# Welcome to PEPit’s documentation!

## PEPit: Performance Estimation in Python

This open source Python library provides a generic way to use PEP framework in Python. Performance estimation problems were introduced in 2014 by Yoel Drori and Marc Teboulle, see [1]. PEPit is mainly based on the formalism and developments from [2, 3] by a subset of the authors of this toolbox. A friendly informal introduction to this formalism is available in this blog post and a corresponding Matlab library is presented in [4] (PESTO).

Website and documentation of PEPit: https://pepit.readthedocs.io/

Source Code (MIT): https://github.com/PerformanceEstimation/PEPit

### Using and citing the toolbox

This code comes jointly with the following reference:

B. Goujaud, C. Moucer, F. Glineur, J. Hendrickx, A. Taylor, A. Dieuleveut (2022).
"PEPit: computer-assisted worst-case analyses of first-order optimization methods in Python."


When using the toolbox in a project, please refer to this note via this Bibtex entry:

@article{pepit2022,
title={{PEPit}: computer-assisted worst-case analyses of first-order optimization methods in {P}ython},
author={Goujaud, Baptiste and Moucer, C\'eline and Glineur, Fran\c{c}ois and Hendrickx, Julien and Taylor, Adrien and Dieuleveut, Aymeric},
journal={arXiv preprint arXiv:2201.04040},
year={2022}
}


### Demo 

This notebook provides a demonstration of how to use PEPit to obtain a worst-case guarantee on a simple algorithm (gradient descent), and a more advanced analysis of three other examples.

### Installation

The library has been tested on Linux and MacOSX. It relies on the following Python modules:

• Numpy

• Scipy

• Cvxpy

• Matplotlib (for the demo notebook)

#### Pip installation

You can install the toolbox through PyPI with:

pip install pepit


pip install -U https://github.com/PerformanceEstimation/PEPit/archive/master.zip # with --user for user install (no root)


#### Post installation check

After a correct installation, you should be able to import the module without errors:

import PEPit


#### Online environment

You can also try the package in this Binder repository.

### Example

The folder Examples contains numerous introductory examples to the toolbox.

Among the other examples, the following code (see GradientMethod) generates a worst-case scenario for iterations of the gradient method, applied to the minimization of a smooth (possibly strongly) convex function f(x). More precisely, this code snippet allows computing the worst-case value of when is generated by gradient descent, and when .

from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction

"""
Consider the convex minimization problem

.. math:: f_\\star \\triangleq \\min_x f(x),

where :math:f is :math:L-smooth and convex.

This code computes a worst-case guarantee for **gradient descent** with fixed step-size :math:\\gamma.
That is, it computes the smallest possible :math:\\tau(n, L, \\gamma) such that the guarantee

.. math:: f(x_n) - f_\\star \\leqslant \\tau(n, L, \\gamma) \\|x_0 - x_\\star\\|^2

is valid, where :math:x_n is the output of gradient descent with fixed step-size :math:\\gamma, and
where :math:x_\\star is a minimizer of :math:f.

In short, for given values of :math:n, :math:L, and :math:\\gamma, :math:\\tau(n, L, \\gamma) is computed as the worst-case
value of :math:f(x_n)-f_\\star when :math:\\|x_0 - x_\\star\\|^2 \\leqslant 1.

**Algorithm**:

.. math:: x_{t+1} = x_t - \\gamma \\nabla f(x_t),

where :math:\\gamma is a step-size.

**Theoretical guarantee**:
When :math:\\gamma \\leqslant \\frac{1}{L}, the **tight** theoretical guarantee can be found in [1, Theorem 3.1]:

.. math:: f(x_n)-f_\\star \\leqslant \\frac{L}{4nL\\gamma+2} \\|x_0-x_\\star\\|^2,

which is tight on some Huber loss functions.

**References**:

[1] Y. Drori, M. Teboulle (2014). Performance of first-order methods for smooth convex minimization: a novel
approach. Mathematical Programming 145(1–2), 451–482.
<https://arxiv.org/pdf/1206.3209.pdf>_

Args:
L (float): the smoothness parameter.
gamma (float): step-size.
n (int): number of iterations.
verbose (int): Level of information details to print.

