Andrew Stirn

Andrew Stirn, Hans-Hermann Wessels, Megan Schertzer et al.

Heteroscedastic regression models a Gaussian variable's mean and variance as a function of covariates. Parametric methods that employ neural networks for these parameter maps can capture complex relationships in the data. Yet, optimizing network parameters via log likelihood gradients can yield suboptimal mean and uncalibrated variance estimates. Current solutions side-step this optimization problem with surrogate objectives or Bayesian treatments. Instead, we make two simple modifications to optimization. Notably, their combination produces a heteroscedastic model with mean estimates that are provably as accurate as those from its homoscedastic counterpart (i.e.~fitting the mean under squared error loss). For a wide variety of network and task complexities, we find that mean estimates from existing heteroscedastic solutions can be significantly less accurate than those from an equivalently expressive mean-only model. Our approach provably retains the accuracy of an equally flexible mean-only model while also offering best-in-class variance calibration. Lastly, we show how to leverage our method to recover the underlying heteroscedastic noise variance.

Andrew Stirn, David A. Knowles

Brittle optimization has been observed to adversely impact model likelihoods for regression and VAEs when simultaneously fitting neural network mappings from a (random) variable onto the mean and variance of a dependent Gaussian variable. Previous works have bolstered optimization and improved likelihoods, but fail other basic posterior predictive checks (PPCs). Under the PPC framework, we propose critiques to test predictive mean and variance calibration and the predictive distribution's ability to generate sensible data. We find that our attractively simple solution, to treat heteroscedastic variance variationally, sufficiently regularizes variance to pass these PPCs. We consider a diverse gamut of existing and novel priors and find our methods preserve or outperform existing model likelihoods while significantly improving parameter calibration and sample quality for regression and VAEs.

Andrew Stirn, Tony Jebara, David A Knowles

We construct a new distribution for the simplex using the Kumaraswamy distribution and an ordered stick-breaking process. We explore and develop the theoretical properties of this new distribution and prove that it exhibits symmetry under the same conditions as the well-known Dirichlet. Like the Dirichlet, the new distribution is adept at capturing sparsity but, unlike the Dirichlet, has an exact and closed form reparameterization--making it well suited for deep variational Bayesian modeling. We demonstrate the distribution's utility in a variety of semi-supervised auto-encoding tasks. In all cases, the resulting models achieve competitive performance commensurate with their simplicity, use of explicit probability models, and abstinence from adversarial training.

Andrew Stirn, Tony Jebara

Thompson sampling, a Bayesian method for balancing exploration and exploitation in bandit problems, has theoretical guarantees and exhibits strong empirical performance in many domains. Traditional Thompson sampling, however, assumes perfect compliance, where an agent's chosen action is treated as the implemented action. This article introduces a stochastic noncompliance model that relaxes this assumption. We prove that any noncompliance in a 2-armed Bernoulli bandit increases existing regret bounds. With our noncompliance model, we derive Thompson sampling variants that explicitly handle both observed and latent noncompliance. With extensive empirical analysis, we demonstrate that our algorithms either match or outperform traditional Thompson sampling in both compliant and noncompliant environments.