The online incremental gradient (or backpropagation) algorithm is widely considered to be the fastest method for solving large-scale neural-network (NN) learning problems. In contrast, we show that an appropriately implemented iterative batch-mode (or block-mode) learning method can be much faster. For example, it is three times faster in the UCI letter classification problem (26 outputs, 16,000 data items, 6,066 parameters with a two-hidden-layer multilayer perceptron) and 353 times faster in a nonlinear regression problem arising in color recipe prediction (10 outputs, 1,000 data items, 2,210 parameters with a neuro-fuzzy modular network). The three principal innovative ingredients in our algorithm are the following: First, we use scaled trust-region regularization with inner-outer it- eration to solve the associated "overdetermined" nonlinear least squares problem, where the inner iteration performs a truncated (or inexact) Newton method. Second, we employ Pearlmutter's implicit sparse Hessian matrix-vector multiply algorithm to con- struct the Krylov subspaces used to solve for the truncated New- ton update. Third, we exploit sparsity (for preconditioning) in the matrices resulting from the NNs having many outputs.

Automatic differentiation---the mechanical transformation of numeric computer programs to calculate derivatives efficiently and accurately---dates to the origin of the computer age. Reverse mode automatic differentiation both antedates and generalizes the method of backwards propagation of errors used in machine learning. Despite this, practitioners in a variety of fields, including machine learning, have been little influenced by automatic differentiation, and make scant use of available tools. Here we review the technique of automatic differentiation, describe its two main modes, and explain how it can benefit machine learning practitioners. To reach the widest possible audience our treatment assumes only elementary differential calculus, and does not assume any knowledge of linear algebra.

The online incremental gradient (or backpropagation) algorithm is widely considered to be the fastest method for solving large-scale neural-network (NN) learning problems. In contrast, we show that an appropriately implemented iterative batch-mode (or block-mode) learning method can be much faster. For example, it is three times faster in the UCI letter classification problem (26 outputs, 16,000 data items, 6,066 parameters with a two-hidden-layer multilayer perceptron) and 353 times faster in a nonlinear regression problem arising in color recipe prediction (10 outputs, 1,000 data items, 2,210 parameters with a neuro-fuzzy modular network). The three principal innovative ingredients in our algorithm are the following: First, we use scaled trust-region regularization with inner-outer iteration tosolve the associated "overdetermined" nonlinear least squares problem, where the inner iteration performs a truncated (or inexact) Newton method. Second, we employ Pearlmutter's implicit sparse Hessian matrix-vector multiply algorithm to construct theKrylov subspaces used to solve for the truncated Newton update. Third, we exploit sparsity (for preconditioning) in the matrices resulting from the NNs having many outputs.

The online incremental gradient (or backpropagation) algorithm is widely considered to be the fastest method for solving large-scale neural-network (NN) learning problems. In contrast, we show that an appropriately implemented iterative batch-mode (or block-mode) learning method can be much faster. For example, it is three times faster in the UCI letter classification problem (26 outputs, 16,000 data items, 6,066 parameters with a two-hidden-layer multilayer perceptron) and 353 times faster in a nonlinear regression problem arising in color recipe prediction (10 outputs, 1,000 data items, 2,210 parameters with a neuro-fuzzy modular network). The three principal innovative ingredients in our algorithm are the following: First, we use scaled trust-region regularization with inner-outer iteration to solve the associated "overdetermined" nonlinear least squares problem, where the inner iteration performs a truncated (or inexact) Newton method. Second, we employ Pearlmutter's implicit sparse Hessian matrix-vector multiply algorithm to construct the Krylov subspaces used to solve for the truncated Newton update. Third, we exploit sparsity (for preconditioning) in the matrices resulting from the NNs having many outputs.

