Riemannian quasi-Newton methods

Manopt.quasi_Newton — Function

quasi_Newton(M, f, grad_f, p; kwargs...)
quasi_Newton!(M, f, grad_f, p; kwargs...)

Perform a quasi Newton iteration to solve

\[\operatorname{arg\,min}_{p ∈ \mathcal M} f(p)\]

with start point p. The iterations can be done in-place of p$=p^{(0)}$. The $k$th iteration consists of

Compute the search direction $η^{(k)} = -\mathcal B_k [\operatorname{grad}f (p^{(k)})]$ or solve $\mathcal H_k [η^{(k)}] = -\operatorname{grad}f (p^{(k)})]$.
Determine a suitable stepsize $α_k$ along the curve $γ(α) = R_{p^{(k)}}(α η^{(k)})$, usually by using WolfePowellLinesearch.
Compute $p^{(k+1)} = R_{p^{(k)}}(α_k η^{(k)})$.
Define $s_k = \mathcal T_{p^{(k)}, α_k η^{(k)}}(α_k η^{(k)})$ and $y_k = \operatorname{grad}f(p^{(k+1)}) - \mathcal T_{p^{(k)}, α_k η^{(k)}}(\operatorname{grad}f(p^{(k)}))$, where $\mathcal T$ denotes a vector transport.
Compute the new approximate Hessian $H_{k+1}$ or its inverse $B_{k+1}$.

Input

M::AbstractManifold: a Riemannian manifold $\mathcal M$
f: a cost function $f: \mathcal M→ ℝ$ implemented as (M, p) -> v
grad_f: the (Riemannian) gradient $\operatorname{grad}f: \mathcal M → T_{p}\mathcal M$ of f as a function (M, p) -> X or a function (M, X, p) -> X computing X in-place
p: a point on the manifold $\mathcal M$

Keyword arguments

basis=DefaultOrthonormalBasis(): basis to use within each of the the tangent spaces to represent the Hessian (inverse) for the cases where it is stored in full (matrix) form.
cautious_update=false: whether or not to use the QuasiNewtonCautiousDirectionUpdate which wraps the direction_upate.
cautious_function=(x) -> x * 1e-4: a monotone increasing function for the cautious update that is zero at $x=0$ and strictly increasing at $0$
direction_update=InverseBFGS(): the AbstractQuasiNewtonUpdateRule to use.
evaluation=AllocatingEvaluation(): specify whether the functions that return an array, for example a point or a tangent vector, work by allocating its result (AllocatingEvaluation) or whether they modify their input argument to return the result therein (InplaceEvaluation). Since usually the first argument is the manifold, the modified argument is the second.For example grad_f(M,p) allocates, but grad_f!(M, X, p) computes the result in-place of X.
initial_operator= initial_scale*Matrix{Float64}(I, n, n): initial matrix to use in case the Hessian (inverse) approximation is stored as a full matrix, that is n=manifold_dimension(M). This matrix is only allocated for the full matrix case. See also initial_scale.
initial_scale=1.0: scale initial s to use in with $\frac{s⟨s_k,y_k⟩_{p_k}}{\lVert y_k\rVert_{p_k}}$ in the computation of the limited memory approach. see also initial_operator
memory_size=20: limited memory, number of $s_k, y_k$ to store. Set to a negative value to use a full memory (matrix) representation
nondescent_direction_behavior=:reinitialize_direction_update: specify how non-descent direction is handled. This can be
- :step_towards_negative_gradient: the direction is replaced with negative gradient, a message is stored.
- :ignore: the verification is not performed, so any computed direction is accepted. No message is stored.
- :reinitialize_direction_update: discards operator state stored in direction update rules.
- any other value performs the verification, keeps the direction but stores a message.
A stored message can be displayed using DebugMessages.
project!=copyto!: for numerical stability it is possible to project onto the tangent space after every iteration. the function has to work inplace of Y, that is (M, Y, p, X) -> Y, where X and Y can be the same memory.
retraction_method=default_retraction_method(M, typeof(p)): a retraction $\operatorname{retr}$ to use, see the section on retractions
stepsize=WolfePowellLinesearch(retraction_method, vector_transport_method): a functor inheriting from Stepsize to determine a step size
stopping_criterion=StopAfterIteration(max(1000, memory_size))|StopWhenGradientNormLess(1e-6): a functor indicating that the stopping criterion is fulfilled
vector_transport_method=default_vector_transport_method(M, typeof(p)): a vector transport $\mathcal T_{⋅←⋅}$ to use, see the section on vector transports

All other keyword arguments are passed to decorate_state! for state decorators or decorate_objective! for objective decorators, respectively.

