Debiasing Mini-Batch Quadratics for Applications in Deep Learning

Quadratic approximations form a fundamental building block of machine learning methods. E.g., second-order optimizers try to find the Newton step into the minimum of a local quadratic proxy to the objective function; and the second-order approximation of a network's loss function can be used to quantify the uncertainty of its outputs via the Laplace approximation. When computations on the entire training set are intractable - typical for deep learning - the relevant quantities are computed on mini-batches. This, however, distorts and biases the shape of the associated *stochastic* quadratic approximations in an intricate way with detrimental effects on applications. In this paper, we (i) show that this bias introduces a systematic error, (ii) provide a theoretical explanation for it, (iii) explain its relevance for second-order optimization and uncertainty quantification via the Laplace approximation in deep learning, and (iv) develop and evaluate debiasing strategies.

Authors

• Lukas Tatzel
• Bálint Mucsányi
• Osane Hackel
• Philipp Hennig

Keywords

• quadratic Taylor approximation
• mini-batching
• second-order optimizers
• conjugate gradients
• uncertainty quantification
• Laplace approximation
• stochastic curvature
• GGN
• KFAC