lpcf 0.1.0

Learning parametric convex functions

LPCF

LPCF stands for learning parametrized convex functions. A parametrized convex function, or PCF, depends on a variable and a parameter, and is convex in the variable for any valid value of the parameter.

LPCF is a framework for fitting a parametrized convex function to some given data that is compatible with disciplined convex programming. This allows to fit a function that can be used in a convex optimization formulation directly to observed or simulated data.

The PCF is represented as a simple neural network whose architecture is designed to ensure disciplined convexity in the variable, for any valid parameter value. After fitting this neural network to triplets of observed (or simulated) values of the function, the variable, and the parameter, the learned PCF can be exported for use in optimization frameworks like CVXPY or JAX.

LPCF supports learning vector functions that depend on multiple variables and parameters. An overview of LPCF can be found in our manuscript.

Installation

LPCF is available on PyPI, and can be installed with

pip install lpcf

LPCF has the following dependencies:

• Python >= 3.9
• jax-sysid >= 1.0.6
• CVXPY >= 1.6.0
• NumPy >= 1.21.6

Example

The following code fits a PCF to observed function values Y, variable values X, and parameter values Theta, and exports the result to CVXPY.

from lpcf.pcf import PCF

# observed data
Y = ...      # shape (N, d)
X = ...      # shape (N, n)
Theta = ...  # shape (N, p)

# fit PCF to data
pcf = PCF()
pcf.fit(Y, X, Theta)

# export PCF to CVXPY
x = cp.Variable((n, 1))
theta = cp.Parameter((p, 1))
pcf_cvxpy = pcf.tocvxpy(x=x, theta=theta)

The CVXPY expression pcf_cvxpy might appear in the objective or the constraints of a CVXPY problem.

Settings

Neural network architecture

The function is approximated as an input-convex main network mapping variables to function values. The weights of the main network are generated by another parameter network, whose inputs are the parameters.

When constructing the PCF object, we allow for a number of customizations to the neural network architecture:

Argument	Description	Type	Default
widths	widths of the main network's hidden layers	array-like	[2((n+d)//2), 2((n+d)//2)]
widths_psi	widths of the parameter network's hidden layers	array-like	[2((p+m)//2), 2((p+m)//2)]
activation	activation function used in the main network	str	'relu'
activation_psi	activation function used in the parameter network	str	'relu'
nonneg	Force the PCF to be nonnegative	Bool	False
increasing	Force the PCF to be increasing	Bool	False
decreasing	Force the PCF to be decreasing	Bool	False
quadratic	Include a convex quadratic term in the PCF	Bool	False
quadratic_r	Include a quadratic term with low-rank + diagonal structure	Bool	False
classification	Use the PCF to solve a classification problem	Bool	False

Note that d is the number of components of the function, n the number of variables, p the number of parameters, and m the number of outputs of the parameter network, i.e., the number of weights of the main network.

Learning configuration

When fitting the PCF to data with its .fit() method, we provide the following options:

Argument	Description	Type	Default
rho_th	regularization on the sum of squared weights of the parameter network	float	1e-8
tau_th	regularization on the sum of absolute weights of the parameter network	float	0
zero_coeff	entries smaller (in abs value) than zero_coeff are zeroed	float	1e-4
cores	number of cores used for parallel training	int	4
seeds	random seeds for training from multiple initial guesses	array-like	max(10, cores)
adam_epochs	number of epochs for running ADAM	int	200
lbfgs_epochs	number of epochs for running L-BFGS-B	int	2000
tune	auto-tune tau_th?	Bool	False
n_folds	number of cross-validation folds when auto-tuning tau_th	int	5
warm_start	warm-start training?	Bool	False

Citing LPCF

Please cite the following paper if you use this software:

@article{SBB25,
    author={Maximilian Schaller and Alberto Bemporad and Stephen Boyd},
    title={Learning Parametric Convex Functions},
    note = {available on arXiv at \url{http://arxiv.org/abs/XXXX.XXXXX}},
    year=2025
}