Factorized TDNN（因子分解TDNN，TDNN-F）

论文

Povey, D., Cheng, G., Wang, Y., Li, K., Xu, H., Yarmohamadi, M., & Khudanpur, S. (2018). Semi-orthogonal low-rank matrix factorization for deep neural networks. In Proceedings of the 19th Annual Conference of the International Speech Communication Association (INTERSPEECH 2018), Hyderabad, India.

Kaldi recipe

swbd/s5c/local/chain/tuning/run_tdnn_7q.sh

   

   

A factorized TDNN has a similar structure as a vanilla TDNN,

except the weight matrices (of the layers) are factorized (using

SVD) into two factors, with one of them constrained to be

semi-orthonormal.

   

   

论文笔记

TDNN又被称为1维CNN（1-d CNNs）。本文提出的TDNN-F，结构与经过SVD分解的TDNN相同。但TDNN-F的训练开始于随机初始化，SVD分解后，其中一个矩阵被限制为半正定的。这对TDNNs以及TDNN-LSTM有实质上的提升。

   

一种减少已训练模型大小的方法是使用奇异值分解（Singular Value Decomposition，SVD）对每个权重矩阵因子分解为两个更小的因子，丢弃更小的奇异值；然后，再次对网络参数进行调优。

   

很明显，直接训练随机初始化后的上述带有线性瓶颈层的网络会更为高效。虽然，有人以这一方法训练成功，但还是出现了训练不稳定的情况。

   

之中的一个矩阵限制为半正定矩阵不会损失任何建模能力；并且，这一限制也符合SVD的结果（即，SVD分解后，其中一个子矩阵也是半正定的）。

   

此外，受到"dense LSTM"的启发，本文使用了跳层连接（skip connection）。这一定程度上类似于残差学习的捷径连接（shortcut connection）和公路连接（highway connection）。

   

前向计算

nnet3/nnet-training.cc
void NnetTrainer::TrainInternal(const NnetExample &eg,

const NnetComputation &computation) {

// note: because we give the 1st arg (nnet_) as a pointer to the

// constructor of 'computer', it will use that copy of the nnet to

// store stats.

NnetComputer computer(config_.compute_config, computation,

nnet_, delta_nnet_);

// give the inputs to the computer object.

computer.AcceptInputs(*nnet_, eg.io);

computer.Run();

   

this->ProcessOutputs(false, eg, &computer);

computer.Run();

   

// If relevant, add in the part of the gradient that comes from L2

// regularization.

ApplyL2Regularization(*nnet_,

GetNumNvalues(eg.io, false) * config_.l2_regularize_factor,

delta_nnet_);

   

// Update the parameters of nnet

bool success = UpdateNnetWithMaxChange(*delta_nnet_, config_.max_param_change,

1.0, 1.0 - config_.momentum, nnet_,

&num_max_change_per_component_applied_, &num_max_change_global_applied_);

   

// Scale down the batchnorm stats (keeps them fresh... this affects what

// happens when we use the model with batchnorm test-mode set).

ScaleBatchnormStats(config_.batchnorm_stats_scale, nnet_);

   

// The following will only do something if we have a LinearComponent

// or AffineComponent with orthonormal-constraint set to a nonzero value.

ConstrainOrthonormal(nnet_);

   

// Scale deta_nnet

if (success)

ScaleNnet(config_.momentum, delta_nnet_);

else

ScaleNnet(0.0, delta_nnet_);

}

  

   

nnet3/nnet-utils.cc
该函数在处理完每个minibatch后被调用一次，用于对参数orthonormal-constraint非零的LinearComponent或继承于AffineComponent的组件实施正交性约束。

对参数矩阵M所做的是将其限制为一个"半正交"矩阵乘以一个常数alpha。也就是说：

   

如果对于某个组件，orthonormal-constraint > 0.0，那么该参数就变为上述alpha。若orthonormal-constraint == 0.0，则什么都不做。若orthonormal-constraint < 0.0，那么使得alpha浮动，也就是说，试图使得M接近于一alpha乘以一个半正交矩阵。
 

