The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, corresponds to the adjacency matrix representation of the Gauss-Markov graph, which encodes conditional independence statements between variables. Minimax results under the spectral norm have previously been established for covariance matrices, both sparse and banded, and for sparse precision matrices. We establish minimax estimation bounds for estimating banded precision matrices under the spectral norm. Our results greatly improve upon the existing bounds; in particular, we find that the minimax rate for estimating banded precision matrices matches that of estimating banded covariance matrices. The key insight in our analysis is that we are able to obtain barely-noisy estimates of $k \times k$ subblocks of the precision matrix by inverting slightly wider blocks of the empirical covariance matrix along the diagonal. Our theoretical results are complemented by experiments demonstrating the sharpness of our bounds.

Indeed, the fact that this parameter is fixed hinders the statistical consistency of the original procedure.

This problem, chordal Markov network structure learning (CMSL), is computationally notoriously challenging; e.g., finding a maximum likelihood chordal Markov network

In this paper, we do so by drawing a novel equivalence between motion planning and the Bayesian active learning paradigm of decision region determination (DRD) . Unfortunately, a straight application of existing methods requires computation exponential in the number of edges in a graph.

In many real-world applications, it is typically the case that the same optimization problem is solved again and again on a regular basis, maintaining the same problem structure but differing in the data.

Sequential decision making problems, such as structured prediction, robotic control, and game playing, require a combination of planning policies and generalisation of those plans.

Since it is generally NPhard [7], much effort has been devoted to the design of polynomial-time approximation algorithms. The greedy algorithm is most favored for its simplicity, which iteratively chooses one item with the largest immediate benefit. Despite the greedy nature, it can perform well in many cases.

The key insight of the approach is that quantization methods, in particular product quantization, perform poorly when there is large variance in the norms of the data points.

Sequential identification questions in game trees with stochastic payoffs arise naturally as robust versions of bandit problems.

We study minimax strategies for the online prediction problem with expert advice.