Source code for ott.solvers.linear.sinkhorn

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from typing import (

import jax
import jax.numpy as jnp
import jax.scipy as jsp
import numpy as np

from ott import utils
from ott.geometry import geometry
from ott.initializers.linear import initializers as init_lib
from ott.math import fixed_point_loop
from ott.math import unbalanced_functions as uf
from ott.math import utils as mu
from ott.problems.linear import linear_problem, potentials
from ott.solvers.linear import acceleration
from ott.solvers.linear import implicit_differentiation as implicit_lib

__all__ = ["Sinkhorn", "SinkhornOutput"]

ProgressCallbackFn_t = Callable[
    [Tuple[np.ndarray, np.ndarray, np.ndarray, "SinkhornState"]], None]

[docs] class SinkhornState(NamedTuple): """Holds the state variables used to solve OT with Sinkhorn.""" potentials: Tuple[jnp.ndarray, ...] errors: Optional[jnp.ndarray] = None old_fus: Optional[jnp.ndarray] = None old_mapped_fus: Optional[jnp.ndarray] = None
[docs] def set(self, **kwargs: Any) -> "SinkhornState": """Return a copy of self, with potential overwrites.""" return self._replace(**kwargs)
[docs] def solution_error( self, ot_prob: linear_problem.LinearProblem, norm_error: Sequence[int], *, lse_mode: bool, parallel_dual_updates: bool, recenter: bool, ) -> jnp.ndarray: """State dependent function to return error.""" fu, gv = self.fu, self.gv if recenter and lse_mode: fu, gv = self.recenter(fu, gv, ot_prob=ot_prob) return solution_error( fu, gv, ot_prob, norm_error=norm_error, lse_mode=lse_mode, parallel_dual_updates=parallel_dual_updates )
[docs] def compute_kl_reg_cost( # noqa: D102 self, ot_prob: linear_problem.LinearProblem, lse_mode: bool ) -> float: return compute_kl_reg_cost(self.fu, self.gv, ot_prob, lse_mode)
[docs] def recenter( self, f: jnp.ndarray, g: jnp.ndarray, ot_prob: linear_problem.LinearProblem, ) -> Tuple[jnp.ndarray, jnp.ndarray]: """Re-center dual potentials. If the ``ot_prob`` is balanced, the ``f`` potential is zero-centered. Otherwise, use prop. 2 of :cite:`sejourne:22` re-center the potentials iff ``tau_a < 1`` and ``tau_b < 1``. Args: f: The first dual potential. g: The second dual potential. ot_prob: Linear OT problem. Returns: The centered potentials. """ if ot_prob.is_balanced: # center the potentials for numerical stability is_finite = jnp.isfinite(f) shift = jnp.sum(jnp.where(is_finite, f, 0.0)) / jnp.sum(is_finite) return f - shift, g + shift if ot_prob.tau_a == 1.0 or ot_prob.tau_b == 1.0: # re-centering wasn't done during the lse-step, ignore return f, g rho_a = uf.rho(ot_prob.epsilon, ot_prob.tau_a) rho_b = uf.rho(ot_prob.epsilon, ot_prob.tau_b) tau = rho_a * rho_b / (rho_a + rho_b) shift = tau * ( mu.logsumexp(-f / rho_a, b=ot_prob.a) - mu.logsumexp(-g / rho_b, b=ot_prob.b) ) return f + shift, g - shift
@property def fu(self) -> jnp.ndarray: """The first dual potential or scaling.""" return self.potentials[0] @property def gv(self) -> jnp.ndarray: """The second dual potential or scaling.""" return self.potentials[1]
def solution_error( f_u: jnp.ndarray, g_v: jnp.ndarray, ot_prob: linear_problem.LinearProblem, *, norm_error: Sequence[int], lse_mode: bool, parallel_dual_updates: bool, ) -> jnp.ndarray: """Given two potential/scaling solutions, computes deviation to optimality. When the ``ot_prob`` problem is balanced and the usual Sinkhorn updates are used, this is simply deviation of the coupling's marginal to ``ot_prob.b``. This is the case because the second (and last) update of the Sinkhorn algorithm equalizes the row marginal of the coupling to ``ot_prob.a``. To simplify the logic, this is parameterized by checking whether `parallel_dual_updates = False`. When that flag is `True`, or when the problem is unbalanced, additional quantities to qualify optimality must be taken into account. Args: f_u: jnp.ndarray, potential or scaling g_v: jnp.ndarray, potential or scaling ot_prob: linear OT problem norm_error: int, p-norm used to compute error. lse_mode: True if log-sum-exp operations, False if kernel vector products. parallel_dual_updates: Whether potentials/scalings were computed in parallel. Returns: a positive number quantifying how far from optimality current solution is. """ if ot_prob.is_balanced and not parallel_dual_updates: return marginal_error( f_u, g_v, ot_prob.b, ot_prob.geom, 0, norm_error, lse_mode ) # In the unbalanced case, we compute the norm of the gradient. # the gradient is equal to the marginal of the current plan minus # the gradient of < z, rho_z(exp^(-h/rho_z) -1> where z is either a or b # and h is either f or g. Note this is equal to z if rho_z → inf, which # is the case when tau_z → 1.0 if lse_mode: grad_a = uf.grad_of_marginal_fit( ot_prob.a, f_u, ot_prob.tau_a, ot_prob.epsilon ) grad_b = uf.grad_of_marginal_fit( ot_prob.b, g_v, ot_prob.tau_b, ot_prob.epsilon ) else: u = ot_prob.geom.potential_from_scaling(f_u) v = ot_prob.geom.potential_from_scaling(g_v) grad_a = uf.grad_of_marginal_fit( ot_prob.a, u, ot_prob.tau_a, ot_prob.epsilon ) grad_b = uf.grad_of_marginal_fit( ot_prob.b, v, ot_prob.tau_b, ot_prob.epsilon ) err = marginal_error(f_u, g_v, grad_a, ot_prob.geom, 1, norm_error, lse_mode) err += marginal_error(f_u, g_v, grad_b, ot_prob.geom, 0, norm_error, lse_mode) return err def marginal_error( f_u: jnp.ndarray, g_v: jnp.ndarray, target: jnp.ndarray, geom: geometry.Geometry, axis: int = 0, norm_error: Sequence[int] = (1,), lse_mode: bool = True ) -> jnp.asarray: """Output how far Sinkhorn solution is w.r.t target. Args: f_u: a vector of potentials or scalings for the first marginal. g_v: a vector of potentials or scalings for the second marginal. target: target marginal. geom: Geometry