Functional brain networks are well described and estimated from data with Gaussian Graphical Models (GGMs), e.g.\ using sparse inverse covariance estimators. Comparing functional connectivity of subjects in two populations calls for comparing these estimated GGMs. Our goal is to identify differences in GGMs known to have similar structure. We characterize the uncertainty of differences with confidence intervals obtained using a parametric distribution on parameters of a sparse estimator. Sparse penalties enable statistical guarantees and interpretable models even in high-dimensional and low-sample settings. Characterizing the distributions of sparse models is inherently challenging as the penalties produce a biased estimator.

Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators. They also include, as special cases, L 2 quantities which are used in many applications. We propose and analyze a family of estimators for Sobolev quantities of unknown probability density functions. We bound the finite-sample bias and variance of our estimators, finding that they are generally minimax rate-optimal. Our estimators are significantly more computationally tractable than previous estimators, and exhibit a statistical/computational trade-off allowing them to adapt to computational constraints.

We propose a novel and efficient algorithm for the collaborative preference completion problem, which involves jointly estimating individualized rankings for a set of entities over a shared set of items, based on a limited number of observed affinity values. Our approach exploits the observation that while preferences are often recorded as numerical scores, the predictive quantity of interest is the underlying rankings. Thus, attempts to closely match the recorded scores may lead to overfitting and impair generalization performance. Instead, we propose an estimator that directly fits the underlying preference order, combined with nuclear norm constraints to encourage low--rank parameters. Besides (approximate) correctness of the ranking order, the proposed estimator makes no generative assumption on the numerical scores of the observations. One consequence is that the proposed estimator can fit any consistent partial ranking over a subset of the items represented as a directed acyclic graph (DAG), generalizing standard techniques that can only fit preference scores.

We propose a novel and efficient algorithm for the collaborative preference completion problem, which involves jointly estimating individualized rankings for a set of entities over a shared set of items, based on a limited number of observed affinity values. Our approach exploits the observation that while preferences are often recorded as numerical scores, the predictive quantity of interest is the underlying rankings. Thus, attempts to closely match the recorded scores may lead to overfitting and impair generalization performance. Instead, we propose an estimator that directly fits the underlying preference order, combined with nuclear norm constraints to encourage low-rank parameters. Besides (approximate) correctness of the ranking order, the proposed estimator makes no generative assumption on the numerical scores of the observations. One consequence is that the proposed estimator can fit any consistent partial ranking over a subset of the items represented as a directed acyclic graph (DAG), generalizing standard techniques that can only fit preference scores. Despite this generality, for supervision representing total or blockwise total orders, the computational complexity of our algorithm is within a log factor of the standard algorithms for nuclear norm regularization based estimates for matrix completion.

We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function f, consistent estimators of the mean embedding of a random variable X lead to consistent estimators of the mean embedding of f(X). For Matern kernels and sufficiently smooth functions we also provide rates of convergence. Our results extend to functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are independent, it is sufficient to have separate estimators of the mean embeddings of their marginal distributions.

We consider massive distributed datasets that consist of elements modeled as keyvalue pairs and the task of computing statistics or aggregates where the contribution of each key is weighted by a function of its frequency (sum of values of its elements). This fundamental problem has a wealth of applications in data analytics and machine learning, in particular, with concave sublinear functions of the frequencies that mitigate the disproportionate effect of keys with high frequency. The family of concave sublinear functions includes low frequency moments ( 1), capping, logarithms, and their compositions. A common approach is to sample keys, ideally, proportionally to their contributions and estimate statistics from the sample. A simple but costly way to do this is by aggregating the data to produce a table of keys and their frequencies, apply our function to the frequency values, and then apply a weighted sampling scheme. Our main contribution is the design of composable sampling sketches that can be tailored to any concave sublinear function of the frequencies. Our sketch structure size is very close to the desired sample size and our samples provide statistical guarantees on the estimation quality that are very close to that of an ideal sample of the same size computed over aggregated data. Finally, we demonstrate experimentally the simplicity and effectiveness of our methods.

