Jed A. Duersch

Jed A. Duersch

Variational inference is an approximation framework for Bayesian inference that seeks to improve quantified uncertainty in predictions by optimizing a simplified distribution over parameters to stand in for the full posterior. Capturing model variations that remain consistent with training data enables more robust predictions by reducing parameter sensitivity. This work introduces a fixed-point optimization for variational inference that is applicable when every feasible log density can be expressed as a linear combination of functions from a given basis. In such cases, the optimizer becomes a fixed-point of projective integral updates. When the basis spans univariate quadratics in each parameter, feasible densities are Gaussian and the projective integral updates yield quasi-Newton variational Bayes (QNVB). Other bases and updates are also possible. As these updates require high-dimensional integration, this work first proposes an efficient quasirandom quadrature sequence for mean-field distributions. Each iterate of the sequence contains two evaluation points that combine to correctly integrate all univariate quadratics and, if the mean-field factors are symmetric, all univariate cubics. More importantly, averaging results over short subsequences achieves periodic exactness on a much larger space of multivariate quadratics. The corresponding variational updates require 4 loss evaluations with standard (not second-order) backpropagation to eliminate error terms from over half of all multivariate quadratic basis functions. This integration technique is motivated by first proposing stochastic blocked mean-field quadratures, which may be useful in other contexts. A PyTorch implementation of QNVB allows for better control over model uncertainty during training than competing methods. Experiments demonstrate superior generalizability for multiple learning problems and architectures.

Jed A. Duersch, Thomas A. Catanach, Niladri Das

Balancing model complexity against the information contained in observed data is the central challenge to learning. In order for complexity-efficient models to exist and be discoverable in high dimensions, we require a computational framework that relates a credible notion of complexity to simple parameter representations. Further, this framework must allow excess complexity to be gradually removed via gradient-based optimization. Our n-ary, or n-argument, activation functions fill this gap by approximating belief functions (probabilistic Boolean logic) using logit representations of probability. Just as Boolean logic determines the truth of a consequent claim from relationships among a set of antecedent propositions, probabilistic formulations generalize predictions when antecedents, truth tables, and consequents all retain uncertainty. Our activation functions demonstrate the ability to learn arbitrary logic, such as the binary exclusive disjunction (p xor q) and ternary conditioned disjunction ( c ? p : q ), in a single layer using an activation function of matching or greater arity. Further, we represent belief tables using a basis that directly associates the number of nonzero parameters to the effective arity of the belief function, thus capturing a concrete relationship between logical complexity and efficient parameter representations. This opens optimization approaches to reduce logical complexity by inducing parameter sparsity.

Niladri Das, Jed A. Duersch, Thomas A. Catanach

In this paper, we address the problem of convergence of sequential variational inference filter (VIF) through the application of a robust variational objective and Hinf-norm based correction for a linear Gaussian system. As the dimension of state or parameter space grows, performing the full Kalman update with the dense covariance matrix for a large scale system requires increased storage and computational complexity, making it impractical. The VIF approach, based on mean-field Gaussian variational inference, reduces this burden through the variational approximation to the covariance usually in the form of a diagonal covariance approximation. The challenge is to retain convergence and correct for biases introduced by the sequential VIF steps. We desire a framework that improves feasibility while still maintaining reasonable proximity to the optimal Kalman filter as data is assimilated. To accomplish this goal, a Hinf-norm based optimization perturbs the VIF covariance matrix to improve robustness. This yields a novel VIF- Hinf recursion that employs consecutive variational inference and Hinf based optimization steps. We explore the development of this method and investigate a numerical example to illustrate the effectiveness of the proposed filter.

Jed A. Duersch, Tommie A. Catanach, Alexander Safonov et al.

Harnessing the local topography of the loss landscape is a central challenge in advanced optimization tasks. By accounting for the effect of potential parameter changes, we can alter the model more efficiently. Contrary to standard assumptions, we find that the Hessian does not always approximate loss curvature well, particularly near gradient discontinuities, which commonly arise in deep learning architectures. We present a new conceptual framework to understand how curvature of expected changes in loss emerges in architectures with many rectified linear units. Each ReLU creates a parameter boundary that, when crossed, induces a pseudorandom gradient perturbation. Our derivations show how these discontinuities combine to form a glass-like structure, similar to amorphous solids that contain microscopic domains of strong, but random, atomic alignment. By estimating the density of the resulting gradient variations, we can bound how the loss may change with parameter movement. Our analysis includes the optimal kernel and sample distribution for approximating glass density from ordinary gradient evaluations. We also derive the optimal modification to quasi-Newton steps that incorporate both glass and Hessian terms, as well as certain exactness properties that are possible with Nesterov-accelerated gradient updates. Our algorithm, Alice, tests these techniques to determine which curvature terms are most impactful for training a given architecture and dataset. Additional safeguards enforce stable exploitation through step bounds that expand on the functionality of Adam. These theoretical and experimental tools lay groundwork to improve future efforts (e.g., pruning and quantization) by providing new insight into the loss landscape.

Jed A. Duersch, Thomas A. Catanach

Bayesian inference provides a uniquely rigorous approach to obtain principled justification for uncertainty in predictions, yet it is difficult to articulate suitably general prior belief in the machine learning context, where computational architectures are pure abstractions subject to frequent modifications by practitioners attempting to improve results. Parsimonious inference is an information-theoretic formulation of inference over arbitrary architectures that formalizes Occam's Razor; we prefer simple and sufficient explanations. Our universal hyperprior assigns plausibility to prior descriptions, encoded as sequences of symbols, by expanding on the core relationships between program length, Kolmogorov complexity, and Solomonoff's algorithmic probability. We then cast learning as information minimization over our composite change in belief when an architecture is specified, training data are observed, and model parameters are inferred. By distinguishing model complexity from prediction information, our framework also quantifies the phenomenon of memorization. Although our theory is general, it is most critical when datasets are limited, e.g. small or skewed. We develop novel algorithms for polynomial regression and random forests that are suitable for such data, as demonstrated by our experiments. Our approaches combine efficient encodings with prudent sampling strategies to construct predictive ensembles without cross-validation, thus addressing a fundamental challenge in how to efficiently obtain predictions from data.