Machine Learning 10-601
Tom M. Mitchell
Machine Learning Department Carnegie Mellon University
February 4, 2015
Today:
•Generative – discriminative classifiers
•Linear regression
•Decomposition of error into bias, variance, unavoidable
Readings:
•Mitchell: “Naïve Bayes and Logistic Regression”
(required)
•Ng and Jordan paper (optional)
•Bishop, Ch 9.1, 9.2 (optional)
Logistic Regression
• Consider learning f: XàY, where
•X is a vector of real-valued features, < X1 … Xn > • Y is boolean
•assume all Xi are conditionally independent given Y •model P(Xi | Y = yk) as Gaussian N(μik,σi)
•model P(Y) as Bernoulli (π)
•Then P(Y|X) is of this form, and we can directly estimate W
•Furthermore, same holds if the Xi are boolean •trying proving that to yourself
•Train by gradient ascent estimation of w’s (no assumptions!)
MLE vs MAP
•Maximum conditional likelihood estimate
•Maximum a posteriori estimate with prior W~N(0,σI)
MAP estimates and Regularization
•Maximum a posteriori estimate with prior W~N(0,σI)
called a “regularization” term
•helps reduce overfitting, especially when training data is sparse
•keep weights nearer to zero (if P(W) is zero mean Gaussian prior), or whatever the prior suggests
•used very frequently in Logistic Regression
Generative vs. Discriminative Classifiers
Training classifiers involves estimating f: XàY, or P(Y|X)
Generative classifiers (e.g., Naïve Bayes)
•Assume some functional form for P(Y), P(X|Y)
•Estimate parameters of P(X|Y), P(Y) directly from training data
•Use Bayes rule to calculate P(Y=y |X= x)
Discriminative classifiers (e.g., Logistic regression)
•Assume some functional form for P(Y|X)
•Estimate parameters of P(Y|X) directly from training data
•NOTE: even though our derivation of the form of P(Y|X) made GNB- style assumptions, the training procedure for Logistic Regression does not!
Use Naïve Bayes or Logisitic Regression?
Consider
•Restrictiveness of modeling assumptions (how well can we learn with infinite data?)
•Rate of convergence (in amount of training data) toward asymptotic (infinite data) hypothesis
–i.e., the learning curve
Naïve Bayes vs Logistic Regression
Consider Y boolean, Xi continuous, X=
Number of parameters: •NB: 4n +1
•LR: n+1
Estimation method:
•NB parameter estimates are uncoupled •LR parameter estimates are coupled
Gaussian Naïve Bayes – Big Picture
assume P(Y=1) = 0.5
Gaussian Naïve Bayes – Big Picture
assume P(Y=1) = 0.5
G.Naïve Bayes vs. Logistic Regression
[Ng & Jordan, 2002]
Recall two assumptions deriving form of LR from GNBayes: 1. Xi conditionally independent of Xk given Y
2. P(Xi | Y = yk) = N(μik,σi), ß not N(μik,σik)
Consider three learning methods: • GNB (assumption 1 only) • GNB2 (assumption 1 and 2)
• LR
Which method works better if we have infinite training data, and… • Both (1) and (2) are satisfied
• Neither (1) nor (2) is satisfied
• (1) is satisfied, but not (2)
G.Naïve Bayes vs. Logistic Regression
[Ng & Jordan, 2002]
Recall two assumptions deriving form of LR from GNBayes: 1. Xi conditionally independent of Xk given Y
2. P(Xi | Y = yk) = N(μik,σi), ß not N(μik,σik)
Consider three learning methods:
• GNB (assumption 1 only) — decision surface can be non-linear • GNB2 (assumption 1 and 2) – decision surface linear
• LR — decision surface linear, trained differently
Which method works better if we have infinite training data, and…
• Both (1) and (2) are satisfied: • Neither (1) nor (2) is satisfied: • (1) is satisfied, but not (2) :
LR = GNB2 = GNB
LR > GNB2, GNB>GNB2
GNB > LR, LR > GNB2
G.Naïve Bayes vs. Logistic Regression
[Ng & Jordan, 2002]
What if we have only finite training data?
