Home > manopt > solvers > steepestdescent > steepestdescent.m

# steepestdescent

## PURPOSE

Steepest descent (gradient descent) minimization algorithm for Manopt.

## SYNOPSIS

function [x, cost, info, options] = steepestdescent(problem, x, options)

## DESCRIPTION

``` Steepest descent (gradient descent) minimization algorithm for Manopt.

function [x, cost, info, options] = steepestdescent(problem)
function [x, cost, info, options] = steepestdescent(problem, x0)
function [x, cost, info, options] = steepestdescent(problem, x0, options)
function [x, cost, info, options] = steepestdescent(problem, [], options)

Apply the steepest descent minimization algorithm to the problem defined
in the problem structure, starting at x0 if it is provided (otherwise, at
a random point on the manifold). To specify options whilst not specifying
an initial guess, give x0 as [] (the empty matrix).

In most of the examples bundled with the toolbox (see link below), the
solver can be replaced by the present one if need be.

The outputs x and cost are the best reached point on the manifold and its
cost. The struct-array info contains information about the iterations:
iter : the iteration number (0 for the initial guess)
cost : cost value
time : elapsed time in seconds
stepsize : norm of the last tangent vector retracted
linesearch : information logged by options.linesearch
And possibly additional information logged by options.statsfun.
For example, type [info.gradnorm] to obtain a vector of the successive

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:

The algorithm terminates if the norm of the gradient drops below this.
maxiter (1000)
The algorithm terminates if maxiter iterations have been executed.
maxtime (Inf)
The algorithm terminates if maxtime seconds elapsed.
minstepsize (1e-10)
The algorithm terminates if the linesearch returns a displacement
vector (to be retracted) smaller in norm than this value.
linesearch (@linesearch or @linesearch_hint)
Function handle to a line search function. The options structure is
passed to the line search too, so you can pass it parameters. See
each line search's documentation for info.
If the problem structure includes a line search hint, then the
default line search used is @linesearch_hint; otherwise
the default is @linesearch.
There are other line search algorithms available in
/manopt/solvers/linesearch/. For example:
- @linesearch_constant
See their documentation with the help command.
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.
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 (3)
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.
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.
For the SD algorithm, a store depth of 2 should always be
sufficient.
hook (none)
A function handle which allows the user to change the current point
x at the beginning of each iteration, before the stopping criterion
is evaluated. See applyHook for help on how to use this option.

## CROSS-REFERENCE INFORMATION

This function calls:
• StoreDB
• applyHook Apply the hook function to possibly replace the current x (for solvers).
• applyStatsfun Apply the statsfun function to a stats structure (for solvers).
• canGetApproxGradient Checks whether an approximate gradient can be computed for this problem.
• canGetCost Checks whether the cost function can be computed for a problem structure.
• canGetGradient Checks whether the gradient can be computed for a problem structure.
• canGetLinesearch Checks whether the problem structure can give a line-search a hint.
• getCostGrad Computes the cost function and the gradient at x in one call if possible.
• getGlobalDefaults Returns a structure with default option values for Manopt.
• mergeOptions Merges two options structures with one having precedence over the other.
• stoppingcriterion Checks for standard stopping criteria, as a helper to solvers.
• approxgradientFD Gradient approx. fnctn handle based on finite differences of the cost.
• linesearch Standard line-search algorithm (step size selection) for descent methods.
• linesearch_hint Armijo line-search based on the line-search hint in the problem structure.
This function is called by:

## SOURCE CODE

