Continuous-time mirror descent (CMD) can be seen as the limit case of the discrete-time MD update when the step-size is infinitesimally small. In this paper, we focus on the geometry of the primal and dual CMD updates and introduce a general framework for reparameterizing one CMD update as another. Specifically, the reparameterized update also corresponds to a CMD, but on the composite loss w.r.t. the new variables, and the original variables are obtained via the reparameterization map. We employ these results to introduce a new family of reparameterizations that interpolate between the two commonly used updates, namely the continuous-time gradient descent (GD) and unnormalized exponentiated gradient (EGU), while extending to many other well-known updates. In particular, we show that for the underdetermined linear regression problem, these updates generalize the known behavior of GD and EGU, and provably converge to the minimum $\mathrm{L}_{2-\tau}$-norm solution for $\tau\in[0,1]$. Our new results also have implications for the regularized training of neural networks to induce sparsity.

Despite breakthrough performance, modern learning models are known to be highly vulnerable to small adversarial perturbations in their inputs. While a wide variety of recent \emph{adversarial training} methods have been effective at improving robustness to perturbed inputs (robust accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard accuracy), leading to a tradeoff between often competing objectives. Complicating matters further, recent empirical evidence suggest that a variety of other factors (size and quality of training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this paper we provide a precise and comprehensive understanding of the role of adversarial training in the context of linear regression with Gaussian features. In particular, we characterize the fundamental tradeoff between the accuracies achievable by any algorithm regardless of computational power or size of the training data. Furthermore, we precisely characterize the standard/robust accuracy and the corresponding tradeoff achieved by a contemporary mini-max adversarial training approach in a high-dimensional regime where the number of data points and the parameters of the model grow in proportion to each other. Our theory for adversarial training algorithms also facilitates the rigorous study of how a variety of factors (size and quality of training data, model overparametrization etc.) affect the tradeoff between these two competing accuracies.

For some or all of the data instances a number of independentworld clustering issues suffer from incomplete data characterization due to losing or absent attributes. Typical clustering approaches cannot be applied directly to such data unless pre-processing by techniques like imputation or marginalization. We have overcome this drawback by utilizing a Sentenced Discrepancy Measure which we refer to as the Attribute Weighted Penalty based Discrepancy (AWPD). Using the AWPD measure, we modified the K-MEANS and Scalable K-MEANS for clustering algorithm and k Nearest Neighbor (kNN) for classification so as to make them directly applicable to datasets with nonexistence attributes. We have presented a detailed theoretical analysis which shows that the new AWPD based K-MEANS, Scalable K-MEANS and kNN algorithm merge into a local prime among the number of iterations is finite. We have reported in depth experiments on numerous benchmark datasets for various forms of Non-Existence showing that the projected clustering and classification techniques usually show better results in comparison to some of the renowned imputation methods that are generally used to process such insufficient data. This technique is designed to trace invaluable data to: directly apply our method on the datasets which have Non-Existence attributes and establish a method for detecting unstructured Non-Existence attributes with the best accuracy rate and minimum cost. Keywords: Clustering, Classification, Non-Existence Attributes, Unstructured Non-Existence, Sentenced Discrepancy Measure (SDM), Attribute Weighted Penalty based Discrepancy (AWPD).

Parameter estimation, statistical tests and confidence sets are the cornerstones of classical statistics that allow scientists to make inferences about the underlying process that generated the observed data. A key question is whether one can still construct hypothesis tests and confidence sets with proper coverage and high power in a so-called likelihood-free inference (LFI) setting; that is, a setting where the likelihood is not explicitly known but one can forward-simulate observable data according to a stochastic model. In this paper, we present $\texttt{ACORE}$ (Approximate Computation via Odds Ratio Estimation), a frequentist approach to LFI that first formulates the classical likelihood ratio test (LRT) as a parametrized classification problem, and then uses the equivalence of tests and confidence sets to build confidence regions for parameters of interest. We also present a goodness-of-fit procedure for checking whether the constructed tests and confidence regions are valid. $\texttt{ACORE}$ is based on the key observation that the LRT statistic, the rejection probability of the test, and the coverage of the confidence set are conditional distribution functions which often vary smoothly as a function of the parameters of interest. Hence, instead of relying solely on samples simulated at fixed parameter settings (as is the convention in standard Monte Carlo solutions), one can leverage machine learning tools and data simulated in the neighborhood of a parameter to improve estimates of quantities of interest. We demonstrate the efficacy of $\texttt{ACORE}$ with both theoretical and empirical results. Our implementation is available on Github.

