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Hierarchical and Permutation-Invariant Feature Transformation Learning via Policy-Guided Embedding Search
Rui Liu, Tao Zhe, Yanyong Huang, Sankha Narayan Guria, Xiao Luo, Wei Fan, Yanjie Fu, Dongjie Wang
TL;DR
Automated feature transformation seeks informative tabular representations, but existing methods struggle to model intricate transformation patterns. PHER combines permutation-invariant hierarchical modeling with policy-guided multi-objective search, and outperforms baselines across 19 datasets.
Problem
Automated feature transformation seeks informative representations to improve tabular predictive performance, but existing methods struggle to model intricate transformation patterns.
Source: §1 · pp. 1–2Method
PHER combines permutation-invariant hierarchical modeling of feature and concept relationships with policy-guided multi-objective search in the learned embedding space.
Source: §6 · p. 9Results
PHER outperforms baseline models on all 19 evaluated datasets across classification and regression tasks.
Source: §4.3.1 · p. 6Takeaways & Limitations
The experiments demonstrate PHER's effectiveness in transforming feature spaces across varied task types.
Source: §4.3.1 · p. 7
Abstract
from arXiv · showhide
Feature transformation improves predictive performance on tabular data by constructing informative abstractions from raw features. Recent generative approaches encode transformation knowledge into continuous embedding spaces for efficient exploration of candidate strategies, but face three key limitations: (1) overlooking hierarchical relationships between low-level features, operations, and high-level abstractions; (2) enforcing order-sensitive embeddings on inherently permutation-invariant transformation sequences, thereby introducing systematic bias; and (3) relying on gradient-based search, which is ill-suited to non-convex transformation spaces. We propose a framework with two complementary components. First, a permutation-invariant hierarchical module captures interactions across features, operations, and abstraction levels, with a self-attention pooling mechanism that maps semantically equivalent structures to consistent embeddings aligned with downstream performance. Second, a policy-guided multi-objective reinforcement learning strategy initializes the search from empirically strong seeds and jointly optimizes predictive accuracy and transformation efficiency. Extensive experiments on diverse tabular benchmarks demonstrate the effectiveness and robustness of our framework against strong baselines. Our code and data are publicly available at: https://github.com/RayLiu1103/PHER.
1 Introduction
PHER addresses limitations in automated feature transformation by combining permutation-invariant hierarchical modeling with policy-guided multi-objective search. Experiments on 19 real-world datasets show superior performance over state-of-the-art feature transformation methods.
1 Introduction
- Automated feature transformation aims to make tabular representations more predictive, addressing deep learning's difficulty with heterogeneous types, varying scales, missing values, and outliers.The introduction also notes that deep models' black-box nature can limit interpretability in data analysis and critical decision-making.Source: §1 · p. 1
- Embedding-based generative approaches overlook hierarchical feature relationships, introduce permutation bias, and use gradient search despite no guarantee that the embedding space is convex.These limitations can weaken feature representations, impede exploration, and trap searches in local optima.Source: §1 · p. 2
- PHER outperforms state-of-the-art feature transformation methods in experiments on 19 real-world datasets.The experiments evaluate the framework's efficiency, resilience, and traceability.Source: §1 · p. 2
- PHER preserves low-level feature interactions and high-level abstractions in a continuous embedding, using self-attention pooling to give permutations of generated concepts identical embeddings.This symmetric pooling is intended to produce a global embedding space that preserves transformation knowledge without permutation bias.Source: §1 · p. 2
- PHER initializes search with top-K sequences ranked by model performance, then uses reinforcement learning to maximize downstream performance while minimizing transformation-sequence length.Seed embeddings are treated as states for exploration, which the authors say reduces the risk of local optima even when the space is non-convex.Source: §1 · p. 2
2 Problem Statement
The problem is to learn feature transformations from records that pair transformation sequences with downstream performance, using permutation-invariant hierarchical embeddings and policy-guided search. The learned embedding is decoded into a transformation sequence intended to maximize downstream task performance.
