For feature selection and related problems, we introduce the notion of classification game, a cooperative game, with features as players and hinge loss based characteristic function and relate a feature's contribution to Shapley value based error apportioning (SVEA) of total training error. Our major contribution is ($\star$) to show that for any dataset the threshold 0 on SVEA value identifies feature subset whose joint interactions for label prediction is significant or those features that span a subspace where the data is predominantly lying. In addition, our scheme ($\star$) identifies the features on which Bayes classifier doesn't depend but any surrogate loss function based finite sample classifier does; this contributes to the excess $0$-$1$ risk of such a classifier, ($\star$) estimates unknown true hinge risk of a feature, and ($\star$) relate the stability property of an allocation and negative valued SVEA by designing the analogue of core of classification game. Due to Shapley value's computationally expensive nature, we build on a known Monte Carlo based approximation algorithm that computes characteristic function (Linear Programs) only when needed. We address the potential sample bias problem in feature selection by providing interval estimates for SVEA values obtained from multiple sub-samples. We illustrate all the above aspects on various synthetic and real datasets and show that our scheme achieves better results than existing recursive feature elimination technique and ReliefF in most cases. Our theoretically grounded classification game in terms of well defined characteristic function offers interpretability and explainability of our framework, including identification of important features.

An essential problem in automated machine learning (AutoML) is that of model selection. A unique challenge in the sequential setting is the fact that the optimal model itself may vary over time, depending on the distribution of features and labels available up to each point in time. In this paper, we propose a novel Bayesian optimization (BO) algorithm to tackle the challenge of model selection in this setting. This is accomplished by treating the performance at each time step as its own black-box function. In order to solve the resulting multiple black-box function optimization problem jointly and efficiently, we exploit potential correlations among black-box functions using deep kernel learning (DKL). To the best of our knowledge, we are the first to formulate the problem of stepwise model selection (SMS) for sequence prediction, and to design and demonstrate an efficient joint-learning algorithm for this purpose. Using multiple real-world datasets, we verify that our proposed method outperforms both standard BO and multi-objective BO algorithms on a variety of sequence prediction tasks.

Dimensionality reduction is an important operation in information visualization, feature extraction, clustering, regression, and classification, especially for processing noisy high dimensional data. However, most existing approaches preserve either the global or the local structure of the data, but not both. Approaches that preserve only the global data structure, such as principal component analysis (PCA), are usually sensitive to outliers. Approaches that preserve only the local data structure, such as locality preserving projections, are usually unsupervised (and hence cannot use label information) and uses a fixed similarity graph. We propose a novel linear dimensionality reduction approach, supervised discriminative sparse PCA with adaptive neighbors (SDSPCAAN), to integrate neighborhood-free supervised discriminative sparse PCA and projected clustering with adaptive neighbors. As a result, both global and local data structures, as well as the label information, are used for better dimensionality reduction. Classification experiments on nine high-dimensional datasets validated the effectiveness and robustness of our proposed SDSPCAAN.

Time Series is a class of data science problems where the primary values of interest are a series of data points measured over a period of time. This notebook aims to provide the basic building blocks of some of the more modern algorithms / techniques (and data!) for solving these types of problems. Is ARIMA the first thing you think of when you hear about time series? It might be time to explore other ventures and methodologies. There is a lot of new innovation and modern techniques being actively developed and some of them are outperforming the traditional ARIMA models.

In data science and machine learning, mathematical skills are as important as programming skills. There are so many good packages that can be used for building predictive models. Thanks to the above-mentioned packages, pretty much everyone with some basic understanding of data science can build a machine learning model. However, it's important that before using these packages, you master the fundamentals of data science, that way you are not using these packages simply as blackbox tools. In this article, we illustrate how in-depth knowledge of the back end of a machine learning algorithm is important and crucial for building efficient and reliable models.

Today I am going to explain the concept of R-squared and adjusted R-squared from the Machine Learning perspective. I'll also show you how to find the R-squared value of your ML model. It acts as an evaluation metric for regression models. To understand it better let me introduce a regression problem. Suppose I'm building a model to predict how many articles I will write in a particular month given the amount of free time I have on that month.

Generative Adversarial Imitation Learning (GAIL) is a powerful and practical approach for learning sequential decision-making policies. Different from Reinforcement Learning (RL), GAIL takes advantage of demonstration data by experts (e.g., human), and learns both the policy and reward function of the unknown environment. Despite the significant empirical progresses, the theory behind GAIL is still largely unknown. The major difficulty comes from the underlying temporal dependency of the demonstration data and the minimax computational formulation of GAIL without convex-concave structure. To bridge such a gap between theory and practice, this paper investigates the theoretical properties of GAIL. Specifically, we show: (1) For GAIL with general reward parameterization, the generalization can be guaranteed as long as the class of the reward functions is properly controlled; (2) For GAIL, where the reward is parameterized as a reproducing kernel function, GAIL can be efficiently solved by stochastic first order optimization algorithms, which attain sublinear convergence to a stationary solution. To the best of our knowledge, these are the first results on statistical and computational guarantees of imitation learning with reward/policy function approximation. Numerical experiments are provided to support our analysis.

Semi-supervised learning is a classification method which makes use of both labeled data and unlabeled data for training. In this paper, we propose a semi-supervised learning algorithm using a Bayesian semi-supervised model. We make a general assumption that the observations will follow two multivariate normal distributions depending on their true labels after the same unknown transformation. We use B-splines to put a prior on the transformation function for each component. To use unlabeled data in a semi-supervised setting, we assume the labels are missing at random. The posterior distributions can then be described using our assumptions, which we compute by the Gibbs sampling technique. The proposed method is then compared with several other available methods through an extensive simulation study. Finally we apply the proposed method in real data contexts for diagnosing breast cancer and classify radar returns. We conclude that the proposed method has better prediction accuracy in a wide variety of cases.

We consider nonconvex-concave minimax problems of the form $\min_{\bf x}\max_{\bf y} f({\bf x},{\bf y})$, where $f$ is strongly-concave in $\bf y$ but possibly nonconvex in $\bf x$. We focus on the stochastic setting, where we can only access an unbiased stochastic gradient estimate of $f$ at each iteration. This formulation includes many machine learning applications as special cases such as adversary training and certifying robustness in deep learning. We are interested in finding an ${\mathcal O}(\varepsilon)$-stationary point of the function $\Phi(\cdot)=\max_{\bf y} f(\cdot, {\bf y})$. The most popular algorithm to solve this problem is stochastic gradient decent ascent, which requires $\mathcal O(\kappa^3\varepsilon^{-4})$ stochastic gradient evaluations, where $\kappa$ is the condition number. In this paper, we propose a novel method called Stochastic Recursive gradiEnt Descent Ascent (SREDA), which estimates gradients more efficiently using variance reduction. This method achieves the best known stochastic gradient complexity of ${\mathcal O}(\kappa^3\varepsilon^{-3})$, and its dependency on $\varepsilon$ is optimal for this problem.

In this section, we will try to implement a very simple example of logisitic regression. For this example, we will try to predict the result(pass/fail) of a student based on his 2 test scores. Notice that the predicted result is a value between 0 and 1, it can be interpreted as the probability that the result is a positive (y 1). In our example, for test scores 60 and 75, there is a 87% probability that the student passes. An additional condition like the following can be used inside the predict function for binary output.