Now according to wiki, I don't need to go through all data points and I can stop when error is small enough. Is it true?
This is especially true when you have a really huge training set and going through all the data points is so expensive. Then, you would check the convergence criterion after K stochastic updates (i.e. after processing K training examples). While it's possible, it doesn't make much sense to do this with a small training set. Another thing people do is randomizing the order in which training examples are processed to avoid having too many correlated examples in a raw which may result in "fake" convergence.
I don't understand what should be the stopping criterion here. If anyone can help with this that would be great.
There are a few options. I recommend trying as many of them and deciding based on empirical results.
- difference in the objective function for the training data is smaller than a threshold.
- difference in the objective function for held-out data (aka. development data, validation data) is smaller than a threshold. The held-out examples should NOT include any of the examples used for training (i.e. for stochastic updates) nor include any of the examples in the test set used for evaluation.
- the total absolute difference in parameters w is smaller than a threshold.
- in 1, 2, and 3 above, instead of specifying a threshold, you could specify a percentage. For example, a reasonable stopping criterion is to stop training when |squared_error(w) - squared_error(previous_w)| < 0.01 * squared_error(previous_w) $$.
- sometimes, we don't care if we have the optimal parameters. We just want to improve the parameters we originally had. In such case, it's reasonable to preset a number of iterations over the training data and stop after that regardless of whether the objective function actually converged.
With this formula - which I used in for loop - is it correct? I believe (w.x_i - y_i) * x_t is my ∆Q(w).
It should be 2 * (w.x_i - y_i) * x_t but it's not a big deal given that you're multiplying by the learning rate alpha anyway.