Morten Hjorth-Jensen
1752 lines
61 KiB
Python
1752 lines
61 KiB
Python
# Logistic Regression
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[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureSeptember18.mp4?vrtx=view-as-webpage)
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## Logistic Regression
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In linear regression our main interest was centered on learning the
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coefficients of a functional fit (say a polynomial) in order to be
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able to predict the response of a continuous variable on some unseen
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data. The fit to the continuous variable $y_i$ is based on some
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independent variables $x_i$. Linear regression resulted in
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analytical expressions for standard ordinary Least Squares or Ridge
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regression (in terms of matrices to invert) for several quantities,
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ranging from the variance and thereby the confidence intervals of the
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optimal parameters $\hat{\beta}$ to the mean squared error. If we can invert
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the product of the design matrices, linear regression gives then a
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simple recipe for fitting our data.
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Classification problems, however, are concerned with outcomes taking
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the form of discrete variables (i.e. categories). We may for example,
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on the basis of DNA sequencing for a number of patients, like to find
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out which mutations are important for a certain disease; or based on
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scans of various patients' brains, figure out if there is a tumor or
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not; or given a specific physical system, we'd like to identify its
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state, say whether it is an ordered or disordered system (typical
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situation in solid state physics); or classify the status of a
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patient, whether she/he has a stroke or not and many other similar
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situations.
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The most common situation we encounter when we apply logistic
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regression is that of two possible outcomes, normally denoted as a
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binary outcome, true or false, positive or negative, success or
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failure etc.
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Logistic regression will also serve as our stepping stone towards
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neural network algorithms and supervised deep learning. For logistic
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learning, the minimization of the cost function leads to a non-linear
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equation in the parameters $\hat{\beta}$. The optimization of the
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problem calls therefore for minimization algorithms. This forms the
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bottle neck of all machine learning algorithms, namely how to find
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reliable minima of a multi-variable function. This leads us to the
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family of gradient descent methods. The latter are the working horses
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of basically all modern machine learning algorithms.
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We note also that many of the topics discussed here on logistic
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regression are also commonly used in modern supervised Deep Learning
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models, as we will see later.
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## Basics
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We consider the case where the dependent variables, also called the
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responses or the outcomes, $y_i$ are discrete and only take values
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from $k=0,\dots,K-1$ (i.e. $K$ classes).
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The goal is to predict the
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output classes from the design matrix $\boldsymbol{X}\in\mathbb{R}^{n\times p}$
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made of $n$ samples, each of which carries $p$ features or predictors. 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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Let us specialize to the case of two classes only, with outputs
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$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a
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credit card user that could default or not on her/his credit card
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debt. That is
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$$
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y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
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$$
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Before moving to the logistic model, let us try to use our linear
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regression model to classify these two outcomes. We could for example
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fit a linear model to the default case if $y_i > 0.5$ and the no
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default case $y_i \leq 0.5$.
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We would then have our
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weighted linear combination, namely
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<!-- Equation labels as ordinary links -->
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<div id="_auto1"></div>
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$$
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\begin{equation}
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\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{\beta} + \boldsymbol{\epsilon},
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\label{_auto1} \tag{1}
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\end{equation}
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$$
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where $\boldsymbol{y}$ is a vector representing the possible outcomes, $\boldsymbol{X}$ is our
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$n\times p$ design matrix and $\boldsymbol{\beta}$ represents our estimators/predictors.
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The main problem with our function is that it takes values on the
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entire real axis. In the case of logistic regression, however, the
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labels $y_i$ are discrete variables. A typical example is the credit
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card data discussed below here, where we can set the state of
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defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons
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in the data set (see the full example below).
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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 values $\{0,1\}$,
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$f(s_i)=sign(s_i)=1$ if $s_i\ge 0$ and 0 if otherwise.
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We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning
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literature. This model is extremely simple. However, in many cases it is more
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favorable to use a ``soft" classifier that outputs
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the probability of a given category. This leads us to the logistic function.
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The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.
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%matplotlib inline
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# Common imports
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import os
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import numpy as np
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import pandas as pd
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import matplotlib.pyplot as plt
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from sklearn.linear_model import LinearRegression, Ridge, Lasso
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from sklearn.model_selection import train_test_split
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from sklearn.utils import resample
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from sklearn.metrics import mean_squared_error
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from IPython.display import display
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from pylab import plt, mpl
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plt.style.use('seaborn')
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mpl.rcParams['font.family'] = 'serif'
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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DATA_ID = "DataFiles/"
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if not os.path.exists(PROJECT_ROOT_DIR):
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os.mkdir(PROJECT_ROOT_DIR)
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if not os.path.exists(FIGURE_ID):
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os.makedirs(FIGURE_ID)
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if not os.path.exists(DATA_ID):
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os.makedirs(DATA_ID)
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def image_path(fig_id):
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return os.path.join(FIGURE_ID, fig_id)
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def data_path(dat_id):
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return os.path.join(DATA_ID, dat_id)
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def save_fig(fig_id):
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plt.savefig(image_path(fig_id) + ".png", format='png')
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infile = open(data_path("chddata.csv"),'r')
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# Read the chd data as csv file and organize the data into arrays with age group, age, and chd
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chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
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chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
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output = chd['CHD']
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age = chd['Age']
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agegroup = chd['Agegroup']
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numberID = chd['ID']
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display(chd)
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plt.scatter(age, output, marker='o')
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plt.axis([18,70.0,-0.1, 1.2])
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plt.xlabel(r'Age')
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plt.ylabel(r'CHD')
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plt.title(r'Age distribution and Coronary heart disease')
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plt.show()
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What we could attempt however is to plot the mean value for each group.
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agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
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group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
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plt.plot(group, agegroupmean, "r-")
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plt.axis([0,9,0, 1.0])
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plt.xlabel(r'Age group')
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plt.ylabel(r'CHD mean values')
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plt.title(r'Mean values for each age group')
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plt.show()
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We are now trying to find a function $f(y\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.
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In standard linear regression with a linear dependence on $x$, we would write this in terms of our model
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$$
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f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
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$$
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This expression implies however that $f(y_i\vert x_i)$ could take any
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value from minus infinity to plus infinity. If we however let
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$f(y\vert y)$ be represented by the mean value, the above example
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shows us that we can constrain the function to take values between
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zero and one, that is we have $0 \le f(y_i\vert x_i) \le 1$. Looking
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at our last curve we see also that it has an S-shaped form. This leads
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us to a very popular model for the function $f$, namely the so-called
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Sigmoid function or logistic model. We will consider this function as
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representing the probability for finding a value of $y_i$ with a given
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$x_i$.
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## The logistic function
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Another widely studied model, is the so-called
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perceptron model, which is an example of a "hard classification" model. We
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will encounter this model when we discuss neural networks as
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well. Each datapoint is deterministically assigned to a category (i.e
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$y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
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classifier that outputs the probability of a given category rather
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than a single value. For example, given $x_i$, the classifier
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outputs the probability of being in a category $k$. Logistic regression
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is the most common example of a so-called soft classifier. In logistic
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regression, the probability that a data point $x_i$
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belongs to a category $y_i=\{0,1\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
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$$
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p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
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$$
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Note that $1-p(t)= p(-t)$.
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## Examples of likelihood functions used in logistic regression and nueral networks
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The following code plots the logistic function, the step function and other functions we will encounter from here and on.
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"""The sigmoid function (or the logistic curve) is a
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function that takes any real number, z, and outputs a number (0,1).
