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function [x, cost, info, options] = rlbfgs(problem, x0, options)
% Riemannian limited memory BFGS solver for smooth objective functions.
% function [x, cost, info, options] = rlbfgs(problem)
% function [x, cost, info, options] = rlbfgs(problem, x0)
% function [x, cost, info, options] = rlbfgs(problem, x0, options)
% function [x, cost, info, options] = rlbfgs(problem, [], options)
% This is a Riemannian limited memory BFGS solver (quasi-Newton method),
% which aims to minimize the cost function in the given problem structure.
% It requires access to the gradient of the cost function.
% Parameter options.memory can be used to specify the number of iterations
% the algorithm remembers and uses to approximate the inverse Hessian of
% the cost. Default value is 30.
% For unlimited memory, set options.memory = Inf.
% For a description of the algorithm and theorems offering convergence
% guarantees, see the references below.
% The initial iterate is x0 if it is provided. Otherwise, a random point on
% the manifold is picked. To specify options whilst not specifying an
% initial iterate, give x0 as [] (the empty matrix).
% The two outputs 'x' and 'cost' are the last reached point on the manifold
% and its cost.
% The output 'info' is a struct-array which contains information about the
% iterations:
% iter (integer)
% The iteration number. The initial guess is 0.
% cost (double)
% The corresponding cost value.
% gradnorm (double)
% The (Riemannian) norm of the gradient.
% time (double)
% The total elapsed time in seconds to reach the corresponding cost.
% stepsize (double)
% The size of the step from the previous to the new iterate.
% accepted (Boolean)
% true if step is accepted in the cautious update. 0 otherwise.
% And possibly additional information logged by options.statsfun.
% For example, type [info.gradnorm] to obtain a vector of the successive
% gradient norms reached at each iteration.
% The options structure is used to overwrite the default values. All
% options have a default value and are hence optional. To force an option
% value, pass an options structure with a field options.optionname, where
% optionname is one of the following and the default value is indicated
% between parentheses:
% tolgradnorm (1e-6)
% The algorithm terminates if the norm of the gradient drops below
% this. For well-scaled problems, a rule of thumb is that you can
% expect to reduce the gradient norm by 8 orders of magnitude
% (sqrt(eps)) compared to the gradient norm at a "typical" point (a
% rough initial iterate for example). Further decrease is sometimes
% possible, but inexact floating point arithmetic will eventually
% limit the final accuracy. If tolgradnorm is set too low, the
% algorithm may end up iterating forever (or at least until another
% stopping criterion triggers).
% maxiter (1000)
% The algorithm terminates if maxiter iterations were executed.
% maxtime (Inf)
% The algorithm terminates if maxtime seconds elapsed.
% minstepsize (1e-10)
% The minimum norm of the tangent vector that points from the current
% point to the next point. If the norm is less than minstepsize, the
% program will terminate.
% memory (30)
% The number of previous iterations the program remembers. This is used
% to approximate the inverse Hessian at the current point. Because of
% difficulty of maintaining a representation of operators in terms of
% coordinates, a recursive method is used. The number of steps in the
% recursion is at most options.memory. This parameter can take any
% integer value >= 0, or Inf, which is taken to be options.maxiter. If
% options.maxiter has value Inf, then it will take value 10000 and a
% warning will be displayed.
% strict_inc_func (@(t) 1e-4*t)
% The Cautious step needs a real function that has value 0 at t = 0,
% and is strictly increasing. See details in Wen Huang's paper
% "A Riemannian BFGS Method without Differentiated Retraction for
% Nonconvex Optimization Problems"
% statsfun (none)
% Function handle to a function that will be called after each
% iteration to provide the opportunity to log additional statistics.
% They will be returned in the info struct. See the generic Manopt
% documentation about solvers for further information. statsfun is
% called with the point x that was reached last.
% stopfun (none)
% Function handle to a function that will be called at each iteration
% to provide the opportunity to specify additional stopping criteria.
% See the generic Manopt documentation about solvers for further
% information.
% verbosity (2)
% Integer number used to tune the amount of output the algorithm
% generates during execution (mostly as text in the command window).
% The higher, the more output. 0 means silent. 3 and above includes a
% display of the options structure at the beginning of the execution.
% debug (false)
% Set to true to allow the algorithm to perform additional
% computations for debugging purposes. If a debugging test fails, you
% will be informed of it, usually via the command window. Be aware
% that these additional computations appear in the algorithm timings
% too, and may interfere with operations such as counting the number
% of cost evaluations, etc. (the debug calls get storedb too).
