Source code for ott.core.sinkhorn

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# Lint as: python3
"""A Jax implementation of the Sinkhorn algorithm."""
from typing import Any, Callable, NamedTuple, Optional, Sequence, Tuple

import jax
import jax.numpy as jnp
import numpy as np

from ott.core import anderson as anderson_lib
from ott.core import fixed_point_loop
from ott.core import implicit_differentiation as implicit_lib
from ott.core import initializers as init_lib
from ott.core import linear_problems
from ott.core import momentum as momentum_lib
from ott.core import unbalanced_functions
from ott.geometry import geometry


class SinkhornState(NamedTuple):
  """Holds the state variables used to solve OT with Sinkhorn."""

  errors: Optional[jnp.ndarray] = None
  fu: Optional[jnp.ndarray] = None
  gv: Optional[jnp.ndarray] = None
  old_fus: Optional[jnp.ndarray] = None
  old_mapped_fus: Optional[jnp.ndarray] = None

  def set(self, **kwargs: Any) -> 'SinkhornState':
    """Return a copy of self, with potential overwrites."""
    return self._replace(**kwargs)

  def solution_error(
      self, ot_prob: linear_problems.LinearProblem, norm_error: Sequence[int],
      lse_mode: bool
  ) -> jnp.ndarray:
    return solution_error(self.fu, self.gv, ot_prob, norm_error, lse_mode)

  def ent_reg_cost(
      self, ot_prob: linear_problems.LinearProblem, lse_mode: bool
  ) -> float:
    return ent_reg_cost(self.fu, self.gv, ot_prob, lse_mode)


def solution_error(
    f_u: jnp.ndarray, g_v: jnp.ndarray, ot_prob: linear_problems.LinearProblem,
    norm_error: Sequence[int], lse_mode: bool
) -> jnp.ndarray:
  """Given two potential/scaling solutions, computes deviation to optimality.

  When the ``ot_prob`` problem is balanced, this is simply deviation to the
  target marginals defined in ``ot_prob.a`` and ``ot_prob.b``. When the problem
  is unbalanced, additional quantities 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 producs.

  Returns:
    a positive number quantifying how far from optimality current solution is.
  """
  if ot_prob.is_balanced:
    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 = unbalanced_functions.grad_of_marginal_fit(
        ot_prob.a, f_u, ot_prob.tau_a, ot_prob.epsilon
    )
    grad_b = unbalanced_functions.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 = unbalanced_functions.grad_of_marginal_fit(
        ot_prob.a, u, ot_prob.tau_a, ot_prob.epsilon
    )
    grad_b = unbalanced_functions.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)
  error = jnp.sum(
      jnp.abs(marginal - target) ** norm_error[:, jnp.newaxis], axis=1
  ) ** (1.0 / norm_error)
  return error


def ent_reg_cost(
    f: jnp.ndarray, g: jnp.ndarray, ot_prob: linear_problems.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 = ot_prob.epsilon * (ot_prob.tau_a / (1 - ot_prob.tau_a))
    div_a = -jnp.sum(
        jnp.where(
            supp_a, ot_prob.a * unbalanced_functions.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 = ot_prob.epsilon * (ot_prob.tau_b / (1 - ot_prob.tau_b))
    div_b = -jnp.sum(
        jnp.where(
            supp_b, ot_prob.b * unbalanced_functions.phi_star(-(g - gb), rho_b),
            0.0
        )
    )

