This paper presents a new multitask learning framework that learns a shared representation among the tasks, incorporating both task and feature clusters. The jointly-induced clusters yield a shared latent subspace where task relationships are learned more effectively and more generally than in state-of-the-art multitask learning methods. The proposed general framework enables the derivation of more specific or restricted state-of-the-art multitask methods. The paper also proposes a highly-scalable multitask learning algorithm, based on the new framework, using conjugate gradient descent and generalized \textit{Sylvester equations}. Experimental results on synthetic and benchmark datasets show that the proposed method systematically outperforms several state-of-the-art multitask learning methods.

Many generative models can be expressed as a differentiable function of random inputs drawn from some simple probability density. This framework includes both deep generative architectures such as Variational Autoencoders and a large class of procedurally defined simulator models. We present a method for performing efficient MCMC inference in such models when conditioning on observations of the model output. For some models this offers an asymptotically exact inference method where Approximate Bayesian Computation might otherwise be employed. We use the intuition that inference corresponds to integrating a density across the manifold corresponding to the set of inputs consistent with the observed outputs. This motivates the use of a constrained variant of Hamiltonian Monte Carlo which leverages the smooth geometry of the manifold to coherently move between inputs exactly consistent with observations. We validate the method by performing inference tasks in a diverse set of models.

Information that is stored in an encrypted format is, by definition, usually not amenable to statistical analysis or machine learning methods. In this paper we present detailed analysis of coordinate and accelerated gradient descent algorithms which are capable of fitting least squares and penalised ridge regression models, using data encrypted under a fully homomorphic encryption scheme. Gradient descent is shown to dominate in terms of encrypted computational speed, and theoretical results are proven to give parameter bounds which ensure correctness of decryption. The characteristics of encrypted computation are empirically shown to favour a non-standard acceleration technique. This demonstrates the possibility of approximating conventional statistical regression methods using encrypted data without compromising privacy.

Neural networks are a powerful class of functions that can be trained with simple gradient descent to achieve state-of-the-art performance on a variety of applications. Despite their practical success, there is a paucity of results that provide theoretical guarantees on why they are so effective. Lying in the center of the problem is the difficulty of analyzing the non-convex loss function with potentially numerous local minima and saddle points. Can neural networks corresponding to the stationary points of the loss function learn the true target function? If yes, what are the key factors contributing to such nice optimization properties? In this paper, we answer these questions by analyzing one-hidden-layer neural networks with ReLU activation, and show that despite the non-convexity, neural networks with diverse units have no spurious local minima. We bypass the non-convexity issue by directly analyzing the first order optimality condition, and show that the loss can be made arbitrarily small if the minimum singular value of the "extended feature matrix" is large enough. We make novel use of techniques from kernel methods and geometric discrepancy, and identify a new relation linking the smallest singular value to the spectrum of a kernel function associated with the activation function and to the diversity of the units. Our results also suggest a novel regularization function to promote unit diversity for potentially better generalization.

I wanted to tell you about an open source forecasting package we just released called Prophet: I thought the readers of your blog might be interested in both the package and the fact that we built it on top of Stan. Under the hood, Prophet uses Stan for optimization (and sampling if the user desires) in order to fit a non-linear additive model and generate uncertainty intervals. The big win for us was that 1) Stan does a great job at letting us separate optimization from the model code and 2) we could share the same core procedure between Python and R implementations. One of the neat things we do is automatically detect changepoints in the time series by specifying a sequence potential parameter changes and shrinking the shifts using a Laplace prior. We also let the user adjust the flexibility of the model by tuning precision of priors, which we think is intuitive for most users.

Bayesian inference is the process of analyzing statistical models with the incorporation of prior knowledge about the model or model parameters. The root of such inference is Bayes' theorem: In this formula mu and tau, sometimes known as hyperparameters, are also known. The following graph shows the prior, likelihood, and posterior for theta. In some simple problems such as the previous normal mean inference example, it is easy to figure out the posterior distribution in a closed form. But in general problems that involve non-conjugate priors, the posterior distributions are difficult or impossible to compute analytically.

In this post, we consider different approaches for time series modeling. The forecasting approaches using linear models, ARIMA alpgorithm, XGBoost machine learning algorithm are described. Results of different model combinations are shown. For probabilistic modeling the approaches using copulas and Bayesian inference are considered. Time series analysis, especially forecasting, is an important problem of modern predictive analytics.

Modern retrieval systems are often driven by an underlying machine learning model. The goal of such systems is to identify and possibly rank the few most relevant items for a given query or context. Thus, such systems are typically evaluated using a ranking-based performance metric such as the area under the precision-recall curve, the $F_\beta$ score, precision at fixed recall, etc. Obviously, it is desirable to train such systems to optimize the metric of interest. In practice, due to the scalability limitations of existing approaches for optimizing such objectives, large-scale retrieval systems are instead trained to maximize classification accuracy, in the hope that performance as measured via the true objective will also be favorable. In this work we present a unified framework that, using straightforward building block bounds, allows for highly scalable optimization of a wide range of ranking-based objectives. We demonstrate the advantage of our approach on several real-life retrieval problems that are significantly larger than those considered in the literature, while achieving substantial improvement in performance over the accuracy-objective baseline.

Surprise describes a range of phenomena from unexpected events to behavioral responses. We propose a measure of surprise and use it for surprise-driven learning. Our surprise measure takes into account data likelihood as well as the degree of commitment to a belief via the entropy of the belief distribution. We find that surprise-minimizing learning dynamically adjusts the balance between new and old information without the need of knowledge about the temporal statistics of the environment. We apply our framework to a dynamic decision-making task and a maze exploration task. Our surprise minimizing framework is suitable for learning in complex environments, even if the environment undergoes gradual or sudden changes and could eventually provide a framework to study the behavior of humans and animals encountering surprising events.

In this paper we study the use of coding techniques to accelerate machine learning (ML). Coding techniques, such as prefix codes, have been extensively studied and used to accelerate low-level data processing primitives such as scans in a relational database system. However, there is little work on how to exploit them to accelerate ML algorithms. In fact, applying coding techniques for faster ML faces a unique challenge: one needs to consider both how the codes fit into the optimization algorithm used to train a model, and the interplay between the model structure and the coding scheme. Surprisingly and intriguingly, our study demonstrates that a slight variant of the classical Lempel-Ziv-Welch (LZW) coding scheme is a good fit for several popular ML algorithms, resulting in substantial runtime savings. Comprehensive experiments on several real-world datasets show that our LZW-based ML algorithms exhibit speedups of up to 31x compared to a popular and state-of-the-art ML library, with no changes to ML accuracy, even though the implementations of our LZW variants are not heavily tuned. Thus, our study reveals a new avenue for accelerating ML algorithms using coding techniques and we hope this opens up a new direction for more research.