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It is useful in neural networks for assigning weights on a relative scale.
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The value z is the weighted sum of parameters involved in the learning algorithm."""
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import numpy
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import matplotlib.pyplot as plt
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import math as mt
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z = numpy.arange(-5, 5, .1)
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sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
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sigma = sigma_fn(z)
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fig = plt.figure()
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ax = fig.add_subplot(111)
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ax.plot(z, sigma)
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ax.set_ylim([-0.1, 1.1])
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ax.set_xlim([-5,5])
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ax.grid(True)
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ax.set_xlabel('z')
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ax.set_title('sigmoid function')
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plt.show()
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"""Step Function"""
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z = numpy.arange(-5, 5, .02)
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step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
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step = step_fn(z)
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fig = plt.figure()
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ax = fig.add_subplot(111)
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ax.plot(z, step)
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ax.set_ylim([-0.5, 1.5])
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ax.set_xlim([-5,5])
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ax.grid(True)
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ax.set_xlabel('z')
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ax.set_title('step function')
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plt.show()
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"""tanh Function"""
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z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
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t = numpy.tanh(z)
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fig = plt.figure()
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ax = fig.add_subplot(111)
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ax.plot(z, t)
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ax.set_ylim([-1.0, 1.0])
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ax.set_xlim([-2*mt.pi,2*mt.pi])
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ax.grid(True)
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ax.set_xlabel('z')
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ax.set_title('tanh function')
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plt.show()
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We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\beta$ in our fitting of the Sigmoid function, that is we define probabilities
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$$
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\begin{align*}
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p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
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p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
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\end{align*}
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$$
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where $\boldsymbol{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$.
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Note that we used
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$$
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p(y_i=0\vert x_i, \boldsymbol{\beta}) = 1-p(y_i=1\vert x_i, \boldsymbol{\beta}).
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$$
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In order to define the total likelihood for all possible outcomes from a
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dataset $\mathcal{D}=\{(y_i,x_i)\}$, with the binary labels
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$y_i\in\{0,1\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle.
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We aim thus at maximizing
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the probability of seeing the observed data. We can then approximate the
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likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is
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$$
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\begin{align*}
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P(\mathcal{D}|\boldsymbol{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\boldsymbol{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]^{1-y_i}\nonumber \\
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\end{align*}
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$$
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from which we obtain the log-likelihood and our **cost/loss** function
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$$
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\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\boldsymbol{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]\right).
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$$
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Reordering the logarithms, we can rewrite the **cost/loss** function as
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$$
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\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
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$$
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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 $\beta$.
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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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$$
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\mathcal{C}(\boldsymbol{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
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$$
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This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression,
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in practice we often 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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The cross entropy is a convex function of the weights $\boldsymbol{\beta}$ and,
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therefore, any local minimizer is a global minimizer.
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Minimizing this
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cost function with respect to the two parameters $\beta_0$ and $\beta_1$ we obtain
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$$
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\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
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$$
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and
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$$
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\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
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$$
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Let us now define a vector $\boldsymbol{y}$ with $n$ elements $y_i$, an
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$n\times p$ matrix $\boldsymbol{X}$ which contains the $x_i$ values and a
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vector $\boldsymbol{p}$ of fitted probabilities $p(y_i\vert x_i,\boldsymbol{\beta})$. We can rewrite in a more compact form the first
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derivative of cost function as
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$$
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\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
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$$
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If we in addition define a diagonal matrix $\boldsymbol{W}$ with elements
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$p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})$, we can obtain a compact expression of the second derivative as
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$$
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\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
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$$
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Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors
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$$
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\log{ \frac{p(\boldsymbol{\beta}\boldsymbol{x})}{1-p(\boldsymbol{\beta}\boldsymbol{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
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$$
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Here we defined $\boldsymbol{x}=[1,x_1,x_2,\dots,x_p]$ and $\boldsymbol{\beta}=[\beta_0, \beta_1, \dots, \beta_p]$ leading to
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$$
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p(\boldsymbol{\beta}\boldsymbol{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
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$$
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Till now we have mainly focused on two classes, the so-called binary
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system. Suppose we wish to extend to $K$ classes. Let us for the sake
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of simplicity assume we have only two predictors. We have then following model
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$$
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\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
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$$
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and
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$$
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\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
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$$
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and so on till the class $C=K-1$ class
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$$
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\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
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$$
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and the model is specified in term of $K-1$ so-called log-odds or
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**logit** transformations.
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In our discussion of neural networks we will encounter the above again
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in terms of a slightly modified function, the so-called **Softmax** function.
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The softmax function is used in various multiclass classification
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methods, such as multinomial logistic regression (also known as
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softmax regression), multiclass linear discriminant analysis, naive
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Bayes classifiers, and artificial neural networks. Specifically, in
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multinomial logistic regression and linear discriminant analysis, the
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input to the function is the result of $K$ distinct linear functions,
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and the predicted probability for the $k$-th class given a sample
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vector $\boldsymbol{x}$ and a weighting vector $\boldsymbol{\beta}$ is (with two
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predictors):
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$$
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p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
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$$
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It is easy to extend to more predictors. The final class is
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$$
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p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
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$$
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and they sum to one. Our earlier discussions were all specialized to
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the case with two classes only. It is easy to see from the above that
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what we derived earlier is compatible with these equations.
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To find the optimal parameters we would typically use a gradient
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descent method. Newton's method and gradient descent methods are
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discussed in the material on [optimization
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methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).
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## Wisconsin Cancer Data
|
||
|
||
We show here how we can use a simple regression case on the breast
|
||
cancer data using Logistic regression as our algorithm for
|
||
classification.
|
||
|
||
import matplotlib.pyplot as plt
|
||
import numpy as np
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.datasets import load_breast_cancer
|
||
from sklearn.linear_model import LogisticRegression
|
||
|
||
# Load the data
|
||
cancer = load_breast_cancer()
|
||
|
||
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
|
||
print(X_train.shape)
|
||
print(X_test.shape)
|
||
# Logistic Regression
|
||
logreg = LogisticRegression(solver='lbfgs')
|
||
logreg.fit(X_train, y_train)
|
||
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
|
||
#now scale the data
|
||
from sklearn.preprocessing import StandardScaler
|
||
scaler = StandardScaler()
|
||
scaler.fit(X_train)
|
||
X_train_scaled = scaler.transform(X_train)
|
||
X_test_scaled = scaler.transform(X_test)
|
||
# Logistic Regression
|
||
logreg.fit(X_train_scaled, y_train)
|
||
print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
|
||
|
||
In addition to the above scores, we could also study the covariance (and the correlation matrix).
|
||
We use **Pandas** to compute the correlation matrix.
