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Minimisation¤

In addition to the following, note that the Optax library offers an extensive collection of minimisers via first-order gradient methods -- as are in widespread use for neural networks. If you would like to use these through the Optimistix API then an optimistix.OptaxMinimiser wrapper is provided.

optimistix.minimise(fn: Callable[[~Y, Any], tuple[Shaped[Array, ''], ~Aux]] | Callable[[~Y, Any], Shaped[Array, '']], solver: optimistix.AbstractMinimiser, y0: ~Y, args: PyTree[Any] = None, options: dict[str, Any] | None = None, *, has_aux: bool = False, max_steps: int | None = 256, adjoint: optimistix.AbstractAdjoint = ImplicitAdjoint(), throw: bool = True, tags: frozenset[object] = frozenset({})) -> optimistix.Solution[~Y, ~Aux] ¤

Minimise a function.

This minimises a nonlinear function fn(y, args) which returns a scalar value.

Arguments:

  • fn: The objective function. This should take two arguments: fn(y, args) and return a scalar.
  • solver: The minimiser solver to use. This should be an optimistix.AbstractMinimiser.
  • y0: An initial guess for what y may be.
  • args: Passed as the args of fn(y, args).
  • options: Individual solvers may accept additional runtime arguments. See each individual solver's documentation for more details.
  • has_aux: If True, then fn may return a pair, where the first element is its function value, and the second is just auxiliary data. Keyword only argument.
  • max_steps: The maximum number of steps the solver can take. Keyword only argument.
  • adjoint: The adjoint method used to compute gradients through the fixed-point solve. Keyword only argument.
  • throw: How to report any failures. (E.g. an iterative solver running out of steps, or encountering divergent iterates.) If True then a failure will raise an error. If False then the returned solution object will have a result field indicating whether any failures occured. (See optimistix.Solution.) Keyword only argument.
  • tags: Lineax tags describing any structure of the Hessian of fn with respect to y. Used with optimistix.ImplicitAdjoint to implement the implicit function theorem as efficiently as possible. Keyword only argument.

Returns:

An optimistix.Solution object.

optimistix.minimise supports any of the following minimisers.

optimistix.AbstractMinimiser

optimistix.AbstractMinimiser ¤

Abstract base class for all minimisers.

init(fn: Callable[[~Y, Any], tuple[~Out, ~Aux]], y: ~Y, args: PyTree, options: dict[str, Any], f_struct: PyTree[jax._src.api.ShapeDtypeStruct], aux_struct: PyTree[jax._src.api.ShapeDtypeStruct], tags: frozenset[object]) -> ~SolverState ¤

Perform all initial computation needed to initialise the solver state.

For example, the optimistix.Chord method computes the Jacobian df/dy with respect to the initial guess y, and then uses it throughout the computation.

Arguments:

  • fn: The function to iterate over. This is expected to take two argumetns fn(y, args) and return a pytree of arrays in the first element, and any auxiliary data in the second argument.
  • y: The value of y at the current (first) iteration.
  • args: Passed as the args of fn(y, args).
  • options: Individual solvers may accept additional runtime arguments. See each individual solver's documentation for more details.
  • f_struct: A pytree of jax.ShapeDtypeStructs of the same shape as the output of fn. This is used to initialise any information in the state which may rely on the pytree structure, array shapes, or dtype of the output of fn.
  • aux_struct: A pytree of jax.ShapeDtypeStructs of the same shape as the auxiliary data returned by fn.
  • tags: exact meaning depends on whether this is a fixed point, root find, least squares, or minimisation problem; see their relevant entry points.

Returns:

A PyTree representing the initial state of the solver.

step(fn: Callable[[~Y, Any], tuple[~Out, ~Aux]], y: ~Y, args: PyTree, options: dict[str, Any], state: ~SolverState, tags: frozenset[object]) -> tuple[~Y, ~SolverState, ~Aux] ¤

Perform one step of the iterative solve.

Arguments:

  • fn: The function to iterate over. This is expected to take two argumetns fn(y, args) and return a pytree of arrays in the first element, and any auxiliary data in the second argument.
  • y: The value of y at the current (first) iteration.
  • args: Passed as the args of fn(y, args).
  • options: Individual solvers may accept additional runtime arguments. See each individual solver's documentation for more details.
  • state: A pytree representing the state of a solver. The shape of this pytree is solver-dependent.
  • tags: exact meaning depends on whether this is a fixed point, root find, least squares, or minimisation problem; see their relevant entry points.

