/// <summary>
/// Tests the gradient reported by <paramref name="f"/>.
/// </summary>
/// <param name="f">Function to test</param>
/// <param name="x">Point at which to test</param>
/// <param name="dir">Direction to test derivative</param>
/// <param name="quiet">Whether to disable output</param>
/// <param name="newGrad">This is a reusable working buffer for intermediate calculations</param>
/// <param name="newX">This is a reusable working buffer for intermediate calculations</param>
/// <returns>Normalized difference between analytic and numeric directional derivative</returns>
public static Float Test(DifferentiableFunction f, ref VBuffer <Float> x, ref VBuffer <Float> dir, bool quiet,
ref VBuffer <Float> newGrad, ref VBuffer <Float> newX)
{
Float normDir = VectorUtils.Norm(dir);
Float val = f(ref x, ref newGrad, null);
Float dirDeriv = VectorUtils.DotProduct(ref newGrad, ref dir);
Float scaledEps = Eps / normDir;
VectorUtils.AddMultInto(ref x, scaledEps, ref dir, ref newX);
Float rVal = f(ref newX, ref newGrad, null);
VectorUtils.AddMultInto(ref x, -scaledEps, ref dir, ref newX);
Float lVal = f(ref newX, ref newGrad, null);
Float numDeriv = (rVal - lVal) / (2 * scaledEps);
Float normDiff = Math.Abs(1 - numDeriv / dirDeriv);
Float diff = numDeriv - dirDeriv;
if (!quiet)
{
Console.WriteLine("{0,-18:0.0000e0}{1,-18:0.0000e0}{2,-15:0.0000e0}{3,0:0.0000e0}", numDeriv, dirDeriv, diff, normDiff);
}
return(normDiff);
}
/// <summary>
/// Tests the gradient using finite differences on each axis in the list
/// </summary>
/// <param name="f">Function to test</param>
/// <param name="x">Point at which to test</param>
/// <param name="coords">List of coordinates to test</param>
public static void TestCoords(DifferentiableFunction f, ref VBuffer <Float> x, IList <int> coords)
{
// REVIEW: Delete this method?
VBuffer <Float> grad = default(VBuffer <Float>);
VBuffer <Float> newGrad = default(VBuffer <Float>);
VBuffer <Float> newX = default(VBuffer <Float>);
Float val = f(ref x, ref grad, null);
Float normX = VectorUtils.Norm(x);
Console.WriteLine(Header);
Random r = new Random(5);
VBuffer <Float> dir = new VBuffer <Float>(x.Length, 1, new Float[] { 1 }, new int[] { 0 });
foreach (int n in coords)
{
dir.Values[0] = n;
VectorUtils.AddMultInto(ref x, Eps, ref dir, ref newX);
Float rVal = f(ref newX, ref newGrad, null);
VectorUtils.AddMultInto(ref x, -Eps, ref dir, ref newX);
Float lVal = f(ref newX, ref newGrad, null);
Float dirDeriv = VectorUtils.DotProduct(ref grad, ref dir);
Float numDeriv = (rVal - lVal) / (2 * Eps);
Float normDiff = Math.Abs(1 - numDeriv / dirDeriv);
Float diff = numDeriv - dirDeriv;
Console.WriteLine("{0,-9}{1,-18:0.0000e0}{2,-18:0.0000e0}{3,-15:0.0000e0}{4,0:0.0000e0}", n, numDeriv, dirDeriv, diff, normDiff);
}
}
/// <summary>
/// Returns true if the norm of the gradient, divided by the value, is less than the tolerance.
/// </summary>
/// <param name="state">current state of the optimzer</param>
/// <param name="message">the current value of the criterion</param>
/// <returns>true iff criterion is less than the tolerance</returns>
public override bool Terminate(Optimizer.OptimizerState state, out string message)
{
var grad = state.Grad;
Float norm = VectorUtils.Norm(grad);
Float val = norm / Math.Abs(state.Value);
message = string.Format("{0,0:0.0000e0}", val);
return(val < _tol);
}
/// <summary>
/// Backtracking line search with Armijo-like condition, from Andrew & Gao
/// </summary>
internal override bool LineSearch(IChannel ch, bool force)
{
Float dirDeriv = -VectorUtils.DotProduct(ref _dir, ref _steepestDescDir);
if (dirDeriv == 0)
{
throw ch.Process(new PrematureConvergenceException(this, "Directional derivative is zero. You may be sitting on the optimum."));
}
// if a non-descent direction is chosen, the line search will break anyway, so throw here
// The most likely reason for this is a bug in your function's gradient computation
// It may also indicate that your function is not convex.