Output

The obtained approximate minimizer $p^*$. To obtain the whole final state of the solver, see get_solver_return for details, especially the return_state= keyword.

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Manopt.quasi_Newton! — Function

quasi_Newton(M, f, grad_f, p; kwargs...)
quasi_Newton!(M, f, grad_f, p; kwargs...)

Perform a quasi Newton iteration to solve

\[\operatorname{arg\,min}_{p ∈ \mathcal M} f(p)\]

with start point p. The iterations can be done in-place of p$=p^{(0)}$. The $k$th iteration consists of

Compute the search direction $η^{(k)} = -\mathcal B_k [\operatorname{grad}f (p^{(k)})]$ or solve $\mathcal H_k [η^{(k)}] = -\operatorname{grad}f (p^{(k)})]$.
Determine a suitable stepsize $α_k$ along the curve $γ(α) = R_{p^{(k)}}(α η^{(k)})$, usually by using WolfePowellLinesearch.
Compute $p^{(k+1)} = R_{p^{(k)}}(α_k η^{(k)})$.
Define $s_k = \mathcal T_{p^{(k)}, α_k η^{(k)}}(α_k η^{(k)})$ and $y_k = \operatorname{grad}f(p^{(k+1)}) - \mathcal T_{p^{(k)}, α_k η^{(k)}}(\operatorname{grad}f(p^{(k)}))$, where $\mathcal T$ denotes a vector transport.
Compute the new approximate Hessian $H_{k+1}$ or its inverse $B_{k+1}$.

Input

M::AbstractManifold: a Riemannian manifold $\mathcal M$
f: a cost function $f: \mathcal M→ ℝ$ implemented as (M, p) -> v
grad_f: the (Riemannian) gradient $\operatorname{grad}f: \mathcal M → T_{p}\mathcal M$ of f as a function (M, p) -> X or a function (M, X, p) -> X computing X in-place
p: a point on the manifold $\mathcal M$

Keyword arguments

basis=DefaultOrthonormalBasis(): basis to use within each of the the tangent spaces to represent the Hessian (inverse) for the cases where it is stored in full (matrix) form.
cautious_update=false: whether or not to use the QuasiNewtonCautiousDirectionUpdate which wraps the direction_upate.
cautious_function=(x) -> x * 1e-4: a monotone increasing function for the cautious update that is zero at $x=0$ and strictly increasing at $0$
direction_update=InverseBFGS(): the AbstractQuasiNewtonUpdateRule to use.
evaluation=AllocatingEvaluation(): specify whether the functions that return an array, for example a point or a tangent vector, work by allocating its result (AllocatingEvaluation) or whether they modify their input argument to return the result therein (InplaceEvaluation). Since usually the first argument is the manifold, the modified argument is the second.For example grad_f(M,p) allocates, but grad_f!(M, X, p) computes the result in-place of X.
initial_operator= initial_scale*Matrix{Float64}(I, n, n): initial matrix to use in case the Hessian (inverse) approximation is stored as a full matrix, that is n=manifold_dimension(M). This matrix is only allocated for the full matrix case. See also initial_scale.
initial_scale=1.0: scale initial s to use in with $\frac{s⟨s_k,y_k⟩_{p_k}}{\lVert y_k\rVert_{p_k}}$ in the computation of the limited memory approach. see also initial_operator
memory_size=20: limited memory, number of $s_k, y_k$ to store. Set to a negative value to use a full memory (matrix) representation
nondescent_direction_behavior=:reinitialize_direction_update: specify how non-descent direction is handled. This can be
- :step_towards_negative_gradient: the direction is replaced with negative gradient, a message is stored.
- :ignore: the verification is not performed, so any computed direction is accepted. No message is stored.
- :reinitialize_direction_update: discards operator state stored in direction update rules.
- any other value performs the verification, keeps the direction but stores a message.
A stored message can be displayed using DebugMessages.
project!=copyto!: for numerical stability it is possible to project onto the tangent space after every iteration. the function has to work inplace of Y, that is (M, Y, p, X) -> Y, where X and Y can be the same memory.
retraction_method=default_retraction_method(M, typeof(p)): a retraction $\operatorname{retr}$ to use, see the section on retractions
stepsize=WolfePowellLinesearch(retraction_method, vector_transport_method): a functor inheriting from Stepsize to determine a step size
stopping_criterion=StopAfterIteration(max(1000, memory_size))|StopWhenGradientNormLess(1e-6): a functor indicating that the stopping criterion is fulfilled
vector_transport_method=default_vector_transport_method(M, typeof(p)): a vector transport $\mathcal T_{⋅←⋅}$ to use, see the section on vector transports

All other keyword arguments are passed to decorate_state! for state decorators or decorate_objective! for objective decorators, respectively.

