Marginal MAP inference involves making MAP predictions in systems defined with latent variables or missing information. It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist. In this work, we generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases. Our method is based on a block coordinate descent algorithm on a new convex decomposition bound, that is guaranteed to converge monotonically, and can be parallelized efficiently. We demonstrate our approach on marginal MAP queries defined on real-world problems from the UAI approximate inference challenge, showing that our framework is faster and more reliable than previous methods.

The false discovery rate (FDR)---the expected fraction of spurious discoveries among all the discoveries---provides a popular statistical assessment of the reproducibility of scientific studies in various disciplines. In this work, we introduce a new method for controlling the FDR in meta-analysis of many decentralized linear models. Our method targets the scenario where many research groups---possibly the number of which is random---are independently testing a common set of hypotheses and then sending summary statistics to a coordinating center in an online manner. Built on the knockoffs framework introduced by Barber and Candes (2015), our procedure starts by applying the knockoff filter to each linear model and then aggregates the summary statistics via one-shot communication in a novel way. This method gives exact FDR control non-asymptotically without any knowledge of the noise variances or making any assumption about sparsity of the signal. In certain settings, it has a communication complexity that is optimal up to a logarithmic factor.

We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation error, leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms. In particular, we show that the upper bound may be obtained from independent and identically distributed samples from a specific non-uniform distribution, while the lower bound if valid for any set of points. Applying our results to kernel-based quadrature, while our results are fairly general, we recover known upper and lower bounds for the special cases of Sobolev spaces. Moreover, our results extend to the more general problem of full function approximations (beyond simply computing an integral), with results in L2- and L$\infty$-norm that match known results for special cases. Applying our results to random features, we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with Lipschitz-continuous losses.

A Kronecker product model is the set of visible marginal probability distributions of an exponential family whose sufficient statistics matrix factorizes as a Kronecker product of two matrices, one for the visible variables and one for the hidden variables. We estimate the dimension of these models by the maximum rank of the Jacobian in the limit of large parameters. The limit is described by the tropical morphism; a piecewise linear map with pieces corresponding to slicings of the visible matrix by the normal fan of the hidden matrix. We obtain combinatorial conditions under which the model has the expected dimension, equal to the minimum of the number of natural parameters and the dimension of the ambient probability simplex. Additionally, we prove that the binary restricted Boltzmann machine always has the expected dimension.

The quality of high-level AI of non-player characters (NPCs) in commercial open-world games (OWGs) has been increasing during the past years. However, due to constraints specific to the game industry, this increase has been slow and it has been driven by larger budgets rather than adoption of new complex AI techniques. Most of the contemporary AI is still expressed as hard-coded scripts. The complexity and manageability of the script codebase is one of the key limiting factors for further AI improvements. In this paper we address this issue. We present behavior objects - a general approach to development of NPC behaviors for large OWGs. Behavior objects are inspired by object-oriented programming and extend the concept of smart objects. Our approach promotes encapsulation of data and code for multiple related behaviors in one place, hiding internal details and embedding intelligence in the environment. Behavior objects are a natural abstraction of five different techniques that we have implemented to manage AI complexity in an upcoming AAA OWG. We report the details of the implementations in the context of behavior trees and the lessons learned during development. Our work should serve as inspiration for AI architecture designers from both the academia and the industry.

Computing the marginal likelihood (ML) of a model requires marginalizing out all of the parameters and latent variables, a difficult high-dimensional summation or integration problem. To make matters worse, it is often hard to measure the accuracy of one's ML estimates. We present bidirectional Monte Carlo, a technique for obtaining accurate log-ML estimates on data simulated from a model. This method obtains stochastic lower bounds on the log-ML using annealed importance sampling or sequential Monte Carlo, and obtains stochastic upper bounds by running these same algorithms in reverse starting from an exact posterior sample. The true value can be sandwiched between these two stochastic bounds with high probability. Using the ground truth log-ML estimates obtained from our method, we quantitatively evaluate a wide variety of existing ML estimators on several latent variable models: clustering, a low rank approximation, and a binary attributes model. These experiments yield insights into how to accurately estimate marginal likelihoods.

The practical performance of online stochastic gradient descent algorithms is highly dependent on the chosen step size, which must be tediously hand-tuned in many applications. The same is true for more advanced variants of stochastic gradients, such as SAGA, SVRG, or AdaGrad. Here we propose to adapt the step size by performing a gradient descent on the step size itself, viewing the whole performance of the learning trajectory as a function of step size. Importantly, this adaptation can be computed online at little cost, without having to iterate backward passes over the full data.

One approach to improving the running time of kernel-based machine learning methods is to build a small sketch of the input and use it in lieu of the full kernel matrix in the machine learning task of interest. Here, we describe a version of this approach that comes with running time guarantees as well as improved guarantees on its statistical performance. By extending the notion of \emph{statistical leverage scores} to the setting of kernel ridge regression, our main statistical result is to identify an importance sampling distribution that reduces the size of the sketch (i.e., the required number of columns to be sampled) to the \emph{effective dimensionality} of the problem. This quantity is often much smaller than previous bounds that depend on the \emph{maximal degrees of freedom}. Our main algorithmic result is to present a fast algorithm to compute approximations to these scores. This algorithm runs in time that is linear in the number of samples---more precisely, the running time is $O(np^2)$, where the parameter $p$ depends only on the trace of the kernel matrix and the regularization parameter---and it can be applied to the matrix of feature vectors, without having to form the full kernel matrix. This is obtained via a variant of length-squared sampling that we adapt to the kernel setting in a way that is of independent interest. Lastly, we provide empirical results illustrating our theory, and we discuss how this new notion of the statistical leverage of a data point captures in a fine way the difficulty of the original statistical learning problem.

We consider the scenario where the parameters of a probabilistic model are expected to vary over time. We construct a novel prior distribution that promotes sparsity and adapts the strength of correlation between parameters at successive timesteps, based on the data. We derive approximate variational inference procedures for learning and prediction with this prior. We test the approach on two tasks: forecasting financial quantities from relevant text, and modeling language contingent on time-varying financial measurements.

Learning a model of perceptual similarity from a collection of objects is a fundamental task in machine learning underlying numerous applications. A common way to learn such a model is from relative comparisons in the form of triplets: responses to queries of the form "Is object a more similar to b than it is to c?". If no consideration is made in the determination of which queries to ask, existing similarity learning methods can require a prohibitively large number of responses. In this work, we consider the problem of actively learning from triplets - finding which queries are most useful for learning. Different from previous active triplet learning approaches, we incorporate auxiliary information into our similarity model and introduce an active learning scheme to find queries that are informative for quickly learning both the relevant aspects of auxiliary data and the directly-learned similarity components. Compared to prior approaches, we show that we can learn just as effectively with much fewer queries. For evaluation, we introduce a new dataset of exhaustive triplet comparisons obtained from humans and demonstrate improved performance for different types of auxiliary information.