//  [python]
def PAC_Score(features_np_all, label_np_all,
                        lda_factor):
  """Compute the PAC_Gauss score with diagonal variance."""
  starttime = time.time()
  nclasses = label_np_all.max()+1
  label_np_all = one_hot(label_np_all)  # [n, v]
  
  mean_feature = np.mean(features_np_all, axis=0, keepdims=True)
  features_np_all -= mean_feature  # [n,k]

  bs = features_np_all.shape[0]
  kd = features_np_all.shape[-1] * nclasses
  ldas2 = lda_factor * bs  # * features_np_all.shape[-1]
  dinv = 1. / float(features_np_all.shape[-1])

  # optimizing log lik + log prior
  def pac_loss_fn(theta):
    theta = np.reshape(theta, [features_np_all.shape[-1] + 1, nclasses])

    w = theta[:features_np_all.shape[-1], :]
    b = theta[features_np_all.shape[-1]:, :]
    logits = np.matmul(features_np_all, w) + b

    log_qz = logits - scipy.special.logsumexp(logits, axis=-1, keepdims=True)
    xent = np.sum(np.sum(
        label_np_all * (np.log(label_np_all + 1e-10) - log_qz), axis=-1)) / bs
    loss = xent + 0.5 * np.sum(np.square(w)) / ldas2
    return loss

  # gradient of xent + l2
  def pac_grad_fn(theta):
    theta = np.reshape(theta, [features_np_all.shape[-1] + 1, nclasses])

    w = theta[:features_np_all.shape[-1], :]
    b = theta[features_np_all.shape[-1]:, :]
    logits = np.matmul(features_np_all, w) + b

    grad_f = scipy.special.softmax(logits, axis=-1)  # [n, k]
    grad_f -= label_np_all
    grad_f /= bs
    grad_w = np.matmul(features_np_all.transpose(), grad_f)  # [d, k]
    grad_w += w / ldas2

    grad_b = np.sum(grad_f, axis=0, keepdims=True)  # [1, k]
    grad = np.ravel(np.concatenate([grad_w, grad_b], axis=0))
    return grad

  # 2nd gradient of theta (elementwise)
  def pac_grad2(theta):
    theta = np.reshape(theta, [features_np_all.shape[-1] + 1, nclasses])

    w = theta[:features_np_all.shape[-1], :]
    b = theta[features_np_all.shape[-1]:, :]
    logits = np.matmul(features_np_all, w) + b

    prob_logits = scipy.special.softmax(logits, axis=-1)  # [n, k]
    grad2_f = prob_logits - np.square(prob_logits)  # [n, k]
    xx = np.square(features_np_all)  # [n, d]

    grad2_w = np.matmul(xx.transpose(), grad2_f)  # [d, k]
    grad2_w += 1. / ldas2
    grad2_b = np.sum(grad2_f, axis=0, keepdims=True)  # [1, k]
    grad2 = np.ravel(np.concatenate([grad2_w, grad2_b], axis=0))
    return grad2

  kernel_shape = [features_np_all.shape[-1], nclasses]
  theta = np.random.normal(size=kernel_shape) * 0.03
  theta_1d = np.ravel(np.concatenate(
      [theta, np.zeros([1, nclasses])], axis=0))

  theta_1d = scipy.optimize.minimize(
      pac_loss_fn, theta_1d, method="L-BFGS-B",
      jac=pac_grad_fn,
      options=dict(maxiter=10), tol=1e-6).x

  pac_opt = pac_loss_fn(theta_1d)
  endtime_opt = time.time()

  h = pac_grad2(theta_1d)
  sigma2_inv = np.sum(h) * ldas2  / kd + 1e-10
  endtime = time.time()

  if lda_factor == 10.:
    s2s = [1000., 100.]
  elif lda_factor == 1.:
    s2s = [100., 10.]
  elif lda_factor == 0.1:
    s2s = [10., 1.]
    
  returnv = []
  for s2_factor in s2s:
    s2 = s2_factor * dinv
    pac_gauss = pac_opt + 0.5 * kd / ldas2 * s2 * np.log(
        sigma2_inv)
    
    # the first item is the pac_gauss metric
    # the second item is the linear metric (without trH)
    returnv += [("pac_gauss_%.1f" % lda_factor, pac_gauss),
                ("time", endtime - starttime),
                ("pac_opt_%.1f" % lda_factor, pac_opt),
                ("time", endtime_opt - starttime)]
  return returnv, theta_1d