Predicting edge weights on graphs has various applications, from transportation systems to social networks. This paper describes a Graph Neural Network (GNN) approach for edge weight prediction with guaranteed coverage. We leverage conformal prediction to calibrate the GNN outputs and produce valid prediction intervals. We handle data heteroscedasticity through error reweighting and Conformalized Quantile Regression (CQR). We compare the performance of our method against baseline techniques on real-world transportation datasets. Our approach has better coverage and efficiency than all baselines and showcases robustness and adaptability.
Conformal prediction (CP) is an emerging uncertainty quantification framework that allows us to construct a prediction set to cover the true label with a pre-specified marginal or conditional probability. Although the valid coverage guarantee has been extensively studied for classification problems, CP often produces large prediction sets which may not be practically useful. This issue is exacerbated for the setting of class-conditional coverage on imbalanced classification tasks with many and/or imbalanced classes. This paper proposes the Rank Calibrated Class-conditional CP (RC3P) algorithm to reduce the prediction set sizes to achieve class-conditional coverage, where the valid coverage holds for each class. In contrast to the standard class-conditional CP (CCP) method that uniformly thresholds the class-wise conformity score for each class, the augmented label rank calibration step allows RC3P to selectively iterate this class-wise thresholding subroutine only for a subset of classes whose class-wise top-k error is small. We prove that agnostic to the classifier and data distribution, RC3P achieves class-wise coverage. We also show that RC3P reduces the size of prediction sets compared to the CCP method. Comprehensive experiments on multiple real-world datasets demonstrate that RC3P achieves class-wise coverage and 26.25% reduction in prediction set sizes on average.
We develop a method to generate prediction sets with a guaranteed coverage rate that is robust to corruptions in the training data, such as missing or noisy variables. Our approach builds on conformal prediction, a powerful framework to construct prediction sets that are valid under the i.i.d assumption. Importantly, naively applying conformal prediction does not provide reliable predictions in this setting, due to the distribution shift induced by the corruptions. To account for the distribution shift, we assume access to privileged information (PI). The PI is formulated as additional features that explain the distribution shift, however, they are only available during training and absent at test time. We approach this problem by introducing a novel generalization of weighted conformal prediction and support our method with theoretical coverage guarantees. Empirical experiments on both real and synthetic datasets indicate that our approach achieves a valid coverage rate and constructs more informative predictions compared to existing methods, which are not supported by theoretical guarantees.
Mixing (or prior) density estimation is an important problem in machine learning and statistics, especially in empirical Bayes $g$-modeling where accurately estimating the prior is necessary for making good posterior inferences. In this paper, we propose neural-$g$, a new neural network-based estimator for $g$-modeling. Neural-$g$ uses a softmax output layer to ensure that the estimated prior is a valid probability density. Under default hyperparameters, we show that neural-$g$ is very flexible and capable of capturing many unknown densities, including those with flat regions, heavy tails, and/or discontinuities. In contrast, existing methods struggle to capture all of these prior shapes. We provide justification for neural-$g$ by establishing a new universal approximation theorem regarding the capability of neural networks to learn arbitrary probability mass functions. To accelerate convergence of our numerical implementation, we utilize a weighted average gradient descent approach to update the network parameters. Finally, we extend neural-$g$ to multivariate prior density estimation. We illustrate the efficacy of our approach through simulations and analyses of real datasets. A software package to implement neural-$g$ is publicly available at https://github.com/shijiew97/neuralG.
Despite the multifaceted recent advances in interventional causal representation learning (CRL), they primarily focus on the stylized assumption of single-node interventions. This assumption is not valid in a wide range of applications, and generally, the subset of nodes intervened in an interventional environment is fully unknown. This paper focuses on interventional CRL under unknown multi-node (UMN) interventional environments and establishes the first identifiability results for general latent causal models (parametric or nonparametric) under stochastic interventions (soft or hard) and linear transformation from the latent to observed space. Specifically, it is established that given sufficiently diverse interventional environments, (i) identifiability up to ancestors is possible using only soft interventions, and (ii) perfect identifiability is possible using hard interventions. Remarkably, these guarantees match the best-known results for more restrictive single-node interventions. Furthermore, CRL algorithms are also provided that achieve the identifiability guarantees. A central step in designing these algorithms is establishing the relationships between UMN interventional CRL and score functions associated with the statistical models of different interventional environments. Establishing these relationships also serves as constructive proof of the identifiability guarantees.
