HINE Parameters X0, X1, ....Xp Output E0, E1,....Ep Error Desired Output D0, D1,...Dp Y0, Y1,...Yp Input w w0 w1 AT&T Laboratories (c) COST FUNCTION Output E0, E1,....Ep Error Desired Output D0, D1,...Dp Y0, Y1,...Yp X0, X1, ....Xp Input Parameters w B R A COMPUTING THE GRADIENT WITH BACKPROPAGATION O = A(I1, I2) dI1 = dO ¶ A ¶ I1 dI2 = dO ¶ A ¶ I2 - The learning machine is composed of modules (e.g. layers) - Each module can do two things: 1- compute its outputs from its inputs (FPROP) 2- compute gradient vectors at its inputs from gradient vectors at its outputs (BPROP) A O, dO I1, dI1 I2, dI2 AT&T Laboratories (c) AN INTERESTING SPECIAL CASE: MULTILAYER NETWORKS X0, X1, ....Xp Output Desired Output D0, D1,...Dp Y0, Y1,...Yp Input || D - Y || 2 2 1 WX F() WX F() Mean Square Error Parameters (weights + biases) w Weight matrix E0, E1,....Ep Sigmoids + Biase

Just storing the Hessian H (the matrix of second derivatives d^2 E/dw_i dw_j of the error E with respect to each pair of weights) of a large neural network is difficult. Since a common use of a large matrix like H is to compute its product with various vectors, we derive a technique that directly calculates Hv, where v is an arbitrary vector. This allows H to be treated as a generalized sparse matrix. To calculate Hv, we first define a differential operator R{f(w)} = (d/dr)f(w + rv)|_{r=0}, note that R{grad_w} = Hv and R{w} = v, and then apply R{} to the equations used to compute grad_w. The result is an exact and numerically stable procedure for computing Hv, which takes about as much computation, and is about as local, as a gradient evaluation. We then apply the technique to backpropagation networks, recurrent backpropagation, and stochastic Boltzmann Machines. Finally, we show that this technique can be used at the heart of many iterative techniques for computing various properties of H, obviating the need for direct methods.

Gain adaptation algorithms for neural networks typically adjust learning rates by monitoring the correlation between successive gradients. Here we discuss the limitations of this approach, and develop an alternative by extending Sutton's work on linear systems to the general, nonlinear case. The resulting online algorithms are computationally little more expensive than other acceleration techniques, do not assume statistical independence between successive training patterns, and do not require an arbitrary smoothing parameter. In our benchmark experiments, they consistently outperform other acceleration methods, and show remarkable robustness when faced with noni. i.d. sampling of the input space.

. The calculation of second derivatives is required by recent training and analyses techniques of connectionist networks, such as the elimination of superfluous weights, and the estimation of confidence intervals both for weights and network outputs. We here review and develop exact and approximate algorithms for calculating second derivatives. For networks with jwj weights, simply writing the full matrix of second derivatives requires O(jwj 2 ) operations. For networks of radial basis units or sigmoid units, exact calculation of the necessary intermediate terms requires of the order of 2h + 2 backward/forward-propagation passes where h is the number of hidden units in the network. We also review and compare three approximations (ignoring some components of the second derivative, numerical differentiation, and scoring). Our algorithms apply to arbitrary activation functions, networks, and error functions (for instance, with connections that skip layers, or radial basis functions, or ...

We consider the problem of developing rapid, stable, and scalable stochastic gradient descent algorithms for optimisation of very large nonlinear systems. Based on earlier work by Orr et al. on adaptive momentum -- an efficient yet extremely unstable stochastic gradient descent algorithm -- we develop a stabilised adaptive momentum algorithm that is suitable for noisy nonlinear optimisation problems. The stability is improved by introducing a forgetting factor 0 < A < I that smoothes the trajectory and enables adaptation in non-stationary environments. The scalability of the new algorithm follows from the fact that at each iteration the multiplication by the curvature matrix can be achieved in O (n) steps using automatic differentiation tools. We illustrate the behaviour of the new algorithm on two examples: a linear neuron with squared loss and highly correlated inputs, and a multilayer perceptron applied to the four regions benchmark task.

We introduce a new method for adapting the step size of each individual weight in a multi-layer perceptron trained by stochastic gradient descent. Our technique derives from the K1 algorithm for linear systems (Sutton, 1992b), which in turn is based on a diagonalized Kalman Filter. We expand upon Sutton's work in two regards: K1 is a) extended to multi-layer perceptrons, and b) made more efficient by linearizing an exponentiation operation. The resulting elk1 (extended, linearized K1) algorithm is computationally little more expensive than alternative proposals (Zimmermann, 1994; Almeida et al., 1997, 1998), and does not require an arbitrary smoothing parameter. In our benchmark experiments, elk1 consistently outperforms these alternatives, as well as stochastic gradient descent with momentum, even when the number of floating-point operations required per weight update is taken into account. Unlike the method of Almeida et al. (1997, 1998), elk1 does not require statistical independence between successive training patterns, and handles large initial learning rates well.