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Uncertainty quantification (UQ) in graph neural networks (GNNs) is crucial in high-stakes domains but remains a significant challenge. In graph settings, message passing often relies on strong assumptions such as exchangeability, which are rarely satisfied in practice. Moreover, achieving reliable UQ typically requires costly resampling or post-hoc calibration. To address these issues, we introduce Quantile-free Prediction Interval GNN (QpiGNN), a framework that builds on quantile regression (QR) to enable GNN-based UQ by directly optimizing coverage and interval width without requiring quantile inputs or post-processing. QpiGNN employs a dual-head architecture that decouples prediction and uncertainty, and is trained with label-only supervision through a quantile-free joint loss. This design allows efficient training and yields robust prediction intervals, with theoretical guarantees of asymptotic coverage and near-optimal width under mild assumptions. Experiments on 19 synthetic and real-world benchmarks show QpiGNN achieves average 22% higher coverage and 50% narrower intervals than baselines, while ensuring efficiency and robustness to noise and structural shifts.

We study the problem of estimating the effect function for a continuous treatment, which maps each treatment value to a population-averaged outcome. A central challenge in this setting is confounding: treatment assignment often depends on covariates, creating selection bias that makes direct regression of the response on treatment unreliable. To address this issue, we propose a two-stage kernel ridge regression method. In the first stage, we learn a model for the response as a function of both treatment and covariates; in the second stage, we use this model to construct pseudo-outcomes that correct for distribution shift, and then fit a second model to estimate the treatment effect. Although the response varies with both treatment and covariates, the induced effect function obtained by averaging over covariates is typically much simpler, and our estimator adapts to this structure. Our optimal learning bounds are achieved without estimating the conditional treatment density, thereby bypassing a major bottleneck in existing methods. Furthermore, we introduce a fully data-driven model selection procedure that achieves provable adaptivity to both the unknown degree of overlap and the spectral decay of the underlying kernel.

Symbolic Regression aims to discover interpretable mathematical expressions from data. Equation Learner (EQL) is a gradient-based method with strong fitting capability and expressive potential, yet it often activates redundant operators as model complexity grows, leading to over-complex expressions and unstable equation recovery. We analyze a gradient residual issue induced by operators that do not vanish at zero, which can prevent the ideal sparse expression from acting as a stable attractor during training and bias training toward unnecessarily complex structures, making exact recovery highly unreliable in practice. To address this, we propose EQL-Z, a structurally controllable symbolic regression framework. EQL-Z enforces zero-point constraints via zero-point consistent operator transformations to eliminate residual gradients on silent paths, and performs a small-to-large structure search that grows depth/width from a compact seed under a complexity-penalized validation score. After selecting a compact structure, we apply BFGS fine-tuning to refine coefficients. Experiments on synthetic and real-world datasets show that EQL-Z substantially improves exact equation recovery and in-/out-of-distribution generalization over vanilla EQL, achieving performance close to leading symbolic regression baselines. Code is available at https://github.com/Caaaa-a/EQL-Z.

Decoding-based regression, which reformulates regression as a sequence generation task, has emerged as a promising paradigm of applying large language models for numerical prediction. However, its progress is hindered by the misalignment between discrete token-level objectives (e.g., cross-entropy) and continuous numerical values. Existing approaches relying on token-level constraints often fail to capture the global magnitude of the target value, limiting their precision and generalization. In this paper, we propose to unlock the potential of decoding-based regression via reinforcement learning. We formulate the generation process as a Markov decision process, utilizing sequence-level rewards to enforce global numerical coherence.Under this framework, we present GenRe$^2$, which combines policy gradient methods and on-policy distillation to provide dense expert supervision while preserving error magnitudes, thereby resolving the temporal credit assignment challenge. Extensive experiments across tabular regression, code metric prediction and generative reward modeling demonstrate that GenRe$^2$ consistently outperforms traditional baselines, establishing a robust paradigm for general-purpose numerical prediction.

