This is a part of a series of blogs where I'll be demonstrating different aspects and the theory of Machine Learning Algorithms by using math and code. This includes the usual modeling structure of the algorithm and the intuition on why and how it works, using Python code. Logistic Regression is one of the first algorithms that is introduced when someone learns about classification. You probably would have read about Regression and the continuous nature of the predictor variable. Classification is done on discrete variables, which means your predictions are finite and class-based like a Yes/No, True/False for binary outcomes.

In this article, we will build a logistic regression model for classifying whether a patient has diabetes or not. The main focus here is that we will only use python to build functions for reading the file, normalizing data, optimizing parameters, and more. So you will be getting in-depth knowledge of how everything from reading the file to make predictions works. If you are new to machine learning, or not familiar with logistic regression or gradient descent, don't worry I'll try my best to explain these in layman's terms. There are more tutorials out there that explain the same concepts.

Learn to build a Polynomial Regression model to predict the values for a non-linear dataset. In this article, we will go through the program for building a Polynomial Regression model based on the non-linear data. In the previous examples of Linear Regression, when the data is plotted on the graph, there was a linear relationship between both the dependent and independent variables. Thus, it was more suitable to build a linear model to get accurate predictions. What if the data points had the following non-linearity making the linear model giving an error in predictions due to non-linearity? In this case, we have to build a polynomial relationship which will accurately fit the data points in the given plot.

I assume you already know what regression is. "Regression is a statistical method used in finance, investing, and other disciplines that attempts to determine the strength and character of the relationship between one dependent variable (usually denoted by Y) and a series of other variables (known as independent variables)." In the most simple terms -- we want to fit a line (or hyperplane) through data points to obtain a line of best fit. The algorithm behind aims to find the line which minimizes the cost function -- typically MSE or RMSE. That's linear regression, but there are other types -- like polynomial regression.

Federated learning (FL) learns a model jointly from a set of participating devices without sharing each other's privately held data. The characteristics of non-iid data across the network, low device participation, and the mandate that data remain private bring challenges in understanding the convergence of FL algorithms, particularly in regards to how convergence scales with the number of participating devices. In this paper, we focus on Federated Averaging (FedAvg)--the most widely used and effective FL algorithm in use today--and provide a comprehensive study of its convergence rate. Although FedAvg has recently been studied by an emerging line of literature, it remains open as to how FedAvg's convergence scales with the number of participating devices in the FL setting--a crucial question whose answer would shed light on the performance of FedAvg in large FL systems. We fill this gap by establishing convergence guarantees for FedAvg under three classes of problems: strongly convex smooth, convex smooth, and overparameterized strongly convex smooth problems. We show that FedAvg enjoys linear speedup in each case, although with different convergence rates. For each class, we also characterize the corresponding convergence rates for the Nesterov accelerated FedAvg algorithm in the FL setting: to the best of our knowledge, these are the first linear speedup guarantees for FedAvg when Nesterov acceleration is used. To accelerate FedAvg, we also design a new momentum-based FL algorithm that further improves the convergence rate in overparameterized linear regression problems. Empirical studies of the algorithms in various settings have supported our theoretical results.

Multiple Linear Regression with scikit-learn In this article, we studied the most fundamental machine learning algorithms i.e. linear regression. We implemented both simple linear regression and multiple linear regression with the help of the Scikit-Learn machine learning library. I hope you guys have enjoyed the reading. Let me know your doubts/suggestions in the comment section. In this 2-hour long project-based course, you will build and evaluate multiple linear regression models using Python.

Economics and social science research often require analyzing datasets of sensitive personal information at fine granularity, with models fit to small subsets of the data. Unfortunately, such fine-grained analysis can easily reveal sensitive individual information. We study algorithms for simple linear regression that satisfy differential privacy, a constraint which guarantees that an algorithm's output reveals little about any individual input data record, even to an attacker with arbitrary side information about the dataset. We consider the design of differentially private algorithms for simple linear regression for small datasets, with tens to hundreds of datapoints, which is a particularly challenging regime for differential privacy. Focusing on a particular application to small-area analysis in economics research, we study the performance of a spectrum of algorithms we adapt to the setting. We identify key factors that affect their performance, showing through a range of experiments that algorithms based on robust estimators (in particular, the Theil-Sen estimator) perform well on the smallest datasets, but that other more standard algorithms do better as the dataset size increases.

This paper studies robust regression for data on Riemannian manifolds. Geodesic regression is the generalization of linear regression to a setting with a manifold-valued dependent variable and one or more real-valued independent variables. The existing work on geodesic regression uses the sum-of-squared errors to find the solution, but as in the classical Euclidean case, the least-squares method is highly sensitive to outliers. In this paper, we use M-type estimators, including the $L_1$, Huber and Tukey biweight estimators, to perform robust geodesic regression, and describe how to calculate the tuning parameters for the latter two. We also show that, on compact symmetric spaces, all M-type estimators are maximum likelihood estimators, and argue for the overall superiority of the $L_1$ estimator over the $L_2$ and Huber estimators on high-dimensional manifolds and over the Tukey biweight estimator on compact high-dimensional manifolds. Results from numerical examples, including analysis of real neuroimaging data, demonstrate the promising empirical properties of the proposed approach.

Researchers often lack the necessary data to credibly estimate racial discrimination in policing. In particular, police administrative records lack information on civilians police observe but do not investigate. In this article, we show that if police racially discriminate when choosing whom to investigate, analyses using administrative records to estimate racial discrimination in police behavior are statistically biased, and many quantities of interest are unidentified--even among investigated individuals--absent strong and untestable assumptions. Using principal stratification in a causal mediation framework, we derive the exact form of the statistical bias that results from traditional estimation. We develop a bias-correction procedure and nonparametric sharp bounds for race effects, replicate published findings, and show the traditional estimator can severely underestimate levels of racially biased policing or mask discrimination entirely.

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