機器學習：從零開始學習梯度下降

coeffs = [2,-5, 4]

def eval_2nd_degree(coeffs, x):    """    Function to return the outputof evaluating a second degree polynomial,    given a specific x value.
    Args:        coeffs: List containingthe coefficients a,b, and c for the polynomial.        x: The input x value tothe polynomial.
    Returns:        y: The correspondingoutput y value for the second degree polynomial.
    """    a = (coeffs[0]*(x*x))    b = coeffs[1]*x    c = coeffs[2]    y = a+b+c    return y

coeffs = [2, -5, 4]x=3eval_2nd_degree(coeffs, x)
7

import numpy as npimport matplotlib.pyplot as plt
hundred_xs=np.random.uniform(-10,10,100)print(hundred_xs)
x_y_pairs = []for x in hundred_xs:    y  =eval_2nd_degree(coeffs, x)    x_y_pairs.append((x,y))
xs = []ys = []for a,b in x_y_pairs:    xs.append(a)    ys.append(b)
plt.figure(figsize=(20,10))plt.plot(xs, ys, 'g+')plt.title('Original data')plt.show()

defeval_2nd_degree_jitter(coeffs, x, j):    """    Function to return the noisy output ofevaluating a second degree polynomial,    given a specific x value. Output values canbe within [y?j,y+j].
    Args:        coeffs: List containing thecoefficients a,b, and c for the polynomial.        x: The input x value to the polynomial.        j: Jitter parameter, to introduce noiseto output y.
    Returns:        y: The corresponding jittered output yvalue for the second degree polynomial.
    """    a = (coeffs[0]*(x*x))    b = coeffs[1]*x    c = coeffs[2]    y = a+b+c    print(y)
    interval = [y-j, y+j]    interval_min = interval[0]    interval_max = interval[1]    print(f"Should get value in the range{interval_min} - {interval_max}")    jit_val = random.random() *interval_max      # Generate a randomnumber in range 0 to interval max
    while interval_min > jit_val:                 # While the random jittervalue is less than the interval min,        jit_val = random.random() *interval_max  # it is not in the rightrange. Re-roll the generator until it                                                 # give a number greater than the interval min.
    return jit_val

7Should get value in the range 3 - 116.233537936801398

x_y_pairs = []for x in hundred_xs:    y  =eval_2nd_degree_jitter(coeffs, x, j)    x_y_pairs.append((x,y))
xs = []ys = []for a,b in x_y_pairs:    xs.append(a)    ys.append(b)
plt.figure(figsize=(20,10))plt.plot(xs, ys, 'g+')plt.title('Original data')plt.show()

rand_coeffs=(random.randrange(-10,10),random.randrange(-10,10),random.randrange(-10,10))rand_coeffs
(7, 6, 3)

y_bar =eval_2nd_degree(rand_coeffs, hundred_xs)
plt.figure(figsize=(20,10))plt.plot(xs, ys, 'g+', label ='original')plt.plot(xs, y_bar, 'ro',label='prediction')plt.title('Original data vsfirst prediction')plt.legend(loc="lowerright")plt.show()

def loss_mse(ys, y_bar):    """    Calculates MSE loss.
    Args:        ys: training data labels        y_bar: prediction labels
    Returns: Calculated MSE loss.    """    return sum((ys - y_bar)*(ys - y_bar)) /len(ys)
  initial_model_loss = loss_mse(ys, y_bar)
  initial_model_loss
47922.39790821987

defcalc_gradient_2nd_poly(rand_coeffs, hundred_xs, ys):    """    calculates the gradient for a second degreepolynomial.
    Args:        coeffs: a,b and c, for a 2nd degreepolynomial [ y = ax^2 + bx + c ]        inputs_x: x input datapoints        outputs_y: actual y output points
    Returns: Calculated gradients for the 2nddegree polynomial, as a tuple of its parts for a,b,c respectively.
    """
    a_s = []    b_s = []    c_s = []

    y_bars = eval_2nd_degree(rand_coeffs,hundred_xs)
    for x,y,y_bar in list(zip(hundred_xs, ys,y_bars)):    # take tuple of (xdatapoint, actual y label, predicted y label)        x_squared = x**2        partial_a = x_squared * (y - y_bar)        a_s.append(partial_a)        partial_b = x * (y-y_bar)        b_s.append(partial_b)        partial_c = (y-y_bar)        c_s.append(partial_c)
    num = [i for i in y_bars]    n = len(num)
    gradient_a = (-2 / n) * sum(a_s)    gradient_b = (-2 / n) * sum(b_s)    gradient_c = (-2 / n) * sum(c_s)    return(gradient_a, gradient_b,gradient_c)   # return calculatedgradients as a a tuple of its 3 parts

