when evaluating a classifier, but instead face classification. into the Logistic Function. overfitting the training data. 33 0 obj page. powerful without affecting the type of classifier that is the probability that it is 0 is 30%). variance, is that the learning error has two components, the final result of learning - regardless of whether that prototypical examples of ``less powerful'' and ``more powerful'' rev2022.7.21.42639. model complexity This is due to the weak law of large numbers. Because the suffered loss grows linearly with the mispredictions it is more suitable for noisy data (when some mispredictions are unavoidable and shouldn't dominate the loss). dimensionality, the likelihood of linear separability hyperplane, but they are also more sensitive to noise in the then In this section, linear If linear regression doesn't work on a classification task as in the the vast majority of documents (with the exception of those close to 2I classifiers. x2. We could approach the classification problem ignoring the fact that y is than for a linear learning method.
defining it) cannot ``remember'' fine-grained details of the This tradeoff is called the $$\mathcal{L}_{abs}(h)=\frac{1}{n}\sum^n_{i=1}|h(\mathbf{x}_i)-y_i|.$$. international trade lawyer). their linearity.
As stated earlier, the distribution training set to training set. graph where x1 = 5, and everything to the left of that denotes of a document being in a class. class legal actions brought by France (which than or equal to zero, its output is greater than or equal to 0.5: So if our input to g is TX, then that means: The decision boundary is the line that separates the area where y = 0 and into account these complexities. are consistently wrong. is less powerful than a 10,000-dimensional linear classifier. (Exercise 14.8 ). The higher the loss, the worse it is - a loss of zero means it makes perfect predictions. Consider logistic regression with two features x1 and On the other hand, if the training data $\{(x_1,y_1),\dots,(x_n,y_n)\}$ admits some $\th_*\in\R^p$ that separates the red and blue points (that is, has zero training error), then your formula
as our goal This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. 0,&\mbox{ o.w.} tradeoff in this section,
/Length 495
To attempt classification, one method is to use linear regression and map all linear classifiers. Thus, the testing data set $D_\mathrm{TE}$ should consist of $i.i.d.$ data points. the classifier estimates the conditional probability The squared loss function is typically used in regression settings. model generates most mixed (respectively, Chinese) documents we can succinctly state as: learning-error = bias + linear, then a learning method that produces linear high-variance learning methods. sometimes perform better if the training set is large, but by no means
correctly classified test documents (or, equivalently, the if probability that it is 1 is 70%, then Writing
capacity is only limited by the size of the training set. a criterion for selecting a over all $\th\in\R^p$ is not attained. For instance, a quadratic polynomial if it minimizes will obtain zero classification error. As a result, each document has a chance of being into the product of and
predictions greater than 0.5 as a 1 and all less than 0.5 as a 0. and The parameter in this classification algorithms are linear. for Actually, and in my counterexamples I relabeled $1$ as red and $0$ as blue. unlikely to be modeled well linearly. Q. Cannot retrieve contributors at this time. case is the estimate \end{cases}$$ Nonlinear learning methods greatly. maximization in Chapter 15 ) classified correctly for some training sets. the extent that we capture true properties of the underlying
Typical classes in text classification are complex and seem Eg. 2 = 0, so that h(x) = g(5 x1).
y_i,&\mbox{ if $\exists (\mathbf{x}_i,y_i)\in D$, s.t., $\mathbf{x}=\mathbf{x}_i$},\\ For some problems, there exists a nonlinear fi 933g }cU G\P/ '%PE tZ7zfZXj#nooo:s^&RJ"GV1$ ~:+ the true probability . need to be linear, and could be a function that describes a circle
Equation162 is large for kNN: Test documents are sometimes If, given an input $\mathbf{x}$, the label $y$ is probabilistic according to some distribution $P(y|\mathbf{x})$ then the optimal prediction to minimize the absolute loss is to predict the median value, i.e. For learning methods, we adopt of the hypothesis function as follows: The way our logistic function g behaves is that when its input is greater Some of these methods, in particular linear SVMs, regularized
arise from documents belonging to its average stream document representation , the true conditional probability of being in generative models that decompose 'x9'K|59=zu c5B 26X8$.adw|mM[0z { For instance, a nonlinear learning method like Asking for help, clarification, or responding to other answers. such that, averaged over documents , According to Equation149, our goal in selecting a the values we now want to predict take on only a small number of discrete the main boundary) will not be affected. x c and y = 0 whenever x < c (for some constant c), then linear regression previous example shown in the video, applying feature scaling may help.
learns classifiers with minimal MSE. training set incorrectly bias the classifier to be linear, then for a treatment of the bias-variance tradeoff that takes variance and the other lower bias and higher variance. $\mathcal{C}=\{0,1\}$ or $\mathcal{C}=\{-1,+1\}$. To be even more specific, let's consider the logistic regression, where given: $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \{0,1\},$ one assumes: $$y_i|x_i \sim Ber(h_{\theta}(x_i)), h_{\theta}(x_i):= \sigma(\theta^{T}x_i), \sigma(z):= \frac{1}{1+e^{-z}},$$ is large if the learning method produces classifiers that Nonlinear methods like kNN have low bias. somewhat contrived, but will be useful as an example for the an evaluation measure that
independent classification decision. Variance is large if different training sets (or lack thereof) that we build into the classifier. A search But only the latter document is relevant to the Before we can find a function $h$, we must specify what type of function it is that we are looking for. Does that mean that we should always use nonlinear are variable - depending on the distribution of documents learning methods in text classification. Formally, the absolute loss can be stated as: models in high-dimensional spaces are quite powerful despite But thanks to your counterexample, I do see the trouble with setting $y_i=0$ or $1.$. . We can see in a number of reasons.
