text: from builtins import range
  import numpy as np
  
  
  
  def affine_forward(x, w, b):
      """
      Computes the forward pass for an affine (fully-connected) layer.
  
      The input x has shape (N, d_1, ..., d_k) and contains a minibatch of N
      examples, where each example x[i] has shape (d_1, ..., d_k). We will
      reshape each input into a vector of dimension D = d_1 * ... * d_k, and
      then transform it to an output vector of dimension M.
  
      Inputs:
      - x: A numpy array containing input data, of shape (N, d_1, ..., d_k)
      - w: A numpy array of weights, of shape (D, M)
      - b: A numpy array of biases, of shape (M,)
  
      Returns a tuple of:
      - out: output, of shape (N, M)
      - cache: (x, w, b)
      """
      out = None
      ###########################################################################
      # TODO: Implement the affine forward pass. Store the result in out. You   #
      # will need to reshape the input into rows.                               #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      x_reshape = x.reshape(x.shape[0],-1)
      b_reshape = b.reshape(1,-1)
      out = np.matmul(x_reshape,w) + b_reshape
      cache = (x, w, b, x_reshape)
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
  
      return out, cache
  
  
  def affine_backward(dout, cache):
      """
      Computes the backward pass for an affine layer.
  
      Inputs:
      - dout: Upstream derivative, of shape (N, M)
      - cache: Tuple of:
        - x: Input data, of shape (N, d_1, ... d_k)
        - w: Weights, of shape (D, M)
        - b: Biases, of shape (M,)
  
      Returns a tuple of:
      - dx: Gradient with respect to x, of shape (N, d1, ..., d_k)
      - dw: Gradient with respect to w, of shape (D, M)
      - db: Gradient with respect to b, of shape (M,)
      """
      x, w, b, x_reshape = cache
      dx, dw, db = None, None, None
      ###########################################################################
      # TODO: Implement the affine backward pass.                               #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      db = np.matmul(dout.T, np.ones(dout.shape[0]))
      dw = np.matmul(x_reshape.T, dout)
      dx = np.matmul(dout, w.T).reshape(x.shape)
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return dx, dw, db
  
  
  def relu_forward(x):
      """
      Computes the forward pass for a layer of rectified linear units (ReLUs).
  
      Input:
      - x: Inputs, of any shape
  
      Returns a tuple of:
      - out: Output, of the same shape as x
      - cache: x
      """
  
      ###########################################################################
      # TODO: Implement the ReLU forward pass.                                  #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      out = np.maximum(x, 0)
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      cache = x
      return out, cache
  
  
  def relu_backward(dout, cache):
      """
      Computes the backward pass for a layer of rectified linear units (ReLUs).
  
      Input:
      - dout: Upstream derivatives, of any shape
      - cache: Input x, of same shape as dout
  
      Returns:
      - dx: Gradient with respect to x
      """
      dx, x = None, cache
      ###########################################################################
      # TODO: Implement the ReLU backward pass.                                 #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      ReLU_mask = (x>0)
  
      dx = ReLU_mask * dout
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return dx
  
  
  def batchnorm_forward(x, gamma, beta, bn_param):
      """
      Forward pass for batch normalization.
  
      During training the sample mean and (uncorrected) sample variance are
      computed from minibatch statistics and used to normalize the incoming data.
      During training we also keep an exponentially decaying running mean of the
      mean and variance of each feature, and these averages are used to normalize
      data at test-time.
  
      At each timestep we update the running averages for mean and variance using
      an exponential decay based on the momentum parameter:
  
      running_mean = momentum * running_mean + (1 - momentum) * sample_mean
      running_var = momentum * running_var + (1 - momentum) * sample_var
  
      Note that the batch normalization paper suggests a different test-time
      behavior: they compute sample mean and variance for each feature using a
      large number of training images rather than using a running average. For
      this implementation we have chosen to use running averages instead since
      they do not require an additional estimation step; the torch7
      implementation of batch normalization also uses running averages.
  
