8.0 \\lambda>8.0 ) over-smooth the attention distribution, reducing the model’s ability to discriminate between important and less relevant variate relationships. The setting λ \\lambda =5.0, as implemented in our model, achieves the optimal trade-off by ensuring stable computation while preserving discriminative attention patterns. This value keeps the exponential inputs within a numerically safe range while allowing sufficient variation in the resulting transport plan."},{"type":"para","text":"Sinkhorn Iteration Count. Our ablation experiments result that set iterations from 1 to 5 shows that the iterations count is not very significant to improve the performance. After iterations arrived 2, the MSE and MAE have little been get better result any more. Adding more iterations will have a negligible effect on improvement, but will significantly increase the computational load, showing a linear growth. This finding aligns with theoretical results showing that Sinkhorn iterations converge rapidly for well-conditioned problems. Fig 6 shows the variation trends of MSE and MAE based on different iteration numbers."},{"type":"heading","level":1,"text":"V Summary"},{"type":"para","text":"Sufficient experimental evaluation shows that MSC-OT achieves state-of-the-art performance across most of datasets for both short-term and long-term multivariate time series forecasting. Ablation studies confirm that multi-scale convolution enhancement and Sinkhorn optimal transport regularization each provide distinct and complementary benefits: convolution captures local structural patterns at multiple granularities, while OT regularization ensures balanced cross-variate information and suppresses the noise. The adaptive fusion mechanism learns to optimally combine these components based on dataset characteristics. MSC-OT exhibits well robustness to outlier noise, validating the theoretical motivation of our MSC-OT approach, establishing MSC-OT as a practical and effective solution for real-world multivariate time series forecasting applications and exploring a new methodological path to optimize attention based on optimal transport."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Transformer [ 20 ] has achieved a success in natural language processing [ 2 ] and computer vision [ 4 ] , emerging as the dominant architecture for sequence modeling tasks. Inspired by this success, Transformer-based models have been extensively adapted for time series forecasting, with different type of innovations including sparse attention mechanisms [ 29 ] , auto-correlation decomposition [ 24 ] , and frequency-enhanced representations [ 30 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"But researchers have recently questioned the validity of conventional Transformer-based forecasters, which typically embed multiple variates of the same time into channels and apply attention on temporal tokens. Considering the numerical but less semantic relationship among time points, simple linear layers have been shown to exceed complicated Transformers on both performance and efficiency [ 27 ] . ITransformer [ 11 ] addressed it by introducing an inverted embedding paradigm that embeds entire time series of each variate as a single token. This type of embedding enable the attention mechanism to explicitly capture cross-variate dependencies rather than temporal dependencies. However, we observe that the inverted embedding paradigm raises a critical limitation: by embedding each variate’s temporal information into a fixed-dimensional token, fine-grained temporal patterns important to prediction may be lost.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Considering these limitations, we focus on how to enhance the inverted embedding framework while preserving its advantages for cross-variate modeling. MSC-OT (Multi-Scale Convolution with Optimal Transport) Attention is proposed, an attention mechanism that addresses the inherent limitations. Technically, we apply parallel 2D convolutions with kernel sizes of 3 × 3 3\\\\times 3 and 5 × 5 5\\\\times 5 to the attention score matrix, capturing local structural patterns at different granularities that reflect compressed temporal information. We formulate attention computation as an entropy-regularized optimal transport problem and employ the Sinkhorn algorithm to achieve doubly-stochastic attention distributions that are inherently robust to anomalous noise. And we introduce an adaptive fusion strategy with learnable weights to dynamically combine base attention, convolution-enhanced scores, and OT-regularized scores based on five datasets. Experimental results show that MSC-OT performs well on five datasets, achieving an improvement compared to several time series models that have been widely cited in research over the past few years, especially compared to time series methods based on other embedding methods, showing a more significant improvement (improve 27 % 27\\\\% compared to Crossformer that based on DSW and TSA; improve 35 % 35\\\\% compared to TimesNet that transforming 1D time series into 2D tensors; improve 19 % 19\\\\% compared to Fedformer that based on frequency-domain sparse representations). The overall structure of MSC-OT is shown in Fig. 1.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note: The code available has been uploaded in GitHub FantaisieDeMickey/msc-ot/.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Related Works\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Deep Learning for Multivariate Time Series Forecasting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Multivariate time series (MTS) forecasting has evolved significantly with the development of deep learning. Early neural approaches employed TCNs [ 1 ] and RNN-based models [ 16 ] [ 15 ] to capture temporal dependencies, treating MTS data as sequences of vectors. LSTnet [ 5 ] combined CNN for cross-dimension dependency and RNN for cross-time dependency. Graph neural network-based methods [ 7 ] [ 25 ] explicitly modeled cross-variate relationships through graph convolution. However, these models face difficulty in capturing long-term dependencies due to the inherent limitations of CNN and RNN architectures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Transformer-based models have emerged as a dominant paradigm for MTS forecasting. Informer [ 29 ] introduced ProbSparse attention to reduce complexity from O ( L 2 ) O(L^{2}) to O ( L l o g L ) O(LlogL) . Autoformer [ 24 ] proposed series decomposition with auto-correlation mechanisms. FEDformer [ 30 ] leveraged frequency-domain sparse representations. Pyraformer [ 10 ] developed pyramidal attention for multi-resolution temporal modeling. However, these methods embed multiple variates at the same timestamp into a single token, focusing primarily on temporal dependency while inadequately capturing cross-variate correlations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent works have shifted focus toward explicit multivariate modeling. Crossformer [ 28 ] proposed Dimension-Segment-Wise embedding and Two-Stage Attention to capture both cross-time and cross-dimension dependencies. PatchTST [ 14 ] adopted channel-independent patching but overlooked inter-variable dependencies. ITransformer [ 11 ] introduced the inverted embedding paradigm that treats each variate’s entire time series as a token, enabling attention to model cross-variate correlations directly. Some models also use novel methods for time series prediction, such as D2Vformer [ 18 ] which devises to leverage timestamp information and feature sequences to generate time-position embeddings, and TimeCMA [ 8 ] which integrates LLM for time series analysis. Our MSC-OT builds upon iTransformer’s inverted framework while addressing its limitations in temporal dependency capturing and noise robustness through multi-scale convolution enhancement and optimal transport regularization.