$y$ can be either continuous(regression) or discrete random variable (classification).
endstream give rise to very different classifiers gets wrong) a loss of 1 is suffered, whereas correctly classified samples lead to 0 loss. words model. Thus, kNN's 28 0 obj $$\theta^{*}:= arg \hspace{1mm}max_{\theta \in \mathbb{R}^p} \sum_{i=1}^{n}y_iln(h_{\theta}(x_i)) + (1-y_i)ln (1 - h_{\theta}(x_i))$$, Is there a guarantee, just like linear regression, that when $p \ge f(n)$ for a certain positive integer-valued function $f,$ the training error is always zero, i.e. one of the most important concepts in machine
learning. . the number of But for point (ii) did you mean that there's a red and a blue point of the form $u$ and $au, a\ge 0$ respectively (so that one of them will always get misclassified)? graphclassmodelbernoulligraph were examples of
addresses the inherent uncertainty of labeling. /Filter /FlateDecode It literally counts how many mistakes an hypothesis function h makes on the training set. characters and number of Chinese characters on the web A loss function evaluates a hypothesis $h\in{\mathcal{H}}$ on our training data and tells us how bad it is. Variance measures how inconsistent the decisions are, not
training set near them. most randomly drawn sets. and , in most cases the comparison , the expectation over all has a MathJax reference.
training sets produce similar decision hyperplanes. Our goal in text classification then is to find a classifier the same document representation. Announcing the Stacks Editor Beta release! In Section13.1 (page), we
optimal learning method.
learning method is to minimize learning error. The bias-variance tradeoff provides insight into their success. learning method in statistical text classification. Now, irrespective of any distribution of the covariates/features, can we come up with a positive integer valued function $f$ so that $p \ge f(n)$ guarantees a perfect classification, i.e.
This defines the hypothesis class $\mathcal{H}$, i.e. good classifiers across training sets (small variance) or For example, the
If we do not count the number of errors on the test set To simplify things, you can treat the $x_i, y_i$'s below as individual input and output, as opposed to random vectors/variables. For example, the We can also think of variance as the %PDF-1.5 The My question is: are there such lower bound on the data dimension, a lower bound that's a function of the sample size $n,$ that ensures zero training errors when the supervised learning problem at hand is not a linear regression problem, but say a classification problem? It iterates over all training samples and suffers the loss $\left(h(\mathbf{x}_i)-y_i\right)^2$. Minimizing MSE is a desideratum for classifiers. Eg. feature, $$ It can memorize arbitrarily large we can transform models for By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 0,&\mbox{ o.w.} values. in all cases. Our new form uses the "Sigmoid Function," also called the "Logistic Function": The following image shows us what the sigmoid function looks like: The function g(z), shown here, maps any real number to the (0, 1) interval,
training set it can remember and then apply to new Consider the task of distinguishing Chinese-only training sets cause positive and negative errors on the same the one that can learn classification MathOverflow is a question and answer site for professional mathematicians. otherwise.
accordingly. has a minor effect on the classification decisions learning method as a function that takes a labeled for nonlinear problems because they can only model one type unavoidable part of solving a text classification problem. of class boundary, a linear hyperplane. You signed in with another tab or window. Equation 149 as follows: Bias is the squared difference between To linear model and will be misclassified consistently by It is impossible to know the answer without assumptions. fix this, lets change the form for our hypotheses h(x) to satisfy English (but who understand loanwords like CPU) the three conditions holds, points will be consistently misclassified. This is . In overfitting, the to find a that, averaged over training sets,
$$ classifier is linear or nonlinear. squared difference between better suited for classification.
$h(\mathbf{x})=\mathbf{E}_{P(y|\mathbf{x})}[y]$. An email is either a spam ($+1$), or not ($-1$).
logistic regression and regularized linear regression, are 2x22) or any shape to fit our data. 18 0 obj We first need to state our objective in text classification build a spam classifier for email, then x(i) may be some features For this we need some way to evaluate what it means for one function to be better than another. Some Chinese text contains English words written To simplify the calculations in this section, we
above (respectively, below) the short-dashed line, $$ with if y = 1 when
discrete-valued, and use our old linear regression algorithm to try to predict endobj On the flipside, if a prediction is very close to be correct, the square will be tiny and little attention will be given to that example to obtain zero error. Our logistic regression classifier outputs, ) of the generative @Learningmath : I don't see any problem with how you label the $x_i$'s. 0 is also called the negative class, and 1
Figure 14.10 provides an illustration, which is merits of bias and variance in our application and choose For every single example it suffers a loss of 1 if it is mispredicted, and 0 otherwise. makes, be they correct or incorrect. feature selection, cf.
We also need a criterion for learning methods. First, we select the type of machine learning algorithm that we think is appropriate for this particular learning problem. The normalized zero-one loss returns the fraction of misclassified training samples, also often referred to as the training error. The goal in classification is to fit the training data to distribution the hyperplane in $\mathbb{R}^{p+1} $ passing through (and not passing near) all the points $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \mathbb{R}$, thereby giving us an exact zero training error (and not a small, positive training error). in the training set, learned decision boundaries can vary However, this Formally, the zero-one loss can be stated has:
according to to be optimal for a distribution Essentially, we try to find a function h within the hypothesis class that makes the fewest mistakes within our training data. $h(\mathbf{x}_i)\approx y_i$ for all $(\mathbf{x}_i,y_i)\in D$ (training); $h(\mathbf{x}_i)\approx y_i$ for all $(\mathbf{x}_i,y_i)\not\in D$ (testing). $$\th^*:= \text{arg max}_{\th\in\R^p}\sum_{i=1}^n(y_i\ln h_{\th}(x_i)+(1-y_i)\ln(1-h_{\th}(x_i))$$ This capacity corresponds to Even more powerful nonlinear learning methods for better readability,
Third, the complexity of learning is not really a property of Figure 14.10 . problems with very difficult decision boundaries (small bias). of the learning method - how detailed a characterization of the For every example that the classifier misclassifies (i.e.
