Recently, I implemented the LR algorithm in Python. The main part of the code is as following(I didn't use mini batch in my code. Instead, I use the whole batch to compute gradients every time):
class Logistic(): def __init__(self): self.w = None self.lr = 10. pass def train(self, xs, ys): m, n = xs.shape ones = np.ones([m, 1]) xs = np.hstack([xs, ones]) ys = np.expand_dims(ys, -1) self.w = np.ones([n+1, 1], dtype=np.float64) * 1.0 epochs = 100 for epoch in xrange(epochs): y_ = self.sigmoid(-np.dot(xs, self.w)) # loss = -1.0/m * np.sum(ys * np.log(y_) + (1 - ys) * np.log(1 - y_)) tmp1 = np.sum(np.log(y_[np.where(ys==1)])) tmp2 = np.sum(np.log(1 - y_[np.where(ys==0)])) loss = - (tmp1 + tmp2) / m print("epoch: %d, loss: %f" % (epoch, loss)) print("y_: %f, %f" % (np.min(y_), np.max(y_))) grad = np.sum((y_ - ys) * xs, axis=0) / m self.w -= self.lr * np.expand_dims(grad, -1) print("grad: %f, %f" % (np.min(grad), np.max(grad))) print("w: %f, %f" % (np.min(self.w), np.max(self.w))) print ""
The dataset I used is MNIST. I marked all digits 0 as class 0, and all other digits as class 1. Then I get this binary classification problem. I test my algorithm with many different learning rate, from 1e-6 to 10, and it turns out all of them produces good results(about 98% accuracy on test set). As far as I know, if the learning rate is to big, LR will not converge. But here although I used very big learning rate, the algorithm still converge to about 98% accuracy. Is there an explanation for this?