private OlsLinearRegressionPredictor TrainCore(IChannel ch, FloatLabelCursor.Factory cursorFactory, int featureCount)
{
Host.AssertValue(ch);
ch.AssertValue(cursorFactory);
int m = featureCount + 1;
// Check for memory conditions first.
if ((long)m * (m + 1) / 2 > int.MaxValue)
{
throw ch.Except("Cannot hold covariance matrix in memory with {0} features", m - 1);
}
// Track the number of examples.
long n = 0;
// Since we are accumulating over many values, we use Double even for the single precision build.
var xty = new Double[m];
// The layout of this algorithm is a packed row-major lower triangular matrix.
var xtx = new Double[m * (m + 1) / 2];
// Build X'X (lower triangular) and X'y incrementally (X'X+=X'X_i; X'y+=X'y_i):
using (var cursor = cursorFactory.Create())
{
while (cursor.MoveNext())
{
var yi = cursor.Label;
// Increment first element of X'y
xty[0] += yi;
// Increment first element of lower triangular X'X
xtx[0] += 1;
var values = cursor.Features.GetValues();
if (cursor.Features.IsDense)
{
int ioff = 1;
ch.Assert(values.Length + 1 == m);
// Increment rest of first column of lower triangular X'X
for (int i = 1; i < m; i++)
{
ch.Assert(ioff == i * (i + 1) / 2);
var val = values[i - 1];
// Add the implicit first bias term to X'X
xtx[ioff++] += val;
// Add the remainder of X'X
for (int j = 0; j < i; j++)
{
xtx[ioff++] += val * values[j];
}
// X'y
xty[i] += val * yi;
}
ch.Assert(ioff == xtx.Length);
}
else
{
var fIndices = cursor.Features.GetIndices();
for (int ii = 0; ii < values.Length; ++ii)
{
int i = fIndices[ii] + 1;
int ioff = i * (i + 1) / 2;
var val = values[ii];
// Add the implicit first bias term to X'X
xtx[ioff++] += val;
// Add the remainder of X'X
for (int jj = 0; jj <= ii; jj++)
{
xtx[ioff + fIndices[jj]] += val * values[jj];
}
// X'y
xty[i] += val * yi;
}
}
n++;
}
ch.Check(n > 0, "No training examples in dataset.");
if (cursor.BadFeaturesRowCount > 0)
{
ch.Warning("Skipped {0} instances with missing features/label during training", cursor.SkippedRowCount);
}
if (_l2Weight > 0)
{
// Skip the bias term for regularization, in the ridge regression case.
// So start at [1,1] instead of [0,0].
// REVIEW: There are two ways to view this, firstly, it is more
// user friendly ot make this scaling factor behave similarly regardless
// of data size, so that if you have the same parameters, you get the same
// model if you feed in your data than if you duplicate your data 10 times.
// This is what I have now. The alternate point of view is to view this
// L2 regularization parameter as providing some sort of prior, in which
// case duplication 10 times should in fact be treated differently! (That
// is, we should not multiply by n below.) Both interpretations seem
// correct, in their way.
Double squared = _l2Weight * _l2Weight * n;
int ioff = 0;
for (int i = 1; i < m; ++i)
{
xtx[ioff += i + 1] += squared;
}
ch.Assert(ioff == xtx.Length - 1);
}
}
if (!(_l2Weight > 0) && n < m)
{
throw ch.Except("Ordinary least squares requires more examples than parameters. There are {0} parameters, but {1} examples. To enable training, use a positive L2 weight so this behaves as ridge regression.", m, n);
}
Double yMean = n == 0 ? 0 : xty[0] / n;
ch.Info("Trainer solving for {0} parameters across {1} examples", m, n);
// Cholesky Decomposition of X'X into LL'
try
{
Mkl.Pptrf(Mkl.Layout.RowMajor, Mkl.UpLo.Lo, m, xtx);
}
catch (DllNotFoundException)
{
// REVIEW: Is there no better way?
throw ch.ExceptNotSupp("The MKL library (libMklImports) or one of its dependencies is missing.");
}
// Solve for beta in (LL')beta = X'y:
Mkl.Pptrs(Mkl.Layout.RowMajor, Mkl.UpLo.Lo, m, 1, xtx, xty, 1);
// Note that the solver overwrote xty so it contains the solution. To be more clear,
// we effectively change its name (through reassignment) so we don't get confused that
// this is somehow xty in the remaining calculation.