object. axis: axis (0 or 1) along which to compute marginal. norm_error: (tuple of int) p's to compute p-norm between marginal/target lse_mode: whether operating on scalings or potentials Returns: Array of floats, quantifying difference between target / marginal. """ if lse_mode: marginal = geom.marginal_from_potentials(f_u, g_v, axis=axis) else: marginal = geom.marginal_from_scalings(f_u, g_v, axis=axis) norm_error = jnp.asarray(norm_error) return jnp.sum( jnp.abs(marginal - target) ** norm_error[:, jnp.newaxis], axis=1 ) ** (1.0 / norm_error) def compute_kl_reg_cost( f: jnp.ndarray, g: jnp.ndarray, ot_prob: linear_problem.LinearProblem, lse_mode: bool ) -> float: r"""Compute objective of Sinkhorn for OT problem given dual solutions. The objective is evaluated for dual solution ``f`` and ``g``, using information contained in ``ot_prob``. The objective is the regularized optimal transport cost (i.e. the cost itself plus entropic and unbalanced terms). Situations where marginals ``a`` or ``b`` in ot_prob have zero coordinates are reflected in minus infinity entries in their corresponding dual potentials. To avoid NaN that may result when multiplying 0's by infinity values, ``jnp.where`` is used to cancel these contributions. Args: f: jnp.ndarray, potential g: jnp.ndarray, potential ot_prob: linear optimal transport problem. lse_mode: bool, whether to compute total mass in lse or kernel mode. Returns: The regularized transport cost. """ supp_a = ot_prob.a > 0 supp_b = ot_prob.b > 0 fa = ot_prob.geom.potential_from_scaling(ot_prob.a) if ot_prob.tau_a == 1.0: div_a = jnp.sum(jnp.where(supp_a, ot_prob.a * (f - fa), 0.0)) else: rho_a = uf.rho(ot_prob.epsilon, ot_prob.tau_a) div_a = -jnp.sum( jnp.where(supp_a, ot_prob.a * uf.phi_star(-(f - fa), rho_a), 0.0) ) gb = ot_prob.geom.potential_from_scaling(ot_prob.b) if ot_prob.tau_b == 1.0: div_b = jnp.sum(jnp.where(supp_b, ot_prob.b * (g - gb), 0.0)) else: rho_b = uf.rho(ot_prob.epsilon, ot_prob.tau_b) div_b = -jnp.sum( jnp.where(supp_b, ot_prob.b * uf.phi_star(-(g - gb), rho_b), 0.0) ) # Using (24) if lse_mode: total_sum = jnp.sum(ot_prob.geom.marginal_from_potentials(f, g)) else: u = ot_prob.geom.scaling_from_potential(f) v = ot_prob.geom.scaling_from_potential(g) total_sum = jnp.sum(ot_prob.geom.marginal_from_scalings(u, v)) return div_a + div_b + ot_prob.epsilon * ( jnp.sum(ot_prob.a) * jnp.sum(ot_prob.b) - total_sum )
[docs] class SinkhornOutput(NamedTuple): """Holds the output of a Sinkhorn solver applied to a problem. Objects of this class contain both solutions and problem definition of a regularized OT problem, along several methods that can be used to access its content, to, for instance, materialize an OT matrix or apply it to a vector (without having to materialize it when not needed). Args: f: dual variables vector of size ``ot.prob.shape[0]`` returned by Sinkhorn g: dual variables vector of size ``ot.prob.shape[1]`` returned by Sinkhorn errors: vector or errors, along iterations. This vector is of size ``max_iterations // inner_iterations`` where those were the parameters passed on to the :class:`~ott.solvers.linear.sinkhorn.Sinkhorn` solver. For each entry indexed at ``i``, ``errors[i]`` can be either a real non-negative value (meaning the algorithm recorded that error at the ``i * inner_iterations`` iteration), a ``jnp.inf`` value (meaning the algorithm computed that iteration but did not compute its error, because, for instance, ``i < min_iterations // inner_iterations``), or a ``-1``, meaning that execution was terminated before that iteration, because the criterion was found to be smaller than ``threshold``. reg_ot_cost: the regularized optimal transport cost. By default this is the linear contribution + KL term. See :attr:`~ott.solvers.linear.sinkhorn.SinkhornOutput.ent_reg_cost`, :attr:`~ott.solvers.linear.sinkhorn.SinkhornOutput.primal_cost` and :attr:`~ott.solvers.linear.sinkhorn.SinkhornOutput.dual_cost` for other objective values. ot_prob: stores the definition of the OT problem, including geometry, marginals, unbalanced regularizers, etc. threshold: convergence threshold used to control the termination of the algorithm. converged: whether the output corresponds to a solution whose error is below the convergence threshold. inner_iterations: number of iterations that were run between two computations of errors. """ potentials: Tuple[jnp.ndarray, ...] errors: Optional[jnp.ndarray] = None reg_ot_cost: Optional[float] = None ot_prob: Optional[linear_problem.LinearProblem] = None threshold: Optional[jnp.ndarray] = None converged: Optional[bool] = None inner_iterations: Optional[int] = None
[docs] def set(self, **kwargs: Any) -> "SinkhornOutput": """Return a copy of self, with potential overwrites.""" return self._replace(**kwargs)
[docs] def set_cost( # noqa: D102 self, ot_prob: linear_problem.LinearProblem, lse_mode: bool, use_danskin: bool ) -> "SinkhornOutput": f = jax.lax.stop_gradient(self.f) if use_danskin else self.f g = jax.lax.stop_gradient(self.g) if use_danskin else self.g return self.set(reg_ot_cost=compute_kl_reg_cost(f, g, ot_prob, lse_mode))
