Meta-analysis aims to generalize results from multiple related statistical analyses through a combined analysis. While the natural outcome of a Bayesian study is a posterior distribution, traditional Bayesian meta-analyses proceed by combining summary statistics (i.e., point-valued estimates) computed from data. In this paper, we develop a framework for combining posterior distributions from multiple related Bayesian studies into a meta-analysis. Importantly, the method is capable of reusing pre-computed posteriors from computationally costly analyses, without needing the implementation details from each study. Besides providing a consensus across studies, the method enables updating the local posteriors post-hoc and therefore refining them by sharing statistical strength between the studies, without rerunning the original analyses. We illustrate the wide applicability of the framework by combining results from likelihood-free Bayesian analyses, which would be difficult to carry out using standard methodology.

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.

Generative Flow Networks (GFlowNets) are a flexible family of amortized samplers trained to generate discrete and compositional objects with probability proportional to a reward function. To this end, they learn a policy function over an intractably large state graph by minimizing a stochastic objective over sampled trajectories. However, learning efficiency is constrained by the model’s ability to rapidly explore diverse high-probability regions during training. To mitigate this issue, recent works have focused on incentivizing the exploration of unvisited and valuable states via curiosity-driven search and self-supervised random network distillation, which tend to waste samples on already well-approximated regions of the state space. In this context, we propose (ACE), a principled algorithm for the effective exploration of novel and high-probability regions when learning GFlowNets. To achieve this, ACE introduces an GFlowNet explicitly trained to search for high-reward states in regions underexplored by the GFlowNet, which learns to sample from the target distribution. Through extensive experiments, we show that ACE consistently and significantly improves upon prior work in terms of approximation accuracy to the target distribution and discovery rate of diverse high-reward states.

Causal discovery (CD) is an important component of many scientific applications, yet most techniques produce unreliable point estimates that often contradict expert knowledge. To mitigate this, recent research has focused on ex-ante incorporation of background knowledge into the CD process, typically under an unrealistic causal sufficiency assumption. When probing experts is costly (e.g., hidden behind expensive LLM APIs), however, ex-post model refinement that maximizes query utility is preferable. Also, when independent experts provide conflicting but better-than-random feedback, a principled aggregation method is required. In this context, we introduce the first CD algorithm that enables (i) distributional inference over ancestral graphs (AGs), which represent causal systems under latent confounding, and (ii) integration of both ex-ante and uncertain ex-post expert knowledge. Briefly, our method is a diversity-seeking reinforcement learning algorithm, termed Ancestral GFlowNet (AGFN), whose policy we iteratively refine based on a Bayesian model of the noisy expert feedback. Importantly, we prove convergence to the true AG given sufficiently accurate responses. Through validation on synthetic and realistic datasets using simulated humans and LLMs, we show AGFN is competitive with or superior to strong baselines in terms of structural Hamming distance and Bayesian Information Criterion.

Conventional wisdom attributes the success of Generative Flow Networks (GFlowNets) to their ability to exploit the compositional structure of the sample space for learning generalizable flow functions (Bengio et al., 2021). Despite the abundance of empirical evidence, formalizing this belief with verifiable non-vacuous statistical guarantees has remained elusive. We address this issue with the first data-dependent generalization bounds for GFlowNets. We also elucidate the negative impact of the state space size on the generalization performance of these models via Azuma-Hoeffding-type oracle PAC-Bayesian inequalities. We leverage our theoretical insights to design a novel distributed learning algorithm for GFlowNets, which we call *Subgraph Asynchronous Learning* (SAL). In a nutshell, SAL utilizes a divide-and-conquer strategy: multiple GFlowNets are trained in parallel on smaller subnetworks of the flow network, and then aggregated with an additional GFlowNet that allocates appropriate flow to each subnetwork. Our experiments with synthetic and real-world problems demonstrate the benefits of SAL over centralized training in terms of mode coverage and distribution matching.

