LLM knowledge for improved Out-of-Distribution Tabular Prediction - A Cross-Validated Regularisation Approach

Tabular prediction has long been dominated by gradient-boosted decision trees and specialized deep tabular models, while large language models (LLMs) remain difficult to make competitive despite their cross-task adaptability and transparent reasoning traces. We address this gap by incorporating tabular structural priors into LLM post-training. Specifically, we propose Permutation Relative Policy Optimization (PRPO), which operationalizes column-permutation invariance through label-preserving column permutations and two-level advantage estimation. This design converts sparse outcome rewards into denser and more stable optimization signals. Extensive experiments on 139 OpenML datasets show that our 8B model reaches a genuinely competitive regime against strong specialized tabular baselines. It achieves strong fully supervised performance, dominates zero-shot settings, and performs on par with 32-shot strong baselines. Moreover, it substantially outperforms much larger general-purpose and reasoning LLMs, including up to a 53.17% improvement over DeepSeek-R1 (685B). These results show that structural-prior RL post-training is an effective route for making LLMs competitive in tabular prediction.

Tabular learning uses structured data to predict target outcomes. Traditionally, this process has relied on labeled data. However, large language models (LLMs) can be used to elicit domain priors based on the task description and feature semantics, thereby enabling predictions without labeled data. We propose Marginal Response Surface Elicitation (MARS), a method that transforms feature-level LLM priors into a reusable, zero-shot tabular classifier. To construct this classifier, MARS selects representative values for each feature from unlabeled data and prompts the LLM to provide corresponding class support scores and feature weights. It then aggregates multiple responses using the median to construct feature response functions, and makes predictions through their weighted sum without further LLM queries. Across eight tabular benchmark tasks, MARS achieves the highest average AUC and AP, outperforming direct prompting by 1.97 and 6.21 percentage points respectively, while substantially reducing end-to-end costs. Evaluations with LLMs of different sizes further demonstrate its predictive advantage over direct prompting.

Large language models (LLMs) have become the default tool for a remarkable range of tasks, yet they have had conspicuously little success at one of the most common machine learning workloads: predictive analytics over tabular data. This gap is the founding premise of the fast-growing field of tabular foundation models, but the question of why generic LLMs fail has remained open. We study a frontier LLM in its purest inference regime - a single generation pass over a prompt containing the full training and test data, with no tools, no agentic scaffolding, and no fine-tuning - and systematically evaluate five hypotheses for the failure: (a) an inability to handle noisy or non-linearly-separable data; (b) the linearised CSV format obscuring column structure; (c) the tokenisation of numeric values; (d) the number of test points classified per query; and (e) the dimensionality of the input. Controlled experiments falsify (a)-(d). Dimensionality, in contrast, is decisive: sweeping random linear projections of thirty-one benchmark datasets, the LLM is the only method among nine whose accuracy decreases as dimensionality grows, while every classical baseline stays flat or improves. A behavioural comparison against 252 configured classical models finds that in two dimensions the LLM predicts like a local, distance-based method (up to 91.6% grid agreement), but in higher dimensions no classical model - even when augmented with tuned, dimension-dependent noise - reproduces its predictions. We do not claim to have identified the internal mechanism; our results show, more modestly, that the LLM's capability dissolves with dimension in a way no noise-corrupted classical learner mimics - which explains why LLMs, so capable elsewhere, keep losing to fifty-year-old baselines on tables, while leaving the mechanism of the prediction as an open question.