      Input:
      - x: Data of shape (N, D)
      - gamma: Scale parameter of shape (D,)
      - beta: Shift paremeter of shape (D,)
      - bn_param: Dictionary with the following keys:
        - mode: 'train' or 'test'; required
        - eps: Constant for numeric stability
        - momentum: Constant for running mean / variance.
        - running_mean: Array of shape (D,) giving running mean of features
        - running_var Array of shape (D,) giving running variance of features
  
      Returns a tuple of:
      - out: of shape (N, D)
      - cache: A tuple of values needed in the backward pass
      """
      mode = bn_param["mode"]
      eps = bn_param.get("eps", 1e-8)
      momentum = bn_param.get("momentum", 0.9)
  
      N, D = x.shape
      running_mean = bn_param.get("running_mean", np.zeros(D, dtype=x.dtype))
      running_var = bn_param.get("running_var", np.zeros(D, dtype=x.dtype))
  
      out, cache = None, None
      if mode == "train":
          #######################################################################
          # TODO: Implement the training-time forward pass for batch norm.      #
          # Use minibatch statistics to compute the mean and variance, use      #
          # these statistics to normalize the incoming data, and scale and      #
          # shift the normalized data using gamma and beta.                     #
          #                                                                     #
          # You should store the output in the variable out. Any intermediates  #
          # that you need for the backward pass should be stored in the cache   #
          # variable.                                                           #
          #                                                                     #
          # You should also use your computed sample mean and variance together #
          # with the momentum variable to update the running mean and running   #
          # variance, storing your result in the running_mean and running_var   #
          # variables.                                                          #
          #                                                                     #
          # Note that though you should be keeping track of the running         #
          # variance, you should normalize the data based on the standard       #
          # deviation (square root of variance) instead!                        #
          # Referencing the original paper (https://arxiv.org/abs/1502.03167)   #
          # might prove to be helpful.                                          #
          #######################################################################
          # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
          mean = np.mean(x, axis=0)
          var = np.var(x, axis=0)
          # normalize x
          x_normal = (x - mean.reshape(1,-1))/(np.sqrt(var.reshape(1,-1))+eps)
  
          out = x_normal* gamma.reshape(1,-1) + beta.reshape(1,-1)
          cache = (x_normal, var, gamma, eps)
  
          running_mean = momentum*running_mean + (1-momentum)*mean
          running_var = momentum*running_var + (1-momentum)*var
          # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
          #######################################################################
          #                           END OF YOUR CODE                          #
          #######################################################################
      elif mode == "test":
          #######################################################################
          # TODO: Implement the test-time forward pass for batch normalization. #
          # Use the running mean and variance to normalize the incoming data,   #
          # then scale and shift the normalized data using gamma and beta.      #
          # Store the result in the out variable.                               #
          #######################################################################
          # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
          # in test mode, normalize x with running_mean and running_var
          x_normal = (x - running_mean.reshape(1,-1)) / (np.sqrt(running_var.reshape(1,-1))+eps)
          out = x_normal * gamma.reshape(1,-1) + beta.reshape(1,-1)
  
  
          # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
          #######################################################################
          #                          END OF YOUR CODE                           #
          #######################################################################
      else:
          raise ValueError('Invalid forward batchnorm mode "%s"' % mode)
  
      # Store the updated running means back into bn_param
      bn_param["running_mean"] = running_mean
      bn_param["running_var"] = running_var
  
      return out, cache
  
  
  def batchnorm_backward(dout, cache):
      """
      Backward pass for batch normalization.
  
      For this implementation, you should write out a computation graph for
      batch normalization on paper and propagate gradients backward through
      intermediate nodes.
  
      Inputs:
      - dout: Upstream derivatives, of shape (N, D)
      - cache: Variable of intermediates from batchnorm_forward.
  