Generative Flow Networks (GFlowNets) are powerful samplers for compositional objects that, by design, sample proportionally to a given non-negative reward. Nonetheless, in practice, they often struggle to explore the reward landscape evenly: trajectories toward easy-to-reach regions dominate training, while hard-to-reach modes receive vanishing or uninformative gradients, leading to poor coverage of high-reward areas. We address this imbalance with Boosted GFlowNets, a method that sequentially trains an ensemble of GFlowNets, each optimizing a residual reward that compensates for the mass already captured by previous models. This residual principle reactivates learning signals in underexplored regions and, under mild assumptions, ensures a monotone non-degradation property: adding boosters cannot worsen the learned distribution and typically improves it. Empirically, Boosted GFlowNets achieve substantially better exploration and sample diversity on multimodal synthetic benchmarks and peptide design tasks, while preserving the stability and simplicity of standard trajectory-balance training.

A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive distribution and enable the use of GP machinery to improve the uncertainty quantification of deep neural networks. In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces. Specifically, we show conditions for when arbitrary-depth NOs with Gaussian-distributed convolution kernels converge to function-valued GPs. Based on this result, we show how to compute the covariance functions of these NO-GPs for two NO parametrizations, including the popular Fourier neural operator (FNO). With this, we compute the posteriors of these GPs in regression scenarios, including PDE solution operators. This work is an important step towards uncovering the inductive biases of current FNO architectures and opens a path to incorporate novel inductive biases for use in kernel-based operator learning methods.

Generative Flow Networks (GFlowNets) are an emerging class of sampling methods for distributions over discrete and compositional objects, e.g., graphs. In spite of their remarkable success in problems such as drug discovery and phylogenetic inference, the question of when and whether GFlowNets learn to sample from the target distribution remains underexplored. To tackle this issue, we first assess the extent to which a violation of the detailed balance of the underlying flow network might hamper the correctness of GFlowNet's sampling distribution. In particular, we demonstrate that the impact of an imbalanced edge on the model's accuracy is influenced by the total amount of flow passing through it and, as a consequence, is unevenly distributed across the network. We also argue that, depending on the parameterization, imbalance may be inevitable. In this regard, we consider the problem of sampling from distributions over graphs with GFlowNets parameterized by graph neural networks (GNNs) and show that the representation limits of GNNs delineate which distributions these GFlowNets can approximate. Lastly, we address these limitations by proposing a theoretically sound and computationally tractable metric for assessing GFlowNets, experimentally showing it is a better proxy for correctness than popular evaluation protocols.

Sheaf diffusion has recently emerged as a promising design pattern for graph representation learning due to its inherent ability to handle heterophilic data and avoid oversmoothing. Meanwhile, cooperative message passing has also been proposed as a way to enhance the flexibility of information diffusion by allowing nodes to independently choose whether to propagate/gather information from/to neighbors. A natural question ensues: is sheaf diffusion capable of exhibiting this cooperative behavior? Here, we provide a negative answer to this question. In particular, we show that existing sheaf diffusion methods fail to achieve cooperative behavior due to the lack of message directionality. To circumvent this limitation, we introduce the notion of cellular sheaves over directed graphs and characterize their in- and out-degree Laplacians. We leverage our construction to propose Cooperative Sheaf Neural Networks (CSNNs). Theoretically, we characterize the receptive field of CSNN and show it allows nodes to selectively attend (listen) to arbitrarily far nodes while ignoring all others in their path, potentially mitigating oversquashing. Our experiments show that CSNN presents overall better performance compared to prior art on sheaf diffusion as well as cooperative graph neural networks.

Generative Flow Networks (GFlowNets) are a novel class of flexible amortized samplers for distributions supported on complex objects (e.g., graphs and sequences), achieving significant success in problems such as combinatorial optimization, drug discovery and natural language processing. Nonetheless, training of GFlowNets is challenging-partly because it relies on estimating high-dimensional integrals, including the log-partition function, via stochastic gradient descent (SGD). In particular, for distributions supported on non-discrete spaces, which have received far less attention from the recent literature, every previously proposed learning objective either depends on estimating a log-partition function or is restricted to on-policy training, which is susceptible to mode-collapse. In this context, inspired by the success of contrastive learning for variational inference, we propose the continuous contrastive loss (CCL) as the first objective function natively enabling off-policy training of continuous GFlowNets without reliance on the approximation of high-dimensional integrals via SGD, extending previous work based on discrete distributions. Additionally, we show that minimizing the CCL objective is empirically effective and often leads to faster training convergence than alternatives.

