The overarching goal of this paper is to derive excess risk bounds for learning from exp-concave loss functions in passive and sequential learning settings. Exp-concave loss functions encompass several fundamental problems in machine learning such as squared loss in linear regression, logistic loss in classification, and negative logarithm loss in portfolio management. In batch setting, we obtain sharp bounds on the performance of empirical risk minimization performed in a linear hypothesis space and with respect to the exp-concave loss functions. We also extend the results to the online setting where the learner receives the training examples in a sequential manner. We propose an online learning algorithm that is a properly modified version of online Newton method to obtain sharp risk bounds. Under an additional mild assumption on the loss function, we show that in both settings we are able to achieve an excess risk bound of $O(d\log n/n)$ that holds with a high probability.

Mixed linear regression involves the recovery of two (or more) unknown vectors from unlabeled linear measurements; that is, where each sample comes from exactly one of the vectors, but we do not know which one. It is a classic problem, and the natural and empirically most popular approach to its solution has been the EM algorithm. As in other settings, this is prone to bad local minima; however, each iteration is very fast (alternating between guessing labels, and solving with those labels). In this paper we provide a new initialization procedure for EM, based on finding the leading two eigenvectors of an appropriate matrix. We then show that with this, a re-sampled version of the EM algorithm provably converges to the correct vectors, under natural assumptions on the sampling distribution, and with nearly optimal (unimprovable) sample complexity. This provides not only the first characterization of EM's performance, but also much lower sample complexity as compared to both standard (randomly initialized) EM, and other methods for this problem.

Accurate calibration of probabilistic predictive models learned is critical for many practical prediction and decision-making tasks. There are two main categories of methods for building calibrated classifiers. One approach is to develop methods for learning probabilistic models that are well-calibrated, ab initio. The other approach is to use some post-processing methods for transforming the output of a classifier to be well calibrated, as for example histogram binning, Platt scaling, and isotonic regression. One advantage of the post-processing approach is that it can be applied to any existing probabilistic classification model that was constructed using any machine-learning method. In this paper, we first introduce two measures for evaluating how well a classifier is calibrated. We prove three theorems showing that using a simple histogram binning post-processing method, it is possible to make a classifier be well calibrated while retaining its discrimination capability. Also, by casting the histogram binning method as a density-based non-parametric binary classifier, we can extend it using two simple non-parametric density estimation methods. We demonstrate the performance of the proposed calibration methods on synthetic and real datasets. Experimental results show that the proposed methods either outperform or are comparable to existing calibration methods.

The least absolute shrinkage and selection operator (lasso) and ridge regression produce usually different estimates although input, loss function and parameterization of the penalty are identical. In this paper we look for ridge and lasso models with identical solution set. It turns out, that the lasso model with shrink vector $\lambda$ and a quadratic penalized model with shrink matrix as outer product of $\lambda$ with itself are equivalent, in the sense that they have equal solutions. To achieve this, we have to restrict the estimates to be positive. This doesn't limit the area of application since we can easily decompose every estimate in a positive and negative part. The resulting problem can be solved with a non negative least square algorithm. Beside this quadratic penalized model, an augmented regression model with positive bounded estimates is developed which is also equivalent to the lasso model, but is probably faster to solve.

We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that 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. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.

We establish minimax risk lower bounds for distributed statistical estimation given a budget $B$ of the total number of bits that may be communicated. Such lower bounds in turn reveal the minimum amount of communication required by any procedure to achieve the classical optimal rate for statistical estimation. We study two classes of protocols in which machines send messages either independently or interactively. The lower bounds are established for a variety of problems, from estimating the mean of a population to estimating parameters in linear regression or binary classification.

Differential privacy is a cryptographically motivated definition of privacy which has gained considerable attention in the algorithms, machine-learning and data-mining communities. While there has been an explosion of work on differentially private machine learning algorithms, a major barrier to achieving end-to-end differential privacy in practical machine learning applications is the lack of an effective procedure for differentially private parameter tuning, or, determining the parameter value, such as a bin size in a histogram, or a regularization parameter, that is suitable for a particular application. In this paper, we introduce a generic validation procedure for differentially private machine learning algorithms that apply when a certain stability condition holds on the training algorithm and the validation performance metric. The training data size and the privacy budget used for training in our procedure is independent of the number of parameter values searched over. We apply our generic procedure to two fundamental tasks in statistics and machine-learning -- training a regularized linear classifier and building a histogram density estimator that result in end-to-end differentially private solutions for these problems.

We establish theoretical results concerning all local optima of various regularized M-estimators, where both loss and penalty functions are allowed to be nonconvex. Our results show that as long as the loss function satisfies restricted strong convexity and the penalty function satisfies suitable regularity conditions, any local optimum of the composite objective function lies within statistical precision of the true parameter vector. Our theory covers a broad class of nonconvex objective functions, including corrected versions of the Lasso for errors-in-variables linear models; regression in generalized linear models using nonconvex regularizers such as SCAD and MCP; and graph and inverse covariance matrix estimation. On the optimization side, we show that a simple adaptation of composite gradient descent may be used to compute a global optimum up to the statistical precision epsilon in log(1/epsilon) iterations, which is the fastest possible rate of any first-order method. We provide a variety of simulations to illustrate the sharpness of our theoretical predictions.

In the high-dimensional regression model a response variable is linearly related to $p$ covariates, but the sample size $n$ is smaller than $p$. We assume that only a small subset of covariates is `active' (i.e., the corresponding coefficients are non-zero), and consider the model-selection problem of identifying the active covariates. A popular approach is to estimate the regression coefficients through the Lasso ($\ell_1$-regularized least squares). This is known to correctly identify the active set only if the irrelevant covariates are roughly orthogonal to the relevant ones, as quantified through the so called `irrepresentability' condition. In this paper we study the `Gauss-Lasso' selector, a simple two-stage method that first solves the Lasso, and then performs ordinary least squares restricted to the Lasso active set. We formulate `generalized irrepresentability condition' (GIC), an assumption that is substantially weaker than irrepresentability. We prove that, under GIC, the Gauss-Lasso correctly recovers the active set.

Motivated by the desire to extend fast randomized techniques to nonlinear $l_p$ regression, we consider a class of structured regression problems. These problems involve Vandermonde matrices which arise naturally in various statistical modeling settings, including classical polynomial fitting problems and recently developed randomized techniques for scalable kernel methods. We show that this structure can be exploited to further accelerate the solution of the regression problem, achieving running times that are faster than input sparsity''. We present empirical results confirming both the practical value of our modeling framework, as well as speedup benefits of randomized regression."