Predictive Entropy Search for Bayesian Optimization with Unknown Constraints

Unknown constraints arise in many types of expensive black-box optimization problems. Several methods have been proposed recently for performing Bayesian optimization with constraints, based on the expected improvement (EI) heuristic. However, EI can lead to pathologies when used with constraints. For example, in the case of decoupled constraints---i.e., when one can independently evaluate the objective or the constraints---EI can encounter a pathology that prevents exploration. Additionally, computing EI requires a current best solution, which may not exist if none of the data collected so far satisfy the constraints. By contrast, information-based approaches do not suffer from these failure modes. In this paper, we present a new information-based method called Predictive Entropy Search with Constraints (PESC). We analyze the performance of PESC and show that it compares favorably to EI-based approaches on synthetic and benchmark problems, as well as several real-world examples. We demonstrate that PESC is an effective algorithm that provides a promising direction towards a unified solution for constrained Bayesian optimization.

Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks

Large multilayer neural networks trained with backpropagation have recently achieved state-of-the-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a large number of hyperparameters to the data, lack of calibrated probabilistic predictions, and a tendency to overfit the training data. In principle, the Bayesian approach to learning neural networks does not have these problems. However, existing Bayesian techniques lack scalability to large dataset and network sizes. In this work we present a novel scalable method for learning Bayesian neural networks, called probabilistic backpropagation (PBP). Similar to classical backpropagation, PBP works by computing a forward propagation of probabilities through the network and then doing a backward computation of gradients. A series of experiments on ten real-world datasets show that PBP is significantly faster than other techniques, while offering competitive predictive abilities. Our experiments also show that PBP provides accurate estimates of the posterior variance on the network weights.

Scalable Bayesian Optimization Using Deep Neural Networks

Bayesian optimization is an effective methodology for the global optimization of functions with expensive evaluations. It relies on querying a distribution over functions defined by a relatively cheap surrogate model. An accurate model for this distribution over functions is critical to the effectiveness of the approach, and is typically fit using Gaussian processes (GPs). However, since GPs scale cubically with the number of observations, it has been challenging to handle objectives whose optimization requires many evaluations, and as such, massively parallelizing the optimization. In this work, we explore the use of neural networks as an alternative to GPs to model distributions over functions. We show that performing adaptive basis function regression with a neural network as the parametric form performs competitively with state-of-the-art GP-based approaches, but scales linearly with the number of data rather than cubically. This allows us to achieve a previously intractable degree of parallelism, which we apply to large scale hyperparameter optimization, rapidly finding competitive models on benchmark object recognition tasks using convolutional networks, and image caption generation using neural language models.

Scalable Bayesian Inference for Excitatory Point Process Networks

Networks capture our intuition about relationships in the world. They describe the friendships between Facebook users, interactions in financial markets, and synapses connecting neurons in the brain. These networks are richly structured with cliques of friends, sectors of stocks, and a smorgasbord of cell types that govern how neurons connect. Some networks, like social network friendships, can be directly observed, but in many cases we only have an indirect view of the network through the actions of its constituents and an understanding of how the network mediates that activity. In this work, we focus on the problem of latent network discovery in the case where the observable activity takes the form of a mutually-excitatory point process known as a Hawkes process. We build on previous work that has taken a Bayesian approach to this problem, specifying prior distributions over the latent network structure and a likelihood of observed activity given this network. We extend this work by proposing a discrete-time formulation and developing a computationally efficient stochastic variational inference (SVI) algorithm that allows us to scale the approach to long sequences of observations. We demonstrate our algorithm on the calcium imaging data used in the Chalearn neural connectomics challenge.

Dependent Multinomial Models Made Easy: Stick Breaking with the Pólya-Gamma Augmentation

Many practical modeling problems involve discrete data that are best represented as draws from multinomial or categorical distributions. For example, nucleotides in a DNA sequence, children's names in a given state and year, and text documents are all commonly modeled with multinomial distributions. In all of these cases, we expect some form of dependency between the draws: the nucleotide at one position in the DNA strand may depend on the preceding nucleotides, children's names are highly correlated from year to year, and topics in text may be correlated and dynamic. These dependencies are not naturally captured by the typical Dirichlet-multinomial formulation. Here, we leverage a logistic stick-breaking representation and recent innovations in P\'olya-gamma augmentation to reformulate the multinomial distribution in terms of latent variables with jointly Gaussian likelihoods, enabling us to take advantage of a host of Bayesian inference techniques for Gaussian models with minimal overhead.