Adaptive Resonance Theory (ART) models allow for categorizing data in a fast and incremental manner. In particular, the Bayesian ART leverages Bayesian methodology to capture complex relationships between categories. While Bayesian methods are well-known for handling uncertainty, Bayesian ART models are at the mercy of a recurring foe: missing data. When data is missing, practitioners must rely on off-the-shelf solutions to impute the data before using Bayesian ART, effectively truncating the uncertainty quantification. To overcome such limitation, we (I) estimate the distribution of missing data entries using a Gaussian mixture model and (II) modify the three steps of Bayesian ART (category choice, matching, and update) to propagate the uncertainty around the missing entries. Experiments in a variety of tabular datasets show that, in general, our novel methodology leads to better results than using off-the-shelf imputation solutions. The performance gap becomes especially noticeable as the number of missing data entries increases.

Generative flow networks (GFlowNets) are powerful samplers for distributions supported in spaces of compositional objects (e.g., sequences and graphs), with applications ranging from the design of biological sequences to causal discovery. However, there are no principled approaches to deal with GFlowNets in federated settings, where the target distribution results from a combination of (possibly sensitive) rewards from different parties. To fill this gap, we propose *federated contrastive GFlowNet* (FC-GFlowNet), a divide-and-conquer framework for federated learning of GFlowNets, requiring a single communication step. First, each client learns a GFlowNet locally to sample proportionally to their reward. Then, the server gathers the local policy networks and aggregates them to enforce *federated balance* (FB), which provably ensures the correctness of FC-GFlowNet. Additionally, our theoretical analysis builds on a novel concept, which we coin *contrastive balance*, that imposes necessary and sufficient conditions for the correctness of general (non-federated) GFlowNets. We empirically attest the performance of FC-GFlowNets in four controlled settings, including grid-world, sequence, and multiset generation, and Bayesian phylogenetic inference. Our experiments also suggest that, in some cases, the contrastive balance objective can accelerate the training of conventional GFlowNets.

Deep kernel learning (DKL) marries the uncertainty quantification of Gaussian processes (GPs) and the representational power of deep neural networks. However, training DKL is challenging and often leads to overfitting. Most notably, DKL often learns “non-local” kernels — incurring spurious correlations. To remedy this issue, we propose using amortized inducing points and a parameter-sharing scheme, which ties together the amortization and DKL networks. This design imposes an explicit dependency between the ELBO’s model fit and capacity terms. In turn, this prevents the former from dominating the optimization procedure and incurring the aforementioned spurious correlations. Extensive experiments show that our resulting method,amortized varitionalDKL (AVDKL), i) consistently outperforms DKL and standard GPs for tabular data; ii) achieves significantly higher accuracy than DKL in node classification tasks; and iii) leads to substantially better accuracy and negative log-likelihood than DKL on CIFAR100.

Deep neural networks are notoriously miscalibrated, i.e., their outputs do not reflect the true probability of the event we aim to predict. While networks for tabular or image data are usually overconfident, recent works have shown that graph neural networks (GNNs) show the opposite behavior for node-level classification. But what happens when we are predicting links? We show that, in this case, GNNs often exhibit a mixed behavior. More specifically, they may be overconfident in negative predictions while being underconfident in positive ones. Based on this observation, we propose IN-N-OUT, the first-ever method to calibrate GNNs for link prediction. IN-N-OUT is based on two simple intuitions: i) attributing true/false labels to an edge while respecting a GNNs prediction should cause but small fluctuations in that edge's embedding; and, conversely, ii) if we label that same edge contradicting our GNN, embeddings should change more substantially. An extensive experimental campaign shows that IN-N-OUT significantly improves the calibration of GNNs in link prediction, consistently outperforming the baselines available – which are not designed for this specific task.

