Hrittik Roy

Marco Miani, Hrittik Roy, Søren Hauberg

Bayesian deep learning all too often underfits so that the Bayesian prediction is less accurate than a simple point estimate. Uncertainty quantification then comes at the cost of accuracy. For linearized models, the null space of the generalized Gauss-Newton matrix corresponds to parameters that preserve the training predictions of the point estimate. We propose to build Bayesian approximations in this null space, thereby guaranteeing that the Bayesian predictive does not underfit. We suggest a matrix-free algorithm for projecting onto this null space, which scales linearly with the number of parameters and quadratically with the number of output dimensions. We further propose an approximation that only scales linearly with parameters to make the method applicable to generative models. An extensive empirical evaluation shows that the approach scales to large models, including vision transformers with 28 million parameters.

Hrittik Roy, Søren Hauberg, Nicholas Krämer

This paper argues that the method of least squares has significant unfulfilled potential in modern machine learning, far beyond merely being a tool for fitting linear models. To release its potential, we derive custom gradients that transform the solver into a differentiable operator, like a neural network layer, enabling many diverse applications. Empirically, we demonstrate: (i) scalability by enforcing weight sparsity on a 50 million parameter model; (ii) imposing conservativeness constraints in score-based generative models; and (iii) hyperparameter tuning of Gaussian processes based on predictive performance. By doing this, our work represents the next iteration in developing differentiable linear-algebra tools and making them widely accessible to machine learning practitioners.

Hrittik Roy, Marco Miani, Carl Henrik Ek et al.

Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i.e. BNNs assign different posterior densities to different parametrizations of identical functions. This creates a fundamental flaw in the application of Bayesian principles as it breaks the correspondence between uncertainty over the parameters with uncertainty over the parametrized function. In this paper, we investigate this issue in the context of the increasingly popular linearized Laplace approximation. Specifically, it has been observed that linearized predictives alleviate the common underfitting problems of the Laplace approximation. We develop a new geometric view of reparametrizations from which we explain the success of linearization. Moreover, we demonstrate that these reparameterization invariance properties can be extended to the original neural network predictive using a Riemannian diffusion process giving a straightforward algorithm for approximate posterior sampling, which empirically improves posterior fit.