Edward

Batch Training

Running algorithms which require the full data set for each update can be expensive when the data is large. In order to scale inferences, we can do batch training. This trains the model using only a subsample of data at a time.

In this tutorial, we extend the supervised learning tutorial, where the task is to infer hidden structure from labeled examples \(\{(x_n, y_n)\}\). An interactive version with Jupyter notebook is available here.

Data

Simulate \(N\) training examples and a fixed number of test examples. Each example is a pair of inputs \(\mathbf{x}_n\in\mathbb{R}^{10}\) and outputs \(y_n\in\mathbb{R}\). They have a linear dependence with normally distributed noise.

def build_toy_dataset(N, w):
  D = len(w)
  x = np.random.normal(0.0, 2.0, size=(N, D))
  y = np.dot(x, w) + np.random.normal(0.0, 0.05, size=N)
  return x, y

N = 10000  # size of training data
M = 128    # batch size during training
D = 10     # number of features

w_true = np.ones(D) * 5
X_train, y_train = build_toy_dataset(N, w_true)
X_test, y_test = build_toy_dataset(235, w_true)

We also define a helper function to select the next batch of data points from the full set of examples. It keeps track of the current batch index and returns the next batch using the function next().

def generator(arrays, batch_size):
  """Generate batches, one with respect to each array's first axis."""
  starts = [0] * len(arrays)  # pointers to where we are in iteration
  while True:
    batches = []
    for i, array in enumerate(arrays):
      start = starts[i]
      stop = start + batch_size
      diff = stop - array.shape[0]
      if diff <= 0:
        batch = array[start:stop]
        starts[i] += batch_size
      else:
        batch = np.concatenate((array[start:], array[:diff]))
        starts[i] = diff
      batches.append(batch)
    yield batches

data = generator([X_train, y_train], M)

We will generate batches from data during inference.

Model

Posit the model as Bayesian linear regression (Murphy, 2012). For a set of \(N\) data points \((\mathbf{X},\mathbf{y})=\{(\mathbf{x}_n, y_n)\}\), the model posits the following distributions:

\[\begin{aligned} p(\mathbf{w}) &= \text{Normal}(\mathbf{w} \mid \mathbf{0}, \sigma_w^2\mathbf{I}), \\[1.5ex] p(b) &= \text{Normal}(b \mid 0, \sigma_b^2), \\ p(\mathbf{y} \mid \mathbf{w}, b, \mathbf{X}) &= \prod_{n=1}^N \text{Normal}(y_n \mid \mathbf{x}_n^\top\mathbf{w} + b, \sigma_y^2).\end{aligned}\]

The latent variables are the linear model’s weights \(\mathbf{w}\) and intercept \(b\), also known as the bias. Assume \(\sigma_w^2,\sigma_b^2\) are known prior variances and \(\sigma_y^2\) is a known likelihood variance. The mean of the likelihood is given by a linear transformation of the inputs \(\mathbf{x}_n\).

Let’s build the model in Edward, fixing \(\sigma_w,\sigma_b,\sigma_y=1\).

X = tf.placeholder(tf.float32, [None, D])
y_ph = tf.placeholder(tf.float32, [None])

w = Normal(loc=tf.zeros(D), scale=tf.ones(D))
b = Normal(loc=tf.zeros(1), scale=tf.ones(1))
y = Normal(loc=ed.dot(X, w) + b, scale=1.0)

Here, we define a placeholder X. During inference, we pass in the value for this placeholder according to batches of data. To enable training with batches of varying size, we don’t fix the number of rows for X and y. (Alternatively, we could fix it to be the batch size if we’re training and testing with a fixed size.)

Inference

We now turn to inferring the posterior using variational inference. Define the variational model to be a fully factorized normal across the weights.

qw = Normal(loc=tf.get_variable("qw/loc", [D]),
            scale=tf.nn.softplus(tf.get_variable("qw/scale", [D])))
qb = Normal(loc=tf.get_variable("qb/loc", [1]),
            scale=tf.nn.softplus(tf.get_variable("qb/scale", [1])))

Run variational inference with the Kullback-Leibler divergence. We use \(5\) latent variable samples for computing black box stochastic gradients in the algorithm. (For more details, see the \(\text{KL}(q\|p)\) tutorial.)

For batch training, we iterate over the number of batches and feed them to the respective placeholder. We set the number of iterations to be the total number of batches for 5 epochs (full passes over the data set).

n_batch = int(N / M)
n_epoch = 5

inference = ed.KLqp({w: qw, b: qb}, data={y: y_ph})
inference.initialize(
    n_iter=n_batch * n_epoch, n_samples=5, scale={y: N / M})
tf.global_variables_initializer().run()

for _ in range(inference.n_iter):
  X_batch, y_batch = next(data)
  info_dict = inference.update({X: X_batch, y_ph: y_batch})
  inference.print_progress(info_dict)

390/390 [100%] ██████████████████████████████ Elapsed: 4s | Loss: 10481.556

When initializing inference, note we scale \(y\) by \(N/M\), so it is as if the algorithm had seen \(N/M\) as many data points per iteration. Algorithmically, this will scale all computation regarding \(y\) by \(N/M\) such as scaling the log-likelihood in a variational method’s objective. (Statistically, this avoids inference being dominated by the prior.)

The loop construction makes training very flexible. For example, we can also try running many updates for each batch.

n_batch = int(N / M)
n_epoch = 1

inference = ed.KLqp({w: qw, b: qb}, data={y: y_ph})
inference.initialize(n_iter=n_batch * n_epoch * 10, n_samples=5, scale={y: N / M})
tf.global_variables_initializer().run()

for _ in range(inference.n_iter // 10):
  X_batch, y_batch = next(data)
  for _ in range(10):
    info_dict = inference.update({X: X_batch, y_ph: y_batch})

  inference.print_progress(info_dict)

770/780 [ 98%] █████████████████████████████  ETA: 0s | Loss: 9760.541

In general, make sure that the total number of training iterations is specified correctly when initializing inference. Otherwise an incorrect number of training iterations can have unintended consequences; for example, ed.KLqp uses an internal counter to appropriately decay its optimizer’s learning rate step size.

Note also that the reported loss value as we run the algorithm corresponds to the computed objective given the current batch and not the total data set. We can instead have it report the loss over the total data set by summing info_dict[’loss’] for each epoch.

Criticism

A standard evaluation for regression is to compare prediction accuracy on held-out “testing” data. We do this by first forming the posterior predictive distribution.

y_post = ed.copy(y, {w: qw, b: qb})
# This is equivalent to
# y_post = Normal(loc=ed.dot(X, qw) + qb, scale=tf.ones(N))

With this we can evaluate various quantities using predictions from the model (posterior predictive).

print("Mean squared error on test data:")
print(ed.evaluate('mean_squared_error', data={X: X_test, y_post: y_test}))

print("Mean absolute error on test data:")
print(ed.evaluate('mean_absolute_error', data={X: X_test, y_post: y_test}))

## Mean squared error on test data:
## 0.00659598
## Mean absolute error on test data:
## 0.0705906

The trained model makes predictions with low error (relative to the magnitude of the output).

Footnotes

Only certain algorithms support batch training such as MAP, KLqp, and SGLD. Also, above we illustrated batch training for models with only global latent variables, which are variables are shared across all data points. For more complex strategies, see the inference data subsampling API.

References