nnet-precondition.cc

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// nnet2/nnet-precondition.cc

// Copyright 2012   Johns Hopkins University (author: Daniel Povey)

// See ../../COPYING for clarification regarding multiple authors
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//  http://www.apache.org/licenses/LICENSE-2.0
//
// THIS CODE IS PROVIDED *AS IS* BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, EITHER EXPRESS OR IMPLIED, INCLUDING WITHOUT LIMITATION ANY IMPLIED
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// MERCHANTABLITY OR NON-INFRINGEMENT.
// See the Apache 2 License for the specific language governing permissions and
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#include "nnet2/nnet-precondition.h"

namespace kaldi {
namespace nnet2 {

void PreconditionDirections(const CuMatrixBase<BaseFloat> &R,
                            double lambda,
                            CuMatrixBase<BaseFloat> *P) {
  
  int32 N = R.NumRows(), D = R.NumCols();
  KALDI_ASSERT(SameDim(R, *P) && N > 0);
  if (N == 1) {
    KALDI_WARN << "Trying to precondition set of only one frames: returning "
               << "unchanged.  Ignore this warning if infrequent.";
    P->CopyFromMat(R);
    return;
  }
  CuMatrixBase<BaseFloat> &Q = *P;
  
  if (N >= D) {
    // Compute G = (\lambda I + 1/(N-1) R^T R)^{-1} by direct inversion.
    // G <-- lambda I.
    CuMatrix<BaseFloat> G(D, D);
    G.AddToDiag(lambda);
    // G += 1.0/(N-1) * R^T R.
    G.SymAddMat2(1.0 / (N-1), R, kTrans, 1.0);
    G.CopyLowerToUpper();
    if (GetVerboseLevel() >= 5 && Rand() % 20 == 0) {
      CuSpMatrix<BaseFloat> tmp(G, kTakeLower);
      SpMatrix<BaseFloat> G_cpu(tmp);
      G_cpu.PrintEigs("G");
    }
    G.SymInvertPosDef();
    // Q <-- R G^T (we just make it transposed as we think
    // it will be slightly faster; it's symmetric).
    Q.AddMatMat(1.0, R, kNoTrans, G, kTrans, 0.0);
  } else {
    // Through a lot of rearrangements, it turns out that
    // if we let  S = (\lambda I + 1/(N-1) R R^T)^{-1}
    // then what we need is
    // Q <-- S R.
    // It is curious and (to me) unexpected that the actual code is basically
    // the same when transposed.
    CuMatrix<BaseFloat> S(N, N);
    // S <-- lambda I.
    S.AddToDiag(lambda);
    // S += (N-1) R R^T.
    // the following function only updates the lower triangle.
    S.SymAddMat2(1.0 / (N-1), R, kNoTrans, 1.0);
    S.CopyLowerToUpper();
    // invert S, so now S = (\lambda I + (N-1) R R^T)^{-1}.
    if (GetVerboseLevel() >= 5 && Rand() % 20 == 0) {
      CuSpMatrix<BaseFloat> tmp(S, kTakeLower);
      SpMatrix<BaseFloat> S_cpu(tmp);
      S_cpu.PrintEigs("S");
    }
    S.SymInvertPosDef();
    Q.AddMatMat(1.0, S, kNoTrans, R, kNoTrans, 0.0);
  }

#if 0  // Old code before it was optimized for CUDA:
  for (int32 n = 0; n < N; n++) {
    CuSubVector<BaseFloat> r(R, n), q(Q, n);
    BaseFloat gamma = VecVec(r, q), // gamma_n = r_n^T q_n.
               beta = 1 + gamma / (N - 1 - gamma);
    if (!(gamma >= 0.0 && beta > 0.0)) {
      KALDI_ERR << "Bad values encountered in preconditioning: gamma = " << gamma
                << ", beta = " << beta;
    }
    // Q and P share the same memory.  The result of the
    // scaling below will be output as P.
    q.Scale(beta);
  }
#else
  CuVector<BaseFloat> gamma(N);
  gamma.AddDiagMatMat(1.0, R, kNoTrans, Q, kTrans, 0.0);
  // at this point, gamma(i) equals the i'th row of R dotted with
  // the i'th row of Q.
  Vector<BaseFloat> cpu_gamma(gamma), cpu_beta(N, kUndefined);
  for (int32 n = 0; n < N; n++) {
    BaseFloat this_gamma = cpu_gamma(n),
        this_beta = 1.0 + this_gamma / (N - 1 - this_gamma);
    if (!(this_gamma >= 0.0 && this_beta > 0.0))
      KALDI_ERR << "Bad values encountered in preconditioning: gamma = "
                << this_gamma << ", beta = " << this_beta;
    cpu_beta(n) = this_beta;
  }
  CuVector<BaseFloat> beta(cpu_beta);
  P->MulRowsVec(beta);
#endif
}


