We study least squares linear regression over $N$ uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features $p$ is at most the sample size $n$, the estimator under consideration coincides with the principal component regression estimator; when $p>n$, the estimator is the least $\ell_2$ norm solution over the selected features. We give an average-case analysis of the out-of-sample prediction error as $p,n,N \to \infty$ with $p/N \to \alpha$ and $n/N \to \beta$, for some constants $\alpha \in [0,1]$ and $\beta \in (0,1)$. In this average-case setting, the prediction error exhibits a `double descent' shape as a function of $p$.

We study the problem of predicting whether the price of the 21 most popular cryptocurrencies (according to coinmarketcap.com) will go up or down on day d, using data up to day d-1. Our C2P2 algorithm is the first algorithm to consider the fact that the price of a cryptocurrency c might depend not only on historical prices, sentiments, global stock indices, but also on the prices and predicted prices of other cryptocurrencies. C2P2 therefore does not predict cryptocurrency prices one coin at a time --- rather it uses similarity metrics in conjunction with collective classification to compare multiple cryptocurrency features to jointly predict the cryptocurrency prices for all 21 coins considered. We show that our C2P2 algorithm beats out a recent competing 2017 paper by margins varying from 5.1-83% and another Bitcoin-specific prediction paper from 2018 by 16%. In both cases, C2P2 is the winner on all cryptocurrencies considered. Moreover, we experimentally show that the use of similarity metrics within our C2P2 algorithm leads to a direct improvement for 20 out of 21 cryptocurrencies ranging from 0.4% to 17.8%. Without the similarity component, C2P2 still beats competitors on 20 out of 21 cryptocurrencies considered. We show that all these results are statistically significant via a Student's t-test with p<1e-5. Check our demo at https://www.cs.dartmouth.edu/dsail/demos/c2p2

Uncoupled regression is the problem to learn a model from unlabeled data and the set of target values while the correspondence between them is unknown. Such a situation arises in predicting anonymized targets that involve sensitive information, e.g., one's annual income. Since existing methods for uncoupled regression often require strong assumptions on the true target function, and thus, their range of applications is limited, we introduce a novel framework that does not require such assumptions in this paper. Our key idea is to utilize pairwise comparison data, which consists of pairs of unlabeled data that we know which one has a larger target value. Such pairwise comparison data is easy to collect, as typically discussed in the learning-to-rank scenario, and does not break the anonymity of data. We propose two practical methods for uncoupled regression from pairwise comparison data and show that the learned regression model converges to the optimal model with the optimal parametric convergence rate when the target variable distributes uniformly. Moreover, we empirically show that for linear models the proposed methods are comparable to ordinary supervised regression with labeled data.

With this summary, we can see important values such as R2, the F-statistic, and many others. You can also analyze a model using a graphical diagnostic such as plotting the residuals against the fitted/predicted values. Above is the fitted versus residual plot for our weight-height dataset, using height as the predictor. For the most part, this plot is random. However, as fitted values increase, so does the range of residuals.

Machine learning and deep learning have been widely embraced, and even more widely misunderstood. In this article, I'd like to step back and explain both machine learning and deep learning in basic terms, discuss some of the most common machine learning algorithms, and explain how those algorithms relate to the other pieces of the puzzle of creating predictive models from historical data. Recall that machine learning is a class of methods for automatically creating models from data. Machine learning algorithms are the engines of machine learning, meaning it is the algorithms that turn a data set into a model. Which kind of algorithm works best (supervised, unsupervised, classification, regression, etc.) depends on the kind of problem you're solving, the computing resources available, and the nature of the data.

This paper examines the assumptions of the derived equivalence between dropout noise injection and $L_2$ regularisation for logistic regression with negative log loss. We show that the approximation method is based on a divergent Taylor expansion, making, subsequent work using this approximation to compare the dropout trained logistic regression model with standard regularisers unfortunately ill-founded to date. Moreover, the approximation approach is shown to be invalid using any robust constraints. We show how this finding extends to general neural network topologies that use a cross-entropy prediction layer.

We construct and analyze active learning algorithms for the problem of binary classification with abstention. We consider three abstention settings: \emph{fixed-cost} and two variants of \emph{bounded-rate} abstention, and for each of them propose an active learning algorithm. All the proposed algorithms can work in the most commonly used active learning models, i.e., \emph{membership-query}, \emph{pool-based}, and \emph{stream-based} sampling. We obtain upper-bounds on the excess risk of our algorithms in a general non-parametric framework and establish their minimax near-optimality by deriving matching lower-bounds. Since our algorithms rely on the knowledge of some smoothness parameters of the regression function, we then describe a new strategy to adapt to these unknown parameters in a data-driven manner. Since the worst case computational complexity of our proposed algorithms increases exponentially with the dimension of the input space, we conclude the paper with a computationally efficient variant of our algorithm whose computational complexity has a polynomial dependence over a smaller but rich class of learning problems.

Instrumental variable regression is a strategy for learning causal relationships in observational data. If measurements of input X and output Y are confounded, the causal relationship can nonetheless be identified if an instrumental variable Z is available that influences X directly, but is conditionally independent of Y given X. The classic two-stage least squares algorithm (2SLS) simplifies the estimation problem by modeling all relationships as linear functions. We propose kernel instrumental variable regression (KIV), a nonparametric generalization of 2SLS, modeling relations among X, Y, and Z as nonlinear functions in reproducing kernel Hilbert spaces (RKHSs). We prove the consistency of KIV under mild assumptions, and derive conditions under which the convergence rate achieves the minimax optimal rate for unconfounded, one-stage RKHS regression. In doing so, we obtain an efficient ratio between training sample sizes used in the algorithm's first and second stages. In experiments, KIV outperforms state of the art alternatives for nonparametric instrumental variable regression.

Estimating regression coefficients using unordered multisets of covariates and responses has been introduced as the regression without correspondence problem. Previous theoretical analysis of the problem has been done in a setting where the responses are a permutation of the regressed covariates. This paper expands the setting by analyzing the problem where they may be missing correspondences and outliers in addition to a permutation action. We term this problem robust regression without correspondence and provide several algorithms for exact and approximate recovery in a noiseless and noisy one-dimensional setting as well as an approximation algorithm for multiple dimensions. The theoretical guarantees of the algorithms are verified in simulation data. We also demonstrate a neuroscience application by obtaining robust point set matchings of the neurons of the model organism Caenorhabditis elegans.

This paper extends the class of ordinal regression models with a structured interpretation of the problem by applying a novel treatment of encoded labels. The net effect of this is to transform the underlying problem from an ordinal regression task to a (structured) classification task which we solve with conditional random fields, thereby achieving a coherent and probabilistic model in which all model parameters are jointly learnt. Importantly, we show that although we have cast ordinal regression to classification, our method still fall within the class of decomposition methods in the ordinal regression ontology. This is an important link since our experience is that many applications of machine learning to healthcare ignores completely the important nature of the label ordering, and hence these approaches should considered naive in this ontology. We also show that our model is flexible both in how it adapts to data manifolds and in terms of the operations that are available for practitioner to execute. Our empirical evaluation demonstrates that the proposed approach overwhelmingly produces superior and often statistically significant results over baseline approaches on forty popular ordinal regression models, and demonstrate that the proposed model significantly out-performs baselines on synthetic and real datasets. Our implementation, together with scripts to reproduce the results of this work, will be available on a public GitHub repository.