      Returns a tuple of:
      - dx: Gradient with respect to inputs x, of shape (N, D)
      - dgamma: Gradient with respect to scale parameter gamma, of shape (D,)
      - dbeta: Gradient with respect to shift parameter beta, of shape (D,)
      """
      dx, dgamma, dbeta = None, None, None
      ###########################################################################
      # TODO: Implement the backward pass for batch normalization. Store the    #
      # results in the dx, dgamma, and dbeta variables.                         #
      # Referencing the original paper (https://arxiv.org/abs/1502.03167)       #
      # might prove to be helpful.                                              #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      # read off cache
      x_normal, var, gamma, eps = cache
      N, D = dout.shape
      # gradient w.r.t. x_normal
      dx_normal = dout * (gamma.reshape(1,-1)/(np.sqrt(var.reshape(1,-1))+eps))
      # gradient w.r.t. gamma
      dgamma = np.sum(dout * x_normal, axis=0)
      # gradient w.r.t. beta
      dbeta = np.sum(dout, axis=0)
  
      # Next, find the derivative of x_normal over x.
      # As an entry in x_normal will have nonzero derivative wrt an entry in x
      # if they are lie on a same column.
      # Then focus on one column, we have two cases:
      # case1 -- the i-th element in x_normal wrt the i-th element in x (using Quotient Rule and simplification)
      """dx_normal_over_dx_case1 = ((N-1.0)/N) * (1-x_normal**2/N) / (np.sqrt(var.reshape(1,-1))+eps)"""
      # case2 -- the i-th element in x_normal wrt the j-th element in x (i != j)
      """dx_normal_over_dx_case2 = (-1.0/N) * (1-x_normal**2/N) / (np.sqrt(var.reshape(1,-1))+eps)"""
      # In summary
      # the NxN coefficient matrix: 1-1/N on diagonal, -1/N otherwise
      coef_matrix = np.identity(N) - np.ones((N,N))*(1.0/N)
  
      # gradient w.r.t. x
      dx = np.matmul(coef_matrix, (1-(x_normal**2)/N) * dx_normal / (np.sqrt(var.reshape(1,-1))+eps))
      print(dx)
  
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
  
      return dx, dgamma, dbeta
  
  
  def batchnorm_backward_alt(dout, cache):
      """
      Alternative backward pass for batch normalization.
  
      For this implementation you should work out the derivatives for the batch
      normalizaton backward pass on paper and simplify as much as possible. You
      should be able to derive a simple expression for the backward pass.
      See the jupyter notebook for more hints.
  
      Note: This implementation should expect to receive the same cache variable
      as batchnorm_backward, but might not use all of the values in the cache.
  
      Inputs / outputs: Same as batchnorm_backward
      """
      dx, dgamma, dbeta = None, None, None
      ###########################################################################
      # TODO: Implement the backward pass for batch normalization. Store the    #
      # results in the dx, dgamma, and dbeta variables.                         #
      #                                                                         #
      # After computing the gradient with respect to the centered inputs, you   #
      # should be able to compute gradients with respect to the inputs in a     #
      # single statement; our implementation fits on a single 80-character line.#
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
  
      return dx, dgamma, dbeta
  
  
  def layernorm_forward(x, gamma, beta, ln_param):
      """
      Forward pass for layer normalization.
  
      During both training and test-time, the incoming data is normalized per data-point,
      before being scaled by gamma and beta parameters identical to that of batch normalization.
  
      Note that in contrast to batch normalization, the behavior during train and test-time for
      layer normalization are identical, and we do not need to keep track of running averages
      of any sort.
  
      Input:
      - x: Data of shape (N, D)
      - gamma: Scale parameter of shape (D,)
      - beta: Shift paremeter of shape (D,)
      - ln_param: Dictionary with the following keys:
          - eps: Constant for numeric stability
  
      Returns a tuple of:
      - out: of shape (N, D)
      - cache: A tuple of values needed in the backward pass
      """
      out, cache = None, None
      eps = ln_param.get("eps", 1e-5)
      ###########################################################################
      # TODO: Implement the training-time forward pass for layer norm.          #
      # Normalize the incoming data, and scale and  shift the normalized data   #
      #  using gamma and beta.                                                  #
      # HINT: this can be done by slightly modifying your training-time         #
      # implementation of  batch normalization, and inserting a line or two of  #
      # well-placed code. In particular, can you think of any matrix            #
      # transformations you could perform, that would enable you to copy over   #
      # the batch norm code and leave it almost unchanged?                      #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return out, cache
  
  
  def layernorm_backward(dout, cache):
      """
      Backward pass for layer normalization.
  
      For this implementation, you can heavily rely on the work you've done already
      for batch normalization.
  
      Inputs:
      - dout: Upstream derivatives, of shape (N, D)
      - cache: Variable of intermediates from layernorm_forward.
  
