CppOptimizationLibrary (Support for TensorFlow)

A header-only library with bindings to C++, TensorFlow and Matlab.

Have you ever looked for a C++ function fminsearch, which is easy to use without adding tons of dependencies and without editing many setting-structs and without dependencies? Solve even your large-scale problems by using TensorFlow+Python to accelerate the minimization. See the TensorFlow-Example.

All solvers are written scratch using Eigen, which means they are very easy to use. Want a full example?

    class Rosenbrock : public Problem<double> {
      public:
        double value(const Vector<double> &x) {
            const double t1 = (1 - x[0]);
            const double t2 = (x[1] - x[0] * x[0]);
            return   t1 * t1 + 100 * t2 * t2;
        }
    };
    int main(int argc, char const *argv[]) {
        Rosenbrock f;
        Vector<double> x(2); x << -1, 2;
        BfgsSolver<Rosenbrock> solver;
        solver.minimize(f, x);
        std::cout << "argmin      " << x.transpose() << std::endl;
        std::cout << "f in argmin " << f(x) << std::endl;
        return 0;
    }

To use another solver, simply replace BfgsSolver by another name. Supported solvers are:

• gradient descent solver (GradientDescentSolver)
• conjugate gradient descent solver (ConjugatedGradientDescentSolver)
• Newton descent solver (NewtonDescentSolver)
• BFGS solver (BfgsSolver)
• L-BFGS solver (LbfgsSolver)
• L-BFGS-B solver (LbfgsbSolver)
• CMAes solver (CMAesSolver)
• Nelder-Mead solver (NelderMeadSolver)

These solvers are tested on the Rosenbrock function from multiple difficult starting points by unit tests using the Google Testing Framework. And yes, you can use them directly in MATLAB. Additional benchmark functions are Beale, GoldsteinPrice, Booth, Matyas, Levi. Note, not all solvers are equivalently good at all problems.

For checking your gradient this library uses high-order central difference. Study the examples for more information about including box-constraints and gradient-information.

Extensive Introduction

There are currently two ways to use this library: directly in your C++ code or in MATLAB by calling the provided mex-File.

TensorFlow

You can use the expressive power of TensorFlow to accelerate the problem evaluation and compute reliable the gradients. You just write the problem in Python:

# y = x'Ax + b'x + c
y = tf.matmul(x, tf.matmul(A, x, transpose_b=True)) + tf.matmul(x, b, transpose_b=True) + c
dx = tf.gradients(y, x)[0]

y = tf.identity(y, name='problem_objective')
dx = tf.identity(dx, name='problem_gradient')

and let TensorFlow figure out how to evaluate this expression and the gradients.

C++

There are several examples within the src/examples directory. These are built into build/bin/examples during make all. Checkout rosenbrock.cpp. Your objective and gradient computations should be stored into a tiny class. The most simple usage is

class YourProblem : public Problem<double> {
  double value(const Vector<double> &x) {}
}

In contrast to previous versions of this library, I switched to classes instead of lambda function. If you poke the examples, you will notice that this is much easier to write and understand. The only method a problem has to provide is the value member, which returns the value of the objective function. For most solvers it should be useful to implement the gradient computation, too. Otherwise the library internally will use finite difference for gradient computations (which is definitely unstable and slow!).

class YourProblem : public Problem<double> {
  double value(const Vector<double> &x) {}
  void gradient(const Vector<double> &x, Vector<double> &grad) {}
}

Notice, the gradient is passed by reference! After defining the problem it can be initialized in your code by:

// init problem
YourProblem f;
// test your gradient ONCE
bool probably_correct = f.checkGradient(x);

By convention, a solver minimizes a given objective function starting in x

// choose a solver
BfgsSolver<YourProblem> solver;
// and minimize the function
solver.minimize(f, x);
double minValue = f(x);