var beta = xty;
xty = null;
// Check that the solution is valid.
for (int i = 0; i < beta.Length; ++i)
{
ch.Check(FloatUtils.IsFinite(beta[i]), "Non-finite values detected in OLS solution");
}
var weights = VBufferUtils.CreateDense <float>(beta.Length - 1);
for (int i = 1; i < beta.Length; ++i)
{
weights.Values[i - 1] = (float)beta[i];
}
var bias = (float)beta[0];
if (!(_l2Weight > 0) && m == n)
{
// We would expect the solution to the problem to be exact in this case.
ch.Info("Number of examples equals number of parameters, solution is exact but no statistics can be derived");
return(new OlsLinearRegressionPredictor(Host, in weights, bias, null, null, null, 1, float.NaN));
}
Double rss = 0; // residual sum of squares
Double tss = 0; // total sum of squares
using (var cursor = cursorFactory.Create())
{
var lrPredictor = new LinearRegressionPredictor(Host, in weights, bias);
var lrMap = lrPredictor.GetMapper <VBuffer <float>, float>();
float yh = default;
while (cursor.MoveNext())
{
var features = cursor.Features;
lrMap(in features, ref yh);
var e = cursor.Label - yh;
rss += e * e;
var ydm = cursor.Label - yMean;
tss += ydm * ydm;
}
}
var rSquared = ProbClamp(1 - (rss / tss));
// R^2 adjusted differs from the normal formula on account of the bias term, by Said's reckoning.
double rSquaredAdjusted;
if (n > m)
{
rSquaredAdjusted = ProbClamp(1 - (1 - rSquared) * (n - 1) / (n - m));
ch.Info("Coefficient of determination R2 = {0:g}, or {1:g} (adjusted)",
rSquared, rSquaredAdjusted);
}
else
{
rSquaredAdjusted = Double.NaN;
}
// The per parameter significance is compute intensive and may not be required for all practitioners.
// Also we can't estimate it, unless we can estimate the variance, which requires more examples than
// parameters.
if (!_perParameterSignificance || m >= n)
{
return(new OlsLinearRegressionPredictor(Host, in weights, bias, null, null, null, rSquared, rSquaredAdjusted));
}
ch.Assert(!Double.IsNaN(rSquaredAdjusted));
var standardErrors = new Double[m];
var tValues = new Double[m];
var pValues = new Double[m];
// Invert X'X:
Mkl.Pptri(Mkl.Layout.RowMajor, Mkl.UpLo.Lo, m, xtx);
var s2 = rss / (n - m); // estimate of variance of y
for (int i = 0; i < m; i++)
{
// Initialize with inverse Hessian.
standardErrors[i] = (Single)xtx[i * (i + 1) / 2 + i];
}
if (_l2Weight > 0)
{
// Iterate through all entries of inverse Hessian to make adjustment to variance.
int ioffset = 1;
float reg = _l2Weight * _l2Weight * n;
for (int iRow = 1; iRow < m; iRow++)
{
for (int iCol = 0; iCol <= iRow; iCol++)
{
var entry = (Single)xtx[ioffset];
var adjustment = -reg * entry * entry;
standardErrors[iRow] -= adjustment;
if (0 < iCol && iCol < iRow)
{
standardErrors[iCol] -= adjustment;
}
ioffset++;
}
}
Contracts.Assert(ioffset == xtx.Length);
}
for (int i = 0; i < m; i++)
{
// sqrt of diagonal entries of s2 * inverse(X'X + reg * I) * X'X * inverse(X'X + reg * I).
standardErrors[i] = Math.Sqrt(s2 * standardErrors[i]);
ch.Check(FloatUtils.IsFinite(standardErrors[i]), "Non-finite standard error detected from OLS solution");
tValues[i] = beta[i] / standardErrors[i];
pValues[i] = (float)MathUtils.TStatisticToPValue(tValues[i], n - m);
ch.Check(0 <= pValues[i] && pValues[i] <= 1, "p-Value calculated outside expected [0,1] range");
}
return(new OlsLinearRegressionPredictor(Host, in weights, bias, standardErrors, tValues, pValues, rSquared, rSquaredAdjusted));
}
