Add helper to build hessian vector product

Co-authored-by: Adrian Seyboldt <[email protected]>

doc/tutorial/gradients.rst

@@ -267,6 +267,16 @@ or, making use of the R-operator:
 >>> f([4, 4], [2, 2])
 array([ 4.,  4.])
 
+There is a builtin helper that uses the first method
+
+>>> x = pt.dvector('x')
+>>> v = pt.dvector('v')
+>>> y = pt.sum(x ** 2)
+>>> Hv = pytensor.gradient.hessian_vector_product(y, x, v)
+>>> f = pytensor.function([x, v], Hv)
+>>> f([4, 4], [2, 2])
+array([ 4.,  4.])
+
 
 Final Pointers
 ==============

pytensor/gradient.py

@@ -2052,6 +2052,83 @@ def hessian(cost, wrt, consider_constant=None, disconnected_inputs="raise"):
     return as_list_or_tuple(using_list, using_tuple, hessians)
 
 
+def hessian_vector_product(cost, wrt, p, **grad_kwargs):
+    """Return the expression of the Hessian times a vector p.
+
+    Notes
+    -----
+    This function uses backward autodiff twice to obtain the desired expression.
+    You may want to manually build the equivalent expression by combining backward
+    followed by forward (if all Ops support it) autodiff.
+    See {ref}`docs/_tutcomputinggrads#Hessian-times-a-Vector` for how to do this.
+
+    Parameters
+    ----------
+    cost: Scalar (0-dimensional) variable.
+    wrt: Vector (1-dimensional tensor) 'Variable' or list of Vectors
+    p: Vector (1-dimensional tensor) 'Variable' or list of Vectors
+        Each vector will be used for the hessp wirt to exach input variable
+    **grad_kwargs:
+        Keyword arguments passed to `grad` function.
+
+    Returns
+    -------
+    :class:` Vector or list of Vectors
+        The Hessian times p of the `cost` with respect to (elements of) `wrt`.
+
+    Examples
+    --------
+
+    >>> import numpy as np
+    >>> from scipy.optimize import minimize
+    >>> from pytensor import function
+    >>> from pytensor.tensor import vector
+    >>> from pytensor.gradient import grad, hessian_vector_product
+    >>>
+    >>> x = vector('x')
+    >>> p = vector('p')
+    >>>
+    >>> rosen = (100 * (x[1:] - x[:-1] ** 2) ** 2 + (1 - x[:-1]) ** 2).sum()
+    >>> rosen_jac = grad(rosen, x)
+    >>> rosen_hessp = hessian_vector_product(rosen, x, p)
+    >>>
+    >>> rosen_fn = function([x], rosen)
+    >>> rosen_jac_fn = function([x], rosen_jac)
+    >>> rosen_hessp_fn = function([x, p], rosen_hessp)
+    >>> x0 = np.array([1.3, 0.7, 0.8, 1.9, 1.2])
+    >>> res = minimize(
+    ...     rosen_fn,
+    ...     x0,
+    ...     method="Newton-CG",
+    ...     jac=rosen_jac_fn,
+    ...     hessp=rosen_hessp_fn,
+    ...     options={"xtol": 1e-8, "disp": True},
+    ... )
+    Optimization terminated successfully.
+             Current function value: 0.000000
+             Iterations: 24
+             Function evaluations: 33
+             Gradient evaluations: 33
+             Hessian evaluations: 66
+    >>> res.x
+    array([1.        , 1.        , 1.        , 0.99999999, 0.99999999])
+    """
+    wrt_list = wrt if isinstance(wrt, Sequence) else [wrt]
+    p_list = p if isinstance(p, Sequence) else [p]
+    grad_wrt_list = grad(cost, wrt=wrt_list, **grad_kwargs)
+    hessian_cost = pytensor.tensor.add(
+        *[
+            (grad_wrt * p).sum()
+            for grad_wrt, p in zip(grad_wrt_list, p_list, strict=True)
+        ]
+    )
+    Hp_list = grad(hessian_cost, wrt=wrt_list, **grad_kwargs)
+
+    if isinstance(wrt, Variable):
+        return Hp_list[0]
+    return Hp_list
+
+
 def _is_zero(x):
     """
     Returns 'yes', 'no', or 'maybe' indicating whether x

tests/test_gradient.py

@@ -2,6 +2,7 @@
 
 import numpy as np
 import pytest
+from scipy.optimize import rosen_hess_prod
 
 import pytensor
 import pytensor.tensor.basic as ptb
@@ -22,6 +23,7 @@
     grad_scale,
     grad_undefined,
     hessian,
+    hessian_vector_product,
     jacobian,
     subgraph_grad,
     zero_grad,
@@ -1081,3 +1083,40 @@ def test_jacobian_disconnected_inputs():
     func_s = pytensor.function([s2], jacobian_s)
     val = np.array(1.0).astype(pytensor.config.floatX)
     assert np.allclose(func_s(val), np.zeros(1))
+
+
+class TestHessianVectorProdudoct:
+    def test_rosen(self):
+        x = vector("x", dtype="float64")
+        rosen = (100 * (x[1:] - x[:-1] ** 2) ** 2 + (1 - x[:-1]) ** 2).sum()
+
+        p = vector("p", dtype="float64")
+        rosen_hess_prod_pt = hessian_vector_product(rosen, wrt=x, p=p)
+
+        x_test = 0.1 * np.arange(9)
+        p_test = 0.5 * np.arange(9)
+        np.testing.assert_allclose(
+            rosen_hess_prod_pt.eval({x: x_test, p: p_test}),
+            rosen_hess_prod(x_test, p_test),
+        )
+
+    def test_multiple_wrt(self):
+        x = vector("x", dtype="float64")
+        y = vector("y", dtype="float64")
+        p_x = vector("p_x", dtype="float64")
+        p_y = vector("p_y", dtype="float64")
+
+        cost = (x**2 - y**2).sum()
+        hessp_x, hessp_y = hessian_vector_product(cost, wrt=[x, y], p=[p_x, p_y])
+
+        hessp_fn = pytensor.function([x, y, p_x, p_y], [hessp_x, hessp_y])
+        test = {
+            # x, y don't matter
+            "x": np.full((3,), np.nan),
+            "y": np.full((3,), np.nan),
+            "p_x": [1, 2, 3],
+            "p_y": [3, 2, 1],
+        }
+        hessp_x_eval, hessp_y_eval = hessp_fn(**test)
+        np.testing.assert_allclose(hessp_x_eval, [2, 4, 6])
+        np.testing.assert_allclose(hessp_y_eval, [-6, -4, -2])