Implementing linear regression is always worse than sklearn

Question

I implemented linear regression with gradient descents in python. To see how good this is, I compared it with the LinearRegression () class scikit-learn. For some reason, sklearn always outperforms my MSE program out of 3 on average (I use the Boston Housing dataset for testing). I understand that I am not currently performing a gradient test to verify convergence, but I allow many iterations and set the learning speed low enough to SHOULD converge. Is there a clear mistake in my implementation of the learning algorithm? Here is my code:

import numpy as np
from sklearn.linear_model import LinearRegression

def getWeights(x):
    lenWeights = len(x[1,:]);
    weights = np.random.rand(lenWeights)
    bias = np.random.random();
    return weights,bias

def train(x,y,weights,bias,maxIter):
    converged = False;
    iterations = 1;
    m = len(x);
    alpha = 0.001;
    while not converged:
            for i in range(len(x)):
                # Dot product of weights and training sample
                hypothesis = np.dot(x[i,:], weights) + bias;
                # Calculate gradient
                error = hypothesis - y[i];
                grad = (alpha * 1/m) * ( error * x[i,:] );
                # Update weights and bias
                weights = weights - grad;
                bias = bias - alpha * error;
                iterations = iterations + 1;

                if iterations > maxIter:
                    converged = True;
                    break

    return weights, bias

def predict(x, weights, bias):
    return np.dot(x,weights) + bias

if __name__ == '__main__':

    data = np.loadtxt('housing.txt');
    x = data[:,:-1];
    y = data[:,-1];
    for i in range(len(x[1,:])):
        x[:,i] = ( (x[:,i] - np.min(x[:,i])) / (np.max(x[:,i]) - np.min(x[:,i])) );

    initialWeights,initialBias = getWeights(x);
    weights,bias = train(x,y,initialWeights,initialBias,55000);
    pred = predict(x, weights,bias);
    MSE = np.mean(abs(pred - y));

    print "This Program MSE: " + str(MSE)

    sklearnModel = LinearRegression();
    sklearnModel = sklearnModel.fit(x,y);
    sklearnModel = sklearnModel.predict(x);

    skMSE = np.mean(abs(sklearnModel - y));

    print "Sklearn MSE: " + str(skMSE)

Answer 1

, . .5*np.mean((pred-y)**2), np.mean(abs(pred - y)).

(SGD) ( ), " ".

SGD - , - , (http://leon.bottou.org/publications/pdf/nips-2007.pdf).

SGD . , , , - alpha/(iterations+1), .

1/m, SGD.

SGD, , , . / ( ), ( ).

Answer 2

. , , , . , l-bfgs, , SGD.

.

http://eli.thegreenplace.net/2014/derivation-of-the-normal-equation-for-linear-regression/.