Output

The obtained approximate minimizer $p^*$. To obtain the whole final state of the solver, see get_solver_return for details, especially the return_state= keyword.

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Background

The aim is to minimize a real-valued function on a Riemannian manifold, that is

\[\min f(x), \quad x ∈ \mathcal{M}.\]

Riemannian quasi-Newtonian methods are as generalizations of their Euclidean counterparts Riemannian line search methods. These methods determine a search direction $η_k ∈ T_{x_k} \mathcal{M}$ at the current iterate $x_k$ and a suitable stepsize $α_k$ along $\gamma(α) = R_{x_k}(α η_k)$, where $R: T \mathcal{M} →\mathcal{M}$ is a retraction. The next iterate is obtained by

\[x_{k+1} = R_{x_k}(α_k η_k).\]

In quasi-Newton methods, the search direction is given by

\[η_k = -{\mathcal{H}_k}^{-1}[\operatorname{grad}f (x_k)] = -\mathcal{B}_k [\operatorname{grad} (x_k)],\]

where $\mathcal{H}_k : T_{x_k} \mathcal{M} →T_{x_k} \mathcal{M}$ is a positive definite self-adjoint operator, which approximates the action of the Hessian $\operatorname{Hess} f (x_k)[⋅]$ and $\mathcal{B}_k = {\mathcal{H}_k}^{-1}$. The idea of quasi-Newton methods is instead of creating a complete new approximation of the Hessian operator $\operatorname{Hess} f(x_{k+1})$ or its inverse at every iteration, the previous operator $\mathcal{H}_k$ or $\mathcal{B}_k$ is updated by a convenient formula using the obtained information about the curvature of the objective function during the iteration. The resulting operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ acts on the tangent space $T_{x_{k+1}} \mathcal{M}$ of the freshly computed iterate $x_{k+1}$. In order to get a well-defined method, the following requirements are placed on the new operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ that is created by an update. Since the Hessian $\operatorname{Hess} f(x_{k+1})$ is a self-adjoint operator on the tangent space $T_{x_{k+1}} \mathcal{M}$, and $\mathcal{H}_{k+1}$ approximates it, one requirement is, that $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ is also self-adjoint on $T_{x_{k+1}} \mathcal{M}$. In order to achieve a steady descent, the next requirement is that $η_k$ is a descent direction in each iteration. Hence a further requirement is that $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ is a positive definite operator on $T_{x_{k+1}} \mathcal{M}$. In order to get information about the curvature of the objective function into the new operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$, the last requirement is a form of a Riemannian quasi-Newton equation:

\[\mathcal{H}_{k+1} [T_{x_k \rightarrow x_{k+1}}({R_{x_k}}^{-1}(x_{k+1}))] = \operatorname{grad}(x_{k+1}) - T_{x_k \rightarrow x_{k+1}}(\operatorname{grad}f(x_k))\]

\[\mathcal{B}_{k+1} [\operatorname{grad}f(x_{k+1}) - T_{x_k \rightarrow x_{k+1}}(\operatorname{grad}f(x_k))] = T_{x_k \rightarrow x_{k+1}}({R_{x_k}}^{-1}(x_{k+1}))\]

where $T_{x_k \rightarrow x_{k+1}} : T_{x_k} \mathcal{M} →T_{x_{k+1}} \mathcal{M}$ and the chosen retraction $R$ is the associated retraction of $T$. Note that, of course, not all updates in all situations meet these conditions in every iteration. For specific quasi-Newton updates, the fulfilment of the Riemannian curvature condition, which requires that

\[g_{x_{k+1}}(s_k, y_k) > 0\]

holds, is a requirement for the inheritance of the self-adjointness and positive definiteness of the $\mathcal{H}_k$ or $\mathcal{B}_k$ to the operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$. Unfortunately, the fulfilment of the Riemannian curvature condition is not given by a step size $\alpha_k > 0$ that satisfies the generalized Wolfe conditions. However, to create a positive definite operator $\mathcal{H}_{k+1}$ or $\mathcal{B}_{k+1}$ in each iteration, the so-called locking condition was introduced in [HGA15], which requires that the isometric vector transport $T^S$, which is used in the update formula, and its associate retraction $R$ fulfil

