Error Functions & Linear Regression (2)
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1 Error Functions & Linear Regression (2) John Kelleher & Brian Mac Namee Machine DIT
2 Overview 1 Introduction Overview 2 Linear Classifiers Threshold Function Perceptron Learning Rule Training/Learning Curve 3 Logistic Regression Logistic Function Gradient Descent 4 Non Linear Models Non Linear Models 5 Summary
3 Linear Classifiers
4 Threshold Function We can use linear functions to do classification as well as regression by creating a decision boundary. If the decision boundary is a straight line (or surface) it is referred to as linearly separator and the classes are said to be linearly separable. In this example (distinguishing between good (1) and faulty (0) machinery using RPM and vibration measurements) the separator can be defined as: vib = rpm or rpm vib = 0
5 Threshold Function Consider the point (700, 483) in relation to this separator: = = 147 Consider the point (720, 201) in relation to this separator: = = 142.6
6 Threshold Function
7 Threshold Function Values for examples above the decision boundary are always negative, while those below the decision boundary are always positive - we can use this as a classifier. Reverting to our previous terminology we have: h w (x) = { 1 if w x 0 0 otherwise
8 Threshold Function
9 Threshold Function We can think of this as a simple threshold on the result of w x as shown below:
10 Perceptron Learning Rule So, how do we find the best values for our weights, w that will minimise the loss on our hypothesis h w (x)? Unfortunately this is a little bit tricky, essentially because our error on a single example is always either 0 or 1. This means that the gradient, or derivative, of the loss is always either 0 or, worse, undefined. Remember when we worked out the weight update rule before w i Loss(w) = w i (y h w (x)) 2 = 2(y h w (x)) w i (y h w (x)) We had to differentiate h w (x) which we could do when it was equal to (w 0 + w 1 x 1 ), but which is not possible with a hard threshold function.
11 Perceptron Learning Rule However, all is not lost. It turns out that there is a simple weight update rule that always converges to a solution as long as the data is linearly separable: w i w i + α(y i h w (x)) x i This is essentially just like our update rule for linear regression and is known as the Perceptron learning rule. Because y and h w (x) are always either 0 or 1 there are three possible outcomes: y = h w (x) The weights are not changed. y = 1 but h w (x) = 0 Each w i is increased if x i is positive and decreased when x i is negative. y = 0 but h w (x) = 1 Each w i is decreased if x i is positive and increased when x i is negative.
12 Perceptron Learning Rule
13 Training/Learning Curve While training will converge using this update rule, the training process isn t particularly pretty, as it thrashes around quite a bit.
14 Training/Learning Curve This situation is made even worse if the classes are not linearly separable, which is very often the case. In this example a good separator could be defined as: rpm = vib or vib rpm = 0
15 Training/Learning Curve In this case the learning curve jumps around even more and may never converge. There are two approaches we can take to addressing this: Decay α Logistic regression
16 Training/Learning Curve Decaying α means reducing the learning rate slightly at each training iteration. If we remember back to our discussion of gradient descent this essentially means taking smaller and smaller weight update steps as the learning process continues. It is typical for α to decay as O(1/t) where t is the iteration number In this case it can be shown that the rule will converge to a minimum-error solution when examples are presented in a random sequence.
17 Training/Learning Curve In the example below α(t) = 1000/( t)
18 Logistic Regression
19 Logistic Function The hard nature of our previous classification threshold caused us an amount of trouble: The hypothesis h w (x) is not differentiable, and in fact discontinuous, which makes learning very unpredictable. The classifier always announces completely confident predictions of 0 or 1 - often a little more subtlety is desirable. These issues can be addressed by using a more sophisticated threshold function (one that is also continuous and differentiable). The logistic function is ideal for this.
20 Logistic Function The logistic function Logistic(z) = 1 1+e z
21 Logistic Function
22 Logistic Function Applying the logistic function to our hypothesis we get h w (x) = Logistic(w x) = 1 1+e w x Let s consider a single example (x, y) again: w i Loss(w) = w i (y h w (x)) 2 = 2(y h w (x)) w i (y h w (x)) (Remember the chain rule)
23 Logistic Function Applying the logistic function to our hypothesis we get h w (x) = Logistic(w x) = 1 1+e w x w i Loss(w) = w i (y h w (x)) 2 = 2(y h w (x)) w i (y h w (x)) = 2(y h w (x)) w i (y Logistic(w x))
24 Logistic Function Applying the logistic function to our hypothesis we get h w (x) = Logistic(w x) = 1 1+e w x w i Loss(w) = w i (y h w (x)) 2 = 2(y h w (x)) w i (y h w (x)) = 2(y h w (x)) w i (y Logistic(w x)) = 2(y h w (x)) w i Logistic(w x) = 2(y h w (x)) w i Logistic(w x) x i w i w x
25 Logistic Function Applying the logistic function to our hypothesis we get h w (x) = Logistic(w x) = 1 1+e w x Luckily the derivative of the logistic function is well known and is : w i Logistic(w x) = Logistic(w x)(1 Logistic(w x))
26 Logistic Function Applying the logistic function to our hypothesis we get h w (x) = Logistic(w x) = 1 1+e w x So: w i Loss(w) = 2(y h w (x)) Logistic(w x)(1 Logistic(w x)) x i
27 Logistic Function Applying the logistic function to our hypothesis we get h w (x) = Logistic(w x) = 1 1+e w x So: w i Loss(w) = 2(y h w (x)) Logistic(w x)(1 Logistic(w x)) x i = 2(y h w (x)) h w (x) (1 h w (x)) x i
28 Logistic Function This means that our weight update rule for logistic regression is: w i w i + α(y h w (x)) h w (x) (1 h w (x)) x i 1 where h w (x) = 1+e w x Which isn t much different to the rule for simple linear regression.
29 Logistic Function The graph below shows the logistic regression learning curve for the linearly separable data set we saw earlier.
30 Logistic Function And the following graph shows the logistic regression learning curve for the non-linearly separable version of the dataset.
31 Logistic Function In the example below α decays according to α(t) = 1000/( t)
32 Non Linear Models
33 Non Linear Models Linear regression models work every well, both for discrete and continuous prediction problems when the underlying models are linear (e.g. linear continuous relationships or linearly separable classification problems). However, sometimes the underlying data will have non-linear relationships.
34 Non Linear Models For example the following dataset shows the relationship between a patient s age and their risk of suffering serious complications from pnuemonia.
35 Non Linear Models While there are many approaches to dealing with these sorts of models, we will look at basis functions which are both a popular and straight-forward solution to this problem. The idea behind basis functions is that we convert each of the inputs in our model using a particular basis function. This keeps our actual model linear but allows more expressiveness. Our linear regression model can be recast using basis functions as follows: y(x) = n j=0 w jφ j (x)
36 Non Linear Models The functions φ j (x) can be any form we like, typical choices include: φ j (x) = x 2 φ j (x) = x 3 x 2 + 4x φ j (x) = log(x) In a multivariate model different parameters can have completely different basis functions.
37 Non Linear Models Example Consider again the example of predicting pnuemonia risk from age. The image below shows the best linear model we can create from the dataset. risk = age
38 Non Linear Models Example However, we can apply the simple basis function: φ(age) = age 2 which then allows us to create the more accurate model: risk = φ(age)
39 Summary
40 1 Introduction Overview 2 Linear Classifiers Threshold Function Perceptron Learning Rule Training/Learning Curve 3 Logistic Regression Logistic Function Gradient Descent 4 Non Linear Models Non Linear Models 5 Summary
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