package edu.stanford.nlp.optimization;
import java.text.DecimalFormat;
import java.text.NumberFormat;
/**
* Conjugate-gradient implementation based on the code in Numerical
* Recipes in C. (See p. 423 and others.) As of now, it requires a
* differentiable function (DiffFunction) as input. Equality
* constraints are supported; inequality constraints may soon be
* added.
* <p/>
* The basic way to use the minimizer is with a null constructor, then
* the simple minimize method:
* <p/>
* <p><code>Minimizer cgm = new CGMinimizer();</code>
* <br><code>DiffFunction df = new SomeDiffFunction();</code>
* <br><code>double tol = 1e-4;</code>
* <br><code>double[] initial = getInitialGuess();</code>
* <br><code>double[] minimum = cgm.minimize(df,tol,initial);</code>
*
* @author <a href="mailto:klein@cs.stanford.edu">Dan Klein</a>
* @version 1.0
* @since 1.0
*/
public class CGMinimizer implements Minimizer<DiffFunction> {
private static NumberFormat nf = new DecimalFormat("0.000E0");
private Function monitor; // = null;
private static final int numToPrint = 5;
private static final boolean simpleGD = false;
private static final boolean checkSimpleGDConvergence = true;
private static final boolean verbose = false;
private boolean silent;
private static final int ITMAX = 2000; // overridden in dbrent(); made bigger
private static final double EPS = 1.0e-30;
private static final int resetFrequency = 10;
static double[] copyArray(double[] a) {
double[] result = new double[a.length];
for (int i = 0; i < a.length; i++) {
result[i] = a[i];
}
return result;
}
// private static String arrayToString(double[] x) {
// return arrayToString(x, x.length);
// }
private static String arrayToString(double[] x, int num) {
StringBuilder sb = new StringBuilder("(");
if (num > x.length) {
num = x.length;
}
for (int j = 0; j < num; j++) {
sb.append(x[j]);
if (j != x.length - 1) {
sb.append(", ");
}
}
if (num < x.length) {
sb.append("...");
}
sb.append(")");
return sb.toString();
}
private static double fabs(double x) {
if (x < 0) {
return -x;
}
return x;
}
private static double fmax(double x, double y) {
if (x < y) {
return y;
}
return x;
}
// private static double fmin(double x, double y) {
// if (x>y)
// return y;
// return x;
// }
private static double sign(double x, double y) {
if (y >= 0.0) {
return fabs(x);
}
return -fabs(x);
}
// private static double arrayMax(double[] x) {
// double max = Double.NEGATIVE_INFINITY;
// for (int i=0; i<x.length; i++) {
// if (max < x[i])
// max = x[i];
// }
// return max;
// }
//
// private static int arrayArgMax(double[] x) {
// double max = Double.NEGATIVE_INFINITY;
// int index = -1;
// for (int i=0; i<x.length; i++) {
// if (max < x[i]) {
// max = x[i];
// index = i;
// }
// }
// return index;
// }
//
// private static double arrayMin(double[] x) {
// double min = Double.POSITIVE_INFINITY;
// for (int i=0; i<x.length; i++) {
// if (min > x[i])
// min = x[i];
// }
// return min;
// }
//
// private static int arrayArgMin(double[] x) {
// double min = Double.POSITIVE_INFINITY;
// int index = -1;
// for (int i=0; i<x.length; i++) {
// if (min > x[i]) {
// min = x[i];
// index = i;
// }
// }
// return index;
// }
static class OneDimDiffFunction {
private DiffFunction function;
private double[] initial;
private double[] direction;
private double[] tempVector;
private double[] vectorOf(double x) {
for (int j = 0; j < initial.length; j++) {
tempVector[j] = initial[j] + x * direction[j];
}
//System.err.println("Tmp "+arrayToString(tempVector,10));
//System.err.println("Dir "+arrayToString(direction,10));
return tempVector;
}
double valueAt(double x) {
