Christian Knoll

Christian Toth, Christian Knoll, Franz Pernkopf et al.

The traditional two-stage approach to causal inference first identifies a single causal model (or equivalence class of models), which is then used to answer causal queries. However, this neglects any epistemic model uncertainty. In contrast, Bayesian causal inference does incorporate epistemic uncertainty into query estimates via Bayesian marginalisation (posterior averaging) over all causal models. While principled, this marginalisation over entire causal models, i.e., both causal structures (graphs) and mechanisms, poses a tremendous computational challenge. In this work, we address this challenge by decomposing structure marginalisation into the marginalisation over (i) causal orders and (ii) directed acyclic graphs (DAGs) given an order. We can marginalise the latter in closed form by limiting the number of parents per variable and utilising Gaussian processes to model mechanisms. To marginalise over orders, we use a sampling-based approximation, for which we devise a novel auto-regressive distribution over causal orders (ARCO). Our method outperforms state-of-the-art in structure learning on simulated non-linear additive noise benchmarks, and yields competitive results on real-world data. Furthermore, we can accurately infer interventional distributions and average causal effects.

Sophie Steger, Christian Knoll, Bernhard Klein et al.

Bayesian inference in function space has gained attention due to its robustness against overparameterization in neural networks. However, approximating the infinite-dimensional function space introduces several challenges. In this work, we discuss function space inference via particle optimization and present practical modifications that improve uncertainty estimation and, most importantly, make it applicable for large and pretrained networks. First, we demonstrate that the input samples, where particle predictions are enforced to be diverse, are detrimental to the model performance. While diversity on training data itself can lead to underfitting, the use of label-destroying data augmentation, or unlabeled out-of-distribution data can improve prediction diversity and uncertainty estimates. Furthermore, we take advantage of the function space formulation, which imposes no restrictions on network parameterization other than sufficient flexibility. Instead of using full deep ensembles to represent particles, we propose a single multi-headed network that introduces a minimal increase in parameters and computation. This allows seamless integration to pretrained networks, where this repulsive last-layer ensemble can be used for uncertainty aware fine-tuning at minimal additional cost. We achieve competitive results in disentangling aleatoric and epistemic uncertainty for active learning, detecting out-of-domain data, and providing calibrated uncertainty estimates under distribution shifts with minimal compute and memory.

Harald Leisenberger, Christian Knoll, Franz Pernkopf

The Bethe free energy approximation provides an effective way for relaxing NP-hard problems of probabilistic inference. However, its accuracy depends on the model parameters and particularly degrades if a phase transition in the model occurs. In this work, we analyze when the Bethe approximation is reliable and how this can be verified. We argue and show by experiment that it is mostly accurate if it is convex on a submanifold of its domain, the 'Bethe box'. For verifying its convexity, we derive two sufficient conditions that are based on the definiteness properties of the Bethe Hessian matrix: the first uses the concept of diagonal dominance, and the second decomposes the Bethe Hessian matrix into a sum of sparse matrices and characterizes the definiteness properties of the individual matrices in that sum. These theoretical results provide a simple way to estimate the critical phase transition temperature of a model. As a practical contribution we propose $\texttt{BETHE-MIN}$, a projected quasi-Newton method to efficiently find a minimum of the Bethe free energy.

Alexander Fuchs, Christian Knoll, Franz Pernkopf

Deep neural networks rely heavily on normalization methods to improve their performance and learning behavior. Although normalization methods spurred the development of increasingly deep and efficient architectures, they also increase the vulnerability with respect to noise and input corruptions. In most applications, however, noise is ubiquitous and diverse; this can often lead to complete failure of machine learning systems as they fail to cope with mismatches between the input distribution during training- and test-time. The most common normalization method, batch normalization, reduces the distribution shift during training but is agnostic to changes in the input distribution during test time. This makes batch normalization prone to performance degradation whenever noise is present during test-time. Sample-based normalization methods can correct linear transformations of the activation distribution but cannot mitigate changes in the distribution shape; this makes the network vulnerable to distribution changes that cannot be reflected in the normalization parameters. We propose an unsupervised non-parametric distribution correction method that adapts the activation distribution of each layer. This reduces the mismatch between the training and test-time distribution by minimizing the 1-D Wasserstein distance. In our experiments, we empirically show that the proposed method effectively reduces the impact of intense image corruptions and thus improves the classification performance without the need for retraining or fine-tuning the model.

Christian Knoll, Adrian Weller, Franz Pernkopf

Belief propagation (BP) is a popular method for performing probabilistic inference on graphical models. In this work, we enhance BP and propose self-guided belief propagation (SBP) that incorporates the pairwise potentials only gradually. This homotopy continuation method converges to a unique solution and increases the accuracy without increasing the computational burden. We provide a formal analysis to demonstrate that SBP finds the global optimum of the Bethe approximation for attractive models where all variables favor the same state. Moreover, we apply SBP to various graphs with random potentials and empirically show that: (i) SBP is superior in terms of accuracy whenever BP converges, and (ii) SBP obtains a unique, stable, and accurate solution whenever BP does not converge.

Christian Knoll, Franz Pernkopf, Dhagash Mehta et al.

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.