为了确保该操作在GPU上的有效性，这里不会使得矩阵M完全正交，只是使其更接近于正交（乘以'orthonormal_constraint'）。在多次iterations后，该操作会使得矩阵M十分接近与正交矩阵。

void ConstrainOrthonormal(Nnet *nnet) {

   

for (int32 c = 0; c < nnet->NumComponents(); c++) {

Component *component = nnet->GetComponent(c);

CuMatrixBase<BaseFloat> *params = NULL;

BaseFloat orthonormal_constraint = 0.0;

   

LinearComponent *lc = dynamic_cast<LinearComponent*>(component);

if (lc != NULL && lc->OrthonormalConstraint() != 0.0) {

orthonormal_constraint = lc->OrthonormalConstraint();

params = &(lc->Params());

}

AffineComponent *ac = dynamic_cast<AffineComponent*>(component);

if (ac != NULL && ac->OrthonormalConstraint() != 0.0) {

orthonormal_constraint = ac->OrthonormalConstraint();

params = &(ac->LinearParams());

}

TdnnComponent *tc = dynamic_cast<TdnnComponent*>(component);

if (tc != NULL && tc->OrthonormalConstraint() != 0.0) {

orthonormal_constraint = tc->OrthonormalConstraint();

params = &(tc->LinearParams());

}

if (orthonormal_constraint == 0.0 || RandInt(0, 3) != 0) {

// For efficiency, only do this every 4 or so minibatches-- it won't have

// time stray far from the constraint in between.

continue;

}

   

int32 rows = params->NumRows(), cols = params->NumCols();

if (rows <= cols) {

ConstrainOrthonormalInternal(orthonormal_constraint, params);

} else {

CuMatrix<BaseFloat> params_trans(*params, kTrans);

ConstrainOrthonormalInternal(orthonormal_constraint, &params_trans);

params->CopyFromMat(params_trans, kTrans);

}

}

}

   

对矩阵M做一个更新，使其更接近与一个正交矩阵（带有正交行的矩阵）乘以'scale'。注意：若'scale'离奇异值太远，则可能会发散。

void ConstrainOrthonormalInternal(BaseFloat scale, CuMatrixBase<BaseFloat> *M) {

KALDI_ASSERT(scale != 0.0);

   

// We'd like to enforce the rows of M to be orthonormal.

// define P = M M^T. If P is unit then M has orthonormal rows.

// We actually want P to equal scale^2 * I, so that M's rows are

// orthogonal with 2-norms equal to 'scale'.

// We (notionally) add to the objective function, the value

// -alpha times the sum of squared elements of Q = (P - scale^2 * I).

int32 rows = M->NumRows(), cols = M->NumCols();

CuMatrix<BaseFloat> M_update(rows, cols);

CuMatrix<BaseFloat> P(rows, rows);

P.SymAddMat2(1.0, *M, kNoTrans, 0.0);

P.CopyLowerToUpper();

   

// The 'update_speed' is a constant that determines how fast we approach a

// matrix with the desired properties (larger -> faster). Larger values will

// update faster but will be more prone to instability. 0.125 (1/8) is the

// value that gives us the fastest possible convergence when we are already

// close to be a semi-orthogonal matrix (in fact, it will lead to quadratic

// convergence).

// See http://www.danielpovey.com/files/2018_interspeech_tdnnf.pdf

// for more details.

BaseFloat update_speed = 0.125;

bool floating_scale = (scale < 0.0);

   

   

if (floating_scale) {

// This (letting the scale "float") is described in Sec. 2.3 of

// http://www.danielpovey.com/files/2018_interspeech_tdnnf.pdf,

// where 'scale' here is written 'alpha' in the paper.

//

// We pick the scale that will give us an update to M that is

// orthogonal to M (viewed as a vector): i.e., if we're doing

// an update M := M + X, then we want to have tr(M X^T) == 0.

// The following formula is what gives us that.

// With P = M M^T, our update formula is doing to be:

// M := M + (-4 * alpha * (P - scale^2 I) * M).

// (The math below explains this update formula; for now, it's

// best to view it as an established fact).

