ICML 2026 Papers

Traditional instrumental variable (IV) estimators cannot accommodate more treatments than instruments, a limitation that is critical for high-dimensional, unstructured data like clinical treatment pathways. Current practice—applying unsupervised dimension reduction before IV estimation—suffers from substantial omitted treatment bias because the representation learning step ignores the instrument. We propose a novel framework that constructs treatment representations by explicitly incorporating instrumental variables. We prove that this instrument-guided approach ensures the identification of optimal outcome-prediction directions even with limited instruments. Validation on large-scale, semi-synthetic clinical data derived from a major hospital, along with other simulations, shows that our approach significantly outperforms conventional two-stage methods.

Although conformal prediction provides robust marginal coverage guarantees, achieving reliable conditional coverage for specific inputs remains challenging. While exact distribution-free conditional coverage is impossible with finite samples, recent work has focused on improving the conditional coverage of standard conformal procedures. Distinct from approaches that target relaxed notions of conditional coverage, we directly target the mean squared error of conditional coverage by refining the quantile regression components that underpin many conformal methods. Leveraging a Taylor expansion, we derive a sharp surrogate objective for quantile regression: a density-weighted pinball loss, where the weights are given by the conditional density of the nonconformity score evaluated at the true quantile. We propose a three-headed quantile network that estimates these weights via finite differences using auxiliary quantile levels at $1-\alpha \pm \delta$, subsequently fine-tuning the central quantile by optimizing the weighted loss. We provide a theoretical analysis with exact non-asymptotic guarantees characterizing the resulting excess risk. Extensive experiments on diverse high-dimensional real-world datasets demonstrate remarkable improvements in conditional coverage performance. We release the code at \url{https://github.com/Cqyiiii/Colorful-Pinball-Conformal-Prediction-CPCP}.

Federated Quantile Regression (FQR) has emerged as a powerful modelling paradigm for estimating conditional quantiles, offering a more comprehensive understanding of response distributions than standard conditional mean regression. However, achieving communication efficiency and optimal statistical guarantees for FQR remains challenging, particularly due to the nonsmooth nature of quantile loss functions and the presence of heterogeneously structured data, where each local agent trains its conditional quantile models with distinct sets of features. In this paper, we propose a data-driven, one-shot weighted ensemble estimator for FQR that incorporates scalable weighting schemes to effectively leverage the partially observed features at each local agent, thereby enjoying both communication efficiency and estimation optimality. Theoretically, we present a unified analysis of the proposed learning procedure, establishing that the resulting estimator exhibits asymptotic normality and attains uniformly minimum variance. Furthermore, we investigate the estimator's sensitivity to perturbations introduced by local agents and derive conditions under which the estimator achieves stability and enjoys strong out-of-sample generalization. Extensive simulations and real data analysis under various scenarios validate the asymptotic normality of our estimator and demonstrate its superior estimation accuracy and uniform convergence compared to several baseline methods across a range of quantile levels.

Symbolic Regression (SR) is the task of finding a closed-form mathematical expression that optimizes some objective. Solving this task is NP-hard. However, SR software routinely discovers accurate, interpretable models without exhaustively searching function space. Motivated by this disconnect between worst-case theory and practical success, we study SR through the lens of \emph{parameterized complexity theory}. In particular, we reanalyze tractability with respect to practically relevant parameters like expression depth, tree size, and number of primitives used. We show that SR is fixed-parameter tractable (FPT) when parameterized by expression depth or tree size, formalizing the tractable regime exploited by bounded-complexity search in popular SR algorithms. However, SR becomes W[1]-hard when parameterized by the number of variables or primitives used, identifying selection as a source of intractability. We further find lower bounds under the exponential time hypothesis, prove approximation hardness, and rule out polynomial kernels for the case where the primitive set is part of the input.

We benchmark time series decomposition as a standalone evaluation task. While decomposition outputs are widely used to interpret trend and periodic structure, their quality is often assessed informally, and no unified benchmark exists for comparing component recovery under controlled generative mechanisms. We introduce a synthetic evaluation suite with explicit trend and cycle taxonomies, a unified interface covering representative decomposition families, and complementary metrics capturing distinct error modes (shape, phase, and spectral fidelity). Across stationary periodic regimes, STL-family methods are near-ceiling; under non-stationary periodicity (frequency drift, regime switching), fixed-period priors induce phase degradation, while subspace/time-frequency methods better preserve seasonal consistency (adaptive spectral methods may require tuning). We further extend the benchmark with a downstream scientific-discovery track—symbolic regression on decomposed components—showing that a decompose-then-regress pipeline materially improves recoverability and reduces expression complexity, linking decomposition quality to structure discovery. Code, result exports, and the web leaderboard are publicly available through the Hugging Face dataset and leaderboard Space.