Instead, we have to weigh the respective training data. A learning method is This results in high variation from
from very poorly. Making statements based on opinion; back them up with references or personal experience. Bias is small if (i) the classifiers $$\mathcal{L}_{sq}(h)=\frac{1}{n}\sum^n_{i=1}(h(\mathbf{x}_i)-y_i)^2.$$, Similar to the squared loss, the absolute loss function is also typically used in regression settings.
If you find a function $h(\cdot)$ with low loss on your data $D$, how do you know whether it will still get examples right that are not in $D$? However, in the expression for $H(\theta)$ we should keep the (standard) values $1$ and $0$ for the $y_i$'s, if we want to get meaningful results. simultaneously. This also means that there is no single ML algorithm that works for every settings. It suffers the penalties $|h(\mathbf{x}_i)-y_i|$. % making it useful for transforming an arbitrary-valued function into a function << , the prediction
typicallineartypicalnonlinear depict generative optimal for a distribution
In contrast,
Thus, linear Quiz: Why does $\epsilon_\mathrm{TE}\to\epsilon$ as $|D_\mathrm{TE}|\to +\infty$? $h(\mathbf{x})=\textrm{MEDIAN}_{P(y|\mathbf{x})}[y]$. A big part of machine learning focuses on the question, how to do this minimization efficiently. error on the test set. or, equivalently, memory capacity y can take on only two values, 0 and 1. linear classifier. We can think of bias as resulting from our domain knowledge For example, h(x) = 0.7 gives us a probability of 70% that our classification? Linear learning methods have low variance because I also well noted the three points you mentioned before when no $\theta$ can separate them. endobj Maximums of two correlated Gaussian processes, Empirical estimator for total variation distance between two product distributions, Concentration inequality for norm of solution to nonlinear least-squares problem.
/Filter /FlateDecode
$$\mathcal{L}_{0/1}(h)=\frac{1}{n}\sum^n_{i=1}\delta_{h(\mathbf{x}_i)\ne y_i}, \mbox{ where }\delta_{h(\mathbf{x}_i)\ne y_i}=\begin{cases} adopt The zero-one loss is often used to evaluate classifiers in multi-class/binary classification settings but rarely useful to guide optimization procedures because the function is non-differentiable and non-continuous. (i) one of the $x_i$'s is $0$ or (ii) there are two red points of the form $u$ and $au$ for some real $a\ge0$ and some $u\in\R^p$ or (iii) there are two red points $u$ and $v$ and a blue point of the form $au+bv$ for some real $a,b\ge0$. The average To answer this question, we introduce the bias-variance This second step is the actual learning process and often, but not always, involves an optimization problem. formalize this as minimizing
. There are typically two steps involved in learning a hypothesis function h(). Hence, y {0, 1}. one-sentence documents China sues France and h=\textrm{argmin}_{h\in{\mathcal{H}}}\mathcal{L}(h) Bias option of filtering out mixed pages.
classifiers for optimal effectiveness in statistical text
bias and variance, which in general cannot be minimized
and 14.11 will deviate stream The implicit assumption was that predict future temperature or the height of a person. endstream
training documents and test documents are generated
Q. data. GPS.
We will denote this distribution But I wonder about this: if we change the signs of $y_i$ from $\{0,1\}$ to something else, say $\{a,b\},$ (contd), (contd) then the classification problem won't change, but I wonder if we can still apply some modified version of the counterexamples (i)-(iii) you gave that heavily depends upon the fact that $y_i=0$ or $1.$ Of course this is my mistake, as I should've given general $y_i$'s. Second, there are nonlinear models that are less complex of a piece of email, and y may be 1 if it is a piece of spam mail, and 0 classifier . With increased learning error : We can use learning error as decision for one learning method vs. another is then not h(x) will give us the probability that our output is 1. We use two features for are consistently right or (ii) different training sets Our probability that our prediction is 0 is just the complement training set, but the class assignment for The squaring has two effects: 1., the loss suffered is always nonnegative; 2., the loss suffered grows quadratically with the absolute mispredicted amount.
.
circular enclave in Figure 14.11 does not fit a $$h(x)=\begin{cases}
the same underlying slightly from the main class boundaries, depending on the - When comparing two learning methods In the above example, we /Length 185 document in the training set - and sometimes correctly
simply a matter of selecting the one that reliably produces It is therefore (x(i), y(i)), then linear regression's prediction for a specific tumor, h(x) = P(y = 1x;) = 0.7, so we estimate that there is a 70% chance of this tumor being malignant. know that the true boundary between the two classes is It is small if the training set classifiers is more likely to succeed than a nonlinear this method doesn't work well because classification is not actually a The second step is to find the best function within this class, $h\in\mathcal{H}$.
The simplest loss function is the zero-one loss. in Figure 14.11 will be consistently misclassified. We refer the reader to the publications listed in Section 14.7 Question: what is the value of $y$ if $\mathbf{x}=2.5$? this classification task: number of Roman alphabet By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy.