  # Using https://arxiv.org/pdf/1910.12958.pdf (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): """Implements the problems.Transport interface, for a Sinkhorn solution.""" f: Optional[jnp.ndarray] = None g: Optional[jnp.ndarray] = None errors: Optional[jnp.ndarray] = None reg_ot_cost: Optional[float] = None ot_prob: Optional[linear_problems.LinearProblem] = None
[docs] def set(self, **kwargs: Any) -> 'SinkhornOutput': """Return a copy of self, with potential overwrites.""" return self._replace(**kwargs)
[docs] def set_cost( self, ot_prob: linear_problems.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=ent_reg_cost(f, g, ot_prob, lse_mode))
@property def linear(self) -> bool: return isinstance(self.ot_prob, linear_problems.LinearProblem) @property def geom(self) -> geometry.Geometry: return self.ot_prob.geom @property def a(self) -> jnp.ndarray: return self.ot_prob.a @property def b(self) -> jnp.ndarray: return self.ot_prob.b @property def linear_output(self) -> bool: return True @property def converged(self) -> bool: if self.errors is None: return False return jnp.logical_and( jnp.sum(self.errors == -1) > 0, jnp.sum(jnp.isnan(self.errors)) == 0 ) @property def scalings(self) -> Tuple[jnp.ndarray, jnp.ndarray]: 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)
[docs] def apply(self, inputs: jnp.ndarray, axis: int = 0) -> jnp.ndarray: """Apply the transport to a ndarray; axis=1 for its transpose.""" try: return self.ot_prob.geom.apply_transport_from_potentials( self.f, self.g, inputs, axis=axis ) except ValueError: u, v = self.scalings return self.ot_prob.geom.apply_transport_from_scalings( u, v, inputs, axis=axis )
[docs] def marginal(self, axis: int) -> jnp.ndarray: 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 transport_mass(self) -> float: """Sum of transport matrix.""" return self.marginal(0).sum()
[docs]@jax.tree_util.register_pytree_node_class class Sinkhorn: """A Sinkhorn solver for linear reg-OT problem implemented as a pytree. A Sinkhorn solver takes a linear OT problem object as an input and returns a SinkhornOutput object that contains all the information required to compute transports. See function ``sinkhorn`` for a wrapper. 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_num_iter 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 ``jax.lax.scan`` loop rather than a custom, unroll-able ``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: a Momentum instance. See ott.core.momentum anderson: an AndersonAcceleration instance. See ott.core.anderson. 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. use_danskin: when ``True``, it is assumed the entropy regularized cost is is evaluated using optimal potentials that are freezed, 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. jit: if True, automatically jits the function upon first call. Should be set to False when used in a function that is jitted by the user, or when computing gradients (in which case the gradient function should be jitted by the user) """ 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[momentum_lib.Momentum] = None, anderson: Optional[anderson_lib.AndersonAcceleration] = None, parallel_dual_updates: bool = False, use_danskin: Optional[bool] = None, implicit_diff: Optional[implicit_lib.ImplicitDiff ] = implicit_lib.ImplicitDiff(), # noqa: E124 potential_initializer: init_lib.SinkhornInitializer = init_lib .DefaultInitializer(), jit: bool = True ): 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 if momentum is not None: self.momentum = momentum_lib.Momentum( momentum.start, momentum.value, self.inner_iterations ) else: self.momentum = momentum_lib.Momentum( inner_iterations=self.inner_iterations ) self.anderson = anderson self.implicit_diff = implicit_diff self.parallel_dual_updates = parallel_dual_updates self.potential_initializer = potential_initializer self.jit = jit # Force implicit_differentiation to True when using Anderson acceleration, # Reset all momentum parameters. if anderson: self.implicit_diff = ( implicit_lib.ImplicitDiff() if self.implicit_diff is None else self.implicit_diff ) self.momentum = momentum_lib.Momentum(start=0, value=1.0) # 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) @property