Regression is a cornerstone of statistics and machine learning, with applications spanning science, engineering, and economics. While quantum algorithms for regression have attracted considerable attention, most existing work has focused on linear regression, leaving many more complex yet practically important variants unexplored. In this work, we present a unified quantum framework for accelerating a broad class of regression tasks---including linear and multiple regression, Lasso, Ridge, Huber, $\ell_p$-, and $\delta_p$-type regressions---achieving up to a quadratic improvement in the number of samples $m$ over the best classical algorithms. This speedup is achieved by a non-trivial quantization of the recent classical breakthrough of Jambulapati et al. (2024), where we construct a full quantum pipeline that strategically employs quantum leverage score approximation to initialize and refine Multiscale Leverage Score Overestimates, enabling efficient importance sampling via the preparation of multiple state copies. For problems of dimension $n$, sparsity $r < n$, and error parameter $\epsilon$, our algorithm solves the problem in $\widetilde{O}(r\sqrt{mn}/\epsilon + \mathrm{poly}(n,1/\epsilon))$ quantum time, demonstrating both the applicability and the efficiency of quantum computing in accelerating regression tasks.

Predicting outputs that are located in non-Euclidean spaces, such as probability distributions, networks, and symmetric positive-definite matrices, is becoming increasingly important in modern data analysis, particularly when inputs are high-dimensional. We propose DeSI (Deep Single-Index Fréchet Regression), a semiparametric framework for regression with metric space-valued outputs and multivariate inputs that assumes a single-index structure for the conditional Fréchet mean. DeSI estimates an interpretable index direction, which quantifies the relative importance of inputs, using a deep neural network, and performs Fréchet regression along the resulting one-dimensional index in the target metric space. This structure mitigates the curse of dimensionality while retaining interpretability, which stands in contrast to standard deep neural networks. We establish theoretical guarantees for DeSI, including uniform approximation and convergence rates, and demonstrate its strong predictive performance through simulations on distributions, networks, and symmetric positive-definite matrices, as well as an application to compositional mood data from New Jersey.

Symbolic regression (SR) aims to discover interpretable analytical expressions that accurately describe observed data. Amortized SR promises to be much more efficient than the predominant genetic programming SR methods, but currently struggles to scale to realistic scientific complexity. We find that a key obstacle is the lack of a fast reduction of equivalent expressions to a concise normalized form. Amortized SR has addressed this with general-purpose Computer Algebra Systems (CAS) like SymPy, but the high computational cost severely limits training and inference speed. We propose **SimpliPy**, a rule-based simplification engine achieving a 100-fold speed-up over SymPy at comparable quality. This enables substantial improvements in amortized SR, including scalability to much larger training sets, more efficient use of the per-expression token budget, and systematic training set decontamination with respect to equivalent test expressions. We demonstrate these advantages in our **Flash-ANSR** framework, which achieves much better accuracy than amortized baselines (NeSymReS, E2E) on the FastSRB benchmark. Moreover, it performs on par with state-of-the-art direct optimization (PySR) while recovering more concise rather than more complex expressions with increasing inference budget.

Reliable uncertainty quantification at unobserved spatial locations is a key challenge in spatial statistics, particularly for complex and heterogeneous datasets. While traditional methods such as Kriging rely on strong distributional assumptions, conformal prediction (CP) offers a distribution-free alternative. However, although non-i.i.d. CP theory is well established for time-series data, a significant gap remains for spatial data, where the lack of a natural ordering and discrete index complicates theoretical guarantees. Existing CP theory for spatial data often relies on exchangeability. We propose Localized Spatial Conformal Prediction (LSCP), a model-agnostic framework that bridges this gap by coupling local quantile regression with conformal calibration. LSCP conditions on spatial neighborhoods to capture local heterogeneity. We show that LSCP retains finite-sample marginal coverage under spatial exchangeability and attains asymptotic conditional coverage under stationarity and spatial mixing. Across synthetic and real datasets, LSCP consistently achieves near-nominal coverage with tighter and more stable prediction intervals than existing methods that fail to capture these spatial dependencies.

Low-precision training is critical for optimizing the trade-off between model quality and training costs, necessitating the joint allocation of model size, dataset size, and numerical precision. While empirical scaling laws suggest that quantization impacts effective model and data capacities or acts as an additive error, the theoretical mechanisms governing these effects remain largely unexplored. In this work, we initiate a theoretical study of scaling laws for low-precision training within a high-dimensional sketched linear regression framework. By analyzing multiplicative (signal-dependent) and additive (signal-independent) quantization, we identify a critical dichotomy in their scaling behaviors. Our analysis reveals that in the worst case, while both schemes introduce an additive error and degrade the effective data size, they exhibit distinct effects on effective model size: multiplicative quantization maintains the full-precision model size, whereas additive quantization reduces the effective model size. Numerical experiments validate our theoretical findings. By rigorously characterizing the complex interplay among model scale, dataset size, and quantization error, our work provides a principled theoretical basis for optimizing training protocols under practical hardware constraints.