Spectral Representations for Convolutional Neural Networks

Discrete Fourier transforms provide a significant speedup in the computation of convolutions in deep learning. In this work, we demonstrate that, beyond its advantages for efficient computation, the spectral domain also provides a powerful representation in which to model and train convolutional neural networks (CNNs). We employ spectral representations to introduce a number of innovations to CNN design. First, we propose spectral pooling, which performs dimensionality reduction by truncating the representation in the frequency domain. This approach preserves considerably more information per parameter than other pooling strategies and enables flexibility in the choice of pooling output dimensionality. This representation also enables a new form of stochastic regularization by randomized modification of resolution. We show that these methods achieve competitive results on classification and approximation tasks, without using any dropout or max-pooling. Finally, we demonstrate the effectiveness of complex-coefficient spectral parameterization of convolutional filters. While this leaves the underlying model unchanged, it results in a representation that greatly facilitates optimization. We observe on a variety of popular CNN configurations that this leads to significantly faster convergence during training.

Early Stopping is Nonparametric Variational Inference

We show that unconverged stochastic gradient descent can be interpreted as a procedure that samples from a nonparametric variational approximate posterior distribution. This distribution is implicitly defined as the transformation of an initial distribution by a sequence of optimization updates. By tracking the change in entropy over this sequence of transformations during optimization, we form a scalable, unbiased estimate of the variational lower bound on the log marginal likelihood. We can use this bound to optimize hyperparameters instead of using cross-validation. This Bayesian interpretation of SGD suggests improved, overfitting-resistant optimization procedures, and gives a theoretical foundation for popular tricks such as early stopping and ensembling. We investigate the properties of this marginal likelihood estimator on neural network models.

Gradient-based Hyperparameter Optimization through Reversible Learning

Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of cross-validation performance with respect to all hyperparameters by chaining derivatives backwards through the entire training procedure. These gradients allow us to optimize thousands of hyperparameters, including step-size and momentum schedules, weight initialization distributions, richly parameterized regularization schemes, and neural network architectures. We compute hyperparameter gradients by exactly reversing the dynamics of stochastic gradient descent with momentum.

A framework for studying synaptic plasticity with neural spike train data

Learning and memory in the brain are implemented by complex, time-varying changes in neural circuitry. The computational rules according to which synaptic weights change over time are the subject of much research, and are not precisely understood. Until recently, limitations in experimental methods have made it challenging to test hypotheses about synaptic plasticity on a large scale. However, as such data become available and these barriers are lifted, it becomes necessary to develop analysis techniques to validate plasticity models. Here, we present a highly extensible framework for modeling arbitrary synaptic plasticity rules on spike train data in populations of interconnected neurons. We treat synaptic weights as a (potentially nonlinear) dynamical system embedded in a fully-Bayesian generalized linear model (GLM). In addition, we provide an algorithm for inferring synaptic weight trajectories alongside the parameters of the GLM and of the learning rules. Using this method, we perform model comparison of two proposed variants of the well-known spike-timing-dependent plasticity (STDP) rule, where nonlinear effects play a substantial role. On synthetic data generated from the biophysical simulator NEURON, we show that we can recover the weight trajectories, the pattern of connectivity, and the underlying learning rules.

Parallel MCMC with Generalized Elliptical Slice Sampling

Probabilistic models are conceptually powerful tools for finding structure in data, but their practical effectiveness is often limited by our ability to perform inference in them. Exact inference is frequently intractable, so approximate inference is often performed using Markov chain Monte Carlo (MCMC). To achieve the best possible results from MCMC, we want to efficiently simulate many steps of a rapidly mixing Markov chain which leaves the target distribution invariant. Of particular interest in this regard is how to take advantage of multi-core computing to speed up MCMC-based inference, both to improve mixing and to distribute the computational load. In this paper, we present a parallelizable Markov chain Monte Carlo algorithm for efficiently sampling from continuous probability distributions that can take advantage of hundreds of cores. This method shares information between parallel Markov chains to build a scale-mixture of Gaussians approximation to the density function of the target distribution. We combine this approximation with a recent method known as elliptical slice sampling to create a Markov chain with no step-size parameters that can mix rapidly without requiring gradient or curvature computations.