Source code for jaxopt._src.broyden

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"""Limited-memory Broyden method"""

from functools import partial

from typing import Any
from typing import Callable
from typing import NamedTuple
from typing import Optional
from typing import Union

from dataclasses import dataclass

import jax
import jax.numpy as jnp

from jaxopt._src import base
from jaxopt._src.backtracking_linesearch import BacktrackingLineSearch
from jaxopt.tree_util import tree_map
from jaxopt.tree_util import tree_vdot
from jaxopt.tree_util import tree_add_scalar_mul
from jaxopt.tree_util import tree_scalar_mul
from jaxopt.tree_util import tree_sub
from jaxopt.tree_util import tree_l2_norm
from jaxopt._src.tree_util import tree_single_dtype


def matvec(d_history, c_history, x, indices):
  # CD^T x
  # here j is the history dimension and i is the space dimension
  dtx = jnp.einsum('j..., ... -> j', d_history[indices], x)
  cdtx = jnp.einsum('j..., j -> ...', c_history[indices], dtx)
  return cdtx

def inv_jacobian_product_leaf(v: jnp.ndarray,
                              d_history: jnp.ndarray,
                              c_history: jnp.ndarray,
                              gamma: float = 1.0,
                              start: int= 0):
  # the computation of the jacobian inverse for
  # the Broyden method
  # this is done using the notations of DEQ
  history_size = len(d_history)

  indices = (start + jnp.arange(history_size)) % history_size

  v = gamma * v + matvec(d_history, c_history, v, indices)
  return v


def inv_jacobian_product(pytree: Any,
                         d_history: Any,
                         c_history: Any,
                         gamma: float = 1.0,
                         start: int = 0):
  """Product between an approximate jacobian inverse and a pytree.

  Histories are pytrees of the same structure as `pytree`.
  Leaves are arrays of shape `(history_size, ...)`, where
  `...` means the same shape as `pytree`'s leaves.

  The notation follows the scipy one.

  Args:
    pytree: pytree to multiply with.
    d_history: pytree with the same structure as `pytree`.
      Leaves contain v variables, i.e., `(x[k] - x[k-1])^T B / ((g[k] - g[k-1])^T (x[k] - x[k-1])^T B)`.
    c_history: pytree with the same structure as `pytree`.
      Leaves contain u variables, i.e., `(x[k] - x[k-1]) - B(g[k] - g[k-1])`.
    gamma: scalar to use for the initial inverse jacobian approximation,
      i.e., `gamma * I`.
    start: starting index in the circular buffer.
  """
  fun = partial(inv_jacobian_product_leaf,
                gamma=gamma,
                start=start)
  return tree_map(fun, pytree, d_history, c_history)

def inv_jacobian_rproduct(pytree: Any,
                          d_history: Any,
                          c_history: Any,
                          gamma: float = 1.0,
                          start: int = 0):
  return inv_jacobian_product(pytree, c_history, d_history, jnp.conjugate(gamma), start)


def init_history(pytree, history_size):
  fun = lambda leaf: jnp.zeros((history_size,) + leaf.shape, dtype=leaf.dtype)
  return tree_map(fun, pytree)


def update_history(history_pytree, new_pytree, last):
  fun = lambda history_array, new_value: history_array.at[last].set(new_value)
  return tree_map(fun, history_pytree, new_pytree)


class BroydenState(NamedTuple):
  """Named tuple containing state information."""
  iter_num: int
  value: float
  stepsize: float
  error: float
  d_history: Any
  c_history: Any
  gamma: jnp.ndarray
  aux: Optional[Any] = None
  failed_linesearch: bool = False

