This study discusses the negative impact of the derivative of the activation functions in the output layer of artificial neural networks, in particular in continual learning. We propose Hebbian descent as a theoretical framework to overcome this limitation, which is implemented through an alternative loss function for gradient descent we refer to as Hebbian descent loss. This loss is effectively the generalized log-likelihood loss and corresponds to an alternative weight update rule for the output layer wherein the derivative of the activation function is disregarded. We show how this update avoids vanishing error signals during backpropagation in saturated regions of the activation functions, which is particularly helpful in training shallow neural networks and deep neural networks where saturating activation functions are only used in the output layer. In combination with centering, Hebbian descent leads to better continual learning capabilities. It provides a unifying perspective on Hebbian learning, gradient descent, and generalized linear models, for all of which we discuss the advantages and disadvantages. Given activation functions with strictly positive derivative (as often the case in practice), Hebbian descent inherits the convergence properties of regular gradient descent. While established pairings of loss and output layer activation function (e.g., mean squared error with linear or cross-entropy with sigmoid/softmax) are subsumed by Hebbian descent, we provide general insights for designing arbitrary loss activation function combinations that benefit from Hebbian descent. For shallow networks, we show that Hebbian descent outperforms Hebbian learning, has a performance similar to regular gradient descent, and has a much better performance than all other tested update rules in continual learning. In combination with centering, Hebbian descent implements a forgetting mechanism that prevents catastrophic interference notably better than the other tested update rules. When training deep neural networks, our experimental results suggest that Hebbian descent has better or similar performance as gradient descent.

This work analyzes centered Restricted Boltzmann Machines (RBMs) and centered Deep Boltzmann Machines (DBMs), where centering is done by subtracting offset values from visible and hidden variables. We show analytically that (i) centered and normal Boltzmann Machines (BMs) and thus RBMs and DBMs are different parameterizations of the same model class, such that any normal BM/RBM/DBM can be transformed to an equivalent centered BM/RBM/DBM and vice versa, and that this equivalence generalizes to artificial neural networks in general, (ii) the expected performance of centered binary BMs/RBMs/DBMs is invariant under simultaneous flip of data and offsets, for any offset value in the range of zero to one, (iii) centering can be reformulated as a different update rule for normal BMs/RBMs/DBMs, and (iv) using the enhanced gradient is equivalent to setting the offset values to the average over model and data mean. Furthermore, we present numerical simulations suggesting that (i) optimal generative performance is achieved by subtracting mean values from visible as well as hidden variables, (ii) centered binary RBMs/DBMs reach significantly higher log-likelihood values than normal binary RBMs/DBMs, (iii) centering variants whose offsets depend on the model mean, like the enhanced gradient, suffer from severe divergence problems, (iv) learning is stabilized if an exponentially moving average over the batch means is used for the offset values instead of the current batch mean, which also prevents the enhanced gradient from severe divergence, (v) on a similar level of log-likelihood values centered binary RBMs/DBMs have smaller weights and bigger bias parameters than normal binary RBMs/DBMs, (vi) centering leads to an update direction that is closer to the natural gradient, which is extremely efficient for training as we show for small binary RBMs, (vii) centering eliminates the need for greedy layer-wise pre-training of DBMs, which often even deteriorates the results independently of whether centering is used or not, and (ix) centering is also beneficial for auto encoders.