For convenience there are some typedefs:

cppoptlib::Vector<T> is a column vector Eigen::Matrix<T, Eigen::Dynamic, 1>;
cppoptlib::Matrix<T> is a matrix        Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic>;

full example

#include <iostream>
#include "../../include/cppoptlib/meta.h"
#include "../../include/cppoptlib/problem.h"
#include "../../include/cppoptlib/solver/bfgssolver.h"

using namespace cppoptlib;
using Eigen::VectorXd;

class Rosenbrock : public Problem<double> {
  public:
    using typename cppoptlib::Problem<double>::Scalar;
    using typename cppoptlib::Problem<double>::TVector;

    double value(const TVector &x) {
        const double t1 = (1 - x[0]);
        const double t2 = (x[1] - x[0] * x[0]);
        return   t1 * t1 + 100 * t2 * t2;
    }
    void gradient(const TVector &x, TVector &grad) {
        grad[0]  = -2 * (1 - x[0]) + 200 * (x[1] - x[0] * x[0]) * (-2 * x[0]);
        grad[1]  = 200 * (x[1] - x[0] * x[0]);
    }
};

int main(int argc, char const *argv[]) {
    Rosenbrock f;
    BfgsSolver<Rosenbrock> solver;
    VectorXd x(2); x << -1, 2;
    solver.minimize(f, x);
    std::cout << "argmin      " << x.transpose() << std::endl;
    std::cout << "f in argmin " << f(x) << std::endl;
    return 0;
}

using box-constraints

The L-BFGS-B algorithm allows to optimize functions with box-constraints, i.e., min_x f(x) s.t. a <= x <= b for some a, b. Given a BoundedProblem-class you can enter these constraints by

// init problem
YourBoundedProblem f;
f.setLowerBound(Vector<double>::Zero(DIM));
f.setUpperBound(Vector<double>::Ones(DIM)*5);
// init solver
cppoptlib::LbfgsbSolver<YourBoundedProblem> solver;
solver.minimize(f, x);

If you do not specify a bound, the algorithm will assume the unbounded case, eg.

// init problem
YourBoundedProblem f;
f.setLowerBound(Vector<double>::Zero(DIM));
// init solver
cppoptlib::LbfgsbSolver<YourBoundedProblem> solver;
solver.minimize(f, x);

will optimize in x s.t. 0 <= x. See src/examples/nonnegls.cpp for an example using L-BFGS-B to solve a bounded problem.

within MATLAB

Simply create a function file for the objective and replace fminsearch or fminunc with cppoptlib. If you want to use symbolic gradient or hessian information see file example.m for details. A basic example would be:

x0 = [-1,2]';
[fx,x] = cppoptlib(x0, @rosenbrock,'gradient',@rosenbrock_grad,'solver','bfgs');
fx     = cppoptlib(x0, @rosenbrock);
fx     = fminsearch(x0, @rosenbrock);

Even optimizing without any gradient information this library outperforms optimization routines from MATLAB on some problems.

Install

Note, this library is header-only, so you just need to add include/cppoptlib to your project without compiling anything and without adding further dependencies. We ship some examples for demonstration purposes and use bazel to compile these examples and unittests. The latest commit using CMake is da314c6581d076e0dbadacdd263aefe4d06a2397.

When using the MATLAB-binding, you need to compile the mex-file. Hereby, open Matlab and run make.m inside the MATLAB folder once. For TensorFlow-Support you need to build TensorFlow-Library from source

user@host $ bazel build -c opt --copt=-mfpmath=both --copt=-msse4.2 --config=cuda //tensorflow:libtensorflow.so
user@host $ bazel build -c opt --copt=-mfpmath=both --copt=-msse4.2 --config=cuda //tensorflow:libtensorflow_cc.so

Benchmarks

Currently, not all solvers are equally good at all objective functions. The file src/test/benchmark.cpp contains some challenging objective functions which are tested by each provided solver. Note, MATLAB will also fail at some objective functions.

Contribute

Make sure that python lint.py does not display any errors and check if travis is happy. It would be great, if some of the Forks-Owner are willing to make pull-request.

References

L-BFGS-B: A LIMITED MEMORY ALGORITHM FOR BOUND CONSTRAINED OPTIMIZATION Richard H. Byrd, Peihuang Lu, Jorge Nocedal and Ciyou Zhu

L-BFGS: Numerical Optimization, 2nd ed. New York: Springer J. Nocedal and S. J. Wright