\[T^{S}{x, ξ_x}(ξ_x) = β T^{R}{x, ξ_x}(ξ_x), \quad β = \frac{\lVert ξ_x \rVert_x}{\lVert T^{R}{x, ξ_x}(ξ_x) \rVert_{R_{x}(ξ_x)}},\]

where $T^R$ is the vector transport by differentiated retraction. With the requirement that the isometric vector transport $T^S$ and its associated retraction $R$ satisfies the locking condition and using the tangent vector

\[y_k = {β_k}^{-1} \operatorname{grad}f(x_{k+1}) - T^{S}{x_k, α_k η_k}(\operatorname{grad}f(x_k)),\]

where

\[β_k = \frac{\lVert α_k η_k \rVert_{x_k}}{\lVert T^{R}{x_k, α_k η_k}(α_k η_k) \rVert_{x_{k+1}}},\]

in the update, it can be shown that choosing a stepsize $α_k > 0$ that satisfies the Riemannian Wolfe conditions leads to the fulfilment of the Riemannian curvature condition, which in turn implies that the operator generated by the updates is positive definite. In the following the specific operators are denoted in matrix notation and hence use $H_k$ and $B_k$, respectively.

Direction updates

In general there are different ways to compute a fixed AbstractQuasiNewtonUpdateRule. In general these are represented by

Manopt.AbstractQuasiNewtonDirectionUpdate — Type

AbstractQuasiNewtonDirectionUpdate

An abstract representation of an Quasi Newton Update rule to determine the next direction given current QuasiNewtonState.

All subtypes should be functors, they should be callable as H(M,x,d) to compute a new direction update.

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Manopt.QuasiNewtonMatrixDirectionUpdate — Type

QuasiNewtonMatrixDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate

The QuasiNewtonMatrixDirectionUpdate represent a quasi-Newton update rule, where the operator is stored as a matrix. A distinction is made between the update of the approximation of the Hessian, $H_k \mapsto H_{k+1}$, and the update of the approximation of the Hessian inverse, $B_k \mapsto B_{k+1}$. For the first case, the coordinates of the search direction $η_k$ with respect to a basis $\{b_i\}_{i=1}^{n}$ are determined by solving a linear system of equations

\[\text{Solve} \quad \hat{η_k} = - H_k \widehat{\operatorname{grad}f(x_k)},\]

where $H_k$ is the matrix representing the operator with respect to the basis $\{b_i\}_{i=1}^{n}$ and $\widehat{\operatorname{grad}} f(p_k)}$ represents the coordinates of the gradient of the objective function $f$ in $x_k$ with respect to the basis $\{b_i\}_{i=1}^{n}$. If a method is chosen where Hessian inverse is approximated, the coordinates of the search direction $η_k$ with respect to a basis $\{b_i\}_{i=1}^{n}$ are obtained simply by matrix-vector multiplication

\[\hat{η_k} = - B_k \widehat{\operatorname{grad}f(x_k)},\]

where $B_k$ is the matrix representing the operator with respect to the basis $\{b_i\}_{i=1}^{n}$ and \widehat{\operatorname{grad}} f(p_k)}. In the end, the search directionη_kis generated from the coordinates\hat{eta_k}and the vectors of the basis\{b_i\}_{i=1}^{n}in both variants. The [AbstractQuasiNewtonUpdateRule](@ref) indicates which quasi-Newton update rule is used. In all of them, the Euclidean update formula is used to generate the matrixH_{k+1}andB_{k+1}, and the basis\{b_i\}_{i=1}^{n}is transported into the upcoming tangent spaceT_{p_{k+1}} \mathcal M`, preferably with an isometric vector transport, or generated there.

Provided functors

(mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction
(η, mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction in-place of η

Fields

basis: an AbstractBasis to use in the tangent spaces
matrix: the matrix which represents the approximating operator.
initial_scale: when initialising the update, a unit matrix is used as initial approximation, scaled by this factor
update: a AbstractQuasiNewtonUpdateRule.
vector_transport_method::AbstractVectorTransportMethodP: a vector transport $\mathcal T_{⋅←⋅}$ to use, see the section on vector transports

Constructor

QuasiNewtonMatrixDirectionUpdate(
    M::AbstractManifold,
    update,
    basis::B=DefaultOrthonormalBasis(),
    m=Matrix{Float64}(I, manifold_dimension(M), manifold_dimension(M));
    kwargs...
)

Keyword arguments

initial_scale=1.0
vector_transport_method=default_vector_transport_method(M, typeof(p)): a vector transport $\mathcal T_{⋅←⋅}$ to use, see the section on vector transports

Generate the Update rule with defaults from a manifold and the names corresponding to the fields.

