Metadata-Version: 2.1
Name: PEPit
Version: 0.0.2a0
Summary: PEPit is a package that allows users to pep their optimization algorithms as easily as they implement them
Home-page: https://github.com/bgoujaud/PEPit
Author: Baptiste Goujaud, Céline Moucer, Julien Hendrickx, Francois Glineur, Adrien Taylor and Aymeric Dieuleveut
Author-email: baptiste.goujaud@gmail.com
License: UNKNOWN
Download-URL: https://github.com/bgoujaud/PEPit/archive/refs/tags/0.0.1.tar.gz
Project-URL: Documentation, https://github.com/bgoujaud/PEPit/docs
Platform: UNKNOWN
Classifier: Programming Language :: Python :: 3
Classifier: License :: OSI Approved :: MIT License
Classifier: Operating System :: OS Independent
Requires-Python: >=3.6
Description-Content-Type: text/markdown
License-File: LICENSE

# PEPit: Performance Estimation in Python

[![PyPI version](https://badge.fury.io/py/PEPit.svg)](https://pypi.python.org/pypi/PEPit/)
[![License](https://img.shields.io/github/license/bgoujaud/PEPit.svg)](https://github.com/bgoujaud/PEPit/blob/master/LICENSE)

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](https://francisbach.com/computer-aided-analyses/)
and a corresponding Matlab library is presented in [4] ([PESTO](https://github.com/AdrienTaylor/Performance-Estimation-Toolbox)).

Website and documentation of PEPit: [https://pepit.readthedocs.io/](https://pepit.readthedocs.io/)

Source Code (MIT): [https://github.com/bgoujaud/PEPit](https://github.com/bgoujaud/PEPit)

## Using and citing the toolbox

This code comes jointly with the following [`reference`](https://arxiv.org/pdf/2201.04040.pdf):

    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:

```bibtex
@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 [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/bgoujaud/PEPit/blob/master/ressources/demo/PEPit_demo.ipynb)


This [notebook](https://github.com/bgoujaud/PEPit/blob/master/ressources/educational/PEPit_demo.ipynb) 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:

```console
pip install pepit
```

or get the very latest version by running:

```console
pip install -U https://github.com/bgoujaud/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:

```python
import PEPit
```

### Online environment
You can also try the package in this Binder repository. [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/bgoujaud/PEPit/HEAD)

## Example

The folder [Examples](https://pepit.readthedocs.io/en/latest/examples.html#) contains numerous introductory examples to the toolbox.

Among the other examples, the following code (see [`GradientMethod`](https://pepit.readthedocs.io/en/latest/examples/a.html#gradient-descent))
generates a worst-case scenario for <img src="https://render.githubusercontent.com/render/math?math=N"> 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 <img src="https://render.githubusercontent.com/render/math?math=f(x_N)-f_\star"> when <img src="https://render.githubusercontent.com/render/math?math=x_N"> is generated by gradient descent, and when <img src="https://render.githubusercontent.com/render/math?math=\|x_0-x_\star\|=1">.

```Python
from PEPit import PEP
from PEPit.functions import SmoothStronglyConvexFunction


def wc_gradient_descent(L, gamma, n, verbose=True):
    """
    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**:
    Gradient descent is described by

    .. 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 1]:

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

    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=True)
        (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: initial conditions (1 constraint(s) added)
        (PEPit) Setting up the problem: interpolation conditions for 1 function(s)
                 function 1 : 30 constraint(s) added
        (PEPit) Compiling SDP
        (PEPit) Calling SDP solver
        (PEPit) Solver status: optimal (solver: MOSEK); optimal value: 0.16666666497937685
        *** 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, param={'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.value(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.value(x) - fs)

    # Solve the PEP
    pepit_tau = problem.solve(verbose=verbose)

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

    # Print conclusion if required
    if verbose:
        print('*** Example file: worst-case performance of gradient descent with fixed step-sizes ***')
        print('\tPEPit guarantee:\t\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=True)

```

### 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](https://pepit.readthedocs.io/en/latest/api/steps.html) (often referred to as "oracles"):

- [Inexact gradient step](https://pepit.readthedocs.io/en/latest/api/steps.html#inexact-gradient-step)
- [Exact line-search step](https://pepit.readthedocs.io/en/latest/api/steps.html#exact-line-search-step)
- [Proximal step](https://pepit.readthedocs.io/en/latest/api/steps.html#proximal-step)
- [Inexact proximal step](https://pepit.readthedocs.io/en/latest/api/steps.html#inexact-proximal-step)
- [Bregman gradient step](https://pepit.readthedocs.io/en/latest/api/steps.html#bregman-gradient-step)
- [Bregman proximal step](https://pepit.readthedocs.io/en/latest/api/steps.html#bregman-proximal-step)
- [Linear optimization step](https://pepit.readthedocs.io/en/latest/api/steps.html#linear-optimization-step)

PEPit provides the following [function classes](https://pepit.readthedocs.io/en/latest/api/functions.html) CNIs:

