神经网络中的SciPy优化警告

Question

我使用SciPy fmin_bfgs()优化接收下一个警告NeuralNetwork.根据Backpropagation算法,一切都应该清晰简单.

1前馈培训示例.
2计算每个单元的误差项.
3累积梯度(对于第一个例子,我正在跳过正则化项).

Starting Loss: 7.26524579601
Check gradient: 2.02493576268
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 5.741300
         Iterations: 3
         Function evaluations: 104
         Gradient evaluations: 92
Trained Loss: 5.74130012926

我刚刚用MATLAB完成了同样的任务,它成功执行fmin了优化功能,但无法理解,我在Python实现中错过了什么.正如你所看到的,甚至scipy.optimize.check_grad回报太大的价值.

def feed_forward(x, theta1, theta2):
    hidden_dot = np.dot(add_bias(x), np.transpose(theta1))
    hidden_p = sigmoid(hidden_dot)

    p = sigmoid(np.dot(add_bias(hidden_p), np.transpose(theta2)))
    return hidden_dot, hidden_p, p


def cost(thetas, x, y, hidden, lam):
    theta1, theta2 = get_theta_from(thetas, x, y, hidden)
    _, _, p = feed_forward(x, theta1, theta2)

    # regularization = (lam / (len(x) * 2)) * (
    #     np.sum(np.square(np.delete(theta1, 0, 1)))
    #     + np.sum(np.square(np.delete(theta2, 0, 1))))

    complete = -1 * np.dot(np.transpose(y), np.log(p)) \
               - np.dot(np.transpose(1 - y), np.log(1 - p))
    return np.sum(complete) / len(x)  # + regularization


def vector(z):
    # noinspection PyUnresolvedReferences
    return np.reshape(z, (np.shape(z)[0], 1))



def gradient(thetas, x, y, hidden, lam):
    theta1, theta2 = get_theta_from(thetas, x, y, hidden)
    hidden_dot, hidden_p, p = feed_forward(x, theta1, theta2)

    error_o = p - y
    error_h = np.multiply(np.dot(
        error_o, np.delete(theta2, 0, 1)), sigmoid_gradient(hidden_dot))

    x = add_bias(x)
    hidden_p = add_bias(hidden_p)

    theta1_grad, theta2_grad = \
        np.zeros(theta1.shape[::-1]), np.zeros(theta2.shape[::-1])
    records = y.shape[0]

    for i in range(records):
        theta1_grad = theta1_grad + np.dot(
            vector(x[i]), np.transpose(vector(error_h[i])))
        theta2_grad = theta2_grad + np.dot(
            vector(hidden_p[i]), np.transpose(vector(error_o[i])))

    theta1_grad = np.transpose(
        theta1_grad / records)  # + (lam / records * theta1)

    theta2_grad = np.transpose(
        theta2_grad / records)  # + (lam / records * theta2)

    return np.append(theta1_grad, theta2_grad)


def get_theta_shapes(x, y, hidden):
    return (hidden, x.shape[1] + 1), \
           (y.shape[1], hidden + 1)


def get_theta_from(thetas, x, y, hidden):
    t1_s, t2_s = get_theta_shapes(x, y, hidden)
    split = t1_s[0] * t1_s[1]

    theta1 = np.reshape(thetas[:split], t1_s)
    theta2 = np.reshape(thetas[split:], t2_s)
    return theta1, theta2



def train(x, y, hidden_size, lam):
    y = get_binary_y(y)

    t1_s, t2_s = get_theta_shapes(x, y, hidden_size)
    thetas = np.append(
        rand_init(t1_s[0], t1_s[1]),
        rand_init(t2_s[0], t2_s[1]))

    initial_cost = cost(thetas, x, y, hidden_size, lam)
    print("Starting Loss: " + str(initial_cost))

    check_grad1 = scipy.optimize.check_grad(
        cost, gradient, thetas, x, y, hidden_size, lam)
    print("Check gradient: " + str(check_grad1))

    trained_theta = scipy.optimize.fmin_bfgs(
        cost, thetas, fprime=gradient, args=(x, y, hidden_size, lam))

    print("Trained Loss: " +
          str(cost(trained_theta, x, y, hidden_size, lam)))

Answer 1

同样，计算存在几个问题，需要解决所有警告并使Scipy优化运行成功，并且与优化功能相同Matlab fminc。（工作示例，您可以在GithubPython上找到）

1.将计算成本更新为正确的成本。在成本函数中按元素相乘。成本的正确解决方案将是（带有正则化项）：

def cost(thetas, x, y, hidden, lam):
    theta1, theta2 = get_theta_from(thetas, x, y, hidden)
    _, _, p = feed_forward(x, theta1, theta2)

    regularization = (lam / (len(x) * 2)) * (
        np.sum(np.square(np.delete(theta1, 0, 1)))
        + np.sum(np.square(np.delete(theta2, 0, 1))))

    complete = np.nan_to_num(np.multiply((-y), np.log(
        p)) - np.multiply((1 - y), np.log(1 - p)))
    avg = np.sum(complete) / len(x)
    return avg + regularization

2.执行此操作后，我们将从上面执行的本例 中收到nan优化Theta项中的值。笔记！这对于意外数字是正确的。Scipynp.nan_to_numMatlab fminc

3.应用正确的正则化，并且不要忘记删除偏差值的正则化。正确的梯度函数应该是这样的：

def gradient(thetas, x, y, hidden, lam):
    theta1, theta2 = get_theta_from(thetas, x, y, hidden)
    hidden_dot, hidden_p, p = feed_forward(x, theta1, theta2)

    error_o = p - y
    error_h = np.multiply(np.dot(
        error_o, theta2),
        sigmoid_gradient(add_bias(hidden_dot)))

    x = add_bias(x)
    error_h = np.delete(error_h, 0, 1)

    theta1_grad, theta2_grad = \
        np.zeros(theta1.shape[::-1]), np.zeros(theta2.shape[::-1])
    records = y.shape[0]

    for i in range(records):
        theta1_grad = theta1_grad + np.dot(
            vector(x[i]), np.transpose(vector(error_h[i])))
        theta2_grad = theta2_grad + np.dot(
            vector(hidden_p[i]), np.transpose(vector(error_o[i])))

    reg_theta1 = theta1.copy()
    reg_theta1[:, 0] = 0

    theta1_grad = np.transpose(
        theta1_grad / records) + ((lam / records) * reg_theta1)

    reg_theta2 = theta2.copy()
    reg_theta2[:, 0] = 0

    theta2_grad = np.transpose(
        theta2_grad / records) + ((lam / records) * reg_theta2)

    return np.append(
        theta1_grad, theta2_grad)