to faster convergence to a better final log-likelihood.

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arXiv2017

Particle Value Functions

The policy gradients of the expected return objective can react slowly to rare rewards. Yet, in some cases agents may wish to emphasize the low or high returns regardless of their probability. Borrowing from the economics and control literature, we review the risk-sensitive value function that arises from an exponential utility and illustrate its effects on an example. This risk-sensitive value function is not always applicable to reinforcement learning problems, so we introduce the particle value function defined by a particle filter over the distributions of an agent's experience, which bounds the risk-sensitive one. We illustrate the benefit of the policy gradients of this objective in Cliffworld.

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arXiv2017

The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables

The reparameterization trick enables optimizing large scale stochastic computation graphs via gradient descent. The essence of the trick is to refactor each stochastic node into a differentiable function of its parameters and a random variable with fixed distribution. After refactoring, the gradients of the loss propagated by the chain rule through the graph are low variance unbiased estimators of the gradients of the expected loss. While many continuous random variables have such reparameterizations, discrete random variables lack useful reparameterizations due to the discontinuous nature of discrete states. In this work we introduce Concrete random variables---continuous relaxations of discrete random variables. The Concrete distribution is a new family of distributions with closed form densities and a simple reparameterization. Whenever a discrete stochastic node of a computation graph can be refactored into a one-hot bit representation that is treated continuously, Concrete stochastic nodes can be used with automatic differentiation to produce low-variance biased gradients of objectives (including objectives that depend on the log-probability of latent stochastic nodes) on the corresponding discrete graph. We demonstrate the effectiveness of Concrete relaxations on density estimation and structured prediction tasks using neural networks.

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arXiv2016

A Poisson process model for Monte Carlo

Simulating samples from arbitrary probability distributions is a major research program of statistical computing. Recent work has shown promise in an old idea, that sampling from a discrete distribution can be accomplished by perturbing and maximizing its mass function. Yet, it has not been clearly explained how this research project relates to more traditional ideas in the Monte Carlo literature. This chapter addresses that need by identifying a Poisson process model that unifies the perturbation and accept-reject views of Monte Carlo simulation. Many existing methods can be analyzed in this framework. The chapter reviews Poisson processes and defines a Poisson process model for Monte Carlo methods. This model is used to generalize the perturbation trick to infinite spaces by constructing Gumbel processes, random functions whose maxima are located at samples over infinite spaces. The model is also used to analyze A* sampling and OS*, two methods from distinct Monte Carlo families.

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arXiv2015

Move Evaluation in Go Using Deep Convolutional Neural Networks

The game of Go is more challenging than other board games, due to the difficulty of constructing a position or move evaluation function. In this paper we investigate whether deep convolutional networks can be used to directly represent and learn this knowledge. We train a large 12-layer convolutional neural network by supervised learning from a database of human professional games. The network correctly predicts the expert move in 55% of positions, equalling the accuracy of a 6 dan human player. When the trained convolutional network was used directly to play games of Go, without any search, it beat the traditional search program GnuGo in 97% of games, and matched the performance of a state-of-the-art Monte-Carlo tree search that simulates a million positions per move.

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arXiv2015

A* Sampling

The problem of drawing samples from a discrete distribution can be converted into a discrete optimization problem. In this work, we show how sampling from a continuous distribution can be converted into an optimization problem over continuous space. Central to the method is a stochastic process recently described in mathematical statistics that we call the Gumbel process. We present a new construction of the Gumbel process and A* sampling, a practical generic sampling algorithm that searches for the maximum of a Gumbel process using A* search. We analyze the correctness and convergence time of A* sampling and demonstrate empirically that it makes more efficient use of bound and likelihood evaluations than the most closely related adaptive rejection sampling-based algorithms.

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arXiv2014

Structured Generative Models of Natural Source Code

We study the problem of building generative models of natural source code (NSC); that is, source code written and understood by humans. Our primary contribution is to describe a family of generative models for NSC that have three key properties: First, they incorporate both sequential and hierarchical structure. Second, we learn a distributed representation of source code elements. Finally, they integrate closely with a compiler, which allows leveraging compiler logic and abstractions when building structure into the model. We also develop an extension that includes more complex structure, refining how the model generates identifier tokens based on what variables are currently in scope. Our models can be learned efficiently, and we show empirically that including appropriate structure greatly improves the models, measured by the probability of generating test programs.

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