Large Language Models (LLMs) offer a promising avenue for scientific discovery, yet their application to symbolic regression is often constrained by inefficient search strategies and coarse feedback signals. Current methods typically guide LLMs using scalar metrics (e.g., global Mean Squared Error), which fail to identify which components of a proposed equation are driving performance or causing error. We introduce \textit{Influence-Guided Symbolic Regression} (IGSR), a method that frames equation discovery as an iterative two-step process combining diverse term generation with rigorous selection: an LLM generates candidate basis functions $\psi_j(\mathbf{x})$ for a linear model, which are then evaluated using granular influence scores $\Delta_j$. These scores quantify each term's marginal contribution to generalization accuracy, enabling an influence-guided pruning process that systematically refines the model structure. Integrating this mechanism into a Monte Carlo Tree Search (MCTS) enables navigating the combinatorial search space while balancing exploration of novel functional forms with exploitation of high-influence components. We demonstrate IGSR's effectiveness on a diverse suite of benchmarks, including LLM-SRBench, pharmacological PKPD models, an epidemiological simulation, and real-world genomic data. Notably, we validate the framework's capacity for genuine discovery in a case study using a high-dimensional biological dataset, in which IGSR identified a novel relationship between DNA methylation and RNA Polymerase II pausing; a hypothesis that was subsequently supported via wet-lab experimentation.

Randomized smoothing has emerged as a scalable technique for certifying the adversarial robustness of classifiers. However, its application to regression remains under-explored and faces unique challenges. Existing regression certificates rely on probabilistic acceptance regions and fail to exploit the local geometry of the function. In this work, we present a novel framework for certified robust regression that addresses these limitations. We derive a prediction-centered certificate that guarantees the stability of the smoothed model’s prediction and ensures practical computability at test time. We investigate several alternatives for constructing these certificates by explicitly incorporating means, variances, and gradients. In particular we demonstrate on the MNIST rotation task that utilizing gradient information yields significantly tighter robustness certificates compared to the current state-of-the-art, $\alpha$-smoothing.

We develop approximation and statistical theory for standard Transformer networks in sequence modeling. Given a sequence-to-sequence target on $[0,1]^{d_x \times n}$ whose entries are $\gamma$-Hölder for $\gamma \in (0,1]$ or belong to a first-order Sobolev class, we establish explicit $L^p$-approximation bounds for all $p \in [1,\infty]$, including the previously elusive endpoint $p=\infty$ under softmax attention. In particular, achieving error $\varepsilon$ in $L^p$-norm requires $\mathcal{O}(\varepsilon^{-d_x n/\gamma})$ parameters for $\gamma$-Hölder targets and $\mathcal{O}(\varepsilon^{-d_x n})$ parameters for Sobolev targets, matching the best known scalings in ambient dimension $d_x n$. We further study nonparametric regression with sequential and dependent observations using Transformer networks. Assuming stationary $\beta$-mixing covariates whose temporal dependence weakens over time, we analyze a sliding-window empirical risk minimization procedure and establish excess-risk guarantees for the Transformer-based estimators. Our analysis clarifies the role of attention and enables extensions beyond softmax.

One of the most common machine learning setups is logistic regression. In many classification models, including neural networks, the final prediction is obtained by applying a logistic link function to a linear score. In binary logistic regression, the feedback can be either soft labels, corresponding to the true conditional probability of the data (as in distillation), or sampled hard labels (taking values $\pm 1$). We point out a fundamental problem that arises even in a particularly favorable setting, where the goal is to learn a noise-free soft target of the form $\sigma(\mathbf{x}^{\top}\mathbf{w}^{\star})$. In the over-constrained case (i.e. the number of samples $n$ exceeds the input dimension $d$) with examples $(\mathbf{x}_i,\sigma(\mathbf{x}_i^{\top}\mathbf{w}^{\star}))$, it is sufficient to recover $\mathbf{w}^{\star}$ and hence achieve the Bayes risk. However, we prove that when the examples are labeled by hard labels $y_i$ sampled from the same conditional distribution $\sigma(\mathbf{x}_i^{\top}\mathbf{w}^{\star})$ and $\mathbf{w}^{\star}$ is $s$-sparse, then rotation-invariant algorithms are provably suboptimal: they incur an excess risk $\Omega\!\left(\frac{d-1}{n}\right)$, while there are simple non-rotation invariant algorithms with excess risk $O(\frac{s\log d}{n})$. The simplest rotation invariant algorithm is gradient descent on the logistic loss (with early stopping). A simple non-rotation-invariant algorithm for sparse targets that achieves the above upper bounds uses gradient descent on the weights $u_i,v_i$, where now the linear weight $w_i$ is reparameterized as $u_iv_i$.