def norm_error(self) -> Tuple[int, ...]: # 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) else: return (self._norm_error,) def __call__( self, ot_prob: linear_problems.LinearProblem, init: Optional[Tuple[Optional[jnp.ndarray], Optional[jnp.ndarray]]] = None ) -> SinkhornOutput: """Main interface to run sinkhorn.""" # noqa: D401 # initialization init_dual_a, init_dual_b = (init if init is not None else (None, None)) if init_dual_a is None: init_dual_a = self.potential_initializer.init_dual_a( ot_problem=ot_prob, lse_mode=self.lse_mode ) if init_dual_b is None: init_dual_b = self.potential_initializer.init_dual_b( ot_problem=ot_prob, lse_mode=self.lse_mode ) # Cancel dual variables for zero weights. init_dual_a = jnp.where( ot_prob.a > 0, init_dual_a, -jnp.inf if self.lse_mode else 0.0 ) init_dual_b = jnp.where( ot_prob.b > 0, init_dual_b, -jnp.inf if self.lse_mode else 0.0 ) run_fn = jax.jit(run) if self.jit else run return run_fn(ot_prob, self, (init_dual_a, init_dual_b)) def tree_flatten(self): 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): return cls(**aux_data, threshold=children[0])
[docs] def lse_step( self, ot_prob: linear_problems.LinearProblem, state: SinkhornState, iteration: int ) -> SinkhornState: """Sinkhorn LSE update.""" w = self.momentum.weight(state, iteration) old_gv = state.gv new_gv = ot_prob.tau_b * ot_prob.geom.update_potential( state.fu, state.gv, jnp.log(ot_prob.b), iteration, axis=0 ) gv = self.momentum(w, state.gv, new_gv, self.lse_mode) new_fu = ot_prob.tau_a * ot_prob.geom.update_potential( state.fu, old_gv if self.parallel_dual_updates else gv, jnp.log(ot_prob.a), iteration, axis=1 ) fu = self.momentum(w, state.fu, new_fu, self.lse_mode) return state.set(fu=fu, gv=gv)
[docs] def kernel_step( self, ot_prob: linear_problems.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(fu=fu, gv=gv)
[docs] def one_iteration( self, ot_prob: linear_problems.LinearProblem, state: SinkhornState, iteration: int, compute_error: bool ) -> SinkhornState: """Carries out 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 = jnp.where( jnp.logical_and(compute_error, iteration >= self.min_iterations), state.solution_error(ot_prob, self.norm_error, self.lse_mode), jnp.inf ) errors = (state.errors.at[iteration // self.inner_iterations, :].set(err)) return state.set(errors=errors)
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_problems.LinearProblem, init: Tuple[jnp.ndarray, jnp.ndarray] ) -> SinkhornState: """Return the initial state of the loop.""" fu, gv = init errors = -jnp.ones((self.outer_iterations, len(self.norm_error)), dtype=fu.dtype) state = SinkhornState(errors=errors, fu=fu, gv=gv) return self.anderson.init_maps(ot_prob, state) if self.anderson else state
[docs] def output_from_state( self, ot_prob: linear_problems.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 <https://en.wikipedia.org/wiki/Danskin%27s_theorem>`_ states that the resulting OT cost as a function of any 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) errors = state.errors[:, 0] return SinkhornOutput(f=f, g=g, errors=errors)
def run( ot_prob: linear_problems.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_problems.LinearProblem, solver: Sinkhorn, init: Tuple[jnp.ndarray, ...] ) -> SinkhornOutput: """Jittable Sinkhorn loop. args contain initialization variables.""" def cond_fn( iteration: int, const: Tuple[linear_problems.LinearProblem, Sinkhorn], state: SinkhornState ) -> bool: _, solver = const return solver._continue(state, iteration) def body_fn( iteration: int, const: Tuple[linear_problems.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_problems.LinearProblem, solver: Sinkhorn, init: Tuple[jnp.ndarray, ...] ) -> Tuple[SinkhornOutput, Tuple[jnp.ndarray, jnp.ndarray, linear_problems.