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mhjensen
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<!-- dom:TITLE: Data Analysis and Machine Learning: Logistic Regression -->\n",
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"# Data Analysis and Machine Learning: Logistic Regression\n",
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"<!-- dom:AUTHOR: Morten Hjorth-Jensen at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University -->\n",
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"<!-- Author: --> \n",
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"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
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"\n",
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"Date: **Sep 17, 2018**\n",
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"\n",
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"Copyright 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
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"\n",
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"\n",
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"\n",
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"\n",
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"<!-- !split -->\n",
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"## Logistic Regression\n",
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"\n",
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"So far we have focused on learning from datasets for which there is a\n",
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"**continuous** output. In linear regression we have been \n",
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"concerned with learning the coefficients of a polynomial to predict\n",
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"the response of a continuous variable $y_i$ on unseen data based on\n",
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"its independent variables ${\\bf x}_i$. \n",
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"\n",
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"Classification problems,\n",
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"however, are concerned with outcomes taking the form of discrete\n",
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"variables (i.e. categories). For example, we may want to detect if\n",
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"there's a cat or a dog in an image. Or given a specific system,\n",
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"we'd like to identify its state, say whether it is an ordered or disordered system (typical situation in solid state physics).\n",
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"(e.g. ordered/disordered). \n",
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"\n",
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"**Logistic regression deals with binary, dichotomous outcomes (e.g. True or\n",
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"False, Success or Failure, etc.). It is worth noting that logistic\n",
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"regression is also commonly used in modern supervised Deep Learning\n",
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"models**, as we will see later.\n",
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"\n",
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"\n",
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"<!-- !split -->\n",
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"## Basics\n",
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"\n",
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"We consider the case where the dependent variables $y_i\\in\\mathbb{Z}$\n",
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"are discrete and only take values from $m=0,\\dots,M-1$ (i.e. $M$\n",
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"classes).\n",
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"\n",
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"The goal is to predict the\n",
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"output classes from the design matrix $X\\in\\mathbb{R}^{n\\times p}$\n",
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"made of $n$ samples, each of which bears $p$ features. The\n",
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"primary goal is to identify the classes to which new unseen samples\n",
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"belong.\n",
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"\n",
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"\n",
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"## Linear classifier\n",
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"\n",
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"Let us start by considering a slightly simpler classifier: a linear classifier that categorizes examples using a weighted linear-combination of the features and an additive offset"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<!-- Equation labels as ordinary links -->\n",
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"<div id=\"_auto1\"></div>\n",
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"\n",
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"$$\n",
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"\\begin{equation}\n",
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"s_i = \\boldsymbol{x}_i^T\\boldsymbol{w} + b_0 \\equiv \\mathbf{x}_i^T\\mathbf{w},\n",
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"\\label{_auto1} \\tag{1}\n",
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"\\end{equation}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"where we use the short-hand notation \n",
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"$\\mathbf{x}_i = (1,\\boldsymbol{x}_i)$ and $\\mathbf{w}_i = (b_0,\\boldsymbol{w}_i)$. \n",
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"\n",
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"## Some selected properties\n",
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"\n",
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"This function takes values on the entire real axis. In the case of\n",
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"logistic regression, however, the labels $y_i$ are discrete\n",
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"variables. One simple way to get a discrete output is to have sign\n",
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"functions that map the output of a linear regressor to $\\{0,1\\}$,\n",
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"$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. Indeed,\n",
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"this is commonly known as the \"perceptron\" in the machine learning\n",
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"literature. This model is extremely simple, and it is favorable in\n",
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"many cases (e.g. noisy data) to have a ``soft\" classifier that outputs\n",
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"the probability of a given category. For example, given\n",
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"$\\mathbf{x}_i$, the classifier outputs the probability of being in\n",
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"category $m$. One such function is the logistic (or sigmoid) function:"
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<!-- Equation labels as ordinary links -->\n",
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"<div id=\"eq:log_fun\"></div>\n",
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"\n",
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"$$\n",
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"\\begin{equation}\n",
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"f(s) = \\frac{1}{1+\\mathrm e^{-s}}.\n",
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"\\label{eq:log_fun} \\tag{2}\n",
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"\\end{equation}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Note that $1-f(s)= f(-s)$, which will be useful shortly. \n",
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"\n",
