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Improved Jacobian Eigen-Analysis Scheme for Accelerating Learning in Feedforward Neural Networks

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Abstract

An important problem in the learning process when training feedforward artificial neural networks is the occurrence of temporary minima which considerably slows down learning convergence. In a series of previous works, we analyzed this problem by deriving a dynamical system model which is valid in the vicinity of temporary minima caused by redundancy of nodes in the hidden layer. We also demonstrated how to incorporate the characteristics of the dynamical model into a constrained optimization algorithm that allows prompt abandonment of temporary minima and acceleration of learning. In this work, we revisit the constrained optimization framework in order to develop a closed-form solution for the evolution of critical dynamical system model parameters during learning in the vicinity of temporary minima. We show that this formalism is equivalent to matrix perturbation theory which was discussed in a previous work, but that the closed-form solution presented in the present paper allows for a weight update rule which is linear to the number of the network’s weights. In terms of computational complexity, this is equivalent to that of the simple back-propagation weight update rule. Simulations demonstrate the computational efficiency and effectiveness of this approach in reducing the time spent in the vicinity of temporary minima as suggested by the analysis.

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Acknowledgments

Nicholas Ampazis would like to express his gratitude to his supervisor late Professor John G. Taylor for the guidance, patience, and continuous encouragement. John has been a source of inspiration to everyone that had the privilege to meet him and has left a shining imprint on the scientific community.

Author information

Authors and Affiliations

  1. Department of Financial and Management Engineering, University of the Aegean, 82100, Chios, Greece

    N. Ampazis & D. Drivaliaris

  2. Institute of Informatics and Telecommunications, National Center for Scientific Research “Demokritos”, Athens, Greece

    S. J. Perantonis

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  1. N. Ampazis
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  2. S. J. Perantonis
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  3. D. Drivaliaris
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Corresponding author

Correspondence to N. Ampazis.

Appendix

Appendix

Let

$$\begin{aligned} {{\varvec{J}}}&= \sum _p(d^{(p)}-y^{(p)})y^{(p)}(1-y^{(p)})f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})(1-f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\\&\quad \left( \begin{array}{cc} (1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})){\varvec{x^{(p)}}}({\varvec{x^{(p)}}})^{T}\nu &{} {\varvec{x^{(p)}}} \\ ({\varvec{x^{(p)}}})^{T} &{} 0\end{array}\right) \end{aligned}$$

with

$$\begin{aligned} y^{(p)}=f(b+K\nu f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})) \end{aligned}$$

and

$$\begin{aligned} {{\varvec{x^{(p)}}}}=\left( \begin{array}{c} x_1^{(p)}\\ x_2^{(p)}\\ \vdots \\ x_N^{(p)} \end{array} \right) \end{aligned}$$

If we let

$$\begin{aligned} A^{(p)}=(d^{(p)}-y^{(p)})y^{(p)}(1-y^{(p)})f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})(1-f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})) \end{aligned}$$

and

$$\begin{aligned} \lambda ^{(p)}=(1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\nu \end{aligned}$$

then, we can write

$$\begin{aligned} {{\varvec{J}}}&= \sum _pA^{(p)}\left( \begin{array}{ccc} \left( \lambda ^{(p)} x_i^{(p)}x_j^{(p)}\right) _{1\le i,j\le N}&{}&{}{{\varvec{x^{(p)}}}}\\ &{}&{}\\ ({\varvec{x^{(p)}}})^{T}&{}&{}0 \end{array} \right) \end{aligned}$$

Let

$$\begin{aligned} {{\varvec{u}}}=\left( \begin{array}{c} u_1\\ u_2\\ \vdots \\ u_N\\ u_{N+1} \end{array} \right) \end{aligned}$$

A simple calculation shows that

$$\begin{aligned} {{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}}&= \displaystyle {\sum _p\left( \sum _{i=1}^Nu_ix_i^{(p)}\right) A^{(p)}\left( \lambda ^{(p)}\left( \sum _{j=1}^Nu_jx_j^{(p)}\right) +2u_{N+1}\right) } \nonumber \\ &= \displaystyle {\sum _p \left({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})A^{(p)}(\lambda ^{(p)}({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})+2u_{N+1}\right)} \end{aligned}$$
(43)

where

$$\begin{aligned} {{\tilde{\varvec{x}^{(p)}}}}=\left( \begin{array}{c} x_1^{(p)}\\ x_2^{(p)}\\ \vdots \\ x_N^{(p)}\\ 0 \end{array} \right) \end{aligned}$$

