View source on GitHub

Computes Jacobian-vector products ("JVP"s) using forward-mode autodiff.

Compare to tf.GradientTape which computes vector-Jacobian products ("VJP"s) using reverse-mode autodiff (backprop). Reverse mode is more attractive when computing gradients of a scalar-valued function with respect to many inputs (e.g. a neural network with many parameters and a scalar loss). Forward mode works best on functions with many outputs and few inputs. Since it does not hold on to intermediate activations, it is much more memory efficient than backprop where it is applicable.

Consider a simple linear regression:

x = tf.constant([[2.0, 3.0], [1.0, 4.0]])
dense = tf.keras.layers.Dense(1)[None, 2])
with tf.autodiff.ForwardAccumulator(
   tangents=tf.constant([[1.], [0.]])) as acc:
  loss = tf.reduce_sum((dense(x) - tf.constant([1., -1.])) ** 2.)
<tf.Tensor: shape=(), dtype=float32, numpy=...>

The example has two variables containing parameters, dense.kernel (2 parameters) and dense.bias (1 parameter). Considering the training data x as a constant, this means the Jacobian matrix for the function mapping from parameters to loss has one row and three columns.

With forwardprop, we specify a length-three vector in advance which multiplies the Jacobian. The primals constructor argument is the parameter (a tf.Tensor or tf.Variable) we're specifying a vector for, and the tangents argument is the "vector" in Jacobian-vector product. If our goal is to compute the entire Jacobian matrix, forwardprop computes one column at a time while backprop computes one row at a time. Since the Jacobian in the linear regression example has only one row, backprop requires fewer invocations:

x = tf.constant([[2.0, 3.0], [1.0, 4.0]])
dense = tf.keras.layers.Dense(1)[None, 2])
loss_fn = lambda: tf.reduce_sum((dense(x) - tf.constant([1., -1.])) ** 2.)
kernel_fprop = []
with tf.autodiff.ForwardAccumulator(
    dense.kernel, tf.constant([[1.], [0.]])) as acc:
with tf.autodiff.ForwardAccumulator(
    dense.kernel, tf.constant([[0.], [1.]])) as acc:
with tf.autodiff.ForwardAccumulator(dense.bias, tf.constant([1.])) as acc:
  bias_fprop = acc.jvp(loss_fn())
with tf.GradientTape() as tape:
  loss = loss_fn()
kernel_grad, bias_grad = tape.gradient(loss, (dense.kernel, dense.bias))
    kernel_grad, tf.stack(kernel_fprop)[:, tf.newaxis])
np.testing.assert_allclose(bias_grad, bias_fprop[tf.newaxis])

Implicit in the tape.gradient call is a length-one vector which left-multiplies the Jacobian, a vector-Jacobian product.

ForwardAccumulator maintains JVPs corresponding primal tensors it is watching, derived from the original primals specified in the constructor. As soon as a primal tensor is deleted, ForwardAccumulator deletes the corresponding JVP.

acc.jvp(x) retrieves acc's JVP corresponding to the primal tensor x. It does not perform any computation. acc.jvp calls can be repeated as long as acc is accessible, whether the context manager is active or not. New JVPs are only computed while the context manager is active.

Note that ForwardAccumulators are always applied in the order their context managers were entered, so inner accumulators will not see JVP computation from outer accumulators. Take higher-order JVPs from outer accumulators:

primal = tf.constant(1.1)
with tf.autodiff.ForwardAccumulator(primal, tf.constant(1.)) as outer:
  with tf.autodiff.ForwardAccumulator(primal, tf.constant(1.)) as inner:
    primal_out = primal ** tf.constant(3.5)
inner_jvp = inner.jvp(primal_out)
inner_jvp  # 3.5 * 1.1 ** 2.5
<tf.Tensor: shape=(), dtype=float32, numpy=4.4417057>
outer.jvp(inner_jvp)  # 3.5 * 2.5 * 1.1 ** 1.5
<tf.Tensor: shape=(), dtype=float32, numpy=10.094786>

Reversing the collection in the last line to instead retrieve inner.jvp(outer.jvp(primal_out)) will not work.

Strict nesting also applies to combinations of ForwardAccumulator and tf.GradientTape. More deeply nested GradientTape objects will ignore the products of outer ForwardAccumulator objects. This allows (for example) memory-efficient forward-over-backward computation of Hessian-vector products, where the inner GradientTape would otherwise hold on to all intermediate JVPs:

v = tf.Variable([1., 2.])
with tf.autodiff.ForwardAccumulator(
    # The "vector" in Hessian-vector product.
    tf.constant([1., 0.])) as acc:
  with tf.GradientTape() as tape:
    y = tf.reduce_sum(v ** 3.)
  backward = tape.gradient(y, v)
backward  # gradient from backprop
<tf.Tensor: shape=(2,), dtype=float32, numpy=array([ 3., 12.], dtype=float32)>
acc.jvp(backward)  # forward-over-backward Hessian-vector product
<tf.Tensor: shape=(2,), dtype=float32, numpy=array([6., 0.], dtype=float32)>

primals A tensor or nested structure of tensors to watch.
tangents A tensor or nested structure of tensors, with the same nesting structure as primals, with each element being a vector with the same size as the corresponding primal element.

ValueError If the same tensor or variable is specified multiple times in primals.



View source

Fetches the Jacobian-vector product computed for primals.

Note that this method performs no computation, and simply looks up a JVP that was already computed (unlike backprop using a tf.GradientTape, where the computation happens on the call to tape.gradient).

primals A watched Tensor or structure of Tensors to fetch the JVPs for.
unconnected_gradients A value which can either hold 'none' or 'zero' and alters the value which will be returned if no JVP was computed for primals. The possible values and effects are detailed in 'tf.UnconnectedGradients' and it defaults to 'none'.

Tensors with the same shapes and dtypes as primals, or None if no JVP is available.


View source


View source