- -1: No verbose at all.
- 0: This example's output.
- 1: This example's output + PEPit information.
- 2: This example's output + PEPit information + CVXPY details.

Returns:
pepit_tau (float): worst-case value
theoretical_tau (float): theoretical value

Example:
>>> L = 3
>>> pepit_tau, theoretical_tau = wc_gradient_descent(L=L, gamma=1 / L, n=4, verbose=1)
(PEPit) Setting up the problem: size of the main PSD matrix: 7x7
(PEPit) Setting up the problem: performance measure is minimum of 1 element(s)
(PEPit) Setting up the problem: Adding initial conditions and general constraints ...
(PEPit) Setting up the problem: initial conditions and general constraints (1 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 1 function(s)
function 1 : Adding 30 scalar constraint(s) ...
function 1 : 30 scalar constraint(s) added
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (solver: SCS); optimal value: 0.16666664596175398
*** Example file: worst-case performance of gradient descent with fixed step-sizes ***
PEPit guarantee:         f(x_n)-f_* <= 0.166667 ||x_0 - x_*||^2
Theoretical guarantee:   f(x_n)-f_* <= 0.166667 ||x_0 - x_*||^2

"""

# Instantiate PEP
problem = PEP()

# Declare a strongly convex smooth function
func = problem.declare_function(SmoothStronglyConvexFunction, mu=0, L=L)

# Start by defining its unique optimal point xs = x_* and corresponding function value fs = f_*
xs = func.stationary_point()
fs = func(xs)

# Then define the starting point x0 of the algorithm
x0 = problem.set_initial_point()

# Set the initial constraint that is the distance between x0 and x^*
problem.set_initial_condition((x0 - xs) ** 2 <= 1)

# Run n steps of the GD method
x = x0
for _ in range(n):
x = x - gamma * func.gradient(x)

# Set the performance metric to the function values accuracy
problem.set_performance_metric(func(x) - fs)

# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(verbose=pepit_verbose)

# Compute theoretical guarantee (for comparison)
theoretical_tau = L / (2 * (2 * n * L * gamma + 1))

# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of gradient descent with fixed step-sizes ***')
print('\tPEPit guarantee:\t f(x_n)-f_* <= {:.6} ||x_0 - x_*||^2'.format(pepit_tau))
print('\tTheoretical guarantee:\t f(x_n)-f_* <= {:.6} ||x_0 - x_*||^2'.format(theoretical_tau))

# Return the worst-case guarantee of the evaluated method (and the reference theoretical value)
return pepit_tau, theoretical_tau

if __name__ == "__main__":
L = 3
pepit_tau, theoretical_tau = wc_gradient_descent(L=L, gamma=1 / L, n=4, verbose=1)


#### Included tools

A lot of common optimization methods can be studied through this framework, using numerous steps and under a large variety of function / operator classes.

PEPit provides the following steps (often referred to as “oracles”):

PEPit provides the following function classes CNIs:

PEPit provides the following operator classes CNIs:

### Authors

This toolbox has been created by

#### Acknowledgments

The authors would like to thank Rémi Flamary for his feedbacks on preliminary versions of the toolbox, as well as for support regarding the continuous integration.

### References

[1] Y. Drori, M. Teboulle (2014). Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming 145(1–2), 451–482.

[2] A. Taylor, J. Hendrickx, F. Glineur (2017). Smooth strongly convex interpolation and exact worst-case performance of first-order methods. Mathematical Programming, 161(1-2), 307-345.

[3] A. Taylor, J. Hendrickx, F. Glineur (2017). Exact worst-case performance of first-order methods for composite convex optimization. SIAM Journal on Optimization, 27(3):1283–1313.

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[5] A. d’Aspremont, D. Scieur, A. Taylor (2021). Acceleration Methods. Foundations and Trends in Optimization: Vol. 5, No. 1-2.

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