```0001 function [x, cost, info, options] = steepestdescent(problem, x, options)
0002 % Steepest descent (gradient descent) minimization algorithm for Manopt.
0003 %
0004 % function [x, cost, info, options] = steepestdescent(problem)
0005 % function [x, cost, info, options] = steepestdescent(problem, x0)
0006 % function [x, cost, info, options] = steepestdescent(problem, x0, options)
0007 % function [x, cost, info, options] = steepestdescent(problem, [], options)
0008 %
0009 % Apply the steepest descent minimization algorithm to the problem defined
0010 % in the problem structure, starting at x0 if it is provided (otherwise, at
0011 % a random point on the manifold). To specify options whilst not specifying
0012 % an initial guess, give x0 as [] (the empty matrix).
0013 %
0014 % In most of the examples bundled with the toolbox (see link below), the
0015 % solver can be replaced by the present one if need be.
0016 %
0017 % The outputs x and cost are the best reached point on the manifold and its
0018 % cost. The struct-array info contains information about the iterations:
0019 %   iter : the iteration number (0 for the initial guess)
0020 %   cost : cost value
0021 %   time : elapsed time in seconds
0023 %   stepsize : norm of the last tangent vector retracted
0024 %   linesearch : information logged by options.linesearch
0025 %   And possibly additional information logged by options.statsfun.
0026 % For example, type [info.gradnorm] to obtain a vector of the successive
0028 %
0029 % The options structure is used to overwrite the default values. All
0030 % options have a default value and are hence optional. To force an option
0031 % value, pass an options structure with a field options.optionname, where
0032 % optionname is one of the following and the default value is indicated
0033 % between parentheses:
0034 %
0036 %       The algorithm terminates if the norm of the gradient drops below this.
0037 %   maxiter (1000)
0038 %       The algorithm terminates if maxiter iterations have been executed.
0039 %   maxtime (Inf)
0040 %       The algorithm terminates if maxtime seconds elapsed.
0041 %   minstepsize (1e-10)
0042 %       The algorithm terminates if the linesearch returns a displacement
0043 %       vector (to be retracted) smaller in norm than this value.
0044 %   linesearch (@linesearch or @linesearch_hint)
0045 %       Function handle to a line search function. The options structure is
0046 %       passed to the line search too, so you can pass it parameters. See
0047 %       each line search's documentation for info.
0048 %       If the problem structure includes a line search hint, then the
0049 %       default line search used is @linesearch_hint; otherwise
0050 %       the default is @linesearch.
0051 %       There are other line search algorithms available in
0052 %       /manopt/solvers/linesearch/. For example:
0054 %       - @linesearch_constant
0055 %       See their documentation with the help command.
0056 %   statsfun (none)
0057 %       Function handle to a function that will be called after each
0058 %       iteration to provide the opportunity to log additional statistics.
0059 %       They will be returned in the info struct. See the generic Manopt
0060 %       documentation about solvers for further information.
0061 %   stopfun (none)
0062 %       Function handle to a function that will be called at each iteration
0063 %       to provide the opportunity to specify additional stopping criteria.
0064 %       See the generic Manopt documentation about solvers for further
0065 %       information.
0066 %   verbosity (3)
0067 %       Integer number used to tune the amount of output the algorithm
0068 %       generates during execution (mostly as text in the command window).
0069 %       The higher, the more output. 0 means silent.
0070 %   storedepth (2)
0071 %       Maximum number of different points x of the manifold for which a
0072 %       store structure will be kept in memory in the storedb for caching.
0073 %       For the SD algorithm, a store depth of 2 should always be
0074 %       sufficient.
0075 %   hook (none)
0076 %       A function handle which allows the user to change the current point
0077 %       x at the beginning of each iteration, before the stopping criterion
0078 %       is evaluated. See applyHook for help on how to use this option.
0079 %
0080 %
0082
0083 % This file is part of Manopt: www.manopt.org.
0084 % Original author: Nicolas Boumal, Dec. 30, 2012.
0085 % Contributors:
0086 % Change log:
0087 %
0088 %   April 3, 2015 (NB):
0089 %       Works with the new StoreDB class system.
0090 %
0091 %   Aug. 2, 2018 (NB):
0092 %       Now using storedb.remove() to keep the cache lean.
0093 %
0094 %   July 19, 2020 (NB):
0095 %       Added support for options.hook.
0096
0097
0098     % Verify that the problem description is sufficient for the solver.
0099     if ~canGetCost(problem)
0100         warning('manopt:getCost', ...
0101                 'No cost provided. The algorithm will likely abort.');
0102     end
0104         % Note: we do not give a warning if an approximate gradient is
0105         % explicitly given in the problem description, as in that case the
0106         % user seems to be aware of the issue.