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B Multi-Scale Convolution for Time Series\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SCINet [ 9 ] exploits the property that temporal relations are largely preserved after downsampling, proposing a recursive downsample-convolve-interact architecture. TimesNet [ 23 ] takes a different approach by transforming 1D time series into 2D tensors based on discovered periodicities. MICN [ 21 ] addresses both local feature extraction and global correlation modeling through a multi-scale branch structure. More recently, MSDCN [ 6 ] proposes shallow multi-scale dilated convolution modules with exponentially growing dilation rates and varying kernel sizes to sample time series at different temporal scales. TimeMixer [ 22 ] proposes a decomposable multiscale mixing architecture that separately processes seasonal and trend components at different temporal scales through MLP-based mixing blocks. ModernTCN [ 12 ] redesigns the time-convolutional-networks(TCN) and shows that large-kernel convolutions can effectively capture long-range temporal dependencies while maintaining computational efficiency. FITS [ 26 ] shows that the method of simple linear interpolation on frequency can take a good forecasting performance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Sinkhorn Optimal Transport in Machine Learning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optimal transport (OT) provides a principled mathematical framework for comparing probability distributions. Cuturi introduced entropic regularization enabling efficient computation through the Sinkhorn-Knopp algorithm [ 3 ] , which alternates between row and column normalizations to converge orders of magnitude faster than linear programming solvers while maintaining full differentiability. This breakthrough has enabled OT integration into deep learning architectures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several works have applied OT principles to attention mechanisms. Sparse Sinkhorn Attention was proposed in employing meta sorting networks to learn latent permutations [ 19 ] , reducing memory complexity through Causal Sinkhorn Balancing and SortCut for dynamic sequence truncation. Sinkformers was introduced in replacing softmax with Sinkhorn normalization to produce doubly-stochastic attention matrices [ 17 ] . Gumbel-Sinkhorn Networks was developed for learning latent permutations through temperature-controlled Sinkhorn operators with Gumbel noise injection [ 13 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike these works that replace the entire attention computation with OT-based formulations, MSC-OT employs Sinkhorn OT as a complementary regularization component that is adaptively fused with base attention and convolution-enhanced scores. This design preserves the discriminative capacity of standard attention while leveraging OT’s inherent denoising capability—the doubly-stochastic constraint prevents any single variate from dominating attention even with anomalously high scores, and the entropy regularization naturally suppresses concentrated distributions sensitive to outliers.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III MSC-OT: Multi-Scale Convolution with Optimal Transport Attention\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section introduces our proposed MSC-OT (Multi-Scale Convolution-Optimal Transport) attention method, which aims to address the limitation of weak local temporal patterns capture capability in inverted embedding strategies in multivariate time series forecasting and achieve better results in noise suppression. We first present a problem description, then detail the various components of the architecture, and finally provide the adaptive score fusion.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A Problem Formulation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a multivariate time series dataset X ∈ ℝ B × T × N X\\\\in\\\\mathbb{R}^{B\\\\times T\\\\times N} , the goal of long-term time series forecasting is to predict the next S time steps: Y ∈ ℝ B × S × N Y\\\\in\\\\mathbb{R}^{B\\\\times S\\\\times N} . Unlike the traditional Transformer architecture, which embeds time unit into all variables, the iTransformer introduces an inverted embedding paradigm that treats the entire time series of each variable as a single token, resulting in an embedding representation H ∈ ℝ B × N × D H\\\\in\\\\mathbb{R}^{B\\\\times N\\\\times D} , where D is the embedding dimension.The inverted embedding does not model temporal dependencies any more, which may lead to the loss of fine-grained temporal local patterns for accurate prediction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B Architecture Overview\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address the limitations, we propose MSC-OT, a novel attention mechanism that integrates two complementary components: (1) Multi-Scale Convolution Enhancement for capturing multi-granularity structural patterns within the attention score matrix, and (2) Sinkhorn Optimal Transport for robust attention weight computation with noise suppression.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C Multi-Scale Convolution Enhancement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although iTransformer takes a significant result by inverted embedding method, the limitation of capturing local structure dependency is idendified. In the iTransformer embedding structure, the time series of each variate 𝐗 : , n ∈ ℝ T \\\\mathbf{X}_{:,n}\\\\in\\\\mathbb{R}^{T} is projected into a D-dimensional variate token by a linear transformation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The linear projection operates as a global receptive field over the entire temporal dimension and good at capture holistic series characteristics (e.g., overall trend, mean amplitude). But it embeds all temporal positions of each variate in a token, and lacks explicit mechanisms to model local temporal structures at multiple scales. But the short-term fluctuations, periodic patterns, and hierarchical temporal dependencies are cannot be ignored.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Addtionally, although the feed-forward network (FFN) in iTransformer is applied to each variate token to learn nonlinear temporal representations, its fully-connected nature still operates on a globally-compressed representation, limiting its ability to preserve and exploit fine-grained temporal dynamics. This architectural choice implicitly assumes that all relevant temporal information can be adequately captured through linear compression followed by nonlinear transformation—an assumption that may not hold for time series exhibiting complex multi-scale temporal structures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our process is that temporal dynamics within each variate should be explicitly modeled. We argue that the quality of multivariate correlation learning in the attention mechanism is fundamentally dependent on the richness of the individual variate representations. If critical local temporal patterns are lost during embedding, the subsequent attention mechanism—regardless of its sophistication—cannot recover this information. Integrating multi-scale convolutional operations within the attention mechanism is proposed to solve this kind of limitation. Given the raw attention scores S = Q K T d k ∈ ℝ B × H × N × N S=\\\\frac{QK^{T}}{\\\\sqrt{d_{k}}}\\\\in\\\\mathbb{R}^{B\\\\times H\\\\times N\\\\times N} , where B B is the batch size, H H is the number of attention heads, and N N is the number of variates, we apply a set of K K convolutional filters with different kernel sizes k 1 k_{1} , k 2 k_{2} , …, k