|
||
|
||
import matplotlib.pyplot as plt
|
||
import numpy as np
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.datasets import load_breast_cancer
|
||
from sklearn.linear_model import LogisticRegression
|
||
cancer = load_breast_cancer()
|
||
import pandas as pd
|
||
# Making a data frame
|
||
cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
|
||
|
||
fig, axes = plt.subplots(15,2,figsize=(10,20))
|
||
malignant = cancer.data[cancer.target == 0]
|
||
benign = cancer.data[cancer.target == 1]
|
||
ax = axes.ravel()
|
||
|
||
for i in range(30):
|
||
_, bins = np.histogram(cancer.data[:,i], bins =50)
|
||
ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)
|
||
ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)
|
||
ax[i].set_title(cancer.feature_names[i])
|
||
ax[i].set_yticks(())
|
||
ax[0].set_xlabel("Feature magnitude")
|
||
ax[0].set_ylabel("Frequency")
|
||
ax[0].legend(["Malignant", "Benign"], loc ="best")
|
||
fig.tight_layout()
|
||
plt.show()
|
||
|
||
import seaborn as sns
|
||
correlation_matrix = cancerpd.corr().round(1)
|
||
# use the heatmap function from seaborn to plot the correlation matrix
|
||
# annot = True to print the values inside the square
|
||
plt.figure(figsize=(15,8))
|
||
sns.heatmap(data=correlation_matrix, annot=True)
|
||
plt.show()
|
||
|
||
In the above example we note two things. In the first plot we display
|
||
the overlap of benign and malignant tumors as functions of the various
|
||
features in the Wisconsing breast cancer data set. We see that for
|
||
some of the features we can distinguish clearly the benign and
|
||
malignant cases while for other features we cannot. This can point to
|
||
us which features may be of greater interest when we wish to classify
|
||
a benign or not benign tumour.
|
||
|
||
In the second figure we have computed the so-called correlation
|
||
matrix, which in our case with thirty features becomes a $30\times 30$
|
||
matrix.
|
||
|
||
We constructed this matrix using **pandas** via the statements
|
||
|
||
cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
|
||
|
||
and then
|
||
|
||
correlation_matrix = cancerpd.corr().round(1)
|
||
|
||
Diagonalizing this matrix we can in turn say something about which
|
||
features are of relevance and which are not. This leads us to
|
||
the classical Principal Component Analysis (PCA) theorem with
|
||
applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)).
|
||
|
||
import matplotlib.pyplot as plt
|
||
import numpy as np
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.datasets import load_breast_cancer
|
||
from sklearn.linear_model import LogisticRegression
|
||
|
||
# Load the data
|
||
cancer = load_breast_cancer()
|
||
|
||
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
|
||
print(X_train.shape)
|
||
print(X_test.shape)
|
||
# Logistic Regression
|
||
logreg = LogisticRegression(solver='lbfgs')
|
||
logreg.fit(X_train, y_train)
|
||
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
|
||
#now scale the data
|
||
from sklearn.preprocessing import StandardScaler
|
||
scaler = StandardScaler()
|
||
scaler.fit(X_train)
|
||
X_train_scaled = scaler.transform(X_train)
|
||
X_test_scaled = scaler.transform(X_test)
|
||
# Logistic Regression
|
||
logreg.fit(X_train_scaled, y_train)
|
||
print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
|
||
|
||
|
||
from sklearn.preprocessing import LabelEncoder
|
||
from sklearn.model_selection import cross_validate
|
||
#Cross validation
|
||
accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score']
|
||
print(accuracy)
|
||
print("Test set accuracy with Logistic Regression and scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
|
||
|
||
|
||
import scikitplot as skplt
|
||
y_pred = logreg.predict(X_test_scaled)
|
||
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
|
||
plt.show()
|
||
y_probas = logreg.predict_proba(X_test_scaled)
|
||
skplt.metrics.plot_roc(y_test, y_probas)
|
||
plt.show()
|
||
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
|
||
plt.show()
|
||
|
||
## Optimization, the central part of any Machine Learning algortithm
|
||
|
||
Almost every problem in machine learning and data science starts with
|
||
a dataset $X$, a model $g(\beta)$, which is a function of the
|
||
parameters $\beta$ and a cost function $C(X, g(\beta))$ that allows
|
||
us to judge how well the model $g(\beta)$ explains the observations
|
||
$X$. The model is fit by finding the values of $\beta$ that minimize
|
||
the cost function. Ideally we would be able to solve for $\beta$
|
||
analytically, however this is not possible in general and we must use
|
||
some approximative/numerical method to compute the minimum.
|
||
|
||
|
||
|
||
## Revisiting our Logistic Regression case
|
||
|
||
In our discussion on Logistic Regression we studied the
|
||
case of
|
||
two classes, with $y_i$ either
|
||
$0$ or $1$. Furthermore we assumed also that we have only two
|
||
parameters $\beta$ in our fitting, that is we
|
||
defined probabilities
|
||
|
||
$$
|
||
\begin{align*}
|
||
p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
|
||
p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
|
||
\end{align*}
|
||
$$
|
||
|
||
where $\boldsymbol{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$.
|
||
|
||
|
||
## The equations to solve
|
||
|
||
Our compact equations used a definition of a vector $\boldsymbol{y}$ with $n$
|
||
elements $y_i$, an $n\times p$ matrix $\boldsymbol{X}$ which contains the
|
||
$x_i$ values and a vector $\boldsymbol{p}$ of fitted probabilities
|
||
$p(y_i\vert x_i,\boldsymbol{\beta})$. We rewrote in a more compact form
|
||
the first derivative of the cost function as
|
||
|
||
$$
|
||
\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
|
||
$$
|
||
|
||
If we in addition define a diagonal matrix $\boldsymbol{W}$ with elements
|
||
$p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})$, we can obtain a compact expression of the second derivative as
|
||
|
||
$$
|
||
\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
|
||
$$
|
||
|
||
This defines what is called the Hessian matrix.
|
||
|
||
|
||
## Solving using Newton-Raphson's method
|
||
|
||
If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
|
||
|
||
Our iterative scheme is then given by
|
||
|
||
$$
|
||
\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T}\right)^{-1}_{\boldsymbol{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\right)_{\boldsymbol{\beta}^{\mathrm{old}}},
|
||
$$
|
||
|
||
or in matrix form as
|
||
|
||
$$
|
||
\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X} \right)^{-1}\times \left(-\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{p}) \right)_{\boldsymbol{\beta}^{\mathrm{old}}}.
|
||
$$
|
||
|
||
The right-hand side is computed with the old values of $\beta$.
|
||
|
||
If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.
|
||
|
||
|
||
|
||
## Brief reminder on Newton-Raphson's method
|
||
|
||
Let us quickly remind ourselves how we derive the above method.
|
||
|
||
Perhaps the most celebrated of all one-dimensional root-finding
|
||
routines is Newton's method, also called the Newton-Raphson
|
||
method. This method requires the evaluation of both the
|
||
function $f$ and its derivative $f'$ at arbitrary points.
|
||
If you can only calculate the derivative
|
||
numerically and/or your function is not of the smooth type, we
|
||
normally discourage the use of this method.
|
||
|
||
|
||
## The equations
|
||
|
||
The Newton-Raphson formula consists geometrically of extending the
|
||
tangent line at a current point until it crosses zero, then setting
|
||
the next guess to the abscissa of that zero-crossing. The mathematics
|
||
behind this method is rather simple. Employing a Taylor expansion for
|
||
$x$ sufficiently close to the solution $s$, we have
|
||
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="eq:taylornr"></div>
|
||
|
||
$$
|
||
f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
|
||
\label{eq:taylornr} \tag{2}
|
||
$$
|
||
|
||
For small enough values of the function and for well-behaved
|
||
functions, the terms beyond linear are unimportant, hence we obtain
|
||
|
||
$$
|
||
f(x)+(s-x)f'(x)\approx 0,
|
||
$$
|
||
|
||
yielding
|
||
|
||
$$
|
||
s\approx x-\frac{f(x)}{f'(x)}.