0109                 'It may be necessary to increase options.tolgradnorm.\n' ...
0110                 'To disable this warning: warning(''off'', ''manopt:getGradient:approx'')']);
0112     end
0113
0114     % Set local defaults here.
0115     localdefaults.minstepsize = 1e-10;
0116     localdefaults.maxiter = 1000;
0118
0119     % Depending on whether the problem structure specifies a hint for
0120     % line-search algorithms, choose a default line-search that works on
0121     % its own (typical) or that uses the hint.
0122     if ~canGetLinesearch(problem)
0123         localdefaults.linesearch = @linesearch;
0124     else
0125         localdefaults.linesearch = @linesearch_hint;
0126     end
0127
0128     % Merge global and local defaults, then merge w/ user options, if any.
0129     localdefaults = mergeOptions(getGlobalDefaults(), localdefaults);
0130     if ~exist('options', 'var') || isempty(options)
0131         options = struct();
0132     end
0133     options = mergeOptions(localdefaults, options);
0134
0135     timetic = tic();
0136
0137     % If no initial point x is given by the user, generate one at random.
0138     if ~exist('x', 'var') || isempty(x)
0139         x = problem.M.rand();
0140     end
0141
0142     % Create a store database and get a key for the current x.
0143     storedb = StoreDB(options.storedepth);
0144     key = storedb.getNewKey();
0145
0146     % Compute objective-related quantities for x.
0149
0150     % Iteration counter.
0151     % At any point, iter is the number of fully executed iterations so far.
0152     iter = 0;
0153
0154     % Save stats in a struct array info, and preallocate.
0155     stats = savestats();
0156     info(1) = stats;
0157     info(min(10000, options.maxiter+1)).iter = [];
0158
0159     if options.verbosity >= 2
0160         fprintf(' iter\t               cost val\t    grad. norm\n');
0161     end
0162
0163     % Start iterating until stopping criterion triggers.
0164     while true
0165
0166         % Display iteration information.
0167         if options.verbosity >= 2
0169         end
0170
0171         % Start timing this iteration.
0172         timetic = tic();
0173
0174         % Apply the hook function if there is one: this allows external code to
0175         % move x to another point. If the point is changed (indicated by a true
0176         % value for the boolean 'hooked'), we update our knowledge about x.
0177         [x, key, info, hooked] = applyHook(problem, x, storedb, key, ...
0178                                                     options, info, iter+1);
0179         if hooked
0182         end
0183
0184         % Run standard stopping criterion checks.
0185         [stop, reason] = stoppingcriterion(problem, x, options, ...
0186                                                              info, iter+1);
0187
0188         % If none triggered, run specific stopping criterion check.
0189         if ~stop && stats.stepsize < options.minstepsize
0190             stop = true;
0191             reason = sprintf(['Last stepsize smaller than minimum '  ...
0192                               'allowed; options.minstepsize = %g.'], ...
0193                               options.minstepsize);
0194         end
0195
0196         if stop
0197             if options.verbosity >= 1
0198                 fprintf([reason '\n']);
0199             end
0200             break;
0201         end
0202
0203         % Pick the descent direction as minus the gradient.
0204         desc_dir = problem.M.lincomb(x, -1, grad);
0205
0206         % Execute the line search.
0207         [stepsize, newx, newkey, lsstats] = options.linesearch( ...
0208                              problem, x, desc_dir, cost, -gradnorm^2, ...
0209                              options, storedb, key);
0210
0211         % Compute the new cost-related quantities for x
0214
0215         % Transfer iterate info, remove cache from previous x.
0216         storedb.removefirstifdifferent(key, newkey);
0217         x = newx;
0218         key = newkey;
0219         cost = newcost;
0222
0223         % Make sure we don't use too much memory for the store database.
0224         storedb.purge();
0225
0226         % iter is the number of iterations we have accomplished.
0227         iter = iter + 1;
0228
0229         % Log statistics for freshly executed iteration.
0230         stats = savestats();
0231         info(iter+1) = stats;
0232
0233     end
0234
0235
0236     info = info(1:iter+1);
0237
0238     if options.verbosity >= 1
0239         fprintf('Total time is %f [s] (excludes statsfun)\n', ...
0240                 info(end).time);
0241     end
0242
0243
0244
0245     % Routine in charge of collecting the current iteration stats.
0246     function stats = savestats()
0247         stats.iter = iter;
0248         stats.cost = cost;
0250         if iter == 0
0251             stats.stepsize = NaN;
0252             stats.time = toc(timetic);
0253             stats.linesearch = [];
0254         else
0255             stats.stepsize = stepsize;
0256             stats.time = info(iter).time + toc(timetic);
0257             stats.linesearch = lsstats;
0258         end
0259         stats = applyStatsfun(problem, x, storedb, key, options, stats);
0260     end
0261
0262 end```

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