The online incremental gradient (or backpropagation) algorithm is widely considered to be the fastest method for solving large-scale neural-network (NN) learning problems. In contrast, we show that an appropriately implemented iterative batch-mode (or block-mode) learning method can be much faster. For example, it is three times faster in the UCI letter classification problem (26 outputs, 16,000 data items, 6,066 parameters with a two-hidden-layer multilayer perceptron) and 353 times faster in a nonlinear regression problem arising in color recipe prediction (10 outputs, 1,000 data items, 2,210 parameters with a neuro-fuzzy modular network). The three principal innovative ingredients in our algorithm are the following: First, we use scaled trust-region regularization with inner-outer iteration to solve the associated "overdetermined" nonlinear least squares problem, where the inner iteration performs a truncated (or inexact) Newton method. Second, we employ Pearlmutter's implicit sparse Hessian matrix-vector multiply algorithm to construct the Krylov subspaces used to solve for the truncated Newton update. Third, we exploit sparsity (for preconditioning) in the matrices resulting from the NNs having many outputs.

We describe a system that localizes a single dipole to reasonable accuracy from noisy magnetoencephalographic (MEG) measurements in real time. At its core is a multilayer perceptron (MLP) trained to map sensor signals and head position to dipole location. Including head position overcomes the previous need to retrain the MLP for each subject and session. The training dataset was generated by mapping randomly chosen dipoles and head positions through an analytic model and adding noise from real MEG recordings. After training, a localization took 0.7 ms with an average error of 0.90 cm. A few iterations of a Levenberg-Marquardt routine using the MLP's output as its initial guess took 15 ms and improved the accuracy to 0.53 cm, only slightly above the statistical limits on accuracy imposed by the noise. We applied these methods to localize single dipole sources from MEG components isolated by blind source separation and compared the estimated locations to those generated by standard manually-assisted commercial software.

We describe a system that localizes a single dipole to reasonable accuracy from noisy magnetoencephalographic (MEG) measurements in real time. At its core is a multilayer perceptron (MLP) trained to map sensor signals and head position to dipole location. Including head position overcomes the previous need to retrain the MLP for each subject and session. The training dataset was generated by mapping randomly chosen dipoles and head positions through an analytic model and adding noise from real MEG recordings. After training, a localization took 0.7 ms with an average error of 0.90 cm. A few iterations of a Levenberg-Marquardt routine using the MLP's output as its initial guess took 15 ms and improved the accuracy to 0.53 cm, only slightly above the statistical limits on accuracy imposed by the noise. We applied these methods to localize single dipole sources from MEG components isolated by blind source separation and compared the estimated locations to those generated by standard manually-assisted commercial software.

We describe a system that localizes a single dipole to reasonable accuracy fromnoisy magnetoencephalographic (MEG) measurements in real time. At its core is a multilayer perceptron (MLP) trained to map sensor signalsand head position to dipole location. Including head position overcomes the previous need to retrain the MLP for each subject and session. Thetraining dataset was generated by mapping randomly chosen dipoles and head positions through an analytic model and adding noise from real MEG recordings. After training, a localization took 0.7 ms with an average error of 0.90 cm. A few iterations of a Levenberg-Marquardt routine using the MLP's output as its initial guess took 15 ms and improved theaccuracy to 0.53 cm, only slightly above the statistical limits on accuracy imposed by the noise. We applied these methods to localize single dipole sources from MEG components isolated by blind source separation and compared the estimated locations to those generated by standard manually-assisted commercial software.

For many problems, the correct behavior of a model depends not only on its input-output mapping but also on properties of its Jacobian matrix, the matrix of partial derivatives of the model's outputs with respect to its inputs. We introduce the J-prop algorithm, an efficient general method for computing the exact partial derivatives of a variety of simple functions of the Jacobian of a model with respect to its free parameters. The algorithm applies to any parametrized feedforward model, including nonlinear regression, multilayer perceptrons, and radial basis function networks.

For many problems, the correct behavior of a model depends not only on its input-output mapping but also on properties of its Jacobian matrix, the matrix of partial derivatives of the model's outputs with respect to its inputs. We introduce the J-prop algorithm, an efficient general method for computing the exact partial derivatives of a variety of simple functions of the Jacobian of a model with respect to its free parameters. The algorithm applies to any parametrized feedforward model, including nonlinear regression, multilayer perceptrons, and radial basis function networks.