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): """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 gr = gr[:2] return ( *solver.implicit_diff.gradient(ot_prob, f, g, solver.lse_mode, gr), 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) def make( tau_a: float = 1.0, tau_b: float = 1.0, threshold: float = 1e-3, norm_error: int = 1, inner_iterations: int = 10, min_iterations: int = 0, max_iterations: int = 2000, momentum: float = 1.0, chg_momentum_from: int = 0, anderson_acceleration: int = 0, refresh_anderson_frequency: int = 1, lse_mode: bool = True, implicit_differentiation: bool = True, implicit_solver_fun=jax.scipy.sparse.linalg.cg, implicit_solver_ridge_kernel: float = 0.0, implicit_solver_ridge_identity: float = 0.0, implicit_solver_symmetric: bool = False, precondition_fun: Optional[Callable[[float], float]] = None, parallel_dual_updates: bool = False, use_danskin: bool = None, potential_initializer: init_lib.SinkhornInitializer = init_lib .DefaultInitializer(), jit: bool = False ) -> Sinkhorn: """For backward compatibility.""" del tau_a, tau_b if not implicit_differentiation: implicit_diff = None else: implicit_diff = implicit_lib.ImplicitDiff( solver_fun=implicit_solver_fun, ridge_kernel=implicit_solver_ridge_kernel, ridge_identity=implicit_solver_ridge_identity, symmetric=implicit_solver_symmetric, precondition_fun=precondition_fun ) if anderson_acceleration > 0: anderson = anderson_lib.AndersonAcceleration( memory=anderson_acceleration, refresh_every=refresh_anderson_frequency ) else: anderson = None return Sinkhorn( lse_mode=lse_mode, threshold=threshold, norm_error=norm_error, inner_iterations=inner_iterations, min_iterations=min_iterations, max_iterations=max_iterations, momentum=momentum_lib.Momentum(start=chg_momentum_from, value=momentum), anderson=anderson, implicit_diff=implicit_diff, parallel_dual_updates=parallel_dual_updates, use_danskin=use_danskin, potential_initializer=potential_initializer, jit=jit )
[docs]def sinkhorn( geom: geometry.Geometry, a: Optional[jnp.ndarray] = None, b: Optional[jnp.ndarray] = None, tau_a: float = 1.0, tau_b: float = 1.0, init_dual_a: Optional[jnp.ndarray] = None, init_dual_b: Optional[jnp.ndarray] = None, **kwargs: Any, ): r"""Jax version of Sinkhorn's algorithm. Solves regularized OT problem using Sinkhorn iterations. 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 exponentials of the former). Calling this function returns therefore a pair of optimal vectors. In addition to these, `sinkhorn` also returns the objective value achieved by these optimal vectors; a vector of size `max_iterations/inner_terations` 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. Some maths: Given a geometry ``geom``, which provides a cost matrix :math:`C` with its regularization parameter :math:`\epsilon`, (resp. 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}{- <a, \phi_a^{*}(-f)> - <b, \phi_b^{*}(-g)> - \epsilon <e^{f/\epsilon}, e^{-C/\epsilon} e^{-g/\epsilon}}> 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/\epsilon}`, .. math:: \arg\max_{u, v >0} - <a,\phi_a^{*}(-\epsilon\log u)> + <b, \phi_b^{*}(-\epsilon\log v)> - <u, K v> Both of these problems corresponds, in their *primal* formulation, to solving the unbalanced optimal transport problem with a variable matrix `P` of size ``n`` x ``m``: .. math:: \arg\min_{P>0} <P,C> -\epsilon \text{KL}(P | ab^T) + \rho_a \text{KL}(P1 | a) + \rho_b \text{KL}(P^T1 | 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} \epsilon KL(P|K) + \rho_a \text{KL}(P1 | a) + \rho_b \text{KL}(P^T1 | 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:`\epsilon \rightarrow 0`, and :math:`\rho_a, \rho_b \rightarrow \infty`. In that case the entropy disappears, whereas the :math:`KL` regularizations 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:`\epsilon > 0`. The *balanced* regularized OT problem is recovered for finite :math:`\epsilon > 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`` := :math:`\rho_a / (\epsilon + \rho_a)` and ``tau_b`` := :math:`\rho_b / (\epsilon + \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 `f`, `g` potentials (or alternatively their parametrization as positive scalings `u`, `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^*` can be recovered from optimal potentials :math:`f^*`, :math:`g^*` or scalings :math:`u^*`, :math:`v^*`, using the geometry's cost or kernel matrix respectively: .. math:: P^* = \exp\left(\frac{f^*\mathbf{1}_m^T + \mathbf{1}_n g^{*T} - C}{\epsilon}\right) \text{ or } P^* = \text{diag}(u^*) K \text{diag}(v^*) By default, the Sinkhorn algorithm solves this dual problem in `f, g` or `u, v` using block coordinate ascent, i.e. devising an update for each `f` and `g` (resp. `u` and `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/\epsilon)`, 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 / \epsilon)` and :math:`v := \exp(g / \epsilon)`. While faster (applying matrices is faster than applying ``lse`` repeatedly over lines), this mode is also less stable numerically, notably for smaller :math:`\epsilon`. 