We study two communication-efficient algorithms for distributed statistical optimization on large-scale data. The first algorithm is an averaging method that distributes the $N$ data samples evenly to $m$ machines, performs separate minimization on each subset, and then averages the estimates. We provide a sharp analysis of this average mixture algorithm, showing that under a reasonable set of conditions, the combined parameter achieves mean-squared error that decays as $\order(N^{-1}+(N/m)^{-2})$. Whenever $m \le \sqrt{N}$, this guarantee matches the best possible rate achievable by a centralized algorithm having access to all $N$ samples. The second algorithm is a novel method, based on an appropriate form of the bootstrap. Requiring only a single round of communication, it has mean-squared error that decays as $\order(N^{-1}+(N/m)^{-3})$, and so is more robust to the amount of parallelization. We complement our theoretical results with experiments on large-scale problems from the Microsoft Learning to Rank dataset.

We consider an abstract class of optimization problems that are parameterized concavely in a single parameter, and show that the solution path along the parameter can always be approximated with accuracy $\varepsilon >0$ by a set of size $O(1/\sqrt{\varepsilon})$. A lower bound of size $\Omega (1/\sqrt{\varepsilon})$ shows that the upper bound is tight up to a constant factor. We also devise an algorithm that calls a step-size oracle and computes an approximate path of size $O(1/\sqrt{\varepsilon})$. Finally, we provide an implementation of the oracle for soft-margin support vector machines, and a parameterized semi-definite program for matrix completion.

Inference on high-order graphical models has become increasingly important in recent years. We consider energies with simple 'sparse' high-order potentials. Previous work in this area uses either specialized message-passing or transforms each high-order potential to the pairwise case. We take a fundamentally different approach, transforming the entire original problem into a comparatively small instance of a submodular vertex-cover problem. These vertex-cover instances can then be attacked by standard pairwise methods, where they run much faster (4--15 times) and are often more effective than on the original problem. We evaluate our approach on synthetic data, and we show that our algorithm can be useful in a fast hierarchical clustering and model estimation framework.

The problem of multiple change point estimation is considered for sequences with unknown number of change points. A consistency framework is suggested that is suitable for highly dependent time-series, and an asymptotically consistent algorithm is proposed. In order for the consistency to be established the only assumption required is that the data is generated by stationary ergodic time-series distributions. No modeling, independence or parametric assumptions are made; the data are allowed to be dependent and the dependence can be of arbitrary form. The theoretical results are complemented with experimental evaluations.

Probabilistic latent variable models are one of the cornerstones of machine learning. They offer a convenient and coherent way to specify prior distributions over unobserved structure in data, so that these unknown properties can be inferred via posterior inference. Such models are useful for exploratory analysis and visualization, for building density models of data, and for providing features that can be used for later discriminative tasks. A significant limitation of these models, however, is that draws from the prior are often highly redundant due to i.i.d. assumptions on internal parameters. For example, there is no preference in the prior of a mixture model to make components non-overlapping, or in topic model to ensure that co-ocurring words only appear in a small number of topics. In this work, we revisit these independence assumptions for probabilistic latent variable models, replacing the underlying i.i.d.\ prior with a determinantal point process (DPP). The DPP allows us to specify a preference for diversity in our latent variables using a positive definite kernel function. Using a kernel between probability distributions, we are able to define a DPP on probability measures. We show how to perform MAP inference with DPP priors in latent Dirichlet allocation and in mixture models, leading to better intuition for the latent variable representation and quantitatively improved unsupervised feature extraction, without compromising the generative aspects of the model.

In many applications classification systems often require in the loop human intervention. In such cases the decision process must be transparent and comprehensible simultaneously requiring minimal assumptions on the underlying data distribution. To tackle this problem, we formulate it as an axis-alligned subspacefinding task under the assumption that query specific information dictates the complementary use of the subspaces. We develop a regression-based approach called RECIP that efficiently solves this problem by finding projections that minimize a nonparametric conditional entropy estimator. Experiments show that the method is accurate in identifying the informative projections of the dataset, picking the correct ones to classify query points and facilitates visual evaluation by users.

Determinantal point processes (DPPs) have recently been proposed as computationally efficientprobabilistic models of diverse sets for a variety of applications, including document summarization, image search, and pose estimation. Many DPP inference operations, including normalization and sampling, are tractable; however, finding the most likely configuration (MAP), which is often required in practice for decoding, is NPhard, so we must resort to approximate inference. This optimization problem, which also arises in experimental design and sensor placement, involves finding the largest principal minor of a positive semidefinite matrix. Because the objective is log-submodular, greedy algorithms have been used in the past with some empirical success; however, these methods only give approximation guarantees in the special case of monotone objectives, which correspond toa restricted class of DPPs. In this paper we propose a new algorithm for approximating the MAP problem based on continuous techniques for submodular functionmaximization. Our method involves a novel continuous relaxation of the log-probability function, which, in contrast to the multilinear extension used for general submodular functions, can be evaluated and differentiated exactly and efficiently. We obtain a practical algorithm with a 1/4-approximation guarantee for a more general class of non-monotone DPPs; our algorithm also extends to MAP inference under complex polytope constraints, making it possible to combine DPPswith Markov random fields, weighted matchings, and other models. We demonstrate that our approach outperforms standard and recent methods on both synthetic and real-world data.

We introduce a new notion of classification accuracy based on the top -quantile values of a scoring function, a relevant criterion in a number of problems arising forsearch engines. We define an algorithm optimizing a convex surrogate of the corresponding loss, and discuss its solution in terms of a set of convex optimization problems.We also present margin-based guarantees for this algorithm based on the top -quantile value of the scores of the functions in the hypothesis set. Finally, we report the results of several experiments in the bipartite setting evaluating the performance of our solution and comparing the results to several other algorithms seeking high precision at the top. In most examples, our solution achieves a better performance in precision at the top.

Discrete mixtures are used routinely in broad sweeping applications ranging from unsupervised settings to fully supervised multi-task learning. Indeed, finite mixtures and infinite mixtures, relying on Dirichlet processes and modifications, have become a standard tool. One important issue that arises in using discrete mixtures is low separation in the components; in particular, different components can be introduced that are very similar and hence redundant. Such redundancy leads to too many clusters that are too similar, degrading performance in unsupervised learning and leading to computational problems and an unnecessarily complex model in supervised settings. Redundancy can arise in the absence of a penalty on components placed close together even when a Bayesian approach is used to learn the number of components. To solve this problem, we propose a novel prior that generates components from a repulsive process, automatically penalizing redundant components. We characterize this repulsive prior theoretically and propose a Markov chain Monte Carlo sampling algorithm for posterior computation. The methods are illustrated using synthetic examples and an iris data set.

We study how to automatically select and adapt multiple abstractions or representations of the world to support model-based reinforcement learning. We address the challenges of transfer learning in heterogeneous environments with varying tasks. We present an efficient, online framework that, through a sequence of tasks, learns a set of relevant representations to be used in future tasks. Without pre-defined mapping strategies, we introduce a general approach to support transfer learning across different state spaces. We demonstrate the potential impact of our system through improved jumpstart and faster convergence to near optimum policy in two benchmark domains.