Interpretable machine learning is essential in high-stakes domains where decision-making requires accountability, transparency, and trust. While rule-based models offer global and exact interpretability, learning rule sets that simultaneously achieve high predictive performance and low, human-understandable complexity remains challenging. To address this, we introduce TT-Sparse, a flexible neural building block that leverages differentiable truth tables as nodes to learn sparse, effective connections. A key contribution of our approach is a new soft TopK operator with straight-through estimation for learning discrete, cardinality-constrained feature selection in an end-to-end differentiable manner. Crucially, the forward pass remains sparse, enabling each node (and the entire model) to be transformed exactly into compact, globally interpretable DNF/CNF Boolean formulas via Quine--McCluskey minimization. Extensive empirical results across 28 datasets spanning binary, multiclass, and regression tasks show that the learned sparse rules exhibit superior predictive performance with lower complexity compared to existing state-of-the-art methods.

Transcendental functions, such as the exponential, are central to scientific computing, yet they cannot be natively calculated by digital hardware. Instead, computers must approximate these functions by combining basic operations, such as $\{+, -, \times, \div\}$, using methods like Taylor series. These methods were developed over centuries by mathematicians, who focused on approaches that could attain arbitrary accuracy. However, computers can handle most applications by using only finite-precision types, like *float32*, where any accuracy beyond the type's precision is effectively discarded. We explore, therefore, whether forgoing arbitrary accuracy can lead to the discovery of more efficient approximations. The evolutionary method of symbolic regression is particularly suitable, as it can search for arbitrary operation combinations and can optimize non-differentiable objectives, such as the number of operations used. Our results show that evolution can discover computer programs that outperform established methods in this setting, despite having no prior mathematical knowledge beyond the calculation of the basic operations. Starting from empty code, symbolic regression constructs programs representing novel mathematical expressions. In particular, we discovered a 10-operation program that approximates the exponential function to 14 significant figures, exceeding the accuracy of previously known approximations of this size by more than 6 orders of magnitude.

Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critical domains. In this work, we propose novel non-asymptotic probabilistic uniform error bounds for kernel-based regression. Compared to related bounds in the literature that are restricted to (conditionally) independent sub-Gaussian noise, our bounds allow to consider a broad class of non-Gaussian distributions, such as sub-Gaussian, bounded, sub-exponential, and variance/moment-bounded noise. Moreover, our results apply to correlated and uncorrelated noise. We compare our proposed error bounds with existing results in terms of the induced uncertainty region and their performance in safe control, demonstrating the tightness of the proposed bounds.

Simulation plays a central role in scientific discovery. In many applications, the bottleneck is no longer running a simulator—it is choosing among large families of plausible simulators, each corresponding to different forward models/hypotheses consistent with observations. Over large model families, classical Bayesian workflows for model-selection are impractical. Furthermore, amortized model-selection methods typically hard-code a fixed model prior—or complexity penalty—at training time, requiring users to commit to a particular parsimony assumption before seeing the data. We introduce PRISM, a simulation-based encoder-decoder that infers a joint posterior over both discrete model structures and associated continuous parameters, while enabling test-time control of model complexity via a tunable model prior that the network is conditioned on. We show that PRISM scales to families with combinatorially many (up to billions of) model instantiations on a synthetic symbolic regression task. As a scientific application, we evaluate PRISM on biophysical modeling for diffusion MRI data, showing the ability to perform model selection across several multi-compartment models, on both synthetic and in-vivo neuroimaging data.

