Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification

This repository is the code for paper Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification. For more information, contact me: toch@di.ku.dk.

Description of files

params_ellips is a folder containing all the parameters of ellipsoids in different cases;

Ellipsoid.m (in Matlab) contains the code for Ellipsoid Model;

solution_fix_point.m (in Matlab) contains the code for fixed-point iteration;

MonDEQ.jl (in Julia) contains the code for Certification and Lipschitz Models;

deq_MNIST_SingleFcNet_m=20.mat contains the parameters of a pre-trained single fully-connected monDEQ;

MNIST_*.mat contains the training and testing MNIST dataset.

attack_*.mat contains several adversarial examples for the first test MNIST example;

border_*.mat contains samples of the image of the input region.

Dependencies

For Julia, it requires to install the following packages: JuMP, LinearAlgebra, MosekTools (need license), MAT, MLDatasets;

For Matlab, it requires to install the CVX package and Mosek solver (need license).

Certification Model

The following code allows us to solve problem (CertMON-i) for the k-th test example in the paper:

test_x, test_y = MNIST.testdata();
vars = matread("deq_MNIST_SingleFcNet_m=20.mat");
A = vars["A"]; B = vars["B"]; U = vars["U"]; u = vars["u"]; C = vars["C"];
p = size(U, 1); m = 20; W = (1-m)*Matrix(I(p))-A'*A+B-B';
me = 0.1307; std = 0.3081; e = 0.1/std;
k = 1; x0 = (vec(test_x[:,:,k]).-me)./std; i = 1;
obj = cert_monDEQ(x0, W, U, u, C[i,:], e; nrm = "linf")

Lipschitz Model

The following code allows us to compute the upper bound of global Lipschitz constant of monDEQ:

vars = matread("deq_MNIST_SingleFcNet_m=20.mat")
A = vars["A"]; B = vars["B"]; U = vars["U"]; u = vars["u"]; C = vars["C"];
p = size(U, 1); m = 20; W = (1-m)*Matrix(I(p))-A'*A+B-B';
std = 0.3081; e = 0.1/std;
obj = lip_monDEQ(W, U, u, C, e; nrm = "linf")

Ellipsoid Model

Run directly the code in the first section of Ellipsoid.m.

Certifying Robustness

For Certification Model, use the following code to compute the ratio of the (first 100) test examples:

test_x, test_y = MNIST.testdata();
vars = matread("deq_MNIST_SingleFcNet_m=20.mat")
A = vars["A"]; B = vars["B"]; U = vars["U"]; u = vars["u"]; C = vars["C"];
p = size(U, 1); m = 20; W = (1-m)*Matrix(I(p))-A'*A+B-B';
n = 100; std = 0.3081; e = 0.1/std;
r = cert(W, U, u, C, n, e, test_x, test_y)

For Lipschitz Model, use the following code to compute the ratio of the (first 100) test examples:

test_x, test_y = MNIST.testdata()
vars = matread("deq_MNIST_SingleFcNet_m=20.mat")
A = vars["A"]; B = vars["B"]; U = vars["U"]; u = vars["u"]; C = vars["C"]; c = vars["c"];
p = size(U, 1); m = 20; W = (1-m)*Matrix(I(p))-A'*A+B-B';
n = 100; std = 0.3081; e = 0.1/std; L = 4.67;
r = cert_lip(W, U, u, C, c, n, e, L, test_x, test_y)

For Ellipsoid Model, run the code in the second section of Ellipsoid.m.