GFlowNets are a promising alternative to MCMC sampling for discrete compositional random variables. Training GFlowNets requires repeated evaluations of the unnormalized target distribution, or reward function. However, for large-scale posterior sampling, this may be prohibitive since it incurs traversing the data several times. Moreover, if the data are distributed across clients, employing standard GFlowNets leads to intensive client-server communication. To alleviate both these issues, we propose _embarrassingly parallel_ GFlowNet (EP-GFlowNet). EP-GFlowNet is a provably correct divide-and-conquer method to sample from product distributions of the form $R(\cdot) \propto R_1(\cdot) ... R_N(\cdot)$ --- e.g., in parallel or federated Bayes, where each $R_n$ is a local posterior defined on a data partition. First, in parallel, we train a local GFlowNet targeting each $R_n$ and send the resulting models to the server. Then, the server learns a global GFlowNet by enforcing our newly proposed _aggregating balance_ condition, requiring a single communication step. Importantly, EP-GFlowNets can also be applied to multi-objective optimization and model reuse. Our experiments illustrate the effectiveness of EP-GFlowNets on multiple tasks, including parallel Bayesian phylogenetics, multi-objective multiset and sequence generation, and federated Bayesian structure learning.

Exception handling (EH) bugs stem from incorrect usage of exception handling mechanisms (EHMs) and often incur severe consequences (e.g., system downtime, data loss, and security risk). Tracking EH bugs is particularly relevant for contemporary systems (e.g., cloud- and AI-based systems), in which the software’s sophisticated logic is an additional threat to the correct use of the EHM. On top of that, bug reporters seldom can tag EH bugs — since it may require an encompassing knowledge of the software’s EH strategy. Surprisingly, to the best of our knowledge, there is no automated procedure to identify EH bugs from report descriptions. First, we aim to evaluate the extent to which Natural Language Processing (NLP) and Machine Learning (ML) can be used to reliably label EH bugs using the text fields from bug reports (e.g., summary, description, and comments). Second, we aim to provide a reliably labeled dataset that the community can use in future endeavors. Overall, we expect our work to raise the community’s awareness regarding the importance of EH bugs. We manually analyzed 4,516 bug reports from the four main components of Apache’s Hadoop project, out of which we labeled ≈ 20% (943) as EH bugs. We also labeled 2,584 non-EH bugs analyzing their bug-fixing code and creating a dataset composed of 7,100 bug reports. Then, we used word embedding techniques (Bag-of-Words and TF-IDF) to summarize the textual fields of bug reports. Subsequently, we used these embeddings to fit five classes of ML methods and evaluate them on unseen data. We also evaluated a pre-trained transformer-based model using the complete textual fields. We have also evaluated whether considering only EH keywords is enough to achieve high predictive performance. Our results show that using a pre-trained DistilBERT with a linear layer trained with our proposed dataset can reasonably label EH bugs, achieving ROC-AUC scores of up to 0.88. The combination of NLP and ML traditional techniques achieved ROC-AUC scores of up to 0.74 and recall up to 0.56. As a sanity check, we also evaluate methods using embeddings extracted solely from keywords. Considering ROC-AUC as the primary concern, for the majority of ML methods tested, the analysis suggests that keywords alone are not sufficient to characterize reports of EH bugs, although this can change based on other metrics (such as recall and precision) or ML methods (e.g., Random Forest). To the best of our knowledge, this is the first study addressing the problem of automatic labeling of EH bugs. Based on our results, we can conclude that the use of ML techniques, specially transformer-base models, sounds promising to automate the task of labeling EH bugs. Overall, we hope (i) that our work will contribute towards raising awareness around EH bugs; and (ii) that our (publicly available) dataset will serve as a benchmarking dataset, paving the way for follow-up works. Additionally, our findings can be used to build tools that help maintainers flesh out EH bugs during the triage process.

