Learning Curves

This is a good technique (a part of Machine Learning Diagnosis)

  • to sanity-check a model
  • to improve performance

A learning curve is a plot where we have two functions of $m$ ($m$ is a set size):

  • training set error $J_{\text{train}}(\theta)$,
  • the cross-validation error $J_{\text{cv}}(\theta)$


We can artificially reduce our training set size.

  • We start from $m = 1$, then $m = 2$ and so on

So suppose we have the following model:

  • $h_{\theta}(x) = \theta_0 + \theta_1 x + \theta_2 x^2$
  • learning-curves-trainingset-red.png
  • for each $m$ we calculate $J_{\text{train}}(\theta)$ and $J_{\text{cv}}(\theta)$ and plot the values
  • learning-curves-grow.png
  • This is the learning curve of the model


Diagnose High Bias (Underfitting)

Suppose we want to fit a straight line to out data:

$h_{\theta}(x) = \theta_0 + \theta_1 x$

As $m$ increases we have pretty same line:

learning-curves-lin.png

If we draw the learning curves, we'll have

learning-curves-lin2.png

So we see that

  • as $m$ grows $J_{\text{cv}}(\theta) \to J_{\text{train}}(\theta)$
  • and both errors are high

$\Rightarrow$ If learning algorithm is suffering from high bias, getting more examples will not help

Diagnose High Variance (Overfitting)

Now suppose we have a model with polynomial of very high order:

$h_{\theta}(x) = \theta_0 + \theta_1 x + \theta_2 x^2 + ... + \theta_{100} x^{100}$

learning-curves-overfit1.png

  • at the beginning we very much overfit
  • as we increase $m$, we still able to fit the data well


learning-curves-overfit2.png

So we can see that as $m$ increases,

  • $J_{\text{train}}(\theta)$ increases (we have more and more data - so it's harder and harder to fit $h_{\theta}(x)$), but it increases very slowly
  • on the other hand, $J_{\text{cv}}(\theta)$ decreases, but also very very slow
  • and there's a huge gap between these 2
  • to fill that gap we need many many more training examples

$\Rightarrow$ if a learning algorithm is suffering from high variance (i.e. it overfits), getting more data is likely to help



See also


Sources

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