/**
* @brief Low level fitting routine for a Gaussian trend filtering problem.
* Function used by tf_admm to fit a Gaussian ADMM trendfilter, or as a
* subproblem by tf_admm_glm when using logistic or poisson losses. Fits
* the solution for a single value of lambda. Most users will want to call
* tf_admm, rather than tf_admm_gauss directly.
*
* @param y a vector of responses
* @param x a vector of response locations; must be in increasing order
* @param w a vector of sample weights
* @param n the length of y, x, and w
* @param k degree of the trendfilter; i.e., k=1 linear
* @param max_iter maximum number of ADMM interations; ignored for k=0
* @param lam the value of lambda
* @param beta allocated space for output coefficents; must pre-fill as it is used in warm start
* @param alpha allocated space for ADMM alpha covariates; must pre-fill as it is used in warm start
* @param u allocated space for ADMM u covariates; must pre-fill as it is used in warm start
* @param obj allocated space to store the objective; will fill at most max_iter elements
* @param iter allocated space to store the number of iterations; will fill just one element
* @param rho tuning parameter for the ADMM algorithm; set to 1 for default
* @param obj_tol stopping criteria tolerance; set to 1e-10 for default
* @param DktDk pointer to the inner product of DktDk
* @param verbose 0/1 flag for printing progress
* @return void
* @see tf_admm
*/
void tf_admm_gauss (double * y, double * x, double * w, int n, int k,
int max_iter, double lam,
double * beta, double * alpha, double * u,
double * obj, int * iter,
double rho, double obj_tol, cs * DktDk, int verbose)
{
int i;
int it;
double *v;
double *z;
double *db;
double loss;
double pen;
cs * kernmat;
gqr * kernmat_qr;
/* Special case for k=0: skip the ADMM algorithm */
if (k==0)
{
/* Use Nick's DP algorithm, weighted version */
tf_dp_weight(n,y,w,lam,beta);
db = (double *) malloc(n*sizeof(double));
/* Compute objective */
loss = 0; pen = 0;
for (i=0; i<n; i++) loss += w[i]*(y[i]-beta[i])*(y[i]-beta[i]);
loss = loss/2;
tf_dx(x,n,k+1,beta,db); /* IMPORTANT: use k+1 here! */
for (i=0; i<n-k-1; i++) pen += fabs(db[i]);
obj[0] = loss+lam*pen;
free(db);
return;
}
/* Otherwise we run our ADMM routine */
/* Construct the kernel matrix and its QR decomposition */
kernmat = scalar_plus_diag(DktDk, rho, w);
kernmat_qr = glmgen_qr(kernmat);
/* Other variables that will be useful during our iterations */
v = (double*) malloc(n*sizeof(double));
z = (double*) malloc(n*sizeof(double));
if (verbose) printf("\nlambda=%0.3e\n",lam);
if (verbose) printf("Iteration\tObjective\tLoss\tPenalty\n");
for(it=0; it < max_iter; it++)
{
/* Update beta: banded linear system (kernel matrix) */
for (i=0; i < n-k; i++) v[i] = alpha[i] + u[i];
tf_dtxtil(x,n,k,v,z);
for (i=0; i<n; i++) beta[i] = w[i]*y[i] + rho*z[i];
/* Solve the least squares problem with sparse QR */
glmgen_qrsol(kernmat_qr, beta);
/* Update alpha: 1d fused lasso
* Build the response vector */
tf_dxtil(x,n,k,beta,v);
for (i=0; i<n-k; i++)
{
z[i] = v[i]-u[i];
}
/* Use Nick's DP algorithm */
tf_dp(n-k,z,lam/rho,alpha);
/* Update u: dual update */
for (i=0; i<n-k; i++)
{
u[i] = u[i]+alpha[i]-v[i];
}
/* Compute loss */
loss = 0;
for (i=0; i<n; i++) loss += w[i]*(y[i]-beta[i])*(y[i]-beta[i]);
loss = loss/2;
/* Compute penalty */
tf_dx(x,n,k+1,beta,z); /* IMPORTANT: use k+1 here! */
pen = 0;
for (i=0; i<n-k-1; i++) pen += fabs(z[i]);
obj[it] = loss+lam*pen;
if (verbose) printf("%i\t%0.3e\t%0.3e\t%0.3e\n",it+1,obj[it],loss,lam*pen);
/* Stop if relative difference of objective values <= obj_tol */
if(it > 0)
{
if( fabs(obj[it] - obj[it-1]) < fabs(obj[it]) * obj_tol ) break;
}
}
*iter = it;
cs_spfree(kernmat);
glmgen_gqr_free(kernmat_qr);
free(v);
free(z);
}
/**
* @brief Low level fitting routine for a Gaussian trend filtering problem.
* Function used by tf_admm to fit a Gaussian ADMM trendfilter, or as a
* subproblem by tf_admm_glm when using logistic or poisson losses. Fits
* the solution for a single value of lambda. Most users will want to call
* tf_admm, rather than tf_admm_gauss directly.
