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
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}
}
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.
The library has been tested on Linux and MacOSX. It relies on the following Python modules:
- Numpy
- Scipy
- Cvxpy
- Matplotlib (for the demo notebook)
You can install the toolbox through PyPI with:
pip install pepitor get the very latest version by running:
pip install -U https://github.com/PerformanceEstimation/PEPit/archive/master.zip # with --user for user install (no root)After a correct installation, you should be able to import the module without errors:
import PEPitYou can also try the package in this Binder repository.
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 SmoothConvexFunction
def wc_gradient_descent(L, gamma, n, wrapper="cvxpy", solver=None, verbose=1):
"""
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 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.
wrapper (str): the name of the wrapper to be used.
solver (str): the name of the solver the wrapper should use.
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 + solver 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, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 7x7
(PEPit) Setting up the problem: performance measure is the 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) Setting up the problem: additional constraints for 0 function(s)
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.1666666649793712
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite up to an error of 6.325029587061441e-10
All the primal scalar constraints are verified up to an error of 6.633613956752438e-09
(PEPit) Dual feasibility check:
The solver found a residual matrix that is positive semi-definite
All the dual scalar values associated with inequality constraints are nonnegative up to an error of 7.0696173743789816e-09
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 7.547098305159261e-08
(PEPit) Final upper bound (dual): 0.16666667331941884 and lower bound (primal example): 0.1666666649793712
(PEPit) Duality gap: absolute: 8.340047652488636e-09 and relative: 5.004028642152831e-08
*** 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 smooth convex function
func = problem.declare_function(SmoothConvexFunction, 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(wrapper=wrapper, solver=solver, 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\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, wrapper="cvxpy", solver=None, verbose=1)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"):
- Inexact gradient step
- Exact line-search step
- Proximal step
- Inexact proximal step
- Bregman gradient step
- Bregman proximal step
- Linear optimization step
- Epsilon-subgradient step
PEPit provides the following function classes CNIs:
- Convex
- Strongly convex
- Smooth
- Convex and smooth
- Strongly convex and smooth
- Convex and Lipschitz continuous
- Convex indicator
- Convex support
- Convex quadratically growing
- Functions verifying restricted secant inequality and upper error bound
PEPit provides the following operator classes CNIs:
- Monotone
- Strongly monotone
- Lipschitz continuous
- Strongly monotone and Lipschitz continuous
- Cocoercive
This toolbox has been created by
All external contributions are welcome. Please read the contribution guidelines.
The contributors to this toolbox are:
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.
[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.
[4] A. Taylor, J. Hendrickx, F. Glineur (2017). Performance Estimation Toolbox (PESTO): automated worst-case analysis of first-order optimization methods. In 56th IEEE Conference on Decision and Control (CDC).
[5] A. d’Aspremont, D. Scieur, A. Taylor (2021). Acceleration Methods. Foundations and Trends in Optimization: Vol. 5, No. 1-2.
[6] O. Güler (1992). New proximal point algorithms for convex minimization. SIAM Journal on Optimization, 2(4):649–664.
[7] Y. Drori (2017). The exact information-based complexity of smooth convex minimization. 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. Optimization Letters, 11(7), 1185-1199.
[9] B.T. Polyak (1964). Some methods of speeding up the convergence of iteration method. URSS Computational Mathematics and Mathematical Physics.
[10] E. Ghadimi, H. R. Feyzmahdavian, M. Johansson (2015). Global convergence of the Heavy-ball method for convex optimization. European Control Conference (ECC).
[11] E. De Klerk, F. Glineur, A. Taylor (2020). Worst-case convergence analysis of inexact gradient and Newton methods through semidefinite programming performance estimation. SIAM Journal on Optimization, 30(3), 2053-2082.
[12] O. Gannot (2021). A frequency-domain analysis of inexact gradient methods. Mathematical Programming.
[13] D. Kim, J. Fessler (2016). Optimized first-order methods for smooth convex minimization. Mathematical Programming 159.1-2: 81-107.
[14] S. Cyrus, B. Hu, B. Van Scoy, L. Lessard (2018). A robust accelerated optimization algorithm for strongly convex functions. American Control Conference (ACC).
[15] Y. Nesterov (2003). Introductory lectures on convex optimization: A basic course. Springer Science & Business Media.
[16] S. Boyd, L. Xiao, A. Mutapcic (2003). Subgradient Methods (lecture notes).
[17] Y. Drori, M. Teboulle (2016). An optimal variant of Kelley's cutting-plane method. Mathematical Programming, 160(1), 321-351.
[18] Van Scoy, B., Freeman, R. A., Lynch, K. M. (2018). The fastest known globally convergent first-order method for minimizing strongly convex functions. IEEE Control Systems Letters, 2(1), 49-54.
[19] P. Patrinos, L. Stella, A. Bemporad (2014). Douglas-Rachford splitting: Complexity estimates and accelerated variants. In 53rd IEEE Conference on Decision and Control (CDC).
[20] Y. Censor, S.A. Zenios (1992). Proximal minimization algorithm with D-functions. Journal of Optimization Theory and Applications, 73(3), 451-464.