// So X (the change in M) is -4 * alpha * (P - scale^2 I) * M,

// where alpha == update_speed / scale^2.

// We want tr(M X^T) == 0. First, forget the -4*alpha, because

// we don't care about constant factors. So we want:

// tr(M * M^T * (P - scale^2 I)) == 0.

// Since M M^T == P, that means:

// tr(P^2 - scale^2 P) == 0,

// or scale^2 = tr(P^2) / tr(P).

// Note: P is symmetric so it doesn't matter whether we use tr(P P) or

// tr(P^T P); we use tr(P^T P) because I believe it's faster to compute.

   

BaseFloat trace_P = P.Trace(), trace_P_P = TraceMatMat(P, P, kTrans);

   

scale = std::sqrt(trace_P_P / trace_P);

   

// The following is a tweak to avoid divergence when the eigenvalues aren't

// close to being the same. trace_P is the sum of eigenvalues of P, and

// trace_P_P is the sum-square of eigenvalues of P. Treat trace_P as a sum

// of positive values, and trace_P_P as their sumsq. Then mean = trace_P /

// dim, and trace_P_P cannot be less than dim * (trace_P / dim)^2,

// i.e. trace_P_P >= trace_P^2 / dim. If ratio = trace_P_P * dim /

// trace_P^2, then ratio >= 1.0, and the excess above 1.0 is a measure of

// how far we are from convergence. If we're far from convergence, we make

// the learning rate slower to reduce the risk of divergence, since the

// update may not be stable for starting points far from equilibrium.

BaseFloat ratio = (trace_P_P * P.NumRows() / (trace_P * trace_P));

KALDI_ASSERT(ratio > 0.999);

if (ratio > 1.02) {

update_speed *= 0.5; // Slow down the update speed to reduce the risk of divergence.

if (ratio > 1.1) update_speed *= 0.5; // Slow it down even more.

}

}

P.AddToDiag(-1.0 * scale * scale);

   

// We may want to un-comment the following code block later on if we have a

// problem with instability in setups with a non-floating orthonormal

// constraint.

/*

if (!floating_scale) {

// This is analogous to the stuff with 'ratio' above, but when we don't have

// a floating scale. It reduces the chances of divergence when we have

// a bad initialization.

BaseFloat error = P.FrobeniusNorm(),

error_proportion = error * error / P.NumRows();

// 'error_proportion' is the sumsq of elements in (P - I) divided by the

// sumsq of elements of I. It should be much less than one (i.e. close to

// zero) if the error is small.

if (error_proportion > 0.02) {

update_speed *= 0.5;

if (error_proportion > 0.1)

update_speed *= 0.5;

}

}

*/

   

if (GetVerboseLevel() >= 1) {

BaseFloat error = P.FrobeniusNorm();

KALDI_VLOG(2) << "Error in orthogonality is " << error;

}

   

// see Sec. 2.2 of http://www.danielpovey.com/files/2018_interspeech_tdnnf.pdf

// for explanation of the 1/(scale*scale) factor, but there is a difference in

// notation; 'scale' here corresponds to 'alpha' in the paper, and

// 'update_speed' corresponds to 'nu' in the paper.

BaseFloat alpha = update_speed / (scale * scale);

   

// At this point, the matrix P contains what, in the math, would be Q =

// P-scale^2*I. The derivative of the objective function w.r.t. an element q(i,j)

// of Q is now equal to -2*alpha*q(i,j), i.e. we could write q_deriv(i,j)

// = -2*alpha*q(i,j) This is also the derivative of the objective function

// w.r.t. p(i,j): i.e. p_deriv(i,j) = -2*alpha*q(i,j).

// Suppose we have define this matrix as 'P_deriv'.

// The derivative of the objective w.r.t M equals

// 2 * P_deriv * M, which equals -4*alpha*(P-scale^2*I)*M.

// (Currently the matrix P contains what, in the math, is P-scale^2*I).

M_update.AddMatMat(-4.0 * alpha, P, kNoTrans, *M, kNoTrans, 0.0);

M->AddMat(1.0, M_update);

}