We study the problem of learning from multiple untrusted data sources, a scenario of increasing practical relevance given the recent emergence of crowdsourcing and collaborative learning paradigms. Specifically, we analyze the situation in which a learning system obtains datasets from multiple sources, some of which might be biased or even adversarially perturbed. It is known that in the single-source case, an adversary with the power to corrupt a fixed fraction of the training data can prevent PAC-learnability, that is, even in the limit of infinitely much training data, no learning system can approach the optimal test error. In this work we show that, surprisingly, the same is not true in the multi-source setting, where the adversary can arbitrarily corrupt a fixed fraction of the data sources. Our main results are a generalization bound that provides finite-sample guarantees for this learning setting, as well as corresponding lower bounds. Besides establishing PAC-learnability our results also show that in a cooperative learning setting sharing data with other parties has provable benefits, even if some participants are malicious.

Feature Selection (FS) is a key task in Machine Learning. It consists in selecting a number of relevant variables for the model construction or data analysis. We present the R package, FSinR, which implements a variety of widely known filter and wrapper methods, as well as search algorithms. Thus, the package provides the possibility to perform the feature selection process, which consists in the combination of a guided search on the subsets of features with the filter or wrapper methods that return an evaluation measure of those subsets. In this article, we also present some examples on the usage of the package and a comparison with other packages available in R that contain methods for feature selection.

Conditional distributions provide a versatile tool for capturing the relationship between a target variable and a conditioning variable (or covariate). The last few decades has seen a broad range of modeling applications across multiple disciplines including econometrics in particular [30, 42], machine learning [14, 40], among others. In many cases, estimating a conditional density function from the observed data is a one of the first crucial steps in the data analysis pipeline. While the task of conditional density estimation has received a considerable attention in the literature, fewer works have investigated the equally important task of evaluating the goodness of fit of a given conditional density model. Several approaches that address the task of conditional model evaluation take the form of a hypothesis test. Given a conditional model, and a joint sample containing realizations of both target variables and covariates, test the null hypothesis stating that the model is correctly specified, against the alternative stating that it is not. The model does not specify the marginal distribution of the covariates. We refer to this task as conditional goodness-of-fit testing. One of the early nonparametric tests is [1], which extended the classic Kolmogorov test to the conditional case.

Deep learning image classifiers usually rely on huge training sets and their training process can be described as learning the similarities and differences among training images. But, images in large training sets are not usually studied from this perspective and fine-level similarities and differences among images is usually overlooked. Some studies aim to identify the influential and redundant training images, but such methods require a model that is already trained on the entire training set. Here, we show that analyzing the contents of large training sets can provide valuable insights about the classification task at hand, prior to training a model on them. We use wavelet decomposition of images and other image processing tools to perform such analysis, with no need for a pre-trained model. This makes the analysis of training sets, straightforward and fast. We show that similar images in standard datasets (such as CIFAR) can be identified in a few seconds, a significant speed-up compared to alternative methods in the literature. We also show that similarities between training and testing images may explain the generalization of models and their mistakes. Finally, we investigate the similarities between images in relation to decision boundaries of a trained model.

We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions the error bound can then be explicitly established.

Classifier calibration does not always go hand in hand with the classifier's ability to separate the classes. There are applications where good classifier calibration, i.e. the ability to produce accurate probability estimates, is more important than class separation. When the amount of data for training is limited, the traditional approach to improve calibration starts to crumble. In this article we show how generating more data for calibration is able to improve calibration algorithm performance in many cases where a classifier is not naturally producing well-calibrated outputs and the traditional approach fails. The proposed approach adds computational cost but considering that the main use case is with small data sets this extra computational cost stays insignificant and is comparable to other methods in prediction time. From the tested classifiers the largest improvement was detected with the random forest and naive Bayes classifiers. Therefore, the proposed approach can be recommended at least for those classifiers when the amount of data available for training is limited and good calibration is essential.