Nicolo Colombo

Nicolo Colombo

We address the problem of making Conformal Prediction (CP) intervals locally adaptive. Most existing methods focus on approximating the object-conditional validity of the intervals by partitioning or re-weighting the calibration set. Our strategy is new and conceptually different. Instead of re-weighting the calibration data, we redefine the conformity measure through a trainable change of variables, $A \to φ_X(A)$, that depends explicitly on the object attributes, $X$. Under certain conditions and if $φ_X$ is monotonic in $A$ for any $X$, the transformations produce prediction intervals that are guaranteed to be marginally valid and have $X$-dependent sizes. We describe how to parameterize and train $φ_X$ to maximize the interval efficiency. Contrary to other CP-aware training methods, the objective function is smooth and can be minimized through standard gradient methods without approximations.

Rui Luo, Nicolo Colombo

Conformal Prediction (CP) is a powerful framework for constructing prediction sets with guaranteed coverage. However, recent studies have shown that integrating confidence calibration with CP can lead to a degradation in efficiency. In this paper, We propose an adaptive approach that considers the classifier's uncertainty and employs entropy-based reweighting to enhance the efficiency of prediction sets for conformal classification. Our experimental results demonstrate that this method significantly improves efficiency.

Lingxuan Tang, Rui Luo, Zhixin Zhou et al.

This paper investigates the application of probabilistic prediction methodologies in route planning within a road network context. Specifically, we introduce the Conformalized Quantile Regression for Graph Autoencoders (CQR-GAE), which leverages the conformal prediction technique to offer a coverage guarantee, thus improving the reliability and robustness of our predictions. By incorporating uncertainty sets derived from CQR-GAE, we substantially improve the decision-making process in route planning under a robust optimization framework. We demonstrate the effectiveness of our approach by applying the CQR-GAE model to a real-world traffic scenario. The results indicate that our model significantly outperforms baseline methods, offering a promising avenue for advancing intelligent transportation systems.

Zheng Zhang, Jie Bao, Zhixin Zhou et al.

Graph Neural Networks (GNNs) excel at modeling relational data but face significant challenges in high-stakes domains due to unquantified uncertainty. Conformal prediction (CP) offers statistical coverage guarantees, but existing methods often produce overly conservative prediction intervals that fail to account for graph heteroscedasticity and structural biases. While residual reweighting CP variants address some of these limitations, they neglect graph topology, cluster-specific uncertainties, and risk data leakage by reusing training sets. To address these issues, we propose Residual Reweighted GNN (RR-GNN), a framework designed to generate minimal prediction sets with provable marginal coverage guarantees. RR-GNN introduces three major innovations to enhance prediction performance. First, it employs Graph-Structured Mondrian CP to partition nodes or edges into communities based on topological features, ensuring cluster-conditional coverage that reflects heterogeneity. Second, it uses Residual-Adaptive Nonconformity Scores by training a secondary GNN on a held-out calibration set to estimate task-specific residuals, dynamically adjusting prediction intervals according to node or edge uncertainty. Third, it adopts a Cross-Training Protocol, which alternates the optimization of the primary GNN and the residual predictor to prevent information leakage while maintaining graph dependencies. We validate RR-GNN on 15 real-world graphs across diverse tasks, including node classification, regression, and edge weight prediction. Compared to CP baselines, RR-GNN achieves improved efficiency over state-of-the-art methods, with no loss of coverage.

Neeraj Mohan Sushma, Yudou Tian, Harshvardhan Mestha et al.

Deep state-space models (Deep SSMs) are becoming popular as effective approaches to model sequence data. They have also been shown to be capable of in-context learning, much like transformers. However, a complete picture of how SSMs might be able to do in-context learning has been missing. In this study, we provide a direct and explicit construction to show that state-space models can perform gradient-based learning and use it for in-context learning in much the same way as transformers. Specifically, we prove that a single structured state-space model layer, augmented with multiplicative input and output gating, can reproduce the outputs of an implicit linear model with least squares loss after one step of gradient descent. We then show a straightforward extension to multi-step linear and non-linear regression tasks. We validate our construction by training randomly initialized augmented SSMs on linear and non-linear regression tasks. The empirically obtained parameters through optimization match the ones predicted analytically by the theoretical construction. Overall, we elucidate the role of input- and output-gating in recurrent architectures as the key inductive biases for enabling the expressive power typical of foundation models. We also provide novel insights into the relationship between state-space models and linear self-attention, and their ability to learn in-context.