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Manopt.QuasiNewtonLimitedMemoryDirectionUpdate — Type

QuasiNewtonLimitedMemoryDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate

This AbstractQuasiNewtonDirectionUpdate represents the limited-memory Riemannian BFGS update, where the approximating operator is represented by $m$ stored pairs of tangent vectors $\{\widehat{s}_i\}_{i=k-m}^{k-1} and \{\widehat{y}_i\}_{i=k-m}^{k-1} in the$k$-th iteration. For the calculation of the search direction$Xk$, the generalisation of the two-loop recursion is used (see [HuangGallivanAbsil:2015](@cite)), since it only requires inner products and linear combinations of tangent vectors in$T{pk}\mathcal M$. For that the stored pairs of tangent vectors$\widehat{s}i, \widehat{y}i$, the gradient$\operatorname{grad} f(pk)$of the objective function$f$in$p_k`` and the positive definite self-adjoint operator

\[\mathcal{B}^{(0)}_k[⋅] = \frac{g_{p_k}(s_{k-1}, y_{k-1})}{g_{p_k}(y_{k-1}, y_{k-1})} \; \mathrm{id}_{T_{p_k} \mathcal{M}}[⋅]\]

are used. The two-loop recursion can be understood as that the InverseBFGS update is executed $m$ times in a row on $\mathcal B^{(0)}_k[⋅]$ using the tangent vectors $\widehat{s}_i,\widehat{y}_i$, and in the same time the resulting operator $\mathcal B^{LRBFGS}_k [⋅]$ is directly applied on $\operatorname{grad}f(x_k)$. When updating there are two cases: if there is still free memory, $k < m$, the previously stored vector pairs $\widehat{s}_i,\widehat{y}_i$ have to be transported into the upcoming tangent space $T_{p_{k+1}}\mathcal M$. If there is no free memory, the oldest pair $\widehat{s}_i,\widehat{y}_i$ has to be discarded and then all the remaining vector pairs $\widehat{s}_i,\widehat{y}_i$ are transported into the tangent space $T_{p_{k+1}}\mathcal M$. After that the new values $s_k = \widehat{s}_k = T^{S}_{x_k, α_k η_k}(α_k η_k)$ and $y_k = \widehat{y}_k$ are stored at the beginning. This process ensures that new information about the objective function is always included and the old, probably no longer relevant, information is discarded.

Provided functors

(mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction
(η, mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction in-place of η

Fields

memory_s; the set of the stored (and transported) search directions times step size $\{\widehat{s}_i\}_{i=k-m}^{k-1}$.
memory_y: set of the stored gradient differences $\{\widehat{y}_i\}_{i=k-m}^{k-1}$.
ξ: a variable used in the two-loop recursion.
ρ; a variable used in the two-loop recursion.
initial_scale: initial scaling of the Hessian
vector_transport_method::AbstractVectorTransportMethodP: a vector transport $\mathcal T_{⋅←⋅}$ to use, see the section on vector transports
message: a string containing a potential warning that might have appeared
project!: a function to stabilize the update by projecting on the tangent space

Constructor

QuasiNewtonLimitedMemoryDirectionUpdate(
    M::AbstractManifold,
    x,
    update::AbstractQuasiNewtonUpdateRule,
    memory_size;
    initial_vector=zero_vector(M,x),
    initial_scale::Real=1.0
    project!=copyto!
)

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Manopt.QuasiNewtonCautiousDirectionUpdate — Type

QuasiNewtonCautiousDirectionUpdate <: AbstractQuasiNewtonDirectionUpdate

These AbstractQuasiNewtonDirectionUpdates represent any quasi-Newton update rule, which are based on the idea of a so-called cautious update. The search direction is calculated as given in QuasiNewtonMatrixDirectionUpdate or QuasiNewtonLimitedMemoryDirectionUpdate, butut the update then is only executed if

\[\frac{g_{x_{k+1}}(y_k,s_k)}{\lVert s_k \rVert^{2}_{x_{k+1}}} ≥ θ(\lVert \operatorname{grad}f(x_k) \rVert_{x_k}),\]