@property def dual_cost(self) -> jnp.ndarray: """Return dual transport cost, without considering regularizer.""" a, b = self.ot_prob.a, self.ot_prob.b dual_cost = jnp.sum(jnp.where(a > 0.0, a * self.f, 0)) dual_cost += jnp.sum(jnp.where(b > 0.0, b * self.g, 0)) return dual_cost @property def primal_cost(self) -> float: """Return transport cost of current transport solution at geometry.""" return self.transport_cost_at_geom(other_geom=self.geom) @property def ent_reg_cost(self) -> float: r"""Entropy regularized cost. This outputs .. math:: \langle P^{\star},C\rangle - \varepsilon H(P^{\star}) + \rho_a\text{KL}(P^{\star} 1|a) + \rho_b\text{KL}(1^T P^{\star}|b), where :math:`P^{\star}, a, b` is the coupling returned by the :class:`~ott.solvers.linear.sinkhorn.Sinkhorn` and the two marginal weight vectors; :math:`\rho_a=\varepsilon \tau_a / (1-\tau_a)` and :math:`\rho_b=\varepsilon \tau_b / (1-\tau_b)` are obtained when the problem is unbalanced from parameters ``tau_a`` and ``tau_b``. Note that the last two terms vanish in the balanced case, when ``tau_a==tau_b==1``. """ ent_a = jnp.sum(jsp.special.entr(self.ot_prob.a)) ent_b = jnp.sum(jsp.special.entr(self.ot_prob.b)) return self.reg_ot_cost - self.geom.epsilon * (ent_a + ent_b) @property def kl_reg_cost(self) -> float: r"""KL regularized OT transport cost. This outputs .. math:: \langle P^{\star}, C \rangle + \varepsilon KL(P^{\star},ab^T) + \rho_a\text{KL}(P^{\star} 1|a) + \rho_b\text{KL}(1^T P^{\star}|b), where :math:`P^{\star}, a, b` are the coupling returned by the :class:`~ott.solvers.linear.sinkhorn.Sinkhorn` algorithm and the two marginal weight vectors, respectively, and :math:`\rho_a=\varepsilon \tau_a / (1-\tau_a)` and :math:`\rho_b=\varepsilon \tau_b / (1-\tau_b)` are obtained when the problem is unbalanced from parameters ``tau_a`` and ``tau_b``. Note that the last two terms vanish in the balanced case, when ``tau_a==tau_b==1``. This quantity coincides with :attr:`reg_ot_cost`, which is computed using dual variables. """ return self.reg_ot_cost
[docs] def transport_cost_at_geom( self, other_geom: geometry.Geometry ) -> jnp.ndarray: r"""Return bare transport cost of current solution at any geometry. In order to compute cost, we check first if the geometry can be converted to a low-rank cost geometry in order to speed up computations, without having to materialize the full cost matrix. If this is not possible, we resort to instantiating both transport matrix and cost matrix. Args: other_geom: geometry whose cost matrix is used to evaluate the transport cost. Returns: the transportation cost at :math:`C`, i.e. :math:`\langle P, C \rangle`. """ # TODO(cuturi): handle online mode for non Euclidean pointcloud geometries. # TODO(michalk8): handle SqEucl point cloud is not converted to LRCGeom if other_geom.can_LRC: geom = other_geom.to_LRCGeometry() return jnp.sum(self.apply(geom.cost_1.T) * geom.cost_2.T) return jnp.sum(self.matrix * other_geom.cost_matrix)
@property def geom(self) -> geometry.Geometry: # noqa: D102 return self.ot_prob.geom @property def a(self) -> jnp.ndarray: # noqa: D102 return self.ot_prob.a @property def b(self) -> jnp.ndarray: # noqa: D102 return self.ot_prob.b @property def n_iters(self) -> int: # noqa: D102 """Returns the total number of iterations that were needed to terminate.""" return jnp.sum(self.errors != -1) * self.inner_iterations @property def scalings(self) -> Tuple[jnp.ndarray, jnp.ndarray]: # noqa: D102 u = self.ot_prob.geom.scaling_from_potential(self.f) v = self.ot_prob.geom.scaling_from_potential(self.g) return u, v @property def matrix(self) -> jnp.ndarray: """Transport matrix if it can be instantiated.""" try: return self.ot_prob.geom.transport_from_potentials(self.f, self.g) except ValueError: return self.ot_prob.geom.transport_from_scalings(*self.scalings) @property def transport_mass(self) -> float: """Sum of transport matrix.""" return self.marginal(0).sum()
[docs] def apply( self, inputs: jnp.ndarray, axis: int = 0, lse_mode: bool = True ) -> jnp.ndarray: """Apply the transport to a ndarray; axis=1 for its transpose.""" geom = self.ot_prob.geom if lse_mode: return geom.apply_transport_from_potentials( self.f, self.g, inputs, axis=axis ) u = geom.scaling_from_potential(self.f) v = geom.scaling_from_potential(self.g) return geom.apply_transport_from_scalings(u, v, inputs, axis=axis)
[docs] def marginal(self, axis: int) -> jnp.ndarray: # noqa: D102 return self.ot_prob.geom.marginal_from_potentials(self.f, self.g, axis=axis)
[docs] def cost_at_geom(self, other_geom: geometry.Geometry) -> float: """Return reg-OT cost for matrix, evaluated at other cost matrix.""" return ( jnp.sum(self.matrix * other_geom.cost_matrix) - self.geom.epsilon * jnp.sum(jax.scipy.special.entr(self.matrix)) )
[docs] def to_dual_potentials(self) -> potentials.EntropicPotentials: """Return the entropic map estimator.""" return potentials.EntropicPotentials(self.f, self.g, self.ot_prob)
@property def f(self) -> jnp.ndarray: """The first dual potential.""" return self.potentials[0] @property def g(self) -> jnp.ndarray: """The second dual potential.""" return self.potentials[1]
[docs] @jax.tree_util.register_pytree_node_class class Sinkhorn: r"""Sinkhorn solver. The Sinkhorn algorithm is a fixed point iteration that solves a regularized optimal transport (reg-OT) problem between two measures. The optimization variables are a pair of vectors (called potentials, or scalings when parameterized as exponential of the former). Calling this function returns therefore a pair of optimal vectors. In addition to these, it also returns the objective value achieved by these optimal vectors; a vector of size ``max_iterations/inner_iterations`` that records the vector of values recorded to monitor convergence, throughout the execution of the algorithm (padded with `-1` if convergence happens before), as well as a boolean to signify whether the algorithm has converged within the number of iterations specified by the user. The reg-OT problem is specified by two measures, of respective sizes ``n`` and ``m``. From the viewpoint of the ``sinkhorn`` function, these two measures are only seen through a triplet (``geom``, ``a``, ``b``), where ``geom`` is a ``Geometry`` object, and ``a`` and ``b`` are weight vectors of respective sizes ``n`` and ``m``. Starting from two initial values for those potentials or scalings (both can be defined by the user by passing value in ``init_dual_a`` or ``init_dual_b``), the Sinkhorn algorithm will use elementary operations that are carried out by the ``geom`` object. Math: Given a geometry ``geom``, which provides a cost matrix :math:`C` with its regularization parameter :math:`\varepsilon`, (or a kernel matrix :math:`K`) the reg-OT problem consists in finding two vectors `f`, `g` of size ``n``, ``m`` that maximize the following criterion. .. math:: \arg\max_{f, g}{- \langle a, \phi_a^{*}(-f) \rangle - \langle b, \phi_b^{*}(-g) \rangle - \varepsilon \langle e^{f/\varepsilon}, e^{-C/\varepsilon} e^{-g/\varepsilon}} \rangle where :math:`\phi_a(z) = \rho_a z(\log z - 1)` is a scaled entropy, and :math:`\phi_a^{*}(z) = \rho_a e^{z/\varepsilon}`, its Legendre transform. That problem can also be written, instead, using positive scaling vectors `u`, `v` of size ``n``, ``m``, handled with the kernel :math:`K := e^{-C/\varepsilon}`, .. math:: \arg\max_{u, v >0} - \langle a,\phi_a^{*}(-\varepsilon\log u) \rangle + \langle b, \phi_b^{*}(-\varepsilon\log v) \rangle - \langle u, K v \rangle Both of these problems corresponds, in their *primal* formulation, to solving the unbalanced optimal transport problem with a variable matrix :math:`P` of size ``n`` x ``m``: .. math:: \arg\min_{P>0} \langle P,C \rangle -\varepsilon \text{KL}(P | ab^T) + \rho_a \text{KL}(P\mathbf{1}_m | a) + \rho_b \text{KL}(P^T \mathbf{1}_n | b) where :math:`KL` is the generalized Kullback-Leibler divergence. The very same primal problem can also be written using a kernel :math:`K` instead of a cost :math:`C` as well: .. math:: \arg\min_{P} \varepsilon \text{KL}(P|K) + \rho_a \text{KL}(P\mathbf{1}_m | a) + \rho_b \text{KL}(P^T \mathbf{1}_n | b) The *original* OT problem taught in linear programming courses is recovered by using the formulation above relying on the cost :math:`C`, and letting :math:`\varepsilon \rightarrow 0`, and :math:`\rho_a, \rho_b \rightarrow \infty`. In that case the entropy disappears, whereas the :math:`KL` regularization above become constraints on the marginals of :math:`P`: This results in a standard min cost flow problem. This problem is not handled for now in this toolbox, which focuses exclusively on the case :math:`\varepsilon > 0`. The *balanced* regularized OT problem is recovered for finite :math:`\varepsilon > 0` but letting :math:`\rho_a, \rho_b \rightarrow \infty`. This problem can be shown to be equivalent to a matrix scaling problem, which can be solved using the Sinkhorn fixed-point algorithm. To handle the case :math:`\rho_a, \rho_b \rightarrow \infty`, the ``sinkhorn`` function uses parameters ``tau_a`` and ``tau_b`` equal respectively to :math:`\rho_a /(\varepsilon + \rho_a)` and :math:`\rho_b / (\varepsilon + \rho_b)` instead. Setting either of these parameters to 1 corresponds to setting the corresponding :math:`\rho_a, \rho_b` to :math:`\infty`. The Sinkhorn algorithm solves the reg-OT problem by seeking optimal :math:`f`, :math:`g` potentials (or alternatively their parameterization as positive scaling vectors :math:`u`, :math:`v`), rather than solving the primal problem in :math:`P`. This is mostly for efficiency (potentials and scalings have a ``n + m`` memory footprint, rather than ``n m`` required to store `P`). This is also because both problems are, in fact, equivalent, since the optimal transport :math:`P^{\star}` can be recovered from optimal potentials :math:`f^{\star}`, :math:`g^{\star}` or scaling :math:`u^{\star}`, :math:`v^{\star}`, using the geometry's cost or kernel matrix respectively: .. math:: P^{\star} = \exp\left(\frac{f^{\star}\mathbf{1}_m^T + \mathbf{1}_n g^{*T}- C}{\varepsilon}\right) \text{ or } P^{\star} = \text{diag}(u^{\star}) K \text{diag}(v^{\star}) By default, the Sinkhorn algorithm solves this dual problem in :math:`f, g` or :math:`u, v` using block coordinate ascent, i.e. devising an update for each :math:`f` and :math:`g` (resp. :math:`u` and :math:`v`) that cancels their respective gradients, one at a time. These two iterations are repeated ``inner_iterations`` times, after which the norm of these gradients will be evaluated and compared with the ``threshold`` value. The iterations are then repeated as long as that error exceeds ``threshold``. Note on Sinkhorn updates: The boolean flag ``lse_mode`` sets whether the algorithm is run in either: - log-sum-exp mode (``lse_mode=True``), in which case it is directly defined in terms of updates to `f` and `g`, using log-sum-exp computations. This requires access to the cost matrix :math:`C`, as it is stored, or possibly computed on the fly by ``geom``. - kernel mode (``lse_mode=False``), in which case it will require access to a matrix vector multiplication operator :math:`z \rightarrow K z`, where :math:`K` is either instantiated from :math:`C` as :math:`\exp(-C/\varepsilon)`, or provided directly. In that case, rather than optimizing on :math:`f` and :math:`g`, it is more convenient to optimize on their so called scaling formulations, :math:`u := \exp(f / \varepsilon)` and :math:`v := \exp(g / \varepsilon)`. While faster (applying matrices is faster than applying ``lse`` repeatedly over lines), this mode is also less stable numerically, notably for smaller :math:`\varepsilon`. In the source code, the variables ``f_u`` or ``g_v`` can be either regarded as potentials (real) or scalings (positive) vectors, depending on the choice of ``lse_mode`` by the user. Once optimization is carried out, we only return dual variables in potential form, i.e. ``f`` and ``g``. In addition to standard Sinkhorn updates, the user can also use