Test Errors

How you can tell that a hypothesis overfits?

  • plotting - not always good

We can split all the data into 2 subsets

  • training set $\approx$ 70% of data, $m$ - number of examples in the training set
  • testing set $\approx$ 30% of data, $m_{\text{test}}$ - number of examples in the testing set

it's better to choose examples for training/testing sets randomly


Error Metrics
Prediction Classification
Example Model Linear Regression Logistic Regression
Test Error $J_{\text{test}}(\theta) = \cfrac{1}{m_{\text{test}}} \sum \text{error} \big(h_{\theta}(x_{\text{test}}^{(i)}), y_{\text{test}}^{(i)} \big)$
$\text{error}(h_{\theta}(x), y)$ Average Square Error
$\text{error}(h_{\theta}(x), y) = \cfrac{1}{2} (h_{\theta}(x) - y)^2$
Misclassification Error
$\text{error}(h_{\theta}(x), y) = \left\{\begin{array}{l} 0 \text { if classification is correct} \\ 1 \text{ otherwise} \end{array}\right.$


Cross-Validation

Generally cross-validation is used to find the best value of some parameter

  • we still have training and test sets
  • but additionally we have a cross-validation set to test the performance of our model depending on the parameter


Motivation: Model Selection Problem

We have a following problem of Model Selection

Suppose we are about to create a model and not sure what degree of polynomial to choose

So the problem

  • So we want to try 10 models
  • let $d$ denote the degree of polynomial
  • $d=1: h_{\theta} = \theta_0 + \theta_1 x$
  • $d=2: h_{\theta} = \theta_0 + \theta_1 x + \theta_2 x^2$
  • ...
  • $d=10: h_{\theta} = \theta_0 + \theta_1 x + \theta_2 x^2 + ... + \theta_{10} x^{10}$

Results

  • Each $d$ will give us a vector (or matrix) $\theta^{(d)}$
  • Now we can try all $d$ models and see which gives the best (lowest) $J_{\text{test}}(\theta^{(i)})$
  • Let's say we decided to choose 5th model


How well this model generalize?

  • the test error is $J_{\text{test}}(\theta^{(5)})$
  • but $J_{\text{test}}(\theta^{(5)})$ is a very optimistic estimate of the generalization error
    • (because the lowest/best error was picked up - so the test error might be biased towards that)
  • i.e. we the extra parameter $d$ was fit to test, and it's not fair to estimate our hypothesis on the test set, because we already used that set to get the best $d$


To address that problem

  • instead of splitting the data set into 2 categories, we split into 3 sets:
  • training set ($\approx$ 60%)
    • $x^{(i)}, y^{(i)}$, total $m$ examples
  • cross-validation set (or cv, $\approx$ 20%)
    • $x_{\text{cv}}^{(i)}, y_{\text{cv}}^{(i)}$, total $m_{\text{cv}}$ examples
  • test set ($\approx$ 20%)
    • $x_{\text{test}}^{(i)}, y_{\text{test}}^{(i)}$, total $m_{\text{test}}$ examples


Now we can define

  • Training error
$J(\theta) = J_{\text{train}}(\theta) = \cfrac{1}{2m} \sum \text{cost}(x^{(i)}, y^{(i)})$
  • Cross-Validation error
$J_{\text{cv}}(\theta) = \cfrac{1}{2m_{\text{cv}}} \sum \text{cost}(x_{\text{cv}}^{(i)}, y_{\text{cv}}^{(i)})$
  • Test Error
$J_{\text{test}}(\theta) = \cfrac{1}{2m_{\text{test}}} \sum \text{cost}(x_{\text{test}}^{(i)}, y_{\text{test}}^{(i)})$


So for Model Selection to fit $d$, we

  • obtain $\theta^{(1)}, ..., \theta^{(d)}$ and select best (lowest) $J_{\text{cv}}(\theta^{(i)})$
  • estimate generalization error for the test set $J_{\text{test}}(\theta^{(i)})$