|
||
$$
|
||
|
||
Having in mind an iterative procedure, it is natural to start iterating with
|
||
|
||
$$
|
||
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
|
||
$$
|
||
|
||
## Simple geometric interpretation
|
||
|
||
The above is Newton-Raphson's method. It has a simple geometric
|
||
interpretation, namely $x_{n+1}$ is the point where the tangent from
|
||
$(x_n,f(x_n))$ crosses the $x$-axis. Close to the solution,
|
||
Newton-Raphson converges fast to the desired result. However, if we
|
||
are far from a root, where the higher-order terms in the series are
|
||
important, the Newton-Raphson formula can give grossly inaccurate
|
||
results. For instance, the initial guess for the root might be so far
|
||
from the true root as to let the search interval include a local
|
||
maximum or minimum of the function. If an iteration places a trial
|
||
guess near such a local extremum, so that the first derivative nearly
|
||
vanishes, then Newton-Raphson may fail totally
|
||
|
||
|
||
|
||
## Extending to more than one variable
|
||
|
||
Newton's method can be generalized to systems of several non-linear equations
|
||
and variables. Consider the case with two equations
|
||
|
||
$$
|
||
\begin{array}{cc} f_1(x_1,x_2) &=0\\
|
||
f_2(x_1,x_2) &=0,\end{array}
|
||
$$
|
||
|
||
which we Taylor expand to obtain
|
||
|
||
$$
|
||
\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1
|
||
\partial f_1/\partial x_1+h_2
|
||
\partial f_1/\partial x_2+\dots\\
|
||
0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1
|
||
\partial f_2/\partial x_1+h_2
|
||
\partial f_2/\partial x_2+\dots
|
||
\end{array}.
|
||
$$
|
||
|
||
Defining the Jacobian matrix $\boldsymbol{J}$ we have
|
||
|
||
$$
|
||
\boldsymbol{J}=\left( \begin{array}{cc}
|
||
\partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\
|
||
\partial f_2/\partial x_1 &\partial f_2/\partial x_2
|
||
\end{array} \right),
|
||
$$
|
||
|
||
we can rephrase Newton's method as
|
||
|
||
$$
|
||
\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)=
|
||
\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+
|
||
\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right),
|
||
$$
|
||
|
||
where we have defined
|
||
|
||
$$
|
||
\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
|
||
-\boldsymbol{J}^{-1}
|
||
\left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
|
||
$$
|
||
|
||
We need thus to compute the inverse of the Jacobian matrix and it
|
||
is to understand that difficulties may
|
||
arise in case $\boldsymbol{J}$ is nearly singular.
|
||
|
||
It is rather straightforward to extend the above scheme to systems of
|
||
more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.
|
||
|
||
|
||
|
||
|
||
## Steepest descent
|
||
|
||
The basic idea of gradient descent is
|
||
that a function $F(\mathbf{x})$,
|
||
$\mathbf{x} \equiv (x_1,\cdots,x_n)$, decreases fastest if one goes from $\bf {x}$ in the
|
||
direction of the negative gradient $-\nabla F(\mathbf{x})$.
|
||
|
||
It can be shown that if
|
||
|
||
$$
|
||
\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k),
|
||
$$
|
||
|
||
with $\gamma_k > 0$.
|
||
|
||
For $\gamma_k$ small enough, then $F(\mathbf{x}_{k+1}) \leq
|
||
F(\mathbf{x}_k)$. This means that for a sufficiently small $\gamma_k$
|
||
we are always moving towards smaller function values, i.e a minimum.
|
||
|
||
|
||
## More on Steepest descent
|
||
|
||
The previous observation is the basis of the method of steepest
|
||
descent, which is also referred to as just gradient descent (GD). One
|
||
starts with an initial guess $\mathbf{x}_0$ for a minimum of $F$ and
|
||
computes new approximations according to
|
||
|
||
$$
|
||
\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0.
|
||
$$
|
||
|
||
The parameter $\gamma_k$ is often referred to as the step length or
|
||
the learning rate within the context of Machine Learning.
|
||
|
||
|
||
## The ideal
|
||
|
||
Ideally the sequence $\{\mathbf{x}_k \}_{k=0}$ converges to a global
|
||
minimum of the function $F$. In general we do not know if we are in a
|
||
global or local minimum. In the special case when $F$ is a convex
|
||
function, all local minima are also global minima, so in this case
|
||
gradient descent can converge to the global solution. The advantage of
|
||
this scheme is that it is conceptually simple and straightforward to
|
||
implement. However the method in this form has some severe
|
||
limitations:
|
||
|
||
In machine learing we are often faced with non-convex high dimensional
|
||
cost functions with many local minima. Since GD is deterministic we
|
||
will get stuck in a local minimum, if the method converges, unless we
|
||
have a very good intial guess. This also implies that the scheme is
|
||
sensitive to the chosen initial condition.
|
||
|
||
Note that the gradient is a function of $\mathbf{x} =
|
||
(x_1,\cdots,x_n)$ which makes it expensive to compute numerically.
|
||
|
||
|
||
|
||
## The sensitiveness of the gradient descent
|
||
|
||
The gradient descent method
|
||
is sensitive to the choice of learning rate $\gamma_k$. This is due
|
||
to the fact that we are only guaranteed that $F(\mathbf{x}_{k+1}) \leq
|
||
F(\mathbf{x}_k)$ for sufficiently small $\gamma_k$. The problem is to
|
||
determine an optimal learning rate. If the learning rate is chosen too
|
||
small the method will take a long time to converge and if it is too
|
||
large we can experience erratic behavior.
|
||
|
||
Many of these shortcomings can be alleviated by introducing
|
||
randomness. One such method is that of Stochastic Gradient Descent
|
||
(SGD), see below.
|
||
|
||
|
||
|
||
## Convex functions
|
||
|
||
Ideally we want our cost/loss function to be convex(concave).
|
||
|
||
First we give the definition of a convex set: A set $C$ in
|
||
$\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and
|
||
all $t \in (0,1)$ , the point $(1 − t)x + ty$ also belongs to
|
||
C. Geometrically this means that every point on the line segment
|
||
connecting $x$ and $y$ is in $C$ as discussed below.
|
||
|
||
The convex subsets of $\mathbb{R}$ are the intervals of
|
||
$\mathbb{R}$. Examples of convex sets of $\mathbb{R}^2$ are the
|
||
regular polygons (triangles, rectangles, pentagons, etc...).
|
||
|
||
|
||
## Convex function
|
||
|
||
**Convex function**: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the function $f: X \rightarrow \mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \in X$ and for all $t \in [0,1]$. If $\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.
|
||
|
||
|
||
## Conditions on convex functions
|
||
|
||
In the following we state first and second-order conditions which
|
||
ensures convexity of a function $f$. We write $D_f$ to denote the
|
||
domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more
|
||
details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004).
|
||
|
||
**First order condition.**
|
||
|
||
Suppose $f$ is differentiable (i.e $\nabla f(x)$ is well defined for
|
||
all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$
|
||
is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
|
||
for all $x,y \in D_f$. This condition means that for a convex function
|
||
the first order Taylor expansion (right hand side above) at any point
|
||
a global under estimator of the function. To convince yourself you can
|
||
make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and
|
||
note that it is always below the graph.
|
||
|
||
|
||
|
||
**Second order condition.**
|
||
|
||
Assume that $f$ is twice
|
||
differentiable, i.e the Hessian matrix exists at each point in
|
||
$D_f$. Then $f$ is convex if and only if $D_f$ is a convex set and its
|
||
Hessian is positive semi-definite for all $x\in D_f$. For a
|
||
single-variable function this reduces to $f''(x) \geq 0$. Geometrically this means that $f$ has nonnegative curvature
|
||
everywhere.