2 Problem Statement
- Given a labeled dataset and mathematical operations, the framework gathers transformation sequences paired with their downstream predictive performance.Each record is denoted (Γ_i, v_i).Source: §2 · p. 2
- Token encoders and decoders model interactions between original features and operations, then aggregate token embeddings into concept embeddings for permutation-invariant global representations.Token reconstruction loss trains the token-level components; a concept-level decoder is also trained to capture patterns among generated concepts.Source: §2 · pp. 2–3
- Policy-guided search explores the global embedding, which the concept- and token-level decoders use to reconstruct a transformation sequence intended to maximize downstream performance M.The target is the optimal embedding G′_opt and its decoded sequence Γ*.Source: §2 · p. 3
3 Methodology
PHER learns hierarchical, permutation-invariant embeddings of feature transformations, then searches that space with a policy-guided multi-objective strategy to find high-performing, shorter transformation sequences.
3.1 Framework Overview
- PHER encodes feature–operation tokens into concepts and permutation-invariant global embeddings, then searches the embedding space and decodes an optimized transformation sequence for the original features.The framework uses token- and concept-level encoders and decoders, with self-attention pooling for global embeddings.Source: §3.1 · p. 3
3.2 Hierarchical Feature Transformation Knowledge Modeling
- PHER models transformation records as sequences paired with task performance and learns token-level feature–operation interactions alongside higher-level concept relationships.The records are collected with an RL-based method [31], in which two agents select features and a third selects operations to maximize downstream performance.Source: §3.2 · p. 3
- Two Transformer encoder layers embed each transformation sequence, and two Transformer decoder layers reconstruct it from token embeddings.The token-level model is trained to reconstruct the input sequence before concept-level training.Source: §3.2 · p. 4
- Self-attention pooling maps a concept set to the same global embedding regardless of concept order, avoiding permutation bias that can fragment equivalent transformations and inflate search.Concepts are composed from feature and operation tokens, recursively aggregating nested transformations in postfix form; learnable seed vectors attend over concept embeddings.Source: §3.2 · p. 4
- Hierarchical training proceeds from token reconstruction to concept reconstruction, then aligns token embeddings with frozen concept-level abstractions to improve end-to-end reconstruction coherence.The concept-level stage freezes the token model and reconstructs concept embeddings; alignment then freezes the concept model while tuning the token encoder and decoder.Source: §3.2 · pp. 4–5
3.3 Policy-guided Multi-objective Search
- PPO explores the non-convex global embedding space from top-K performance-ranked records, balancing downstream task performance against transformation-sequence length.Its weighted reward combines the change in downstream task performance with a normalized sequence-length penalty.Source: §3.3 · p. 5
- After policy convergence, PHER decodes enhanced embeddings into candidate transformation sequences and selects the sequence with the highest evaluated downstream performance.The actor–critic search is trained with PPO’s clipped policy objective and critic value prediction.Source: §3.3 · p. 5
4 Experiments · 4.1 Datasets and Evaluation Metrics
Experiments evaluate PHER across classification and regression datasets using a shared Random Forest protocol, repeated hold-out testing, and task-specific metrics.
4.1 Datasets and Evaluation Metrics
- PHER is evaluated on 19 public datasets: 14 classification tasks and 5 regression tasks from UCI, LibSVM, Kaggle, and OpenML.Source: §4.1 · p. 5
- Random Forest serves as the unified downstream model, with performance evaluated by five-fold cross-validation on each training partition.Source: §4.1 · p. 5
- Each dataset is split randomly into 80% training and 20% testing data under a hold-out evaluation protocol.Source: §4.1 · p. 5
- The hold-out experiments are repeated five times with different random seeds, and results are reported as the mean and standard deviation.Source: §4.1 · pp. 5–6
- Regression metrics are 1-RAE, 1-MAE, 1-MSE, and 1-RMSE; classification metrics are F1-score, Precision, Recall, and ROC/AUC.Source: §4.1 · p. 5
- The reported classification and regression comparisons use F1-Score and 1-RAE, respectively, with standard deviation computed across five independent runs.Source: §4.1 · p. 6
4.2 Baseline Models
The section compares PHER with nine feature-transformation methods and introduces three PHER variants to assess the framework’s technical components.