Offline reinforcement learning (RL) can be used to improve future performance by leveraging historical data. There exist many different algorithms for offline RL, and it is well recognized that these algorithms, and their hyperparameter settings, can lead to decision policies with substantially differing performance. This prompts the need for pipelines that allow practitioners to systematically perform algorithmhyperparameter selection for their setting. Critically, in most real-world settings, this pipeline must only involve the use of historical data. Inspired by statistical model selection methods for supervised learning, we introduce a task-and methodagnostic pipeline for automatically training, comparing, selecting, and deploying the best policy when the provided dataset is limited in size.

We consider the problem of estimating a structured multivariate density, subject to Markov conditions implied by an undirected graph. In the worst case, without Markovian assumptions, this problem suffers from the curse of dimensionality. Our main result shows how the curse of dimensionality can be avoided or greatly alleviated under the Markov property, and applies to arbitrary graphs. While existing results along these lines focus on sparsity or manifold assumptions, we introduce a new graphical quantity called "graph resilience" and show how it controls the sample complexity. Surprisingly, although one might expect the sample complexity of this problem to scale with local graph parameters such as the degree, this turns out not to be the case. Through explicit examples, we compute uniform deviation bounds and illustrate how the curse of dimensionality in density estimation can thus be circumvented. Notable examples where the rate improves substantially include sequential, hierarchical, and spatial data.

The most prevalent approach for estimating endogenous regression models with instruments is assuming low-dimensional linear relationships, i.e. h The coefficient in the final regression is taken to be the estimate of . Then a 2SLS estimation method is applied on these transformed feature spaces. The authors show asymptotic consistency of the resulting estimator, assuming that the approximation error goes to zero. Subsequently, they also estimate the function m(z) =E[y h(x) | z] based on another growing sieve. Though it may seem at first that the approach in that paper and ours are quite distinct, the population limit of our objective function coincides with theirs. To see this, consider the simplified version of our estimator presented in (6), where the function classes are already norm-constrained and no norm based regularization is imposed. Moreover, for a moment consider the population version of this estimator, i.e. min max (h, f) kfk Thus in the population limit and without norm regularization on the test function f, our criterion is equivalent to the minimum distance criterion analyzed in Chen and Pouzo [2012]. Another point of similarity is that we prove convergence of the estimator in terms of the pseudo-metric, the projected MSE defined in Section 4 of Chen and Pouzo [2012] - and like that paper we require additional conditions to relate the pseudo-metric to the true MSE. The present paper differs in a number of ways: (i) the finite sample criterion is different; (ii) we prove our results using localized Rademacher analysis which allows for weaker assumptions; (iii) we consider a broader range of estimation approaches than linear sieves, necessitating more of a focus on optimization. Digging into the second point, Chen and Pouzo [2012] take a more traditional parameter recovery approach which requires several minimum eigenvalue conditions and several regularity conditions to be satisfied for their estimation rate to hold (see e.g. This is analogous to a mean squared error proof in an exogenous linear regression setting, that requires the minimum eigenvalue of the feature co-variance to be bounded away from zero. Moreover, such parameter recovery methods seem limited to the growing sieve approach, since only then one has a clear finite dimensional parameter vector to work on for each fixed n.

In this paper, we propose a novel conceptual framework to detect outliers using optimal transport with a concave cost function. Conventional outlier detection approaches typically use a two-stage procedure: first, outliers are detected and removed, and then estimation is performed on the cleaned data. However, this approach does not inform outlier removal with the estimation task, leaving room for improvement. To address this limitation, we propose an automatic outlier rectification mechanism that integrates rectification and estimation within a joint optimization framework. We take the first step to utilize the optimal transport distance with a concave cost function to construct a rectification set in the space of probability distributions. Then, we select the best distribution within the rectification set to perform the estimation task. Notably, the concave cost function we introduced in this paper is the key to making our estimator effectively identify the outlier during the optimization process. We demonstrate the effectiveness of our approach over conventional approaches in simulations and empirical analyses for mean estimation, least absolute regression, and the fitting of option implied volatility surfaces.