% storedepth (2)
% Maximum number of different points x of the manifold for which a
% store structure will be kept in memory in the storedb for caching.
% If memory usage is an issue, you may try to lower this number.
% Profiling may then help to investigate if a performance hit was
% incurred as a result.
% Please cite the Manopt paper as well as the research paper:
% @InBook{Huang2016,
% title = {A {R}iemannian {BFGS} Method for Nonconvex Optimization Problems},
% author = {Huang, W. and Absil, P.-A. and Gallivan, K.A.},
% year = {2016},
% publisher = {Springer International Publishing},
% editor = {Karas{\"o}zen, B{\"u}lent and Manguo{\u{g}}lu, Murat and Tezer-Sezgin, M{\"u}nevver and G{\"o}ktepe, Serdar and U{\u{g}}ur, {\"O}m{\"u}r},
% address = {Cham},
% booktitle = {Numerical Mathematics and Advanced Applications ENUMATH 2015},
% pages = {627--634},
% doi = {10.1007/978-3-319-39929-4_60}
% }
% We point out that, at the moment, this implementation of RLBFGS can be
% slower than the implementation in ROPTLIB by Wen Huang et al. referenced
% above. For the purpose of comparing to their work, please use their
% implementation.
% This file is part of Manopt:
% Original author: Changshuo Liu, July 19, 2017.
% Contributors: Nicolas Boumal
% Change log:
% Nov. 27, 2017 (Wen Huang):
% Changed the default strict_inc_func to @(t) 1e-4*t from @(t) t.
% Jan. 18, 2018 (NB):
% Corrected a bug related to the way the line search hint was defined
% by default.
% Aug. 2, 2018 (NB):
% Using the new storedb.remove features to keep storedb lean, and
% reduced the default value of storedepth from 30 to 2 as a result.
% Verify that the problem description is sufficient for the solver.
if ~canGetCost(problem)
warning('manopt:getCost', ...
'No cost provided. The algorithm will likely abort.');
if ~canGetGradient(problem) && ~canGetApproxGradient(problem)
% Note: we do not give a warning if an approximate gradient is
% explicitly given in the problem description, as in that case the user
% seems to be aware of the issue.
warning('manopt:getGradient:approx', ...
['No gradient provided. Using an FD approximation instead (slow).\n' ...
'It may be necessary to increase options.tolgradnorm.\n' ...
'To disable this warning: warning(''off'', ''manopt:getGradient:approx'')']);
problem.approxgrad = approxgradientFD(problem);
% This solver uses linesearch_hint as a line search algorithm. By
% default, try a step size of 1, so that if the BFGS approximation of
% the Hessian (or inverse Hessian) is good, then the iteration is close
% to a Newton step.
if ~canGetLinesearch(problem)
problem.linesearch = @(x, d) 1;
% Local defaults for the program
localdefaults.minstepsize = 1e-10;
localdefaults.maxiter = 1000;
localdefaults.tolgradnorm = 1e-6;
localdefaults.memory = 30;
localdefaults.strict_inc_func = @(t) 1e-4*t;
localdefaults.ls_max_steps = 25;
localdefaults.storedepth = 2;
% Merge global and local defaults, then merge w/ user options, if any.
localdefaults = mergeOptions(getGlobalDefaults(), localdefaults);
if ~exist('options', 'var') || isempty(options)
options = struct();
options = mergeOptions(localdefaults, options);
% To make sure memory in range [0, Inf)
options.memory = max(options.memory, 0);
if options.memory == Inf
if isinf(options.maxiter)
options.memory = 10000;
warning('rlbfgs:memory', ['options.memory and options.maxiter' ...
' are both Inf; options.memory has been changed to 10000.']);
options.memory = options.maxiter;
M = problem.M;
timetic = tic();
% Create a random starting point if no starting point is provided.
if ~exist('x0', 'var')|| isempty(x0)
xCur = M.rand();
xCur = x0;
% Create a store database and get a key for the current x
storedb = StoreDB(options.storedepth);
key = storedb.getNewKey();
% __________Initialization of variables______________
% Number of iterations since the last restart
k = 0;
% Total number of BFGS iterations
iter = 0;
% This cell stores step vectors which point from x_{t} to x_{t+1} for t
% indexing the last iterations, capped at options.memory.