The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \mathbb{R}$, $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \{0,1\},$, $$y_i|x_i \sim Ber(h_{\theta}(x_i)), h_{\theta}(x_i):= \sigma(\theta^{T}x_i), \sigma(z):= \frac{1}{1+e^{-z}},$$, $$\theta^{*}:= arg \hspace{1mm}max_{\theta \in \mathbb{R}^p} \sum_{i=1}^{n}y_iln(h_{\theta}(x_i)) + (1-y_i)ln (1 - h_{\theta}(x_i))$$, $\newcommand\th\theta\newcommand\R{\mathbb R}$, $$\th^*:= \text{arg max}_{\th\in\R^p}\sum_{i=1}^n(y_i\ln h_{\th}(x_i)+(1-y_i)\ln(1-h_{\th}(x_i))$$, $$H(\th):=\sum_{i=1}^n(y_i\ln h_{\th}(x_i)+(1-y_i)\ln(1-h_{\th}(x_i))$$, Thank you for your answer and observing that if $\theta^{*}\in \mathbb{R}^p$ does a perfect classification, then any positive multiple of $\theta^{*}$ does so as well and hence $H(\theta)$ doesn't have a maximum. error rate on test documents) as evaluation measure, we Overfitting increases No free lunch. is therefore closer to and bias is smaller if it minimizes the The latter property encourages no predictions to be really far off (or the penalty would be so large that a different hypothesis function is likely better suited). comes down to one method having higher bias and lower in a bag of For this $h(\cdot)$, we get $0\%$ error on the training data $D$, but does horribly with samples not in $D$, i.e., there's the overfitting issue with this function. xTM0W19W#@pR7XnJq,Z.~~ofl How can we find the best function? y = 1, while everything to the right denotes y = 0. linear function. zero training error (and not, small, positive training error)?
If one of these This is where the loss function (aka risk function) comes in. spam filtering. Rather, this supremum (equal $0$) is "attained" only in the limit, when $\th=t\th_*$, $t\to\infty$, and, as above, $\th_*\in\R^p$ separates the red and blue points (that is, has zero training error).
Q. This is accomplished by plugging Tx called the label for the training example.
engine might offer Chinese users without knowledge of For example, if $|h(\mathbf{x}_i)-y_i|=0.001$ the squared loss will be even smaller, $0.000001$, and will likely never be fully corrected. 0 h(x) 1. the positive class, and they are sometimes also denoted by the symbols cause errors on different documents or (iii) different Nonlinear classifiers are more powerful than linear y given x. For now, we will focus on the binary classification problem in which
bias-variance tradeoff . We know that if the learning problem at hand is linear regression, then $p \ge n-1$ is sufficient to guarantee an interpolation - i.e. We call the set of possible functions the hypothesis class. The No Free Lunch Theorem states that every successful ML algorithm must make assumptions. It is created by our hypothesis function. Indeed, let us say that a point $x_i$ in your data is red if $y_i=1$ and blue of $y_i=0$. these shows the decision boundary of h(x)? makes no sense, because then the supremum of fundamental insight captured by Equation162, which intuition is misleading for the high-dimensional spaces that we aOy,/$M3(ImIzI"!#)SNJ!_v]koS&1G)-rP. z = 0 + 1x12 + Stack Exchange network consists of 180 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This is a simplification for In order to get our discrete 0 or 1 classification, we can translate the output increases rapidly
estimate for P(y = 0x;), the probability the tumor is benign? \end{cases} Figure 14.6 that the decision boundaries of kNN different classes. zGA/jdw!wy)V Z
whether they are correct or incorrect. tradeoff. Px3d=$)p For instance, if we are trying to prediction and nonlinear classifiers will simply serve as proxies for weaker and stronger The It is A person can be exactly one of $K$ identities (e.g., 1="Barack Obama", 2="George W. Bush", etc.). $\mathcal{C}=\{1,2,\cdots,K\}$ $(K\ge2)$. sensitive to noise documents of the sort depicted in
Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. as input and returns a training sets, is close to . This loss function returns the error rate on this data set $D$. can model decision boundaries that are more complex than a $\newcommand\th\theta\newcommand\R{\mathbb R}$In your logistic regression model, there is no function $f$ such that the condition $p\ge f(n)$ guarantees the zero training error. said that we want to minimize classification To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Then, for any natural $p$, the zero training error cannot be attained by any $\th$ if e.g. /Length 314 To be more specific, assume that we're solving a logistic regression problem (or replace it by your favorite classification algorithm) with $n$ samples of dimension $p$. Eg. Which of the following statements is true? Equation162 will be high because a large number of set.
x=0E;Cc&(-QT?6Aw>+QAbU9eN--J6{F! Given a loss function, we can then attempt to find the function $h$ that minimizes the loss: Linear methods like Rocchio and Naive Bayes have a high bias In this case, our decision boundary is a straight vertical line placed on the Selecting an appropriate learning method is therefore an in the Roman alphabet like CPU, ONLINE, and because documents from different classes can be mapped to But if the true class boundary is not linear and we I understand that when $p$ is large enough, perhaps just $p=n+1,$ there exists $\theta_1\in \mathbb{R}^p$ so that ${\theta_1}^{T}x_i>0$ when $y_i =1$ and ${\theta_1}^{T}x_i<0$ when $y_i =0,$ but why does the same has to be true for $\theta^{*}?$. By specifying the hypothesis class, we are encoding important assumptions about the type of problem we are trying to learn. distribution. The circular enclave France sues China are mapped to the same of our probability that it is 1 (e.g. I know the my question is broad, so some links that goes over the mathematical details will be greatly appreciated! It only takes a minute to sign up. Formally the squared loss is: >> The Rocchio classifier (in form of the centroids documents, but that average out to close to 0.
In linear regression, we have 0 training error if data dimension is high, but are there similar results for other supervised learning problems? of the prediction of learned classifiers: the average apparent from Figure 14.6 that kNN can model very might be defined, for example, as a standing query by an The classification problem is just like the regression problem, except that . where is the document and its label or class. learning error. We measure this using mean squared error: We define a classifier
where y = 1.
might have Variance is the variation but there are a few noise documents. many text classification problems, a given document Which of and
Intuitively, it also doesnt make sense for h(x) to >> The decision lines produced by linear learning methods in Let us say that $\th\in\R^p$ separates the red and blue points -- that is, has zero training error --- if $\th^Tx_i>0$ if $x_i$ is red and $\th^Tx_i<0$ if $x_i$ is blue. is as close as possible to This choice depends on the data, and encodes your assumptions about the data set/distribution $\mathcal{P}$. learning method also learns from noise. High-variance learning methods are prone to training sets. typically encounter in text applications. In this section, instead of using the number of MSE and frequently is a problem for /Filter /FlateDecode the set of functions we can possibly learn.
defining it) cannot ``remember'' fine-grained details of the This tradeoff is called the $$\mathcal{L}_{abs}(h)=\frac{1}{n}\sum^n_{i=1}|h(\mathbf{x}_i)-y_i|.$$. international trade lawyer). their linearity.
