Support Recovery for the Drift Coefficient of High-Dimensional Diffusions

Abstract-Consider the problem of learning the drift coefficient of a p-dimensional stochastic differential equation from a sample path of length T . We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem when both p and T can tend to infinity. In particular, we prove a general lower bound on the sample-complexity T by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a p × p matrix which describes which degrees of freedom interact under the dynamics. In this case, we analyze a 1-regularized least squares estimator and prove an upper bound on T that nearly matches the lower bound on specific classes of sparse matrices

The set of solutions of random XORSAT formulae

The XOR-satisfiability (XORSAT) problem requires finding an assignment of $n$ Boolean variables that satisfy $m$ exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing $n$ variables and $m$ clauses of size $k$. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as $k$-satisfiability ($k$-SAT), hypergraph bicoloring and graph coloring. For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random $k$-XORSAT. In particular, we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is intimately tied to the construction of specific subgraphs of the hypergraph associated with an instance of $k$-XORSAT. In order to study such subgraphs, we establish novel local weak convergence results for them.Comment: Published at http://dx.doi.org/10.1214/14-AAP1060 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Transportation system reconstruction

2035-03-17 [Patent Duration]Current Assignees: Urban Engines Inc Google LLCA system for reconstructing vehicle itinerary include a processor and a memory storing instructions, implemented by the processor, to cluster historical trip records into a plurality of clusters, each of the plurality of clusters including a set of historical trip records that describe events occurring within a predetermined time range at one location; identify a sequence of clusters that includes a cluster at each location; and estimate an itinerary for a vehicle based on the sequence of clusters and constraint data describing physical constraints, the itinerary for the vehicle describing a sequence of arrival and departure times at a sequence of locations for the vehicle

Reinforcement Learning, Bit by Bit

Reinforcement learning agents have demonstrated remarkable achievements in simulated environments. Data efficiency poses an impediment to carrying this success over to real environments. The design of data-efficient agents calls for a deeper understanding of information acquisition and representation. We develop concepts and establish a regret bound that together offer principled guidance. The bound sheds light on questions of what information to seek, how to seek that information, and it what information to retain. To illustrate concepts, we design simple agents that build on them and present computational results that demonstrate improvements in data efficiency

Epistemic Neural Networks

Intelligence relies on an agent's knowledge of what it does not know. This capability can be assessed based on the quality of joint predictions of labels across multiple inputs. Conventional neural networks lack this capability and, since most research has focused on marginal predictions, this shortcoming has been largely overlooked. We introduce the epistemic neural network (ENN) as an interface for models that represent uncertainty as required to generate useful joint predictions. While prior approaches to uncertainty modeling such as Bayesian neural networks can be expressed as ENNs, this new interface facilitates comparison of joint predictions and the design of novel architectures and algorithms. In particular, we introduce the epinet: an architecture that can supplement any conventional neural network, including large pretrained models, and can be trained with modest incremental computation to estimate uncertainty. With an epinet, conventional neural networks outperform very large ensembles, consisting of hundreds or more particles, with orders of magnitude less computation. We demonstrate this efficacy across synthetic data, ImageNet, and some reinforcement learning tasks. As part of this effort we open-source experiment code