http://modelingsocialdata.org

1. Model complexity and generalization
APAM E4990
Modeling Social Data
Jake Hofman
Columbia University
March 3, 2017
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 1 / 10

2. Overfitting (a la xkcd)
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 2 / 10

3. Overfitting (a la xkcd)
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 3 / 10

4. Complexity
Our models should be complex enough to explain the past, but
simple enough to generalize to the future
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 4 / 10

5. Bias-variance tradeoff
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 5 / 10

6. Bias-variance tradeoff
38 2. Overview of Supervised Learning
High Bias
Low Variance
Low Bias
High Variance
PredictionError
Model Complexity
Training Sample
Test Sample
Low High
FIGURE 2.11. Test and training error as a function of model complexity.
be close to f(x0). As k grows, the neighbors are further away, and then
anything can happen.
The variance term is simply the variance of an average here, and de-
creases as the inverse of k. So as k varies, there is a bias–variance tradeoff.
Simple models may be “wrong” (high bias), but fits don’t vary a
lot with different samples of training data (low variance)
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 6 / 10

7. Bias-variance tradeoff
38 2. Overview of Supervised Learning
High Bias
Low Variance
Low Bias
High Variance
PredictionError
Model Complexity
Training Sample
Test Sample
Low High
FIGURE 2.11. Test and training error as a function of model complexity.
be close to f(x0). As k grows, the neighbors are further away, and then
anything can happen.
The variance term is simply the variance of an average here, and de-
creases as the inverse of k. So as k varies, there is a bias–variance tradeoff.
Flexible models can capture more complex relationships (low bias),
but are also sensitive to noise in the training data (high variance)
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 6 / 10

8. Bigger models = Better models
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 7 / 10

9. Cross-validation
set error of the final chosen model will underestimate the true test error,
sometimes substantially.
It is difficult to give a general rule on how to choose the number of
observations in each of the three parts, as this depends on the signal-to-
noise ratio in the data and the training sample size. A typical split might
be 50% for training, and 25% each for validation and testing:
TestTrain Validation TestTrain Validation TestValidationTrain Validation TestTrain
The methods in this chapter are designed for situations where there is
insufficient data to split it into three parts. Again it is too difficult to give
a general rule on how much training data is enough; among other things,
this depends on the signal-to-noise ratio of the underlying function, and
the complexity of the models being fit to the data.
• Randomly split our data into three sets
• Fit models on the training set
• Use the validation set to find the best model
• Quote final performance of this model on the test set
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 8 / 10

10. K-fold cross-validation
Estimates of generalization error from one train / validation split
can be noisy, so shuffle data and average over K distinct validation
partitions instead
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 9 / 10

11. K-fold cross-validation: pseudocode
(randomly) divide the data into K parts
for each model
for each of the K folds
train on everything but one fold
measure the error on the held out fold
store the training and validation error
compute and store the average error across all folds
pick the model with the lowest average validation error
evaluate its performance on a final, held out test set
Jake Hofman (Columbia University) Model complexity and generalization March 3, 2017 10 / 10