|
||
|
||
|
||
|
||
This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
|
||
|
||
|
||
## More on convex functions
|
||
|
||
The next result is of great importance to us and the reason why we are
|
||
going on about convex functions. In machine learning we frequently
|
||
have to minimize a loss/cost function in order to find the best
|
||
parameters for the model we are considering.
|
||
|
||
Ideally we want the
|
||
global minimum (for high-dimensional models it is hard to know
|
||
if we have local or global minimum). However, if the cost/loss function
|
||
is convex the following result provides invaluable information:
|
||
|
||
**Any minimum is global for convex functions.**
|
||
|
||
Consider the problem of finding $x \in \mathbb{R}^n$ such that $f(x)$
|
||
is minimal, where $f$ is convex and differentiable. Then, any point
|
||
$x^*$ that satisfies $\nabla f(x^*) = 0$ is a global minimum.
|
||
|
||
|
||
|
||
This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.
|
||
|
||
|
||
## Some simple problems
|
||
|
||
1. Show that $f(x)=x^2$ is convex for $x \in \mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \in D_f$ and any $\lambda \in [0,1]$ $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
|
||
|
||
2. Using the second order condition show that the following functions are convex on the specified domain.
|
||
|
||
* $f(x) = e^x$ is convex for $x \in \mathbb{R}$.
|
||
|
||
* $g(x) = -\ln(x)$ is convex for $x \in (0,\infty)$.
|
||
|
||
|
||
3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \in \mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.
|
||
|
||
4. A norm is any function that satisfy the following properties
|
||
|
||
* $f(\alpha x) = |\alpha| f(x)$ for all $\alpha \in \mathbb{R}$.
|
||
|
||
* $f(x+y) \leq f(x) + f(y)$
|
||
|
||
* $f(x) \leq 0$ for all $x \in \mathbb{R}^n$ with equality if and only if $x = 0$
|
||
|
||
|
||
Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
|
||
|
||
|
||
|
||
## Friday September 25
|
||
|
||
[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember25.mp4?vrtx=view-as-webpage) and [link to handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember25.pdf).
|
||
|
||
|
||
|
||
## Standard steepest descent
|
||
|
||
|
||
Before we proceed, we would like to discuss the approach called the
|
||
**standard Steepest descent** (different from the above steepest descent discussion), which again leads to us having to be able
|
||
to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
|
||
|
||
[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)
|
||
for finding solutions of non-linear problems is based on the theory
|
||
of conjugate gradients for linear systems of equations. It belongs to
|
||
the class of iterative methods for solving problems from linear
|
||
algebra of the type
|
||
|
||
$$
|
||
\boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}.
|
||
$$
|
||
|
||
In the iterative process we end up with a problem like
|
||
|
||
$$
|
||
\boldsymbol{r}= \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x},
|
||
$$
|
||
|
||
where $\boldsymbol{r}$ is the so-called residual or error in the iterative process.
|
||
|
||
When we have found the exact solution, $\boldsymbol{r}=0$.
|
||
|
||
|
||
## Gradient method
|
||
|
||
The residual is zero when we reach the minimum of the quadratic equation
|
||
|
||
$$
|
||
P(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T\boldsymbol{b},
|
||
$$
|
||
|
||
with the constraint that the matrix $\boldsymbol{A}$ is positive definite and
|
||
symmetric. This defines also the Hessian and we want it to be positive definite.
|
||
|
||
|
||
|
||
## Steepest descent method
|
||
|
||
We denote the initial guess for $\boldsymbol{x}$ as $\boldsymbol{x}_0$.
|
||
We can assume without loss of generality that
|
||
|
||
$$
|
||
\boldsymbol{x}_0=0,
|
||
$$
|
||
|
||
or consider the system
|
||
|
||
$$
|
||
\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
|
||
$$
|
||
|
||
instead.
|
||
|
||
|
||
|
||
## Steepest descent method
|
||
One can show that the solution $\boldsymbol{x}$ is also the unique minimizer of the quadratic form
|
||
|
||
$$
|
||
f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
|
||
$$
|
||
|
||
This suggests taking the first basis vector $\boldsymbol{r}_1$ (see below for definition)
|
||
to be the gradient of $f$ at $\boldsymbol{x}=\boldsymbol{x}_0$,
|
||
which equals
|
||
|
||
$$
|
||
\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
|
||
$$
|
||
|
||
and
|
||
$\boldsymbol{x}_0=0$ it is equal $-\boldsymbol{b}$.
|
||
|
||
|
||
|
||
|
||
## Final expressions
|
||
We can compute the residual iteratively as
|
||
|
||
$$
|
||
\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1},
|
||
$$
|
||
|
||
which equals
|
||
|
||
$$
|
||
\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{r}_k),
|
||
$$
|
||
|
||
or
|
||
|
||
$$
|
||
(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{r}_k,
|
||
$$
|
||
|
||
which gives
|
||
|
||
$$
|
||
\alpha_k = \frac{\boldsymbol{r}_k^T\boldsymbol{r}_k}{\boldsymbol{r}_k^T\boldsymbol{A}\boldsymbol{r}_k}
|
||
$$
|
||
|
||
leading to the iterative scheme
|
||
|
||
$$
|
||
\boldsymbol{x}_{k+1}=\boldsymbol{x}_k-\alpha_k\boldsymbol{r}_{k},
|
||
$$
|
||
|
||
## Steepest descent example
|
||
|
||
import numpy as np
|
||
import numpy.linalg as la
|
||
|
||
import scipy.optimize as sopt
|
||
|
||
import matplotlib.pyplot as pt
|
||
from mpl_toolkits.mplot3d import axes3d
|
||
|
||
def f(x):
|
||
return 0.5*x[0]**2 + 2.5*x[1]**2
|
||
|
||
def df(x):
|
||
return np.array([x[0], 5*x[1]])
|
||
|
||
fig = pt.figure()
|
||
ax = fig.gca(projection="3d")
|
||
|
||
xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
|
||
fmesh = f(np.array([xmesh, ymesh]))
|
||
ax.plot_surface(xmesh, ymesh, fmesh)
|
||
|
||
And then as countor plot
|
||
|
||
pt.axis("equal")
|
||
pt.contour(xmesh, ymesh, fmesh)
|
||
guesses = [np.array([2, 2./5])]
|
||
|
||
Find guesses
|
||
|
||
x = guesses[-1]
|
||
s = -df(x)
|
||
|
||
Run it!
|
||
|
||
def f1d(alpha):
|
||
return f(x + alpha*s)
|
||
|
||
alpha_opt = sopt.golden(f1d)
|
||
next_guess = x + alpha_opt * s
|
||
guesses.append(next_guess)
|
||
print(next_guess)
|
||
|
||
What happened?
|
||
|
||
pt.axis("equal")
|
||
pt.contour(xmesh, ymesh, fmesh, 50)
|
||
it_array = np.array(guesses)
|
||
pt.plot(it_array.T[0], it_array.T[1], "x-")
|
||
|
||
## Conjugate gradient method
|
||
In the CG method we define so-called conjugate directions and two vectors
|
||
$\boldsymbol{s}$ and $\boldsymbol{t}$
|
||
are said to be
|
||
conjugate if
|
||
|
||
$$
|
||
\boldsymbol{s}^T\boldsymbol{A}\boldsymbol{t}= 0.
|
||
$$
|
||
|
||
The philosophy of the CG method is to perform searches in various conjugate directions
|
||
of our vectors $\boldsymbol{x}_i$ obeying the above criterion, namely
|
||
|
||
$$
|
||
\boldsymbol{x}_i^T\boldsymbol{A}\boldsymbol{x}_j= 0.
