Single Index Models (SIMs) are simple yet flexible semi-parametric models for classification and regression. Response variables are modeled as a nonlinear, monotonic function of a linear combination of features. Estimation in this context requires le
arning both the feature weights, and the nonlinear function. While methods have been described to learn SIMs in the low dimensional regime, a method that can efficiently learn SIMs in high dimensions has not been forthcoming. We propose three variants of a computationally and statistically efficient algorithm for SIM inference in high dimensions. We establish excess risk bounds for the proposed algorithms and experimentally validate the advantages that our SIM learning methods provide relative to Generalized Linear Model (GLM) and low dimensional SIM based learning methods.
We consider the problem of learning convex aggregation of models, that is as good as the best convex aggregation, for the binary classification problem. Working in the stream based active learning setting, where the active learner has to make a decis
ion on-the-fly, if it wants to query for the label of the point currently seen in the stream, we propose a stochastic-mirror descent algorithm, called SMD-AMA, with entropy regularization. We establish an excess risk bounds for the loss of the convex aggregate returned by SMD-AMA to be of the order of $Oleft(sqrt{frac{log(M)}{{T^{1-mu}}}}right)$, where $muin [0,1)$ is an algorithm dependent parameter, that trades-off the number of labels queried, and excess risk.
This paper proposes a fast and accurate method for sparse regression in the presence of missing data. The underlying statistical model encapsulates the low-dimensional structure of the incomplete data matrix and the sparsity of the regression coeffic
ients, and the proposed algorithm jointly learns the low-dimensional structure of the data and a linear regressor with sparse coefficients. The proposed stochastic optimization method, Sparse Linear Regression with Missing Data (SLRM), performs an alternating minimization procedure and scales well with the problem size. Large deviation inequalities shed light on the impact of the various problem-dependent parameters on the expected squared loss of the learned regressor. Extensive simulations on both synthetic and real datasets show that SLRM performs better than competing algorithms in a variety of contexts.
In this paper we propose a multi-armed bandit inspired, pool based active learning algorithm for the problem of binary classification. By carefully constructing an analogy between active learning and multi-armed bandits, we utilize ideas such as lowe
r confidence bounds, and self-concordant regularization from the multi-armed bandit literature to design our proposed algorithm. Our algorithm is a sequential algorithm, which in each round assigns a sampling distribution on the pool, samples one point from this distribution, and queries the oracle for the label of this sampled point. The design of this sampling distribution is also inspired by the analogy between active learning and multi-armed bandits. We show how to derive lower confidence bounds required by our algorithm. Experimental comparisons to previously proposed active learning algorithms show superior performance on some standard UCI datasets.
In this paper we address the problem of pool based active learning, and provide an algorithm, called UPAL, that works by minimizing the unbiased estimator of the risk of a hypothesis in a given hypothesis space. For the space of linear classifiers an
d the squared loss we show that UPAL is equivalent to an exponentially weighted average forecaster. Exploiting some recent results regarding the spectra of random matrices allows us to establish consistency of UPAL when the true hypothesis is a linear hypothesis. Empirical comparison with an active learner implementation in Vowpal Wabbit, and a previously proposed pool based active learner implementation show good empirical performance and better scalability.