K k_{K} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where Conv2d(·; k) denotes a 2D convolution operation with k × k k\\\\times k size kernel. In our experiments, we set K = 2 K=2 scales with kernel sizes 3 and 5, designed to capture multi-granularity structural patterns interactions. The multi-scale features are then aggregated through simple averaging:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This averaging strategy ensures numerical stability while preserving the complementary information captured at different scales. The convolution weights are initialized with small constant values to prevent disrupting the original attention patterns during early training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-D Sinkhorn Optimal Transport on Attention\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A single outlier can significantly influence the attention distribution, which means standard softmax attention are susceptible to outliers and noise in the input data. This phenomenon is particularly obvious in MTS. The data commonly used in MTS experiments usually contains measurement noise, missing values, and special points. It maybe cause the model to overemphasize special features and making the result offset to expected.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optimal Transport (OT) theory offers a simple method to solve this problem. It makes the model more focused on the overall distribution and partially ignoring outliers by transforming the attention mechanism to a optimal transport problem. It is aimed to find an allocation matrix that minimizes the cost of ”transferring” the query to the key while satisfying marginal constraints. The entropy regularization term in the Sinkhorn algorithm, an optimal transport algorithm, naturally smooths the final attention weights, effectively suppressing the impact of outliers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We formulate the attention computation as an entropy-regularized optimal transport problem. Given the attention score matrix S S , we compute the transport plan P P by solving:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where C = − S C=-S is the cost matrix (negative scores represent low transportation cost), 𝒰 ( r , c ) = { P ∣ P 𝟏 = r , P T 𝟏 = c } \\\\mathcal{U}(r,c)=\\\\{P\\\\mid P\\\\mathbf{1}=r,\\\\,P^{T}\\\\mathbf{1}=c\\\\} is the set of valid transport plans with marginals r r and c c (both uniform distributions in our case), λ \\\\lambda is the regularization strength, and H ( P ) = − ∑ i j P i j log P i j H(P)=-\\\\sum_{ij}P_{ij}\\\\log P_{ij} is the entropy of the transport plan. The Sinkhorn algorithm provides an efficient iterative solution by alternating between row and column normalization. Starting with the kernel matrix K = exp ( S λ ) K=\\\\exp\\\\left(\\\\frac{S}{\\\\lambda}\\\\right) , we iterate:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The final transport plan is computed as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In our experiments, we use λ = 5.0 \\\\lambda=5.0 , striking a balance between computational efficiency and solution quality. The larger λ l value ensures numerical stability by preventing extreme values in the exponential computation. The key advantage of Sinkhorn regularization is that its inherent noising suppression capability. The doubly-stochastic constraint forces the attention weights to be distributed across multiple keys, so that any noise value can be prevented from dominating the attention even if it produces an high score. Additionally, the entropy regularization term penalizes concentrated attention distributions, promoting a smoother attention pattern that is less sensitive to individual outliers. The OT effect on attention is illustrated in Fig 3.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-E Adaptive Score Fusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We combine the base attention scores S S , the convolution-enhanced scores S conv S_{\\\\text{conv}} , and the OT-regularized scores P P through a learnable weighted fusion:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where α \\\\alpha , β \\\\beta , γ \\\\gamma are learnable parameters initialized to 0.7, 0.2, and 0.1, respectively. We normalize the weights using softmax for ensuring stable training and prevent any component from dominating:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The initial weight distribution reflects our design idea: the base attention scores provide the main signal, the MSC enhancement offers complementary local structure information, and the OT regularization is used to keep stable and suppress the noise. During training, the model can adaptively adjust these weights based on the characteristics of the dataset.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-F Numerical Stability Considerations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Real-world time-series data often contains extreme values, which can lead to numerical instability in attention computation. We use two component-class methods (score for pruning and Sinkhorn kernel stabilization) to safeguard our experiments:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Score Pruning: The base attention scores are processed using a tanh activation function and scaled to the range [ − 8 , 8 ] [-8,8] to prevent extreme values from causing overflow in subsequent exponential operations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sinkhorn Kernel Stabilization: Before computing the kernel matrix K, we subtract the maximum value and prune the input values to the range [ − 50 , 50 ] [-50,50] . And we clamp intermediate results during iteration to prevent division by zero.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-G Scope of Applicability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MSC-OT is specifically designed for the inverted embedding method in MTS forecasting. The effectiveness of our method comes from its ability to compensate for the loss of temporal dependency information caused by inverted embedding methods. When the MSC-OT method is applied to traditional time series embedding methods (where each time step of all variables is treated as an embedded label), its advantages are reduced.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Experiments Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated the MSC-OT model using five widely used benchmark datasets, including ECL, ETT, Exchange, Traffic and Weather, where ETT consists of 4 subsets (ETTh1, ETTh2, ETTm1 and ETTm2). These datasets cover different application domains and exhibit varying characteristics in terms of dimensionality, temporal granularity, and sequence length.