Modeling Social Data, Lecture 8: Regression, Part 2

Transcript

1. Model complexity and generalization
   APAM E4990
   Modeling Social Data
   Jake Hofman
   Columbia University
   March 15, 2019
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2. Overfitting (a la xkcd)
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3. Overfitting (a la xkcd)
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4. Complexity
   Our models should be complex enough to explain the past, but
   simple enough to generalize to the future
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5. Bias-variance tradeoff
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6. Bias-variance tradeoff
   38 2. Overview of Supervised Learning
   High Bias
   Low Variance
   Low Bias
   High Variance
   Prediction Error
   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)
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7. Bias-variance tradeoff
   38 2. Overview of Supervised Learning
   High Bias
   Low Variance
   Low Bias
   High Variance
   Prediction Error
   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)
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8. Bigger models = Better models
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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:
   Test
   Train Validation Test
   Train Validation Test
   Validation
   Train Validation Test
   Train
   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
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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
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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
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