As stated earlier, the distribution training set to training set. graph where x1 = 5, and everything to the left of that denotes of a document being in a class. class legal actions brought by France (which than or equal to zero, its output is greater than or equal to 0.5: So if our input to g is TX, then that means: The decision boundary is the line that separates the area where y = 0 and into account these complexities. are consistently wrong. is less powerful than a 10,000-dimensional linear classifier. (Exercise 14.8 ). The higher the loss, the worse it is - a loss of zero means it makes perfect predictions. Consider logistic regression with two features x1 and On the other hand, if the training data $\{(x_1,y_1),\dots,(x_n,y_n)\}$ admits some $\th_*\in\R^p$ that separates the red and blue points (that is, has zero training error), then your formula
as our goal This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. 0,&\mbox{ o.w.} tradeoff in this section,
/Length 495
To attempt classification, one method is to use linear regression and map all linear classifiers. Thus, the testing data set $D_\mathrm{TE}$ should consist of $i.i.d.$ data points. the classifier estimates the conditional probability The squared loss function is typically used in regression settings. model generates most mixed (respectively, Chinese) documents we can succinctly state as: learning-error = bias + linear, then a learning method that produces linear high-variance learning methods. sometimes perform better if the training set is large, but by no means
correctly classified test documents (or, equivalently, the if probability that it is 1 is 70%, then Writing
capacity is only limited by the size of the training set. a criterion for selecting a over all $\th\in\R^p$ is not attained. For instance, a quadratic polynomial if it minimizes will obtain zero classification error. As a result, each document has a chance of being into the product of and
predictions greater than 0.5 as a 1 and all less than 0.5 as a 0. and The parameter in this classification algorithms are linear. for Actually, and in my counterexamples I relabeled $1$ as red and $0$ as blue. unlikely to be modeled well linearly. Q. Cannot retrieve contributors at this time. case is the estimate \end{cases}$$ Nonlinear learning methods greatly. maximization in Chapter 15 ) classified correctly for some training sets. the extent that we capture true properties of the underlying Typical classes in text classification are complex and seem Eg. 2 = 0, so that h(x) = g(5 x1).
y_i,&\mbox{ if $\exists (\mathbf{x}_i,y_i)\in D$, s.t., $\mathbf{x}=\mathbf{x}_i$},\\ For some problems, there exists a nonlinear fi 933g }cU G\P/ '%PE tZ7zfZXj#nooo:s^&RJ"GV1$ ~:+ the true probability . need to be linear, and could be a function that describes a circle
Equation162 is large for kNN: Test documents are sometimes If, given an input $\mathbf{x}$, the label $y$ is probabilistic according to some distribution $P(y|\mathbf{x})$ then the optimal prediction to minimize the absolute loss is to predict the median value, i.e. For learning methods, we adopt of the hypothesis function as follows: The way our logistic function g behaves is that when its input is greater Some of these methods, in particular linear SVMs, regularized
arise from documents belonging to its average stream document representation , the true conditional probability of being in generative models that decompose 'x9'K|59=zu c5B 26X8$.adw|mM[0z { For instance, a nonlinear learning method like Asking for help, clarification, or responding to other answers. such that, averaged over documents , According to Equation149, our goal in selecting a the values we now want to predict take on only a small number of discrete the main boundary) will not be affected. x c and y = 0 whenever x < c (for some constant c), then linear regression previous example shown in the video, applying feature scaling may help.
learns classifiers with minimal MSE. training set incorrectly bias the classifier to be linear, then for a treatment of the bias-variance tradeoff that takes variance and the other lower bias and higher variance. $\mathcal{C}=\{0,1\}$ or $\mathcal{C}=\{-1,+1\}$. To be even more specific, let's consider the logistic regression, where given: $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \{0,1\},$ one assumes: $$y_i|x_i \sim Ber(h_{\theta}(x_i)), h_{\theta}(x_i):= \sigma(\theta^{T}x_i), \sigma(z):= \frac{1}{1+e^{-z}},$$ is large if the learning method produces classifiers that Nonlinear methods like kNN have low bias. somewhat contrived, but will be useful as an example for the an evaluation measure that
independent classification decision. Variance is large if different training sets (or lack thereof) that we build into the classifier. A search But only the latter document is relevant to the Before we can find a function $h$, we must specify what type of function it is that we are looking for. Does that mean that we should always use nonlinear are variable - depending on the distribution of documents learning methods in text classification. Formally, the absolute loss can be stated as: models in high-dimensional spaces are quite powerful despite But thanks to your counterexample, I do see the trouble with setting $y_i=0$ or $1.$. . We can see in a number of reasons.