      Returns a tuple of:
      - dx: Gradient with respect to inputs x, of shape (N, D)
      - dgamma: Gradient with respect to scale parameter gamma, of shape (D,)
      - dbeta: Gradient with respect to shift parameter beta, of shape (D,)
      """
      dx, dgamma, dbeta = None, None, None
      ###########################################################################
      # TODO: Implement the backward pass for layer norm.                       #
      #                                                                         #
      # HINT: this can be done by slightly modifying your training-time         #
      # implementation of batch normalization. The hints to the forward pass    #
      # still apply!                                                            #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return dx, dgamma, dbeta
  
  
  def dropout_forward(x, dropout_param):
      """
      Performs the forward pass for (inverted) dropout.
  
      Inputs:
      - x: Input data, of any shape
      - dropout_param: A dictionary with the following keys:
        - p: Dropout parameter. We keep each neuron output with probability p.
        - mode: 'test' or 'train'. If the mode is train, then perform dropout;
          if the mode is test, then just return the input.
        - seed: Seed for the random number generator. Passing seed makes this
          function deterministic, which is needed for gradient checking but not
          in real networks.
  
      Outputs:
      - out: Array of the same shape as x.
      - cache: tuple (dropout_param, mask). In training mode, mask is the dropout
        mask that was used to multiply the input; in test mode, mask is None.
  
      NOTE: Please implement **inverted** dropout, not the vanilla version of dropout.
      See http://cs231n.github.io/neural-networks-2/#reg for more details.
  
      NOTE 2: Keep in mind that p is the probability of **keep** a neuron
      output; this might be contrary to some sources, where it is referred to
      as the probability of dropping a neuron output.
      """
      p, mode = dropout_param["p"], dropout_param["mode"]
      if "seed" in dropout_param:
          np.random.seed(dropout_param["seed"])
  
      mask = None
      out = None
  
      if mode == "train":
          #######################################################################
          # TODO: Implement training phase forward pass for inverted dropout.   #
          # Store the dropout mask in the mask variable.                        #
          #######################################################################
          # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
          pass
  
          # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
          #######################################################################
          #                           END OF YOUR CODE                          #
          #######################################################################
      elif mode == "test":
          #######################################################################
          # TODO: Implement the test phase forward pass for inverted dropout.   #
          #######################################################################
          # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
          pass
  
          # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
          #######################################################################
          #                            END OF YOUR CODE                         #
          #######################################################################
  
      cache = (dropout_param, mask)
      out = out.astype(x.dtype, copy=False)
  
      return out, cache
  
  
  def dropout_backward(dout, cache):
      """
      Perform the backward pass for (inverted) dropout.
  
      Inputs:
      - dout: Upstream derivatives, of any shape
      - cache: (dropout_param, mask) from dropout_forward.
      """
      dropout_param, mask = cache
      mode = dropout_param["mode"]
  
      dx = None
      if mode == "train":
          #######################################################################
          # TODO: Implement training phase backward pass for inverted dropout   #
          #######################################################################
          # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
          pass
  
          # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
          #######################################################################
          #                          END OF YOUR CODE                           #
          #######################################################################
      elif mode == "test":
          dx = dout
      return dx
  
  
  def conv_forward_naive(x, w, b, conv_param):
      """
      A naive implementation of the forward pass for a convolutional layer.
  
      The input consists of N data points, each with C channels, height H and
      width W. We convolve each input with F different filters, where each filter
      spans all C channels and has height HH and width WW.
  
      Input:
      - x: Input data of shape (N, C, H, W)
      - w: Filter weights of shape (F, C, HH, WW)
      - b: Biases, of shape (F,)
      - conv_param: A dictionary with the following keys:
        - 'stride': The number of pixels between adjacent receptive fields in the
          horizontal and vertical directions.
        - 'pad': The number of pixels that will be used to zero-pad the input.
  
  
      During padding, 'pad' zeros should be placed symmetrically (i.e equally on both sides)
      along the height and width axes of the input. Be careful not to modfiy the original
      input x directly.
  
