Efficient Matrix-Free Approximations of Second-Order Information, with Applications to Pruning and Optimization

• URL: http://arxiv.org/abs/2107.03356v3
• Date: Fri, 9 Jul 2021 09:38:58 GMT
• Title: Efficient Matrix-Free Approximations of Second-Order Information, with Applications to Pruning and Optimization
• Authors: Elias Frantar, Eldar Kurtic, Dan Alistarh
• Abstract summary: We investigate matrix-free, linear-time approaches for estimating Inverse-Hessian Vector Products (IHVPs) These algorithms yield state-of-the-art results for network pruning and optimization with lower computational overhead relative to existing second-order methods.
• Score: 16.96639526117016
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Efficiently approximating local curvature information of the loss function is a key tool for optimization and compression of deep neural networks. Yet, most existing methods to approximate second-order information have high computational or storage costs, which can limit their practicality. In this work, we investigate matrix-free, linear-time approaches for estimating Inverse-Hessian Vector Products (IHVPs) for the case when the Hessian can be approximated as a sum of rank-one matrices, as in the classic approximation of the Hessian by the empirical Fisher matrix. We propose two new algorithms as part of a framework called M-FAC: the first algorithm is tailored towards network compression and can compute the IHVP for dimension $d$, if the Hessian is given as a sum of $m$ rank-one matrices, using $O(dm^2)$ precomputation, $O(dm)$ cost for computing the IHVP, and query cost $O(m)$ for any single element of the inverse Hessian. The second algorithm targets an optimization setting, where we wish to compute the product between the inverse Hessian, estimated over a sliding window of optimization steps, and a given gradient direction, as required for preconditioned SGD. We give an algorithm with cost $O(dm + m^2)$ for computing the IHVP and $O(dm + m^3)$ for adding or removing any gradient from the sliding window. These two algorithms yield state-of-the-art results for network pruning and optimization with lower computational overhead relative to existing second-order methods. Implementations are available at [10] and [18].

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