4.2 Baseline Models
- The first four baselines are RDG, ERG, LDA, and AFAT [10].RDG randomly forms feature-operation-feature records; ERG expands features with operations before selecting informative features; LDA learns latent topic representations for dimensionality reduction; AFAT iteratively transforms and selects features.Source: §4.2 · p. 6
- The remaining baselines are NFS, TTG [16], GRFG [31], DIFER, and MOAT.NFS uses an RL-trained recurrent controller; TTG explores transformation paths on a graph; GRFG uses three reinforced agents with feature grouping; DIFER applies greedy gradient-based search to embedded sequences; MOAT uses postfix embeddings, gradient ascent, and beam search.Source: §4.2 · p. 6
- Three PHER variants isolate the concept encoder-decoder, permutation invariance, and RL search components.PHER−c removes the concept encoder and decoder; PHER−p replaces their permutation-invariant layers with permutation-sensitive self-attention; PHER−g replaces RL search with a Genetic Algorithm.Source: §4.2 · p. 6
4.3 Performance Evaluation … 4.3.4 Search Seeds.
PHER outperforms baseline models across 19 datasets, and the evaluation attributes its results to hierarchical, permutation-invariant representations and policy-guided search. Ablations, classifier tests, seed comparisons, and scalability results examine the contributions and robustness of these components.
4.3 Performance Evaluation
- Across 19 datasets, PHER outperformed the other baseline models on all datasets, evaluated with F1-score for classification and 1-RAE for regression; experiments were independently run five times with different random seeds.Source: §4.3.1 · p. 6
4.3.2 Ablation Study.
- On SpamBase, PHER outperformed baseline methods across Random Forest, XGBoost, KNN, and Decision Tree downstream models.The proposed data collector uses validation-guided exploration rather than a specific classifier, producing transformation records that support cross-classifier generalization.Source: Figure 4 · p. 7, §4.3.3 · p. 7
4.3.1 Overall Performance.
- PHER’s hierarchical encoder captures low- and high-level relationships, permutation-invariant encoding removes order sensitivity, and policy-guided RL searches the non-convex embedding space.The paper says these mechanisms build an informative global embedding space and help the agent find higher-quality regions than gradient-based methods.Source: §4.3.1 · p. 7
4.3.2 Ablation Study.
- PHER consistently outperformed variants that removed its concept encoder-decoder, used permutation-sensitive attention, or replaced policy-guided search with a genetic algorithm.The ablation results support the contributions of hierarchical modeling, permutation-invariant encoding, and RL exploration without convexity assumptions.Source: §4.3.2 · p. 7
4.3.3 Robustness Check.
- On eight randomly chosen datasets, search initialized with high-quality seeds consistently outperformed random initialization in average F1-score over five runs.Informative seeds anchor the search in promising embedding regions and narrow its scope, while random starts increase the risk of local optima and unstable, suboptimal sequences.Source: §4.3.4 · pp. 7–8, Figure 5 · p. 7
4.3.4 Search Seeds.
- Compared with MOAT, PHER produced significantly shorter transformation sequences and consistently outperformed it across all datasets.The paper attributes this combination of performance and compactness to jointly maximizing task performance and minimizing sequence length.Source: Figure 6 · p. 7, §4.3.5 · p. 8
4.3.5 Model Scalability. … 4.3.8 Traceability Case Study.
The experiments examine permutation robustness, hyperparameter sensitivity, and traceability; the reported findings include permutation-invariant embeddings, stable performance across reward trade-offs, and informative PHER-generated wine features.
4.3.5 Model Scalability.
- Across four datasets, the experiment selected five samples per dataset and generated 20 randomly shuffled concept-embedding variants for each sample.Original and permuted sets were encoded and visualized with t-SNE.Source: §4.3.6 · p. 8
4.3.8 Traceability Case Study.
- Approximately 70% of crucial features in PHER’s generated Wine Quality Red feature set are newly generated, and the new feature space improves downstream ML performance by 23.7%.Figure 9 compares traceability on the original feature set and selected feature subset.Source: Figure 9 · p. 8, §4.3.8 · p. 9
4.3.6 Permutation Sensitivity.
- Permuted concept orders cluster around their original embeddings, confirming that self-attention pooling eliminates permutation bias through content-based weighting and commutative aggregation.The attention weights depend on similarity to fixed learnable seed queries, not input positions.Source: §4.3.6 · p. 8