[21] E. Ryu, S. Boyd (2016). A primer on monotone operator methods. Applied and Computational Mathematics 15(1), 3-43.
[22] E. Ryu, A. Taylor, C. Bergeling, P. Giselsson (2020). Operator splitting performance estimation: Tight contraction factors and optimal parameter selection. SIAM Journal on Optimization, 30(3), 2251-2271.
[23] P. Giselsson, and S. Boyd (2016). Linear convergence and metric selection in Douglas-Rachford splitting and ADMM. IEEE Transactions on Automatic Control, 62(2), 532-544.
[24] M .Frank, P. Wolfe (1956). An algorithm for quadratic programming. Naval research logistics quarterly, 3(1-2), 95-110.
[25] M. Jaggi (2013). Revisiting Frank-Wolfe: Projection-free sparse convex optimization. In 30th International Conference on Machine Learning (ICML).
[26] A. Auslender, M. Teboulle (2006). Interior gradient and proximal methods for convex and conic optimization. SIAM Journal on Optimization 16.3 (2006): 697-725.
[27] H.H. Bauschke, J. Bolte, M. Teboulle (2017). A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications. Mathematics of Operations Research, 2017, vol. 42, no 2, p. 330-348
[28] R. Dragomir, A. Taylor, A. d’Aspremont, J. Bolte (2021). Optimal complexity and certification of Bregman first-order methods. Mathematical Programming, 1-43.
[29] A. Taylor, J. Hendrickx, F. Glineur (2018). Exact worst-case convergence rates of the proximal gradient method for composite convex minimization. Journal of Optimization Theory and Applications, 178(2), 455-476.
[30] B. Polyak (1987). Introduction to Optimization. Optimization Software New York.
[31] L. Lessard, B. Recht, A. Packard (2016). Analysis and design of optimization algorithms via integral quadratic constraints. SIAM Journal on Optimization 26(1), 57–95.
[32] D. Davis, W. Yin (2017). A three-operator splitting scheme and its optimization applications. Set-valued and variational analysis, 25(4), 829-858.
[33] Taylor, A. B. (2017). Convex interpolation and performance estimation of first-order methods for convex optimization. PhD Thesis, UCLouvain.
[34] H. Abbaszadehpeivasti, E. de Klerk, M. Zamani (2021). The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions. arXiv 2104.05468.
[35] J. Bolte, S. Sabach, M. Teboulle, Y. Vaisbourd (2018). First order methods beyond convexity and Lipschitz gradient continuity with applications to quadratic inverse problems. SIAM Journal on Optimization, 28(3), 2131-2151.
[36] A. Defazio (2016). A simple practical accelerated method for finite sums. Advances in Neural Information Processing Systems (NIPS), 29, 676-684.
[37] A. Defazio, F. Bach, S. Lacoste-Julien (2014). SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives. In Advances in Neural Information Processing Systems (NIPS).
[38] B. Hu, P. Seiler, L. Lessard (2020). Analysis of biased stochastic gradient descent using sequential semidefinite programs. Mathematical programming.
[39] A. Taylor, F. Bach (2019). Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functions. Conference on Learning Theory (COLT).
[40] D. Kim (2021). Accelerated proximal point method for maximally monotone operators. Mathematical Programming, 1-31.
[41] W. Moursi, L. Vandenberghe (2019). Douglas–Rachford Splitting for the Sum of a Lipschitz Continuous and a Strongly Monotone Operator. Journal of Optimization Theory and Applications 183, 179–198.
[42] G. Gu, J. Yang (2020). Tight sublinear convergence rate of the proximal point algorithm for maximal monotone inclusion problem. SIAM Journal on Optimization, 30(3), 1905-1921.
[43] B. Halpern (1967). Fixed points of nonexpanding maps. American Mathematical Society, 73(6), 957–961.
[44] F. Lieder (2021). On the convergence rate of the Halpern-iteration. Optimization Letters, 15(2), 405-418.
[45] F. Lieder (2018). Projection Based Methods for Conic Linear Programming Optimal First Order Complexities and Norm Constrained Quasi Newton Methods. PhD thesis, HHU Düsseldorf.
[46] Y. Nesterov (1983). A method for solving the convex programming problem with convergence rate O(1/k^2). In Dokl. akad. nauk Sssr (Vol. 269, pp. 543-547).
[47] N. Bansal, A. Gupta (2019). Potential-function proofs for gradient methods. Theory of Computing, 15(1), 1-32.
[48] M. Barre, A. Taylor, F. Bach (2021). A note on approximate accelerated forward-backward methods with absolute and relative errors, and possibly strongly convex objectives. arXiv:2106.15536v2.
[49] J. Eckstein and W. Yao (2018). Relative-error approximate versions of Douglas–Rachford splitting and special cases of the ADMM. Mathematical Programming, 170(2), 417-444.
[50] M. Barré, A. Taylor, A. d’Aspremont (2020). Complexity guarantees for Polyak steps with momentum. In Conference on Learning Theory (COLT).