Nicolo Colombo

In the Quantum Supremacy regime, quantum computers may overcome classical machines on several tasks if we can estimate, mitigate, or correct unavoidable hardware noise. Estimating the error requires classical simulations, which become unfeasible in the Quantum Supremacy regime. We leverage Machine Learning data-driven approaches and Conformal Prediction, a Machine Learning uncertainty quantification tool known for its mild assumptions and finite-sample validity, to find theoretically valid upper bounds of the fidelity between noiseless and noisy outputs of quantum devices. Under reasonable extrapolation assumptions, the proposed scheme applies to any Quantum Computing hardware, does not require modeling the device's noise sources, and can be used when classical simulations are unavailable, e.g. in the Quantum Supremacy regime.

Rui Luo, Nicolo Colombo

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.

Nicolo Colombo

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.

Nicolo Colombo, Yang Gao

We propose a new gradient-based approach for extracting sub-architectures from a given large model. Contrarily to existing pruning methods, which are unable to disentangle the network architecture and the corresponding weights, our architecture-pruning scheme produces transferable new structures that can be successfully retrained to solve different tasks. We focus on a transfer-learning setup where architectures can be trained on a large data set but very few data points are available for fine-tuning them on new tasks. We define a new gradient-based algorithm that trains architectures of arbitrarily low complexity independently from the attached weights. Given a search space defined by an existing large neural model, we reformulate the architecture search task as a complexity-penalized subset-selection problem and solve it through a two-temperature relaxation scheme. We provide theoretical convergence guarantees and validate the proposed transfer-learning strategy on real data.

Yang Gao, Nicolo Colombo, Wei Wang

Adapting pre-trained neural models to downstream tasks has become the standard practice for obtaining high-quality models. In this work, we propose a novel model adaptation paradigm, adapting by pruning, which prunes neural connections in the pre-trained model to optimise the performance on the target task; all remaining connections have their weights intact. We formulate adapting-by-pruning as an optimisation problem with a differentiable loss and propose an efficient algorithm to prune the model. We prove that the algorithm is near-optimal under standard assumptions and apply the algorithm to adapt BERT to some GLUE tasks. Results suggest that our method can prune up to 50% weights in BERT while yielding similar performance compared to the fine-tuned full model. We also compare our method with other state-of-the-art pruning methods and study the topological differences of their obtained sub-networks.

Nicolo Colombo, Yang Gao

The history of deep learning has shown that human-designed problem-specific networks can greatly improve the classification performance of general neural models. In most practical cases, however, choosing the optimal architecture for a given task remains a challenging problem. Recent architecture-search methods are able to automatically build neural models with strong performance but fail to fully appreciate the interaction between neural architecture and weights. This work investigates the problem of disentangling the role of the neural structure and its edge weights, by showing that well-trained architectures may not need any link-specific fine-tuning of the weights. We compare the performance of such weight-free networks (in our case these are binary networks with {0, 1}-valued weights) with random, weight-agnostic, pruned and standard fully connected networks. To find the optimal weight-agnostic network, we use a novel and computationally efficient method that translates the hard architecture-search problem into a feasible optimization problem.More specifically, we look at the optimal task-specific architectures as the optimal configuration of binary networks with {0, 1}-valued weights, which can be found through an approximate gradient descent strategy. Theoretical convergence guarantees of the proposed algorithm are obtained by bounding the error in the gradient approximation and its practical performance is evaluated on two real-world data sets. For measuring the structural similarities between different architectures, we use a novel spectral approach that allows us to underline the intrinsic differences between real-valued networks and weight-free architectures.

Nicolo Colombo, Vladimir Vovk

Efficiency criteria for conformal prediction, such as \emph{observed fuzziness} (i.e., the sum of p-values associated with false labels), are commonly used to \emph{evaluate} the performance of given conformal predictors. Here, we investigate whether it is possible to exploit efficiency criteria to \emph{learn} classifiers, both conformal predictors and point classifiers, by using such criteria as training objective functions. The proposed idea is implemented for the problem of binary classification of hand-written digits. By choosing a 1-dimensional model class (with one real-valued free parameter), we can solve the optimization problems through an (approximate) exhaustive search over (a discrete version of) the parameter space. Our empirical results suggest that conformal predictors trained by minimizing their observed fuzziness perform better than conformal predictors trained in the traditional way by minimizing the \emph{prediction error} of the corresponding point classifier. They also have a reasonable performance in terms of their prediction error on the test set.