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use two widely used metrics to quantitatively evaluate forecast performance: Mean Squared Error (MSE) and Mean Absolute Error (MAE). MSE penalizes larger forecast errors more severely and is therefore more sensitive to outliers; while MAE provides a more balanced assessment of average forecast accuracy. For both metrics, lower values indicate better forecast performance. We compare MSC-OT with the recent-year MTS models including iTansformer, RLinear, PatchTST, Crossformer, TimesNet, SCINet, FEDformer and Autoformer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MSC-OT the model is architecture consists of two encoder layers and one decoder layer, with a model dimension of 512, eight attention heads, a feedforward dimension of 2048, and a dropout rate of 0.1. For the core attention module of MSC-OT, we configured multi-scale convolutions with kernel sizes of 3 × 3 3\\\\times 3 and 5 × 5 5\\\\times 5 to capture local patterns at different granularities. To ensure training stability, the convolution weights were initialized at a constant value of 0.1. The optimal sinkhorn transport regularization uses an entropy regularization coefficient λ = 5.0 \\\\lambda=5.0 and performs two matrix scaling iterations, which we found sufficient to obtain a near-optimal birandom distribution while maintaining computational efficiency. The adaptive fusion mechanism initializes the learnable weights as follows: base attention score α = 0.7 \\\\alpha=0.7 , convolutional enhancement score β = 0.2 \\\\beta=0.2 , and OT regularization score γ = 0.1 \\\\gamma=0.1 , using softmax normalization to ensure the sum of the weights is 1. The bound sets as t a n h tan{h} and the scaling factor sets as 8.0, that constrains the attention score within a reasonable range, cropping the score to the range [-20.0, 20.0] before OT calculation, and using a safety index limited to the range [-50, 50]. The model is optimized using the Adam optimizer with a learning rate of 1 × 10 − 4 1\\\\times 10^{-4} , a batch size of 32, and early stopping with a patience value of 3 epochs. The input sequence length is fixed at 96, and prediction ranges of 96, 192, 336, and 720 are used for long-term prediction evaluation. All experiments are repeated 3 times with different random seeds, and the average performance is reported.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-B Main Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We present comprehensive multivariate prediction results for prediction timeframes ranging from 96 to 720 time steps, enabling a comprehensive evaluation of both short-term and long-term prediction scenarios.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The experiments show that MSC-OT has the best MSE result on three datasets including ECL, Traffic and Weather, and has suboptimal result of MSE on Exchange. Meanwhile, the MSC-OT has best MAE result on four datasets except for ETT. Experimental results show that MSC-OT outperforms transformer-based multivariate time series models developed in recent years. The result is illustrated in Table 1.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Meanwhile, we also obtained the weight results of each component after adaptive weight fusion. This shows that the weights initially set to [0.7, 0.2, 0.1], after adaptive learning, generally account for about 50% of the MSC-OT weights across all datasets, indicating that they do indeed have a significant impact on the Attention mechanism and have an optimization effect on datasets of different lengths. Table 2 shows the detailed results of weights of different lengths across the four datasets. Fig. 4 shows a comparison of the proportions of the three weights after averaging across the datasets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-C Ablation Experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ablation experiments are conducted to validate the effectiveness of each proposed component and understand their individual and combined contributions. The experiments systematically separated the effects of multi-scale convolutional enhancement, Sinkhorn optimal transport regularization, and their synergistic combination.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sinkhorn Optimal Transport Regularization Analysis. We analyze the OT regularization component by comparing variants with and without the Sinkhorn algorithm. Excluding OT regularization leads to a measurable performance degradation, especially on high-dimensional datasets. The dual stochastic attention distribution enforced by Sinkhorn iterations ensures that each variable contributes information and receives information from other variables in a balanced manner. OT regularization effectively mitigates this problem by guaranteeing that the row and column sums of the attention matrix converge to a uniform distribution. Furthermore, as described in our methodology, the OT formula suppresses the influence of anomalous noise by treating attention computation as an optimal transmission problem, where extreme values are naturally suppressed through an entropy regularization term controlled by λ \\\\lambda .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Synergistic Effect of Combined Components. Ablation experiments show that multi-scale convolution and Sinkhorn OT regularization have a synergistic effect, instead of any independent one. The ablation experiment result shows a complementary relationship: multi-scale convolution captures local structural patterns and OT regularization ensures a balance in information propagation, so that optimizing the global distributed attention. This combined mechanism allows MSC-OT to simultaneously capture local patterns and balance global information, achieving robust and accurate predictions. The ablation experiment result on ECL is illustrated in Fig 5.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Adaptive Fusion Strategy Analysis. We further investigated the adaptive fusion mechanism by comparing the learned weights with fixed equal weights ( α = β = γ = 1 3 \\\\alpha=\\\\beta=\\\\gamma{}=\\\\frac{1}{3} ). The results show that the learnable fusion strategy consistently outperforms fixed weights, indicating that the optimal combination of base attention, convolutional augmentation scores, and OT regularization scores varies across datasets and is best determined through end-to-end learning. This adaptive behavior validates our design using learnable fusion weights with softmax normalization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We investigate the sensitivity of MSC-OT to key hyperparameters, focusing on the design choices reflected in our implementation. OT Regularization Strength ( λ \\\\lambda ). The entropy regularization coefficient λ \\\\lambda controls the smoothness of the optimal transport plan. We evaluate λ \\\\lambda values ranging from 1.0 to 10.0. Smaller values ( λ < 3.0 \\\\lambda<3.0 ) lead to numerical instability. Larger values ( λ > 8.0 \\\\lambda>8.0 ) over-smooth the attention distribution, reducing the model’s ability to discriminate between important and less relevant variate relationships. The setting λ \\\\lambda =5.0, as implemented in our model, achieves the optimal trade-off by ensuring stable computation while preserving discriminative attention patterns. This value keeps the exponential inputs within a numerically safe range while allowing sufficient variation in the resulting transport plan.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sinkhorn Iteration Count. Our ablation experiments result that set iterations from 1 to 5 shows that the iterations count is not very significant to improve the performance. After iterations arrived 2, the MSE and MAE have little been get better result any more. Adding more iterations will have a negligible effect on improvement, but will significantly increase the computational load, showing a linear growth. This finding aligns with theoretical results showing that Sinkhorn iterations converge rapidly for well-conditioned problems. Fig 6 shows the variation trends of MSE and MAE based on different iteration numbers.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Summary\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sufficient experimental evaluation shows that MSC-OT achieves state-of-the-art performance across most of datasets for both short-term and long-term multivariate time series forecasting. Ablation studies confirm that multi-scale convolution enhancement and Sinkhorn optimal transport regularization each provide distinct and complementary benefits: convolution captures local structural patterns at multiple granularities, while OT regularization ensures balanced cross-variate information and suppresses the noise. The adaptive fusion mechanism learns to optimally combine these components based on dataset characteristics. MSC-OT exhibits well robustness to outlier noise, validating the theoretical motivation of our MSC-OT approach, establishing MSC-OT as a practical and effective solution for real-world multivariate time series forecasting applications and exploring a new methodological path to optimize attention based on optimal transport.