Randomized self-reductions (RSRs) express $f(x)$ using $f$ evaluated at random correlated points, enabling self-correcting programs, instance-hiding protocols, and applications in complexity theory and cryptography. Yet discovering RSRs has required manual expert derivation for over 40 years, limiting their practical use. We present Bitween for automated RSR learning. First, we formalize RSR learning with sample complexity analysis under correlated sampling. Second, we develop Vanilla Bitween, which integrates multiple backends (linear regression, genetic programming, symbolic regression, and mixed-integer programming). The linear regression backend outperforms the others, discovering RSRs for 43 of 80 functions (54%) in RSR-Bench, our benchmark suite, including the first known reduction for sigmoid. Third, we introduce Agentic Bitween, a neuro-symbolic approach where LLM agents propose novel query functions beyond the fixed set ($x+r$, $x-r$, $x \cdot r$, $x$, $r$) in prior work. Agentic Bitween discovers RSRs for 64 of 80 functions (80%), outperforming pure neural baselines in both RSR discovery and verification accuracy.

Existing symbolic regression approaches primarily focus on learning explicit input-output mappings, often neglecting relational structures among data instances. This paper introduces Contrastive Symbolic Regression (CSR), a feature-construction-based symbolic regression approach that integrates evolutionary feature construction with contrastive learning to shape a representation space where geometric proximity reflects similarity in the target space. CSR employs a contrastive objective that optimizes a linear transformation of constructed features, with a closed-form solution for aligning the feature space with the target space. The constructed features are applied to K-nearest neighbor regression, where we propose an efficient leave-one-out cross-validation (LOOCV) method that addresses standard LOOCV's computational expense and adaptively selects the neighborhood size, along with a linear-rank weighted K-nearest neighbor variant for faithful assessment of representation quality during evolution. A determinantal point process-based ensemble selection mechanism further enhances robustness by jointly considering model quality and diversity. Extensive experiments on 58 real-world regression datasets demonstrate that CSR consistently surpasses both traditional symbolic regression and modern machine learning counterparts, highlighting CSR as a promising direction for interpretable and effective regression modeling.

A key challenge in probabilistic regression is ensuring that predictive distributions accurately reflect true empirical uncertainty. Minimizing overall prediction error often encourages models to prioritize informativeness over calibration, producing narrow but overconfident predictions. However, in safety-critical settings, trustworthy uncertainty estimates are often more valuable than narrow intervals. Realizing the problem, several recent works have focused on post-hoc corrections; however, existing methods either rely on weak notions of calibration (such as PIT uniformity) or impose restrictive parametric assumptions on the nature of the error. To address these limitations, we propose a novel nonparametric re-calibration algorithm based on conditional kernel mean embeddings, capable of correcting calibration error without restrictive modeling assumptions. For efficient inference with real-valued targets, we introduce a novel characteristic kernel over distributions that can be evaluated in $\mathcal{O}(n \log n)$ time for empirical distributions of size $n$. We demonstrate that our method consistently outperforms prior re-calibration approaches across a diverse set of regression benchmarks and model classes.

Transformers have demonstrated remarkable in-context learning (ICL) capabilities. The strong ICL performance of transformers is commonly believed to arise from their ability to implicitly execute certain algorithms on the context, thereby enhancing prediction and generation. In this work, we investigate how transformers with softmax attention perform in-context learning on linear classification data. We first construct a class of multi-layer transformers that can perform in-context logistic regression, with each layer exactly performing one step of normalized gradient descent on an in-context loss. Then, we show that our constructed transformer can be obtained through (i) training a single self-attention layer supervised by one-step gradient descent, and (ii) recurrently applying the trained layer to obtain a looped model. Training convergence guarantees of the self-attention layer and out-of-distribution generalization guarantees of the looped model are provided. Our results advance the theoretical understanding of ICL mechanism by showcasing how softmax transformers can effectively act as in-context learners.