Nicolo Colombo

We address the problem of merging graph and feature-space information while learning a metric from structured data. Existing algorithms tackle the problem in an asymmetric way, by either extracting vectorized summaries of the graph structure or adding hard constraints to feature-space algorithms. Following a different path, we define a metric regression scheme where we train metric-constrained linear combinations of dissimilarity matrices. The idea is that the input matrices can be pre-computed dissimilarity measures obtained from any kind of available data (e.g. node attributes or edge structure). As the model inputs are distance measures, we do not need to assume the existence of any underlying feature space. Main challenge is that metric constraints (especially positive-definiteness and sub-additivity), are not automatically respected if, for example, the coefficients of the linear combination are allowed to be negative. Both positive and sub-additive constraints are linear inequalities, but the computational complexity of imposing them scales as O(D3), where D is the size of the input matrices (i.e. the size of the data set). This becomes quickly prohibitive, even when D is relatively small. We propose a new graph-based technique for optimizing under such constraints and show that, in some cases, our approach may reduce the original computational complexity of the optimization process by one order of magnitude. Contrarily to existing methods, our scheme applies to any (possibly non-convex) metric-constrained objective function.

Nicolo Colombo, Ricardo Silva, Soong M Kang et al.

We consider problems in which a system receives external \emph{perturbations} from time to time. For instance, the system can be a train network in which particular lines are repeatedly disrupted without warning, having an effect on passenger behavior. The goal is to predict changes in the behavior of the system at particular points of interest, such as passenger traffic around stations at the affected rails. We assume that the data available provides records of the system functioning at its "natural regime" (e.g., the train network without disruptions) and data on cases where perturbations took place. The inference problem is how information concerning perturbations, with particular covariates such as location and time, can be generalized to predict the effect of novel perturbations. We approach this problem from the point of view of a mapping from the counterfactual distribution of the system behavior without disruptions to the distribution of the disrupted system. A variant on \emph{distribution regression} is developed for this setup.

Yin Cheng Ng, Nicolo Colombo, Ricardo Silva

We propose a data-efficient Gaussian process-based Bayesian approach to the semi-supervised learning problem on graphs. The proposed model shows extremely competitive performance when compared to the state-of-the-art graph neural networks on semi-supervised learning benchmark experiments, and outperforms the neural networks in active learning experiments where labels are scarce. Furthermore, the model does not require a validation data set for early stopping to control over-fitting. Our model can be viewed as an instance of empirical distribution regression weighted locally by network connectivity. We further motivate the intuitive construction of the model with a Bayesian linear model interpretation where the node features are filtered by an operator related to the graph Laplacian. The method can be easily implemented by adapting off-the-shelf scalable variational inference algorithms for Gaussian processes.

Nicolo Colombo, Nikos Vlassis

We consider the problem of approximate joint triangularization of a set of noisy jointly diagonalizable real matrices. Approximate joint triangularizers are commonly used in the estimation of the joint eigenstructure of a set of matrices, with applications in signal processing, linear algebra, and tensor decomposition. By assuming the input matrices to be perturbations of noise-free, simultaneously diagonalizable ground-truth matrices, the approximate joint triangularizers are expected to be perturbations of the exact joint triangularizers of the ground-truth matrices. We provide a priori and a posteriori perturbation bounds on the `distance' between an approximate joint triangularizer and its exact counterpart. The a priori bounds are theoretical inequalities that involve functions of the ground-truth matrices and noise matrices, whereas the a posteriori bounds are given in terms of observable quantities that can be computed from the input matrices. From a practical perspective, the problem of finding the best approximate joint triangularizer of a set of noisy matrices amounts to solving a nonconvex optimization problem. We show that, under a condition on the noise level of the input matrices, it is possible to find a good initial triangularizer such that the solution obtained by any local descent-type algorithm has certain global guarantees. Finally, we discuss the application of approximate joint matrix triangularization to canonical tensor decomposition and we derive novel estimation error bounds.