We want to calculate

$$\begin{aligned} \frac{\partial }{\partial \nu }({{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}})\quad \hbox { and }\quad \frac{\partial }{\partial {\varvec{\omega }}}({{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}}) \end{aligned}$$

Recall that

$$\begin{aligned} \frac{\text{d}f}{\text{d}t}(t)=f(t)(1-f(t))=f(t)-f^2(t) \end{aligned}$$
(44)

Using simple properties of the derivatives and (44), we get that

$$\begin{aligned} \frac{\partial y^{(p)}}{\partial \nu }=K(y^{(p)}-(y^{(p)})^2)f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}) \end{aligned}$$

which, using again simple properties of the derivatives, gives

$$\begin{aligned}&\displaystyle {\frac{\partial }{\partial \nu }}\left(d^{(p)}y^{(p)}-(1+d^{(p)})(y^{(p)}\right)^2+(y^{(p)})^3)\nonumber \\ &\quad = K\left(d^{(p)}-2(1+d^{(p)})y^{(p)} +3(y^{(p)})^2\right)\left(y^{(p)}-(y^{(p)}\right)^2)f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}) \end{aligned}$$

which implies that

$$\begin{aligned} \displaystyle {\frac{\partial A^{(p)}}{\partial \nu }}&= K \left(d^{(p)}-2(1+d^{(p)})y^{(p)}+3(y^{(p)})^2\right)\left(y^{(p)}-(y^{(p)})^2\right)\nonumber \\ &\quad f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})) \end{aligned}$$
(45)

On the other hand, we obviously have that

$$\begin{aligned} \frac{\partial \lambda ^{(p)}}{\partial \nu } =1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}) \end{aligned}$$
(46)

Using (45) and (46) and simple properties of the derivatives, we get that

$$\begin{aligned} \displaystyle {\frac{\partial }{\partial \nu }({{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}})}&= \displaystyle {\sum _p} \left({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})(y^{(p)}-(y^{(p)}\right)^2(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\nonumber \\&\qquad ((d^{(p)}-y^{(p)})(1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})\nonumber \\&\quad +K(d^{(p)}-2(1+d^{(p)})y^{(p)}+3(y^{(p)})^2)f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})\nonumber \\&\qquad (\nu (1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})+2u_{N+1})) \end{aligned}$$
(47)

Let

$$\begin{aligned} \delta ^{(p)}&= (y^{(p)}-(y^{(p)})^2)(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\end{aligned}$$
(48)
$$\begin{aligned} \rho ^{(p)}&= K (d^{(p)}-2(1+d^{(p)})y^{(p)}+3(y^{(p)})^2)f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})\nonumber \\&\quad (\nu (1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})+2u_{N+1}) \end{aligned}$$
(49)

Combining (47), (48), and (49), we get that

$$\begin{aligned}&\displaystyle {\frac{\partial }{\partial \nu }({{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}})}= \displaystyle {\sum _p}({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})\delta ^{(p)}\nonumber \\&\quad ((d^{(p)}-y^{(p)})(1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})+\rho ^{(p)}) \end{aligned}$$
(50)

For the evaluation of \(\frac{\partial }{\partial {\varvec{\omega }}}({{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}})\) , we have:

Using simple properties of the derivatives and (44), we get that

$$\begin{aligned} \frac{\partial f}{\partial {\varvec{\omega }}}({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})=(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})){{\varvec{x^{(p)}}}} \end{aligned}$$
(51)

which, using again simple properties of the derivatives and (44), gives

$$\begin{aligned} \frac{\partial y^{(p)}}{\partial {\varvec{\omega }}}=K\nu (y^{(p)}-(y^{(p)})^2)(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})){{\varvec{x^{(p)}}}} \end{aligned}$$

which implies that

$$\begin{aligned}&\displaystyle {\frac{\partial }{\partial {\varvec{\omega }}}}(d^{(p)}y^{(p)}-(1+d^{(p)})(y^{(p)})^2+(y^{(p)})^3)\nonumber \\&\quad = K\nu (d^{(p)}-2(1+d^{(p)})y^{(p)}+3(y^{(p)})^2) (y^{(p)}\nonumber \\&\qquad -(y^{(p)})^2)(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})){{\varvec{x^{(p)}}}} \end{aligned}$$
(52)

