Contents
function [x,out] = lbreg_accelerated(A,b,alpha,opts)
% lbreg_accelerated: linearized Bregman iteration with Nesterov's acceleration % minimize |x|_1 + 1/(2*alpha) |x|_2^2 % subject to Ax = b % % input: % A: constraint matrix % b: constraint vector % alpha: smoothing parameter, typical value: 1 to 10 times estimated norm(x,inf) % opts. % lip: the estimated Lipschitz constrant of the dual objective, default: alpha*normest(A*A',1e-2) % tol: stop iteration once norm(Ax-b)<tol*norm(b), default: 1e-4 % maxit: max number of iterations, default: 3000 % maxT: max running time in second, default: 1000 % x_ref: if provided, norm(x^k - x_ref) is computed and saved in out.hist_err, default: [] % % output: % x: solution vector % out. % iter: number of iterations % hist_obj: objective value at each iteration % hist_res: |Ax-b|_2 at each iteration % hist_err: if opts.x_ref is provided, contains norm(x^k - x_ref); otherwise, will be set to [] % % Algorithm: % Linearized Bregman is the dual gradient ascent iteration. % The dual problem objective is: % b'y - alpha/2 |shrink(A'y,1)|^2, where shrink(z,1) = z - proj_[-1,1](z) = sign(z).*max(abs(z)-1,0) % % Let y be the dual variable. The gradient ascent iteration is % y^{k+1} = y^k + stepsize (b - alpha A shrink(A'y^k,1)); % Primal variable x is obtained as x^k = alpha shrink(A'y^k,1) % % The accelerated gradient ascent iteration is % beta^k = (1-theta^k)*(sqrt(theta^k*theta^k+4) - theta^k)/2; (extroplation weight computation) % z^{k+1} = y^k + stepsize (b - alpha A shrink(A'y^k,1)); (gradient step, we choose stepsize = 1 / lip) % y^{k+1} = z^{k+1} + beta^k (z^{k+1} - z^k); (extrapolation step) % theta^{k+1} = theta^k*(sqrt(theta^k*theta^k+4) - theta^k)/2; (update of theta) % % How to set alpha: % There exists alpha0 so that any alpha >= alpha0 gives the solution to minimize |x|_1 subject to Ax = b. % The alpha depends on the data, but a typical value is 1 to 10 times the estimate of norm(solution_x,inf) % % How is the algorithm stopped: see "% stopping" below % % The code implements the Nesterov's acceleration algorithm for Linearized Bregman described in % B. Huang, S. Ma, and D. Goldfarb, Accelerated Linearized Bregman Method, J. Sci. Comput, 2012. DOI: 10.1007/s10915-012-9592-9 % % More information can be found at % http://www.caam.rice.edu/~optimization/linearized_bregman
Parameters and defaults
if isfield(opts,'lip'), lip = opts.lip; else lip = alpha*normest(A*A',1e-2); end if isfield(opts,'tol'), tol = opts.tol; else tol = 1e-4; end if isfield(opts,'maxit'), maxit = opts.maxit; else maxit = 500; end if isfield(opts,'maxT'), maxT = opts.maxT; else maxT = 1e3; end if isfield(opts,'x_ref'), x_ref = opts.x_ref; else x_ref = []; out.out.hist_err = []; end
Data preprocessing and initialization
m = size(A,1); y = zeros(m,1); % variable y in Nesterov's method z = zeros(m,1); % variable x in Nesterov's method res = b; % residual (b - Ax) norm_b = norm(b); shrink = @(z) sign(z).*max(0,abs(z)-1); theta = 1; % extrapolation auxilary parameter, initialized to 1
Main iterations
start_time = tic; for k = 1:maxit % --- extrapolation parameter --- beta = (1-theta)*(sqrt(theta*theta+4) - theta)/2; % computes beta_k in P.80 (2.2.9) of Nesterov's 2004 textbook % --- y-update --- z_new = y + res/lip; % step 1a in P.80 of Nesterov's 2004 textbook y = z_new + beta*(z_new - z); % step 1b in P.80 of Nesterov's 2004 textbook, extrapolation z = z_new; % --- x-update --- x = alpha * shrink(y.'*A).'; Ax = A*x; res = b - Ax; % res will be used in next y-update % --- theta-update --- theta = theta*(sqrt(theta*theta+4) - theta)/2; % computes alpha_{k+1} in P.80 (2.2.9) of Nesterov's 2004 textbook % --- diagnostics, reporting, stopping checks --- % reporting out.hist_obj(k) = b.'*y - norm(x)^2/alpha/2; % dual objective out.hist_res(k) = norm(res); % residual size |b - Ax|_2 if ~isempty(x_ref); out.hist_err(k) = norm(x - x_ref); end % stopping if toc(start_time) > maxT; break; end; % running time check if k > 1 && norm(res) < tol*norm_b; break; end % relative primal residual check end out.iter = k;