      Returns a tuple of:
      - out: Output data, of shape (N, F, H', W') where H' and W' are given by
        H' = 1 + (H + 2 * pad - HH) / stride
        W' = 1 + (W + 2 * pad - WW) / stride
      - cache: (x, w, b, conv_param)
      """
      out = None
      ###########################################################################
      # TODO: Implement the convolutional forward pass.                         #
      # Hint: you can use the function np.pad for padding.                      #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      cache = (x, w, b, conv_param)
      return out, cache
  
  
  def conv_backward_naive(dout, cache):
      """
      A naive implementation of the backward pass for a convolutional layer.
  
      Inputs:
      - dout: Upstream derivatives.
      - cache: A tuple of (x, w, b, conv_param) as in conv_forward_naive
  
      Returns a tuple of:
      - dx: Gradient with respect to x
      - dw: Gradient with respect to w
      - db: Gradient with respect to b
      """
      dx, dw, db = None, None, None
      ###########################################################################
      # TODO: Implement the convolutional backward pass.                        #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return dx, dw, db
  
  
  def max_pool_forward_naive(x, pool_param):
      """
      A naive implementation of the forward pass for a max-pooling layer.
  
      Inputs:
      - x: Input data, of shape (N, C, H, W)
      - pool_param: dictionary with the following keys:
        - 'pool_height': The height of each pooling region
        - 'pool_width': The width of each pooling region
        - 'stride': The distance between adjacent pooling regions
  
      No padding is necessary here. Output size is given by
  
      Returns a tuple of:
      - out: Output data, of shape (N, C, H', W') where H' and W' are given by
        H' = 1 + (H - pool_height) / stride
        W' = 1 + (W - pool_width) / stride
      - cache: (x, pool_param)
      """
      out = None
      ###########################################################################
      # TODO: Implement the max-pooling forward pass                            #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      cache = (x, pool_param)
      return out, cache
  
  
  def max_pool_backward_naive(dout, cache):
      """
      A naive implementation of the backward pass for a max-pooling layer.
  
      Inputs:
      - dout: Upstream derivatives
      - cache: A tuple of (x, pool_param) as in the forward pass.
  
      Returns:
      - dx: Gradient with respect to x
      """
      dx = None
      ###########################################################################
      # TODO: Implement the max-pooling backward pass                           #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return dx
  
  
  def spatial_batchnorm_forward(x, gamma, beta, bn_param):
      """
      Computes the forward pass for spatial batch normalization.
  
      Inputs:
      - x: Input data of shape (N, C, H, W)
      - gamma: Scale parameter, of shape (C,)
      - beta: Shift parameter, of shape (C,)
      - bn_param: Dictionary with the following keys:
        - mode: 'train' or 'test'; required
        - eps: Constant for numeric stability
        - momentum: Constant for running mean / variance. momentum=0 means that
          old information is discarded completely at every time step, while
          momentum=1 means that new information is never incorporated. The
          default of momentum=0.9 should work well in most situations.
        - running_mean: Array of shape (D,) giving running mean of features
        - running_var Array of shape (D,) giving running variance of features
  
      Returns a tuple of:
      - out: Output data, of shape (N, C, H, W)
      - cache: Values needed for the backward pass
      """
      out, cache = None, None
  
      ###########################################################################
      # TODO: Implement the forward pass for spatial batch normalization.       #
      #                                                                         #
      # HINT: You can implement spatial batch normalization by calling the      #
      # vanilla version of batch normalization you implemented above.           #
      # Your implementation should be very short; ours is less than five lines. #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
  
      return out, cache
  
  
  def spatial_batchnorm_backward(dout, cache):
      """
      Computes the backward pass for spatial batch normalization.
  
      Inputs:
      - dout: Upstream derivatives, of shape (N, C, H, W)
      - cache: Values from the forward pass
  
      Returns a tuple of:
      - dx: Gradient with respect to inputs, of shape (N, C, H, W)
      - dgamma: Gradient with respect to scale parameter, of shape (C,)
      - dbeta: Gradient with respect to shift parameter, of shape (C,)
      """
      dx, dgamma, dbeta = None, None, None
  
      ###########################################################################
      # TODO: Implement the backward pass for spatial batch normalization.      #
      #                                                                         #
      # HINT: You can implement spatial batch normalization by calling the      #
      # vanilla version of batch normalization you implemented above.           #
      # Your implementation should be very short; ours is less than five lines. #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
  
      return dx, dgamma, dbeta
  
  
  def spatial_groupnorm_forward(x, gamma, beta, G, gn_param):
      """
      Computes the forward pass for spatial group normalization.
      In contrast to layer normalization, group normalization splits each entry
      in the data into G contiguous pieces, which it then normalizes independently.
      Per feature shifting and scaling are then applied to the data, in a manner identical to that of batch normalization and layer normalization.
  