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 ater ``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`` using, by default, implicit differentiation of the optimality conditions (``implicit_differentiation`` set to ``True``). This choice has two consequences. - 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 backprop 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 recomputation 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: geom: a Geometry object. a: [num_a,] or [batch, num_a] weights. b: [num_b,] or [batch, num_b] weights. tau_a: ratio rho/(rho+eps) between KL divergence regularizer to first marginal and itself + epsilon regularizer used in the unbalanced formulation. tau_b: ratio rho/(rho+eps) between KL divergence regularizer to first marginal and itself + epsilon regularizer used in the unbalanced formulation. init_dual_a: optional initialization for potentials/scalings w.r.t. first marginal (``a``) of reg-OT problem. init_dual_b: optional initialization for potentials/scalings w.r.t. second marginal (``b``) of reg-OT problem. 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_num_iter 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 ``jax.lax.scan`` loop rather than a custom, unroll-able ``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: a float in [0,2]. chg_momentum_from: if positive, momentum is recomputed using the adaptive rule provided in :cite:`lehmann:21` after that number of iterations. anderson_acceleration: int, if 0 (default), no acceleration. If positive, use Anderson acceleration on the dual sinkhorn (in log/potential form), as described `here <https://en.wikipedia.org/wiki/Anderson_acceleration>`_ and advocated in :cite:`chizat:20`, with a memory of size equal to ``anderson_acceleration``. In that case, differentiation is necessarily handled implicitly (``implicit_differentiation`` is set to ``True``) and all ``momentum`` related parameters are ignored. refresh_anderson_frequency: int, when using ``anderson_acceleration``, recompute direction periodically every int sinkhorn iterations. lse_mode: ``True`` for log-sum-exp computations, ``False`` for kernel multiplication. implicit_differentiation: ``True`` if using implicit differentiation, ``False`` if unrolling Sinkhorn iterations. linear_solve_kwargs: parametrization of linear solver when using implicit differentiation. Arguments currently accepted appear in the optional arguments of ``apply_inv_hessian``, namely ``linear_solver_fun``, a Callable that specifies the linear solver, as well as ``ridge_kernel`` and ``ridge_identity``, to be added to enforce stability of linear solve. parallel_dual_updates: updates potentials or scalings in parallel if True, sequentially (in Gauss-Seidel fashion) if False. use_danskin: when ``True``, it is assumed the entropy regularized cost is 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. jit: if True, automatically jits the function upon first call. Should be set to False when used in a function that is jitted by the user, or when computing gradients (in which case the gradient function should be jitted by the user). kwargs: Additional keyword arguments (see above). Returns: a ``SinkhornOutput`` named tuple. The tuple contains two optimal potential vectors ``f`` and ``g``, the objective ``reg_ot_cost`` evaluated at those solutions, an array of ``errors`` to monitor convergence every ``inner_iterations`` and a flag ``converged`` that is ``True`` if the algorithm has converged within the number of iterations that was predefined by the user. """ sink = make(**kwargs) ot_prob = linear_problems.LinearProblem(geom, a, b, tau_a, tau_b) return sink(ot_prob, (init_dual_a, init_dual_b))