Bayesian brain theory suggests that the brain employs generative models to understand the external world.

Recently, researchers have found that representations learned by large-scale pre-trained language models are useful in various downstream tasks.

Bayesian brain theory suggests that the brain employs generative models to understand the external world.

The lasso and elastic net linear regression models impose a double-exponential prior distribution on the model parameters to achieve regression shrinkage and variable selection, allowing the inference of robust models from large data sets. However, there has been limited success in deriving estimates for the full posterior distribution of regression coefficients in these models, due to a need to evaluate analytically intractable partition function integrals. Here, the Fourier transform is used to express these integrals as complex-valued oscillatory integrals over regression frequencies. This results in an analytic expansion and stationary phase approximation for the partition functions of the Bayesian lasso and elastic net, where the non-differentiability of the double-exponential prior has so far eluded such an approach. Use of this approximation leads to highly accurate numerical estimates for the expectation values and marginal posterior distributions of the regression coefficients, and allows for Bayesian inference of much higher dimensional models than previously possible.

Several recent works in machine learning have focused on evaluating the test-time robustness of a classifier: how well the classifier performs not just on the target domain it was trained upon, but upon perturbed examples. In these settings, the focus has largely been on two extremes of robustness: the robustness to perturbations drawn from within some distribution (i.e., robustness to random perturbations), and the robustness to the perturbation in some set (i.e., adversarial robustness). In this paper, we argue that a sliding scale between these two extremes provides a valuable additional metric by which to gauge robustness. Specifically, we illustrate that each of these two extremes is naturally characterized by a (functional) q-norm over perturbation space, with q=1 corresponding to robustness to random perturbations and q=\infty corresponding to adversarial perturbations. We then present the main technical contribution of our paper: a method for efficiently estimating the value of these norms by interpreting them as the partition function of a particular distribution, then using path sampling with MCMC methods to estimate this partition function (either traditional Metropolis-Hastings for non-differentiable perturbations, or Hamiltonian Monte Carlo for differentiable perturbations). We show that our approach provides substantially better estimates than simple random sampling of the actual "intermediate-q" robustness of both standard, data-augmented, and adversarially-trained classifiers, illustrating a clear tradeoff between classifiers that optimize different metrics.

We present a novel method for reducing the computational complexity of rigorously estimating the partition functions of Gibbs (or Boltzmann) distributions, which arise ubiquitously in probabilistic graphical models. A major obstacle to applying the Gibbs distribution in practice is the need to estimate their partition function (normalizing constant). The state of the art in addressing this problem is multi-stage algorithms which consist of a cooling schedule and a mean estimator in each step of the schedule. While the cooling schedule in these algorithms is adaptive, the mean estimate computations use MCMC as a black-box to draw approximately-independent samples. Here we develop a doubly adaptive approach, combining the adaptive cooling schedule with an adaptive MCMC mean estimator, whose number of Markov chain steps adapts dynamically to the underlying chain. Through rigorous theoretical analysis, we prove that our method outperforms the state of the art algorithms in several factors: (1) The computational complexity of our method is smaller; (2) Our method is less sensitive to loose bounds on mixing times, an inherent components in these algorithms; and (3) The improvement obtained by our method is particularly significant in the most challenging regime of high precision estimates. We demonstrate the advantage of our method in experiments run on classic factor graphs, such as voting models and Ising models.

Probabilistic inference in pairwise Markov Random Fields (MRFs), i.e. computing the partition function or computing a MAP estimate of the variables, is a foundational problem in probabilistic graphical models. Semidefinite programming relaxations have long been a theoretically powerful tool for analyzing properties of probabilistic inference, but have not been practical owing to the high computational cost of typical solvers for solving the resulting SDPs. In this paper, we propose an efficient method for computing the partition function or MAP estimate in a pairwise MRF by instead exploiting a recently proposed coordinate-descent-based fast semidefinite solver. We also extend semidefinite relaxations from the typical binary MRF to the full multi-class setting, and develop a compact semidefinite relaxation that can again be solved efficiently using the solver. We show that the method substantially outperforms (both in terms of solution quality and speed) the existing state of the art in approximate inference, on benchmark problems drawn from previous work. We also show that our approach can scale to large MRF domains such as fully-connected pairwise CRF models used in computer vision.