return function.valueAt(vectorOf(x));
}
double derivativeAt(double x) {
double[] g = function.derivativeAt(vectorOf(x));
double d = 0.0;
for (int j = 0; j < g.length; j++) {
d += g[j] * direction[j];
}
return d;
}
OneDimDiffFunction(DiffFunction function, double[] initial, double[] direction) {
this.function = function;
this.initial = copyArray(initial);
this.direction = copyArray(direction);
this.tempVector = new double[function.domainDimension()];
}
} // end class OneDimDiffFunction
// constants
private static final double GOLD = 1.618034;
private static final double GLIMIT = 100.0;
private static final double TINY = 1.0e-20;
private static Triple mnbrak(Triple abc, OneDimDiffFunction function) {
// inputs
double ax = abc.a;
double fa = function.valueAt(ax);
double bx = abc.b;
double fb = function.valueAt(bx);
if (fb > fa) {
// swap
double temp = fa;
fa = fb;
fb = temp;
temp = ax;
ax = bx;
bx = temp;
}
// guess cx
double cx = bx + GOLD * (bx - ax);
double fc = function.valueAt(cx);
// loop until we get a bracket
while (fb > fc) {
double r = (bx - ax) * (fb - fc);
double q = (bx - cx) * (fb - fa);
double u = bx - ((bx - cx) * q - (bx - ax) * r) / (2.0 * sign(fmax(fabs(q - r), TINY), q - r));
double fu;
double ulim = bx + GLIMIT * (cx - bx);
if ((bx - u) * (u - cx) > 0.0) {
fu = function.valueAt(u);
if (fu < fc) {
//Ax = new Double(bx);
//Bx = new Double(u);
//Cx = new Double(cx);
//System.err.println("\nReturning3: a="+bx+" ("+fb+") b="+u+"("+fu+") c="+cx+" ("+fc+")");
return new Triple(bx, u, cx);
} else if (fu > fb) {
//Cx = new Double(u);
//Ax = new Double(ax);
//Bx = new Double(bx);
//System.err.println("\nReturning2: a="+ax+" ("+fa+") b="+bx+"("+fb+") c="+u+" ("+fu+")");
return new Triple(ax, bx, u);
}
u = cx + GOLD * (cx - bx);
fu = function.valueAt(u);
} else if ((cx - u) * (u - ulim) > 0.0) {
fu = function.valueAt(u);
if (fu < fc) {
bx = cx;
cx = u;
u = cx + GOLD * (cx - bx);
fb = fc;
fc = fu;
fu = function.valueAt(u);
}
} else if ((u - ulim) * (ulim - cx) >= 0.0) {
u = ulim;
fu = function.valueAt(u);
} else {
u = cx + GOLD * (cx - bx);
fu = function.valueAt(u);
}
ax = bx;
bx = cx;
cx = u;
fa = fb;
fb = fc;
fc = fu;
}
//System.err.println("\nReturning: a="+ax+" ("+fa+") b="+bx+"("+fb+") c="+cx+" ("+fc+")");
return new Triple(ax, bx, cx);
}
private static double dbrent(OneDimDiffFunction function, double ax, double bx, double cx) {
// constants
final boolean dbVerbose = false;
final int ITMAX = 100;
final double TOL = 1.0e-4;
boolean ok1, ok2;
double d = 0.0, d1, d2, du, e = 0.0;
double fu, olde, tol1, tol2, u, u1, u2, xm;
double a = (ax < cx ? ax : cx);
double b = (ax > cx ? ax : cx);
double x = bx;
double v = bx;
double w = bx;
double fx = function.valueAt(x);
double fv = fx;
double fw = fx;
double dx = function.derivativeAt(x);
double dv = dx;
double dw = dx;
for (int iteration = 0; iteration < ITMAX; iteration++) {
//System.err.println("dbrent "+iteration+" x "+x+" fx "+fx);
xm = 0.5 * (a + b);
tol1 = TOL * fabs(x); //+ZEPS (was 1e-10);
tol2 = 2.0 * tol1;
if (fabs(x - xm) <= (tol2 - 0.5 * (b - a))) {
if (dbVerbose) {
System.err.println("dbrent returning because min is cornered " + a + " (" + function.valueAt(a) + ") ~ " + x + " (" + fx + ") " + b + " (" + function.valueAt(b) + ")");
}
return x;
}
if (fabs(e) > tol1) {
d1 = 2.0 * (b - a);
d2 = d1;
if (dw != dx) {
d1 = (w - x) * dx / (dx - dw);
}
if (dv != dx) {
d2 = (v - x) * dx / (dx - dv);
}
u1 = x + d1;
u2 = x + d2;
ok1 = ((a - u1) * (u1 - b) > 0.0 && dx * d1 <= 0.0);
ok2 = ((a - u2) * (u2 - b) > 0.0 && dx * d2 <= 0.0);
olde = e;
e = d;
if (ok1 || ok2) {
if (ok1 && ok2) {
