Tabular Regression

Smooth-basis models such as Chebyshev polynomial regressors and radial basis function (RBF) networks are well established in numerical analysis. Their continuously differentiable prediction surfaces suit surrogate optimisation, sensitivity analysis, and other settings where the response varies gradually with inputs. Despite these properties, smooth models seldom appear in tabular regression, where tree ensembles dominate. We ask whether they can compete, benchmarking models across 55 regression datasets organised by application domain. We develop an anisotropic RBF network with data-driven centre placement and gradient-based width optimisation, a ridge-regularised Chebyshev polynomial regressor, and a smooth-tree hybrid (Chebyshev model tree); all three are released as scikit-learn-compatible packages. We benchmark these against tree ensembles, a pre-trained transformer, and standard baselines, evaluating accuracy alongside generalisation behaviour. The transformer ranks first on accuracy across a majority of datasets, but its GPU dependence, inference latency, and dataset-size limits constrain deployment in the CPU-based settings common across applied science and industry. Among CPU-viable models, smooth models and tree ensembles are statistically tied on accuracy, but the former tend to exhibit tighter generalisation gaps. We recommend routinely including smooth-basis models in the candidate pool, particularly when downstream use benefits from tighter generalisation and gradually varying predictions.

Standard regression methods typically optimize a single pointwise objective, such as mean squared error, which conflates the learning of ordering with the learning of scale. This coupling renders models vulnerable to outliers and heavy-tailed noise. We propose CAIRO (Calibrate After Initial Rank Ordering), a framework that decouples regression into two distinct stages. In the first stage, we learn a scoring function by minimizing a scale-invariant ranking loss; in the second, we recover the target scale via isotonic regression. We theoretically characterize a class of "Optimal-in-Rank-Order" objectives -- including variants of RankNet and Gini covariance -- and prove that they recover the ordering of the true conditional mean under mild assumptions. We further show that subsequent monotone calibration recovers the true regression function at the population level and mathematically guarantees that finite-sample predictions are strictly auto-calibrated. Empirically, CAIRO combines the representation learning of neural networks with the robustness of rank-based statistics. It matches the performance of state-of-the-art tree ensembles on tabular benchmarks and significantly outperforms standard regression objectives in regimes with heavy-tailed or heteroskedastic noise.

Neural networks (NNs) achieve outstanding performance in many domains; however, their decision processes are often opaque and their inference can be computationally expensive in resource-constrained environments. We recently proposed Differentiable Logic Networks (DLNs) to address these issues for tabular classification based on relaxing discrete logic into a differentiable form, thereby enabling gradient-based learning of networks built from binary logic operations. DLNs offer interpretable reasoning and substantially lower inference cost. We extend the DLN framework to supervised tabular regression. We first redesign the final output layer (the SumLayer) to support continuous targets. More critically, we find the original two-phase training procedure used for classification is suboptimal for regression, and thus develop a unified, single-stage optimization procedure. We also demonstrate that temperature annealing of the network's differentiable relaxations is decisive for achieving stable convergence and high accuracy. We evaluate the resulting model on 15 public regression benchmarks, comparing it with modern neural networks and classical regression baselines. Regression DLNs match or exceed baseline accuracy while preserving interpretability and fast inference. Our results show that DLNs are a viable, cost-effective alternative for regression tasks, especially where model transparency and computational efficiency are important.

Utilities publish Estimated Times of Restoration (ETRs) for customer-facing storm outages, and their accuracy governs whether customers can make sound decisions about food, medical equipment, and relocation. Prior work treats ETR as static tabular regression in which each outage contributes one record, discarding the fact that every development of an outage, from crew assignment through dispatch, suspension, damage assessment and partial restoration, is recorded as a revision. We reformulate ETR prediction as longitudinal tabular regression and introduce a Longitudinal Tabular Transformer (LTT), an axial-attention model that consumes the revisions preceding a prediction and issues a refined estimate at every one. On 242{,}928 storm-attributed outages from a cohort of 526{,}468 filtered events and 10.0 million revisions at six operating companies, LTT reduces customer-weighted asymmetric error at all six, by a median of 36.9,% against the estimates the utilities published during the same storms and 11.3,% against the strongest learned baseline at each. It is the only method improving on the incumbent's satisfaction impact at all six companies while also reducing root mean squared error at all six. Stratification by revision index shows that LTT error is largest at the first revision, where no history is available, and falls monotonically as revisions accumulate.

I show that ordinary least squares (OLS) predictions can be rewritten as the output of a restricted attention module, akin to those forming the backbone of large language models. The connection comes from viewing OLS as a similarity-based prediction rule in a learned embedding space. In this representation, least squares does not estimate coefficients per se. Instead, it selects an embedding that minimizes squared prediction error by matching training and test vectors through inner products. This maps directly onto the query-key-value structure of attention mechanisms. I then discuss extensions to dimensionality reduction, nonlinearity, and time series econometrics. Monte Carlo simulations and real-data experiments on UCI/OpenML benchmarks show that nonlinear Attention Regression performs competitively against standard machine learning baselines. In the reverse direction, I replace the attention sublayer of a transformer for tabular data with an explicit regression on polynomial features. The resulting model performs comparably to the standard transformer at a fraction of its parameter count.