is satisfied, where $θ$ is a monotone increasing function satisfying $θ(0) = 0$ and $θ$ is strictly increasing at $0$. If this is not the case, the corresponding update is skipped, which means that for QuasiNewtonMatrixDirectionUpdate the matrix $H_k$ or $B_k$ is not updated. The basis $\{b_i\}^{n}_{i=1}$ is nevertheless transported into the upcoming tangent space $T_{x_{k+1}} \mathcal{M}$, and for QuasiNewtonLimitedMemoryDirectionUpdate neither the oldest vector pair $\{ \widetilde{s}_{k−m}, \widetilde{y}_{k−m}\}$ is discarded nor the newest vector pair $\{ \widetilde{s}_{k}, \widetilde{y}_{k}\}$ is added into storage, but all stored vector pairs $\{ \widetilde{s}_i, \widetilde{y}_i\}_{i=k-m}^{k-1}$ are transported into the tangent space $T_{x_{k+1}} \mathcal{M}$. If InverseBFGS or InverseBFGS is chosen as update, then the resulting method follows the method of [HAG18], taking into account that the corresponding step size is chosen.

Provided functors

(mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction
(η, mp::AbstractManoptproblem, st::QuasiNewtonState) -> η to compute the update direction in-place of η

Fields

update: an AbstractQuasiNewtonDirectionUpdate
θ: a monotone increasing function satisfying $θ(0) = 0$ and $θ$ is strictly increasing at $0$.

Constructor

QuasiNewtonCautiousDirectionUpdate(U::QuasiNewtonMatrixDirectionUpdate; θ = identity)
QuasiNewtonCautiousDirectionUpdate(U::QuasiNewtonLimitedMemoryDirectionUpdate; θ = identity)

Generate a cautious update for either a matrix based or a limited memory based update rule.

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Manopt.initialize_update! — Function

initialize_update!(s::AbstractQuasiNewtonDirectionUpdate)

Initialize direction update. By default no change is made.

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initialize_update!(d::QuasiNewtonLimitedMemoryDirectionUpdate)

Initialize the limited memory direction update by emptying the memory buffers.

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Hessian update rules

Using

Manopt.update_hessian! — Function

update_hessian!(d::AbstractQuasiNewtonDirectionUpdate, amp, st, p_old, k)

update the Hessian within the QuasiNewtonState st given a AbstractManoptProblem amp as well as the an AbstractQuasiNewtonDirectionUpdate d and the last iterate p_old. Note that the current (kth) iterate is already stored in get_iterate(st).

See also AbstractQuasiNewtonUpdateRule and its subtypes for the different rules that are available within d.

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the following update formulae for either $H_{k+1}$ or $B_{k+1}$ are available.

Manopt.AbstractQuasiNewtonUpdateRule — Type

AbstractQuasiNewtonUpdateRule

Specify a type for the different AbstractQuasiNewtonDirectionUpdates, that is for a QuasiNewtonMatrixDirectionUpdate there are several different updates to the matrix, while the default for QuasiNewtonLimitedMemoryDirectionUpdate the most prominent is InverseBFGS.

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Manopt.BFGS — Type

BFGS <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian BFGS update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{H}_k^\mathrm{BFGS}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{BFGS}_{k+1} = \widetilde{H}^\mathrm{BFGS}_k + \frac{y_k y^{\mathrm{T}}_k }{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{H}^\mathrm{BFGS}_k s_k s^{\mathrm{T}}_k \widetilde{H}^\mathrm{BFGS}_k }{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{BFGS}_k s_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

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Manopt.DFP — Type

DFP <: AbstractQuasiNewtonUpdateRule

indicates in an AbstractQuasiNewtonDirectionUpdate that the Riemannian DFP update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{H}_k^\mathrm{DFP}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{DFP}_{k+1} = \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{y_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} \Bigr) \widetilde{H}^\mathrm{DFP}_k \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{s_k y^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} \Bigr) + \frac{y_k y^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

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Manopt.Broyden — Type

Broyden <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian Broyden update is used in the Riemannian quasi-Newton method, which is as a convex combination of BFGS and DFP.

Denote by $\widetilde{H}_k^\mathrm{Br}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{Br}_{k+1} = \widetilde{H}^\mathrm{Br}_k - \frac{\widetilde{H}^\mathrm{Br}_k s_k s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k}{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k} + \frac{y_k y^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} + φ_k s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k \Bigl( \frac{y_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{H}^\mathrm{Br}_k s_k}{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k} \Bigr) \Bigl( \frac{y_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{H}^\mathrm{Br}_k s_k}{s^{\mathrm{T}}_k \widetilde{H}^\mathrm{Br}_k s_k} \Bigr)^{\mathrm{T}}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively, and $φ_k$ is the Broyden factor which is :constant by default but can also be set to :Davidon.