      Inputs:
      - x: Input data of shape (N, C, H, W)
      - gamma: Scale parameter, of shape (C,)
      - beta: Shift parameter, of shape (C,)
      - G: Integer mumber of groups to split into, should be a divisor of C
      - gn_param: Dictionary with the following keys:
        - eps: Constant for numeric stability
  
      Returns a tuple of:
      - out: Output data, of shape (N, C, H, W)
      - cache: Values needed for the backward pass
      """
      out, cache = None, None
      eps = gn_param.get("eps", 1e-5)
      ###########################################################################
      # TODO: Implement the forward pass for spatial group normalization.       #
      # This will be extremely similar to the layer norm implementation.        #
      # In particular, think about how you could transform the matrix so that   #
      # the bulk of the code is similar to both train-time batch normalization  #
      # and layer normalization!                                                #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return out, cache
  
  
  def spatial_groupnorm_backward(dout, cache):
      """
      Computes the backward pass for spatial group normalization.
  
      Inputs:
      - dout: Upstream derivatives, of shape (N, C, H, W)
      - cache: Values from the forward pass
  
      Returns a tuple of:
      - dx: Gradient with respect to inputs, of shape (N, C, H, W)
      - dgamma: Gradient with respect to scale parameter, of shape (C,)
      - dbeta: Gradient with respect to shift parameter, of shape (C,)
      """
      dx, dgamma, dbeta = None, None, None
  
      ###########################################################################
      # TODO: Implement the backward pass for spatial group normalization.      #
      # This will be extremely similar to the layer norm implementation.        #
      ###########################################################################
      # *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
  
      pass
  
      # *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
      ###########################################################################
      #                             END OF YOUR CODE                            #
      ###########################################################################
      return dx, dgamma, dbeta
  
  
  def svm_loss(x, y):
      """
      Computes the loss and gradient using for multiclass SVM classification.
  
      Inputs:
      - x: Input data, of shape (N, C) where x[i, j] is the score for the jth
        class for the ith input.
      - y: Vector of labels, of shape (N,) where y[i] is the label for x[i] and
        0 <= y[i] < C
  
      Returns a tuple of:
      - loss: Scalar giving the loss
      - dx: Gradient of the loss with respect to x
      """
      N = x.shape[0]
      correct_class_scores = x[np.arange(N), y]
      margins = np.maximum(0, x - correct_class_scores[:, np.newaxis] + 1.0)
      margins[np.arange(N), y] = 0
      loss = np.sum(margins) / N
      num_pos = np.sum(margins > 0, axis=1)
      dx = np.zeros_like(x)
      dx[margins > 0] = 1
      dx[np.arange(N), y] -= num_pos
      dx /= N
      return loss, dx
  
  
  def softmax_loss(x, y):
      """
      Computes the loss and gradient for softmax classification.
  
      Inputs:
      - x: Input data, of shape (N, C) where x[i, j] is the score for the jth
        class for the ith input.
      - y: Vector of labels, of shape (N,) where y[i] is the label for x[i] and
        0 <= y[i] < C
  
      Returns a tuple of:
      - loss: Scalar giving the loss
      - dx: Gradient of the loss with respect to x
      """
      shifted_logits = x - np.max(x, axis=1, keepdims=True)
      Z = np.sum(np.exp(shifted_logits), axis=1, keepdims=True)
      log_probs = shifted_logits - np.log(Z)
      probs = np.exp(log_probs)
      N = x.shape[0]
      loss = -np.sum(log_probs[np.arange(N), y]) / N
      dx = probs.copy()
      dx[np.arange(N), y] -= 1
      dx /= N
      return loss, dx
  