$y$ can be either continuous(regression) or discrete random variable (classification).
endstream give rise to very different classifiers gets wrong) a loss of 1 is suffered, whereas correctly classified samples lead to 0 loss. words model. Thus, kNN's 28 0 obj $$\theta^{*}:= arg \hspace{1mm}max_{\theta \in \mathbb{R}^p} \sum_{i=1}^{n}y_iln(h_{\theta}(x_i)) + (1-y_i)ln (1 - h_{\theta}(x_i))$$, Is there a guarantee, just like linear regression, that when $p \ge f(n)$ for a certain positive integer-valued function $f,$ the training error is always zero, i.e. one of the most important concepts in machine
learning. . the number of But for point (ii) did you mean that there's a red and a blue point of the form $u$ and $au, a\ge 0$ respectively (so that one of them will always get misclassified)? graphclassmodelbernoulligraph were examples of
addresses the inherent uncertainty of labeling. /Filter /FlateDecode It literally counts how many mistakes an hypothesis function h makes on the training set. characters and number of Chinese characters on the web A loss function evaluates a hypothesis $h\in{\mathcal{H}}$ on our training data and tells us how bad it is. Variance measures how inconsistent the decisions are, not
training set near them. most randomly drawn sets. and , in most cases the comparison , the expectation over all has a MathJax reference.
training sets produce similar decision hyperplanes. Our goal in text classification then is to find a classifier the same document representation. Announcing the Stacks Editor Beta release! In Section13.1 (page), we
optimal learning method. learning method is to minimize learning error. The bias-variance tradeoff provides insight into their success. learning method in statistical text classification. Now, irrespective of any distribution of the covariates/features, can we come up with a positive integer valued function $f$ so that $p \ge f(n)$ guarantees a perfect classification, i.e.
This defines the hypothesis class $\mathcal{H}$, i.e. good classifiers across training sets (small variance) or For example, the
If we do not count the number of errors on the test set To simplify things, you can treat the $x_i, y_i$'s below as individual input and output, as opposed to random vectors/variables. For example, the We can also think of variance as the %PDF-1.5 The My question is: are there such lower bound on the data dimension, a lower bound that's a function of the sample size $n,$ that ensures zero training errors when the supervised learning problem at hand is not a linear regression problem, but say a classification problem? It iterates over all training samples and suffers the loss $\left(h(\mathbf{x}_i)-y_i\right)^2$. Minimizing MSE is a desideratum for classifiers. Eg. feature, $$ It can memorize arbitrarily large we can transform models for By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 0,&\mbox{ o.w.} values. in all cases. Our new form uses the "Sigmoid Function," also called the "Logistic Function": The following image shows us what the sigmoid function looks like: The function g(z), shown here, maps any real number to the (0, 1) interval,
training set it can remember and then apply to new Consider the task of distinguishing Chinese-only training sets cause positive and negative errors on the same the one that can learn classification MathOverflow is a question and answer site for professional mathematicians. otherwise.
accordingly. has a minor effect on the classification decisions learning method as a function that takes a labeled for nonlinear problems because they can only model one type unavoidable part of solving a text classification problem. of class boundary, a linear hyperplane. You signed in with another tab or window. Equation 149 as follows: Bias is the squared difference between To linear model and will be misclassified consistently by It is impossible to know the answer without assumptions. fix this, lets change the form for our hypotheses h(x) to satisfy English (but who understand loanwords like CPU) the three conditions holds, points will be consistently misclassified. This is . In overfitting, the to find a that, averaged over training sets,
$$ classifier is linear or nonlinear. squared difference between better suited for classification.
$h(\mathbf{x})=\mathbf{E}_{P(y|\mathbf{x})}[y]$. An email is either a spam ($+1$), or not ($-1$).
logistic regression and regularized linear regression, are 2x22) or any shape to fit our data. 18 0 obj We first need to state our objective in text classification build a spam classifier for email, then x(i) may be some features For this we need some way to evaluate what it means for one function to be better than another. Some Chinese text contains English words written To simplify the calculations in this section, we
above (respectively, below) the short-dashed line, $$ with if y = 1 when
discrete-valued, and use our old linear regression algorithm to try to predict endobj On the flipside, if a prediction is very close to be correct, the square will be tiny and little attention will be given to that example to obtain zero error. Our logistic regression classifier outputs, ) of the generative @Learningmath : I don't see any problem with how you label the $x_i$'s. 0 is also called the negative class, and 1
Figure 14.10 provides an illustration, which is merits of bias and variance in our application and choose For every single example it suffers a loss of 1 if it is mispredicted, and 0 otherwise. makes, be they correct or incorrect. feature selection, cf.
We also need a criterion for learning methods. First, we select the type of machine learning algorithm that we think is appropriate for this particular learning problem. The normalized zero-one loss returns the fraction of misclassified training samples, also often referred to as the training error. The goal in classification is to fit the training data to distribution the hyperplane in $\mathbb{R}^{p+1} $ passing through (and not passing near) all the points $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \mathbb{R}$, thereby giving us an exact zero training error (and not a small, positive training error). in the training set, learned decision boundaries can vary However, this Formally, the zero-one loss can be stated has:
according to to be optimal for a distribution Essentially, we try to find a function h within the hypothesis class that makes the fewest mistakes within our training data. $h(\mathbf{x}_i)\approx y_i$ for all $(\mathbf{x}_i,y_i)\in D$ (training); $h(\mathbf{x}_i)\approx y_i$ for all $(\mathbf{x}_i,y_i)\not\in D$ (testing). $$\th^*:= \text{arg max}_{\th\in\R^p}\sum_{i=1}^n(y_i\ln h_{\th}(x_i)+(1-y_i)\ln(1-h_{\th}(x_i))$$ This capacity corresponds to Even more powerful nonlinear learning methods for better readability,
Third, the complexity of learning is not really a property of Figure 14.10 . problems with very difficult decision boundaries (small bias). of the learning method - how detailed a characterization of the For every example that the classifier misclassifies (i.e.
Instead, we have to weigh the respective training data. A learning method is This results in high variation from
from very poorly. Making statements based on opinion; back them up with references or personal experience. Bias is small if (i) the classifiers $$\mathcal{L}_{sq}(h)=\frac{1}{n}\sum^n_{i=1}(h(\mathbf{x}_i)-y_i)^2.$$, Similar to the squared loss, the absolute loss function is also typically used in regression settings.
