Principal component regression (PCR) is a two-stage procedure: the first stage performs principal component analysis (PCA) and the second stage constructs a regression model whose explanatory variables are replaced by principal components obtained by the first stage. Since PCA is performed by using only explanatory variables, the principal components have no information about the response variable. To address the problem, we propose a one-stage procedure for PCR in terms of singular value decomposition approach. Our approach is based upon two loss functions, a regression loss and a PCA loss, with sparse regularization. The proposed method enables us to obtain principal component loadings that possess information about both explanatory variables and a response variable. An estimation algorithm is developed by using alternating direction method of multipliers. We conduct numerical studies to show the effectiveness of the proposed method.

Multinomial Logistic Regression is a well-studied tool for classification and has been widely used in fields like image processing, computer vision and, bioinformatics, to name a few. Under a supervised classification scenario, a Multinomial Logistic Regression model learns a weight vector to differentiate between any two classes by optimizing over the likelihood objective. With the advent of big data, the inundation of data has resulted in large dimensional weight vector and has also given rise to a huge number of classes, which makes the classical methods applicable for model estimation not computationally viable. To handle this issue, we here propose a parallel iterative algorithm: Parallel Iterative Algorithm for MultiNomial LOgistic Regression (PIANO) which is based on the Majorization Minimization procedure, and can parallely update each element of the weight vectors. Further, we also show that PIANO can be easily extended to solve the Sparse Multinomial Logistic Regression problem - an extensively studied problem because of its attractive feature selection property. In particular, we work out the extension of PIANO to solve the Sparse Multinomial Logistic Regression problem with l1 and l0 regularizations. We also prove that PIANO converges to a stationary point of the Multinomial and the Sparse Multinomial Logistic Regression problems. Simulations were conducted to compare PIANO with the existing methods, and it was found that the proposed algorithm performs better than the existing methods in terms of speed of convergence.

We introduce Deep Sigma Point Processes, a class of parametric models inspired by the compositional structure of Deep Gaussian Processes (DGPs). Deep Sigma Point Processes (DSPPs) retain many of the attractive features of (variational) DGPs, including mini-batch training and predictive uncertainty that is controlled by kernel basis functions. Importantly, since DSPPs admit a simple maximum likelihood inference procedure, the resulting predictive distributions are not degraded by any posterior approximations. In an extensive empirical comparison on univariate and multivariate regression tasks we find that the resulting predictive distributions are significantly better calibrated than those obtained with other probabilistic methods for scalable regression, including variational DGPs--often by as much as a nat per datapoint.

\textit{SummerTime} seeks to summarize globally time series signals and provides a fixed-length, robust summarization of the variable-length time series. Many classical machine learning methods for classification and regression depend on data instances with a fixed number of features. As a result, those methods cannot be directly applied to variable-length time series data. One common approach is to perform classification over a sliding window on the data and aggregate the decisions made at local sections of the time series in some way, through majority voting for classification or averaging for regression. The downside to this approach is that minority local information is lost in the voting process and averaging assumes that each time series measurement is equal in significance. Also, since time series can be of varying length, the quality of votes and averages could vary greatly in cases where there is a close voting tie or bimodal distribution of regression domain. Summarization conducted by the \textit{SummerTime} method will be a fixed-length feature vector which can be used in-place of the time series dataset for use with classical machine learning methods. We use Gaussian Mixture models (GMM) over small same-length disjoint windows in the time series to group local data into clusters. The time series' rate of membership for each cluster will be a feature in the summarization. The model is naturally capable of converging to an appropriate cluster count. We compare our results to state-of-the-art studies in physical activity classification and show high-quality improvement by classifying with only the summarization. Finally, we show that regression using the summarization can augment energy expenditure estimation, producing more robust and precise results.

Recent advances in machine learning highlight successes on a small set of tasks where a large number of labeled examples have been collected and exploited. These include image classification with 1.2 million labeled examples Deng et al. (2009) and French-English machine translation with 40 million paired sentences Bojar et al. (2014). For common tasks, however, collecting clean labels is costly, as they require human expertise (as in medical imaging) or physical interactions (as in robotics), for example. Thus collected real-world datasets follow a long-tailed distribution, in which a dominant set of tasks only have a small number of training examples Wang et al. (2017). Inspired by human ingenuity in quickly solving novel problems by leveraging prior experience, meta-learning approaches aim to jointly learn from past experience to quickly adapt to new tasks with little available data Schmidhuber (1987); Thrun & Pratt (2012). This has had a significant impact in few-shot supervised learning, where each task is associated with only a few training examples. By leveraging structural similarities among those tasks, one can achieve accuracy far greater than what can be achieved for each task in isolation Finn et al. (2017); Ravi & Larochelle (2016); Koch et al. (2015); Oreshkin et al. (2018); Triantafillou et al. (2019); Rusu et al. (2018). The success of such approaches hinges on the following fundamental question: When can we jointly train small data tasks to achieve the accuracy of large data tasks? We investigate this tradeoff under a canonical scenario where the tasks are linear regressions in d-dimensions and the regression parameters are drawn i.i.d.