Recent advances in large language models (LLMs) for code applications have demonstrated remarkable zero-shot fluency and instruction following on challenging code related tasks ranging from test case generation to self-repair. Unsurprisingly, however, models struggle to compose syntactically valid programs in programming languages unrepresented in pre-training, referred to as very low-resource Programming Languages (VLPLs). VLPLs appear in crucial settings, including domain-specific languages for internal tools, tool-chains for legacy languages, and formal verification frameworks. Inspired by a technique called natural programming elicitation, we propose designing an intermediate language that LLMs "naturally" know how to use and which can be automatically compiled to a target VLPL. When LLMs generate code that lies outside of this intermediate language, we use compiler techniques to repair the code into programs in the intermediate language. Overall, we introduce \emph{synthetic programming elicitation and compilation} (SPEAC), an approach that enables LLMs to generate syntactically valid code even for VLPLs. We empirically evaluate the performance of SPEAC in a case study for the UCLID5 formal verification language and find that, compared to existing retrieval and fine-tuning baselines, SPEAC produces syntactically correct programs more frequently and without sacrificing semantic correctness.
In 1986, Robert LaLonde published an article comparing nonexperimental estimates to experimental benchmarks (LaLonde 1986). He concluded that the nonexperimental methods at the time could not systematically replicate experimental benchmarks, casting doubt on their credibility. Following LaLonde's critical assessment, there have been significant methodological advances and practical changes, including (i) an emphasis on the unconfoundedness assumption separated from functional form considerations, (ii) a focus on the importance of overlap in covariate distributions, (iii) the introduction of propensity score-based methods leading to doubly robust estimators, (iv) methods for estimating and exploiting treatment effect heterogeneity, and (v) a greater emphasis on validation exercises to bolster research credibility. To demonstrate the practical lessons from these advances, we reexamine the LaLonde data. We show that modern methods, when applied in contexts with sufficient covariate overlap, yield robust estimates for the adjusted differences between the treatment and control groups. However, this does not imply that these estimates are causally interpretable. To assess their credibility, validation exercises (such as placebo tests) are essential, whereas goodness-of-fit tests alone are inadequate. Our findings highlight the importance of closely examining the assignment process, carefully inspecting overlap, and conducting validation exercises when analyzing causal effects with nonexperimental data.
Conformal Prediction (CP) algorithms estimate the uncertainty of a prediction model by calibrating its outputs on labeled data. The same calibration scheme usually applies to any model and data without modifications. The obtained prediction intervals are valid by construction but could be inefficient, i.e. unnecessarily big, if the prediction errors are not uniformly distributed over the input space. We present a general scheme to localize the intervals by training the calibration process. The standard prediction error is replaced by an optimized distance metric that depends explicitly on the object attributes. Learning the optimal metric is equivalent to training a Normalizing Flow that acts on the joint distribution of the errors and the inputs. Unlike the Error Reweighting CP algorithm of Papadopoulos et al. (2008), the framework allows estimating the gap between nominal and empirical conditional validity. The approach is compatible with existing locally-adaptive CP strategies based on re-weighting the calibration samples and applies to any point-prediction model without retraining.
Constructing valid prediction intervals rather than point estimates is a well-established approach for uncertainty quantification in the regression setting. Models equipped with this capacity output an interval of values in which the ground truth target will fall with some prespecified probability. This is an essential requirement in many real-world applications where simple point predictions' inability to convey the magnitude and frequency of errors renders them insufficient for high-stakes decisions. Quantile regression is a leading approach for obtaining such intervals via the empirical estimation of quantiles in the (non-parametric) distribution of outputs. This method is simple, computationally inexpensive, interpretable, assumption-free, and effective. However, it does require that the specific quantiles being learned are chosen a priori. This results in (a) intervals that are arbitrarily symmetric around the median which is sub-optimal for realistic skewed distributions, or (b) learning an excessive number of intervals. In this work, we propose Relaxed Quantile Regression (RQR), a direct alternative to quantile regression based interval construction that removes this arbitrary constraint whilst maintaining its strengths. We demonstrate that this added flexibility results in intervals with an improvement in desirable qualities (e.g. mean width) whilst retaining the essential coverage guarantees of quantile regression.
Unlike traditional statistical methods, Conformal Prediction (CP) allows for the determination of valid and accurate confidence levels associated with individual predictions based only on exchangeability of the data. We here introduce a new feature selection method that takes advantage of the CP framework. Our proposal, named Conformal Recursive Feature Elimination (CRFE), identifies and recursively removes features that increase the non-conformity of a dataset. We also present an automatic stopping criterion for CRFE, as well as a new index to measure consistency between subsets of features. CRFE selections are compared to the classical Recursive Feature Elimination (RFE) method on several multiclass datasets by using multiple partitions of the data. The results show that CRFE clearly outperforms RFE in half of the datasets, while achieving similar performance in the rest. The automatic stopping criterion provides subsets of effective and non-redundant features without computing any classification performance.
