hex.glm.LSMSolver Maven / Gradle / Ivy
package hex.glm;
import hex.glm.Gram.Cholesky;
import java.util.Arrays;
import jsr166y.CountedCompleter;
import water.H2O;
import water.Iced;
import water.Key;
import water.MemoryManager;
/**
* Distributed least squares solvers
* @author tomasnykodym
*
*/
public abstract class LSMSolver extends Iced{
public enum LSMSolverType {
AUTO, // AUTO: (len(beta) < 1000)?ADMM:GenGradient
ADMM,
GenGradient
}
double _lambda;
final double _alpha;
public Key _jobKey;
public String _id;
public LSMSolver(double lambda, double alpha){
_lambda = lambda;
_alpha = alpha;
}
public final double [] grad(Gram gram, double [] beta, double [] xy){
double [] grad = gram.mul(beta);
for(int i = 0; i < grad.length; ++i)
grad[i] -= xy[i];
return grad;
}
public static void subgrad(final double alpha, final double lambda, final double [] beta, final double [] grad){
if(beta == null)return;
final double l1pen = lambda*alpha;
for(int i = 0; i < grad.length-1; ++i) {// add l2 reg. term to the gradient
if(beta[i] < 0) grad[i] -= l1pen;
else if(beta[i] > 0) grad[i] += l1pen;
else grad[i] = LSMSolver.shrinkage(grad[i], l1pen);
}
}
/**
* @param xy - guassian: -X'y binomial: -(1/4)X'(XB + (y-p)/(p*1-p))
* @param yy - < y,y > /2
* @param newBeta - resulting vector of coefficients
* @return true if converged
*
*/
public abstract boolean solve(Gram gram, double [] xy, double yy, double [] newBeta);
protected boolean _converged;
public final boolean converged(){return _converged;}
public static class LSMSolverException extends RuntimeException {
public LSMSolverException(String msg){super(msg);}
}
public abstract String name();
protected static double shrinkage(double x, double kappa) {
double sign = x < 0?-1:1;
double sx = x*sign;
if(sx <= kappa) return 0;
return sign*(sx - kappa);
// return Math.max(0, x - kappa) - Math.max(0, -x - kappa);
}
/**
* Compute least squares objective function value:
* lsm_obj(beta) = 0.5*(y - X*b)'*(y - X*b) + l1 + l2
* = 0.5*y'y - (X'y)'*b + 0.5*b'*X'X*b) + l1 + l2
* l1 = alpha*lambda_value*l1norm(beta)
* l2 = (1-alpha)*lambda_value*l2norm(beta)/2
* @param xy: X'y
* @param yy: 0.5*y'y
* @param beta: b (vector of coefficients)
* @param xb: X'X*beta
* @return 0.5*(y - X*b)'*(y - X*b) + l1 + l2
*/
protected double objectiveVal(double[] xy, double yy, double[] beta, double [] xb) {
double res = lsm_objectiveVal(xy,yy,beta, xb);
double l1 = 0, l2 = 0;
for(int i = 0; i < beta.length; ++i){
l1 += Math.abs(beta[i]);
l2 += beta[i]*beta[i];
}
return res + _alpha*_lambda*l1 + 0.5*(1-_alpha)*_lambda*l2;
}
/**
* Compute the LSM objective.
*
* lsm_obj(beta) = 0.5 * (y - X*b)' * (y - X*b)
* = 0.5 * y'y - (X'y)'*b + 0.5*b'*X'X*b)
* = 0.5yy + b*(0.5*X'X*b - X'y)
* @param xy X'y
* @param yy y'y
* @param beta
* @param xb X'X*beta
* @return
*/
protected double lsm_objectiveVal(double[] xy, double yy, double[] beta, double [] xb) {
double res = 0.5*yy;
for(int i = 0; i < xb.length; ++i)
res += beta[i]*(0.5*xb[i] - xy[i]);
return res;
}
static final double[] mul(double[][] X, double[] y, double[] z) {
final int M = X.length;
final int N = y.length;
for( int i = 0; i < M; ++i ) {
z[i] = X[i][0] * y[0];
for( int j = 1; j < N; ++j )
z[i] += X[i][j] * y[j];
}
return z;
}
static final double[] mul(double[] x, double a, double[] z) {
for( int i = 0; i < x.length; ++i )
z[i] = a * x[i];
return z;
}
static final double[] plus(double[] x, double[] y, double[] z) {
for( int i = 0; i < x.length; ++i )
z[i] = x[i] + y[i];
return z;
}
static final double[] minus(double[] x, double[] y, double[] z) {
for( int i = 0; i < x.length; ++i )
z[i] = x[i] - y[i];
return z;
}
static final double[] shrink(double[] x, double[] z, double kappa) {
for( int i = 0; i < x.length - 1; ++i )
z[i] = shrinkage(x[i], kappa);
z[x.length - 1] = x[x.length - 1]; // do not penalize intercept!
