使用许多多项式的梯度下降不会收敛

Question

context: I am trying to create a generic function to optimize the cost of any regression problem using polynomial regression (of any specified degree). I am trying to fit my model to the load_boston dataset (with the house price as the label and 13 features).

I used multiple degrees of polynomials, and multiple learning rates and epochs (with gradient descent) and the MSE is coming out to be so high even on the training dataset (I am using 100% of the data to train the model, and I am checking the cost on the same data, but the MSE cost is still very high).

import tensorflow as tf
from sklearn.datasets import load_boston

def polynomial(x, coeffs):
    y = 0
    for i in range(len(coeffs)):
        y += coeffs[i]*x**i
    return y

def initial_parameters(dimensions, data_type, degree): # list number of dims/features and degree
    thetas = [tf.Variable(0, dtype=data_type)] # the constant theta/bias
    for i in range(degree):
        thetas.append(tf.Variable( tf.zeros([dimensions, 1], dtype=data_type)))
    return thetas

def regression_error(x, y, thetas):
    hx = thetas[0] # constant thetas - no need to have 1 for each variable (e.g x^0*th + y^0*th...)
    for i in range(1, len(thetas)):
        hx = tf.add(hx, tf.matmul( tf.pow(x, i), thetas[i]))
    return tf.reduce_mean(tf.squared_difference(hx, y))

def polynomial_regression(x, y, data_type, degree, learning_rate, epoch): #features=dimensions=variables
    thetas = initial_parameters(x.shape[1], data_type, degree)
    cost = regression_error(x, y, thetas)
    init = tf.initialize_all_variables()
    optimizer = tf.train.GradientDescentOptimizer(learning_rate).minimize(cost)
    with tf.Session() as sess:
        sess.run(init)
        for epoch in range(epoch): 
            sess.run(optimizer)
        return cost.eval()

x, y = load_boston(True) # yes just use the entire dataset
for deg in range(1, 2):
    for lr in range(-8, -5):
        error = polynomial_regression(x, y, tf.float64, deg, 10**lr, 100 )
        print (deg, lr, error)

It outputs 97.3 even though most of the labels are around 30 (degree = 1, learning rate = 10^-6). what is wrong with the code?

Answer 1

The problem is that the different features are on different orders of magnitude and hence are not compatible with the learning rate which is the same for all features. Even more, when using a non-zero variable initialization, one has to make sure that these initial values are as well compatible with the feature values.

In [1]: from sklearn.datasets import load_boston

In [2]: x, y = load_boston(True)

In [3]: x.std(axis=0)
Out[3]: 
array([8.58828355e+00, 2.32993957e+01, 6.85357058e+00, 2.53742935e-01,
       1.15763115e-01, 7.01922514e-01, 2.81210326e+01, 2.10362836e+00,
       8.69865112e+00, 1.68370495e+02, 2.16280519e+00, 9.12046075e+01,
       7.13400164e+00])

In [4]: x.mean(axis=0)
Out[4]: 
array([3.59376071e+00, 1.13636364e+01, 1.11367787e+01, 6.91699605e-02,
       5.54695059e-01, 6.28463439e+00, 6.85749012e+01, 3.79504269e+00,
       9.54940711e+00, 4.08237154e+02, 1.84555336e+01, 3.56674032e+02,
       1.26530632e+01])

A common approach is to normalize the input data (eg to have zero mean and unit variance) and to choose the initial weights randomly (eg normal distribution, std.dev. = 1). sklearn.preprocessing offers various functionality for these cases.

• PolynomialFeatures can be used to generate the polynomial features automatically.
• StandardScaler scales the data to zero mean and unit variance.
• pipeline.Pipeline can be used for convenience to combine these preprocessing steps.

The polynomial_regression function then reduces to:

pipeline = Pipeline([
    ('poly', PolynomialFeatures(degree)),
    ('scaler', StandardScaler())
])
x = pipeline.fit_transform(x)
thetas = tf.Variable(tf.random_normal([x.shape[1], 1], dtype=data_type))
cost = tf.reduce_mean(tf.squared_difference(tf.matmul(x, thetas), y))

# Perform variable initialization and optimizer instantiation here.
# Run optimization over epochs.