\"\n }\n]","isConversational":false}}},{"rowIdx":378,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.11963"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Evaluating Reliability in Machine Learning Models for Early Chronic Kidney Disease Prediction: A Systematic Review of Data Leakage and Predictor Stability"},"authors":{"kind":"list like","value":["Mashrul Hossain","Nafesa Kibria","Fahim Shahriar"],"string":"[\n \"Mashrul Hossain\",\n \"Nafesa Kibria\",\n \"Fahim Shahriar\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.11963v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Evaluating Reliability in Machine Learning Models for Early"},{"type":"para","text":"Chronic Kidney Disease Prediction: A Systematic Review of"},{"type":"para","text":"Data Leakage and Predictor Stability"},{"type":"para","text":"Mashrul Hossain, Nafesa Kibria, Fahim Shahriar"},{"type":"para","text":"mashrul16hossain@gmail.com, nafesa220128@gmail.com, fahimshahriar1306@gmail.com"},{"type":"para","text":"East West University, Dhaka, Bangladesh"},{"type":"para","text":"ABSTRACT - The early detection of Chronic Kidney Disease using machine learning has attracted significant interest in healthcare-related computer science. Despite rapid advancements in this field, many reported studies remain inconsistent and potentially misleading. A significant drawback is the lack of organized evaluation regarding methodological concerns. Key issues include data leakage, limited access to temporal patient records and inconsistency in reported clinical indicators. This research offers a systematic literature review of existing CKD prediction studies using interpretable machine learning techniques, where nineteen relevant studies were selected via systematic searches across major academic databases. To assess methodological reliability, this study introduces a structured taxonomy of information leakage and a quantitative leakage scoring framework to systematically evaluate reliability across CKD prediction studies. The analysis reveals a strong relationship between leakage and inflated performance. Here, High leakage-studies report an average accuracy of 95.48%, compared to 80.2% for leakage-free studies, reflecting an increase of approximately 15.28%. Furthermore, a cross-study feature stability analysis shows that only a small subset of predictors is consistently reproducible, with over 80% lacking reliability. Overall, the findings suggest that many reported performance improvements stem from methodological limitations rather than true predictive capability."},{"type":"para","text":"Keywords: Chronic Kidney Disease, Machine Learning, Data Leakage, Metrics, Feature Analysis, Biomarkers"},{"type":"heading","level":1,"text":"1. Introduction"},{"type":"para","text":"Chronic Kidney Disease (CKD) is a progressive condition characterized by gradual decline in kidney function, affecting more than 850 million individuals worldwide [ 2 ] [ 24 ] . Early detection of CKD is crucial because symptoms often appear only in advanced stages when treatment options quite limited. Early identification of CKD is therefore essential to delay disease progression, manage associated comorbidities and reduce the long-term economic burden on healthcare systems [ 8 ] [ 22 ] . In recent years, machine learning (ML) has emerged as a promising approach for analyzing complex clinical datasets and identifying patterns associated with the early onset of CKD that may not be detected through conventional statistical techniques [ 5 ] [ 20 ] ."},{"type":"para","text":"Machine Learning models can evaluate various types of patient data, such as lab results, demographic information and medical history, to assess a person’s likelihood of developing CKD. These forecasting systems assist healthcare providers in identifying high-risk patients sooner and facilitating preventive treatment strategies before significant kidney damage occurs [ 8 ] [ 20 ] . However, numerous critical challenges continue to restrict the real world implementation of ML driven diagnostic systems in healthcare. A well known problem is the restricted interpretability associated with numerous high-performing models, such as deep neural networks and complex ensemble techniques [ 8 ] [ 10 ] . Clinicians frequently require clear justifications for algorithmic predictions to trust them and integrate them into their medical decision making. Also, methodological limitations such as dataset bias, limited external validation and small sample sizes from individual centers continue to affect the reliability and generalizability of many existing studies [ 5 ] [ 20 ] [ 32 ] ."},{"type":"para","text":"Recent studies on early CKD prediction indicate a gradual shift from traditional statistical approaches to more advanced machine learning methods. Logistic regression is frequently utilized as a straightforward baseline model, while ensemble methods based on trees such as Random Forest, XGBoost and CatBoost which are typically applied to improve prediction accuracy [ 5 ] [ 4 ] [ 19 ] . These models usually depend on several widely cited clinical indicators, such as serum creatinine concentrations, estimated glomerular filtration rate (eGFR), hemoglobin concentrations, markers of albuminuria and demographic variables including age or hypertension history [ 5 ] [ 2 ] [ 33 ] [ 19 ] . Multiple studies demonstrate exceptionally high diagnostic precision and robust area-under the curve (AUC) metrics while evaluating their models on selected datasets. Consequently, models that achieve almost flawless outcomes in controlled studies may not function as effectively in real healthcare settings, where data quality, patient diversity and clinical variability are considerably higher [ 20 ] [ 1 ] ."},{"type":"para","text":"Despite the rapid advancements in machine learning research for early CKD prediction, several methodological issues remain insufficiently addressed in the literature. A key issue in CKD detection studies is the information leakage, which occurs when predictors contain information directly tied to the outcome label used during model training. For example, variables such as eGFR or serum creatinine are sometimes used as predictors even when CKD labels are derived from the same measurements, which can lead to overly optimistic model performance [ 17 ] . Again, the predictors which is identified as important often differ across studies, suggesting that variability in feature significance is based on feature selection technique and dataset traits [ 17 ] . Although previous studies on CKD prediction mainly compared model performance, they lack a unified framework to systematically detect information leakage or evaluate cross-study feature stability. This limitation highlights the need for a comprehensive evaluation of methodological validity, consistency of predictors and clinical relevance in existing CKD studies."},{"type":"para","text":"Research Questions : • How much does the data leakage influence ML models effectiveness?"},{"type":"para","text":"• Which clinical features show consistent stability in early CKD prediction studies and which predictors are dataset specific?"},{"type":"para","text":"Objectives : This study addresses the mentioned challenges by providing a structured analysis of methodological approaches in machine learning based CKD prediction research. The primary objectives of this study are outlined as follows: • A categorization of information leakage in CKD machine learning study, classifying direct leakage, proxy leakage and temporal leakage to provide more consistent experimental design in future studies. • Cross-study feature stability analysis in three steps to identify clinical predictors that remain consistent across multiple populations and those that seems to be dataset-specific predictors."