4.3.7 Hyperparameter Sensitivity Analysis.
- On SpectF, PHER’s clipping ratio 𝜖 and reward trade-off 𝜆 were varied from 0.1 to 0.9 to assess hyperparameter sensitivity.𝜖 controls policy-update magnitude, while 𝜆 balances task performance and transformation sequence length.Source: §4.3.7 · p. 8
- 𝜖=0.2 and 𝜆=0.9 yield optimal performance; results remain relatively stable across a broad 𝜆 range, with slight gains as 𝜆 approaches 0.9.The paper notes that 𝜖=0.2 is widely used in standard reinforcement-learning benchmarks [27].Source: §4.3.7 · p. 9
4.3.8 Traceability Case Study.
- For red wine quality, PHER identifies [alcohol] as the original set’s most important feature and derives informative composites that support tracing transformed features to domain-relevant analytical rules.The paper attributes this capability to hierarchical modeling and policy-guided multi-objective search.Source: §4.3.8 · p. 9
5 Related Works
Prior AFT methods use expansion and pruning, iterative optimization, or broader AutoML searches, but face limits in transformation depth, search cost, or representation. PHER addresses representation and search limitations with hierarchical permutation-invariant modeling and multi-objective policy-guided search.
Related Works
- Expansion-reduction methods [10] [19] enumerate transformed features and prune them, but shallow compositions and heuristic selection hinder complex transformations.These approaches aim to improve feature expressiveness and downstream prediction by expanding the original tabular feature space.Source: §5 · p. 9
- Evolution-evaluation methods [16] [31] iteratively generate and select features, but discrete decisions and combinatorial search drive high cost and unstable optimization.They use evolutionary algorithms or reinforcement learning to retain transformations that improve validation performance.Source: §5 · p. 9
- AutoML approaches [30] [37] jointly search transformation strategies and model configurations; embeds RL-collected sequences for gradient-ascent beam search.The cited approach represents sequences with postfix expressions, but is limited by overlooking hierarchical knowledge, order-sensitive encoding, and a convexity assumption.Source: §5 · p. 9
- PHER combines hierarchical permutation-invariant modeling with multi-objective policy-guided search to capture token- and concept-level knowledge and find sequences without assuming embedding-space convexity.Its modeling mitigates order sensitivity among concepts and creates an unbiased global embedding space.Source: §5 · p. 9
6 Conclusion Remarks
PHER combines permutation-invariant hierarchical feature modeling with multi-objective policy-guided search to explore feature-transformation embeddings. Its experiments highlight the roles of hierarchical interactions, order invariance, and robust exploration, while future work targets efficiency and scalability.
6 Conclusion Remarks
- PHER combines permutation-invariant hierarchical modeling with multi-objective policy-guided search to explore learned feature-transformation embeddings.The modeling module uses token-level and concept-level encoder-decoder models, while self-attention pooling symmetrically scores generated concepts to ensure permutation invariance.Source: §6 · p. 9
- Experiments indicate that hierarchical modeling captures meaningful interactions and enhances the expressivity of the learned embedding space.The module preserves transformation knowledge at both the feature-operation token level and the generated concept level.Source: §6 · p. 9
- The permutation-invariant module mitigates order sensitivity, stabilizing embedding-space learning and search.Source: §6 · p. 9
- Policy-guided reinforcement-learning search supports effective exploration, mitigating local-optimum convergence risk and improving search robustness.The multi-objective strategy explores the learned embedding space without relying on convexity assumptions.Source: §6 · p. 9
- Future work could improve PHER's computational efficiency and scalability through lightweight concept modeling or refinements to embedding-space reinforcement-learning search.Source: §6 · p. 9
GenAI Usage Disclosure
Generative AI was used only for language polishing and auxiliary code assistance; the authors created, verified, and approved the research and manuscript.
GenAI Usage Disclosure
- Generative AI assisted only with language polishing, grammar refinement, and code editing and debugging suggestions; authors created and approved all research and final manuscript content.The authors also created, verified, and approved the experimental designs, code, data processing, and analyses.Source: GenAI Usage Disclosure · p. 10