diffcp is a Python package for computing the derivative of a convex cone program, with respect to its problem data. The derivative is implemented as an abstract linear map, with methods for its forward application and its adjoint.
The implementation is based on the calculations in our paper Differentiating through a cone program.
diffcp is available on PyPI, as a source distribution. Install it with
pip install diffcpYou will need a C++11-capable compiler to build diffcp.
diffcp requires:
- NumPy >= 1.15
- SciPy >= 1.10
- SCS >= 2.0.2
- pybind11 >= 2.4
- threadpoolctl >= 1.1
- ECOS >= 2.0.10
- Clarabel >= 0.5.1
- Python >= 3.7
diffcp uses Eigen; Eigen operations can be automatically vectorized by compilers. To enable vectorization, install with
MARCH_NATIVE=1 pip install diffcpOpenMP can be enabled by passing extra arguments to your compiler. For example, on linux, you can tell gcc to activate the OpenMP extension by specifying the flag "-fopenmp":
OPENMP_FLAG="-fopenmp" pip install diffcpTo enable both vectorization and OpenMP (on linux), use
MARCH_NATIVE=1 OPENMP_FLAG="-fopenmp" pip install diffcpdiffcp differentiates through a primal-dual cone program pair. The primal problem must be expressed as
minimize c'x
subject to Ax + s = b
s in K
where x and s are variables, A, b and c are the user-supplied problem data, and K is a user-defined convex cone. The corresponding dual problem is
minimize b'y
subject to A'y + c == 0
y in K^*
with dual variable y.
diffcp exposes the function
solve_and_derivative(A, b, c, cone_dict, warm_start=None, solver=None, **kwargs).This function returns a primal-dual solution x, y, and s, along with
functions for evaluating the derivative and its adjoint (transpose).
These functions respectively compute right and left multiplication of the derivative
of the solution map at A, b, and c by a vector.
The solver argument determines which solver to use; the available solvers
are solver="SCS", solver="ECOS", and solver="Clarabel".
If no solver is specified, diffcp will choose the solver itself.
In the case that the problem is not solved, i.e. the solver fails for some reason, we will raise
a SolverError Exception.
The arguments A, b, and c correspond to the problem data of a cone program.
Amust be a SciPy sparse CSC matrix.bandcmust be NumPy arrays.cone_dictis a dictionary that defines the convex coneK.warm_startis an optional tuple(x, y, s)at which to warm-start. (Note: this is only available for the SCS solver).**kwargsare keyword arguments to forward to the solver (e.g.,verbose=False).
These inputs must conform to the SCS convention for problem data. The keys in cone_dict correspond to the cones, with
diffcp.ZEROfor the zero cone,diffcp.POSfor the positive orthant,diffcp.SOCfor a product of SOC cones,diffcp.PSDfor a product of PSD cones, anddiffcp.EXPfor a product of exponential cones.
The values in cone_dict denote the sizes of each cone; the values of diffcp.SOC, diffcp.PSD, and diffcp.EXP should be lists. The order of the rows of A must match the ordering of the cones given above. For more details, consult the SCS documentation.
The function solve_and_derivative returns a tuple
(x, y, s, derivative, adjoint_derivative)-
x,y, andsare a primal-dual solution. -
derivativeis a function that applies the derivative at(A, b, c)to perturbationsdA,db,dc. It has the signaturederivative(dA, db, dc) -> dx, dy, ds, wheredAis a SciPy sparse CSC matrix with the same sparsity pattern asA, anddbanddcare NumPy arrays.dx,dy, anddsare NumPy arrays, approximating the change in the primal-dual solution due to the perturbation. -
adjoint_derivativeis a function that applies the adjoint of the derivative to perturbationsdx,dy,ds. It has the signatureadjoint_derivative(dx, dy, ds) -> dA, db, dc, wheredx,dy, anddsare NumPy arrays.
import numpy as np
from scipy import sparse
import diffcp
cone_dict = {
diffcp.ZERO: 3,
diffcp.POS: 3,
diffcp.SOC: [5]
}
m = 3 + 3 + 5
n = 5
A, b, c = diffcp.utils.random_cone_prog(m, n, cone_dict)
x, y, s, D, DT = diffcp.solve_and_derivative(A, b, c, cone_dict)
# evaluate the derivative
nonzeros = A.nonzero()
data = 1e-4 * np.random.randn(A.size)
dA = sparse.csc_matrix((data, nonzeros), shape=A.shape)
db = 1e-4 * np.random.randn(m)
dc = 1e-4 * np.random.randn(n)
dx, dy, ds = D(dA, db, dc)
# evaluate the adjoint of the derivative
dx = c
dy = np.zeros(m)
ds = np.zeros(m)
dA, db, dc = DT(dx, dy, ds)For more examples, including the SDP example described in the paper, see the examples directory.
If you wish to cite diffcp, please use the following BibTex:
@article{diffcp2019,
author = {Agrawal, A. and Barratt, S. and Boyd, S. and Busseti, E. and Moursi, W.},
title = {Differentiating through a Cone Program},
journal = {Journal of Applied and Numerical Optimization},
year = {2019},
volume = {1},
number = {2},
pages = {107--115},
}
@misc{diffcp,
author = {Agrawal, A. and Barratt, S. and Boyd, S. and Busseti, E. and Moursi, W.},
title = {{diffcp}: differentiating through a cone program, version 1.0},
howpublished = {\url{https://github.com/cvxgrp/diffcp}},
year = 2019
}
The following thesis concurrently derived the mathematics behind differentiating cone programs.
@phdthesis{amos2019differentiable,
author = {Brandon Amos},
title = {{Differentiable Optimization-Based Modeling for Machine Learning}},
school = {Carnegie Mellon University},
year = 2019,
month = May,
}
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