Compute hessian with respect to several variables

问题:

Computing Hessian in tensorflow is quite easy:

x = tf.Variable([1., 1., 1.], dtype=tf.float32, name="x")
f = (x[0] + x[1] ** 2 + x[0] * x[1] + x[2]) ** 2
hessian = tf.hessians(f, x)

This correctly returns

[[ 8., 20.,  4.],
   [20., 34.,  6.],
   [ 4.,  6.,  2.]]

In my real case instead of using one single variable x holding three values, I need to split it in two variables: x (holding the first two) and y (holding the last one).

x = tf.Variable([1., 1.], dtype=tf.float32, name="x")
y = tf.Variable([1.], dtype=tf.float32, name="y")
f = (x[0] + x[1] ** 2 + x[0] * x[1] + y) ** 2

I tried a naive

hessian = tf.hessians(f, [x, y])

but I get: [[ 8., 20.], [20., 34.]], [[2.]]

I also tried:

xy = tf.concat([x, y], axis=-1)

but when defining the hessian

hessian = tf.hessians(f, xy)

I get a very bad error:

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/op_def_library.py in _apply_op_helper(self, op_type_name, name, **keywords)
    510                 as_ref=input_arg.is_ref,
--> 511                 preferred_dtype=default_dtype)
    512           except TypeError as err:

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/ops.py in internal_convert_to_tensor(value, dtype, name, as_ref, preferred_dtype, ctx, accept_symbolic_tensors)
   1174     if ret is None:
-> 1175       ret = conversion_func(value, dtype=dtype, name=name, as_ref=as_ref)
   1176 

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/constant_op.py in _constant_tensor_conversion_function(v, dtype, name, as_ref)
    303   _ = as_ref
--> 304   return constant(v, dtype=dtype, name=name)
    305 

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/constant_op.py in constant(value, dtype, shape, name)
    244   return _constant_impl(value, dtype, shape, name, verify_shape=False,
--> 245                         allow_broadcast=True)
    246 

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/constant_op.py in _constant_impl(value, dtype, shape, name, verify_shape, allow_broadcast)
    282           value, dtype=dtype, shape=shape, verify_shape=verify_shape,
--> 283           allow_broadcast=allow_broadcast))
    284   dtype_value = attr_value_pb2.AttrValue(type=tensor_value.tensor.dtype)

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/tensor_util.py in make_tensor_proto(values, dtype, shape, verify_shape, allow_broadcast)
    453     if values is None:
--> 454       raise ValueError("None values not supported.")
    455     # if dtype is provided, forces numpy array to be the type

ValueError: None values not supported.

During handling of the above exception, another exception occurred:

ValueError                                Traceback (most recent call last)
~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/op_def_library.py in _apply_op_helper(self, op_type_name, name, **keywords)
    524               observed = ops.internal_convert_to_tensor(
--> 525                   values, as_ref=input_arg.is_ref).dtype.name
    526             except ValueError as err:

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/ops.py in internal_convert_to_tensor(value, dtype, name, as_ref, preferred_dtype, ctx, accept_symbolic_tensors)
   1174     if ret is None:
-> 1175       ret = conversion_func(value, dtype=dtype, name=name, as_ref=as_ref)
   1176 

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/constant_op.py in _constant_tensor_conversion_function(v, dtype, name, as_ref)
    303   _ = as_ref
--> 304   return constant(v, dtype=dtype, name=name)
    305 

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/constant_op.py in constant(value, dtype, shape, name)
    244   return _constant_impl(value, dtype, shape, name, verify_shape=False,
--> 245                         allow_broadcast=True)
    246 

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/constant_op.py in _constant_impl(value, dtype, shape, name, verify_shape, allow_broadcast)
    282           value, dtype=dtype, shape=shape, verify_shape=verify_shape,
--> 283           allow_broadcast=allow_broadcast))
    284   dtype_value = attr_value_pb2.AttrValue(type=tensor_value.tensor.dtype)

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/tensor_util.py in make_tensor_proto(values, dtype, shape, verify_shape, allow_broadcast)
    453     if values is None:
--> 454       raise ValueError("None values not supported.")
    455     # if dtype is provided, forces numpy array to be the type

ValueError: None values not supported.

During handling of the above exception, another exception occurred:

ValueError                                Traceback (most recent call last)
<ipython-input-358-70bce7e5d400> in <module>
      3 f = (x[0] + x[1] ** 2 + x[0] * x[1] + y) ** 2
      4 xy = tf.concat([x, y], axis=-1)
----> 5 hessian = tf.hessians(f, xy)

~/venv3/lib/python3.7/site-packages/tensorflow/python/ops/gradients_impl.py in hessians(ys, xs, name, colocate_gradients_with_ops, gate_gradients, aggregation_method)
   1405   for gradient, x in zip(_gradients, xs):
   1406     # change shape to one-dimension without graph branching
-> 1407     gradient = array_ops.reshape(gradient, [-1])
   1408 
   1409     # Declare an iterator and tensor array loop variables for the gradients.