heavy-ball type updates using a ``momentum`` parameter in ]0,2[. We also implement a strategy that tries to set that parameter adaptively at ``chg_momentum_from`` iterations, as a function of progress in the error, as proposed in the literature. Another upgrade to the standard Sinkhorn updates provided to the users lies in using Anderson acceleration. This can be parameterized by setting the otherwise null ``anderson`` to a positive integer. When selected,the algorithm will recompute, every ``refresh_anderson_frequency`` (set by default to 1) an extrapolation of the most recently computed ``anderson`` iterates. When using that option, notice that differentiation (if required) can only be carried out using implicit differentiation, and that all momentum related parameters are ignored. The ``parallel_dual_updates`` flag is set to ``False`` by default. In that setting, ``g_v`` is first updated using the latest values for ``f_u`` and ``g_v``, before proceeding to update ``f_u`` using that new value for ``g_v``. When the flag is set to ``True``, both ``f_u`` and ``g_v`` are updated simultaneously. Note that setting that choice to ``True`` requires using some form of averaging (e.g. ``momentum=0.5``). Without this, and on its own ``parallel_dual_updates`` won't work. Differentiation: The optimal solutions ``f`` and ``g`` and the optimal objective (``reg_ot_cost``) outputted by the Sinkhorn algorithm can be differentiated w.r.t. relevant inputs ``geom``, ``a`` and ``b``. In the default setting, implicit differentiation of the optimality conditions (``implicit_diff`` not equal to ``None``), this has two consequences, treating ``f`` and ``g`` differently from ``reg_ot_cost``. - The termination criterion used to stop Sinkhorn (cancellation of gradient of objective w.r.t. ``f_u`` and ``g_v``) is used to differentiate ``f`` and ``g``, given a change in the inputs. These changes are computed by solving a linear system. The arguments starting with ``implicit_solver_*`` allow to define the linear solver that is used, and to control for two types or regularization (we have observed that, depending on the architecture, linear solves may require higher ridge parameters to remain stable). The optimality conditions in Sinkhorn can be analyzed as satisfying a ``z=z'`` condition, which are then differentiated. It might be beneficial (e.g., as in :cite:`cuturi:20a`) to use a preconditioning function ``precondition_fun`` to differentiate instead ``h(z) = h(z')``. - The objective ``reg_ot_cost`` returned by Sinkhorn uses the so-called envelope (or Danskin's) theorem. In that case, because it is assumed that the gradients of the dual variables ``f_u`` and ``g_v`` w.r.t. dual objective are zero (reflecting the fact that they are optimal), small variations in ``f_u`` and ``g_v`` due to changes in inputs (such as ``geom``, ``a`` and ``b``) are considered negligible. As a result, ``stop_gradient`` is applied on dual variables ``f_u`` and ``g_v`` when evaluating the ``reg_ot_cost`` objective. Note that this approach is `invalid` when computing higher order derivatives. In that case the ``use_danskin`` flag must be set to ``False``. An alternative yet more costly way to differentiate the outputs of the Sinkhorn iterations is to use unrolling, i.e. reverse mode differentiation of the Sinkhorn loop. This is possible because Sinkhorn iterations are wrapped in a custom fixed point iteration loop, defined in ``fixed_point_loop``, rather than a standard while loop. This is to ensure the end result of this fixed point loop can also be differentiated, if needed, using standard JAX operations. To ensure differentiability, the ``fixed_point_loop.fixpoint_iter_backprop`` loop does checkpointing of state variables (here ``f_u`` and ``g_v``) every ``inner_iterations``, and backpropagates automatically, block by block, through blocks of ``inner_iterations`` at a time. Note: * The Sinkhorn algorithm may not converge within the maximum number of iterations for possibly several reasons: 1. the regularizer (defined as ``epsilon`` in the geometry ``geom`` object) is too small. Consider either switching to ``lse_mode=True`` (at the price of a slower execution), increasing ``epsilon``, or, alternatively, if you are unable or unwilling to increase ``epsilon``, either increase ``max_iterations`` or ``threshold``. 2. the probability weights ``a`` and ``b`` do not have the same total mass, while using a balanced (``tau_a=tau_b=1.0``) setup. Consider either normalizing ``a`` and ``b``, or set either ``tau_a`` and/or ``tau_b<1.0``. 3. OOMs issues may arise when storing either cost or kernel matrices that are too large in ``geom``. In the case where, the ``geom`` geometry is a ``PointCloud``, some of these issues might be solved by setting the ``online`` flag to ``True``. This will trigger a re-computation on the fly of the cost/kernel matrix. * The weight vectors ``a`` and ``b`` can be passed on with coordinates that have zero weight. This is then handled by relying on simple arithmetic for ``inf`` values that will likely arise (due to :math:`\log 0` when ``lse_mode`` is ``True``, or divisions by zero when ``lse_mode`` is ``False``). Whenever that arithmetic is likely to produce ``NaN`` values (due to ``-inf * 0``, or ``-inf - -inf``) in the forward pass, we use ``jnp.where`` conditional statements to carry ``inf`` rather than ``NaN`` values. In the reverse mode differentiation, the inputs corresponding to these 0 weights (a location `x`, or a row in the corresponding cost/kernel matrix), and the weight itself will have ``NaN`` gradient values. This is reflects that these gradients are undefined, since these points were not considered in the optimization and have