A unified theory of adaptive stochastic gradient descent as Bayesian filtering
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There are a diverse array of schemes for adaptive stochastic gradient descent for optimizing neural networks, from fully factorised methods with and without momentum (e.g. RMSProp and ADAM), to Kronecker factored methods that consider the Hessian for a full weight matrix. However, these schemes have been derived and justified using a wide variety of mathematical approaches, and as such, there is no unified theory of adaptive stochastic gradients descent methods. Here, we provide such a theory by showing that many successful adaptive stochastic gradient descent schemes emerge by considering a filtering-based inference in a Bayesian optimization problem. In particular, we use backpropagated gradients to compute a Gaussian posterior over the optimal neural network parameters, given the data minibatches seen so far. Our unified theory is able to give some guidance to practitioners on how to choose between the large number of available optimization methods. In the fully factorised setting, we recover RMSProp and ADAM under different priors, along with additional improvements such as Nesterov acceleration and AdamW. Moreover, we obtain new recommendations, including the possibility of combining RMSProp and ADAM updates. In the Kronecker factored setting, we obtain a adaptive natural gradient adaptation scheme that is derived specifically for the minibatch setting. Furthermore, under a modified prior, we obtain a Kronecker factored analogue of RMSProp or ADAM, that preconditions the gradient by whitening (i.e. by multiplying by the square root of the Hessian, as in RMSProp/ADAM). Our work raises the hope that it is possible to achieve unified theoretical understanding of empirically successful adaptive gradient descent schemes for neural networks.
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