d = (fabs(d1) < fabs(d2) ? d1 : d2);
} else if (ok1) {
d = d1;
} else {
d = d2;
}
if (fabs(d) <= fabs(0.5 * olde)) {
u = x + d;
if (u - a < tol2 || b - u < tol2) {
d = sign(tol1, xm - x);
}
} else {
e = (dx >= 0.0 ? a - x : b - x);
d = 0.5 * e;
}
} else {
e = (dx >= 0.0 ? a - x : b - x);
d = 0.5 * e;
}
} else {
e = (dx >= 0.0 ? a - x : b - x);
d = 0.5 * e;
}
if (fabs(d) >= tol1) {
u = x + d;
fu = function.valueAt(u);
} else {
u = x + sign(tol1, d);
fu = function.valueAt(u);
if (fu > fx) {
if (dbVerbose) {
System.err.println("dbrent returning because derivative is broken");
}
return x;
}
}
du = function.derivativeAt(u);
if (fu <= fx) {
if (u >= x) {
a = x;
} else {
b = x;
}
v = w;
fv = fw;
dv = dw;
w = x;
fw = fx;
dw = dx;
x = u;
fx = fu;
dx = du;
} else {
if (u < x) {
a = u;
} else {
b = u;
}
if (fu <= fw || w == x) {
v = w;
fv = fw;
dv = dw;
w = u;
fw = fu;
dw = du;
} else if (fu < fv || v == x || v == w) {
v = u;
fv = fu;
dv = du;
}
}
}
// dan's addition:
if (fx < function.valueAt(0.0)) {
return x;
}
if (dbVerbose) {
System.err.println("Warning: exiting dbrent because ITMAX exceeded!");
}
return 0.0;
}
private static class Triple {
public double a;
public double b;
public double c;
public Triple(double a, double b, double c) {
this.a = a;
this.b = b;
this.c = c;
}
}
//public double lastXx = 1.0;
double[] lineMinimize(DiffFunction function, double[] initial, double[] direction) {
// make a 1-dim function along the direction line
// THIS IS A HACK (but it's the NRiC peoples' hack)
OneDimDiffFunction oneDim = new OneDimDiffFunction(function, initial, direction);
// do a 1-dim line min on this function
//Double Ax = new Double(0.0);
//Double Xx = new Double(1.0);
//Double Bx = new Double(0.0);
// bracket the extreme pt
double guess = 0.01;
//System.err.println("Current "+oneDim.valueAt(0)+" nudge "+(oneDim.smallestZeroPositiveLocation()*1e-2)+" "+oneDim.valueAt(oneDim.smallestZeroPositiveLocation()*1e-5));
if (!silent) {
System.err.print("[");
}
Triple bracketing = mnbrak(new Triple(0, guess, 0), oneDim);
if (!silent) {
System.err.print("]");
}
double ax = bracketing.a;
double xx = bracketing.b;
double bx = bracketing.c;
//lastXx = xx;
// CHECK FOR END OF WORLD
if (!(ax <= xx && xx <= bx) && !(bx <= xx && xx <= ax)) {
System.err.println("Bad bracket order!");
}
if (verbose) {
System.err.println("Bracketing found: " + ax + " " + xx + " " + bx);
System.err.println("Bracketing found: " + oneDim.valueAt(ax) + " " + oneDim.valueAt(xx) + " " + oneDim.valueAt(bx));
//System.err.println("Bracketing found: "+arrayToString(oneDim.vectorOf(ax),3)+" "+arrayToString(oneDim.vectorOf(xx),3)+" "+arrayToString(oneDim.vectorOf(bx),3));
}
// find the extreme pt
if (!silent) {
System.err.print("<");
}
double xmin = dbrent(oneDim, ax, xx, bx);
if (!silent) {
System.err.print(">");
}
// return the full vector
//System.err.println("Went "+xmin+" during lineMinimize");
return oneDim.vectorOf(xmin);
}
public double[] minimize(DiffFunction function, double functionTolerance, double[] initial) {
return minimize(function, functionTolerance, initial, ITMAX);
}
public double[] minimize(DiffFunction dfunction, double functionTolerance, double[] initial, int maxIterations) {
// check for derivatives
int dimension = dfunction.domainDimension();
//lastXx = 1.0;
// evaluate function
double fp = dfunction.valueAt(initial);
if (verbose) {
System.err.println("Initial: " + fp);
}
double[] xi = copyArray(dfunction.derivativeAt(initial));
if (verbose) {
System.err.println("Initial at: " + arrayToString(initial, numToPrint));
System.err.println("Initial deriv: " + arrayToString(xi, numToPrint));