},{"type":"heading","level":1,"text":"2. Related Works"},{"type":"para","text":"Detecting of Chronic Kidney Disease early is vital as it can slow down the disease progression. CKD doesn’t show symptoms in initial stages, so most cases are found in primary care settings [ 8 , 1 ] . Machine learning interpretable frameworks has emerged as a significant tool to sort complex clinical unorganized data and uncover patterns for risk detection that experienced clinicians may miss [ 20 , 1 ] . Early CKD prediction studies relied on classical algorithms, such as Logistic Regression, Support Vector Machines and Decision Trees [ 30 , 29 ] . These models generalizability was severely limited because of small, single-center datasets and minimal data quality management [ 9 , 1 ] . Specifically, the high performance scores often reported on the UCI benchmark serve as evidence of overfitting, raising doubts about model reliability in real-world healthcare settings. In real-world with real diversity in patients and clinical chaos, these models would likely stumble. [ 20 , 1 ] . There is methodological agreement regarding the importance of essential biomarkers such as serum creatinine and estimated glomerular filtration rate (eGFR), often identified through feature selection methods like Recursive Feature Elimination (RFE), Principal Component Analysis (PCA) or LASSO [ 6 , 1 ] . Furthermore, common predictors are albuminuria, patient age and hemoglobin levels [ 17 , 3 ] . A core methodological concern is the instability of leading predictors across studies. These variations often reflect dataset biases and regional demographic differences such as varying environmental factors and health system constraints [ 17 , 15 ] ."},{"type":"para","text":"A critical limitation identified is the widespread presence of information leakage. Numerous studies include pathology-test-based markers, such as serum creatinine or eGFR, as input features for model training even though those same exact tests are used to define CKD in the first place [ 17 ] . This practice introduces circular reasoning, where model ends up just matching the abnormal values instead of actually predicting anything useful, which results in misleading performance metrics that render the model redundant for clinical screening [ 17 , 29 ] . To address this issue, recent work focuses on leakage-safe preprocessing pipelines, ensuring that operations like scaling, outlier capping and class balancing are performed strictly within the training split [ 2 ] . Also, Performance evaluation has shifted towards metrics that handle imbalanced data better, like AUC-ROC, F1-score and Recall [ 5 , 17 ] ."},{"type":"heading","level":1,"text":"3. Methodology"},{"type":"heading","level":2,"text":"3.1 Literature Search Strategy"},{"type":"para","text":"A detailed search was conducted to identify studies on the early detection of Chronic Kidney Disease utilizing machine learning methods. Searching through many academic databases, such as IEEE Xplore, PubMed and Google Scholar, several papers were selected in the first stage."},{"type":"para","text":"The search utilized different combinations of particular keywords. Essential terms included: “CKD\" or Chronic Kidney Disease” or “CKD early detection” or “machine learning” or “ML”, “clinical indicators” or “clinical attributes”. These were combined into queries such as (“CKD” AND “machine learning” AND “early detection”) and (“chronic kidney disease” AND “clinical predictors”)."},{"type":"para","text":"The search focused on research published from 2021 to 2025 to include recent advancements in machine learning. Additionally, backward and forward citation tracking was conducted to discover relevant studies beyond the original search results."},{"type":"heading","level":2,"text":"3.2 Study Selection Process"},{"type":"para","text":"The selection of studies was conducted using a systematic method based on the PRISMA framework. Studies were first collected from various databases employing the specified search strategy. Duplicate entries were eliminated and the remaining documents were subjected to title and abstract screening to evaluate their relevance. Potentially relevant studies were then subjected to full-text review. During this stage, papers were assessed based on their emphasis on early-stage CKD detection, application of machine learning methods, inclusion of clinical predictors and incorporation of interpretability or explainability techniques. A total of 494 records were identified. After screening, reducing irrelevant studies, 19 studies were selected. Figure 1 is the visual representation of the study selection process. This structural filtering ensures that only relevant and high-quality studies are considered."},{"type":"heading","level":2,"text":"3.3 Inclusion and Exclusion Criteria"},{"type":"para","text":"Explicit inclusion and exclusion criteria were defined to maintain consistency during study selection."},{"type":"heading","level":3,"text":"3.3.1 Inclusion Criteria"},{"type":"para","text":"• Focused on ML-based early CKD prediction • Addressed early stage prediction • Utilized clinical features (laboratory or demographic data) • Published in English within (2021-2025)"},{"type":"heading","level":3,"text":"3.3.2 Exclusion Criteria"},{"type":"para","text":"• Focused only on medical imaging without predictors • Addressed other diseases (cardiovascular, diabetes, etc) • Focused on late-stage or general CKD detection • Lacked clinical predictor analysis"},{"type":"heading","level":2,"text":"3.4 Datasets Overview"},{"type":"para","text":"Datasets used in early chronic kidney disease (CKD) research range from relatively small hospital-based clinical records to very large national repositories such as the UK Biobank, which contains health information from more than half"},{"type":"para","text":"a million participants [ 11 ] . Among the datasets commonly used for benchmarking machine learning models, the UCI Machine Learning Repository remains one of the most widely adopted sources for algorithm evaluation and comparison [ 17 , 13 , 16 ] . At the same time, recent studies increasingly emphasize community-based screening datasets, particularly in South Asian regions, to support early detection strategies in low-resource settings where access to advanced laboratory testing is limited [ 17 , 27 ] ."},{"type":"para","text":"Real-world clinical datasets contain high-dimensional feature spaces, irregular time intervals between medical observations and substantial class imbalance, where CKD-positive cases represent a minority of the population [ 5 , 13 , 25 ] . Longitudinal Electronic Medical Record (EMR) data further highlight the dynamic nature of disease progression. For instance, evidence suggests that approximately 20.3% of diabetic patients showing early kidney impairment develop CKD within six months [ 5 ] . Most predictive modeling studies rely on datasets containing roughly 24 to 36 clinical attributes, which include demographic factors as well as biochemical markers such as serum creatinine and estimated glomerular filtration rate (eGFR) [ 13 , 33 , 34 ] ."},{"type":"para","text":"However, widely used benchmark datasets may not fully represent the demographic diversity of global populations. As a result, several studies validate their models using independent cohorts collected from countries such as Bangladesh, Korea, China and the United Arab Emirates [ 17 , 21 , 35 , 10 ] . National health surveys like NHANES are also frequently used because they follow standardized data collection procedures and incorporate additional lifestyle variables, including dietary patterns and sleep habits, which can influence kidney health outcomes [ 33 , 34 ] . Table 1 summarizes the datasets used across reviewed studies."