void PreconditionDirectionsAlpha(
    const CuMatrixBase<BaseFloat> &R,
    double alpha,
    CuMatrixBase<BaseFloat> *P) {
  KALDI_ASSERT(alpha > 0.0);
  // probably does not really make sense.
  double t = TraceMatMat(R, R, kTrans), floor = 1.0e-20;
  if (t < floor) {
    KALDI_WARN << "Flooring trace from " << t
               << " to " << floor;
    t = floor;
  }
  double lambda = t * alpha / R.NumRows() / R.NumCols();
  // see the extended comment below for an explanation of this.
  if (lambda <= 0.0) {
    // This should never really happen, it would probably indicate a bug
    // in the calling code.
    KALDI_WARN << "Zero or negative lambda in PreconditionDirectionsAlpha.";
    lambda = 1.0e-10;
  }
  PreconditionDirections(R, lambda, P);
}


void PreconditionDirectionsAlphaRescaled(
    const CuMatrixBase<BaseFloat> &R,
    double alpha,
    CuMatrixBase<BaseFloat> *P) {
  KALDI_ASSERT(alpha > 0.0); // alpha > 1.0
  // probably does not really make sense.
  double t = TraceMatMat(R, R, kTrans), floor = 1.0e-20;
  if (t == 0.0) {
    P->CopyFromMat(R);
    return;
  }
  if (t < floor) {
    KALDI_WARN << "Flooring trace from " << t
               << " to " << floor;
    t = floor;
  }
  double lambda = t * alpha / R.NumRows() / R.NumCols();
  // see the extended comment below for an explanation of this.
  KALDI_ASSERT(lambda != 0.0);
  PreconditionDirections(R, lambda, P);
  double p_trace = TraceMatMat(*P, *P, kTrans),
      rescale = sqrt(t / p_trace);
  KALDI_ASSERT(p_trace != 0.0);
  P->Scale(rescale);
}


} // namespace nnet2
} // namespace kaldi

/*
  Notes for an idea on preconditioning.
  update is of form:
     params += learning_rate * input_row * output_deriv'
  want to precondition by fisher-like matrix in each of (the input dim and the
  output dim).
  [note: in this method we'll pretend the chunk-weights are all one.
   It shouldn't really matter, it's only preconditioning.]

   The first observation is, if we do:

    params += learning_rate * S * input_row * output_deriv' * T

   for any positive definite S and T that we choose (well, perhaps we have
   to ensure their eigenvalues are bounded in some way, but we'll bother with
   that later),  then we'll still get convergence.  But S and T cannot be
   functions of the current sample, the one that creates "input_row" and
   "output_deriv", or this introduces a bias.

   We can view it as a preconditioning of the vectorized form of the
   transformation matrix.

   For a Fisher-like preconditioning, we can precondition using
   the inverse of the scatter of the other features in the batch.
   For the input_row, call this r_j.

   Let the total scatter be

    S =  \sum_n r_n r_n^T
  where the sum is taken over the minibatch, and
   S_n = S - r_n  r_n^T
  i.e. the scatter with this sample removed.
  Let F_n be the normalized version of this, dividing by the #samples.
   F_n = 1/(N-1) S_n
  where N is the minibatch size (so N-1 is excluding the current sample).
 We're going to want to invert F_n, so we need to make it positive definite.

  We're going to define G_n as a smoothed form of the estimated Fisher matrix
  for this batch:
   G_n = F_n + \lambda_n I
  where I is the identity.  A suitable formula for \lambda_n is to define
  a small constant \alpha (say, \alpha=0.1), and let
  
   \lambda_n =  (\alpha/dim(F)) trace(F_n) .

  In practice (although we lost strict convergence guarantees) it will be easier
  to set a global \lambda, to:

   \lambda  =  (\alpha/dim(S)) trace(S)
            = (\alpha/(R.NumRows()*R.NumCols()) * trace(R^T R)).
  
  This is an easy way to set it.  Let's define P_n as the inverse of G_n.  This
  is what we'll be multiplying the input values by:

    P_n = G_n^{-1} = (F_n + \lambda_n I)^{-1}

  First, let's define an uncorrected "global" Fisher matrix
    F = (1/(N-1)) S_n,
  and G = F^{-1}.
  If we let R be the matrix each of whose rows is one of the r_n,
  then
    S = R^T R, and
   F = 1/(N-1) R^T R

           G = (F + \lambda I)^{-1}
             = (1/(N-1) R^T R + \lambda I)^{-1}
Using the Woodbury formula,
     G  = (1/\lambda) I  - (1/\lambda^2) R^T M R
where
  M = ((N-1) I + 1/\lambda R R^T)^{-1}
(and this inversion for M is actually done as an inversion, in a lower
 dimension such as 250, versus the actual dimension which might be 1000).