Traditionally, there are two models on differential privacy: the central model and the local model. The central model focuses on the machine learning model and the local model focuses on the training data. In this paper, we study the \textit{input perturbation} method in differentially private empirical risk minimization (DP-ERM), preserving privacy of the central model. By adding noise to the original training data and training with the `perturbed data', we achieve ($\epsilon$,$\delta$)-differential privacy on the final model, along with some kind of privacy on the original data. We observe that there is an interesting connection between the local model and the central model: the perturbation on the original data causes the perturbation on the gradient, and finally the model parameters. This observation means that our method builds a bridge between local and central model, protecting the data, the gradient and the model simultaneously, which is more superior than previous central methods. Detailed theoretical analysis and experiments show that our method achieves almost the same (or even better) performance as some of the best previous central methods with more protections on privacy, which is an attractive result. Moreover, we extend our method to a more general case: the loss function satisfies the Polyak-Lojasiewicz condition, which is more general than strong convexity, the constraint on the loss function in most previous work.

We address the problem of the achievable regret rates with online logistic regression. We derive lower bounds with logarithmic regret under $L_1$, $L_2$, and $L_\infty$ constraints on the parameter values. The bounds are dominated by $d/2 \log T$, where $T$ is the horizon and $d$ is the dimensionality of the parameter space. We show their achievability for $d=o(T^{1/3})$ in all these cases with Bayesian methods, that achieve them up to a $d/2 \log d$ term. Interesting different behaviors are shown for larger dimensionality. Specifically, on the negative side, if $d = \Omega(\sqrt{T})$, any algorithm is guaranteed regret of $\Omega(d \log T)$ (greater than $\Omega(\sqrt{T})$) under $L_\infty$ constraints on the parameters (and the example features). On the positive side, under $L_1$ constraints on the parameters, there exist algorithms that can achieve regret that is sub-linear in $d$ for the asymptotically larger values of $d$. For $L_2$ constraints, it is shown that for large enough $d$, the regret remains linear in $d$ but no longer logarithmic in $T$. Adapting the redundancy-capacity theorem from information theory, we demonstrate a principled methodology based on grids of parameters to derive lower bounds. Grids are also utilized to derive some upper bounds. Our results strengthen results by Kakade and Ng (2005) and Foster et al. (2018) for upper bounds for this problem, introduce novel lower bounds, and adapt a methodology that can be used to obtain such bounds for other related problems. They also give a novel characterization of the asymptotic behavior when the dimension of the parameter space is allowed to grow with $T$. They additionally establish connections to the information theory literature, demonstrating that the actual regret for logistic regression depends on the richness of the parameter class, where even within this problem, richer classes lead to greater regret.

We present a method for financial time series forecasting using representation learning techniques. Recent progress on deep autoregressive models has shown their ability to capture long-term dependencies of the sequence data. However, the shortage of available financial data for training will make the deep models susceptible to the overfitting problem. In this paper, we propose a neural-network-powered conditional mutual information (CMI) estimator for learning representations for the forecasting task. Specifically, we first train an encoder to maximize the mutual information between the latent variables and the label information conditioned on the encoded observed variables. Then the features extracted from the trained encoder are used to learn a subsequent logistic regression model for predicting time series movements. Our proposed estimator transforms the CMI maximization problem to a classification problem whether two encoded representations are sampled from the same class or not. This is equivalent to perform pairwise comparisons of the training datapoints, and thus, improves the generalization ability of the deep autoregressive model. Empirical experiments indicate that our proposed method has the potential to advance the state-of-the-art performance.

Land-use regression (LUR) models are important for the assessment of air pollution concentrations in areas without measurement stations. While many such models exist, they often use manually constructed features based on restricted, locally available data. Thus, they are typically hard to reproduce and challenging to adapt to areas beyond those they have been developed for. In this paper, we advocate a paradigm shift for LUR models: We propose the Data-driven, Open, Global (DOG) paradigm that entails models based on purely data-driven approaches using only openly and globally available data. Progress within this paradigm will alleviate the need for experts to adapt models to the local characteristics of the available data sources and thus facilitate the generalizability of air pollution models to new areas on a global scale. In order to illustrate the feasibility of the DOG paradigm for LUR, we introduce a deep learning model called MapLUR. It is based on a convolutional neural network architecture and is trained exclusively on globally and openly available map data without requiring manual feature engineering. We compare our model to state-of-the-art baselines like linear regression, random forests and multi-layer perceptrons using a large data set of modeled $\text{NO}_2$ concentrations in Central London. Our results show that MapLUR significantly outperforms these approaches even though they are provided with manually tailored features. Furthermore, we illustrate that the automatic feature extraction inherent to models based on the DOG paradigm can learn features that are readily interpretable and closely resemble those commonly used in traditional LUR approaches.

The information bottleneck principle provides an information-theoretic method for representation learning, by training an encoder to retain all information which is relevant for predicting the label while minimizing the amount of other, excess information in the representation. The original formulation, however, requires labeled data to identify the superfluous information. In this work, we extend this ability to the multi-view unsupervised setting, where two views of the same underlying entity are provided but the label is unknown. This enables us to identify superfluous information as that not shared by both views. A theoretical analysis leads to the definition of a new multi-view model that produces state-of-the-art results on the Sketchy dataset and label-limited versions of the MIR-Flickr dataset. We also extend our theory to the single-view setting by taking advantage of standard data augmentation techniques, empirically showing better generalization capabilities when compared to common unsupervised approaches for representation learning.