% That is, it stores up to options.memory of the most recent step
% vectors. However, the implementation below does not need step vectors
% in their respective tangent spaces at x_{t}'s. Rather, it requires
% them transported to the current point's tangent space by vector
% tranport. For details regarding the requirements on the the vector
% tranport, see the reference paper by Huang et al.
% In this implementation, those step vectors are iteratively
% transported to the current point's tangent space after every
% iteration. Thus, at every iteration, vectors in sHistory are in the
% current point's tangent space.
sHistory = cell(1, options.memory);
% This cell stores the differences for latest t's of the gradient at
% x_{t+1} and the gradient at x_{t}, transported to x_{t+1}'s tangent
% space. The memory is also capped at options.memory.
yHistory = cell(1, options.memory);
% rhoHistory{t} stores the reciprocal of the inner product between
% sHistory{t} and yHistory{t}.
rhoHistory = cell(1, options.memory);
% Scaling of direction given by getDirection for acceptable step
alpha = 1;
% Scaling of initial matrix, Barzilai-Borwein.
scaleFactor = 1;
% Norm of the step
stepsize = 1;
% Stores whether the step is accepted by the cautious update check.
accepted = true;
% Query the cost function and its gradient
[xCurCost, xCurGradient] = getCostGrad(problem, xCur, storedb, key);
xCurGradNorm = M.norm(xCur, xCurGradient);
% Line-search statistics for recording in info.
lsstats = [];
% Flag to control restarting scheme to avoid infinite loops (see below)
ultimatum = false;
% Save stats in a struct array info, and preallocate.
stats = savestats();
info(1) = stats;
info(min(10000, options.maxiter+1)).iter = [];
if options.verbosity >= 2
fprintf(' iter cost val grad. norm alpha\n');
% Main iteration
while true
% Display iteration information
if options.verbosity >= 2
fprintf('%5d %+.16e %.8e %.4e\n', ...
iter, xCurCost, xCurGradNorm, alpha);
% Start timing this iteration
timetic = tic();
% Run standard stopping criterion checks
[stop, reason] = stoppingcriterion(problem, xCur, options, ...
info, iter+1);
% If none triggered, run specific stopping criterion check
if ~stop
if stats.stepsize < options.minstepsize
% To avoid infinite loop and to push the search further
% in case BFGS approximation of Hessian is off towards
% the end, we erase the memory by setting k = 0;
% In this way, it starts off like a steepest descent.
% If even steepest descent does not work, then it is
% hopeless and we will terminate.
if ~ultimatum
if options.verbosity >= 2
fprintf(['stepsize is too small, restarting ' ...
'the bfgs procedure at the current point.\n']);
k = 0;
ultimatum = true;
stop = true;
reason = sprintf(['Last stepsize smaller than ' ...
'minimum allowed; options.minstepsize = %g.'], ...
% We are not in trouble: lift the ultimatum if it was on.
ultimatum = false;
if stop
if options.verbosity >= 1
fprintf([reason '\n']);
% Compute BFGS direction
p = getDirection(M, xCur, xCurGradient, sHistory,...
yHistory, rhoHistory, scaleFactor, min(k, options.memory));
% Execute line-search
[stepsize, xNext, newkey, lsstats] = ...
linesearch_hint(problem, xCur, p, xCurCost, ...
M.inner(xCur, xCurGradient, p), ...
options, storedb, key);
% Record the BFGS step-multiplier alpha which was effectively
% selected. Toward convergence, we hope to see alpha = 1.
alpha = stepsize/M.norm(xCur, p);
step = M.lincomb(xCur, alpha, p);
% Query cost and gradient at the candidate new point.
[xNextCost, xNextGrad] = getCostGrad(problem, xNext, storedb, newkey);
% Compute sk and yk
sk = M.transp(xCur, xNext, step);
yk = M.lincomb(xNext, 1, xNextGrad, ...
-1, M.transp(xCur, xNext, xCurGradient));
% Computation of the BFGS step is invariant under scaling of sk and
% yk by a common factor. For numerical reasons, we scale sk and yk
% so that sk is a unit norm vector.
norm_sk = M.norm(xNext, sk);
sk = M.lincomb(xNext, 1/norm_sk, sk);
yk = M.lincomb(xNext, 1/norm_sk, yk);
inner_sk_yk = M.inner(xNext, sk, yk);
inner_sk_sk = M.norm(xNext, sk)^2; % ensures nonnegativity
% If the cautious step is accepted (which is the intended
% behavior), we record sk, yk and rhok and need to do some
% housekeeping. If the cautious step is rejected, these are not
% recorded. In all cases, xNext is the next iterate: the notion of
% accept/reject here is limited to whether or not we keep track of
% sk, yk, rhok to update the BFGS operator.
cap = options.strict_inc_func(xCurGradNorm);
if inner_sk_sk ~= 0 && (inner_sk_yk / inner_sk_sk) >= cap
accepted = true;
rhok = 1/inner_sk_yk;
scaleFactor = inner_sk_yk / M.norm(xNext, yk)^2;
% Time to store the vectors sk, yk and the scalar rhok.