},{"type":"heading","level":2,"text":"3.5 Preprocessing Techniques"},{"type":"para","text":"Data preprocessing is a crucial step in preparing healthcare datasets for machine learning analysis, as raw clinical data often contain inconsistencies and missing values [ 10 , 34 ] . One of the most common challenges involves handling incomplete records, which may arise from random data loss or manual entry errors during clinical documentation [ 16 , 13 ] . To address this issue, researchers frequently apply imputation techniques such as Multivariate Imputation by Chained Equations (MICE), K-nearest neighbor (KNN) imputation or median replacement to reduce the influence of missing information while maintaining robustness against outliers [ 17 , 13 , 33 ] ."},{"type":"para","text":"Additionally to handle temporal variability, longitudinal medical records are frequently summarized within a specific observation window using statistical measures such as the mean, quartiles or standard deviation [ 5 ] . Feature scaling is also routinely performed to ensure consistent data ranges across variables, typically through Min–Max normalization or Z-score standardization [ 17 , 13 , 27 , 16 ] . Moreover, techniques for identifying and correcting outliers like interquartile range (IQR) are employed to enhance model stability and prediction accuracy [ 27 , 13 ] . Due to the common issue of class imbalance in CKD datasets, methods such as the Synthetic Minority Over-sampling Technique (SMOTE) or random oversampling are frequently employed to equilibrate the distribution between healthy and CKD-affected cases [ 13 , 25 , 11 ] ."},{"type":"heading","level":2,"text":"3.6 Feature Selection Methods"},{"type":"para","text":"Feature selection is equally crucial in determining the most informative clinical predictors. These techniques are typically categorized into statistical filter methods, model-based wrapper approaches and regularization strategies [ 17 , 27 ] . Tests like Pearson correlation, ANOVA and the Mann–Whitney U test are frequently utilized to identify variables that significantly vary between CKD and non-CKD populations [ 17 , 13 , 11 ] . Wrapper-based methods like Recursive Feature Elimination with Cross-Validation (RFECV) and Sequential Feature Selection (SFS) have demonstrated the ability to discover smaller subsets of features that often outperform models trained on the complete set of variables [ 17 , 27 ] ."},{"type":"heading","level":1,"text":"4. Data Leakage Evaluation"},{"type":"para","text":"The following section provides a structured evaluation of data leakage in early chronic kidney disease (CKD) prediction studies. To ensure consistent evaluation across studies, three types of data leakage are defined using the following criteria : Type 1—A feature is considered Direct Leakage (DL) if it is included in diagnostic criteria for defining CKD. Here, the criteria consist of estimated glomerular filtration rate, serum creatinine and albuminuria. Including these as predictive features may lead to circular reasoning or may even hamper the model’s performance, because these features directly impact the clinical understanding of CKD."},{"type":"para","text":"Type 2—Proxy Leakage (PL) occurs when a feature is not part of the diagnostic criteria but is statistically or clinically redundant with one. In this study, proxy leakage is identified using a hybrid criterion: (i) strong statistical correlation with diagnostic biomarkers ( | r | ≥ 0.85 |r|\\geq 0.85 ) or (ii) clinical or physiological dependency on the same underlying renal function pathway, even when pairwise correlations are moderate due to dataset-specific factors. The ( | r | ≥ 0.85 |r|\\geq 0.85 ) threshold is commonly adopted as a conservative indicator of strong association in clinical datasets."},{"type":"para","text":"Type 3—Temporal leakage occurs when predictors gathered after the outcome event are utilized in training the model. Examples include medications after diagnosis, dialysis treatments or subsequent interventions. Since these variables are unavailable at prediction time, their inclusion leads to unrealistic model performance, a phenomenon widely recognized in machine-learning evaluation pipelines [ 18 ] . Table 2 presents an overview of the definitions and detection criteria for every leakage category utilized in this analysis."},{"type":"para","text":"To quantify the extent of leakage in each study, a unified leakage score is defined as:"},{"type":"para","text":"where L i L_{i} represents the total leakage score for study i i , while D L i DL_{i} , P L i PL_{i} and T L i TL_{i} denote direct, proxy and temporal leakage components, respectively."},{"type":"para","text":"The individual leakage components are defined as:"},{"type":"para","text":"A correlation of | r | ≥ 0.85 |r|\\geq 0.85 corresponds to a Variance Inflation Factor exceeding 6.7 6.7 ( V I F = 1 / ( 1 − r 2 ) VIF=1/(1-r^{2}) ), indicating near-collinear redundancy with diagnostic biomarkers and warranting full proxy leakage assignment ( P L = 2 PL=2 ) [ 12 , 23 ] . The intermediate range ( 0.60 ≤ | r | < 0.85 0.60\\leq|r|<0.85 , V I F = 1.6 VIF=1.6 – 6.7 6.7 ) captures moderate statistical dependence that may not individually constitute leakage but creates meaningful predictive overlap, particularly when multiple such variables are included simultaneously [ 31 ] ( P L = 1 PL=1 ), consistent with evidence that partial information leakage can still inflate model performance [ 18 ] . Variables with | r | < 0.60 |r|<0.60 are treated as statistically independent of diagnostic criteria ( P L = 0 PL=0 ), following conventional interpretations of correlation strength where values near r = 0.50 r=0.50 represent large effects in behavioral and clinical data [ 7 ] ."},{"type":"para","text":"Based on the relative severity of each leakage type, weights are assigned as:"},{"type":"para","text":"Thus, the final leakage scoring function is:"},{"type":"para","text":"Leakage Classification Rule"},{"type":"para","text":"To facilitate interpretation, studies are categorized according to their leakage score:"},{"type":"para","text":"Figure 2 presents a conceptual comparison between conventional and leakage-aware CKD prediction workflows. The figure highlights the importance of appropriate feature handling and leakage control to ensure reliable model evaluation."},{"type":"heading","level":2,"text":"4.1 Effect of Data Leakage on Model Performance"},{"type":"para","text":"The analysis reveals a systematic relationship between information leakage and excessive model performance in early CKD prediction studies. By applying the proposed three-part taxonomy which are Direct Leakage (DL), Proxy Leakage (PL) and Temporal Leakage (TL), a consistent pattern emerges across the reviewed literature. Studies that incorporate features directly embedded in diagnostic criteria, such as eGFR, serum creatinine or albuminuria, almost always achieve disproportionately high reported accuracies. This is not surprising, as these variables are not merely predictive signals but are, in fact, definitional components of the outcome itself. Consequently, models in these settings are not learning latent disease patterns but are effectively reconstructing the diagnostic rule. This phenomenon is reflected across a large proportion of studies [ 8 , 9 , 28 , 25 , 16 , 13 , 35 ] which receive the maximum leakage score due to the simultaneous presence of direct and proxy leakage. Importantly, even in the absence of temporal leakage, the coexistence of direct and proxy leakage is sufficient to produce near-ceiling performance metrics."