Using simple properties of the derivatives and (51), we get that

$$\begin{aligned} \displaystyle {\frac{\partial }{\partial {\varvec{\omega }}}}(f({{\varvec{\omega }}} \cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))&= (1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})\nonumber \\&\quad -f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})){{\varvec{x^{(p)}}}} \end{aligned}$$
(53)

Using simple properties of the derivatives and (52) and (53), we get that

$$\begin{aligned}\displaystyle {\frac{\partial A^{(p)}}{\partial {\varvec{\omega }}}}&=(y^{(p)}-(y^{(p)})^2)(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))((d^{(p)}-y^{(p)})(1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\nonumber \\&\quad +K\nu (d^{(p)}-2(1+d^{(p)})y^{(p)}+3(y^{(p)})^2)(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))){{\varvec{x^{(p)}}}} \end{aligned}$$
(54)

On the other hand, using simple properties of the derivatives and (51), we get that

$$\begin{aligned} \frac{\partial \lambda ^{(p)}}{\partial {\varvec{\omega }}} =-2\nu (f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})){{\varvec{x^{(p)}}}} \end{aligned}$$
(55)

Using (54) and (55) and simple properties of the derivatives, we get that

$$\begin{aligned} \displaystyle {\frac{\partial }{\partial {\varvec{\omega }}}({{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}})}&= \displaystyle {\sum _p}({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})(y^{(p)}-(y^{(p)})^2)(f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})-f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\nonumber \\&\quad (K(d^{(p)}-2(1+d^{(p)})y^{(p)}+3(y^{(p)})^2)f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})\nonumber \\&\quad (\nu (1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})+2u_{N+1}) \nu (1-f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\nonumber \\&\quad +(d^{(p)}-y^{(p)}) (\nu (1-6f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})+6f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})\nonumber \\&\quad +2(1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))u_{N+1})){{\varvec{x^{(p)}}}} \end{aligned}$$
(56)

Combining (48), (49) and (56), we get that

$$\begin{aligned} \displaystyle {\frac{\partial }{\partial {\varvec{\omega }}}({{\varvec{u}}}^T{{\varvec{J}}}{{\varvec{u}}})}&= \displaystyle {\sum _p}({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})\delta ^{(p)}(\rho ^{(p)}\nu (1-f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))\nonumber \\&\quad +(d^{(p)}-y^{(p)}) (\nu (1-6f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}})+6f^2({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))({{\varvec{u}}}^T\cdot {{\tilde{\varvec{x}^{(p)}}}})\nonumber \\&\quad +2(1-2f({{\varvec{\omega }}}\cdot {{\varvec{x^{(p)}}}}))u_{N+1})){{\varvec{x^{(p)}}}} \end{aligned}$$
(57)

For a Jacobian matrix of size \(Q \times Q\), it follows that there are Q derivatives to evaluate. By simply taking the expression for the differential of Eq. (27) and evaluating it at small perturbations by finite differences, we would have to evaluate Q such terms (one for each component of the state vector \({{\varvec{\varOmega }}}\)), each requiring O(Q) operations. Thus, the total computational effort required to evaluate all the derivatives scales as \(O(Q^2)\).

Equations (50) and (57) allow the derivatives to be propagated back and evaluated in O(Q) operations similarly to the back-propagation algorithm which requires O(W) operations, where W is the number of the network’s weights [9]. This follows from the fact that both the forward phase [evaluation of Eq. (43)] and the backward propagation phases are O(Q), and the evaluation of the derivatives require O(Q) operations. Thus the closed-form solution has reduced the computational complexity from \(O(Q^2)\) of the finite differences method to O(Q) for each input vector, which results in a significant computational gain.

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Ampazis, N., Perantonis, S.J. & Drivaliaris, D. Improved Jacobian Eigen-Analysis Scheme for Accelerating Learning in Feedforward Neural Networks. Cogn Comput 7, 86–102 (2015). https://doi.org/10.1007/s12559-014-9263-2

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