Differentiable convex optimization layers

# cvxpylayers

cvxpylayers is a Python library for constructing differentiable convex optimization layers in PyTorch, JAX, and TensorFlow using CVXPY. A convex optimization layer solves a parametrized convex optimization problem in the forward pass to produce a solution. It computes the derivative of the solution with respect to the parameters in the backward pass.

This library accompanies our NeurIPS 2019 paper on differentiable convex optimization layers. For an informal introduction to convex optimization layers, see our blog post.

Our package uses CVXPY for specifying parametrized convex optimization problems.

## Installation

Use the package manager pip to install cvxpylayers.

pip install cvxpylayers


Our package includes convex optimization layers for PyTorch, JAX, and TensorFlow 2.0; the layers are functionally equivalent. You will need to install PyTorch, JAX, or TensorFlow separately, which can be done by following the instructions on their websites.

cvxpylayers has the following dependencies:

## Usage

Below are usage examples of our PyTorch, JAX, and TensorFlow layers. Note that the parametrized convex optimization problems must be constructed in CVXPY, using DPP.

### PyTorch

import cvxpy as cp
import torch
from cvxpylayers.torch import CvxpyLayer

n, m = 2, 3
x = cp.Variable(n)
A = cp.Parameter((m, n))
b = cp.Parameter(m)
constraints = [x >= 0]
objective = cp.Minimize(0.5 * cp.pnorm(A @ x - b, p=1))
problem = cp.Problem(objective, constraints)
assert problem.is_dpp()

cvxpylayer = CvxpyLayer(problem, parameters=[A, b], variables=[x])
A_tch = torch.randn(m, n, requires_grad=True)
b_tch = torch.randn(m, requires_grad=True)

# solve the problem
solution, = cvxpylayer(A_tch, b_tch)

# compute the gradient of the sum of the solution with respect to A, b
solution.sum().backward()


Note: CvxpyLayer cannot be traced with torch.jit.

### JAX

import cvxpy as cp
import jax
from cvxpylayers.jax import CvxpyLayer

n, m = 2, 3
x = cp.Variable(n)
A = cp.Parameter((m, n))
b = cp.Parameter(m)
constraints = [x >= 0]
objective = cp.Minimize(0.5 * cp.pnorm(A @ x - b, p=1))
problem = cp.Problem(objective, constraints)
assert problem.is_dpp()

cvxpylayer = CvxpyLayer(problem, parameters=[A, b], variables=[x])
key = jax.random.PRNGKey(0)
key, k1, k2 = jax.random.split(key, 3)
A_jax = jax.random.normal(k1, shape=(m, n))
b_jax = jax.random.normal(k2, shape=(m,))

solution, = cvxpylayer(A_jax, b_jax)

# compute the gradient of the summed solution with respect to A, b
dcvxpylayer = jax.grad(lambda A, b: sum(cvxpylayer(A, b)[0]), argnums=[0, 1])


Note: CvxpyLayer cannot be traced with the JAX jit or vmap operations.

### TensorFlow 2

import cvxpy as cp
import tensorflow as tf
from cvxpylayers.tensorflow import CvxpyLayer

n, m = 2, 3
x = cp.Variable(n)
A = cp.Parameter((m, n))
b = cp.Parameter(m)
constraints = [x >= 0]
objective = cp.Minimize(0.5 * cp.pnorm(A @ x - b, p=1))
problem = cp.Problem(objective, constraints)
assert problem.is_dpp()

cvxpylayer = CvxpyLayer(problem, parameters=[A, b], variables=[x])
A_tf = tf.Variable(tf.random.normal((m, n)))
b_tf = tf.Variable(tf.random.normal((m,)))

with tf.GradientTape() as tape:
# solve the problem, setting the values of A, b to A_tf, b_tf
solution, = cvxpylayer(A_tf, b_tf)
summed_solution = tf.math.reduce_sum(solution)
# compute the gradient of the summed solution with respect to A, b


Note: CvxpyLayer cannot be traced with tf.function.

