# Copyright OTT-JAX
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import dataclasses
from typing import TYPE_CHECKING, Any, Callable, Dict, Optional, Tuple
import jax
import jax.numpy as jnp
from ott import utils
from ott.math import unbalanced_functions as uf
if TYPE_CHECKING:
from ott.problems.linear import linear_problem
LinOp_t = Callable[[jnp.ndarray], jnp.ndarray]
Solver_t = Callable[[LinOp_t, jnp.ndarray, Optional[LinOp_t], bool],
jnp.ndarray]
__all__ = ["ImplicitDiff", "solve_jax_cg"]
[docs]@utils.register_pytree_node
class ImplicitDiff:
"""Implicit differentiation of Sinkhorn algorithm.
Args:
solver: Callable to compute the solution to a linear problem. The callable
expects a linear function, a vector, optionally another linear function
that implements the transpose of that function, and a boolean flag to
specify symmetry. This solver is by default one of :class:`lineax.CG` or
:class:`lineax.NormalCG` solvers, if the package can be imported, as
described in :func:`~ott.solvers.linear.lineax_implicit.solve_lineax`.
The :mod:`jax` alternative is described in
:func:`~ott.solvers.linear.implicit_differentiation.solve_jax_cg`.
Note that `lineax` solvers handle better poorly conditioned problems,
which arise typically when differentiating the solutions of balanced OT
problems (when ``tau_a==tau_b==1.0``). Relying on
:func:`~ott.solvers.linear.implicit_differentiation.solve_jax_cg`
for such cases might require hand-tuning ridge parameters,
in particular ``ridge_kernel`` and ``ridge_identity`` as described in its
doc. These parameters can be passed using ``solver_kwargs`` below.
solver_kwargs: keyword arguments passed on to the solver.
symmetric: flag used to figure out whether the linear system solved in the
implicit function theorem is symmetric or not. This happens when
``tau_a==tau_b``, and when ``a == b``, or the precondition_fun
is the identity. The flag is False by default, and is also tested against
``tau_a==tau_b``. It needs to be set manually by the user in the more
favorable case where the system is guaranteed to be symmetric.
precondition_fun: Function used to precondition, on both sides, the linear
system derived from first-order conditions of the regularized OT problem.
That linear system typically involves an equality between marginals (or
simple transform of these marginals when the problem is unbalanced) and
another function of the potentials. When that function is specified, that
function is applied on both sides of the equality, before being further
differentiated to provide the Jacobians needed for implicit function
theorem differentiation.
"""
solver: Optional[Solver_t] = None
solver_kwargs: Optional[Dict[str, Any]] = None
symmetric: bool = False
precondition_fun: Optional[Callable[[jnp.ndarray], jnp.ndarray]] = None
[docs] def solve(
self,
gr: Tuple[jnp.ndarray, jnp.ndarray],
ot_prob: "linear_problem.LinearProblem",
f: jnp.ndarray,
g: jnp.ndarray,
lse_mode: bool,
) -> jnp.ndarray:
r"""Apply minus inverse of [hessian ``reg_ot_cost`` w.r.t. ``f``, ``g``].
This function is used to carry out implicit differentiation of ``sinkhorn``
outputs, notably optimal potentials ``f`` and ``g``. That differentiation
requires solving a linear system, using (and inverting) the Jacobian of
(preconditioned) first-order conditions w.r.t. the reg-OT problem.
Given a ``precondition_fun``, written here for short as :math:`h`,
the first order conditions for the dual energy
.. math::
E(K, \epsilon, a, b, f, g) :=- <a,\phi_a^{*}(-f)> + <b,
\phi_b^{*}(-g)> - \langle\exp^{f/\epsilon}, K \exp^{g/\epsilon}>
form the basis of the Sinkhorn algorithm. To differentiate optimal solutions
to that problem, we exploit the fact that :math:`h(\nabla E = 0)` and
differentiate that identity to recover variations (Jacobians) of optimal
solutions :math:`f^\star, g^\star$` as a function of changes in the inputs.
The Jacobian of :math:`h(\nabla_{f,g} E = 0)` is a linear operator which, if
it were to be instantiated as a matrix, would be of size
:math:`(n+m) \times (n+m)`. When :math:`h` is the identity, that matrix is
the Hessian of :math:`E`, is symmetric and negative-definite
(:math:`E` is concave) and is structured as :math:`[A, B; B^T, D]`. More
generally, for other functions :math:`h`, the Jacobian of these
preconditioned
first order conditions is no longer symmetric (except if ``a==b``), and
has now a structure as :math:`[A, B; C, D]`. That system can
be still inverted more generic solvers. By default, :math:`h = \epsilon
\log`, as proposed in :cite:`cuturi:20a`.
