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--[[ FISTA with backtracking line search
- `f` : smooth function
- `g` : non-smooth function
- `pl` : minimizer of intermediate problem Q(x,y)
- `xinit` : initial point
- `params` : table of parameters (**optional**)
- `params.L` : 1/(step size) for ISTA/FISTA iteration (0.1)
- `params.Lstep` : step size multiplier at each iteration (1.5)
- `params.maxiter` : max number of iterations (50)
- `params.maxline` : max number of line search iterations per iteration (20)
- `params.errthres`: Error thershold for convergence check (1e-4)
- `params.doFistaUpdate` : true : use FISTA, false: use ISTA (true)
- `params.verbose` : store each iteration solution and print detailed info (false)
On output, `params` will contain these additional fields that can be reused.
- `params.L` : last used L value will be written.
These are temporary storages needed by the algo and if the same params object is
passed a second time, these same storages will be used without new allocation.
- `params.xkm` : previous iterarion point
- `params.y` : fista iteration
- `params.ply` : ply = pl(y - 1/L grad(f))
Returns the solution x and history of {function evals, number of line search ,...}
Algorithm is published in
@article{beck-fista-09,
Author = {Beck, Amir and Teboulle, Marc},
Journal = {SIAM J. Img. Sci.},
Number = {1},
Pages = {183--202},
Title = {A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems},
Volume = {2},
Year = {2009}}
]]
function optim.FistaLS(f, g, pl, xinit, params)
local params = params or {}
local L = params.L or 0.1
local Lstep = params.Lstep or 1.5
local maxiter = params.maxiter or 50
local maxline = params.maxline or 20
local errthres = params.errthres or 1e-4
local doFistaUpdate = params.doFistaUpdate
local verbose = params.verbose
-- temporary allocations
params.xkm = params.xkm or torch.Tensor()
params.y = params.y or torch.Tensor()
params.ply = params.ply or torch.Tensor()
local xkm = params.xkm -- previous iteration
local y = params.y -- fista iteration
local ply = params.ply -- soft shrinked y
-- we start from all zeros
local xk = xinit
xkm:resizeAs(xk):zero()
ply:resizeAs(xk):zero()
y:resizeAs(xk):zero()
local history = {} -- keep track of stuff
local niter = 0 -- number of iterations done
local converged = false -- are we done?
local tk = 1 -- momentum param for FISTA
local tkp = 0
local gy = g(y)
local fval = math.huge -- fval = f+g
while not converged and niter < maxiter do
-- run through smooth function (code is input, input is target)
-- get derivatives from smooth function
local fy,gfy = f(y,'dx')
--local gfy = f(y)
local fply = 0
local gply = 0
local Q = 0
----------------------------------------------
-- do line search to find new current location starting from fista loc
local nline = 0
local linesearchdone = false
while not linesearchdone do
-- take a step in gradient direction of smooth function
ply:copy(y)
ply:add(-1/L,gfy)
-- and solve for minimum of auxiliary problem
pl(ply,L)
-- this is candidate for new current iteration
xk:copy(ply)
-- evaluate this point F(ply)
fply = f(ply)
-- ply - y
ply:add(-1, y)
-- <ply-y , \Grad(f(y))>
local Q2 = gfy:dot(ply)
-- L/2 ||beta-y||^2
local Q3 = L/2 * ply:dot(ply)
-- Q(beta,y) = F(y) + <beta-y , \Grad(F(y))> + L/2||beta-y||^2 + G(beta)
Q = fy + Q2 + Q3
if verbose then
print(string.format('nline=%d L=%g fply=%g Q=%g fy=%g Q2=%g Q3=%g',nline,L,fply,Q,fy,Q2,Q3))
end
-- check if F(beta) < Q(pl(y),\t)
if fply <= Q then --and Fply + Gply <= F then
-- now evaluate G here
linesearchdone = true
elseif nline >= maxline then
linesearchdone = true
xk:copy(xkm) -- if we can't find a better point, current iter = previous iter
--print('oops')
else
L = L * Lstep
end
nline = nline + 1
end
-- end line search
---------------------------------------------
---------------------------------------------
-- FISTA
---------------------------------------------
if doFistaUpdate then
-- do the FISTA step
tkp = (1 + math.sqrt(1 + 4*tk*tk)) / 2
-- x(k-1) = x(k-1) - x(k)
xkm:add(-1,xk)
-- y(k+1) = x(k) + (1-t(k)/t(k+1))*(x(k-1)-x(k))
y:copy(xk)
y:add( (1-tk)/tkp , xkm)
-- store for next iterations
-- x(k-1) = x(k)
xkm:copy(xk)
else
y:copy(xk)
end
-- t(k) = t(k+1)
tk = tkp
fply = f(y)
gply = g(y)
if verbose then
print(string.format('iter=%d eold=%g enew=%g',niter,fval,fply+gply))
end
niter = niter + 1
-- bookeeping
fval = fply + gply
history[niter] = {}
history[niter].nline = nline
history[niter].L = L
history[niter].F = fval
history[niter].Fply = fply
history[niter].Gply = gply
history[niter].Q = Q
params.L = L
if verbose then
history[niter].xk = xk:clone()
history[niter].y = y:clone()
end
-- are we done?
if niter > 1 and math.abs(history[niter].F - history[niter-1].F) <= errthres then
converged = true
xinit:copy(y)
return y,history
end
if niter >= maxiter then
xinit:copy(y)
return y,history
end
--if niter > 1 and history[niter].F > history[niter-1].F then
--print(niter, 'This was supposed to be a convex function, we are going up')
--converged = true
--return xk,history
--end
end
error('not supposed to be here')
end
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