Obtaining labels can be costly and time-consuming. Active learning allows a learning algorithm to intelligently query samples to be labeled for efficient learning. Fisher information ratio (FIR) has been used as an objective for selecting queries in active learning. However, little is known about the theory behind the use of FIR for active learning. There is a gap between the underlying theory and the motivation of its usage in practice. In this paper, we attempt to fill this gap and provide a rigorous framework for analyzing existing FIR-based active learning methods. In particular, we show that FIR can be asymptotically viewed as an upper bound of the expected variance of the log-likelihood ratio. Additionally, our analysis suggests a unifying framework that not only enables us to make theoretical comparisons among the existing querying methods based on FIR, but also allows us to give insight into the development of new active learning approaches based on this objective.

In analyzing high-dimensional models, sparsity of the model parameter is a common but often undesirable assumption. In this paper, we study the following two-sample testing problem: given two samples generated by two high-dimensional linear models, we aim to test whether the regression coefficients of the two linear models are identical. We propose a framework named TIERS (short for TestIng Equality of Regression Slopes), which solves the two-sample testing problem without making any assumptions on the sparsity of the regression parameters. TIERS builds a new model by convolving the two samples in such a way that the original hypothesis translates into a new moment condition. A self-normalization construction is then developed to form a moment test. We provide rigorous theory for the developed framework. Under very weak conditions of the feature covariance, we show that the accuracy of the proposed test in controlling Type I errors is robust both to the lack of sparsity in the features and to the heavy tails in the error distribution, even when the sample size is much smaller than the feature dimension. Moreover, we discuss minimax optimality and efficiency properties of the proposed test. Simulation analysis demonstrates excellent finite-sample performance of our test. In deriving the test, we also develop tools that are of independent interest. The test is built upon a novel estimator, called Auto-aDaptive Dantzig Selector (ADDS), which not only automatically chooses an appropriate scale of the error term but also incorporates prior information. To effectively approximate the critical value of the test statistic, we develop a novel high-dimensional plug-in approach that complements the recent advances in Gaussian approximation theory.

This image comes from Xkcd, a webcomic of romance, sarcasm, math, and language. Created by Randall Munroe, he is a CNU graduate with a degree in physics. Before starting xkcd, he worked on robots at NASA's Langley Research Center in Virginia.

We propose a novel kernel based post selection inference (PSI) algorithm, which can not only handle non-linearity in data but also structured output such as multi-dimensional and multi-label outputs. Specifically, we develop a PSI algorithm for independence measures, and propose the Hilbert-Schmidt Independence Criterion (HSIC) based PSI algorithm (hsicInf). The novelty of the proposed algorithm is that it can handle non-linearity and/or structured data through kernels. Namely, the proposed algorithm can be used for wider range of applications including nonlinear multi-class classification and multi-variate regressions, while existing PSI algorithms cannot handle them. Through synthetic experiments, we show that the proposed approach can find a set of statistically significant features for both regression and classification problems. Moreover, we apply the hsicInf algorithm to a real-world data, and show that hsicInf can successfully identify important features.

A couple of weeks ago I wrote an article on simple linear regression, which I would recommend reading before proceeding to read this one. Machine learning is a very interesting topic and I have been studying it on my free time. I hope this article sparks your interest in the subject or helps continue fuel it. In simple linear regression there is a one-to-one relationship between the input variable and the output variable. But in multiple linear regression, as the name implies there is a many-to-one relationship, instead of just using one input variable, you use several.

Principal component regression (PCR) is a widely used two-stage procedure: principal component analysis (PCA), followed by regression in which the selected principal components are regarded as new explanatory variables in the model. Note that PCA is based only on the explanatory variables, so the principal components are not selected using the information on the response variable. In this paper, we propose a one-stage procedure for PCR in the framework of generalized linear models. The basic loss function is based on a combination of the regression loss and PCA loss. An estimate of the regression parameter is obtained as the minimizer of the basic loss function with a sparse penalty. We call the proposed method sparse principal component regression for generalized linear models (SPCR-glm). Taking the two loss function into consideration simultaneously, SPCR-glm enables us to obtain sparse principal component loadings that are related to a response variable. However, a combination of loss functions may cause a parameter identification problem, but this potential problem is avoided by virtue of the sparse penalty. Thus, the sparse penalty plays two roles in this method. The parameter estimation procedure is proposed using various update algorithms with the coordinate descent algorithm. We apply SPCR-glm to two real datasets, doctor visits data and mouse consomic strain data. SPCR-glm provides more easily interpretable principal component (PC) scores and clearer classification on PC plots than the usual PCA.

Linear regression is one of the first things you should try if you're modeling a linear relationship (actually, non-linear relationships too!). It's fairly simple, and probably the first thing to learn when tackling machine learning.

Now that we have training and test datasets sampled, initially preprocessed and available in the data grid, we can close Web Notebook and start experimenting with different techniques and algorithms by submitting Spark applications. For our first baseline approach let's take a single feature device_conn_type and logistic regression algorithm: We will explain a little bit more what happens here. At first, we load the training dataset from the data grid, which we prepared and saved earlier with Web Notebook. Then we use StringIndexer and OneHotEncoder to map a column of categories to a column of binary vectors. For example, with 4 categories of device_conn_type, an input value of the second category would map to an output vector of [0.0, 1.0, 0.0, 0.0, 0.0].

Having tried logistic regression the first time around, I moved on to decision trees and KNN. But unfortunately, those models performed horribly and had to be scrapped. Random Forest seemed to be the buzz word around the Kaggle forums, so I obviously had to try it out next. I took a couple of days to read up on it, worked out a few examples on my own before re-taking a stab at the titanic dataset. The'caret' package is a beauty.

In computer science, a logistic model tree (LMT) is a classification model with an associated supervised training algorithm that combines logistic regression (LR) and decision tree learning.[1][2] Logistic model trees are based on the earlier idea of a model tree: a decision tree that has linear regression models at its leaves to provide a piecewise linear regression model (where ordinary decision trees with constants at their leaves would produce a piecewise constant model).[1] In the logistic variant, the LogitBoost algorithm is used to produce an LR model at every node in the tree; the node is then split using the C4.5 criterion. Each LogitBoost invocation is warm-started[vague] from its results in the parent node. Finally, the tree is pruned.[3]