Bayesian inference in the space of functions has been an important topic for Bayesian modeling in the past. In this paper, we propose a new solution to this problem called Functional Variational Inference (FVI). In FVI, we minimize a divergence in function space between the variational distribution and the posterior process.

Bayesian inference in the space of functions has been an important topic for Bayesian modeling in the past. In this paper, we propose a new solution to this problem called Functional Variational Inference (FVI). In FVI, we minimize a divergence in function space between the variational distribution and the posterior process. This is done by using as functional variational family a new class of flexible distributions called Stochastic Process Generators (SPGs), which are cleverly designed so that the functional ELBO can be estimated efficiently using analytic solutions and mini-batch sampling. FVI can be applied to stochastic process priors when random function samples from those priors are available. Our experiments show that FVI consistently outperforms weight-space and function space VI methods on several tasks, which validates the effectiveness of our approach.

A.1 Proof of Proposition 1 and 3 To prove Proposition 1, we first need the following lemma: Readers may refer to [47] for the proof of this lemma. Let's first consider the left handside, The first inequality is due to information processing inequality. The compactness assumption in Proposition 2 seems restrictive, since BNNs with Gaussian priors on weights will break the compactness assumption. Indeed, the assumptions in proposition 2 are merely sufficient conditions. In this section, we discuss the non-parametric counter part of Proposition 2, i.e., is the grid functional KL between a parametric model and a Gaussian process is still finite?

Bayesian inference in the space of functions has been an important topic for Bayesian modeling in the past. In this paper, we propose a new solution to this problem called Functional V ariational Inference (FVI). In FVI, we minimize a divergence in function space between the variational distribution and the posterior process.

Bayesian inference in the space of functions has been an important topic for Bayesian modeling in the past. In this paper, we propose a new solution to this problem called Functional Variational Inference (FVI). In FVI, we minimize a divergence in function space between the variational distribution and the posterior process. This is done by using as functional variational family a new class of flexible distributions called Stochastic Process Generators (SPGs), which are cleverly designed so that the functional ELBO can be estimated efficiently using analytic solutions and mini-batch sampling. FVI can be applied to stochastic process priors when random function samples from those priors are available.

Bayesian neural networks (BNN) promise to combine the predictive performance of neural networks with principled uncertainty modeling important for safety-critical systems and decision making. However, posterior uncertainty estimates depend on the choice of prior, and finding informative priors in weight-space has proven difficult. This has motivated variational inference (VI) methods that pose priors directly on the function generated by the BNN rather than on weights. In this paper, we address a fundamental issue with such function-space VI approaches pointed out by Burt et al. (2020), who showed that the objective function (ELBO) is negative infinite for most priors of interest. Our solution builds on generalized VI (Knoblauch et al., 2019) with the regularized KL divergence (Quang, 2019) and is, to the best of our knowledge, the first well-defined variational objective for function-space inference in BNNs with Gaussian process (GP) priors. Experiments show that our method incorporates the properties specified by the GP prior on synthetic and small real-world data sets, and provides competitive uncertainty estimates for regression, classification and out-of-distribution detection compared to BNN baselines with both function and weight-space priors.

Fitted value iteration (FVI) with ordinary least squares regression is known to diverge. We present a new method, "Expansion-Constrained Ordinary Least Squares" (ECOLS), that produces a linear approximation but also guarantees convergence when used with FVI. To ensure convergence, we constrain the least squares regression operator to be a non-expansion in the -norm. We show that the space of function approximators that satisfy this constraint is more rich than the space of "averagers," we prove a minimax property of the ECOLS residual error, and we give an efficient algorithm for computing the coefficients of ECOLS based on constraint generation. We illustrate the algorithmic convergence of FVI with ECOLS in a suite of experiments, and discuss its properties.

Fitted value iteration (FVI) with ordinary least squares regression is known to diverge. We present a new method, "Expansion-Constrained Ordinary Least Squares" (ECOLS), that produces a linear approximation but also guarantees convergence when used with FVI. To ensure convergence, we constrain the least squares regression operator to be a non-expansion in the infinity-norm. We show that the space of function approximators that satisfy this constraint is more rich than the space of "averagers," we prove a minimax property of the ECOLS residual error, and we give an efficient algorithm for computing the coefficients of ECOLS based on constraint generation. We illustrate the algorithmic convergence of FVI with ECOLS in a suite of experiments, and discuss its properties.

Fitted value iteration (FVI) with ordinary least squares regression is known to diverge. We present a new method, "Expansion-Constrained Ordinary Least Squares" (ECOLS), that produces a linear approximation but also guarantees convergence when used with FVI. To ensure convergence, we constrain the least squares regression operator to be a non-expansion in the infinity-norm. We show that the space of function approximators that satisfy this constraint is more rich than the space of "averagers," we prove a minimax property of the ECOLS residual error, and we give an efficient algorithm for computing the coefficients of ECOLS based on constraint generation. We illustrate the algorithmic convergence of FVI with ECOLS in a suite of experiments, and discuss its properties.