  num_fun_eval: int = 0
  num_linesearch_iter: int = 0


[docs]@dataclass(eq=False) class Broyden(base.IterativeSolver): """Limited-memory Broyden solver. This method is a quasi-Newton approach to root finding. While similar to L-BFGS in spirit, it is not applied in the same situations: indeed, because the function whose root we are looking for is not necessarily a gradient, its Jacobian (i.e. its Hessian in the optimization case) is not necessarily symmetric. As a consequence, we cannot include symmetry in the secant conditions defining the updates of the Broyden matrices, and therefore the resulting Jacobian approximation is not symmetric, while it is for L-BFGS. Another consequence is that each Broyden update is of rank-1 while it is rank-2 for L-BFGS. Attributes: fun: a function of the form ``fun(x, *args, **kwargs)``. has_aux: whether ``fun`` outputs auxiliary data or not. If ``has_aux`` is False, ``fun`` is expected to be scalar-valued. If ``has_aux`` is True, then we have one of the following two cases. At each iteration of the algorithm, the auxiliary outputs are stored in ``state.aux``. maxiter: maximum number of Broyden iterations. tol: tolerance of the stopping criterion. stepsize: a stepsize to use (if <= 0, use backtracking line search), or a callable specifying the **positive** stepsize to use at each iteration. linesearch: the type of line search to use: for now only "backtracking" for backtracking line search is available. stop_if_linesearch_fails: whether to stop iterations if the line search fails. When True, this matches the behavior of core JAX. maxls: maximum number of iterations to use in the line search. decrease_factor: factor by which to decrease the stepsize during line search (default: 0.8). increase_factor: factor by which to increase the stepsize during line search (default: 1.5). max_stepsize: upper bound on stepsize. min_stepsize: lower bound on stepsize. history_size: size of the memory to use. gamma: the initialization of the inverse Jacobian is going to be gamma * I. implicit_diff: whether to enable implicit diff or autodiff of unrolled iterations. implicit_diff_solve: the linear system solver to use. jit: whether to JIT-compile the optimization loop (default: "auto"). unroll: whether to unroll the optimization loop (default: "auto"). verbose: whether to print error on every iteration or not. Warning: verbose=True will automatically disable jit. Reference: Charles G. Broyden. A Class of Methods for Solving Nonlinear Simultaneous Equations. Equation (4.5) (page 581). """ fun: Callable has_aux: bool = False maxiter: int = 500 tol: float = 1e-3 stepsize: Union[float, Callable] = 0.0 linesearch: str = "backtracking" stop_if_linesearch_fails: bool = False condition: str = "wolfe" maxls: int = 15 decrease_factor: float = 0.8 increase_factor: float = 1.5 max_stepsize: float = 1.0 # FIXME: should depend on whether float32 or float64 is used. min_stepsize: float = 1e-6 history_size: int = None gamma: float = 1.0 implicit_diff: bool = True implicit_diff_solve: Optional[Callable] = None jit: base.AutoOrBoolean = "auto" unroll: base.AutoOrBoolean = "auto" verbose: bool = False def _cond_fun(self, inputs): _, state = inputs[0] if self.verbose: print("error:", state.error) # We continue the optimization loop while the error tolerance is not met and, # either failed linesearch is disallowed or linesearch hasn't failed. return (state.error > self.tol) & jnp.logical_or(not self.stop_if_linesearch_fails, ~state.failed_linesearch)
[docs] def init_state(self, init_params: Any, *args, **kwargs) -> BroydenState: """Initialize the solver state. Args: init_params: pytree containing the initial parameters. *args: additional positional arguments to be passed to ``fun``. **kwargs: additional keyword arguments to be passed to ``fun``. Returns: state """ if isinstance(init_params, base.OptStep): # `init_params` can either be a pytree or an OptStep object state_kwargs = dict( d_history=init_params.state.d_history, c_history=init_params.state.c_history, gamma=init_params.state.gamma, iter_num=init_params.state.iter_num, stepsize=init_params.state.stepsize, ) init_params = init_params.params dtype = tree_single_dtype(init_params) else: dtype = tree_single_dtype(init_params) state_kwargs = dict( d_history=init_history(init_params, self.history_size), c_history=init_history(init_params, self.history_size), gamma=jnp.asarray(self.gamma, dtype=dtype), iter_num=jnp.asarray(0), stepsize=jnp.asarray(self.max_stepsize, dtype=dtype), ) value, aux = self._value_with_aux(init_params, *args, **kwargs) return BroydenState(value=value, error=jnp.asarray(jnp.inf), **state_kwargs, aux=aux, failed_linesearch=jnp.asarray(False), num_fun_eval=jnp.array(1, base.NUM_EVAL_DTYPE), num_linesearch_iter=jnp.array(0, base.NUM_EVAL_DTYPE) )