|
||
$$
|
||
|
||
Two vectors are conjugate if they are orthogonal with respect to
|
||
this inner product. Being conjugate is a symmetric relation: if $\boldsymbol{s}$ is conjugate to $\boldsymbol{t}$, then $\boldsymbol{t}$ is conjugate to $\boldsymbol{s}$.
|
||
|
||
|
||
|
||
|
||
## Conjugate gradient method
|
||
An example is given by the eigenvectors of the matrix
|
||
|
||
$$
|
||
\boldsymbol{v}_i^T\boldsymbol{A}\boldsymbol{v}_j= \lambda\boldsymbol{v}_i^T\boldsymbol{v}_j,
|
||
$$
|
||
|
||
which is zero unless $i=j$.
|
||
|
||
|
||
|
||
|
||
|
||
## Conjugate gradient method
|
||
Assume now that we have a symmetric positive-definite matrix $\boldsymbol{A}$ of size
|
||
$n\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector
|
||
|
||
$$
|
||
\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+\alpha_i\boldsymbol{p}_{i}.
|
||
$$
|
||
|
||
We assume that $\boldsymbol{p}_{i}$ is a sequence of $n$ mutually conjugate directions.
|
||
Then the $\boldsymbol{p}_{i}$ form a basis of $R^n$ and we can expand the solution
|
||
$ \boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}$ in this basis, namely
|
||
|
||
$$
|
||
\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i \boldsymbol{p}_i.
|
||
$$
|
||
|
||
## Conjugate gradient method
|
||
The coefficients are given by
|
||
|
||
$$
|
||
\mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}.
|
||
$$
|
||
|
||
Multiplying with $\boldsymbol{p}_k^T$ from the left gives
|
||
|
||
$$
|
||
\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{p}_i= \boldsymbol{p}_k^T \boldsymbol{b},
|
||
$$
|
||
|
||
and we can define the coefficients $\alpha_k$ as
|
||
|
||
$$
|
||
\alpha_k = \frac{\boldsymbol{p}_k^T \boldsymbol{b}}{\boldsymbol{p}_k^T \boldsymbol{A} \boldsymbol{p}_k}
|
||
$$
|
||
|
||
## Conjugate gradient method and iterations
|
||
|
||
If we choose the conjugate vectors $\boldsymbol{p}_k$ carefully,
|
||
then we may not need all of them to obtain a good approximation to the solution
|
||
$\boldsymbol{x}$.
|
||
We want to regard the conjugate gradient method as an iterative method.
|
||
This will us to solve systems where $n$ is so large that the direct
|
||
method would take too much time.
|
||
|
||
We denote the initial guess for $\boldsymbol{x}$ as $\boldsymbol{x}_0$.
|
||
We can assume without loss of generality that
|
||
|
||
$$
|
||
\boldsymbol{x}_0=0,
|
||
$$
|
||
|
||
or consider the system
|
||
|
||
$$
|
||
\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
|
||
$$
|
||
|
||
instead.
|
||
|
||
|
||
|
||
|
||
|
||
## Conjugate gradient method
|
||
One can show that the solution $\boldsymbol{x}$ is also the unique minimizer of the quadratic form
|
||
|
||
$$
|
||
f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
|
||
$$
|
||
|
||
This suggests taking the first basis vector $\boldsymbol{p}_1$
|
||
to be the gradient of $f$ at $\boldsymbol{x}=\boldsymbol{x}_0$,
|
||
which equals
|
||
|
||
$$
|
||
\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
|
||
$$
|
||
|
||
and
|
||
$\boldsymbol{x}_0=0$ it is equal $-\boldsymbol{b}$.
|
||
The other vectors in the basis will be conjugate to the gradient,
|
||
hence the name conjugate gradient method.
|
||
|
||
|
||
|
||
|
||
|
||
## Conjugate gradient method
|
||
Let $\boldsymbol{r}_k$ be the residual at the $k$-th step:
|
||
|
||
$$
|
||
\boldsymbol{r}_k=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k.
|
||
$$
|
||
|
||
Note that $\boldsymbol{r}_k$ is the negative gradient of $f$ at
|
||
$\boldsymbol{x}=\boldsymbol{x}_k$,
|
||
so the gradient descent method would be to move in the direction $\boldsymbol{r}_k$.
|
||
Here, we insist that the directions $\boldsymbol{p}_k$ are conjugate to each other,
|
||
so we take the direction closest to the gradient $\boldsymbol{r}_k$
|
||
under the conjugacy constraint.
|
||
This gives the following expression
|
||
|
||
$$
|
||
\boldsymbol{p}_{k+1}=\boldsymbol{r}_k-\frac{\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{r}_k}{\boldsymbol{p}_k^T\boldsymbol{A}\boldsymbol{p}_k} \boldsymbol{p}_k.
|
||
$$
|
||
|
||
## Conjugate gradient method
|
||
We can also compute the residual iteratively as
|
||
|
||
$$
|
||
\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1},
|
||
$$
|
||
|
||
which equals
|
||
|
||
$$
|
||
\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{p}_k),
|
||
$$
|
||
|
||
or
|
||
|
||
$$
|
||
(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{p}_k,
|
||
$$
|
||
|
||
which gives
|
||
|
||
$$
|
||
\boldsymbol{r}_{k+1}=\boldsymbol{r}_k-\boldsymbol{A}\boldsymbol{p}_{k},
|
||
$$
|
||
|
||
## Revisiting our first homework
|
||
|
||
We will use linear regression as a case study for the gradient descent
|
||
methods. Linear regression is a great test case for the gradient
|
||
descent methods discussed in the lectures since it has several
|
||
desirable properties such as:
|
||
|
||
1. An analytical solution (recall homework set 1).
|
||
|
||
2. The gradient can be computed analytically.
|
||
|
||
3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates
|
||
|
||
We revisit an example similar to what we had in the first homework set. We had a function of the type
|
||
|
||
x = 2*np.random.rand(m,1)
|
||
y = 4+3*x+np.random.randn(m,1)
|
||
|
||
with $x_i \in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\cal {N}(0,1)$.
|
||
The linear regression model is given by
|
||
|
||
$$
|
||
h_\beta(x) = \boldsymbol{y} = \beta_0 + \beta_1 x,
|
||
$$
|
||
|
||
such that
|
||
|
||
$$
|
||
\boldsymbol{y}_i = \beta_0 + \beta_1 x_i.
|
||
$$
|
||
|
||
## Gradient descent example
|
||
|
||
Let $\mathbf{y} = (y_1,\cdots,y_n)^T$, $\mathbf{\boldsymbol{y}} = (\boldsymbol{y}_1,\cdots,\boldsymbol{y}_n)^T$ and $\beta = (\beta_0, \beta_1)^T$
|
||
|
||
It is convenient to write $\mathbf{\boldsymbol{y}} = X\beta$ where $X \in \mathbb{R}^{100 \times 2} $ is the design matrix given by (we keep the intercept here)
|
||
|
||
$$
|
||
X \equiv \begin{bmatrix}
|
||
1 & x_1 \\
|
||
\vdots & \vdots \\
|
||
1 & x_{100} & \\
|
||
\end{bmatrix}.
|
||
$$
|
||
|
||
The cost/loss/risk function is given by (
|
||
|
||
$$
|
||
C(\beta) = \frac{1}{n}||X\beta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2\right]
|
||
$$
|
||
|
||
and we want to find $\beta$ such that $C(\beta)$ is minimized.