In both cases :math:`A` and :math:`D` are diagonal matrices, equal to the
row and
column marginals respectively, multiplied by the derivatives of :math:`h`
evaluated at those marginals, corrected (if handling the unbalanced case)
by the second derivative of the part of the objective that ties potentials
to the marginals (terms in ``phi_star``). When :math:`h` is the identity,
:math:`B` and :math:`B^T` are equal respectively to the OT matrix and its
transpose, i.e. :math:`n \times m` and :math:`m \times n` matrices.
When :math:`h` is not the identity, :math:`B` (resp. :math:`C`) is equal
to the OT matrix (resp. its transpose), rescaled on the left by the
application elementwise of :math:`h'` to the row (respectively column)
marginal sum of the transport.
Note that we take great care in not instantiating these transport
matrices, to rely instead on calls to the ``app_transport`` method from the
``Geometry`` object ``geom`` (which will either use potentials or scalings,
depending on ``lse_mode``)
The Jacobian's diagonal + off-diagonal blocks structure allows to exploit
Schur complements. Depending on the sizes involved, it is better to
instantiate the Schur complement of the first or of the second diagonal
block.
These linear systems are solved using the user-defined ``solver``, using
by default :mod:`lineax` solvers when available, or falling back on
:mod:`jax` when not.
Args:
gr: 2-tuple, (vector of size ``n``, vector of size ``m``).
ot_prob: the instantiation of the regularized transport problem.
f: potential, w.r.t marginal a.
g: potential, w.r.t marginal b.
lse_mode: bool, log-sum-exp mode if True, kernel else.
Returns:
A tuple of two vectors, of the same size as ``gr``.
"""
solver = _get_solver() if self.solver is None else self.solver
solver_kwargs = {} if self.solver_kwargs is None else self.solver_kwargs
geom = ot_prob.geom
marginal_a, marginal_b, app_transport = (
ot_prob.get_transport_functions(lse_mode)
)
if self.precondition_fun is None:
precond_fun = lambda x: geom.epsilon * jnp.log(x)
symmetric = False
else:
precond_fun = self.precondition_fun
symmetric = self.symmetric
derivative = jax.vmap(jax.grad(precond_fun))
n, m = geom.shape
# pylint: disable=g-long-lambda
vjp_fg = lambda z: app_transport(
f, g, z * derivative(marginal_b(f, g)), axis=1
) / geom.epsilon
vjp_gf = lambda z: app_transport(
f, g, z * derivative(marginal_a(f, g)), axis=0
) / geom.epsilon
if not symmetric:
vjp_fgt = lambda z: app_transport(
f, g, z, axis=0
) * derivative(marginal_b(f, g)) / geom.epsilon
vjp_gft = lambda z: app_transport(
f, g, z, axis=1
) * derivative(marginal_a(f, g)) / geom.epsilon
diag_hess_a = (
marginal_a(f, g) * derivative(marginal_a(f, g)) / geom.epsilon +
uf.diag_jacobian_of_marginal_fit(
ot_prob.a, f, ot_prob.tau_a, geom.epsilon, derivative
)
)
diag_hess_b = (
marginal_b(f, g) * derivative(marginal_b(f, g)) / geom.epsilon +
uf.diag_jacobian_of_marginal_fit(
ot_prob.b, g, ot_prob.tau_b, geom.epsilon, derivative
)
)
n, m = geom.shape
# TODO(cuturi) consider materializing linear operator schur if size allows.
# Forks on using Schur complement of either A or D, depending on size.
if n > m: # if n is bigger, run m x m linear system.
inv_vjp_ff = lambda z: z / diag_hess_a
vjp_gg = lambda z: z * diag_hess_b
schur = lambda z: vjp_gg(z) - vjp_gf(inv_vjp_ff(vjp_fg(z)))
if not symmetric:
schur_t = lambda z: vjp_gg(z) - vjp_fgt(inv_vjp_ff(vjp_gft(z)))
else:
schur_t = None
res = gr[1] - vjp_gf(inv_vjp_ff(gr[0]))
sch = solver(schur, res, schur_t, symmetric, **solver_kwargs)
vjp_gr_f = inv_vjp_ff(gr[0] - vjp_fg(sch))
vjp_gr_g = sch
else:
vjp_ff = lambda z: z * diag_hess_a
inv_vjp_gg = lambda z: z / diag_hess_b
schur = lambda z: vjp_ff(z) - vjp_fg(inv_vjp_gg(vjp_gf(z)))
if not symmetric:
schur_t = lambda z: vjp_ff(z) - vjp_gft(inv_vjp_gg(vjp_fgt(z)))
else:
schur_t = None
res = gr[0] - vjp_fg(inv_vjp_gg(gr[1]))
sch = solver(schur, res, schur_t, symmetric, **solver_kwargs)
vjp_gr_g = inv_vjp_gg(gr[1] - vjp_gf(sch))
vjp_gr_f = sch
return jnp.concatenate((-vjp_gr_f, -vjp_gr_g))
[docs] def first_order_conditions(
self, prob, f: jnp.ndarray, g: jnp.ndarray, lse_mode: bool
):
r"""Compute vector of first order conditions for the reg-OT problem.