[51] D. Kim, J. Fessler (2017). On the convergence analysis of the optimized gradient method. Journal of Optimization Theory and Applications, 172(1), 187-205.
[52] Steven Diamond and Stephen Boyd (2016). CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research (JMLR) 17.83.1--5 (2016).
[53] Agrawal, Akshay and Verschueren, Robin and Diamond, Steven and Boyd, Stephen (2018). A rewriting system for convex optimization problems. Journal of Control and Decision (JCD) 5.1.42--60 (2018).
[54] Adrien Taylor, Bryan Van Scoy, Laurent Lessard (2018). Lyapunov Functions for First-Order Methods: Tight Automated Convergence Guarantees. International Conference on Machine Learning (ICML).
[55] C. Guille-Escuret, B. Goujaud, A. Ibrahim, I. Mitliagkas (2022). Gradient Descent Is Optimal Under Lower Restricted Secant Inequality And Upper Error Bound.
[56] B. Goujaud, A. Taylor, A. Dieuleveut (2022). Optimal first-order methods for convex functions with a quadratic upper bound.
[57] W. Su, S. Boyd, E. J. Candès (2016). A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights. In the Journal of Machine Learning Research (JMLR).
[58] C. Moucer, A. Taylor, F. Bach (2022). A systematic approach to Lyapunov analyses of continuous-time models in convex optimization.
[59] A. C. Wilson, B. Recht, M. I. Jordan (2021). A Lyapunov analysis of accelerated methods in optimization. In the Journal of Machine Learning Reasearch (JMLR), 22(113):1−34, 2021.
[60] J.M. Sanz-Serna and K. C. Zygalakis (2021) The connections between Lyapunov functions for some optimization algorithms and differential equations. In SIAM Journal on Numerical Analysis, 59 pp 1542-1565.
[61] D. Scieur, V. Roulet, F. Bach and A. D'Aspremont (2017). Integration methods and accelerated optimization algorithms. In Advances in Neural Information Processing Systems (NIPS).
[62] J. Park, E. Ryu (2022). Exact Optimal Accelerated Complexity for Fixed-Point Iterations. In 39th International Conference on Machine Learning (ICML).
[63] M. Barre, A. Taylor, F. Bach (2020). Principled analyses and design of first-order methods with inexact proximal operators.
[64] Y. Drori and A. Taylor (2020). Efficient first-order methods for convex minimization: a constructive approach. Mathematical Programming 184 (1), 183-220.
[65] Z. Shi, R. Liu (2016). Better worst-case complexity analysis of the block coordinate descent method for large scale machine learning. In 2017 16th IEEE International Conference on Machine Learning and Applications (ICMLA).
[66] R.D. Millán, M.P. Machado (2019). Inexact proximal epsilon-subgradient methods for composite convex optimization problems. Journal of Global Optimization 75.4 (2019): 1029-1060.
[67] M. Kirszbraun (1934). Uber die zusammenziehende und Lipschitzsche transformationen. Fundamenta Mathematicae, 22 (1934).
[68] F.A. Valentine (1943). On the extension of a vector function so as to preserve a Lipschitz condition. Bulletin of the American Mathematical Society, 49 (2).
[69] F.A. Valentine (1945). A Lipschitz condition preserving extension for a vector function. American Journal of Mathematics, 67(1).
[70] M. Kirszbraun (1934). Uber die zusammenziehende und Lipschitzsche transformationen. Fundamenta Mathematicae, 22 (1934).
[71] F.A. Valentine (1943). On the extension of a vector function so as to preserve a Lipschitz condition. Bulletin of the American Mathematical Society, 49 (2).
[72] F.A. Valentine (1945). A Lipschitz condition preserving extension for a vector function. American Journal of Mathematics, 67(1).
[73] E. Gorbunov, A. Taylor, S. Horváth, G. Gidel (2023). Convergence of proximal point and extragradient-based methods beyond monotonicity: the case of negative comonotonicity. International Conference on Machine Learning.
[74] A. Brøndsted, R.T. Rockafellar. On the subdifferentiability of convex functions. Proceedings of the American Mathematical Society 16(4), 605–611 (1965)
[75] R.T. Rockafellar (1976). Monotone operators and the proximal point algorithm. SIAM journal on control and optimization, 14(5), 877-898.
[76] R.D. Monteiro, B.F. Svaiter (2013). An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods. SIAM Journal on Optimization, 23(2), 1092-1125.
[77] S. Salzo, S. Villa (2012). Inexact and accelerated proximal point algorithms. Journal of Convex analysis, 19(4), 1167-1192.
[78] H. H. Bauschke and P. L. Combettes (2017). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer New York, 2nd ed.
[79] J. M. Altschuler, P. A. Parrilo (2023). Acceleration by Stepsize Hedging I: Multi-Step Descent and the Silver Stepsize Schedule. arXiv preprint arXiv:2309.07879.
[80] J. M. Altschuler, P. A. Parrilo (2023). Acceleration by Stepsize Hedging II: Silver Stepsize Schedule for Smooth Convex Optimization. arXiv preprint arXiv:2309.16530.
[81] 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.