- [Convex](https://pepit.readthedocs.io/en/latest/api/functions.html#convex)
- [Strongly convex](https://pepit.readthedocs.io/en/latest/api/functions.html#strongly-convex)
- [Smooth](https://pepit.readthedocs.io/en/latest/api/functions.html#smooth)
- [Convex and smooth](https://pepit.readthedocs.io/en/latest/api/functions.html#convex-and-smooth)
- [Strongly convex and smooth](https://pepit.readthedocs.io/en/latest/api/functions.html#strongly-convex-and-smooth)
- [Convex and Lipschitz continuous](https://pepit.readthedocs.io/en/latest/api/functions.html#convex-and-lipschitz-continuous)
- [Convex indicator](https://pepit.readthedocs.io/en/latest/api/functions.html#convex-indicator)

PEPit provides the following [operator classes](https://pepit.readthedocs.io/en/latest/api/operators.html) CNIs:

- [Monotone](https://pepit.readthedocs.io/en/latest/api/operators.html#monotone)
- [Strongly monotone](https://pepit.readthedocs.io/en/latest/api/operators.html#strongly-monotone)
- [Lipschitz continuous](https://pepit.readthedocs.io/en/latest/api/operators.html#lipschitz-continuous)
- [Strongly monotone and Lipschitz continuous](https://pepit.readthedocs.io/en/latest/api/operators.html#strongly-monotone-and-lipschitz-continuous)
- [Cocoercive](https://pepit.readthedocs.io/en/latest/api/operators.html#cocoercive)


## Authors

This toolbox has been created by

- [**Baptiste Goujaud**]() (main contributor #1) 
- [**Céline Moucer**]() (main contributor #2)
- [**Julien Hendrickx**](https://perso.uclouvain.be/julien.hendrickx/index.html) (project supervision)
- [**François Glineur**](https://perso.uclouvain.be/francois.glineur/) (project supervision)
- [**Adrien Taylor**](http://www.di.ens.fr/~ataylor/) (contributor & main project supervision)
- [**Aymeric Dieuleveut**](http://www.cmap.polytechnique.fr/~aymeric.dieuleveut/) (contributor & main project supervision)

### Acknowledgments

The authors would like to thank [**Rémi Flamary**](https://remi.flamary.com/)
for his feedbacks on preliminary versions of the toolbox,
as well as for support regarding the continuous integration.

## Contributions

All external contributions are welcome.
Please read the [contribution guidelines](https://pepit.readthedocs.io/en/latest/contributing.html).

## References

[1] Y. Drori, M. Teboulle (2014).
[Performance of first-order methods for smooth convex minimization: a novel approach.](https://arxiv.org/pdf/1206.3209.pdf)
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.](https://arxiv.org/pdf/1502.05666.pdf)
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.](https://arxiv.org/pdf/1512.07516.pdf)
SIAM Journal on Optimization, 27(3):1283–1313.

[4] A. Taylor, J. Hendrickx, F. Glineur (2017).
[Performance Estimation Toolbox (PESTO): automated worst-case analysis of first-order optimization methods.](https://github.com/AdrienTaylor/Performance-Estimation-Toolbox)
In 56th IEEE Conference on Decision and Control (CDC).

[5] A. d’Aspremont, D. Scieur, A. Taylor (2021).
[Acceleration Methods.](https://arxiv.org/pdf/2101.09545.pdf)
Foundations and Trends in Optimization: Vol. 5, No. 1-2.

[6] O. Güler (1992).
[New proximal point algorithms for convex minimization.](https://epubs.siam.org/doi/abs/10.1137/0802032?mobileUi=0)
SIAM Journal on Optimization, 2(4):649–664.

[7] Y. Drori (2017).
[The exact information-based complexity of smooth convex minimization.](https://arxiv.org/pdf/1606.01424.pdf)
Journal of Complexity, 39, 1-16.

[8] E. De Klerk, F. Glineur, A. Taylor (2017).
[On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions.](https://link.springer.com/content/pdf/10.1007/s11590-016-1087-4.pdf)
Optimization Letters, 11(7), 1185-1199.

[9] B.T. Polyak (1964).
[Some methods of speeding up the convergence of iteration method.](https://www.sciencedirect.com/science/article/pii/0041555364901375)
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[Global convergence of the Heavy-ball method for convex optimization.](https://arxiv.org/pdf/1412.7457.pdf)
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[11] E. De Klerk, F. Glineur, A. Taylor (2020).
[Worst-case convergence analysis of inexact gradient and Newton methods through semidefinite programming performance estimation.](https://arxiv.org/pdf/1709.05191.pdf)
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[22] E. Ryu, A. Taylor, C. Bergeling, P. Giselsson (2020).
[Operator splitting performance estimation: Tight contraction factors and optimal parameter selection.](https://arxiv.org/pdf/1812.00146.pdf)
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[38] B. Hu, P. Seiler, L. Lessard (2020).
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[PEPit: computer-assisted worst-case analyses of first-order optimization methods in Python.](https://arxiv.org/pdf/2201.04040.pdf)