Tuning Learning Parameter

General algorithm

  • Split data set into
    • Learning set
    • Validation set
    • Test set
  • use validation set for tuning control parameters
  • use test set only for final evaluation

Let $\gamma$ be the control parameter

  • for every possible value of $\gamma$
    • build a classification model $C$ using the learning set
    • use the validation set to estimate the expected error rate of $C$
  • select the optimal value of the control parameter
    • that is the value of $\gamma$ s.t. the error rate is minimal


Regularization Parameter $\lambda$

Suppose now we're fitting a model with high-order polynomial

  • $h_{\theta}(x) = \theta_0 + \theta_1 x + ... + \theta_4 x^4$
  • to prevent overfitting we use regularization
  • $J(\theta) = \cfrac{1}{m} \sum \text{cost}(h_{\theta}(x^{(i)}), y^{(i)}) + \cfrac{\lambda}{2m} \sum_{j=1}^{m} \theta_j^2$


diagnosis-regularization.png

  • (a) If $\lambda$ if large (say $\lambda = 10000$), all $\theta$ are penalized and $\theta_1 \approx \theta_2 \approx ... \approx 0$, $h_{\theta}(x) \approx \theta_0$
  • (b) if $\lambda$ is intermediate, we fit well
  • (c) if $\lambda$ is small (close to 0) we fit too well, i.e. we overfit


How can we chose good $\lambda$? Let's define

  • $J_{\text{train}}(\theta) = \cfrac{1}{m} \sum \text{cost}(h_{\theta}(x^{(i)}), y^{(i)})$ (same as $J(\theta)$, but without regularization)
  • $J_{\text{cv}}(\theta)$ and $J_{\text{test}}(\theta)$ - same, but on cross-validation and test datasets respectively

Now we

  • Choose a range of possible values for $\lambda$ (say 0, 0.01, 0.02, 0.04, ..., 10.24) - that gives us 12 models to checks
  • For each $\lambda^{(i)}$,
    • calculate $\theta^{(i)}$,
    • calculate $J_{\text{cv}}(\theta^{(i)})$,
    • and take $\lambda^{(i)}$ with lowest $J_{\text{cv}}(\theta^{(i)})$


After that we report the test error $J_{\text{test}}(\theta^{(i)})$


K-Fold Cross-Validation

$K$-Fold Cross-Validation If we want to reduce variability in the data

  • we can perform multiple rounds of cross-validation using different partitions
  • and then average the results over all the rounds

We're given a dataset $S$ sampled from the population $D$

  • we partition $S$ into $K$ equal disjoint subsets $(T_1, ..., T_K)$ (typically 5-10 subsets)
  • then perform $K$ steps, and at step $k$ do:
    • use $R_k = S - T_k$ as the training set
    • build classifier $C_k$ using $R_k$
    • use $T_k$ as the test test, compute error $\delta_k = \text{error}(C_k, T_k)$
  • let $\delta^* = \cfrac{1}{K} \sum_{k = 1}^K \delta_k$
    • this is the expected error rate

k-fold-cv.png

  • note that there's only two subsets at each iteration
    • and they are used to estimate the true error


Tuning Learning Parameter

Choosing best value for parameter $\gamma$ with $K$-Fold Cross-Validation

  • you have to put some data aside before training your classifier
    • i.e. split your data into learning set and test set
  • for every possible value of $\gamma$
    • use $K$-fold Cross-Validation to estimate the expected error rate
    • use the learning set as the set $S$ (and do $K$-fold Cross-Validation only on it)
  • select the optimal value of $\gamma$
  • and then use the test set for the final evaluation


Stratified K-Fold Cross-Validation

What if we want to preserve the class distribution over $K$ runs?

  • then for each $T_i$ pick up the same proportion of labels as in the original dataset
  • k-fold-cv-strat.png
  • it's very important if the test distribution is not uniform


See Also

Sources

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