Let's assume \lambda is a constant, i.e. there is no \lambda_n.
We can get it from the previous minibatch.

 We want to compute

    G_n = F_n^{-1} = (F - 1/(N-1) r_n r_n^T)^{-1}

 and using the Sherman-Morrison formula, this may be written as:

   G_n = G  +  \alpha_n q_n q_n^T  # Caution: \alpha_n has nothing to do with \alpha.

 where q_n = G r_n, and

 \alpha_n =  1/( (N-1) (1 - 1/(N-1) r_n^T q_n) )
          =  1 / (N - 1 - r_n^T q_n)

  We'll want to compute this efficiently.  For each r_n we'll want to compute

 p_n =  G_n r_n

 which will correspond to the direction we update in.
 We'll use

  p_n = G r_n + \alpha_n q_n q_n^T r_n

  and since q_n = G r_n, both terms in this equation point in
  the same direction, and we can write this as:

  p_n = \beta_n q_n,

  where, defining \gamma_n = r_n^T q_n, we have

  \beta_n = 1 + \gamma_n \alpha_n 
          = 1  +  \gamma_n / ((N-1) (1 - \gamma_n/(N-1)))
          = 1  +  \gamma_n / (N - 1 - \gamma_n)
  
*/

/*

  SUMMARY:
   let the input features (we can extend these with a 1 for the bias term) be
   a matrix R, each row of which corresponds to a training example r_n

   The dimension of R is N x D, where N is the minibatch size and D is the
   dimension of the input to this layer of the network.

   We'll be computing a matrix P, each row p_n of which will be the corresponding
   row r_n of R, multiplied by a positive definite preconditioning matrix G_n.
   [we can check that for each i, p_n^T r_n >= 0].
   The following computation obtains P:

   \lambda <-- (\alpha/N) \trace(R R^T).   # 0 < \alpha <= 1 is a global constant, e.g.
                                           # \alpha = 0.1, but should try different
                                           # values, this will be important (note: if the
                                           # minibatch size is >= the dimension (N >= D),
                                           # then we can let \alpha be quite small, e.g.
                                           # 0.001.

   if N >= D, then
     # compute G by direct inversion.
     G <-- (\lambda I  +  1/(N-1) R^T R)^{-1}
     Q <-- R G.
   else   # number of samples is less than dimension, use
          # morrison-Woodbury formula, it's more efficient.
      # We'd first compute:
      # L <-- ((N-1) I + 1/\lambda R R^T)
      # (note: L is something that appears in the morrison-Woodbury expansion of G)
      # M <-- L^{-1}
      # Note: G is  1/\lambda I  -  (1/\lambda^2) R^T M R
      # We're doing Q <-- R G, which is:
      # Q <-- 1/\lambda R - (1/\lambda^2) R (R^T M R)
      # It's more efficient in this case to left-multiply R
      # by something, i.e. bracket as:
      # Q <-- 1/\lambda R - (1/\lambda^2) (R R^T M) R
      # so let's write it as
      # Q <-- S R, with
      # S = 1/\lambda I - 1/\lambda^2 R R^T M
      #   = 1/\lambda (I - 1/\lambda R R^T M)
      # Now, -1/\lambda R R^T = (N-1) I - L, and L M = I, so
      # S = 1/\lambda (I  + ((N-1) I - L) M)
      #   = (N-1)/\lambda M
      # and we can get rid of that scalar earlier on:
      # if we let L' = \lambda/(N-1) L, so that
      # L' = (lambda I + 1/(N-1) R R^T)
      # then
      # S = (\lambda I + 1/(N-1) R R^T)^{-1}. 

      S <-- (\lambda I + 1/(N-1) R R^T)^{-1}.
      Q <-- S R
   fi

   Here, we're right multiplying each row r_n of r by the symmetric matrix G, to get
   the corresponding row q_n of q.  Note: in practice Q will be the same memory as P.
   Next we work out for each n:
     \gamma_n = r_n^T q_n     # This should be nonnegative!  Check this.
      \beta_n = 1  +  \gamma_n / (N - 1 - \gamma_n)  # This should be positive; check this.
  For each n, we'll do (for the corresponding rows of P and Q):
     p_n <-- \beta_n q_n.
  In practice, we'd do this computation in-place, with P and Q using the
  same memory.

  If we're being paranoid, we should verify that

   p_n = (\lambda I  +  1/(N-1) \sum_{m != n} r_n r_n^T)^{-1} r_n.

  This is exact mathematically, but there could be differences due to roundoff,
  and if \alpha is quite small, these differences could be substantial.
  
 */