It is nice to have logistic regression on your resume, as many jobs request it, especially in some fields such as biostatistics. And if you learned the details during your college classes, good for you. However, for a beginner, this is not the first thing you should learn. In my career, being an isolated statistician (working with marketing guys, sales people, or engineers) in many of my roles, I had the flexibility to choose which tools and methodology to use. Many practitioners today are in a similar environment.

Interest in derivative-free optimization (DFO) and "evolutionary strategies" (ES) has recently surged in the Reinforcement Learning (RL) community, with growing evidence that they match state of the art methods for policy optimization tasks. However, blackbox DFO methods suffer from high sampling complexity since they require a substantial number of policy rollouts for reliable updates. They can also be very sensitive to noise in the rewards, actuators or the dynamics of the environment. In this paper we propose to replace the standard ES derivative-free paradigm for RL based on simple reward-weighted averaged random perturbations for policy updates, that has recently become a subject of voluminous research, by an algorithm where gradients of blackbox RL functions are estimated via regularized regression methods. In particular, we propose to use L1/L2 regularized regression-based gradient estimation to exploit sparsity and smoothness, as well as LP decoding techniques for handling adversarial stochastic and deterministic noise. Our methods can be naturally aligned with sliding trust region techniques for efficient samples reuse to further reduce sampling complexity. This is not the case for standard ES methods requiring independent sampling in each epoch. We show that our algorithms can be applied in locomotion tasks, where training is conducted in the presence of substantial noise, e.g. for learning in sim transferable stable walking behaviors for quadruped robots or training quadrupeds how to follow a path. We further demonstrate our methods on several $\mathrm{OpenAI}$ $\mathrm{Gym}$ $\mathrm{Mujoco}$ RL tasks. We manage to train effective policies even if up to $25\%$ of all measurements are arbitrarily corrupted, where standard ES methods produce sub-optimal policies or do not manage to learn at all. Our empirical results are backed by theoretical guarantees.

This blog is the first in a series about regression analysis in RAPIDS, an open GPU data science platform. There are many varieties of regression techniques, and we're working to include them all in RAPIDS. In this blog edition, I use Ordinary Least Squares (OLS) and Ridge regression to choose a model to predict Washington, D.C. bikeshare rentals¹. I want to take a moment to tell the origin story of regression analysis, which will explain why it has that name. I believe that of all the common machine learning techniques (K-means, kNN, PCA), "regression analysis" has the most opaque name.

In this paper, we analyze a fast parallel algorithm to efficiently select and build a set of $k$ random variables from a large set of $n$ candidate elements. This combinatorial optimization problem can be viewed in the context of feature selection for the prediction of a response variable. Using the adaptive sampling technique, which has recently been shown to exponentially speed up submodular maximization algorithms, we propose a new parallelizable algorithm that dramatically speeds up previous selection algorithms by reducing the number of rounds from $\mathcal O(k)$ to $\mathcal O(\log n)$ for objectives that do not conform to the submodularity property. We introduce a new metric to quantify the closeness of the objective function to submodularity and analyze the performance of adaptive sampling under this regime. We also conduct experiments on synthetic and real datasets and show that the empirical performance of adaptive sampling on not-submodular objectives greatly outperforms its theoretical lower bound. Additionally, the empirical running time drastically improved in all experiments without comprising the terminal value, showing the practicality of adaptive sampling.

A more formal discussion explanatory variables is of high practical importance is provided in Section 2. in many disciplines. Recent work exploits stability of regression coefficients or invariance However most state of the art methods suffer from scalability properties of models across different experimental problems since they scan all potential subsets of variables conditions for reconstructing the full causal and test whether the conditional distribution of Y given graph. These approaches generally do not scale a subset of variables is invariant across all environments well with the number of the explanatory variables (Peters et al., 2016) . This search is hence exponential in and are difficult to extend to nonlinear relationships. the number of covariates; the methods, while maintaining Contrary to existing work, we propose an appealing theoretical guarantees, are thus already computationally approach which even works for observational data hard for graphs of ten variables, and get infeasible alone, while still offering theoretical guarantees for larger graphs, unless one resorts to heuristic procedures.

This paper presents a novel data-driven approach for predicting the number of vegetation-related outages that occur in power distribution systems on a monthly basis. In order to develop an approach that is able to successfully fulfill this objective, there are two main challenges that ought to be addressed. The first challenge is to define the extent of the target area. An unsupervised machine learning approach is proposed to overcome this difficulty. The second challenge is to correctly identify the main causes of vegetation-related outages and to thoroughly investigate their nature. In this paper, these outages are categorized into two main groups: growth-related and weather-related outages, and two types of models, namely time series and non-linear machine learning regression models are proposed to conduct the prediction tasks, respectively. Moreover, various features that can explain the variability in vegetation-related outages are engineered and employed. Actual outage data, obtained from a major utility in the U.S., in addition to different types of weather and geographical data are utilized to build the proposed approach. Finally, by utilizing various time series models and machine learning methods, a comprehensive case study is carried out to demonstrate how the proposed approach can be used to successfully predict the number of vegetation-related outages and to help decision-makers to detect vulnerable zones in their systems.