~/venv3/lib/python3.7/site-packages/tensorflow/python/ops/gen_array_ops.py in reshape(tensor, shape, name)
   7178   try:
   7179     _, _, _op = _op_def_lib._apply_op_helper(
-> 7180         "Reshape", tensor=tensor, shape=shape, name=name)
   7181   except (TypeError, ValueError):
   7182     result = _dispatch.dispatch(

~/venv3/lib/python3.7/site-packages/tensorflow/python/framework/op_def_library.py in _apply_op_helper(self, op_type_name, name, **keywords)
    527               raise ValueError(
    528                   "Tried to convert '%s' to a tensor and failed. Error: %s" %
--> 529                   (input_name, err))
    530             prefix = ("Input '%s' of '%s' Op has type %s that does not match" %
    531                       (input_name, op_type_name, observed))

ValueError: Tried to convert 'tensor' to a tensor and failed. Error: None values not supported.


1

​

回答1:

EDIT: Here is a more fleshed out solution, essentially the same but for an arbitrary number of variables. Also I have added the option of using Python or TensorFlow loops for the Jacobian. Note the code assumes all variables are 1D tensors.

from itertools import combinations, count
import tensorflow as tf

def jacobian(y, x, tf_loop=False):
    # If the shape of Y is fully defined you can choose between a
    # Python-level or TF-level loop to make the Jacobian matrix
    # If the shape of Y is not fully defined you must use TF loop
    # In both cases it is just a matter of stacking gradients for each Y
    if tf_loop or y.shape.num_elements() is None:
        i = tf.constant(0, dtype=tf.int32)
        y_size = tf.size(y)
        rows = tf.TensorArray(dtype=y.dtype, size=y_size, element_shape=x.shape)
        _, rows = tf.while_loop(
            lambda i, rows: i < y_size,
            lambda i, rows: [i + 1, rows.write(i, tf.gradients(y[i], x)[0])],
            [i, rows])
        return rows.stack()
    else:
        return tf.stack([tf.gradients(y[i], x)[0]
                         for i in range(y.shape.num_elements())], axis=0)

def hessian_multivar(ys, xs, tf_loop=False):
    # List of list of pieces of the Hessian matrix
    hessian_pieces = [[None] * len(xs) for _ in xs]
    # Hessians with respect to each x (diagonal pieces of the full Hessian)
    for i, h in enumerate(tf.hessians(ys, xs)):
        hessian_pieces[i][i] = h
    # First-order derivatives
    xs_grad = tf.gradients(ys, xs)
    # Pairwise second order derivatives as Jacobian matrices
    for (i1, (x1, g1)), (i2, (x2, g2)) in combinations(zip(count(), zip(xs, xs_grad)), 2):
        # Derivates in both orders
        hessian_pieces[i1][i2] = jacobian(g1, x2, tf_loop=tf_loop)
        hessian_pieces[i2][i1] = jacobian(g2, x1, tf_loop=tf_loop)
    # Concatenate everything together
    return tf.concat([tf.concat(hp, axis=1) for hp in hessian_pieces], axis=0)

# Test it with three variables
with tf.Graph().as_default():
    x = tf.Variable([1., 1.], dtype=tf.float32, name="x")
    y = tf.Variable([1.], dtype=tf.float32, name="y")
    z = tf.Variable([1., 1.], dtype=tf.float32, name="z")
    f = (x[0] + x[1] ** 2 + x[0] * x[1] + y + x * y * z) ** 2
    hessian = hessian_multivar(f, [x, y, z])
    init_op = tf.global_variables_initializer()
    with tf.Session() as sess:
        sess.run(init_op)
        print(sess.run(hessian))

Output:

[[26. 54. 30. 16.  4.]
 [54. 90. 38.  6. 18.]
 [30. 38. 16. 14. 14.]
 [16.  6. 14.  2.  0.]
 [ 4. 18. 14.  0.  2.]]

---

I'm not sure if there can be a "good" way of doing that with the current API. Obviously, you can compute the Hessian matrix elements by yourself... It is not very elegant and probably not the fastest solution either, but here is how it might be done in your example:

import tensorflow as tf

x = tf.Variable([1., 1.], dtype=tf.float32, name="x")
y = tf.Variable([1.], dtype=tf.float32, name="y")
f = (x[0] + x[1] ** 2 + x[0] * x[1] + y) ** 2
# X and Y pieces of Hessian
hx, hy = tf.hessians(f, [x, y])
# First-order X and Y derivatives
gx, gy = tf.gradients(f, [x, y])
# Remanining elements of Hessian can be computed as Jacobian matrices with
# X, Y and first-order derivatives. However TensorFlow does not implement this
# (https://github.com/tensorflow/tensorflow/issues/675)
# So you have to build it "by hand"
hxy = [tf.gradients(gx[i], y)[0] for i in range(x.shape.num_elements())]
hxy = tf.concat(hxy, axis=0)
# Here since Y has one element only it is easier
hyx, = tf.gradients(gy, x)
# Combine pieces of Hessian
h1 = tf.concat([hx, tf.expand_dims(hxy, 1)], axis=1)
h2 = tf.concat([tf.expand_dims(hyx, 0), hy], axis=1)
hessian = tf.concat([h1, h2], axis=0)
# Test it
init_op = tf.global_variables_initializer()
with tf.Session() as sess:
    sess.run(init_op)
    print(sess.run(hessian))

Output:

[[ 8. 20.  4.]
 [20. 34.  6.]
 [ 4.  6.  2.]]