[docs] def update(self, params: Any, state: BroydenState, *args, **kwargs) -> base.OptStep: """Performs one iteration of Broyden. Args: params: pytree containing the parameters. state: named tuple containing the solver state. *args: additional positional arguments to be passed to ``fun``. **kwargs: additional keyword arguments to be passed to ``fun``. Returns: (params, state) """ if isinstance(params, base.OptStep): params = params.params start = state.iter_num % self.history_size value = state.value jac_prod_kwargs = dict( d_history=state.d_history, c_history=state.c_history, gamma=state.gamma, start=start, ) jac_prod = partial( inv_jacobian_product, **jac_prod_kwargs, ) jac_rprod = partial( inv_jacobian_rproduct, **jac_prod_kwargs, ) product = jac_prod(pytree=value) descent_direction = tree_scalar_mul(-1.0, product) use_linesearch = not isinstance(self.stepsize, Callable) and self.stepsize <= 0 if use_linesearch: if self.linesearch == "backtracking": # we need to build the function used for the line search # which is going to be the squared norm of the original function # as in scipy https://github.com/scipy/scipy/blob/main/scipy/optimize/_nonlin.py#L278 # we then need to check if the gradient can be obtained with jax # and if not we can build it in the same fashion as scipy # https://github.com/scipy/scipy/blob/main/scipy/optimize/_nonlin.py#L285 def ls_fun_with_aux(params, *args, **kwargs): f, aux = self._value_with_aux(params, *args, **kwargs) norm_squared = tree_l2_norm(f, squared=True) return norm_squared, (f, aux) # here we need a check if the function is not smooth ls_fun_with_aux_and_grad = jax.value_and_grad(ls_fun_with_aux, has_aux=True) ls = BacktrackingLineSearch(fun=ls_fun_with_aux_and_grad, value_and_grad=True, maxiter=self.maxls, decrease_factor=self.decrease_factor, max_stepsize=self.max_stepsize, condition=self.condition, jit=self.jit, unroll=self.unroll, has_aux=True, tol=1e-2) init_stepsize = jnp.where(state.stepsize <= self.min_stepsize, # If stepsize became too small, we restart it. self.max_stepsize, # Else, we increase a bit the previous one. state.stepsize * self.increase_factor) new_stepsize, ls_state = ls.run(init_stepsize, params, value, None, descent_direction, fun_args=args, fun_kwargs=kwargs) new_value, new_aux = ls_state.aux new_params = ls_state.params new_num_linesearch_iter = state.num_linesearch_iter + ls_state.iter_num new_num_fun_eval = state.num_fun_eval + ls_state.num_fun_eval failed_linesearch = ls_state.failed else: raise ValueError("Invalid name in 'linesearch' option.") else: # without line search if isinstance(self.stepsize, Callable): new_stepsize = self.stepsize(state.iter_num) else: new_stepsize = self.stepsize failed_linesearch = False new_params = tree_add_scalar_mul(params, new_stepsize, descent_direction) new_value, new_aux = self._value_with_aux(new_params, *args, **kwargs) new_num_fun_eval = state.num_fun_eval + 1 new_num_linesearch_iter = state.num_linesearch_iter delta_x = tree_sub(new_params, params) v = jac_rprod(delta_x) delta_g = tree_sub(new_value, value) denom = 1 / tree_vdot(v, delta_g) d = tree_scalar_mul(denom, v) c = tree_sub(delta_x, jac_prod(delta_g)) last = (start + self.history_size) % self.history_size d_history = update_history(state.d_history, d, last) c_history = update_history(state.c_history, c, last) new_state = BroydenState(iter_num=state.iter_num + 1, value=new_value, stepsize=jnp.asarray(new_stepsize), error=tree_l2_norm(new_value), d_history=d_history, c_history=c_history, gamma=state.gamma, aux=new_aux, num_fun_eval=new_num_fun_eval, num_linesearch_iter=new_num_linesearch_iter, failed_linesearch=failed_linesearch) return base.OptStep(params=new_params, state=new_state)
[docs] def optimality_fun(self, params, *args, **kwargs): """Optimality function mapping compatible with ``@custom_root``.""" value = self._value_fun(params, *args, **kwargs) return value
def _value_fun(self, params, *args, **kwargs): if isinstance(params, base.OptStep): params = params.params value, _ = self._value_with_aux(params, *args, **kwargs) return value def __post_init__(self): if self.has_aux: fun_ = self.fun else: fun_ = lambda p, *a, **kw: (self.fun(p, *a, **kw), None) self._value_with_aux = fun_ self.reference_signature = self.fun if self.history_size is None: self.history_size = self.maxiter