While discriminative classifiers often yield strong predictive performance, missing feature values at prediction time can still be a challenge. Classifiers may not behave as expected under certain ways of substituting the missing values, since they inherently make assumptions about the data distribution they were trained on. In this paper, we propose a novel framework that classifies examples with missing features by computing the expected prediction on a given feature distribution. We then use geometric programming to learn a naive Bayes distribution that embeds a given logistic regression classifier and can efficiently take its expected predictions. Empirical evaluations show that our model achieves the same performance as the logistic regression with all features observed, and outperforms standard imputation techniques when features go missing during prediction time. Furthermore, we demonstrate that our method can be used to generate 'sufficient explanations' of logistic regression classifications, by removing features that do not affect the classification.

Regression problems that have closed-form solutions are well understood and can be easily implemented when the dataset is small enough to be all loaded into the RAM. Challenges arise when data is too big to be stored in RAM to compute the closed form solutions. Many techniques were proposed to overcome or alleviate the memory barrier problem but the solutions are often local optimal. In addition, most approaches require accessing the raw data again when updating the models. Parallel computing clusters are also expected if multiple models need to be computed simultaneously. We propose multiple learning approaches that utilize an array of sufficient statistics (SS) to address this big data challenge. This memory oblivious approach breaks the memory barrier when computing regressions with closed-form solutions, including but not limited to linear regression, weighted linear regression, linear regression with Box-Cox transformation (Box-Cox regression) and ridge regression models. The computation and update of the SS array can be handled at per row level or per mini-batch level. And updating a model is as easy as matrix addition and subtraction. Furthermore, multiple SS arrays for different models can be easily computed simultaneously to obtain multiple models at one pass through the dataset. We implemented our approaches on Spark and evaluated over the simulated datasets. Results showed our approaches can achieve closed-form solutions of multiple models at the cost of half training time of the traditional methods for a single model.

Decision trees and logistic regression are one of the most popular and well-known machine learning algorithms, frequently used to solve a variety of real-world problems. Stability of learning algorithms is a powerful tool to analyze their performance and sensitivity and subsequently allow researchers to draw reliable conclusions. The stability of these two algorithms has remained obscure. To that end, in this paper, we derive two stability notions for decision trees and logistic regression: hypothesis and pointwise hypothesis stability. Additionally, we derive these notions for L2-regularized logistic regression and confirm existing findings that it is uniformly stable. We show that the stability of decision trees depends on the number of leaves in the tree, i.e., its depth, while for logistic regression, it depends on the smallest eigenvalue of the Hessian matrix of the cross-entropy loss. We show that logistic regression is not a stable learning algorithm. We construct the upper bounds on the generalization error of all three algorithms. Moreover, we present a novel stability measuring framework that allows one to measure the aforementioned notions of stability. The measures are equivalent to estimates of expected loss differences at an input example and then leverage bootstrap sampling to yield statistically reliable estimates. Finally, we apply this framework to the three algorithms analyzed in this paper to confirm our theoretical findings and, in addition, we discuss the possibilities of developing new training techniques to optimize the stability of logistic regression, and hence decrease its generalization error.

Differential privacy is becoming a standard notion for performing privacy-preserving machine learning over sensitive data. It provides formal guarantees, in terms of the privacy budget, $\epsilon$, on how much information about individual training records is leaked by the model. While the privacy budget is directly correlated to the privacy leakage, the calibration of the privacy budget is not well understood. As a result, many existing works on privacy-preserving machine learning select large values of $\epsilon$ in order to get acceptable utility of the model, with little understanding of the concrete impact of such choices on meaningful privacy. Moreover, in scenarios where iterative learning procedures are used which require privacy guarantees for each iteration, relaxed definitions of differential privacy are often used which further tradeoff privacy for better utility. In this paper, we evaluate the impacts of these choices on privacy in experiments with logistic regression and neural network models. We quantify the privacy leakage in terms of advantage of the adversary performing inference attacks and by analyzing the number of members at risk for exposure. Our main findings are that current mechanisms for differential privacy for machine learning rarely offer acceptable utility-privacy tradeoffs: settings that provide limited accuracy loss provide little effective privacy, and settings that provide strong privacy result in useless models. Open source code is available at https://github.com/bargavj/EvaluatingDPML.