calc_grad= calc_gradient_2nd_poly(rand_coeffs, hundred_xs, ys)
lr =0.0001a_new= rand_coeffs[0] - lr * calc_grad[0]b_new= rand_coeffs[1] - lr * calc_grad[1]c_new= rand_coeffs[2] - lr * calc_grad[2]
new_model_coeffs= (a_new, b_new, c_new)print(f"Newmodel coeffs: {new_model_coeffs}")print("")
#updatewith these new coeffs:new_y_bar= eval_2nd_degree(new_model_coeffs, hundred_xs)updated_model_loss= loss_mse(ys, new_y_bar)
print(f"Nowhave smaller model loss: {updated_model_loss} vs {original_model_loss}")

New model coeffs: 5.290395171471687 5.903335222089396 2.9704266522693037Now have smaller model loss: 23402.14716735533 vs 47922.39790821987

plt.figure(figsize=(20,10))plt.plot(xs, ys, 'g+', label ='original model')plt.plot(xs, y_bar, 'ro', label= 'first prediction')plt.plot(xs, new_y_bar, 'b.',label = 'updated prediction')plt.title('Original model vs1st prediction vs updated prediction with lower loss')plt.legend(loc="lower right")plt.show()

defcalc_gradient_2nd_poly_for_GD(coeffs, inputs_x, outputs_y, lr):    """    calculates the gradient for a second degreepolynomial.
    Args:        coeffs: a,b and c, for a 2nd degreepolynomial [ y = ax^2 + bx + c ]        inputs_x: x input datapoints        outputs_y: actual y output points        lr: learning rate
    Returns: Calculated gradients for the 2nddegree polynomial, as a tuple of its parts for a,b,c respectively.
    """    a_s = []    b_s = []    c_s = []
    y_bars = eval_2nd_degree(coeffs, inputs_x)
    for x,y,y_bar in list(zip(inputs_x,outputs_y, y_bars)):    # take tuple of(x datapoint, actual y label, predicted y label)        x_squared = x**2               partial_a = x_squared * (y - y_bar)        a_s.append(partial_a)        partial_b = x * (y-y_bar)        b_s.append(partial_b)        partial_c = (y-y_bar)        c_s.append(partial_c)
    num = [i for i in y_bars]    n = len(num)
    gradient_a = (-2 / n) * sum(a_s)    gradient_b = (-2 / n) * sum(b_s)    gradient_c = (-2 / n) * sum(c_s)

    a_new = coeffs[0] - lr * gradient_a    b_new = coeffs[1] - lr * gradient_b    c_new = coeffs[2] - lr * gradient_c
    new_model_coeffs = (a_new, b_new, c_new)
    #update with these new coeffs:    new_y_bar = eval_2nd_degree(new_model_coeffs,inputs_x)
    updated_model_loss = loss_mse(outputs_y,new_y_bar)    return updated_model_loss,new_model_coeffs, new_y_bar

def gradient_descent(epochs,lr):    """    Perform gradient descent for a seconddegree polynomial.
    Args:        epochs: number of iterations to performof finding new coefficients and updatingt loss.        lr: specified learning rate
    Returns: Tuple containing (updated_model_loss,new_model_coeffs, new_y_bar predictions, saved loss updates)
    """    losses = []    rand_coeffs_to_test = rand_coeffs    for i in range(epochs):        loss =calc_gradient_2nd_poly_for_GD(rand_coeffs_to_test, hundred_xs, ys, lr)       rand_coeffs_to_test = loss[1]        losses.append(loss[0])    print(losses)    return loss[0], loss[1], loss[2],losses  #(updated_model_loss,new_model_coeffs, new_y_bar, saved loss updates)

GD = gradient_descent(1500,0.0001)
plt.figure(figsize=(20,10))plt.plot(xs, ys, 'g+', label ='original')plt.plot(xs, GD[2], 'b.', label= 'final_prediction')plt.title('Original vs Finalprediction after Gradient Descent')plt.legend(loc="lowerright")plt.show()

print(f"FinalCoefficients predicted: {GD[1]}")print(f"OriginalCoefficients: {coeffs}")

plt.figure(figsize=(20,10))plt.plot(GD[3], 'b-', label ='loss')plt.title('Loss over 1500iterations')plt.legend(loc="lowerright")plt.xlabel('Iterations')plt.ylabel('MSE')plt.show()