The output of this vector should be close to zero at optimality.
Upon completion of the Sinkhorn forward pass, its norm (using the norm
parameter defined using ``norm_error``) should be below the threshold
parameter.
This error will be itself assumed to be close to zero when using implicit
differentiation.
Args:
prob: definition of the linear optimal transport problem.
f: jnp.ndarray, first potential
g: jnp.ndarray, second potential
lse_mode: bool
Returns:
a jnp.ndarray of size (size of ``n + m``) quantifying deviation to
optimality for variables ``f`` and ``g``.
"""
geom = prob.geom
marginal_a, marginal_b, _ = prob.get_transport_functions(lse_mode)
grad_a = uf.grad_of_marginal_fit(prob.a, f, prob.tau_a, geom.epsilon)
grad_b = uf.grad_of_marginal_fit(prob.b, g, prob.tau_b, geom.epsilon)
if self.precondition_fun is None:
precond_fun = lambda x: geom.epsilon * jnp.log(x)
else:
precond_fun = self.precondition_fun
result_a = jnp.where(
prob.a > 0,
precond_fun(marginal_a(f, g)) - precond_fun(grad_a), 0.0
)
result_b = jnp.where(
prob.b > 0,
precond_fun(marginal_b(f, g)) - precond_fun(grad_b), 0.0
)
return jnp.concatenate((result_a, result_b))
[docs] def gradient(
self, prob: "linear_problem.LinearProblem", f: jnp.ndarray,
g: jnp.ndarray, lse_mode: bool, gr: Tuple[jnp.ndarray, jnp.ndarray]
) -> "linear_problem.LinearProblem":
"""Apply VJP to recover gradient in reverse mode differentiation."""
# Applies first part of vjp to gr: inverse part of implicit function theorem
vjp_gr = self.solve(gr, prob, f, g, lse_mode)
# Instantiates vjp of first order conditions of the objective, as a
# function of geom, a and b parameters (against which we differentiate)
foc_prob = lambda prob: self.first_order_conditions(prob, f, g, lse_mode)
# Carries pullback onto original inputs, here geom, a and b.
_, pull_prob = jax.vjp(foc_prob, prob)
return pull_prob(vjp_gr)
[docs] def replace(self, **kwargs: Any) -> "ImplicitDiff": # noqa: D102
return dataclasses.replace(self, **kwargs)
[docs]def solve_jax_cg(
lin: LinOp_t,
b: jnp.ndarray,
lin_t: Optional[LinOp_t] = None,
symmetric: bool = False,
ridge_identity: float = 0.0,
ridge_kernel: float = 0.0,
**kwargs: Any
) -> jnp.ndarray:
"""Wrapper around JAX native linear solvers.
Args:
lin: Linear operator
b: vector. Returned `x` is such that `lin(x)=b`
lin_t: Linear operator, corresponding to transpose of `lin`.
symmetric: whether `lin` is symmetric.
ridge_kernel: promotes zero-sum solutions. Only use if `tau_a = tau_b = 1.0`
ridge_identity: handles rank deficient transport matrices (this happens
typically when rows/cols in cost/kernel matrices are collinear, or,
equivalently when two points from either measure are close).
kwargs: arguments passed to :func:`~jax.scipy.sparse.linalg.cg`
"""
op = lin if symmetric else lambda x: lin_t(lin(x))
if ridge_kernel > 0.0 or ridge_identity > 0.0:
lin_reg = lambda x: op(x) + ridge_kernel * jnp.sum(x) + ridge_identity * x
else:
lin_reg = op
return jax.scipy.sparse.linalg.cg(lin_reg, b, **kwargs)[0]
def _get_solver() -> Solver_t:
"""Get lineax solver when possible, default to jax.scipy else."""
try:
from ott.solvers.linear import lineax_implicit
return lineax_implicit.solve_lineax
except ImportError:
return solve_jax_cg
