我正在尝试使用MXNet的梯度下降优化器来最小化功能。 Tensorflow中的等效示例将是：

import tensorflow as tf

x = tf.Variable(2, name='x', dtype=tf.float32)
log_x = tf.log(x)
log_x_squared = tf.square(log_x)

optimizer = tf.train.GradientDescentOptimizer(0.5)
train = optimizer.minimize(log_x_squared)

init = tf.initialize_all_variables()

def optimize():
  with tf.Session() as session:
    session.run(init)
    print("starting at", "x:", session.run(x), "log(x)^2:", session.run(log_x_squared))
    for step in range(10):  
      session.run(train)
      print("step", step, "x:", session.run(x), "log(x)^2:", session.run(log_x_squared))

我不确定如何在MXNet中完成相同的任务。优化器API 文档似乎没有等效的方法。到目前为止，这是我尝试过的。主要的困惑在于需要传递训练数据：

import mxnet as mx

x = mx.sym.Variable('data')
log_x = mx.sym.log(x)
log_x_squared = mx.sym.square(log_x)

mod = mx.mod.Module(log_x_squared)  # Create a module where the loss function
                                    # is the one we want to optimize
mod.bind(data_shapes=[('data', (1,1))])  # ?? not sure if this is correct - we
                                         # are saying our input is a scalar
mod.init_params()
mod.init_optimizer()  # SGD is default

mod.fit()  # ?? must pass data_iter to fit

似乎应该以某种方式将x变量作为data_iter反馈回来，但我不知道该如何完成。

更新：感谢kevinthesun的出色回答！这是建立在单个隐藏层神经网络之上的工作最小化例程：

import mxnet as mx
import numpy as np


def minimize(objective_function,
             initial_params,
             max_iters=1000,
             optimizer='sgd',
             optimizer_params=(('learning_rate', 0.1),),
             tol=1e-8):

    class InitialParam(mx.init.Initializer):

        def __init__(self, vals):
            super(InitialParam, self).__init__()
            self._vals = vals

        def _init_weight(self, _, arr):
            arr[:] = self._vals.asnumpy()[:, np.newaxis]


    x = mx.sym.Variable('data')
    params_len = initial_params.shape[0]
    fc = mx.sym.FullyConnected(data=x, name='fc1',
                               num_hidden=params_len,
                               no_bias=True)

    # Passing the FullyConnected layer into the objective function
    # is difficult to manipulate. If the fully connected layer represents
    # [x, y] for optimizing a 2 dimensional function f(x, y) it is easier
    # to work with x, and y. So we split the fully connected layer into a
    # number of symbols for each parameter:
    param_syms = []
    for i in range(params_len):
        ps = mx.sym.slice(fc, begin=(0, i), end=(1, i + 1))
        param_syms.append(ps)

    # The loss function for the network is our objective function.
    loss = mx.sym.MakeLoss(objective_function(param_syms))
    mod = mx.mod.Module(loss)

    mod.bind(data_shapes=[('data', (1,))])
    mod.init_params(InitialParam(initial_params))
    mod.init_optimizer(optimizer=optimizer,
                       optimizer_params=optimizer_params)

    (o_name, o_shape), = mod.output_shapes

    i = 0
    params = initial_params
    old_val = np.full(o_shape, np.nan)
    while i < max_iters:
        mod.forward_backward(mx.io.DataBatch(
            data=[mx.nd.ones((1,))])) 
        mod.update()
        params = mod.get_params()[0]['fc1_weight']
        val = mod.get_outputs()[0].asnumpy()
        if np.allclose(old_val, val, atol=tol):
            print 'Function value: {}'.format(val)
            print 'Iterations: {}'.format(i)
            return params

        old_val = val
        i += 1

    return params

并使用它：

def my_func(x):
    return (x[0] + 1) ** 2

p = minimize(my_func, mx.nd.array([1.0]))
p.asnumpy()

>>> array([[-0.99999988]], dtype=float32)

另一个：

def my_func(x):
    return (x[0] + 1) ** 2 + (x[1] - 2) ** 2 + (x[2] + 3) ** 2

p = minimize(my_func, mx.nd.array([1.0, 1.5, 2.0]))
p.asnumpy()

>>> array([[-0.99996436],
           [ 1.99999106],
           [-2.99991083]], dtype=float32)