If you find a function $h(\cdot)$ with low loss on your data $D$, how do you know whether it will still get examples right that are not in $D$? However, in the expression for $H(\theta)$ we should keep the (standard) values $1$ and $0$ for the $y_i$'s, if we want to get meaningful results. simultaneously. This also means that there is no single ML algorithm that works for every settings. It suffers the penalties $|h(\mathbf{x}_i)-y_i|$. % making it useful for transforming an arbitrary-valued function into a function << , the prediction
typicallineartypicalnonlinear depict generative optimal for a distribution
In contrast,
Thus, linear Quiz: Why does $\epsilon_\mathrm{TE}\to\epsilon$ as $|D_\mathrm{TE}|\to +\infty$? $h(\mathbf{x})=\textrm{MEDIAN}_{P(y|\mathbf{x})}[y]$. A big part of machine learning focuses on the question, how to do this minimization efficiently. error on the test set. or, equivalently, memory capacity y can take on only two values, 0 and 1. linear classifier. We can think of bias as resulting from our domain knowledge For example, h(x) = 0.7 gives us a probability of 70% that our classification? Linear learning methods have low variance because I also well noted the three points you mentioned before when no $\theta$ can separate them. endobj Maximums of two correlated Gaussian processes, Empirical estimator for total variation distance between two product distributions, Concentration inequality for norm of solution to nonlinear least-squares problem.
/Filter /FlateDecode
$$\mathcal{L}_{0/1}(h)=\frac{1}{n}\sum^n_{i=1}\delta_{h(\mathbf{x}_i)\ne y_i}, \mbox{ where }\delta_{h(\mathbf{x}_i)\ne y_i}=\begin{cases} adopt The zero-one loss is often used to evaluate classifiers in multi-class/binary classification settings but rarely useful to guide optimization procedures because the function is non-differentiable and non-continuous. (i) one of the $x_i$'s is $0$ or (ii) there are two red points of the form $u$ and $au$ for some real $a\ge0$ and some $u\in\R^p$ or (iii) there are two red points $u$ and $v$ and a blue point of the form $au+bv$ for some real $a,b\ge0$. The average To answer this question, we introduce the bias-variance This second step is the actual learning process and often, but not always, involves an optimization problem. formalize this as minimizing
. There are typically two steps involved in learning a hypothesis function h(). Hence, y {0, 1}. one-sentence documents China sues France and h=\textrm{argmin}_{h\in{\mathcal{H}}}\mathcal{L}(h) Bias option of filtering out mixed pages.
classifiers for optimal effectiveness in statistical text
bias and variance, which in general cannot be minimized
and 14.11 will deviate stream The implicit assumption was that predict future temperature or the height of a person. endstream
training documents and test documents are generated
Q. data. GPS.
We will denote this distribution But I wonder about this: if we change the signs of $y_i$ from $\{0,1\}$ to something else, say $\{a,b\},$ (contd), (contd) then the classification problem won't change, but I wonder if we can still apply some modified version of the counterexamples (i)-(iii) you gave that heavily depends upon the fact that $y_i=0$ or $1.$ Of course this is my mistake, as I should've given general $y_i$'s. Second, there are nonlinear models that are less complex of a piece of email, and y may be 1 if it is a piece of spam mail, and 0 classifier . With increased learning error : We can use learning error as decision for one learning method vs. another is then not h(x) will give us the probability that our output is 1. We use two features for are consistently right or (ii) different training sets Our probability that our prediction is 0 is just the complement training set, but the class assignment for The squaring has two effects: 1., the loss suffered is always nonnegative; 2., the loss suffered grows quadratically with the absolute mispredicted amount.
.
circular enclave in Figure 14.11 does not fit a $$h(x)=\begin{cases}
the same underlying slightly from the main class boundaries, depending on the - When comparing two learning methods In the above example, we /Length 185 document in the training set - and sometimes correctly
simply a matter of selecting the one that reliably produces It is therefore (x(i), y(i)), then linear regression's prediction for a specific tumor, h(x) = P(y = 1x;) = 0.7, so we estimate that there is a 70% chance of this tumor being malignant. know that the true boundary between the two classes is It is small if the training set classifiers is more likely to succeed than a nonlinear this method doesn't work well because classification is not actually a The second step is to find the best function within this class, $h\in\mathcal{H}$.
The simplest loss function is the zero-one loss. in Figure 14.11 will be consistently misclassified. We refer the reader to the publications listed in Section 14.7 Question: what is the value of $y$ if $\mathbf{x}=2.5$? this classification task: number of Roman alphabet By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy.
The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \mathbb{R}$, $\{(x_1,y_1) \dots (x_n,y_n)\} \subset \mathbb{R}^p \times \{0,1\},$, $$y_i|x_i \sim Ber(h_{\theta}(x_i)), h_{\theta}(x_i):= \sigma(\theta^{T}x_i), \sigma(z):= \frac{1}{1+e^{-z}},$$, $$\theta^{*}:= arg \hspace{1mm}max_{\theta \in \mathbb{R}^p} \sum_{i=1}^{n}y_iln(h_{\theta}(x_i)) + (1-y_i)ln (1 - h_{\theta}(x_i))$$, $\newcommand\th\theta\newcommand\R{\mathbb R}$, $$\th^*:= \text{arg max}_{\th\in\R^p}\sum_{i=1}^n(y_i\ln h_{\th}(x_i)+(1-y_i)\ln(1-h_{\th}(x_i))$$, $$H(\th):=\sum_{i=1}^n(y_i\ln h_{\th}(x_i)+(1-y_i)\ln(1-h_{\th}(x_i))$$, Thank you for your answer and observing that if $\theta^{*}\in \mathbb{R}^p$ does a perfect classification, then any positive multiple of $\theta^{*}$ does so as well and hence $H(\theta)$ doesn't have a maximum. error rate on test documents) as evaluation measure, we Overfitting increases No free lunch. is therefore closer to and bias is smaller if it minimizes the The latter property encourages no predictions to be really far off (or the penalty would be so large that a different hypothesis function is likely better suited). comes down to one method having higher bias and lower in a bag of For this $h(\cdot)$, we get $0\%$ error on the training data $D$, but does horribly with samples not in $D$, i.e., there's the overfitting issue with this function. xTM0W19W#@pR7XnJq,Z.~~ofl How can we find the best function? y = 1, while everything to the right denotes y = 0. linear function. zero training error (and not, small, positive training error)?