Dynamic graphs provide a flexible data abstraction for modelling many sorts of real-world systems, such as transport, trade, and social networks. Graph neural networks (GNNs) are powerful tools allowing for different kinds of prediction and inference on these systems, but getting a handle on uncertainty, especially in dynamic settings, is a challenging problem. In this work we propose to use a dynamic graph representation known in the tensor literature as the unfolding, to achieve valid prediction sets via conformal prediction. This representation, a simple graph, can be input to any standard GNN and does not require any modification to existing GNN architectures or conformal prediction routines. One of our key contributions is a careful mathematical consideration of the different inference scenarios which can arise in a dynamic graph modelling context. For a range of practically relevant cases, we obtain valid prediction sets with almost no assumptions, even dispensing with exchangeability. In a more challenging scenario, which we call the semi-inductive regime, we achieve valid prediction under stronger assumptions, akin to stationarity. We provide real data examples demonstrating validity, showing improved accuracy over baselines, and sign-posting different failure modes which can occur when those assumptions are violated.
While existing depression prediction methods based on deep learning show promise, their practical application is hindered by the lack of trustworthiness, as these deep models are often deployed as black box models, leaving us uncertain on the confidence of their predictions. For high-risk clinical applications like depression prediction, uncertainty quantification is essential in decision-making. In this paper, we introduce conformal depression prediction (CDP), a depression prediction method with uncertainty quantification based on conformal prediction (CP), giving valid confidence intervals with theoretical coverage guarantees for the model predictions. CDP is a plug-and-play module that requires neither model retraining nor an assumption about the depression data distribution. As CDP provides only an average coverage guarantee across all inputs rather than per-input performance guarantee, we further propose CDP-ACC, an improved conformal prediction with approximate conditional coverage. CDP-ACC firstly estimates the prediction distribution through neighborhood relaxation, and then introduces a conformal score function by constructing nested sequences, so as to provide a tighter prediction interval adaptive to specific input. We empirically demonstrate the application of CDP in uncertainty-aware facial depression prediction, as well as the effectiveness and superiority of CDP-ACC on the AVEC 2013 and AVEC 2014 datasets. Our code is publicly available at https://github.com/PushineLee/CDP.
Conformal Prediction (CP) is a popular uncertainty quantification method that provides distribution-free, statistically valid prediction sets, assuming that training and test data are exchangeable. In such a case, CP's prediction sets are guaranteed to cover the (unknown) true test output with a user-specified probability. Nevertheless, this guarantee is violated when the data is subjected to adversarial attacks, which often result in a significant loss of coverage. Recently, several approaches have been put forward to recover CP guarantees in this setting. These approaches leverage variations of randomised smoothing to produce conservative sets which account for the effect of the adversarial perturbations. They are, however, limited in that they only support $\ell^2$-bounded perturbations and classification tasks. This paper introduces VRCP (Verifiably Robust Conformal Prediction), a new framework that leverages recent neural network verification methods to recover coverage guarantees under adversarial attacks. Our VRCP method is the first to support perturbations bounded by arbitrary norms including $\ell^1$, $\ell^2$, and $\ell^\infty$, as well as regression tasks. We evaluate and compare our approach on image classification tasks (CIFAR10, CIFAR100, and TinyImageNet) and regression tasks for deep reinforcement learning environments. In every case, VRCP achieves above nominal coverage and yields significantly more efficient and informative prediction regions than the SotA.
Machine learning algorithms have become indispensable in today's world. They support and accelerate the way we make decisions based on the data at hand. This acceleration means that data structures that were valid at one moment could no longer be valid in the future. With these changing data structures, it is necessary to adapt machine learning (ML) systems incrementally to the new data. This is done with the use of online learning or continuous ML technologies. While deep learning technologies have shown exceptional performance on predefined datasets, they have not been widely applied to online, streaming, and continuous learning. In this retrospective of our tutorial titled Opportunities and Challenges of Online Deep Learning held at ECML PKDD 2023, we provide a brief overview of the opportunities but also the potential pitfalls for the application of neural networks in online learning environments using the frameworks River and Deep-River.
Directed acyclic graph (DAG) learning is a central task in structure discovery and causal inference. Although the field has witnessed remarkable advances over the past few years, it remains statistically and computationally challenging to learn a single (point estimate) DAG from data, let alone provide uncertainty quantification. We address the difficult task of quantifying graph uncertainty by developing a Bayesian variational inference framework based on novel, provably valid distributions that have support directly on the space of sparse DAGs. These distributions, which we use to define our prior and variational posterior, are induced by a projection operation that maps an arbitrary continuous distribution onto the space of sparse weighted acyclic adjacency matrices. While this projection is combinatorial, it can be solved efficiently using recent continuous reformulations of acyclicity constraints. We empirically demonstrate that our method, ProDAG, can outperform state-of-the-art alternatives in both accuracy and uncertainty quantification.
