sporco.fista.fista¶
Base classes for FISTA algorithms
Classes
FISTA (xshape, dtype[, opt]) |
Base class for Fast Iterative Shrinkage/Thresholding algorithm (FISTA) algorithms [4]. |
FISTADFT (xshape, dtype[, opt]) |
Base class for FISTA algorithms with gradients and updates computed in the frequency domain. |
Class Descriptions¶
- class
sporco.fista.fista.
FISTA
(xshape, dtype, opt=None)[source]¶Bases:
sporco.common.IterativeSolver
Base class for Fast Iterative Shrinkage/Thresholding algorithm (FISTA) algorithms [4]. A robust variant [15] is also supported.
Solve optimisation problems of the form
\[\mathrm{argmin}_{\mathbf{x}} \; f(\mathbf{x}) + g(\mathbf{x}) \;\;,\]where \(f, g\) are convex functions and \(f\) is smooth.
This class is intended to be a base class of other classes that specialise to specific optimisation problems.
After termination of the
solve
method, attributeitstat
is a list of tuples representing statistics of each iteration. The default fields of the named tupleIterationStats
are:
Iter
: Iteration number
ObjFun
: Objective function value
FVal
: Value of smooth objective function component \(f\)
GVal
: Value of objective function component \(g\)
F_Btrack
: Value of objective function \(f + g\) (see Sec. 2.2 of [4])
Q_Btrack
: Value of Quadratic approximation \(Q_L\) (see Sec. 2.3 of [4])
IterBtrack
: Number of iterations in backtracking
Rsdl
: Residual
L
: Inverse of gradient step parameter
Time
: Cumulative run time
Parameters:
- xshape : tuple of ints
Shape of working variable X
- dtype : data-type
Data type for working variables (overridden by ‘DataType’ option)
- opt :
FISTA.Options
objectAlgorithm options
- class
Options
(opt=None)[source]¶Bases:
sporco.cdict.ConstrainedDict
ADMM algorithm options.
Options:
FastSolve
: Flag determining whether non-essential computation is skipped. WhenFastSolve
isTrue
andVerbose
isFalse
, the functional value and related iteration statistics are not computed. IfFastSolve
isTrue
residuals are also not calculated, in which case the residual-based stopping method is also disabled, with the number of iterations determined only byMaxMainIter
.
Verbose
: Flag determining whether iteration status is displayed.
StatusHeader
: Flag determining whether status header and separator are displayed.
DataType
: Specify data type for solution variables, e.g.np.float32
.
X0
: Initial value for X variable.
Callback
: Callback function to be called at the end of every iteration.
MaxMainIter
: Maximum main iterations.
IterTimer
: Label of the timer to use for iteration times.
RelStopTol
: Relative convergence tolerance for fixed point residual (see Sec. 4.3 of [22]).
L
: Inverse of gradient step parameter \(L\).
AutoStop
: Options for adaptive stoping strategy (fixed point residual, see Sec. 4.3 of [22]).
Enabled
: Flag determining whether the adaptive stopping relative parameter strategy is enabled.
Tau0
: numerator in adaptive criterion (\(\tau_0\) in [22]).
BackTrack
: Options for adaptive L strategy (backtracking, see Sec. 4 of [4] or Robust Fista in [15]).
Enabled
: Flag determining whether adaptive inverse step size parameter strategy is enabled. When true, backtracking in Sec. 4 of [4] is used. In combination with theRobust
flag it enables the backtracking strategy in [15].
Robust
: Flag determining if the robust FISTA update is to be applied as in [15].
gamma_d
: Multiplier applied to decrease L when backtracking in robust FISTA (\(\gamma_d\) in [15]).
gamma_u
: Multiplier applied to increase L when backtracking in standard FISTA (corresponding to \(\eta\) in [4]) or in robust FISTA (corresponding Total \(\gamma_u\) in [15]).
MaxIter
: Maximum iterations of updating L when backtracking.
Parameters:
- opt : dict or None, optional (default None)
FISTA algorithm options
fwiter
= 4¶Field width for iteration count display column
fpothr
= 2¶Field precision for other display columns
itstat_fields_objfn
= ('ObjFun', 'FVal', 'GVal')¶Fields in IterationStats associated with the objective function; see
eval_objfun
itstat_fields_alg
= ('Rsdl', 'F_Btrack', 'Q_Btrack', 'IterBTrack', 'L')¶Fields in IterationStats associated with the specific solver algorithm
itstat_fields_extra
= ()¶Non-standard fields in IterationStats; see
itstat_extra
hdrtxt_objfn
= ('Fnc', 'f', 'g')¶Display column headers associated with the objective function; see
eval_objfun
hdrval_objfun
= {'Fnc': 'ObjFun', 'f': 'FVal', 'g': 'GVal'}¶Dictionary mapping display column headers in
hdrtxt_objfn
to IterationStats entries
solve
()[source]¶Start (or re-start) optimisation. This method implements the framework for the iterations of a FISTA algorithm. There is sufficient flexibility in overriding the component methods that it calls that it is usually not necessary to override this method in derived clases.