therefore no impact on the output. Args: lse_mode: ``True`` for log-sum-exp computations, ``False`` for kernel multiplication. threshold: tolerance used to stop the Sinkhorn iterations. This is typically the deviation between a target marginal and the marginal of the current primal solution when either or both tau_a and tau_b are 1.0 (balanced or semi-balanced problem), or the relative change between two successive solutions in the unbalanced case. norm_error: power used to define p-norm of error for marginal/target. inner_iterations: the Sinkhorn error is not recomputed at each iteration but every ``inner_iterations`` instead. min_iterations: the minimum number of Sinkhorn iterations carried out before the error is computed and monitored. max_iterations: the maximum number of Sinkhorn iterations. If ``max_iterations`` is equal to ``min_iterations``, Sinkhorn iterations are run by default using a :func:`jax.lax.scan` loop rather than a custom, unroll-able :func:`jax.lax.while_loop` that monitors convergence. In that case the error is not monitored and the ``converged`` flag will return ``False`` as a consequence. momentum: Momentum instance. anderson: AndersonAcceleration instance. implicit_diff: instance used to solve implicit differentiation. Unrolls iterations if None. parallel_dual_updates: updates potentials or scalings in parallel if True, sequentially (in Gauss-Seidel fashion) if False. recenter_potentials: Whether to re-center the dual potentials. If the problem is balanced, the ``f`` potential is zero-centered for numerical stability. Otherwise, use the approach of :cite:`sejourne:22` to achieve faster convergence. Only used when ``lse_mode = True`` and ``tau_a < 1`` and ``tau_b < 1``. use_danskin: when ``True``, it is assumed the entropy regularized cost is evaluated using optimal potentials that are frozen, i.e. whose gradients have been stopped. This is useful when carrying out first order differentiation, and is only valid (as with ``implicit_differentiation``) when the algorithm has converged with a low tolerance. initializer: how to compute the initial potentials/scalings. This refers to a few possible classes implemented following the template in :class:`~ott.initializers.linear.SinkhornInitializer`. progress_fn: callback function which gets called during the Sinkhorn iterations, so the user can display the error at each iteration, e.g., using a progress bar. See :func:`~ott.utils.default_progress_fn` for a basic implementation. kwargs_init: keyword arguments when creating the initializer. """ def __init__( self, lse_mode: bool = True, threshold: float = 1e-3, norm_error: int = 1, inner_iterations: int = 10, min_iterations: int = 0, max_iterations: int = 2000, momentum: Optional[acceleration.Momentum] = None, anderson: Optional[acceleration.AndersonAcceleration] = None, parallel_dual_updates: bool = False, recenter_potentials: bool = False, use_danskin: Optional[bool] = None, implicit_diff: Optional[implicit_lib.ImplicitDiff ] = implicit_lib.ImplicitDiff(), # noqa: B008 initializer: Union[Literal["default", "gaussian", "sorting", "subsample"], init_lib.SinkhornInitializer] = "default", progress_fn: Optional[ProgressCallbackFn_t] = None, kwargs_init: Optional[Mapping[str, Any]] = None, ): self.lse_mode = lse_mode self.threshold = threshold self.inner_iterations = inner_iterations self.min_iterations = min_iterations self.max_iterations = max_iterations self._norm_error = norm_error self.anderson = anderson self.implicit_diff = implicit_diff if momentum is not None: self.momentum = acceleration.Momentum( momentum.start, momentum.error_threshold, momentum.value, self.inner_iterations ) else: # Use no momentum if using Anderson or unrolling. if self.anderson is not None or self.implicit_diff is None: self.momentum = acceleration.Momentum( inner_iterations=self.inner_iterations ) else: # no momentum self.momentum = acceleration.Momentum() self.parallel_dual_updates = parallel_dual_updates self.recenter_potentials = recenter_potentials self.initializer = initializer self.progress_fn = progress_fn self.kwargs_init = {} if kwargs_init is None else kwargs_init # Force implicit_differentiation to True when using Anderson acceleration, # Reset all momentum parameters to default (i.e. no momentum) if anderson: self.implicit_diff = ( implicit_lib.ImplicitDiff() if self.implicit_diff is None else self.implicit_diff ) self.momentum = acceleration.Momentum( inner_iterations=self.inner_iterations ) # By default, use Danskin theorem to differentiate # the objective when using implicit_lib. self.use_danskin = ((self.implicit_diff is not None) if use_danskin is None else use_danskin) def __call__( self, ot_prob: linear_problem.LinearProblem, init: Tuple[Optional[jnp.ndarray], Optional[jnp.ndarray]] = (None, None), rng: Optional[jax.Array] = None, ) -> SinkhornOutput: """Run Sinkhorn algorithm. Args: ot_prob: Linear OT problem. init: Initial dual potentials/scalings f_u and g_v, respectively. Any `None` values will be initialized using the initializer. rng: Random number generator key for stochastic initialization. Returns: The Sinkhorn output. """ rng = utils.default_prng_key(rng) initializer = self.create_initializer() init_dual_a, init_dual_b = initializer( ot_prob, *init, lse_mode=self.lse_mode, rng=rng ) return run(ot_prob, self, (init_dual_a, init_dual_b))