|
||
|
||
|
||
## The derivative of the cost/loss function
|
||
|
||
Computing $\partial C(\beta) / \partial \beta_0$ and $\partial C(\beta) / \partial \beta_1$ we can show that the gradient can be written as
|
||
|
||
$$
|
||
\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
|
||
\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
|
||
\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}),
|
||
$$
|
||
|
||
where $X$ is the design matrix defined above.
|
||
|
||
|
||
## The Hessian matrix
|
||
The Hessian matrix of $C(\beta)$ is given by
|
||
|
||
$$
|
||
\boldsymbol{H} \equiv \begin{bmatrix}
|
||
\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
|
||
\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
|
||
\end{bmatrix} = \frac{2}{n}X^T X.
|
||
$$
|
||
|
||
This result implies that $C(\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.
|
||
|
||
|
||
|
||
|
||
|
||
## Simple program
|
||
|
||
We can now write a program that minimizes $C(\beta)$ using the gradient descent method with a constant learning rate $\gamma$ according to
|
||
|
||
$$
|
||
\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots
|
||
$$
|
||
|
||
We can use the expression we computed for the gradient and let use a
|
||
$\beta_0$ be chosen randomly and let $\gamma = 0.001$. Stop iterating
|
||
when $||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8}$. **Note that the code below does not include the latter stop criterion**.
|
||
|
||
And finally we can compare our solution for $\beta$ with the analytic result given by
|
||
$\beta= (X^TX)^{-1} X^T \mathbf{y}$.
|
||
|
||
|
||
## Gradient Descent Example
|
||
|
||
Here our simple example
|
||
|
||
|
||
# Importing various packages
|
||
from random import random, seed
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from mpl_toolkits.mplot3d import Axes3D
|
||
from matplotlib import cm
|
||
from matplotlib.ticker import LinearLocator, FormatStrFormatter
|
||
import sys
|
||
|
||
# the number of datapoints
|
||
n = 100
|
||
x = 2*np.random.rand(n,1)
|
||
y = 4+3*x+np.random.randn(n,1)
|
||
|
||
X = np.c_[np.ones((n,1)), x]
|
||
# Hessian matrix
|
||
H = (2.0/n)* X.T @ X
|
||
# Get the eigenvalues
|
||
EigValues, EigVectors = np.linalg.eig(H)
|
||
print(EigValues)
|
||
|
||
beta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y
|
||
print(beta_linreg)
|
||
beta = np.random.randn(2,1)
|
||
|
||
eta = 1.0/np.max(EigValues)
|
||
Niterations = 1000
|
||
|
||
for iter in range(Niterations):
|
||
gradient = (2.0/n)*X.T @ (X @ beta-y)
|
||
beta -= eta*gradient
|
||
|
||
print(beta)
|
||
xnew = np.array([[0],[2]])
|
||
xbnew = np.c_[np.ones((2,1)), xnew]
|
||
ypredict = xbnew.dot(beta)
|
||
ypredict2 = xbnew.dot(beta_linreg)
|
||
plt.plot(xnew, ypredict, "r-")
|
||
plt.plot(xnew, ypredict2, "b-")
|
||
plt.plot(x, y ,'ro')
|
||
plt.axis([0,2.0,0, 15.0])
|
||
plt.xlabel(r'$x$')
|
||
plt.ylabel(r'$y$')
|
||
plt.title(r'Gradient descent example')
|
||
plt.show()
|
||
|
||
## And a corresponding example using **scikit-learn**
|
||
|
||
# Importing various packages
|
||
from random import random, seed
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.linear_model import SGDRegressor
|
||
|
||
n = 100
|
||
x = 2*np.random.rand(n,1)
|
||
y = 4+3*x+np.random.randn(n,1)
|
||
|
||
X = np.c_[np.ones((n,1)), x]
|
||
beta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
|
||
print(beta_linreg)
|
||
sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
|
||
sgdreg.fit(x,y.ravel())
|
||
print(sgdreg.intercept_, sgdreg.coef_)
|
||
|
||
## Gradient descent and Ridge
|
||
|
||
We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\beta$,
|
||
|
||
$$
|
||
C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0.
|
||
$$
|
||
|
||
In order to minimize $C_{\text{ridge}}(\beta)$ using GD we only have adjust the gradient as follows
|
||
|
||
$$
|
||
\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
|
||
\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
|
||
\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta).
|
||
$$
|
||
|
||
We can easily extend our program to minimize $C_{\text{ridge}}(\beta)$ using gradient descent and compare with the analytical solution given by
|
||
|
||
$$
|
||
\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}.
|
||
$$
|
||
|
||
## Program example for gradient descent with Ridge Regression
|
||
|
||
from random import random, seed
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from mpl_toolkits.mplot3d import Axes3D
|
||
from matplotlib import cm
|
||
from matplotlib.ticker import LinearLocator, FormatStrFormatter
|
||
import sys
|
||
|
||
# the number of datapoints
|
||
n = 100
|
||
x = 2*np.random.rand(n,1)
|
||
y = 4+3*x+np.random.randn(n,1)
|
||
|
||
X = np.c_[np.ones((n,1)), x]
|
||
XT_X = X.T @ X
|
||
|
||
#Ridge parameter lambda
|
||
lmbda = 0.001
|
||
Id = lmbda* np.eye(XT_X.shape[0])
|
||
|
||
beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y
|
||
print(beta_linreg)
|
||
# Start plain gradient descent
|
||
beta = np.random.randn(2,1)
|
||
|
||
eta = 0.1
|
||
Niterations = 100
|
||
|
||
for iter in range(Niterations):
|
||
gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta
|
||
beta -= eta*gradients
|
||
|
||
print(beta)
|
||
ypredict = X @ beta
|
||
ypredict2 = X @ beta_linreg
|
||
plt.plot(x, ypredict, "r-")
|
||
plt.plot(x, ypredict2, "b-")
|
||
plt.plot(x, y ,'ro')
|
||
plt.axis([0,2.0,0, 15.0])
|
||
plt.xlabel(r'$x$')
|
||
plt.ylabel(r'$y$')
|
||
plt.title(r'Gradient descent example for Ridge')
|
||
plt.show()
|
||
|
||
## Using gradient descent methods, limitations
|
||
|
||
* **Gradient descent (GD) finds local minima of our function**. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
|
||
|
||
* **GD is sensitive to initial conditions**. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.
|
||
|
||
* **Gradients are computationally expensive to calculate for large datasets**. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm.
|
||
|
||
* **GD is very sensitive to choices of learning rates**. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape.
|
||
|
||
* **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.
|
||
|
||
* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.
|
||
|
||
## Stochastic Gradient Descent
|
||
|
||
Stochastic gradient descent (SGD) and variants thereof address some of
|
||
the shortcomings of the Gradient descent method discussed above.
|
||
|
||
The underlying idea of SGD comes from the observation that the cost
|
||
function, which we want to minimize, can almost always be written as a
|
||
sum over $n$ data points $\{\mathbf{x}_i\}_{i=1}^n$,
|
||
|
||
$$
|
||
C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i,
|
||
\mathbf{\beta}).
|
||
$$
|
||
|
||
## Computation of gradients
|
||
|
||
This in turn means that the gradient can be
|
||
computed as a sum over $i$-gradients
|
||
|
||
$$
|
||
\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i,
|
||
\mathbf{\beta}).
