Marginalising out uncertain quantities within the internal representations or parameters of neural networks is of central importance for a wide range of learning techniques, such as empirical, variational or full Bayesian methods. We set out to generalise fast dropout (Wang & Manning, 2013) to cover a wider variety of noise processes in neural networks. This leads to an efficient calculation of the marginal likelihood and predictive distribution which evades sampling and the consequential increase in training time due to highly variant gradient estimates. This allows us to approximate variational Bayes for the parameters of feed-forward neural networks. Inspired by the minimum description length principle, we also propose and experimentally verify the direct optimisation of the regularised predictive distribution. The methods yield results competitive with previous neural network based approaches and Gaussian processes on a wide range of regression tasks.

In this paper, we study a classification problem in which sample labels are randomly corrupted. In this scenario, there is an unobservable sample with noise-free labels. However, before being observed, the true labels are independently flipped with a probability $\rho\in[0,0.5)$, and the random label noise can be class-conditional. Here, we address two fundamental problems raised by this scenario. The first is how to best use the abundant surrogate loss functions designed for the traditional classification problem when there is label noise. We prove that any surrogate loss function can be used for classification with noisy labels by using importance reweighting, with consistency assurance that the label noise does not ultimately hinder the search for the optimal classifier of the noise-free sample. The other is the open problem of how to obtain the noise rate $\rho$. We show that the rate is upper bounded by the conditional probability $P(y|x)$ of the noisy sample. Consequently, the rate can be estimated, because the upper bound can be easily reached in classification problems. Experimental results on synthetic and real datasets confirm the efficiency of our methods.

This report is concerned with the Mondrian process and its applications in machine learning. The Mondrian process is a guillotine-partition-valued stochastic process that possesses an elegant self-consistency property. The first part of the report uses simple concepts from applied probability to define the Mondrian process and explore its properties. The Mondrian process has been used as the main building block of a clever online random forest classification algorithm that turns out to be equivalent to its batch counterpart. We outline a slight adaptation of this algorithm to regression, as the remainder of the report uses regression as a case study of how Mondrian processes can be utilized in machine learning. In particular, the Mondrian process will be used to construct a fast approximation to the computationally expensive kernel ridge regression problem with a Laplace kernel. The complexity of random guillotine partitions generated by a Mondrian process and hence the complexity of the resulting regression models is controlled by a lifetime hyperparameter. It turns out that these models can be efficiently trained and evaluated for all lifetimes in a given range at once, without needing to retrain them from scratch for each lifetime value. This leads to an efficient procedure for determining the right model complexity for a dataset at hand. The limitation of having a single lifetime hyperparameter will motivate the final Mondrian grid model, in which each input dimension is endowed with its own lifetime parameter. In this model we preserve the property that its hyperparameters can be tweaked without needing to retrain the modified model from scratch.

Approximate Bayesian computation (ABC) using a sequential Monte Carlo method provides a comprehensive platform for parameter estimation, model selection and sensitivity analysis in differential equations. However, this method, like other Monte Carlo methods, incurs a significant computational cost as it requires explicit numerical integration of differential equations to carry out inference. In this paper we propose a novel method for circumventing the requirement of explicit integration by using derivatives of Gaussian processes to smooth the observations from which parameters are estimated. We evaluate our methods using synthetic data generated from model biological systems described by ordinary and delay differential equations. Upon comparing the performance of our method to existing ABC techniques, we demonstrate that it produces comparably reliable parameter estimates at a significantly reduced execution time.

In this paper, we propose a generalized scale mixture family of distributions, namely the Power Exponential Scale Mixture (PESM) family, to model the sparsity inducing priors currently in use for sparse signal recovery (SSR). We show that the successful and popular methods such as LASSO, Reweighted $\ell_1$ and Reweighted $\ell_2$ methods can be formulated in an unified manner in a maximum a posteriori (MAP) or Type I Bayesian framework using an appropriate member of the PESM family as the sparsity inducing prior. In addition, exploiting the natural hierarchical framework induced by the PESM family, we utilize these priors in a Type II framework and develop the corresponding EM based estimation algorithms. Some insight into the differences between Type I and Type II methods is provided and of particular interest in the algorithmic development is the Type II variant of the popular and successful reweighted $\ell_1$ method. Extensive empirical results are provided and they show that the Type II methods exhibit better support recovery than the corresponding Type I methods.

We highlight a pitfall when applying stochastic variational inference to general Bayesian networks. For global random variables approximated by an exponential family distribution, natural gradient steps, commonly starting from a unit length step size, are averaged to convergence. This useful insight into the scaling of initial step sizes is lost when the approximation factorizes across a general Bayesian network, and care must be taken to ensure practical convergence. We experimentally investigate how much of the baby (well-scaled steps) is thrown out with the bath water (exact gradients).

This paper studies active posterior estimation in a Bayesian setting when the likelihood is expensive to evaluate. Existing techniques for posterior estimation are based on generating samples representative of the posterior. Such methods do not consider efficiency in terms of likelihood evaluations. In order to be query efficient we treat posterior estimation in an active regression framework.  We propose two myopic query strategies to choose where to evaluate the likelihood and implement them using Gaussian processes. Via experiments on a series of synthetic and real examples we demonstrate that our approach is significantly more query efficient than existing techniques and other heuristics for posterior estimation.

Most multi-label domains lack an authoritative taxonomy. Therefore, different taxonomies are commonly used in the same domain, which results in complications. Although this situation occurs frequently, there has been little study of it using a principled statistical approach. Given that (1) different taxonomies used in the same domain are generally founded on the same latent semantic space, where each possible label set in a taxonomy denotes a single semantic concept, and that (2) crowdsourcing is beneficial in identifying relationships between semantic concepts and instances at low cost, we proposed a novel probabilistic cascaded method for establishing a semantic matching function in a crowdsourcing setting that maps label sets in one (source) taxonomy to label sets in another (target) taxonomy in terms of the semantic distances between them. The established function can be used to detect the associated label set in the target taxonomy for an instance directly from its associated label set in the source taxonomy without any extra effort. Experimental results on real-world data (emotion annotations for narrative sentences) demonstrated that the proposed method can robustly establish semantic matching functions exhibiting satisfactory performance from a limited number of crowdsourced annotations.

This paper revisits the problem of learning a k-CNF Boolean function from examples, for fixed k, in the context of online learning under the logarithmic loss. We give a Bayesian interpretation to one of Valiant’s classic PAC learning algorithms, which we then build upon to derive three efficient, online, probabilistic, supervised learning algorithms for predicting the output of an unknown k-CNF Boolean function. We analyze the loss of our methods, and show that the cumulative log-loss can be upper bounded by a polynomial function of the size of each example.

Graphical models, such as Bayesian Networks and Markov networks play an important role in artificial intelligence and machine learning. Inference is a central problem to be solved on these networks. This, and other problems on these graph models are often known to be hard to solve in general, but tractable on graphs with bounded Treewidth. Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models. In this paper, we study the approximability of a number of graph problems: Treewidth and Pathwidth of graphs, Minimum Fill-In, and a variety of different graph layout problems such as Minimum Cut Linear Arrangement. We show that, assuming Small Set Expansion Conjecture, all of these problems are NP-hard to approx- imate to within any constant factor in polynomial time.