ch.Check(dirDeriv < 0, "L-BFGS chose a non-descent direction.");
Float alpha = (Iter == 1 ? (1 / VectorUtils.Norm(_dir)) : 1);
GetNextPoint(alpha);
Float unnormCos = VectorUtils.DotProduct(ref _steepestDescDir, ref _newX) - VectorUtils.DotProduct(ref _steepestDescDir, ref _x);
if (unnormCos < 0)
{
VBufferUtils.ApplyWith(ref _steepestDescDir, ref _dir,
(int ind, Float sdVal, ref Float dirVal) =>
{
if (sdVal * dirVal < 0 && ind >= _biasCount)
{
dirVal = 0;
}
});
GetNextPoint(alpha);
unnormCos = VectorUtils.DotProduct(ref _steepestDescDir, ref _newX) - VectorUtils.DotProduct(ref _steepestDescDir, ref _x);
}
int i = 0;
while (true)
{
Value = Eval(ref _newX, ref _newGrad);
GradientCalculations++;
if (Value <= LastValue - Gamma * unnormCos)
{
return(true);
}
++i;
if (!force && i == MaxLineSearch)
{
return(false);
}
alpha *= (Float)0.25;
GetNextPoint(alpha);
unnormCos = VectorUtils.DotProduct(ref _steepestDescDir, ref _newX) - VectorUtils.DotProduct(ref _steepestDescDir, ref _x);
}
}
public static void Main(string[] argv)
{
RunTest(QuadTest);
RunTest(LogTest);
VBuffer <Float> grad = VBufferUtils.CreateEmpty <Float>(2);
int n = 0;
bool print = false;
DTerminate term =
(ref VBuffer <Float> x) =>
{
QuadTest2D(ref x, ref grad);
Float norm = VectorUtils.Norm(grad);
if (++n % 1000 == 0 || print)
{
Console.WriteLine("{0}\t{1}", n, norm);
}
return(norm < 1e-5);
};
SgdOptimizer sgdo = new SgdOptimizer(term, SgdOptimizer.RateScheduleType.Constant, false, 100, 1, (Float)0.99);
VBuffer <Float> init;
CreateWrapped(out init, 0, 0);
VBuffer <Float> ans = default(VBuffer <Float>);
sgdo.Minimize(StochasticQuadTest2D, ref init, ref ans);
QuadTest2D(ref ans, ref grad);
Console.WriteLine(VectorUtils.Norm(grad));
Console.WriteLine();
Console.WriteLine();
n = 0;
GDOptimizer gdo = new GDOptimizer(term, null, true);
print = true;
CreateWrapped(out init, 0, 0);
gdo.Minimize(QuadTest2D, ref init, ref ans);
QuadTest2D(ref ans, ref grad);
Console.WriteLine(VectorUtils.Norm(grad));
}
/// <summary>
/// An implementation of the line search for the Wolfe conditions, from Nocedal & Wright
/// </summary>
internal virtual bool LineSearch(IChannel ch, bool force)
{
Contracts.AssertValue(ch);
Float dirDeriv = VectorUtils.DotProduct(ref _dir, ref _grad);
if (dirDeriv == 0)
{
throw ch.Process(new PrematureConvergenceException(this, "Directional derivative is zero. You may be sitting on the optimum."));
}
// if a non-descent direction is chosen, the line search will break anyway, so throw here
// The most likely reasons for this is a bug in your function's gradient computation,
ch.Check(dirDeriv < 0, "L-BFGS chose a non-descent direction.");
Float c1 = (Float)1e-4 * dirDeriv;
Float c2 = (Float)0.9 * dirDeriv;
Float alpha = (Iter == 1 ? (1 / VectorUtils.Norm(_dir)) : 1);
PointValueDeriv last = new PointValueDeriv(0, LastValue, dirDeriv);
PointValueDeriv aLo = new PointValueDeriv();
PointValueDeriv aHi = new PointValueDeriv();
// initial bracketing phase
while (true)
{
VectorUtils.AddMultInto(ref _x, alpha, ref _dir, ref _newX);
if (EnforceNonNegativity)
{
VBufferUtils.Apply(ref _newX, delegate(int ind, ref Float newXval)
{
if (newXval < 0.0)
{
newXval = 0;
}
});
}
Value = Eval(ref _newX, ref _newGrad);
GradientCalculations++;
if (Float.IsPositiveInfinity(Value))
{
alpha /= 2;
continue;
}
if (!FloatUtils.IsFinite(Value))
{
throw ch.Except("Optimizer unable to proceed with loss function yielding {0}", Value);
}
dirDeriv = VectorUtils.DotProduct(ref _dir, ref _newGrad);
PointValueDeriv curr = new PointValueDeriv(alpha, Value, dirDeriv);
if ((curr.V > LastValue + c1 * alpha) || (last.A > 0 && curr.V >= last.V))
{
aLo = last;
aHi = curr;
break;
}
else if (Math.Abs(curr.D) <= -c2)
{
return(true);
}
else if (curr.D >= 0)
{
aLo = curr;
aHi = last;
break;
}
last = curr;
if (alpha == 0)
{
alpha = Float.Epsilon; // Robust to divisional underflow.
