mhjensen
286 lines
9.6 KiB
HTML
286 lines
9.6 KiB
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Logistic Regression', 2, None, '___sec0'),
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('Basics', 2, None, '___sec1'),
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('Linear classifier', 2, None, '___sec2'),
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('Some selected properties', 2, None, '___sec3'),
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('The cross-entropy as a cost function for logistic regression',
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2,
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None,
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'___sec4'),
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('Maximum likelihood', 2, None, '___sec5'),
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('Minimizing the cross entropy', 2, None, '___sec6')]}
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<!-- ------------------- main content ---------------------- -->
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<center><h1>Data Analysis and Machine Learning: Logistic Regression</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Sep 17, 2018</h4></center> <!-- date -->
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<br>
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<p>
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<!-- !split -->
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<h2 id="___sec0">Logistic Regression </h2>
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<p>
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So far we have focused on learning from datasets for which there is a
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<b>continuous</b> output. In linear regression we have been
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concerned with learning the coefficients of a polynomial to predict
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the response of a continuous variable \( y_i \) on unseen data based on
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its independent variables \( {\bf x}_i \).
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<p>
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Classification problems,
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however, are concerned with outcomes taking the form of discrete
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variables (i.e. categories). For example, we may want to detect if
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there's a cat or a dog in an image. Or given a specific system,
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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).
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(e.g. ordered/disordered).
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<p>
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<b>Logistic regression deals with binary, dichotomous outcomes (e.g. True or
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False, Success or Failure, etc.). It is worth noting that logistic
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regression is also commonly used in modern supervised Deep Learning
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models</b>, as we will see later.
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<p>
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<!-- !split -->
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<h2 id="___sec1">Basics </h2>
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<p>
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We consider the case where the dependent variables \( y_i\in\mathbb{Z} \)
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are discrete and only take values from \( m=0,\dots,M-1 \) (i.e. \( M \)
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classes).
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<p>
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The goal is to predict the
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output classes from the design matrix \( X\in\mathbb{R}^{n\times p} \)
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made of \( n \) samples, each of which bears \( p \) features. The
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primary goal is to identify the classes to which new unseen samples
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belong.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec2">Linear classifier </h2>
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<p>
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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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$$
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\begin{equation}
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s_i = \boldsymbol{x}_i^T\boldsymbol{w} + b_0 \equiv \mathbf{x}_i^T\mathbf{w},
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\label{_auto1}
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\end{equation}
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$$
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where we use the short-hand notation
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\( \mathbf{x}_i = (1,\boldsymbol{x}_i) \) and \( \mathbf{w}_i = (b_0,\boldsymbol{w}_i) \).
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec3">Some selected properties </h2>
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<p>
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This function takes values on the entire real axis. In the case of
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logistic regression, however, the labels \( y_i \) are discrete
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variables. One simple way to get a discrete output is to have sign
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functions that map the output of a linear regressor to \( \{0,1\} \),
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\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise. Indeed,
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this is commonly known as the "perceptron" in the machine learning
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literature. This model is extremely simple, and it is favorable in
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many cases (e.g. noisy data) to have a ``soft" classifier that outputs
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the probability of a given category. For example, given
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\( \mathbf{x}_i \), the classifier outputs the probability of being in
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category \( m \). One such function is the logistic (or sigmoid) function:
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$$
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\begin{equation}
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f(s) = \frac{1}{1+\mathrm e^{-s}}.
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\label{eq:log_fun}
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\end{equation}
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$$
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Note that \( 1-f(s)= f(-s) \), which will be useful shortly.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec4">The cross-entropy as a cost function for logistic regression </h2>
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<p>
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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 \( m \).
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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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$$
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\begin{eqnarray}
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P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& \frac{1}{1+\mathrm{e}^{-\mathbf{x}^T_i\mathbf{w}}},\nonumber\\
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P(y_i=0|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& 1 - P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)},
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\end{eqnarray}
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$$
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where \( \boldsymbol{\theta}=\mathbf{w} \) are the weights we wish to learn from the data.
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<p>
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Notice that in terms of the logistic function, we can write
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$$
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P(y_i=1) =f(\mathbf{x}_i^T\mathbf{w})=1-P(y_i=0).
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$$
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<p>
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<!-- !split -->
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<h2 id="___sec5">Maximum likelihood </h2>
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<p>
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We now define the cost function for logistic regression using Maximum
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Likelihood Estimation (MLE). Recall, that in MLE we choose parameters
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to maximize the probability of seeing the observed data. Consider a
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dataset \( \mathcal{D}=\{(y_i,\boldsymbol{x}_i)\} \) with binary labels
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\( y_i\in\{0,1\} \) where the data points are drawn independently. The
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likelihood of the seeing the data under our model is just:
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$$
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\begin{align}
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P(\mathcal{D}|\mathbf{w})& = \prod_{i=1}^n \left[f(\mathbf{x}_i^T\mathbf{w})\right]^{y_i}\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]^{1-y_i}\nonumber \\
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\label{_auto2}
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\end{align}
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$$
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from which we can readily compute the log-likelihood:
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$$
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\begin{equation}
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l(\mathbf{w}) = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].
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\label{_auto3}
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\end{equation}
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$$
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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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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$$
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\hat{\mathbf{w}} = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].
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$$
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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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$$
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\begin{eqnarray}
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\mathcal{C}(\mathbf{w}) &=& - l(\mathbf{w}) \\
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&=& \sum_{i=1}^n -y_i\log f(\mathbf{x}_i^T\mathbf{w}) - (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].\nonumber
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\end{eqnarray}
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$$
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This equation is known in statistics as the \emph{cross entropy}. Finally, we note that just as in linear regression,
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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.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec6">Minimizing the cross entropy </h2>
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<p>
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The cross entropy is a convex function of the weights \( \mathbf{w} \) and,
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therefore, any local minimizer is a global minimizer. Minimizing this
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cost function leads to the following equation
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$$
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\begin{equation}
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\boldsymbol{0}=\boldsymbol{\nabla} \mathcal{C}(\mathbf{w}) = \sum_{i=1}^n\left[f(\mathbf{x}_i^T\mathbf{w})-y_i\right]\mathbf{x}_i,
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\label{_auto4}
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\end{equation}
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$$
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<p>
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where we made use of the logistic function identity \( \partial_z f(z) =
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f(z)[1-f(z)] \). This equation defines a transcendental equation for
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\( \mathbf{w} \), the solution of which, unlike linear regression, cannot
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be written in a closed form.
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Here we need gradient descent methods!
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<!-- ------------------- end of main content --------------- -->
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<center style="font-size:80%">
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<!-- copyright --> © 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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