\(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\W}[1]{ \; #1 \; }\)

lasso.cpp

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Lasso on Fixed Effects: Example and Test

Model

We are given a set of times \(\{ t_i \W{:} i = 0 , \ldots , N-1 \}\) and

\begin{eqnarray} q( \theta, s ) & = & \theta_0 (s / N) + \theta_1 \sin ( 2 \pi s ) + \theta_2 \cos ( 2 \pi s ) \\ z_i & = & q( \theta , t_i ) + e_i \\ \B{p} ( e_i | \theta ) & \sim & \B{N} ( 0, \sigma ) \end{eqnarray}

The idea in Lasso is that one or more of the components of \(\theta\) are zero and using the Laplace prior we can recover this fact. We use \(\B{L} ( \mu , \sigma )\) to denote the Laplace distribution with mean \(\mu\) and standard deviation \(\sigma\).

\[\B{p} ( \theta ) \sim \B{L} ( 0 , \delta )\]

The corresponding fixed likelihood g(theta) is

\[g( \theta ) = \sum_{i=0}^{N-1} \left[ \log ( \sigma \sqrt{2 \pi} ) + \left( \frac{ z_i - q( \theta , t_i ) }{2 \sigma} \right)^2 \right] + \sum_{j=0}^2 \left[ \log \left( \delta \sqrt{2} \right) + \sqrt{2} \; \left| \frac{\theta_j}{\delta} \right| \right]\]

# include <cppad/cppad.hpp>
# include <cppad/mixed/cppad_mixed.hpp>
# include <cppad/mixed/manage_gsl_rng.hpp>
# include <gsl/gsl_randist.h>

namespace {
    using CppAD::log;
    using CppAD::AD;
    //
    using CppAD::mixed::d_sparse_rcv;
    using CppAD::mixed::a1_double;
    using CppAD::mixed::d_vector;
    using CppAD::mixed::a1_vector;
    //
    class mixed_derived : public cppad_mixed {
    private:
        size_t                n_fixed_;
        double                sigma_;
        double                delta_;
        const d_vector&       t_;
        const d_vector&       z_;
    public:
        // constructor
        mixed_derived(
            size_t                 n_fixed        ,
            size_t                 n_random       ,
            double                 sigma          ,
            double                 delta          ,
            const d_vector&        t              ,
            const d_vector&        z              ) :
            cppad_mixed(n_fixed, n_random)  ,
            n_fixed_(n_fixed)               ,
            sigma_(sigma)                   ,
            delta_(delta)                   ,
            t_(t)                           ,
            z_(z)
        {   assert(n_fixed == 3);
            assert( t.size() == z.size() );
        }
        // implementation of fix_likelihood as p(z|theta) * p(theta)
        virtual a1_vector fix_likelihood(
            const a1_vector&         fixed_vec  )
        {   size_t N = t_.size();

            // initialize log-density
            a1_vector vec(1 + n_fixed_);
            vec[0] = a1_double(0.0);

            // compute this factors once
            a1_double   pi     = a1_double( 4.0 * CppAD::atan(1.0) );
            a1_double sqrt_2   = a1_double( CppAD::sqrt( 2.0 ) );
            // a1_double sqrt_2pi = CppAD::sqrt( 2.0 * pi );

            // Data terms p(z|theta)
            for(size_t i = 0; i < N; i++)
            {   a1_double q_i   = 0.0;
                q_i += fixed_vec[0] * t_[i] / double(N);
                 q_i += fixed_vec[1] * sin( 2.0 * pi * t_[i] );
                 q_i += fixed_vec[2] * cos( 2.0 * pi * t_[i] );
                a1_double res  = ( z_[i] - q_i ) / sigma_;
                vec[0] += res * res / 2.0;
                // following term does not depend on the fixed effects
                // vec[0] += log(sigma_ * sqrt_2pi);
            }

            // Prior terms p(theta)
            for(size_t j = 0; j < n_fixed_; j++)
            {   // following term does not depend on the fixed effects
                // vec[0] += log( delta_ * sqrt_2 );
                vec[1 + j] = sqrt_2 * fixed_vec[j] / delta_;
            }
            return vec;
        }
    };
}

bool lasso_xam(void)
{
    bool   ok         = true;
    double inf         = std::numeric_limits<double>::infinity();
    // size_t random_seed = CppAD::mixed::new_gsl_rng(0);
    CppAD::mixed::new_gsl_rng(0);
    gsl_rng* rng       = CppAD::mixed::get_gsl_rng();

    // fixed effects
    size_t n_fixed  = 3;
    d_vector
        fixed_lower(n_fixed), fixed_in(n_fixed), fixed_upper(n_fixed);
    for(size_t j = 0; j < n_fixed; j++)
    {   fixed_lower[j] = - inf;
        fixed_in[j]    = 0.0;
        fixed_upper[j] = inf;
    }
    //
    // no random effects
    size_t n_random = 0;
    d_vector random_in(0);
    d_vector random_lower(n_random), random_upper(n_random);
    std::string random_ipopt_options = "";
    //
    // no constraints
    d_vector fix_constraint_lower(0), fix_constraint_upper(0);
    //
    size_t n_data = 50;
    double sigma  = 0.1;
    double pi     = 4.0 * std::atan(1.0);
    d_vector z(n_data), t(n_data);
    for(size_t i = 0; i < n_data; i++)
    {   t[i] = double(i) / double(n_data - 1) - 0.5;
        //
        // simulation theta_0 = 0, theta_1 = 1, theta_2 = 0
        double q_i = 0.0 * t[i] / double(n_data);
        q_i       += 1.0 * sin(2.0 * pi *t[i]);
        q_i       += 0.0 * cos(2.0 * pi *t[i]);
        double e_i = gsl_ran_gaussian(rng, sigma);
        z[i]       = q_i + e_i;
    }

    // object that is derived from cppad_mixed
    double delta     = 0.002;
    mixed_derived mixed_object(n_fixed, n_random, sigma, delta, t, z);
    mixed_object.initialize(fixed_in, random_in);

    // optimize the fixed effects using quasi-Newton method
    std::string fixed_ipopt_options =
        "Integer print_level               0\n"
        "String  sb                        yes\n"
        "String  derivative_test           adaptive\n"
        "String  derivative_test_print_all yes\n"
        "Numeric tol                       1e-8\n"
        "Integer max_iter                  15\n"
    ;
    d_vector fixed_scale = fixed_in;
    CppAD::mixed::fixed_solution solution = mixed_object.optimize_fixed(
        fixed_ipopt_options,
        random_ipopt_options,
        fixed_lower,
        fixed_upper,
        fix_constraint_lower,
        fix_constraint_upper,
        fixed_scale,
        fixed_in,
        random_lower,
        random_upper,
        random_in
    );
    d_vector fixed_out = solution.fixed_opt;
    //
    // coefficients that should be zero
    ok &= fabs( fixed_out[0] ) <= 5e-8;
    ok &= fabs( fixed_out[2] ) <= 5e-8;
    //
    // non-zero coefficient has shrunk (due to prior)
    ok &= fixed_out[1] < 1.0;
    ok &= 0.5 < fixed_out[1];
    //
    if( ! ok )
        std::cout << "\nfixed_out = " << fixed_out << "\n";
    //
    CppAD::mixed::free_gsl_rng();
    return ok;
}