All-at-once optimization for CP tensor decomposition (with Poblano)
We explain how to use cp_opt with the Poblano toolbox. The default is to use L-BFGS-B (not Poblabo), which is described here.
Contents
Poblano Optimization Toolbox
Check that you have Poblano 1.1 installed. The output of your 'ver' command should look something like the following.
ver
---------------------------------------------------------------------------------------------------- MATLAB Version: 9.2.0.556344 (R2017a) MATLAB License Number: 192525 Operating System: Microsoft Windows 10 Enterprise Version 10.0 (Build 14393) Java Version: Java 1.7.0_60-b19 with Oracle Corporation Java HotSpot(TM) 64-Bit Server VM mixed mode ---------------------------------------------------------------------------------------------------- MATLAB Version 9.2 (R2017a) Parallel Computing Toolbox Version 6.10 (R2017a) Poblano Toolbox (Sandia National Labs) Version 1.1 Statistics and Machine Learning Toolbox Version 11.1 (R2017a) Tensor Toolbox (Sandia National Labs) Version 3.0-dev
Create an example problem.
Create an example 50 x 40 x 30 tensor with rank 5 and add 10% noise.
R = 5; info = create_problem('Size', [50 40 30], 'Num_Factors', R, 'Noise', 0.10); X = info.Data; M_true = info.Soln;
Create initial guess using 'nvecs'
M_init = create_guess('Data', X, 'Num_Factors', R, ... 'Factor_Generator', 'nvecs');
Set up the optimization parameters
It's genearlly a good idea to consider the parameters of the optimization method. The default options may be either too stringent or not stringent enough. The most important options to consider are detailed here.
% Get the defaults ncg_opts = ncg('defaults'); % Tighten the stop tolerance (norm of gradient). This is often too large. ncg_opts.StopTol = 1.0e-6; % Tighten relative change in function value tolearnce. This is often too large. ncg_opts.RelFuncTol = 1.0e-20; % Increase the number of iterations. ncg_opts.MaxIters = 10^4; % Only display every 10th iteration ncg_opts.DisplayIters = 10; % Display the final set of options ncg_opts
ncg_opts =
struct with fields:
Display: 'iter'
DisplayIters: 10
LineSearch_ftol: 1.0000e-04
LineSearch_gtol: 0.0100
LineSearch_initialstep: 1
LineSearch_maxfev: 20
LineSearch_method: 'more-thuente'
LineSearch_stpmax: 1.0000e+15
LineSearch_stpmin: 1.0000e-15
LineSearch_xtol: 1.0000e-15
MaxFuncEvals: 10000
MaxIters: 10000
RelFuncTol: 1.0000e-20
RestartIters: 20
RestartNW: 0
RestartNWTol: 0.1000
StopTol: 1.0000e-06
TraceFunc: 0
TraceFuncEvals: 0
TraceGrad: 0
TraceGradNorm: 0
TraceRelFunc: 0
TraceX: 0
Update: 'PR'
Call the cp_opt method
Here is an example call to the cp_opt method. By default, each iteration prints the least squares fit function value (being minimized) and the norm of the gradient. The meaning of any line search warnings can be checked via doc cvsrch.
[M,~,output] = cp_opt(X, R, 'init', M_init, ... 'opt', 'ncg', 'opt_options', ncg_opts);
Iter FuncEvals F(X) ||G(X)||/N
------ --------- ---------------- ----------------
0 1 28323.90709757 0.49199185
10 73 318.63828960 0.34043521
20 126 276.86091334 0.02452650
30 164 276.07707790 0.00756378
40 196 275.99604005 0.00081675
50 216 275.99505143 0.00011444
60 236 275.99503708 0.00003744
70 256 275.99503597 0.00001239
80 276 275.99503582 0.00000121
81 278 275.99503582 0.00000072
Check the output
It's important to check the output of the optimization method. In particular, it's worthwhile to check the exit flag. A zero (0) indicates successful termination with the gradient smaller than the specified StopTol, and a three (3) indicates a successful termination where the change in function value is less than RelFuncTol. The meaning of any other flags can be checked via doc poblano_params.
exitflag = output.ExitFlag
exitflag =
0
The fit is the percentage of the data that is explained by the model. Because we have noise, we do not expect the fit to be perfect.
fit = output.Fit
fit = 99.0205
Evaluate the output
We can "score" the similarity of the model computed by CP and compare that with the truth. The score function on ktensor's gives a score in [0,1] with 1 indicating a perfect match. Because we have noise, we do not expect the fit to be perfect. See doc score for more details.
scr = score(M,M_true)
scr =
0.9960
Overfitting example
Consider the case where we don't know R in advance. We might guess too high. Here we show a case where we guess R+1 factors rather than R.
% Generate initial guess of the corret size M_plus_init = create_guess('Data', X, 'Num_Factors', R+1, ... 'Factor_Generator', 'nvecs');
% Loosen the stop tolerance (norm of gradient).
ncg_opts.StopTol = 1.0e-2;
% Run the algorithm [M_plus,~,output] = cp_opt(X, R+1, 'init', M_plus_init, ... 'opt', 'ncg', 'opt_options', ncg_opts); exitflag = output.ExitFlag fit = output.Fit
Iter FuncEvals F(X) ||G(X)||/N
------ --------- ---------------- ----------------
0 1 28324.22221703 0.41000218
10 73 318.70871296 0.28371386
20 126 276.85068977 0.02067054
28 158 276.04553324 0.00988816
exitflag =
0
fit =
99.0203
% Check the answer (1 is perfect)
scr = score(M_plus, M_true)
scr =
0.9926