[docs] def lse_step( self, ot_prob: linear_problem.LinearProblem, state: SinkhornState, iteration: int ) -> SinkhornState: """Sinkhorn LSE update.""" def k(tau_i: float, tau_j: float) -> float: num = -tau_j * (tau_a - 1) * (tau_b - 1) * (tau_i - 1) denom = (tau_j - 1) * (tau_a * (tau_b - 1) + tau_b * (tau_a - 1)) return num / denom def xi(tau_i: float, tau_j: float) -> float: k_ij = k(tau_i, tau_j) return k_ij / (1.0 - k_ij) def smin( potential: jnp.ndarray, marginal: jnp.ndarray, tau: float ) -> float: rho = uf.rho(ot_prob.epsilon, tau) return -rho * mu.logsumexp(-potential / rho, b=marginal) # only for an unbalanced problems with `tau_{a,b} < 1` recenter = ( self.recenter_potentials and ot_prob.tau_a < 1.0 and ot_prob.tau_b < 1.0 ) w = self.momentum.weight(state, iteration) tau_a, tau_b = ot_prob.tau_a, ot_prob.tau_b old_fu, old_gv = state.fu, state.gv if recenter: k11, k22 = k(tau_a, tau_a), k(tau_b, tau_b) xi12, xi21 = xi(tau_a, tau_b), xi(tau_b, tau_a) # update g potential new_gv = tau_b * ot_prob.geom.update_potential( old_fu, old_gv, jnp.log(ot_prob.b), iteration, axis=0 ) if recenter: new_gv -= k22 * smin(old_fu, ot_prob.a, tau_a) new_gv += xi21 * smin(new_gv, ot_prob.b, tau_b) gv = self.momentum(w, old_gv, new_gv, self.lse_mode) if not self.parallel_dual_updates: old_gv = gv # update f potential new_fu = tau_a * ot_prob.geom.update_potential( old_fu, old_gv, jnp.log(ot_prob.a), iteration, axis=1 ) if recenter: new_fu -= k11 * smin(old_gv, ot_prob.b, tau_b) new_fu += xi12 * smin(new_fu, ot_prob.a, tau_a) fu = self.momentum(w, old_fu, new_fu, self.lse_mode) return state.set(potentials=(fu, gv))
[docs] def kernel_step( self, ot_prob: linear_problem.LinearProblem, state: SinkhornState, iteration: int ) -> SinkhornState: """Sinkhorn multiplicative update.""" w = self.momentum.weight(state, iteration) old_gv = state.gv new_gv = ot_prob.geom.update_scaling( state.fu, ot_prob.b, iteration, axis=0 ) ** ot_prob.tau_b gv = self.momentum(w, state.gv, new_gv, self.lse_mode) new_fu = ot_prob.geom.update_scaling( old_gv if self.parallel_dual_updates else gv, ot_prob.a, iteration, axis=1 ) ** ot_prob.tau_a fu = self.momentum(w, state.fu, new_fu, self.lse_mode) return state.set(potentials=(fu, gv))
[docs] def one_iteration( self, ot_prob: linear_problem.LinearProblem, state: SinkhornState, iteration: int, compute_error: bool ) -> SinkhornState: """Carries out one Sinkhorn iteration. Depending on lse_mode, these iterations can be either in: - log-space for numerical stability. - scaling space, using standard kernel-vector multiply operations. Args: ot_prob: the transport problem definition state: SinkhornState named tuple. iteration: the current iteration of the Sinkhorn loop. compute_error: flag to indicate this iteration computes/stores an error Returns: The updated state. """ # When running updates in parallel (Gauss-Seidel mode), old_g_v will be # used to update f_u, rather than the latest g_v computed in this loop. # Unused otherwise. if self.anderson: state = self.anderson.update(state, iteration, ot_prob, self.lse_mode) if self.lse_mode: # In lse_mode, run additive updates. state = self.lse_step(ot_prob, state, iteration) else: state = self.kernel_step(ot_prob, state, iteration) if self.anderson: state = self.anderson.update_history(state, ot_prob, self.lse_mode) # re-computes error if compute_error is True, else set it to inf. err = jax.lax.cond( jnp.logical_or( iteration == self.max_iterations - 1, jnp.logical_and(compute_error, iteration >= self.min_iterations) ), lambda state, prob: state.solution_error( prob, self.norm_error, lse_mode=self.lse_mode, parallel_dual_updates=self.parallel_dual_updates, recenter=self.recenter_potentials )[0], lambda *_: jnp.inf, state, ot_prob, ) errors =[iteration // self.inner_iterations, :].set(err) state = state.set(errors=errors) if self.progress_fn is not None: jax.debug.callback( self.progress_fn, (iteration, self.inner_iterations, self.max_iterations, state) ) return state
def _converged(self, state: SinkhornState, iteration: int) -> bool: err = state.errors[iteration // self.inner_iterations - 1, 0] return jnp.logical_and(iteration > 0, err < self.threshold) def _diverged(self, state: SinkhornState, iteration: int) -> bool: err = state.errors[iteration // self.inner_iterations - 1, 0] return jnp.logical_not(jnp.isfinite(err)) def _continue(self, state: SinkhornState, iteration: int) -> bool: """Continue while not(converged) and not(diverged).""" return jnp.logical_and( jnp.logical_not(self._diverged(state, iteration)), jnp.logical_not(self._converged(state, iteration)) ) @property def outer_iterations(self) -> int: """Upper bound on number of times inner_iterations are carried out. This integer can be used to set constant array sizes to track the algorithm progress, notably errors. """ return np.ceil(self.max_iterations / self.inner_iterations).astype(int)
[docs] def init_state( self, ot_prob: linear_problem.LinearProblem, init: Tuple[jnp.ndarray, jnp.ndarray] ) -> SinkhornState: """Return the initial state of the loop.""" errors = -jnp.ones((self.outer_iterations, len(self.norm_error))) state = SinkhornState(init, errors=errors) return self.anderson.init_maps(ot_prob, state) if self.anderson else state