return z;
}
public static final class ADMMSolver extends LSMSolver {
//public static final double DEFAULT_LAMBDA = 1e-5;
public static final double DEFAULT_ALPHA = 0.5;
public double [] _wgiven;
public double _proximalPenalty;
final public double _gradientEps;
private static final double GLM1_RHO = 1.0e-3;
public double gerr = Double.POSITIVE_INFINITY;
public int iterations = 0;
public long decompTime;
public boolean normalize() {return _lambda != 0;}
public double _addedL2;
public ADMMSolver (double lambda, double alpha, double gradEps) {
super(lambda,alpha);
_gradientEps = gradEps;
}
public ADMMSolver (double lambda, double alpha, double gradEps,double addedL2) {
super(lambda,alpha);
_addedL2 = addedL2;
_gradientEps = gradEps;
}
public static class NonSPDMatrixException extends LSMSolverException {
public NonSPDMatrixException(){super("Matrix is not SPD, can't solve without regularization\n");}
public NonSPDMatrixException(Gram grm){
super("Matrix is not SPD, can't solve without regularization\n" + grm);
}
}
@Override
public boolean solve(Gram gram, double [] xy, double yy, double[] z) {
return solve(gram, xy, yy, z, Double.POSITIVE_INFINITY);
}
private static double l1_norm(double [] v){
double res = 0;
for(double d:v)res += Math.abs(d);
return res;
}
private static double l2_norm(double [] v){
double res = 0;
for(double d:v)res += d*d;
return res;
}
private double converged(Gram g, double [] beta, double [] xy){
double [] grad = grad(g,beta,xy);
subgrad(_alpha,_lambda,beta,grad);
double err = 0;
for(double d:grad)
if(d > err)err = d;
else if(d < -err)err = -d;
return err;
}
private double getGrad(Gram gram, double [] beta, double [] xy){
double [] g = grad(gram,beta,xy);
double err = 0;
for(double d3:g)
if(d3 > err)err = d3;
else if(d3 < -err)err = -d3;
return err;
}
public ParallelSolver parSolver(Gram gram, double [] xy, double [] res, double rho, int iBlock, int rBlock){
return new ParallelSolver(gram, xy, res, rho,iBlock, rBlock);
}
public final class ParallelSolver extends H2O.H2OCountedCompleter {
final Gram gram;
final double rho;
final double kappa;
double _bestErr = Double.POSITIVE_INFINITY;
double _lastErr = Double.POSITIVE_INFINITY;
final double [] xy;
double [] _xyPrime;
double _orlx;
int _k;
final double [] u;
final double [] z;
Cholesky chol;
final double d;
int _iter;
final int N;
final int max_iter;
final int round;
final int _iBlock;
final int _rBlock;
private ParallelSolver(Gram g, double [] xy, double [] res, double rho, int iBlock, int rBlock){
_iBlock = iBlock;
_rBlock = rBlock;
gram = g; this.xy = xy; this.z = res;;
N = xy.length;
d = gram._diagAdded;
this.rho = rho;
u = MemoryManager.malloc8d(N);
kappa = _lambda*_alpha/rho;
max_iter = (int)(10000*(250.0/(1+xy.length)));
round = Math.max(20,(int)(max_iter*0.01));
_k = round;
}
@Override
public void compute2() {
Arrays.fill(z, 0);
if(_lambda>0 || _addedL2 > 0)
gram.addDiag(_lambda*(1-_alpha) + _addedL2);
if(_alpha > 0 && _lambda > 0)
gram.addDiag(rho);
if(_proximalPenalty > 0 && _wgiven != null){
gram.addDiag(_proximalPenalty, true);
for(int i = 0; i < xy.length; ++i)
xy[i] += _proximalPenalty*_wgiven[i];
}
int attempts = 0;
long t1 = System.currentTimeMillis();
chol = gram.cholesky(null,true,_id);
long t2 = System.currentTimeMillis();
while(!chol.isSPD() && attempts < 10){
if(_addedL2 == 0) _addedL2 = 1e-5;
else _addedL2 *= 10;
++attempts;
gram.addDiag(_addedL2); // try to add L2 penalty to make the Gram issp
gram.cholesky(chol);
}
decompTime = (t2-t1);
if(!chol.isSPD())
throw new NonSPDMatrixException(gram);
if(_alpha == 0 || _lambda == 0){ // no l1 penalty
