This code comes jointly with the following reference.

[1] Taylor, Adrien B., Julien M. Hendrickx, and François Glineur. "Performance Estimation Toolbox (PESTO): automated worst-case analysis of first-order optimization methods." Proceedings of the 56th IEEE Conference on Decision and Control (CDC 2017).

Date: May 2021

Version: May 2021

• Adrien Taylor (author)
• Julien Hendrickx (author)
• François Glineur (author)
• Sébastien Colla (contributor)

The toolbox implements the performance estimation approach as developped in the following works:

[2] Taylor, Adrien B., Julien M. Hendrickx, and François Glineur. "Smooth strongly convex interpolation and exact worst-case performance of first-order methods." Mathematical Programming 161.1-2 (2017): 307-345.

[3] Taylor, Adrien B., Julien M. Hendrickx, and François Glineur. "Exact worst-case performance of first-order methods for composite convex optimization." SIAM Journal on Optimization 27.3 (2017): 1283-1313

Note that the approach of using semidefinite programming for obtaining worst-case guarantees was originally introduced in

[4] Drori, Yoel, and Marc Teboulle. "Performance of first-order methods for smooth convex minimization: a novel approach." Mathematical Programming 145.1-2 (2014): 451-482

The document PESTO_CDC2017_FINAL contains the reference paper for the toolbox. This paper contains a simplified and quick general introduction to the theory underlying the toolbox, and to its use.

The document UserGuide contains more detailed information and examples about the use of the toolbox.

The general purpose of the toolbox is to help the researchers producing worst-case guarantees for their favorite first-order methods.

Note: This code requires YALMIP along with a suitable SDP solver (e.g., Sedumi, SDPT3, Mosek).

Setup with dependencies: guidelines can be found on the wiki

Once YALMIP and the SDP solver installed (type 'yalmiptest' for checking the installation went fine); the toolbox can simply be installed by typing

Install_PESTO

in the Matlab prompt (which only adds the required folders to your Matlab path). You can now execute the demo files for a step by step introduction to the toolbox.

Numerical efficiency Note that it is in general not reasonable to solve 'pure' PEPs (not involving any relaxation) with more than 300 iterations (in the base case: unconstrained minimization), see our numerical tests on the wiki.

Run all examples You can run all examples by typing

test_install

in the matlab prompt. Alternatively, the command

test_install(1)

allows to go a bit more quietly through them.

The folder Examples contains numerous introductory examples to the toolbox.

The files demo summarizes the currently available step-by-step demonstration files and their targets. As an example, the first demonstration file can be executed by typing

demo1

in the prompt.

Among the other examples, the following code (see GradientMethod) generates a worst-case scenario for the gradient method applied to a smooth (possibly strongly) convex function.

% In this example, we use a fixed-step gradient method for
% solving the L-smooth convex minimization problem
%   min_x F(x); for notational convenience we denote xs=argmin_x F(x).
%
% We show how to compute the worst-case value of F(xN)-F(xs) when xN is
% obtained by doing N steps of the gradient method starting with an initial
% iterate satisfying ||x0-xs||<=1.


% (0) Initialize an empty PEP
P=pep();

% (1) Set up the objective function
param.mu=0;	% Strong convexity parameter
param.L=1;      % Smoothness parameter

F=P.DeclareFunction('SmoothStronglyConvex',param); % F is the objective function

% (2) Set up the starting point and initial condition
x0=P.StartingPoint();		  % x0 is some starting point
[xs,fs]=F.OptimalPoint(); 	  % xs is an optimal point, and fs=F(xs)
P.InitialCondition((x0-xs)^2<=1); % Add an initial condition ||x0-xs||^2<= 1

% (3) Algorithm
h=1/param.L;		% step size
N=10;		% number of iterations

x=x0;
for i=1:N
    x=x-h*F.gradient(x);
end
xN=x;

% (4) Set up the performance measure
fN=F.value(xN);                % g=grad F(x), f=F(x)
P.PerformanceMetric(fN-fs); % Worst-case evaluated as F(x)-F(xs)

% (5) Solve the PEP
P.solve()

% (6) Evaluate the output
double(fN-fs)   % worst-case objective function accuracy

% The result should be
% param.L/2/(2*N+1)

The authors would like to thank Francois Gonze from UCLouvain and Yoel Drori from Google Inc. for their feedbacks on preliminary versions of the toolbox.

Additional material was incorporated thanks to:

• Ernest Ryu (SNU), Carolina Bergeling (Lund), and Pontus Giselsson (Lund) (monotone operators and splitting methods),
• Francis Bach (Inria & ENS Paris) (stochastic methods, potential functions, and inexact proximal operators),
• Radu-Alexandru Dragomir (ENS Paris & TSE), Alexandre d’Aspremont (CNRS & ENS Paris), and Jérôme Bolte (TSE) (Bregman divergences and new notions of smoothness).
• Mathieu Barré (Inria & ENS Paris), and Alexandre d’Aspremont (CNRS & ENS Paris) (adaptive methods, and inexact proximal operators).

Last but not least, we thank Loic Estève (Inria) for technical support.