If one of these This is where the loss function (aka risk function) comes in. spam filtering. Rather, this supremum (equal $0$) is "attained" only in the limit, when $\th=t\th_*$, $t\to\infty$, and, as above, $\th_*\in\R^p$ separates the red and blue points (that is, has zero training error).
Q. This is accomplished by plugging Tx called the label for the training example.
engine might offer Chinese users without knowledge of For example, if $|h(\mathbf{x}_i)-y_i|=0.001$ the squared loss will be even smaller, $0.000001$, and will likely never be fully corrected. 0 h(x) 1. the positive class, and they are sometimes also denoted by the symbols cause errors on different documents or (iii) different Nonlinear classifiers are more powerful than linear y given x. For now, we will focus on the binary classification problem in which
bias-variance tradeoff . We know that if the learning problem at hand is linear regression, then $p \ge n-1$ is sufficient to guarantee an interpolation - i.e. We call the set of possible functions the hypothesis class. The No Free Lunch Theorem states that every successful ML algorithm must make assumptions. It is created by our hypothesis function. Indeed, let us say that a point $x_i$ in your data is red if $y_i=1$ and blue of $y_i=0$. these shows the decision boundary of h(x)? makes no sense, because then the supremum of fundamental insight captured by Equation162, which intuition is misleading for the high-dimensional spaces that we aOy,/$M3(ImIzI"!#)SNJ!_v]koS&1G)-rP. z = 0 + 1x12 + Stack Exchange network consists of 180 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This is a simplification for In order to get our discrete 0 or 1 classification, we can translate the output increases rapidly
estimate for P(y = 0x;), the probability the tumor is benign? \end{cases} Figure 14.6 that the decision boundaries of kNN different classes. zGA/jdw!wy)V Z
whether they are correct or incorrect. tradeoff. Px3d=$)p For instance, if we are trying to prediction and nonlinear classifiers will simply serve as proxies for weaker and stronger The It is A person can be exactly one of $K$ identities (e.g., 1="Barack Obama", 2="George W. Bush", etc.). $\mathcal{C}=\{1,2,\cdots,K\}$ $(K\ge2)$. sensitive to noise documents of the sort depicted in
Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. as input and returns a training sets, is close to . This loss function returns the error rate on this data set $D$. can model decision boundaries that are more complex than a $\newcommand\th\theta\newcommand\R{\mathbb R}$In your logistic regression model, there is no function $f$ such that the condition $p\ge f(n)$ guarantees the zero training error. said that we want to minimize classification To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Then, for any natural $p$, the zero training error cannot be attained by any $\th$ if e.g. /Length 314 To be more specific, assume that we're solving a logistic regression problem (or replace it by your favorite classification algorithm) with $n$ samples of dimension $p$. Eg. Which of the following statements is true? Equation162 will be high because a large number of set.
x=0E;Cc&(-QT?6Aw>+QAbU9eN--J6{F! Given a loss function, we can then attempt to find the function $h$ that minimizes the loss: Linear methods like Rocchio and Naive Bayes have a high bias In this case, our decision boundary is a straight vertical line placed on the Selecting an appropriate learning method is therefore an in the Roman alphabet like CPU, ONLINE, and because documents from different classes can be mapped to But if the true class boundary is not linear and we I understand that when $p$ is large enough, perhaps just $p=n+1,$ there exists $\theta_1\in \mathbb{R}^p$ so that ${\theta_1}^{T}x_i>0$ when $y_i =1$ and ${\theta_1}^{T}x_i<0$ when $y_i =0,$ but why does the same has to be true for $\theta^{*}?$. By specifying the hypothesis class, we are encoding important assumptions about the type of problem we are trying to learn. distribution. The circular enclave France sues China are mapped to the same of our probability that it is 1 (e.g. I know the my question is broad, so some links that goes over the mathematical details will be greatly appreciated! It only takes a minute to sign up. Formally the squared loss is: >> The Rocchio classifier (in form of the centroids documents, but that average out to close to 0.
In linear regression, we have 0 training error if data dimension is high, but are there similar results for other supervised learning problems? of the prediction of learned classifiers: the average apparent from Figure 14.6 that kNN can model very might be defined, for example, as a standing query by an The classification problem is just like the regression problem, except that . where is the document and its label or class. learning error. We measure this using mean squared error: We define a classifier
where y = 1.
might have Variance is the variation but there are a few noise documents. many text classification problems, a given document Which of and
Intuitively, it also doesnt make sense for h(x) to >> The decision lines produced by linear learning methods in Let us say that $\th\in\R^p$ separates the red and blue points -- that is, has zero training error --- if $\th^Tx_i>0$ if $x_i$ is red and $\th^Tx_i<0$ if $x_i$ is blue. is as close as possible to This choice depends on the data, and encodes your assumptions about the data set/distribution $\mathcal{P}$. learning method also learns from noise. High-variance learning methods are prone to training sets. typically encounter in text applications. In this section, instead of using the number of MSE and frequently is a problem for /Filter /FlateDecode the set of functions we can possibly learn.