Ridge regression using stochastic gradient descent in Python

Question

I am trying to implement a solution to Ridge regression in Python using Stochastic gradient descent as the solver. My code for SGD is as follows:

def fit(self, X, Y):
    # Convert to data frame in case X is numpy matrix
    X = pd.DataFrame(X)

    # Prepend a column of 1s to the data for the intercept
    X.insert(0, 'intercept', np.array([1.0]*X.shape[0]))

    # Find dimensions of train
    m, d = X.shape

    # Initialize weights to random
    beta = self.initializeRandomWeights(d)
    beta_prev = None

    epochs = 0
    prev_error = None
    while (epochs < self.nb_epochs):
        print("## Epoch: " + str(epochs))
        indices = range(0, m)
        shuffle(indices)
        for i in indices:   # Pick a training example from a randomly shuffled set
            beta_prev = beta
            xi = X.iloc[i]
            errori = sum(beta*xi) - Y[i]        # Error[i] = sum(beta*x) - y = error of ith training example
            gradient_vector = xi*errori + self.l*beta_prev
            beta = beta_prev - self.alpha*gradient_vector
        epochs += 1

The data I'm testing this on is not normalized and my implementation always ends up with all the weights being Infinity, even though I initialize the weights vector to low values. Only when I set the learning rate alpha to a very small value ~1e-8, the algorithm ends up with valid values of the weights vector.

My understanding is that normalizing/scaling input features only helps reduce convergence time. But the algorithm should not fail to converge as a whole if the features are not normalized. Is my understanding correct?