Returns:

A 3-tuple containing the new y value in the first element, the next solver state in the second element, and the aux output of fn(y, args) in the third element.

terminate(fn: Callable[[~Y, Any], tuple[~Out, ~Aux]], y: ~Y, args: PyTree, options: dict[str, Any], state: ~SolverState, tags: frozenset[object]) -> tuple[Bool[Array, ''], optimistix.RESULTS] ¤

Determine whether or not to stop the iterative solve.

Arguments:

  • fn: The function to iterate over. This is expected to take two argumetns fn(y, args) and return a pytree of arrays in the first element, and any auxiliary data in the second argument.
  • y: The value of y at the current iteration.
  • args: Passed as the args of fn(y, args).
  • options: Individual solvers may accept additional runtime arguments. See each individual solver's documentation for more details.
  • state: A pytree representing the state of a solver. The shape of this pytree is solver-dependent.
  • tags: exact meaning depends on whether this is a fixed point, root find, least squares, or minimisation problem; see their relevant entry points.

Returns:

A 2-tuple containing a bool indicating whether or not to stop iterating in the first element, and an optimistix.RESULTS object in the second element.

postprocess(fn: Callable[[~Y, Any], tuple[~Out, ~Aux]], y: ~Y, aux: ~Aux, args: PyTree, options: dict[str, Any], state: ~SolverState, tags: frozenset[object], result: optimistix.RESULTS) -> tuple[~Y, ~Aux, dict[str, Any]] ¤

Any final postprocessing to perform on the result of the solve.

Arguments:

  • fn: The function to iterate over. This is expected to take two argumetns fn(y, args) and return a pytree of arrays in the first element, and any auxiliary data in the second argument.
  • y: The value of y at the last iteration.
  • aux: The auxiliary output at the last iteration.
  • args: Passed as the args of fn(y, args).
  • options: Individual solvers may accept additional runtime arguments. See each individual solver's documentation for more details.
  • state: A pytree representing the final state of a solver. The shape of this pytree is solver-dependent.
  • tags: exact meaning depends on whether this is a fixed point, root find, least squares, or minimisation problem; see their relevant entry points.
  • result: as returned by the final call to terminate.

Returns:

A 3-tuple of:

  • final_y: the final y to return as the solution of the solve.
  • final_aux: the final aux to return as the auxiliary output of the solve.
  • stats: any additional information to place in the sol.stats dictionary.

Info

Most solvers will not need to use this, so that this method may be defined as: 

def postprocess(self, fn, y, aux, args, options, state, tags, result):
    return y, aux, {}

optimistix.AbstractGradientDescent

optimistix.AbstractGradientDescent(optimistix.AbstractMinimiser) ¤

The gradient descent method for unconstrained minimisation.

At every step, this algorithm performs a line search along the steepest descent direction. You should subclass this to provide it with a particular choice of line search. (E.g. optimistix.GradientDescent uses a simple learning rate step.)

Subclasses must provide the following abstract attributes, with the following types:

  • rtol: float
  • atol: float
  • norm: Callable[[PyTree], Scalar]
  • descent: AbstractDescent
  • search: AbstractSearch

Supports the following options:

  • autodiff_mode: whether to use forward- or reverse-mode autodifferentiation to compute the gradient. Can be either "fwd" or "bwd". Defaults to "bwd", which is usually more efficient. Changing this can be useful when the target function does not support reverse-mode automatic differentiation.

optimistix.GradientDescent(optimistix.AbstractGradientDescent) ¤

Classic gradient descent with a learning rate learning_rate.