This paper introduces a conformal inference method to evaluate uncertainty in classification by generating prediction sets with valid coverage conditional on adaptively chosen features. These features are carefully selected to reflect potential model limitations or biases. This can be useful to find a practical compromise between efficiency -- by providing informative predictions -- and algorithmic fairness -- by ensuring equalized coverage for the most sensitive groups. We demonstrate the validity and effectiveness of this method on simulated and real data sets.
We introduce a data-driven approach to computing finite bisimulations for state transition systems with very large, possibly infinite state space. Our novel technique computes stutter-insensitive bisimulations of deterministic systems, which we characterize as the problem of learning a state classifier together with a ranking function for each class. Our procedure learns a candidate state classifier and candidate ranking functions from a finite dataset of sample states; then, it checks whether these generalise to the entire state space using satisfiability modulo theory solving. Upon the affirmative answer, the procedure concludes that the classifier constitutes a valid stutter-insensitive bisimulation of the system. Upon a negative answer, the solver produces a counterexample state for which the classifier violates the claim, adds it to the dataset, and repeats learning and checking in a counterexample-guided inductive synthesis loop until a valid bisimulation is found. We demonstrate on a range of benchmarks from reactive verification and software model checking that our method yields faster verification results than alternative state-of-the-art tools in practice. Our method produces succinct abstractions that enable an effective verification of linear temporal logic without next operator, and are interpretable for system diagnostics.
Identifying the causal pathways of unfairness is a critical objective for improving policy design and algorithmic decision-making. Prior work in causal fairness analysis often requires knowledge of the causal graph, hindering practical applications in complex or low-knowledge domains. Moreover, global discovery methods that learn causal structure from data can display unstable performance on finite samples, preventing robust fairness conclusions. To mitigate these challenges, we introduce local discovery for direct discrimination (LD3): a method that uncovers structural evidence of direct unfairness by identifying the causal parents of an outcome variable. LD3 performs a linear number of conditional independence tests relative to variable set size, and allows for latent confounding under the sufficient condition that all parents of the outcome are observed. We show that LD3 returns a valid adjustment set (VAS) under a new graphical criterion for the weighted controlled direct effect, a qualitative indicator of direct discrimination. LD3 limits unnecessary adjustment, providing interpretable VAS for assessing unfairness. We use LD3 to analyze causal fairness in two complex decision systems: criminal recidivism prediction and liver transplant allocation. LD3 was more time-efficient and returned more plausible results on real-world data than baselines, which took 46$\times$ to 5870$\times$ longer to execute.
Graph Neural Networks have achieved remarkable accuracy in semi-supervised node classification tasks. However, these results lack reliable uncertainty estimates. Conformal prediction methods provide a theoretical guarantee for node classification tasks, ensuring that the conformal prediction set contains the ground-truth label with a desired probability (e.g., 95%). In this paper, we empirically show that for each node, aggregating the non-conformity scores of nodes with the same label can improve the efficiency of conformal prediction sets while maintaining valid marginal coverage. This observation motivates us to propose a novel algorithm named Similarity-Navigated Adaptive Prediction Sets (SNAPS), which aggregates the non-conformity scores based on feature similarity and structural neighborhood. The key idea behind SNAPS is that nodes with high feature similarity or direct connections tend to have the same label. By incorporating adaptive similar nodes information, SNAPS can generate compact prediction sets and increase the singleton hit ratio (correct prediction sets of size one). Moreover, we theoretically provide a finite-sample coverage guarantee of SNAPS. Extensive experiments demonstrate the superiority of SNAPS, improving the efficiency of prediction sets and singleton hit ratio while maintaining valid coverage.
The development of Artificial Intelligence for healthcare is of great importance. Models can sometimes achieve even superior performance to human experts, however, they can reason based on spurious features. This is not acceptable to the experts as it is expected that the models catch the valid patterns in the data following domain expertise. In the work, we analyse whether Deep Learning (DL) models for vision follow the histopathologists' practice so that when diagnosing a part of a lesion, they take into account also the surrounding tissues which serve as context. It turns out that the performance of DL models significantly decreases when the amount of contextual information is limited, therefore contextual information is valuable at prediction time. Moreover, we show that the models sometimes behave in an unstable way as for some images, they change the predictions many times depending on the size of the context. It may suggest that partial contextual information can be misleading.