}
else
{
alpha *= 2;
}
}
Float minChange = (Float)0.01;
int maxSteps = 10;
// this loop is the "zoom" procedure described in Nocedal & Wright
for (int step = 0; ; ++step)
{
if (step == maxSteps && !force)
{
return(false);
}
PointValueDeriv left = aLo.A < aHi.A ? aLo : aHi;
PointValueDeriv right = aLo.A < aHi.A ? aHi : aLo;
if (left.D > 0 && right.D < 0)
{
// interpolating cubic would have max in range, not min (can this happen?)
// set a to the one with smaller value
alpha = aLo.V < aHi.V ? aLo.A : aHi.A;
}
else
{
alpha = CubicInterp(aLo, aHi);
if (Float.IsNaN(alpha) || Float.IsInfinity(alpha))
{
alpha = (aLo.A + aHi.A) / 2;
}
}
// this is to ensure that the new point is within bounds
// and that the change is reasonably sized
Float ub = (minChange * left.A + (1 - minChange) * right.A);
if (alpha > ub)
{
alpha = ub;
}
Float lb = (minChange * right.A + (1 - minChange) * left.A);
if (alpha < lb)
{
alpha = lb;
}
VectorUtils.AddMultInto(ref _x, alpha, ref _dir, ref _newX);
if (EnforceNonNegativity)
{
VBufferUtils.Apply(ref _newX, delegate(int ind, ref Float newXval)
{
if (newXval < 0.0)
{
newXval = 0;
}
});
}
Value = Eval(ref _newX, ref _newGrad);
GradientCalculations++;
if (!FloatUtils.IsFinite(Value))
{
throw ch.Except("Optimizer unable to proceed with loss function yielding {0}", Value);
}
dirDeriv = VectorUtils.DotProduct(ref _dir, ref _newGrad);
PointValueDeriv curr = new PointValueDeriv(alpha, Value, dirDeriv);
if ((curr.V > LastValue + c1 * alpha) || (curr.V >= aLo.V))
{
if (aHi.A == curr.A)
{
if (force)
{
throw ch.Process(new PrematureConvergenceException(this, "Step size interval numerically zero."));
}
else
{
return(false);
}
}
aHi = curr;
}
else if (Math.Abs(curr.D) <= -c2)
{
return(true);
}
else
{
if (curr.D * (aHi.A - aLo.A) >= 0)
{
aHi = aLo;
}
if (aLo.A == curr.A)
{
if (force)
{
throw ch.Process(new PrematureConvergenceException(this, "Step size interval numerically zero."));
}
else
{
return(false);
}
}
aLo = curr;
}
}
}
/// <summary>
/// Tests the gradient reported by f.
/// </summary>
/// <param name="f">function to test</param>
/// <param name="x">point at which to test</param>
/// <param name="quiet">If false, outputs detailed info.</param>
/// <returns>maximum normalized difference between analytic and numeric directional derivative over multiple tests</returns>
public static Float Test(DifferentiableFunction f, ref VBuffer <Float> x, bool quiet)
{
// REVIEW: Delete this method?
VBuffer <Float> grad = default(VBuffer <Float>);
VBuffer <Float> newGrad = default(VBuffer <Float>);
VBuffer <Float> newX = default(VBuffer <Float>);
Float normX = VectorUtils.Norm(x);
f(ref x, ref grad, null);
if (!quiet)
{
Console.WriteLine(Header);
}
Float maxNormDiff = Float.NegativeInfinity;
int numIters = Math.Min((int)x.Length, 10);
int maxDirCount = Math.Min((int)x.Length / 2, 100);
for (int n = 1; n <= numIters; n++)
{
int dirCount = Math.Min(n * 10, maxDirCount);
List <int> indices = new List <int>(dirCount);
List <Float> values = new List <Float>(dirCount);
for (int i = 0; i < dirCount; i++)
{
int index = _r.Next((int)x.Length);
while (indices.IndexOf(index) >= 0)
{
index = _r.Next((int)x.Length);
}
indices.Add(index);
values.Add(SampleFromGaussian(_r));
}
VBuffer <Float> dir = new VBuffer <Float>(x.Length, values.Count, values.ToArray(), indices.ToArray());
Float norm = VectorUtils.Norm(dir);
VectorUtils.ScaleBy(ref dir, 1 / norm);
VectorUtils.AddMultInto(ref x, Eps, ref dir, ref newX);
Float rVal = f(ref newX, ref newGrad, null);
VectorUtils.AddMultInto(ref x, -Eps, ref dir, ref newX);
Float lVal = f(ref newX, ref newGrad, null);
Float dirDeriv = VectorUtils.DotProduct(ref grad, ref dir);
Float numDeriv = (rVal - lVal) / (2 * Eps);
Float normDiff = Math.Abs(1 - numDeriv / dirDeriv);
Float diff = numDeriv - dirDeriv;
if (!quiet)
{
Console.WriteLine("{0,-9}{1,-18:0.0000e0}{2,-18:0.0000e0}{3,-15:0.0000e0}{4,0:0.0000e0}", n, numDeriv, dirDeriv, diff, normDiff);
}
maxNormDiff = Math.Max(maxNormDiff, normDiff);
}
return(maxNormDiff);
}