Supports the following options:

  • autodiff_mode: whether to use forward- or reverse-mode autodifferentiation to compute the gradient. Can be either "fwd" or "bwd". Defaults to "bwd", which is usually more efficient. Changing this can be useful when the target function does not support reverse-mode automatic differentiation.
__init__(learning_rate: float, rtol: float, atol: float, norm: Callable[[PyTree], Shaped[Array, '']] = <function max_norm>) ¤

Arguments:

  • learning_rate: Specifies a constant learning rate to use at each step.
  • rtol: Relative tolerance for terminating the solve.
  • atol: Absolute tolerance for terminating the solve.
  • norm: The norm used to determine the difference between two iterates in the convergence criteria. Should be any function PyTree -> Scalar. Optimistix includes three built-in norms: optimistix.max_norm, optimistix.rms_norm, and optimistix.two_norm.
optimistix.AbstractBFGS

optimistix.AbstractBFGS(optimistix.AbstractMinimiser) ¤

Abstract BFGS (Broyden--Fletcher--Goldfarb--Shanno) minimisation algorithm.

This is a quasi-Newton optimisation algorithm, whose defining feature is the way it progressively builds up a Hessian approximation using multiple steps of gradient information.

This abstract version may be subclassed to choose alternative descent and searches.

Supports the following options:

  • autodiff_mode: whether to use forward- or reverse-mode autodifferentiation to compute the gradient. Can be either "fwd" or "bwd". Defaults to "bwd", which is usually more efficient. Changing this can be useful when the target function does not support reverse-mode automatic differentiation.

optimistix.BFGS(optimistix.AbstractBFGS) ¤

BFGS (Broyden--Fletcher--Goldfarb--Shanno) minimisation algorithm.

This is a quasi-Newton optimisation algorithm, whose defining feature is the way it progressively builds up a Hessian approximation using multiple steps of gradient information.

Supports the following options:

  • autodiff_mode: whether to use forward- or reverse-mode autodifferentiation to compute the gradient. Can be either "fwd" or "bwd". Defaults to "bwd", which is usually more efficient. Changing this can be useful when the target function does not support reverse-mode automatic differentiation.
__init__(rtol: float, atol: float, norm: Callable[[PyTree], Shaped[Array, '']] = <function max_norm>, use_inverse: bool = True, verbose: frozenset[str] = frozenset({})) ¤

Arguments:

  • rtol: Relative tolerance for terminating the solve.
  • atol: Absolute tolerance for terminating the solve.
  • norm: The norm used to determine the difference between two iterates in the convergence criteria. Should be any function PyTree -> Scalar. Optimistix includes three built-in norms: optimistix.max_norm, optimistix.rms_norm, and optimistix.two_norm.
  • use_inverse: The BFGS algorithm involves computing matrix-vector products of the form B^{-1} g, where B is an approximation to the Hessian of the function to be minimised. This means we can either (a) store the approximate Hessian B, and do a linear solve on every step, or (b) store the approximate Hessian inverse B^{-1}, and do a matrix-vector product on every step. Option (a) is generally cheaper for sparse Hessians (as the inverse may be dense). Option (b) is generally cheaper for dense Hessians (as matrix-vector products are cheaper than linear solves). The default is (b), denoted via use_inverse=True. Note that this is incompatible with line search methods like optimistix.ClassicalTrustRegion, which use the Hessian approximation B as part of their own computations.
  • verbose: Whether to print out extra information about how the solve is proceeding. Should be a frozenset of strings, specifying what information to print. Valid entries are step_size, loss, y. For example verbose=frozenset({"step_size", "loss"}).

optimistix.OptaxMinimiser(optimistix.AbstractMinimiser) ¤

A wrapper to use Optax first-order gradient-based optimisers with optimistix.minimise.

__init__(optim, rtol: float, atol: float, norm: Callable[[PyTree], Shaped[Array, '']] = <function max_norm>, verbose: frozenset[str] = frozenset({})) ¤

Arguments:

  • optim: The Optax optimiser to use.
  • rtol: Relative tolerance for terminating the solve. Keyword only argument.
  • atol: Absolute tolerance for terminating the solve. Keyword only argument.
  • norm: The norm used to determine the difference between two iterates in the convergence criteria. Should be any function PyTree -> Scalar. Optimistix includes three built-in norms: optimistix.max_norm, optimistix.rms_norm, and optimistix.two_norm. Keyword only argument.
  • verbose: Whether to print out extra information about how the solve is proceeding. Should be a frozenset of strings, specifying what information to print out. Valid entries are step, loss, y. For example verbose=frozenset({"step", "loss"}).

optim in optimistix.OptaxMinimiser is an instance of an Optax minimiser. For example, correct usage is optimistix.OptaxMinimiser(optax.adam(...), ...), not optimistix.OptaxMinimiser(optax.adam, ...).

optimistix.NonlinearCG(optimistix.AbstractGradientDescent) ¤

The nonlinear conjugate gradient method.