Logistic regression is widely used in applications; however, when the dimension scales with the sample size, theory reveals that the asymptotic behavior of common M-estimators depends on nonzero bias and variance factors, which are functions of the signal strength. To leverage the theory to design valid statistical methodologies, it is essential to obtain accurate estimates of the signal strength. In this work, we utilize a data-splitting strategy to efficiently estimate the signal strength. To alleviate issues caused by separable data, we analyze the exact asymptotics of an M-estimator with a data-driven, non-decomposable regularizer that adapts to the true covariance structure. We justify the validity of our method through both theoretical analysis and numerical experiments.

Symbolic regression aims to discover compact, interpretable mathematical expressions from data, but neural generation is challenging because expressions are tree-structured. Existing neural methods often linearize expression trees into token sequences, facilitating autoregressive modeling but obscuring hierarchical relations and complicating structure-dependent constraint enforcement. We propose GCN-SR, a graph-based symbolic regression framework that generates expressions in an explicit tree-aligned form, making structural context available during decoding. To enable batched generation over variable-topology expressions, we introduce Symbolic Perfect Binary Trees (SPBTs), a fixed-topology scaffold that preserves tree hierarchy while supporting graph-based node-attribute prediction. We further introduce Similarity-Weighted Policy Gradient (SWPG) to incorporate genetic programming (GP) refinement without directly imitating GP-refined elites; instead, refined expressions construct similarity-weighted rewards for samples drawn by the current generator. Experiments on standard symbolic regression benchmarks and ablations show that GCN-SR consistently improves exact recovery over strong neural and hybrid baselines under matched evaluation budgets.

Cross-view geo-localization (CVGL) aims to retrieve the corresponding satellite image given a street query and is critical for autonomous navigation. Although recent methods perform well on benchmarks, they often fail to generalize to unseen environments. A key limitation is the use of contrastive learning, which assigns equal labels to all negative samples and induces similarity-amplified repulsion. But should all negatives be treated equally? In CVGL, semi-positive samples that are geographically proximate to the positive often share important semantic cues. Treating them as ordinary negatives forces the model to overfit noise, leading to a collapse in generalization. To address this issue, we propose an uncertainty-aware framework grounded in Deep Evidential Regression (DER), modeling the Normal-Inverse-Gamma (NIG) distribution as a conjugate prior to quantify environmental complexity $u$ in a single forward pass. The estimated $u$ adaptively softens labels for hard negatives in Soft InfoNCE, mitigating excessive repulsion on semi-positive samples. An Uncertainty Head with cls-to-spatial cross-attention and attention statistics is designed to accurately fit the NIG distribution. Extensive experiments demonstrate state-of-the-art performance, including an average 18\% R@1 improvement in zero-shot cross-dataset transfer, filling the critical gap between laboratory benchmarks and robust real-world deployment.

Analytical solutions to differential equations offer exact, interpretable insight but are rarely available because discovering them requires expert intuition or exhaustive search of combinatorial spaces. We introduce SIGS, a neuro-symbolic framework for equation-driven closed-form solution discovery. SIGS uses a context-free grammar to generate mathematically valid and physically meaningful building blocks, with a user-specified Ansatz prescribing how these blocks combine, embeds them into a topology-regularised continuous latent manifold, and searches this manifold in two stages: structure selection followed by coefficient refinement using gradient descent, scoring candidates only against the PDE residual and prescribed boundary and initial conditions. This design unifies symbolic reasoning with numerical optimization; the grammar constrains candidate solution blocks to be proper by construction, while the latent search makes exploration tractable and data-free. SIGS is the first neuro-symbolic method to (i) recover analytical solutions for coupled nonlinear PDE systems, (ii) discover equivalent symbolic forms when the grammar lacks the natural primitives, and (iii) produce accurate symbolic approximations for PDEs lacking known closed-form solutions. Overall, SIGS improves over existing symbolic methods by orders of magnitude in both accuracy and runtime across standard PDE benchmarks.