Generative Flow Networks (GFlowNets) are amortized samplers of unnormalized distributions over compositional objects with applications to causal discovery, NLP, and drug design. Recently, it was shown that GFlowNets can be framed as a hierarchical variational inference (HVI) method for discrete distributions. Despite this equivalence, attempts to train GFlowNets using traditional divergence measures as learning objectives were unsuccessful. Instead, current approaches for training these models rely on minimizing the log-squared difference between a proposal (forward policy) and a target (backward policy) distributions. In this work, we first formally extend the relationship between GFlowNets and HVI to distributions on arbitrary measurable topological spaces. Then, we empirically show that the ineffectiveness of divergence-based learning of GFlowNets is due to large gradient variance of the corresponding stochastic objectives. To address this issue, we devise a collection of provably variance-reducing control variates for gradient estimation based on the REINFORCE leave-one-out estimator. Our experimental results suggest that the resulting algorithms often accelerate training convergence when compared against previous approaches. All in all, our work contributes by narrowing the gap between GFlowNet training and HVI, paving the way for algorithmic advancements inspired by the divergence minimization viewpoint.

Topological deep learning (TDL) has recently emerged as a family of neural networks for data on topological domains (e.g., graphs and cellular complexes). In the context of graph data, TDL provides a recipe for leveraging high-order information, e.g., from cliques, to boost the expressiveness of graph neural networks. This additional power comes at a computational cost, stemming from message passing between higher-order structures. In this paper, we alleviate the computing toll of TDL by proposing SimpleXCN, a linear convolutional model for simplicial complexes. SimpleXCN derives naturally from removing non-linearities of base TDL architectures. Despite its simplicity, SimpleXCN remains competitive with established topological neural networks in node classification tasks. Our experiments show a reduction of approximately 25% in time and 10 times in the number of parameters, and consequently a significant reduction in memory usage of up to 50% compared to other SNNs.

Bayes' rule naturally allows for inference refinement in a streaming fashion, without the need to recompute posteriors from scratch whenever new data arrives. In principle, Bayesian streaming is straightforward: we update our prior with the available data and use the resulting posterior as a prior when processing the next data chunk. In practice, however, this recipe entails i) approximating an intractable posterior at each time step; and ii) encapsulating results appropriately to allow for posterior propagation. For continuous state spaces, variational inference (VI) is particularly convenient due to its scalability and the tractability of variational posteriors, For discrete state spaces, however, state-of-the-art VI results in analytically intractable approximations that are ill-suited for streaming settings. To enable streaming Bayesian inference over discrete parameter spaces, we propose streaming Bayes GFlowNets (abbreviated as SB-GFlowNets) by leveraging the recently proposed GFlowNets --- a powerful class of amortized samplers for discrete compositional objects. Notably, SB-GFlowNet approximates the initial posterior using a standard GFlowNet and subsequently updates it using a tailored procedure that requires only the newly observed data. Our case studies in linear preference learning and phylogenetic inference showcase the effectiveness of SB-GFlowNets in sampling from an unnormalized posterior in a streaming setting. As expected, we also observe that SB-GFlowNets is significantly faster than repeatedly training a GFlowNet from scratch to sample from the full posterior.

Explaining node predictions in graph neural networks (GNNs) often boils down to finding graph substructures that preserve predictions. Finding these structures usually implies back-propagating through the GNN, bonding the complexity (e.g., number of layers) of the GNN to the cost of explaining it. This naturally begs the question: Can we break this bond by explaining a simpler surrogate GNN? To answer the question, we propose Distill n' Explain (DnX). First, DnX learns a surrogate GNN via knowledge distillation. Then, DnX extracts node or edge-level explanations by solving a simple convex program. We also propose FastDnX, a faster version of DnX that leverages the linear decomposition of our surrogate model. Experiments show that DnX and FastDnX often outperform state-of-the-art GNN explainers while being orders of magnitude faster. Additionally, we support our empirical findings with theoretical results linking the quality of the surrogate model (i.e., distillation error) to the faithfulness of explanations.