code: #!/usr/bin/env python
  
  # Save parameters every a few SGD iterations as fail-safe
  SAVE_PARAMS_EVERY = 5000
  
  import glob
  import random
  import numpy as np
  import os.path as op
  import pickle
  
  
  def load_saved_params():
      """
      A helper function that loads previously saved parameters and resets
      iteration start.
      """
      st = 0
      for f in glob.glob("saved_params_*.npy"):
          iter = int(op.splitext(op.basename(f))[0].split("_")[2])
          if (iter > st):
              st = iter
  
      if st > 0:
          with open("saved_params_%d.npy" % st, "r") as f:
              params = pickle.load(f)
              state = pickle.load(f)
          return st, params, state
      else:
          return st, None, None
  
  
  def save_params(iter, params):
      with open("saved_params_%d.npy" % iter, "w") as f:
          pickle.dump(params, f)
          pickle.dump(random.getstate(), f)
  
  
  def sgd(f, x0, step, iterations, postprocessing=None, useSaved=False,
          PRINT_EVERY=10):
      """ Stochastic Gradient Descent
  
      Implement the stochastic gradient descent method in this function.
  
      Arguments:
      f -- the function to optimize, it should take a single
           argument and yield two outputs, a cost and the gradient
           with respect to the arguments
      x0 -- the initial point to start SGD from
      step -- the step size for SGD
      iterations -- total iterations to run SGD for
      postprocessing -- postprocessing function for the parameters
                        if necessary. In the case of word2vec we will need to
                        normalize the word vectors to have unit length.
      PRINT_EVERY -- specifies how many iterations to output loss
  
      Return:
      x -- the parameter value after SGD finishes
      """
  
      # Anneal learning rate every several iterations
      ANNEAL_EVERY = 20000
  
      if useSaved:
          start_iter, oldx, state = load_saved_params()
          if start_iter > 0:
              x0 = oldx
              step *= 0.5 ** (start_iter / ANNEAL_EVERY)
  
          if state:
              random.setstate(state)
      else:
          start_iter = 0
  
      x = x0
  
      if not postprocessing:
          postprocessing = lambda x: x
  
      expcost = None
  
      for iter in range(start_iter + 1, iterations + 1):
          # Don't forget to apply the postprocessing after every iteration!
          # You might want to print the progress every few iterations.
  
          cost = None
          ### YOUR CODE HERE
          cost, grad = f(x)
          x -= step * grad
          postprocessing(x)
          ### END YOUR CODE
  
          if iter % PRINT_EVERY == 0:
              if not expcost:
                  expcost = cost
              else:
                  expcost = .95 * expcost + .05 * cost
              print("iter %d: %f" % (iter, expcost))
  
          if iter % SAVE_PARAMS_EVERY == 0 and useSaved:
              save_params(iter, x)
  
          if iter % ANNEAL_EVERY == 0:
              step *= 0.5
  
      return x
  
  
  def sanity_check():
      quad = lambda x: (np.sum(x ** 2), x * 2)
  
      print("Running sanity checks...")
      t1 = sgd(quad, 0.5, 0.01, 1000, PRINT_EVERY=100)
      print("test 1 result:", t1)
      assert abs(t1) <= 1e-6
  
      t2 = sgd(quad, 0.0, 0.01, 1000, PRINT_EVERY=100)
      print("test 2 result:", t2)
      assert abs(t2) <= 1e-6
  
      t3 = sgd(quad, -1.5, 0.01, 1000, PRINT_EVERY=100)
      print("test 3 result:", t3)
      assert abs(t3) <= 1e-6
  
      print("")
  
  
  def your_sanity_checks():
      """
      Use this space add any additional sanity checks by running:
          python q3_sgd.py
      This function will not be called by the autograder, nor will
      your additional tests be graded.
      """
      print("Running your sanity checks...")
      ### YOUR CODE HERE
      # raise NotImplementedError
      ### END YOUR CODE
  
  
  if __name__ == "__main__":
      sanity_check()
      your_sanity_checks()
  
apis: numpy.sum