*
* @param x a vector of data locations; must be in increasing order
* @param y a vector of responses
* @param w a vector of sample weights
* @param n the length of x, y, and w
* @param k polynomial degree of the fitted trend; i.e., k=1 for linear
* @param max_iter maximum number of ADMM interations; ignored for k=0
* @param lam the value of lambda
* @param df allocated space for df value at the solution
* @param beta allocated space for output coefficents; must pre-fill as it is used in warm start
* @param alpha allocated space for ADMM alpha variable; must pre-fill as it is used in warm start
* @param u allocated space for ADMM u variable; must pre-fill as it is used in warm start
* @param obj allocated space to store the objective; will fill at most max_iter elements
* @param iter allocated space to store the number of iterations; will fill just one element
* @param rho tuning parameter for the ADMM algorithm; set to 1 for default
* @param obj_tol stopping criteria tolerance; set to 1e-10 for default
* @param DktDk pointer to the inner product of DktDk
* @param verbose 0/1 flag for printing progress
* @return void
* @see tf_admm
*/
void tf_admm_gauss (double * x, double * y, double * w, int n, int k,
int max_iter, double lam, int * df,
double * beta, double * alpha, double * u,
double * obj, int * iter,
double rho, double obj_tol, cs * DktDk, int verbose)
{
int i;
int d;
int it, itbest;
double *v;
double *z;
double *betabest;
double *alphabest;
double descent;
double variation;
cs * kernmat;
gqr * kernmat_qr;
/* Special case for k=0: skip the ADMM algorithm */
if (k==0)
{
/* Use Nick's DP algorithm, weighted version */
tf_dp_weight(n,y,w,lam,beta);
/* Compute df value */
d = 1;
for (i=0; i<n-1; i++) if (beta[i] != beta[i+1]) d += 1;
*df = d;
/* Compute objective */
v = (double *) malloc(n*sizeof(double));
obj[0] = tf_obj_gauss(x,y,w,n,k,lam,beta,v);
free(v);
return;
}
/* Otherwise we run our ADMM routine */
/* Construct the kernel matrix and its QR decomposition */
kernmat = scalar_plus_diag(DktDk, rho, w);
kernmat_qr = glmgen_qr(kernmat);
/* Other variables that will be useful during our iterations */
v = (double*) malloc(n*sizeof(double));
z = (double*) malloc(n*sizeof(double));
betabest = (double*) malloc(n*sizeof(double));
alphabest = (double*) malloc(n*sizeof(double));
if (verbose) printf("\nlambda=%0.3e\n",lam);
if (verbose) printf("Iteration\tObjective\n");
itbest = 0;
obj[0] = tf_obj_gauss(x,y,w,n,k,lam,beta,v);
memcpy(betabest, beta, n * sizeof(double));
memcpy(alphabest, alpha, n * sizeof(double));
for (it=0; it < max_iter; it++)
{
/* Update beta: banded linear system (kernel matrix) */
for (i=0; i < n-k; i++) v[i] = alpha[i] + u[i];
tf_dtxtil(x,n,k,v,z);
for (i=0; i<n; i++) beta[i] = w[i]*y[i] + rho*z[i];
/* Solve the least squares problem with sparse QR */
glmgen_qrsol(kernmat_qr, beta);
/* Update alpha: 1d fused lasso
* Build the response vector */
tf_dxtil(x,n,k,beta,v);
for (i=0; i<n-k; i++) z[i] = v[i]-u[i];
/* Use Nick's DP algorithm */
tf_dp(n-k,z,lam/rho,alpha);
/* Update u: dual update */
for (i=0; i<n-k; i++) u[i] = u[i]+alpha[i]-v[i];
/* Compute objective */
obj[it+1] = tf_obj_gauss(x,y,w,n,k,lam,beta,z);
if (verbose) printf("%i\t%0.3e\n",it+1,obj[it]);
/* Stop if relative difference of objective values < obj_tol */
descent = obj[itbest] - obj[it+1];
if ( descent > 0 )
{
memcpy(betabest, beta, n * sizeof(double));
memcpy(alphabest, alpha, n * sizeof(double));
itbest = it+1;
}
if (it >= 10)
{
variation = 0;
for (i=0; i < 10; i++ )
variation += fabs(obj[it+1-i] - obj[it-i]);
//variation = fabs(obj[it+1] - obj[it]) + fabs(obj[it] - obj[it-1]) + fabs(obj[it-1] - obj[it-2]);
if (variation < fabs(obj[itbest]) * 10 * obj_tol)
break;
}
}
memcpy(beta, betabest, n * sizeof(double));
memcpy(alpha, alphabest, n * sizeof(double));
*iter = it;
if (verbose)
printf("itbest = %d it = %d obj[0]= %f obj.best = %f\n",
itbest, it, obj[0], obj[itbest]);
/* Compute final df value, based on alpha */
d = k+1;
for (i=0; i<n-k-1; i++) if (alpha[i] != alpha[i+1]) d += 1;
*df = d;
cs_spfree(kernmat);
glmgen_gqr_free(kernmat_qr);
free(v);
free(z);
free(betabest);
free(alphabest);
}