% Remember: we need to transport all vectors to the most
% current tangent space.
% If we are out of memory
if k >= options.memory
% sk and yk are saved from 1 to the end with the most
% current recorded to the rightmost hand side of the cells
% that are occupied. When memory is full, do a shift so
% that the rightmost is earliest and replace it with the
% most recent sk, yk.
for i = 2 : options.memory
sHistory{i} = M.transp(xCur, xNext, sHistory{i});
yHistory{i} = M.transp(xCur, xNext, yHistory{i});
if options.memory > 1
sHistory = sHistory([2:end, 1]);
yHistory = yHistory([2:end, 1]);
rhoHistory = rhoHistory([2:end 1]);
if options.memory > 0
sHistory{options.memory} = sk;
yHistory{options.memory} = yk;
rhoHistory{options.memory} = rhok;
% If we are not out of memory
for i = 1:k
sHistory{i} = M.transp(xCur, xNext, sHistory{i});
yHistory{i} = M.transp(xCur, xNext, yHistory{i});
sHistory{k+1} = sk;
yHistory{k+1} = yk;
rhoHistory{k+1} = rhok;
k = k + 1;
% The cautious step is rejected: we do not store sk, yk, rhok but
% we still need to transport stored vectors to the new tangent
% space.
accepted = false;
for i = 1 : min(k, options.memory)
sHistory{i} = M.transp(xCur, xNext, sHistory{i});
yHistory{i} = M.transp(xCur, xNext, yHistory{i});
% Update variables to new iterate.
storedb.removefirstifdifferent(key, newkey);
xCur = xNext;
key = newkey;
xCurGradient = xNextGrad;
xCurGradNorm = M.norm(xNext, xNextGrad);
xCurCost = xNextCost;
% iter is the number of iterations we have accomplished.
iter = iter + 1;
% Make sure we don't use too much memory for the store database
% (this is independent from the BFGS memory.)
% Log statistics for freshly executed iteration
stats = savestats();
info(iter+1) = stats;
% Housekeeping before we return
info = info(1:iter+1);
x = xCur;
cost = xCurCost;
if options.verbosity >= 1
fprintf('Total time is %f [s] (excludes statsfun)\n', ...
% Routine in charge of collecting the current iteration stats
function stats = savestats()
stats.iter = iter;
stats.cost = xCurCost;
stats.gradnorm = xCurGradNorm;
if iter == 0
stats.stepsize = NaN;
stats.time = toc(timetic);
stats.accepted = NaN;
stats.stepsize = stepsize;
stats.time = info(iter).time + toc(timetic);
stats.accepted = accepted;
stats.linesearch = lsstats;
stats = applyStatsfun(problem, xCur, storedb, key, options, stats);
% BFGS step, see Wen's paper for details. This functon takes in a tangent
% vector g, and applies an approximate inverse Hessian P to it to get Pg.
% Then, -Pg is returned.
% Theory requires the vector transport to be isometric and to satisfy the
% locking condition (see paper), but these properties do not seem to be
% crucial in practice. If your manifold provides M.isotransp, it may be
% good to do M.transp = M.isotransp; after loading M with a factory.
% This implementation operates in the tangent space of the most recent
% point since all vectors in sHistory and yHistory have been transported
% there.
function dir = getDirection(M, xCur, xCurGradient, sHistory, yHistory, ...
rhoHistory, scaleFactor, k)
q = xCurGradient;
inner_s_q = zeros(1, k);
for i = k : -1 : 1
inner_s_q(1, i) = rhoHistory{i} * M.inner(xCur, sHistory{i}, q);
q = M.lincomb(xCur, 1, q, -inner_s_q(1, i), yHistory{i});
r = M.lincomb(xCur, scaleFactor, q);
for i = 1 : k
omega = rhoHistory{i} * M.inner(xCur, yHistory{i}, r);
r = M.lincomb(xCur, 1, r, inner_s_q(1, i)-omega, sHistory{i});
dir = M.lincomb(xCur, -1, r);