If option
Verbose
isTrue
, the progress of the optimisation is displayed at every iteration. At termination of this method, attributeitstat
is a list of tuples representing statistics of each iteration, unless optionFastSolve
isTrue
and optionVerbose
isFalse
.Attribute
timer
is an instance ofutil.Timer
that provides the following labelled timers:
init
: Time taken for object initialisation by__init__
solve
: Total time taken by call(s) tosolve
solve_wo_func
: Total time taken by call(s) tosolve
, excluding time taken to compute functional value and related iteration statistics
solve_wo_rsdl
: Total time taken by call(s) tosolve
, excluding time taken to compute functional value and related iteration statistics as well as time take to compute residuals
solve_wo_btrack
: Total time taken by call(s) tosolve
, excluding time taken to compute functional value and related iteration statistics as well as time take to compute residuals and implementedBackTrack
mechanism
combination_step
()[source]¶Build next update by a smart combination of previous updates. (standard FISTA [4]).
standard_backtrack
()[source]¶Estimate step size L by computing a linesearch that guarantees that F <= Q according to the standard FISTA backtracking strategy in [4]. This also updates variable Y.
robust_backtrack
()[source]¶Estimate step size L by computing a linesearch that guarantees that F <= Q according to the robust FISTA backtracking strategy in [15]. This also updates all the supporting variables.
eval_linear_approx
(Dxy, gradY)[source]¶Compute term \(\langle \nabla f(\mathbf{y}), \mathbf{x} - \mathbf{y} \rangle\) that is part of the quadratic function \(Q_L\) used for backtracking.
- classmethod
hdrval
()[source]¶Construct dictionary mapping display column title to IterationStats entries.
eval_objfn
()[source]¶Compute components of objective function as well as total contribution to objective function.
display_start
()[source]¶Set up status display if option selected. NB: this method assumes that the first entry is the iteration count and the last is the L value.
display_status
(fmtstr, itst)[source]¶Display current iteration status as selection of fields from iteration stats tuple.
obfn_f
(X)[source]¶Compute \(f(\mathbf{x})\) component of FISTA objective function.
Overriding this method is required (even if
eval_objfun
is overriden, since this method is required for backtracking).
- class
sporco.fista.fista.
FISTADFT
(xshape, dtype, opt=None)[source]¶Bases:
sporco.fista.fista.FISTA
Base class for FISTA algorithms with gradients and updates computed in the frequency domain.
Solve optimisation problems of the form
\[\mathrm{argmin}_{\mathbf{x}} \; f(\mathbf{x}) + g(\mathbf{x}) \;\;,\]where \(f, g\) are convex functions and \(f\) is smooth.
This class specialises class FISTA, but remains a base class for other classes that specialise to specific optimisation problems.
Parameters:
- xshape : tuple of ints
Shape of working variable X (the primary variable)
- dtype : data-type
Data type for working variables
- opt :
FISTADFT.Options
objectAlgorithm options
- class
Options
(opt=None)[source]¶Bases:
sporco.fista.fista.Options
FISTADFT algorithm options.
Options include all of those defined in
FISTA.Options
.
Parameters:
- opt : dict or None, optional (default None)
FISTADFT algorithm options
postinitialization_backtracking_DFT
()[source]¶Computes variables needed for backtracking when the updates are made in the DFT. (This requires the variables in DFT to have been initialized).
zzfinit
()[source]¶Return initialiser for working variable ZZ in frequency domain (required for robust FISTA [15]).
proximal_step
(gradf=None)[source]¶Compute proximal update (gradient descent + constraint). Variables are mapped back and forth between input and frequency domains.
combination_step
()[source]¶Update auxiliary state by a smart combination of previous updates in the frequency domain (standard FISTA [4]).
eval_linear_approx
(Dxy, gradY)[source]¶Compute term \(\langle \nabla f(\mathbf{y}), \mathbf{x} - \mathbf{y} \rangle\) (in frequency domain) that is part of the quadratic function \(Q_L\) used for backtracking. Since this class computes the backtracking in the DFT, it is important to preserve the DFT scaling.