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High-dimensional regression with noisy and missing data: Provable guarantees with non-convexity
TLDR
This work is able to both analyze the statistical error associated with any global optimum, and prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers.
Regularized M-estimators with nonconvexity: statistical and algorithmic theory for local optima
TLDR
Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, it is proved that any stationary point of the composite objective function will lie within statistical precision of the underlying parameter vector.
Statistical consistency and asymptotic normality for high-dimensional robust M-estimators
  • Po-Ling Loh
  • Mathematics, Computer Science
    ArXiv
  • 1 January 2015
TLDR
This work establishes a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution, and analysis of the local curvature of the loss function has useful consequences for optimization when the robust regression function and/or regularizer is nonconvex and the objective function possesses stationary points outside the local region.
High-dimensional learning of linear causal networks via inverse covariance estimation
TLDR
It is shown that when the error variances are known or estimated to close enough precision, the true DAG is the unique minimizer of the score computed using the reweighted squared l2-loss.
Support recovery without incoherence: A case for nonconvex regularization
TLDR
The primal-dual witness proof method may be used to establish variable selection consistency and $\ell_\infty$-bounds for sparse regression problems, even when the loss function and/or regularizer are nonconvex.
Generalization Error Bounds for Noisy, Iterative Algorithms
In statistical learning theory, generalization error is used to quantify the degree to which a supervised machine learning algorithm may overfit to training data. Recent work [Xu and Raginsky (2017)]
Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
TLDR
For certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph, and nonasymptotic guarantees for graph selection methods are provided.
Information-theoretic bounds for exact recovery in weighted stochastic block models using the Renyi divergence
TLDR
The main result, characterizing the precise boundary between success and failure of maximum likelihood estimation when edge weights are drawn from discrete distributions, involves the Renyi divergence of order $\frac{1}{2}$ between the distributions of within-community and between-community edges.
Adversarial Risk Bounds for Binary Classification via Function Transformation
TLDR
This work derives new bounds for a notion of adversarial risk, characterizing the robustness of binary classifiers and neural network classifiers, and introduces transformations with the property that the risk of the transformed functions upper-bounds the adversarialrisk of the original functions.
Confidence Sets for the Source of a Diffusion in Regular Trees
  • Justin Khim, Po-Ling Loh
  • Mathematics, Computer Science
    IEEE Transactions on Network Science and…
  • 19 October 2015
TLDR
A probabilistic analysis of Pólya urns corresponding to the number of uninfected neighbors in specific subtrees of the infection tree is provided to provide an example illustrating the shortcomings of source estimation techniques in settings where the underlying graph is asymmetric.
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