They converge at different rates to their asymptotic (∞ data) error
Let refer to expected error of learning algorithm A after n training examples
Let d be the number of features:
So, GNB requires n = O(log d) to converge, but LR requires n = O(d)
Some experiments from UCI data sets
[Ng & Jordan, 2002]
Naïve Bayes vs. Logistic Regression
The bottom line:
GNB2 and LR both use linear decision surfaces, GNB need not
Given infinite data, LR is better than GNB2 because training procedure does not make assumptions 1 or 2 (though our derivation of the form of P(Y|X) did).
But GNB2 converges more quickly to its perhaps-less-accurate asymptotic error
And GNB is both more biased (assumption1) and less (no assumption 2) than LR, so either might beat the other
What you should know:
•Logistic regression
–Functional form follows from Naïve Bayes assumptions •For Gaussian Naïve Bayes assuming variance σi,k = σi
•For discrete-valued Naïve Bayes too
–But training procedure picks parameters without the conditional independence assumption
–MCLE training: pick W to maximize P(Y | X, W)
–MAP training: pick W to maximize P(W | X,Y) •regularization: e.g., P(W) ~ N(0,σ)
•helps reduce overfitting
•Gradient ascent/descent
–General approach when closed-form solutions for MLE, MAP are
unavailable
•Generative vs. Discriminative classifiers –Bias vs. variance tradeoff
Regression
So far, we’ve been interested in learning P(Y|X) where Y has discrete values (called ‘classification’)
What if Y is continuous? (called ‘regression’)
•predict weight from gender, height, age, …
•predict Google stock price today from Google, Yahoo, MSFT prices yesterday
•predict each pixel intensity in robot’s current camera image, from previous image and previous action
Regression
Wish to learn f:XàY, where Y is real, given {
Approach:
1.choose some parameterized form for P(Y|X; θ) ( θ is the vector of parameters)
2.derive learning algorithm as MCLE or MAP estimate for θ
1. Choose parameterized form for P(Y|X; θ)
Y
X
Assume Y is some deterministic f(X), plus random noise
where
Therefore Y is a random variable that follows the distribution and the expected value of y for any given x is f(x)
Consider Linear Regression
E.g., assume f(x) is linear function of x
Notation: to make our parameters explicit, let’s write
Training Linear Regression
How can we learn W from the training data?
Training Linear Regression
How can we learn W from the training data?
Learn Maximum Conditional Likelihood Estimate!
where
Training Linear Regression
Learn Maximum Conditional Likelihood Estimate where
Training Linear Regression
Learn Maximum Conditional Likelihood Estimate where
so:
Training Linear Regression
Learn Maximum Conditional Likelihood Estimate Can we derive gradient descent rule for training?
How about MAP instead of MLE estimate?
Regression – What you should know
Under general assumption
1.MLE corresponds to minimizing sum of squared prediction errors
2.MAP estimate minimizes SSE plus sum of squared weights
3.Again, learning is an optimization problem once we choose our objective function
•maximize data likelihood
•maximize posterior prob of W
4.Again, we can use gradient descent as a general learning algorithm
•as long as our objective fn is differentiable wrt W
•though we might learn local optima ins
5.Almost nothing we said here required that f(x) be linear in x
Bias/Variance Decomposition of Error
Bias and Variance
given some estimator Y for some parameter θ, we define
the bias of estimator Y =
the variance of estimator Y =
e.g., define Y as the MLE estimator for probability of heads, based on n independent coin flips
biased or unbiased?
variance decreases as sqrt(1/n)
Bias – Variance decomposition of error
Reading: Bishop chapter 9.1, 9.2
•Consider simple regression problem f:XàY y = f(x) + ε
noise N(0,σ) deterministic
What are sources of prediction error?
learned estimate of f(x)
Sources of error
•What if we have perfect learner, infinite data?
– Our learned h(x) satisfies h(x)=f(x) – Still have remaining, unavoidable error
σ2
Sources of error
•What if we have only n training examples? •What is our expected error
– Taken over random training sets of size n, drawn from distribution D=p(x,y)
Sources of error
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