Bayesian Neural Networks (BNNs) place priors over the parameters in a neural network. Inference in BNNs, however, is difficult; all inference methods for BNNs are approximate. In this work, we empirically compare the quality of predictive uncertainty estimates for 10 common inference methods on both regression and classification tasks. Our experiments demonstrate that commonly used metrics (e.g. test log-likelihood) can be misleading. Our experiments also indicate that inference innovations designed to capture structure in the posterior do not necessarily produce high quality posterior approximations.
In literature, posteriors for Bayesian Neural Network models obtained by Hamiltonian Monte Carlo (HMC) @cite_12 are frequently used as ground truth. However, HMC scales poorly on high dimensional parameter space and large datasets @cite_0 @cite_4 . Mini-batched versions of HMC, such as Stochastic Gradient Langevin Dynamics (SGLD) @cite_0 and Stochastic Gadient HMC @cite_5 , have been introduced to address the issue of scalability. However, these methods still suffer from lower mixing rate and are not theoretically guaranteed to converge to the true posterior when model assumptions are not met (e.g. when the true model of the gradient noise is not well-estimated).
@cite_0 @cite_5 @cite_4 @cite_12
2167433878 2144193737 1719489212
Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system--such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate computed from a subset of the data. In this paper, we explore the properties of such a stochastic gradient HMC approach. Surprisingly, the natural implementation of the stochastic approximation can be arbitrarily bad. To address this problem we introduce a variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution. Results on simulated data validate our theory. We also provide an application of our methods to a classification task using neural networks and to online Bayesian matrix factorization. Hamiltonian Monte Carlo (HMC) is a widely deployed method to sample from high-dimensional distributions in Statistics and Machine learning. HMC is known to run very efficiently in practice and its popular second-order "leapfrog" implementation has long been conjectured to run in @math gradient evaluations. Here we show that this conjecture is true when sampling from strongly log-concave target distributions that satisfy a weak third-order regularity property associated with the input data. Our regularity condition is weaker than the Lipschitz Hessian property and allows us to show faster convergence bounds for a much larger class of distributions than would be possible with the usual Lipschitz Hessian constant alone. Important distributions that satisfy our regularity condition include posterior distributions used in Bayesian logistic regression for which the data satisfies an "incoherence" property. Our result compares favorably with the best available bounds for the class of strongly log-concave distributions, which grow like @math gradient evaluations with the dimension. Moreover, our simulations on synthetic data suggest that, when our regularity condition is satisfied, leapfrog HMC performs better than its competitors -- both in terms of accuracy and in terms of the number of gradient evaluations it requires. We obtain several quantitative bounds on the mixing properties of the Hamiltonian Monte Carlo (HMC) algorithm for a strongly log-concave target distribution @math on @math , showing that HMC mixes quickly in this setting. One of our main results is a dimension-free bound on the mixing of an "ideal" HMC chain, which is used to show that the usual leapfrog implementation of HMC can sample from @math using only @math gradient evaluations. This dependence on dimension is sharp, and our results significantly extend and improve previous quantitative bounds on the mixing of HMC. The Jacobi metric introduced in mathematical physics can be used to analyze Hamiltonian Monte Carlo (HMC). In a geometrical setting, each step of HMC corresponds to a geodesic on a Riemannian manifold with a Jacobi metric. Our calculation of the sectional curvature of this HMC manifold allows us to see that it is positive in cases such as sampling from a high dimensional multivariate Gaussian. We show that positive curvature can be used to prove theoretical concentration results for HMC Markov chains.