|
||
$$
|
||
|
||
Stochasticity/randomness is introduced by only taking the
|
||
gradient on a subset of the data called minibatches. If there are $n$
|
||
data points and the size of each minibatch is $M$, there will be $n/M$
|
||
minibatches. We denote these minibatches by $B_k$ where
|
||
$k=1,\cdots,n/M$.
|
||
|
||
|
||
## SGD example
|
||
As an example, suppose we have $10$ data points $(\mathbf{x}_1,\cdots, \mathbf{x}_{10})$
|
||
and we choose to have $M=5$ minibathces,
|
||
then each minibatch contains two data points. In particular we have
|
||
$B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 =
|
||
(\mathbf{x}_9,\mathbf{x}_{10})$. Note that if you choose $M=1$ you
|
||
have only a single batch with all data points and on the other extreme,
|
||
you may choose $M=n$ resulting in a minibatch for each datapoint, i.e
|
||
$B_k = \mathbf{x}_k$.
|
||
|
||
The idea is now to approximate the gradient by replacing the sum over
|
||
all data points with a sum over the data points in one the minibatches
|
||
picked at random in each gradient descent step
|
||
|
||
$$
|
||
\nabla_{\beta}
|
||
C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i,
|
||
\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta
|
||
c_i(\mathbf{x}_i, \mathbf{\beta}).
|
||
$$
|
||
|
||
## The gradient step
|
||
|
||
Thus a gradient descent step now looks like
|
||
|
||
$$
|
||
\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i,
|
||
\mathbf{\beta})
|
||
$$
|
||
|
||
where $k$ is picked at random with equal
|
||
probability from $[1,n/M]$. An iteration over the number of
|
||
minibathces (n/M) is commonly referred to as an epoch. Thus it is
|
||
typical to choose a number of epochs and for each epoch iterate over
|
||
the number of minibatches, as exemplified in the code below.
|
||
|
||
|
||
## Simple example code
|
||
|
||
import numpy as np
|
||
|
||
n = 100 #100 datapoints
|
||
M = 5 #size of each minibatch
|
||
m = int(n/M) #number of minibatches
|
||
n_epochs = 10 #number of epochs
|
||
|
||
j = 0
|
||
for epoch in range(1,n_epochs+1):
|
||
for i in range(m):
|
||
k = np.random.randint(m) #Pick the k-th minibatch at random
|
||
#Compute the gradient using the data in minibatch Bk
|
||
#Compute new suggestion for
|
||
j += 1
|
||
|
||
Taking the gradient only on a subset of the data has two important
|
||
benefits. First, it introduces randomness which decreases the chance
|
||
that our opmization scheme gets stuck in a local minima. Second, if
|
||
the size of the minibatches are small relative to the number of
|
||
datapoints ($M < n$), the computation of the gradient is much
|
||
cheaper since we sum over the datapoints in the $k-th$ minibatch and not
|
||
all $n$ datapoints.
|
||
|
||
|
||
## When do we stop?
|
||
|
||
A natural question is when do we stop the search for a new minimum?
|
||
One possibility is to compute the full gradient after a given number
|
||
of epochs and check if the norm of the gradient is smaller than some
|
||
threshold and stop if true. However, the condition that the gradient
|
||
is zero is valid also for local minima, so this would only tell us
|
||
that we are close to a local/global minimum. However, we could also
|
||
evaluate the cost function at this point, store the result and
|
||
continue the search. If the test kicks in at a later stage we can
|
||
compare the values of the cost function and keep the $\beta$ that
|
||
gave the lowest value.
|
||
|
||
|
||
## Slightly different approach
|
||
|
||
Another approach is to let the step length $\gamma_j$ depend on the
|
||
number of epochs in such a way that it becomes very small after a
|
||
reasonable time such that we do not move at all.
|
||
|
||
As an example, let $e = 0,1,2,3,\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \cdot m + i$ where $m$ is the number of minibatches and $i=0,\cdots,m-1$. Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in *time* $t$.
|
||
|
||
In this way we can fix the number of epochs, compute $\beta$ and
|
||
evaluate the cost function at the end. Repeating the computation will
|
||
give a different result since the scheme is random by design. Then we
|
||
pick the final $\beta$ that gives the lowest value of the cost
|
||
function.
|
||
|
||
import numpy as np
|
||
|
||
def step_length(t,t0,t1):
|
||
return t0/(t+t1)
|
||
|
||
n = 100 #100 datapoints
|
||
M = 5 #size of each minibatch
|
||
m = int(n/M) #number of minibatches
|
||
n_epochs = 500 #number of epochs
|
||
t0 = 1.0
|
||
t1 = 10
|
||
|
||
gamma_j = t0/t1
|
||
j = 0
|
||
for epoch in range(1,n_epochs+1):
|
||
for i in range(m):
|
||
k = np.random.randint(m) #Pick the k-th minibatch at random
|
||
#Compute the gradient using the data in minibatch Bk
|
||
#Compute new suggestion for beta
|
||
t = epoch*m+i
|
||
gamma_j = step_length(t,t0,t1)
|
||
j += 1
|
||
|
||
print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
|
||
|
||
## Program for stochastic gradient
|
||
|
||
# Importing various packages
|
||
from math import exp, sqrt
|
||
from random import random, seed
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.linear_model import SGDRegressor
|
||
|
||
m = 100
|
||
x = 2*np.random.rand(m,1)
|
||
y = 4+3*x+np.random.randn(m,1)
|
||
|
||
X = np.c_[np.ones((m,1)), x]
|
||
theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
|
||
print("Own inversion")
|
||
print(theta_linreg)
|
||
sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
|
||
sgdreg.fit(x,y.ravel())
|
||
print("sgdreg from scikit")
|
||
print(sgdreg.intercept_, sgdreg.coef_)
|
||
|
||
|
||
theta = np.random.randn(2,1)
|
||
eta = 0.1
|
||
Niterations = 1000
|
||
|
||
|
||
for iter in range(Niterations):
|
||
gradients = 2.0/m*X.T @ ((X @ theta)-y)
|
||
theta -= eta*gradients
|
||
print("theta from own gd")
|
||
print(theta)
|
||
|
||
xnew = np.array([[0],[2]])
|
||
Xnew = np.c_[np.ones((2,1)), xnew]
|
||
ypredict = Xnew.dot(theta)
|
||
ypredict2 = Xnew.dot(theta_linreg)
|
||
|
||
|
||
n_epochs = 50
|
||
t0, t1 = 5, 50
|
||
def learning_schedule(t):
|
||
return t0/(t+t1)
|
||
|
||
theta = np.random.randn(2,1)
|
||
|
||
for epoch in range(n_epochs):
|
||
for i in range(m):
|
||
random_index = np.random.randint(m)
|
||
xi = X[random_index:random_index+1]
|
||
yi = y[random_index:random_index+1]
|
||
gradients = 2 * xi.T @ ((xi @ theta)-yi)
|
||
eta = learning_schedule(epoch*m+i)
|
||
theta = theta - eta*gradients
|
||
print("theta from own sdg")
|
||
print(theta)
|
||
|
||
plt.plot(xnew, ypredict, "r-")
|
||
plt.plot(xnew, ypredict2, "b-")
|
||
plt.plot(x, y ,'ro')
|
||
plt.axis([0,2.0,0, 15.0])
|
||
plt.xlabel(r'$x$')
|
||
plt.ylabel(r'$y$')
|
||
plt.title(r'Random numbers ')
|
||
plt.show()
|
||
|
||
**Challenge**: try to write a similar code for a Logistic Regression case. |