Constructor

Broyden(φ, update_rule::Symbol = :constant)

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Manopt.SR1 — Type

SR1 <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian SR1 update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{H}_k^\mathrm{SR1}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[H^\mathrm{SR1}_{k+1} = \widetilde{H}^\mathrm{SR1}_k + \frac{ (y_k - \widetilde{H}^\mathrm{SR1}_k s_k) (y_k - \widetilde{H}^\mathrm{SR1}_k s_k)^{\mathrm{T}} }{ (y_k - \widetilde{H}^\mathrm{SR1}_k s_k)^{\mathrm{T}} s_k }\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

This method can be stabilized by only performing the update if denominator is larger than $r\lVert s_k\rVert_{x_{k+1}}\lVert y_k - \widetilde{H}^\mathrm{SR1}_k s_k \rVert_{x_{k+1}}$ for some $r>0$. For more details, see Section 6.2 in [NW06].

Constructor

SR1(r::Float64=-1.0)

Generate the SR1 update.

source

Manopt.InverseBFGS — Type

InverseBFGS <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the inverse Riemannian BFGS update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{B}_k^\mathrm{BFGS}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{BFGS}_{k+1} = \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{s_k y^{\mathrm{T}}_k }{s^{\mathrm{T}}_k y_k} \Bigr) \widetilde{B}^\mathrm{BFGS}_k \Bigl( \mathrm{id}_{T_{x_{k+1}} \mathcal{M}} - \frac{y_k s^{\mathrm{T}}_k }{s^{\mathrm{T}}_k y_k} \Bigr) + \frac{s_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

source

Manopt.InverseDFP — Type

InverseDFP <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the inverse Riemannian DFP update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{B}_k^\mathrm{DFP}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{DFP}_{k+1} = \widetilde{B}^\mathrm{DFP}_k + \frac{s_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{B}^\mathrm{DFP}_k y_k y^{\mathrm{T}}_k \widetilde{B}^\mathrm{DFP}_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{DFP}_k y_k}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

source

Manopt.InverseBroyden — Type

InverseBroyden <: AbstractQuasiNewtonUpdateRule

Indicates in AbstractQuasiNewtonDirectionUpdate that the Riemannian Broyden update is used in the Riemannian quasi-Newton method, which is as a convex combination of InverseBFGS and InverseDFP.

Denote by $\widetilde{H}_k^\mathrm{Br}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{Br}_{k+1} = \widetilde{B}^\mathrm{Br}_k - \frac{\widetilde{B}^\mathrm{Br}_k y_k y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k} + \frac{s_k s^{\mathrm{T}}_k}{s^{\mathrm{T}}_k y_k} + φ_k y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k \Bigl( \frac{s_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{B}^\mathrm{Br}_k y_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k} \Bigr) \Bigl( \frac{s_k}{s^{\mathrm{T}}_k y_k} - \frac{\widetilde{B}^\mathrm{Br}_k y_k}{y^{\mathrm{T}}_k \widetilde{B}^\mathrm{Br}_k y_k} \Bigr)^{\mathrm{T}}\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively, and $φ_k$ is the Broyden factor which is :constant by default but can also be set to :Davidon.

Constructor

InverseBroyden(φ, update_rule::Symbol = :constant)

source

Manopt.InverseSR1 — Type

InverseSR1 <: AbstractQuasiNewtonUpdateRule

indicates in AbstractQuasiNewtonDirectionUpdate that the inverse Riemannian SR1 update is used in the Riemannian quasi-Newton method.

Denote by $\widetilde{B}_k^\mathrm{SR1}$ the operator concatenated with a vector transport and its inverse before and after to act on $x_{k+1} = R_{x_k}(α_k η_k)$. Then the update formula reads

\[B^\mathrm{SR1}_{k+1} = \widetilde{B}^\mathrm{SR1}_k + \frac{ (s_k - \widetilde{B}^\mathrm{SR1}_k y_k) (s_k - \widetilde{B}^\mathrm{SR1}_k y_k)^{\mathrm{T}} }{ (s_k - \widetilde{B}^\mathrm{SR1}_k y_k)^{\mathrm{T}} y_k }\]

where $s_k$ and $y_k$ are the coordinate vectors with respect to the current basis (from QuasiNewtonState) of

\[T^{S}_{x_k, α_k η_k}(α_k η_k) \quad\text{and}\quad \operatorname{grad}f(x_{k+1}) - T^{S}_{x_k, α_k η_k}(\operatorname{grad}f(x_k)) ∈ T_{x_{k+1}} \mathcal{M},\]

respectively.