### Log-log convex programs

Starting with version 0.1.3, cvxpylayers can also differentiate through log-log convex programs (LLCPs), which generalize geometric programs. Use the keyword argument gp=True when constructing a CvxpyLayer for an LLCP. Below is a simple usage example

import cvxpy as cp
import torch
from cvxpylayers.torch import CvxpyLayer

x = cp.Variable(pos=True)
y = cp.Variable(pos=True)
z = cp.Variable(pos=True)

a = cp.Parameter(pos=True, value=2.)
b = cp.Parameter(pos=True, value=1.)
c = cp.Parameter(value=0.5)

objective_fn = 1/(x*y*z)
objective = cp.Minimize(objective_fn)
constraints = [a*(x*y + x*z + y*z) <= b, x >= y**c]
problem = cp.Problem(objective, constraints)
assert problem.is_dgp(dpp=True)

layer = CvxpyLayer(problem, parameters=[a, b, c],
variables=[x, y, z], gp=True)
a_tch = torch.tensor(a.value, requires_grad=True)
b_tch = torch.tensor(b.value, requires_grad=True)
c_tch = torch.tensor(c.value, requires_grad=True)

x_star, y_star, z_star = layer(a_tch, b_tch, c_tch)
sum_of_solution = x_star + y_star + z_star
sum_of_solution.backward()


## Solvers

At this time, we support two open-source solvers: SCS and ECOS. SCS can be used to solve any problem expressible in CVXPY; ECOS can be used to solve problems that don't use the positive semidefinite or exponential cone (this roughly means that if you have positive semidefinite matrices or use atoms like cp.log, ECOS cannot be used to solve your problem via cvxpylayers). By default, cvxpylayers uses SCS to solve the problem.

### Using ECOS

First make sure that you have cvxpylayers and diffcp up to date, by running

pip install --upgrade cvxpylayers diffcp


Then, to use ECOS, you would pass the solver_args argument to the layer:

solution = layer(*params, solver_args={"solve_method": "ECOS"})


### Passing arguments to the solvers

One can pass arguments to both SCS and ECOS by adding the argument as a key-value pair in the solver_args argument. For example, to increase the tolerance of SCS to 1e-8 one would write:

layer(*parameters, solver_args={"eps": 1e-8})


If SCS is not converging, we highly recommend switching to ECOS (if possible), and if not, using the following arguments to SCS:

solver_args={"eps": 1e-8, "max_iters": 10000, "acceleration_lookback": 0}


## Examples

Our examples subdirectory contains simple applications of convex optimization layers in IPython notebooks.

## Contributing

Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

Please make sure to update tests as appropriate.

Please lint the code with flake8.

pip install flake8  # if not already installed
flake8


### Running tests

cvxpylayers uses the pytest framework for running tests. To install pytest, run:

pip install pytest


Execute the tests from the main directory of this repository with:

pytest cvxpylayers/{torch,jax,tensorflow}


## Projects using cvxpylayers

Below is a list of projects using cvxpylayers. If you have used cvxpylayers in a project, you're welcome to make a PR to add it to this list.

cvxpylayers carries an Apache 2.0 license.

## Citing

If you use cvxpylayers for research, please cite our accompanying NeurIPS paper:

@inproceedings{cvxpylayers2019,
author={Agrawal, A. and Amos, B. and Barratt, S. and Boyd, S. and Diamond, S. and Kolter, Z.},
title={Differentiable Convex Optimization Layers},
booktitle={Advances in Neural Information Processing Systems},
year={2019},
}


If you use cvxpylayers to differentiate through a log-log convex program, please cite the accompanying paper:

@article{agrawal2020differentiating,
title={Differentiating through log-log convex programs},
author={Agrawal, Akshay and Boyd, Stephen},
journal={arXiv},
archivePrefix={arXiv},
eprint={2004.12553},
primaryClass={math.OC},
year={2020},
}


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