}
// make some vectors
double[] g = new double[dimension];
double[] h = new double[dimension];
double[] p = new double[dimension];
for (int j = 0; j < dimension; j++) {
g[j] = -xi[j];
xi[j] = g[j];
h[j] = g[j];
p[j] = initial[j];
}
// iterations
boolean simpleGDStep = false;
for (int iterations = 1; iterations < maxIterations; iterations++) {
if (!silent) {
System.err.print("Iter " + iterations + " ");
}
// do a line min along descent direction
//System.err.println("Minimizing from ("+p[0]+","+p[1]+") along ("+xi[0]+","+xi[1]+")\n");
if (verbose) {
System.err.println("Minimizing along " + arrayToString(xi, numToPrint));
}
//System.err.println("Current is "+fp);
double[] p2 = lineMinimize(dfunction, p, xi);
double fp2 = dfunction.valueAt(p2);
//System.err.println("Result is "+fp2+" (from "+fp+") at ("+p2[0]+","+p2[1]+")\n");
if (verbose) {
System.err.println("Result is " + fp2 + " after " + iterations);
System.err.println("Result at " + arrayToString(p2, numToPrint));
}
//System.err.print(fp2+"|"+(int)(Math.log((fabs(fp2-fp)+1e-100)/(fabs(fp)+fabs(fp2)+1e-100))/Math.log(10)));
if (!silent) {
System.err.printf(" %s (delta: %s)\n",
nf.format(fp2), nf.format(fp-fp2));
}
if (monitor != null) {
double monitorReturn = monitor.valueAt(p2);
if (monitorReturn < functionTolerance) {
return p2;
}
}
// check convergence
if (2.0 * fabs(fp2 - fp) <= functionTolerance * (fabs(fp2) + fabs(fp) + EPS)) {
// convergence
if (!checkSimpleGDConvergence || simpleGDStep || simpleGD) {
return p2;
}
simpleGDStep = true;
//System.err.println("Switched to GD for a step.");
} else {
//if (!simpleGD)
//System.err.println("Switching to CGD.");
simpleGDStep = false;
}
// shift variables
for (int j = 0; j < dimension; j++) {
xi[j] = p2[j] - p[j];
p[j] = p2[j];
}
fp = fp2;
// find the new gradient
xi = copyArray(dfunction.derivativeAt(p));
//System.err.print("mx "+arrayMax(xi)+" mn "+arrayMin(xi));
if (!simpleGDStep && !simpleGD && (iterations % resetFrequency != 0)) {
// do the magic -- part i
// (calculate some dot products we'll need)
double dgg = 0.0;
double gg = 0.0;
for (int j = 0; j < dimension; j++) {
// g dot g
gg += g[j] * g[j];
// grad dot grad
// FR method is:
// dgg += x[j]*x[j];
// PR method is:
dgg += (xi[j] + g[j]) * xi[j];
}
// check for miraculous convergence
if (gg == 0.0) {
return p;
}
// magic part ii
// (update the sequence in a way that tries to preserve conjugacy)
double gam = dgg / gg;
for (int j = 0; j < dimension; j++) {
g[j] = -xi[j];
h[j] = g[j] + gam * h[j];
xi[j] = h[j];
}
} else {
// miraculous simpleGD convergence
double xixi = 0.0;
for (int j = 0; j < dimension; j++) {
xixi += xi[j] * xi[j];
}
// reset cgd
for (int j = 0; j < dimension; j++) {
g[j] = -xi[j];
xi[j] = g[j];
h[j] = g[j];
}
if (xixi == 0.0) {
return p;
}
}
}
// too many iterations
System.err.println("Warning: exiting minimize because ITER exceeded!");
return p;
}
/**
* Basic constructor, use this.
*/
public CGMinimizer() {
this(true);
}
/**
* Pass in <code>false</code> to get per-iteration progress reports
* (to stderr).
*
* @param silent a <code>boolean</code> value
*/
public CGMinimizer(boolean silent) {
this.silent = silent;
}
/**
* Perform minimization with monitoring. After each iteration,
* monitor.valueAt(x) gets called, with the double array <code>x</code>
* being that iteration's ending point. A return <code><
* tol</code> forces convergence (terminates the CG procedure).
* Specially for Kristina.
*
* @param monitor a <code>Function</code> value
*/
public CGMinimizer(Function monitor) {
this();
this.monitor = monitor;
}
}
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