This method can be stabilized by only performing the update if denominator is larger than $r\lVert y_k\rVert_{x_{k+1}}\lVert s_k - \widetilde{H}^\mathrm{SR1}_k y_k \rVert_{x_{k+1}}$ for some $r>0$. For more details, see Section 6.2 in [NW06].

Constructor

InverseSR1(r::Float64=-1.0)

Generate the InverseSR1.

source

State

The quasi Newton algorithm is based on a DefaultManoptProblem.

Manopt.QuasiNewtonState — Type

QuasiNewtonState <: AbstractManoptSolverState

The AbstractManoptSolverState represent any quasi-Newton based method and stores all necessary fields.

Fields

direction_update: an AbstractQuasiNewtonDirectionUpdate rule.
η: the current update direction
nondescent_direction_behavior: a Symbol to specify how to handle direction that are not descent ones.
nondescent_direction_value: the value from the last inner product from checking for descent directions
p::P: a point on the manifold $\mathcal M$storing the current iterate
p_old: the last iterate
sk: the current step
yk: the current gradient difference
retraction_method::AbstractRetractionMethod: a retraction $\operatorname{retr}$ to use, see the section on retractions
stepsize::Stepsize: a functor inheriting from Stepsize to determine a step size
stop::StoppingCriterion: a functor indicating that the stopping criterion is fulfilled
X::T: a tangent vector at the point $p$ on the manifold $\mathcal M$storing the gradient at the current iterate
X_old: the last gradient

Constructor

QuasiNewtonState(M::AbstractManifold, p; kwargs...)

Generate the Quasi Newton state on the manifold M with start point p.

Keyword arguments

direction_update=QuasiNewtonLimitedMemoryDirectionUpdate(M, p, InverseBFGS(), 20; vector_transport_method=vector_transport_method)
stopping_criterion=[StopAfterIteration9(@ref)(1000)|StopWhenGradientNormLess(1e-6): a functor indicating that the stopping criterion is fulfilled
retraction_method=default_retraction_method(M, typeof(p)): a retraction $\operatorname{retr}$ to use, see the section on retractions
stepsize=default_stepsize(M, QuasiNewtonState): a functor inheriting from Stepsize to determine a step size
vector_transport_method=default_vector_transport_method(M, typeof(p)): a vector transport $\mathcal T_{⋅←⋅}$ to use, see the section on vector transports
X=zero_vector(M, p): a tangent vector at the point $p$ on the manifold $\mathcal M$to specify the representation of a tangent vector

See also

quasi_Newton

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Technical details

The quasi_Newton solver requires the following functions of a manifold to be available

A retract!(M, q, p, X); it is recommended to set the default_retraction_method to a favourite retraction. If this default is set, a retraction_method= does not have to be specified.
A vector_transport_to!M, Y, p, X, q); it is recommended to set the default_vector_transport_method to a favourite retraction. If this default is set, a vector_transport_method= or vector_transport_method_dual= (for $\mathcal N$) does not have to be specified.
By default quasi Newton uses ArmijoLinesearch which requires max_stepsize(M) to be set and an implementation of inner(M, p, X).
the norm as well, to stop when the norm of the gradient is small, but if you implemented inner, the norm is provided already.
A copyto!(M, q, p) and copy(M,p) for points and similarly copy(M, p, X) for tangent vectors.
By default the tangent vector storing the gradient is initialized calling zero_vector(M,p).

Most Hessian approximations further require get_coordinates(M, p, X, b) with respect to the AbstractBasis b provided, which is DefaultOrthonormalBasis by default from the basis= keyword.

Literature

[HAG18]: W. Huang, P.-A. Absil and K. A. Gallivan. A Riemannian BFGS method without differentiated retraction for nonconvex optimization problems. SIAM Journal on Optimization 28, 470–495 (2018).
[HGA15]: W. Huang, K. A. Gallivan and P.-A. Absil. A Broyden class of quasi-Newton methods for Riemannian optimization. SIAM Journal on Optimization 25, 1660–1685 (2015).
[NW06]: J. Nocedal and S. J. Wright. Numerical Optimization. 2 Edition (Springer, New York, 2006).