Large Language Models (LLMs) can propose natural-language rules, circumventing the reliance on a predefined predicate space in traditional rule learning. However, existing LLM-based methods often neglect the global interactions among rules, and the potential of using fine-grained rule importance scores to calibrate neuro-symbolic reasoning remains underexplored. To address this gap, we introduce RLIE, a framework that integrates LLMs with probabilistic modeling to learn weighted rule sets in four stages: (1) Rule generation: proposing and filtering candidate rules via LLMs; (2) Logistic regression: learning sparse, calibrated weights for global rule selection; (3) Iterative refinement: revising the rule set with error-driven hard examples; and (4) Evaluation: validating the learned system via comparative inference paradigms. Across multiple real-world datasets and LLM backbones, our learned weighted rules achieve superior stability and accuracy, whereas rule-injection prompting yields mixed results and often degrades performance. These results suggest LLMs excel at semantic rule discovery but are less reliable at controlled probabilistic aggregation. Our findings highlight both the promise and the limits of LLMs for inductive reasoning, motivating a principled integration with classic probabilistic rule combination for reliable neuro-symbolic reasoning.

This paper studies active regression for single-index models under general $\ell_p$-loss with an unknown $1$-Lipschitz link function $f$, formulated as $\min_{f,x} \Vert f(Ax)-b\Vert_p^p$ with full access to $A$ but coordinate-query access to $b$. Prior work established upper bounds for known link functions for all $p\geq 1$ and for unknown link functions only in the $p=2$ case, together with lower bounds for $p\leq 2$. This work addresses the more challenging setting of unknown link functions and general $p \geq 1$. A non-adaptive sampling algorithm is presented that achieves a $(1+\epsilon)$-approximation using $O(d^{p/2\vee 1}/\epsilon^{p\vee 2}\text{poly}\log(n/\epsilon))$ queries. Nearly tight lower bounds are also established for non-adaptive queries when $p>2$. These results close much of the remaining gap in active $\ell_p$ regression for single-index models.

We study a multi-factor block model for variable clustering and connect it to regularized subspace clustering through a distributionally robust version of nodewise regression. To solve the latter problem, we derive a convex relaxation, provide a data-driven approach for selecting the size of the robust region, and develop an ADMM algorithm for efficient implementation. We validate our method in extensive numerical studies and demonstrate its superior performance.

Symbolic regression (SR) traditionally balances accuracy and complexity, implicitly assuming that simpler formulas are structurally more rational. We argue that this assumption is insufficient: existing algorithms often exploit this metric to discover accurate and compact but structurally irrational formulas that are numerically ill-conditioned and physically inexplicable. Inspired by the structural stability of real physical laws, we propose the Effective Information Criterion (EIC) to quantify formula rationality. EIC models formulas as information channels and measures the amplification of inherent rounding noise during recursive calculation, effectively distinguishing physically plausible structures from pathological ones without relying on ground truth. Our analysis reveals a stark structural stability gap between human-derived equations and SR-discovered results. By integrating EIC into SR workflows, we provide explicit structural guidance: for heuristic search, EIC steers algorithms toward stable regions to yield superior Pareto frontiers; for generative models, EIC-based filtering improves pre-training sample efficiency by 2–4 times and boosts generalization $R^2$ by 22.4\%. Finally, an extensive study with 108 human experts shows that EIC aligns with human preferences in 70\% of cases, validating structural stability as a critical prerequisite for human-perceived interpretability. We release our code at https://github.com/tsinghua-fib-lab/EIC

We study *grokking* — the onset of generalization long after overfitting — in a classical ridge regression setting. We prove end-to-end grokking results for learning over-parameterized linear regression models using gradient descent with weight decay. Specifically, we prove that the following stages occur: (i) the model overfits the training data early during training; (ii) poor generalization persists long after overfitting has manifested; and (iii) the generalization error eventually becomes arbitrarily small. Moreover, we show, both theoretically and empirically, that grokking can be amplified or eliminated in a principled manner through proper hyperparameter tuning. To the best of our knowledge, these are the first rigorous quantitative bounds on the generalization delay (which we refer to as the "grokking time") in terms of training hyperparameters. Lastly, going beyond the linear setting, we empirically demonstrate that our quantitative bounds also capture the behavior of grokking on non-linear neural networks. Our results suggest that grokking is not an inherent failure mode of deep learning, but rather a consequence of specific training conditions, and thus does not require fundamental changes to the model architecture or learning algorithm to avoid.