System.arraycopy(xy, 0, z, 0, xy.length);
chol.parSolver(this,z,_iBlock,_rBlock).fork();
return;
}
gerr = Double.POSITIVE_INFINITY;
_xyPrime = xy.clone();
_orlx = 1.8; // over-relaxation
// first compute the x update
// add rho*(z-u) to A'*y
new ADMMIteration(this).fork();
}
@Override public void onCompletion(CountedCompleter caller){
gram.addDiag(-gram._diagAdded + d);
assert gram._diagAdded == d;
}
private final class ADMMIteration extends CountedCompleter {
final long t1;
public ADMMIteration(H2O.H2OCountedCompleter cmp){super(cmp); t1 = System.currentTimeMillis();}
@Override public void compute(){
++_iter;
final double [] xyPrime = _xyPrime;
// first compute the x update
// add rho*(z-u) to A'*y
for( int j = 0; j < N-1; ++j )xyPrime[j] = xy[j] + rho*(z[j] - u[j]);
xyPrime[N-1] = xy[N-1];
// updated x
chol.parSolver(this,xyPrime,_iBlock,_rBlock).fork();
}
@Override
public void onCompletion(CountedCompleter caller) {
final double [] xyPrime = _xyPrime;
final double orlx = _orlx;
// compute u and z updateADMM
for( int j = 0; j < N-1; ++j ) {
double x_hat = xyPrime[j];
x_hat = x_hat * orlx + (1 - orlx) * z[j];
z[j] = shrinkage(x_hat + u[j], kappa);
u[j] += x_hat - z[j];
}
z[N-1] = xyPrime[N-1];
if(_iter == _k) {
double[] grad = grad(gram, z, xy);
subgrad(_alpha, _lambda, z, grad);
for (int x = 0; x < grad.length - 1; ++x) {
if (gerr < grad[x] || gerr < -grad[x])
gerr = grad[x];
}
if (gerr < 9e-4)
return;
// if(grad < bestErr){
// bestErr = err;
// System.arraycopy(z,0,res,0,z.length);
// if(err < _gradientEps)
// break;
// } else {
// boolean allzeros = true;
// for (int x = 0; allzeros && x < z.length - 1; ++x)
// allzeros = z[x] == 0;
// if (!allzeros) { // only want this check if we're past the warm up period (there can be many iterations with all zeros!)
// // did not converge, check if we can converge in reasonable time
// if (diff < 1e-4) // we won't ever converge with this setup (maybe change rho and try again?)
// break;
// orlx = (1 + 15 * orlx) * 0.0625;
// } else
// orlx = 1.8;
// }
// lastErr = err;
_k += round;
}
if(_iter < max_iter){
getCompleter().addToPendingCount(1);
new ADMMIteration((H2O.H2OCountedCompleter)getCompleter()).fork();
}
}
}
}
final static double RELTOL = 1e-4;
public boolean solve(Gram gram, double [] xy, double yy, final double[] z, final double rho) {
gerr = 0;
double d = gram._diagAdded;
final int N = xy.length;
Arrays.fill(z, 0);
if(_lambda>0 || _addedL2 > 0)
gram.addDiag(_lambda*(1-_alpha) + _addedL2);
if(_alpha > 0 && _lambda > 0)
gram.addDiag(rho);
if(_proximalPenalty > 0 && _wgiven != null){
gram.addDiag(_proximalPenalty, true);
xy = xy.clone();
for(int i = 0; i < xy.length; ++i)
xy[i] += _proximalPenalty*_wgiven[i];
}
int attempts = 0;
long t1 = System.currentTimeMillis();
Cholesky chol = gram.cholesky(null,true,_id);
long t2 = System.currentTimeMillis();
while(!chol.isSPD() && attempts < 10){
if(_addedL2 == 0) _addedL2 = 1e-5;
else _addedL2 *= 10;
++attempts;
gram.addDiag(_addedL2); // try to add L2 penalty to make the Gram issp
gram.cholesky(chol);
}
decompTime = (t2-t1);
if(!chol.isSPD())
throw new NonSPDMatrixException(gram);
if(_alpha == 0 || _lambda == 0){ // no l1 penalty
System.arraycopy(xy, 0, z, 0, xy.length);
chol.solve(z);
gram.addDiag(-gram._diagAdded + d);
return true;
}
double[] u = MemoryManager.malloc8d(N);
double [] xyPrime = xy.clone();
double kappa = _lambda*_alpha/rho;
int i;
int max_iter = Math.max(500,(int)(50000.0/(1+(xy.length >> 3))));
double orlx = 1.8; // over-relaxation
double reltol = RELTOL;
for(i = 0; i < max_iter; ++i ) {