Supports the following options:

  • autodiff_mode: whether to use forward- or reverse-mode autodifferentiation to compute the gradient. Can be either "fwd" or "bwd". Defaults to "bwd", which is usually more efficient. Changing this can be useful when the target function does not support reverse-mode automatic differentiation.
__init__(rtol: float, atol: float, norm: Callable[[PyTree[Array]], Shaped[Array, '']] = <function max_norm>, method: Callable[[~Y, ~Y, ~Y], Shaped[Array, '']] = <function polak_ribiere>, search: optimistix.AbstractSearch[~Y, optimistix._search.FunctionInfo.EvalGrad, optimistix._search.FunctionInfo.Eval, Any] = BacktrackingArmijo(decrease_factor=0.5, slope=0.1)) ¤

Arguments:

optimistix.NonlinearCG supports several different methods for computing its β parameter. If you are trying multiple solvers to see which works best on your problem, then you may wish to try all four versions of nonlinear CG. These can each be passed as NonlinearCG(..., method=...).

optimistix.polak_ribiere(grad_vector: ~Y, grad_prev: ~Y, y_diff_prev: ~Y) -> Shaped[Array, ''] ¤

The Polak--Ribière formula for β. Used with optimistix.NonlinearCG and optimistix.NonlinearCGDescent.

optimistix.fletcher_reeves(grad: ~Y, grad_prev: ~Y, y_diff_prev: ~Y) -> Shaped[Array, ''] ¤

The Fletcher--Reeves formula for β. Used with optimistix.NonlinearCG and optimistix.NonlinearCGDescent.

optimistix.hestenes_stiefel(grad: ~Y, grad_prev: ~Y, y_diff_prev: ~Y) -> Shaped[Array, ''] ¤

The Hestenes--Stiefel formula for β. Used with optimistix.NonlinearCG and optimistix.NonlinearCGDescent.

optimistix.dai_yuan(grad: ~Y, grad_prev: ~Y, y_diff_prev: ~Y) -> Shaped[Array, ''] ¤

The Dai--Yuan formula for β. Used with optimistix.NonlinearCG and optimistix.NonlinearCGDescent.

optimistix.NelderMead(optimistix.AbstractMinimiser) ¤

The Nelder-Mead minimisation algorithm. (Downhill simplex derivative-free method.)

This algorithm is notable in that it only uses function evaluations, and does not need gradient evaluations.

This is usually an "algorithm of last resort". Gradient-based algorithms are usually much faster, and are more likely to converge to a minima.

Comparable to scipy.optimize.minimize(method="Nelder-Mead").

__init__(rtol: float, atol: float, norm: Callable[[PyTree], Shaped[Array, '']] = <function max_norm>, rdelta: float = 0.05, adelta: float = 0.00025) ¤

Arguments:

  • rtol: Relative tolerance for terminating the solve.
  • atol: Absolute tolerance for terminating the solve.
  • norm: The norm used to determine the difference between two iterates in the convergence criteria. Should be any function PyTree -> Scalar. Optimistix includes three built-in norms: optimistix.max_norm, optimistix.rms_norm, and optimistix.two_norm.
  • rdelta: Nelder-Mead creates an initial simplex by appending a scaled identity matrix to y. The ith element of this matrix is rdelta * y_i + adelta. That is, this is the relative size for creating the initial simplex.
  • adelta: Nelder-Mead creates an initial simplex by appending a scaled identity matrix to y. The ith element of this matrix is rdelta * y_i + adelta. That is, this is the absolute size for creating the initial simplex.

optimistix.BestSoFarMinimiser(optimistix.AbstractMinimiser) ¤

Wraps another minimiser, to return the best-so-far value. That is, it makes a copy of the best y seen, and returns that.

__init__(solver: optimistix.AbstractMinimiser[~Y, tuple[Shaped[Array, ''], ~Aux], Any]) ¤

Arguments:

  • solver: the minimiser to wrap.