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"## The cross-entropy as a cost function for logistic regression\n",
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"\n",
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"The perceptron is an example of a ``hard classification\": each datapoint is deterministically assigned to a category (i.e $y_i=0$ or $y_i=1$). In many cases, it is favorable to have a \"soft\" classifier that outputs the probability of a given category rather than a single value. For example, given $\\mathbf{x}_i$, the classifier outputs the probability of being in category $m$. \n",
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"Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point $\\boldsymbol{x}_i$ belongs to a category $y_i=\\{0,1\\}$ is is given by"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$\n",
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"\\begin{eqnarray}\n",
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"P(y_i=1|\\boldsymbol{x}_i,\\boldsymbol{\\theta)} &=& \\frac{1}{1+\\mathrm{e}^{-\\mathbf{x}^T_i\\mathbf{w}}},\\nonumber\\\\\n",
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"P(y_i=0|\\boldsymbol{x}_i,\\boldsymbol{\\theta)} &=& 1 - P(y_i=1|\\boldsymbol{x}_i,\\boldsymbol{\\theta)},\n",
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"\\end{eqnarray}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"where $\\boldsymbol{\\theta}=\\mathbf{w}$ are the weights we wish to learn from the data. \n",
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"\n",
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"\n",
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"Notice that in terms of the logistic function, we can write"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$\n",
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"P(y_i=1) =f(\\mathbf{x}_i^T\\mathbf{w})=1-P(y_i=0).\n",
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"$$"
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]
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},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<!-- !split -->\n",
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"## Maximum likelihood\n",
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"\n",
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"We now define the cost function for logistic regression using Maximum\n",
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"Likelihood Estimation (MLE). Recall, that in MLE we choose parameters\n",
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"to maximize the probability of seeing the observed data. Consider a\n",
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"dataset $\\mathcal{D}=\\{(y_i,\\boldsymbol{x}_i)\\}$ with binary labels\n",
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"$y_i\\in\\{0,1\\}$ where the data points are drawn independently. The\n",
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"likelihood of the seeing the data under our model is just:"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$\n",
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"P(\\mathcal{D}|\\mathbf{w}) = \\prod_{i=1}^n \\left[f(\\mathbf{x}_i^T\\mathbf{w})\\right]^{y_i}\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right]^{1-y_i}\\nonumber\n",
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"$$"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<!-- Equation labels as ordinary links -->\n",
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"<div id=\"_auto2\"></div>\n",
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"\n",
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"$$\n",
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"\\begin{equation} \n",
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"\\label{_auto2} \\tag{3}\n",
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"\\end{equation}\n",
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"$$"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"from which we can readily compute the log-likelihood:"
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<!-- Equation labels as ordinary links -->\n",
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"<div id=\"_auto3\"></div>\n",
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"\n",
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"$$\n",
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"\\begin{equation}\n",
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"l(\\mathbf{w}) = \\sum_{i=1}^n y_i\\log f(\\mathbf{x}_i^T\\mathbf{w}) + (1-y_i)\\log\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right].\n",
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"\\label{_auto3} \\tag{4}\n",
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"\\end{equation}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\theta$"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$\n",
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"\\hat{\\mathbf{w}} = \\sum_{i=1}^n y_i\\log f(\\mathbf{x}_i^T\\mathbf{w}) + (1-y_i)\\log\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right].\n",
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"$$"
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]
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},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$\n",
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"\\begin{eqnarray}\n",
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"\\mathcal{C}(\\mathbf{w}) &=& - l(\\mathbf{w}) \\\\\n",
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"&=& \\sum_{i=1}^n -y_i\\log f(\\mathbf{x}_i^T\\mathbf{w}) - (1-y_i)\\log\\left[1-f(\\mathbf{x}_i^T\\mathbf{w})\\right].\\nonumber\n",
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"\\end{eqnarray}\n",
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"$$"
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"This equation is known in statistics as the \\emph{cross entropy}. Finally, we note that just as in linear regression, \n",
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"in practice we usually supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n",
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"\n",
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"## Minimizing the cross entropy\n",
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"\n",
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"The cross entropy is a convex function of the weights $\\mathbf{w}$ and,\n",
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"therefore, any local minimizer is a global minimizer. Minimizing this\n",
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"cost function leads to the following equation"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<!-- Equation labels as ordinary links -->\n",
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"<div id=\"_auto4\"></div>\n",
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"\n",
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"$$\n",
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"\\begin{equation}\n",
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"\\boldsymbol{0}=\\boldsymbol{\\nabla} \\mathcal{C}(\\mathbf{w}) = \\sum_{i=1}^n\\left[f(\\mathbf{x}_i^T\\mathbf{w})-y_i\\right]\\mathbf{x}_i,\n",
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"\\label{_auto4} \\tag{5}\n",
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"\\end{equation}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"where we made use of the logistic function identity $\\partial_z f(z) =\n",
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"f(z)[1-f(z)]$. This equation defines a transcendental equation for\n",
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"$\\mathbf{w}$, the solution of which, unlike linear regression, cannot\n",
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"be written in a closed form. \n",
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"Here we need gradient descent methods!"
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]
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}
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],
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"metadata": {},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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