long tX = System.currentTimeMillis();
// first compute the x update
// add rho*(z-u) to A'*y
for( int j = 0; j < N-1; ++j )
xyPrime[j] = xy[j] + rho*(z[j] - u[j]);
xyPrime[N-1] = xy[N-1];
// updated x
chol.solve(xyPrime);
// compute u and z updateADMM
double rnorm = 0, snorm = 0, unorm = 0, xnorm = 0;
for( int j = 0; j < N-1; ++j ) {
double x = xyPrime[j];
double zold = z[j];
double x_hat = x * orlx + (1 - orlx) * zold;
z[j] = shrinkage(x_hat + u[j], kappa);
u[j] += x_hat - z[j];
double r = xyPrime[j] - z[j];
double s = z[j] - zold;
rnorm += r*r;
snorm += s*s;
xnorm += x*x;
unorm += u[j]*u[j];
}
z[N-1] = xyPrime[N-1];
if(rnorm < reltol*xnorm && snorm < reltol*unorm){
gerr = 0;
double [] grad = grad(gram,z,xy);
subgrad(_alpha,_lambda,z,grad);
for(int x = 0; x < grad.length-1; ++x){
if(gerr < grad[x]) gerr = grad[x];
else if(gerr < -grad[x]) gerr = -grad[x];
}
if(gerr < 1e-4 || reltol <= 1e-6)break;
while(rnorm < reltol*xnorm && snorm < reltol*unorm)
reltol *= .1;
}
if(i % 20 == 0)
orlx = (1 + 15 * orlx) * 0.0625;
}
gram.addDiag(-gram._diagAdded + d);
assert gram._diagAdded == d;
iterations = i;
return _converged = (gerr < _gradientEps);
}
@Override
public String name() {return "ADMM";}
}
public static final class ProxSolver extends LSMSolver {
public ProxSolver(double lambda, double alpha){super(lambda,alpha);}
/**
* @param newB
* @param oldObj
* @param oldB
* @param
* @param t
* @return
*/
private static final double f_hat(double [] newB,double oldObj, double [] oldB,double [] xb, double [] xy, double t){
double res = oldObj;
double l2 = 0;
for(int i = 0; i < newB.length; ++i){
double diff = newB[i] - oldB[i];
res += (xb[i]-xy[i])*diff;
l2 += diff*diff;
}
return res + 0.25*l2/t;
}
private double penalty(double [] beta){
double l1 = 0,l2 = 0;
for(int i = 0; i < beta.length; ++i){
l1 += Math.abs(beta[i]);
l2 += beta[i]*beta[i];
}
return _lambda*(_alpha*l1 + (1-_alpha)*l2*0.5);
}
private static double betaDiff(double [] b1, double [] b2){
double res = 0;
for(int i = 0; i < b1.length; ++i)
Math.max(res, Math.abs(b1[i] - b2[i]));
return res;
}
@Override
public boolean solve(Gram gram, double [] xy, double yy, double[] beta) {
ADMMSolver admm = new ADMMSolver(_lambda,_alpha,1e-2);
if(gram != null)return admm.solve(gram,xy,yy,beta);
Arrays.fill(beta,0);
long t1 = System.currentTimeMillis();
final double [] xb = gram.mul(beta);
double objval = objectiveVal(xy,yy,beta,xb);
final double [] newB = MemoryManager.malloc8d(beta.length);
final double [] newG = MemoryManager.malloc8d(beta.length);
double step = 1;
final double l1pen = _lambda*_alpha;
final double l2pen = _lambda*(1-_alpha);
double lsmobjval = lsm_objectiveVal(xy,yy,beta,xb);
boolean converged = false;
final int intercept = beta.length-1;
int iter = 0;
MAIN:
while(!converged && iter < 1000) {
++iter;
step = 1;
while(step > 1e-12){ // line search
double l2shrink = 1/(1+step*l2pen);
double l1shrink = l1pen*step;
for(int i = 0; i < beta.length-1; ++i)
newB[i] = l2shrink*shrinkage((beta[i]-step*(xb[i]-xy[i])),l1shrink);
newB[intercept] = beta[intercept] - step*(xb[intercept]-xy[intercept]);
gram.mul(newB, newG);
double newlsmobj = lsm_objectiveVal(xy, yy, newB,newG);
double fhat = f_hat(newB,lsmobjval,beta,xb,xy,step);
if(newlsmobj <= fhat){
lsmobjval = newlsmobj;
converged = betaDiff(beta,newB) < 1e-6;
System.arraycopy(newB,0,beta,0,newB.length);
System.arraycopy(newG,0,xb,0,newG.length);
continue MAIN;
} else step *= 0.8;
}
converged = true;
}
return converged;
}
public String name(){return "ProximalGradientSolver";}
}
}
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