[docs] def output_from_state( self, ot_prob: linear_problem.LinearProblem, state: SinkhornState ) -> SinkhornOutput: """Create an output from a loop state. Note: When differentiating the regularized OT cost, and assuming Sinkhorn has run to convergence, Danskin's (or the envelope) `theorem <>`_ :cite:`danskin:67,bertsekas:71` states that the resulting OT cost as a function of the inputs (``geometry``, ``a``, ``b``) behaves locally as if the dual optimal potentials were frozen and did not vary with those inputs. Notice this is only valid, as when using ``implicit_differentiation`` mode, if the Sinkhorn algorithm outputs potentials that are near optimal. namely when the threshold value is set to a small tolerance. The flag ``use_danskin`` controls whether that assumption is made. By default, that flag is set to the value of ``implicit_differentiation`` if not specified. If you wish to compute derivatives of order 2 and above, set ``use_danskin`` to ``False``. Args: ot_prob: the transport problem. state: a SinkhornState. Returns: A SinkhornOutput. """ geom = ot_prob.geom f = state.fu if self.lse_mode else geom.potential_from_scaling(state.fu) g = state.gv if self.lse_mode else geom.potential_from_scaling(state.gv) if self.recenter_potentials: f, g = state.recenter(f, g, ot_prob=ot_prob) # By convention, the algorithm is said to have converged if the algorithm # has not nan'ed during iterations (notice some errors might be infinite, # this convention is used when the error is not recomputed), and if the # last recorded error is lower than the threshold. Note that this will be # the case if either the algorithm terminated earlier (in which case the # last state.errors[-1] = -1 by convention) or if the algorithm carried out # the maximal number of iterations and its last recorded error (at -1 # position) is lower than the threshold. converged = jnp.logical_and( jnp.logical_not(jnp.any(jnp.isnan(state.errors))), state.errors[-1] < self.threshold )[0] return SinkhornOutput((f, g), errors=state.errors[:, 0], threshold=jnp.array(self.threshold), converged=converged, inner_iterations=self.inner_iterations)
@property def norm_error(self) -> Tuple[int, ...]: """Powers used to compute the p-norm between marginal/target.""" # To change momentum adaptively, one needs errors in ||.||_1 norm. # In that case, we add this exponent to the list of errors to compute, # notably if that was not the error requested by the user. if self.momentum and self.momentum.start > 0 and self._norm_error != 1: return self._norm_error, 1 return self._norm_error, # TODO(michalk8): in the future, enforce this (+ in GW) via abstract method
[docs] def create_initializer(self) -> init_lib.SinkhornInitializer: # noqa: D102 if isinstance(self.initializer, init_lib.SinkhornInitializer): return self.initializer if self.initializer == "default": return init_lib.DefaultInitializer() if self.initializer == "gaussian": return init_lib.GaussianInitializer() if self.initializer == "sorting": return init_lib.SortingInitializer(**self.kwargs_init) if self.initializer == "subsample": return init_lib.SubsampleInitializer(**self.kwargs_init) raise NotImplementedError( f"Initializer `{self.initializer}` is not yet implemented." )
def tree_flatten(self): # noqa: D102 aux = vars(self).copy() aux["norm_error"] = aux.pop("_norm_error") aux.pop("threshold") return [self.threshold], aux @classmethod def tree_unflatten(cls, aux_data, children): # noqa: D102 return cls(**aux_data, threshold=children[0])
def run( ot_prob: linear_problem.LinearProblem, solver: Sinkhorn, init: Tuple[jnp.ndarray, ...] ) -> SinkhornOutput: """Run loop of the solver, outputting a state upgraded to an output.""" iter_fun = _iterations_implicit if solver.implicit_diff else iterations out = iter_fun(ot_prob, solver, init) # Be careful here, the geom and the cost are injected at the end, where it # does not interfere with the implicit differentiation. out = out.set_cost(ot_prob, solver.lse_mode, solver.use_danskin) return out.set(ot_prob=ot_prob) def iterations( ot_prob: linear_problem.LinearProblem, solver: Sinkhorn, init: Tuple[jnp.ndarray, ...] ) -> SinkhornOutput: """Jittable Sinkhorn loop. args contain initialization variables.""" def cond_fn( iteration: int, const: Tuple[linear_problem.LinearProblem, Sinkhorn], state: SinkhornState ) -> bool: _, solver = const return solver._continue(state, iteration) def body_fn( iteration: int, const: Tuple[linear_problem.LinearProblem, Sinkhorn], state: SinkhornState, compute_error: bool ) -> SinkhornState: ot_prob, solver = const return solver.one_iteration(ot_prob, state, iteration, compute_error) # Run the Sinkhorn loop. Choose either a standard fixpoint_iter loop if # differentiation is implicit, otherwise switch to the backprop friendly # version of that loop if unrolling to differentiate. if solver.implicit_diff: fix_point = fixed_point_loop.fixpoint_iter else: fix_point = fixed_point_loop.fixpoint_iter_backprop const = ot_prob, solver state = solver.init_state(ot_prob, init) state = fix_point( cond_fn, body_fn, solver.min_iterations, solver.max_iterations, solver.inner_iterations, const, state ) return solver.output_from_state(ot_prob, state) def _iterations_taped( ot_prob: linear_problem.LinearProblem, solver: Sinkhorn, init: Tuple[jnp.ndarray, ...] ) -> Tuple[SinkhornOutput, Tuple[jnp.ndarray, jnp.ndarray, linear_problem.LinearProblem, Sinkhorn]]: """Run forward pass of the Sinkhorn algorithm storing side information.""" state = iterations(ot_prob, solver, init) return state, (state.f, state.g, ot_prob, solver) def _iterations_implicit_bwd(res, gr: SinkhornOutput): """Run Sinkhorn in backward mode, using implicit differentiation. Args: res: residual data sent from fwd pass, used for computations below. In this case consists in the output itself, as well as inputs against which we wish to differentiate. gr: gradients w.r.t outputs of fwd pass, here w.r.t size f, g, errors. Note that differentiability w.r.t. errors is not handled, and only f, g is considered. Returns: a tuple of gradients: PyTree for geom, one jnp.ndarray for each of a and b. """ f, g, ot_prob, solver = res out = solver.implicit_diff.gradient( ot_prob, f, g, solver.lse_mode, gr.potentials ) return *out, None, None # sets threshold, norm_errors, geom, a and b to be differentiable, as those are # non-static. Only differentiability w.r.t. geom, a and b will be used. _iterations_implicit = jax.custom_vjp(iterations) _iterations_implicit.defvjp(_iterations_taped, _iterations_implicit_bwd)