},{"type":"para","text":"Table 3 demonstrates a continuously increasing trend in reported accuracy as the severity of data leakage increases across models. In the clean configuration (no leakage), models achieved accuracy ranging from 92.50% (AdaBoost) to 97.50% (LightGBM). Under the proxy leakage condition, where features correlated with diagnostic criteria, all models exhibited measurable performance gains, with AdaBoost showing the most dramatic improvement of +5.00% in accuracy and +6.29% in F1-score. Under the direct leakage condition, where definitional components of CKD were included as predictors, all models achieved near-perfect or perfect accuracy (98.75–100.00%) and AUC (100.00%), regardless of their baseline performance. Here, Figure 3 summarizes the overall workflow of the proposed leakage scoring framework."},{"type":"heading","level":2,"text":"4.2 Proxy Correlation Undermining Predictive Learning"},{"type":"para","text":"A detailed examination of proxy leakage shows how machine learning models can generate overly optimistic performance in chronic kidney disease (CKD) prediction tasks. Proxy leakage occurs when non-diagnostic variables are highly correlated ( | r | ≥ 0.85 |r|\\geq 0.85 ) with established diagnostic biomarkers or clinically redundant with diagnostic biomarkers which enables the model to implicitly reconstruct the target label without learning independent predictive patterns. In clinical datasets, variables such as Blood Urea Nitrogen (BUN), Cystatin C and hemoglobin often demonstrate significant correlations with primary indicators like estimated glomerular filtration rate (eGFR) and serum creatinine. Although these variables represent different clinical measurements, their statistical connections allow models to avoid authentic predictive learning by leveraging overlapping data. This phenomenon is demonstrated in studies such as [ 27 ] and [ 21 ] , where moderate leakage correlates with high accuracy, suggesting that even limited proxy leakage can greatly inflate reported performance."},{"type":"para","text":"Importantly, proxy leakage in clinical datasets is not always reflected through high pairwise correlation alone. In several cases (UCI-based studies), individual correlations with diagnostic biomarkers remain moderate ( | r | ≈ 0.5 |r|\\approx 0.5 – 0.6 0.6 ), yet multiple physiologically linked variables (blood urea nitrogen, hemoglobin, packed cell volume) are simultaneously included. These variables collectively encode the same underlying renal dysfunction, forming a multivariate proxy structure. As a result, PL=2 is assigned in such cases to reflect cumulative redundancy, even in the absence of a single dominant correlation."},{"type":"para","text":"Due to limited public availability of all datasets used in the reviewed studies (Table 4 ), correlation analysis was conducted on representative datasets (UCI CKD, Enam Medical, NHANES and Abu Dhabi cohorts). For studies utilizing identical or derived datasets, proxy leakage labels were assigned based on these empirical correlations combined with reported feature sets."},{"type":"para","text":"Apart from performance inflation, proxy leakage affects the learning behavior of machine learning models. Tree-based ensemble methods, including XGBoost and Random Forest, are particularly sensitive due to their split-based optimization process. During training, these models select features that maximize information gain and highly correlated predictors are often interchangeable in this process. As a result, proxy variables can repeatedly appear in decision splits, reinforcing the same underlying signal across the ensemble [ 10 , 11 , 16 , 21 , 27 ] . This leads to models that appear to learn clinically meaningful relationships but are actually exploiting statistical redundancy. Consequently, variables such as hemoglobin and blood urea nitrogen may receive high importance even though they primarily reflect renal impairment already encoded by creatinine or eGFR [ 27 , 20 , 6 ] ."},{"type":"para","text":"Rankings of feature importance under proxy leakage might not represent independent risk factors but rather serve as substitute indicators of diagnostic criteria. Numerous studies recognize hemoglobin and packed cell volume as important predictors [ 6 , 20 , 27 , 28 ] , but these factors are physiologically related to kidney function and highly correlated with creatinine-based metrics, resulting in erroneous conclusions where models seem to indicate causal relationships but actually reflect disease status. This likewise diminishes external validity, since models developed on datasets with robust internal correlations frequently perform poorly on external cohorts having different correlation patterns [ 6 , 2 , 14 , 28 ] . This hybrid formulation ensures that proxy leakage is not underestimated in heterogeneous clinical datasets where statistical correlation alone may fail to capture underlying physiological dependencies."},{"type":"heading","level":2,"text":"4.3 Statistical Evidence of Leakage-Induced Inflation"},{"type":"para","text":"The aggregate performance comparison in Table 5 provides clear evidence of leakage-induced inflation. This highlights the importance of rigorous leakage auditing before evaluating model performance or reporting results in early CKD prediction."},{"type":"para","text":"The statistical significance (p = 0.005) of Table 5 indicates that the observed difference is very unlikely to be explained solely by sampling variability. The consistent inflation of model performance due to leakage indicates a methodological bias instead of authentic predictive capability."},{"type":"para","text":"Figure 4 represents a gradual rise in reported accuracy as the severity of data leakage increases across multiple studies. Models classified under the high leakage category consistently report inflated performance, indicating potential overestimation of true predictive ability. In contrast, leakage-free studies exhibit relatively lower and more varied accuracy values, indicating more authentic model performance."},{"type":"para","text":"Besides statistical significance, the size of the observed difference indicates a considerable practical effect. An average inflation exceeding 15.2% implies that models utilizing predictors prone to leakage could greatly exaggerate their diagnostic ability in practical scenarios. Importantly, most of the studies where leakage was identified fall into the moderate leakage category, indicating that even small amounts of leakage can influence the reported performance of machine learning models, where inconsistent handling of leakage across studies undermines the reliability of cross-study performance comparisons and further complicates fair benchmarking of models.These findings collectively indicate that to have a fair comparison between the published performance benchmarks, we first need to audit and standardize how leakage is handled."},{"type":"heading","level":1,"text":"5. Cross-Study Feature Stability Analysis"},{"type":"para","text":"The growing use of machine learning models to predict Chronic Kidney Disease (CKD) has resulted in a fragmented and often inconsistent understanding of key factors of the disease. Top predictors of CKD differ across studies, making it difficult to detect characteristics that indicate generalizable risk signals. This inconsistency raises an essential question: Which predictors are actually reliable across different methods and geographies? which result from dataset-specific traits, methodological biases?"},{"type":"para","text":"To address this problem, a systematic cross-study feature stability analysis was performed on nineteen CKD prediction studies. Specifically, 28 standardized attributes (Table 6 ) were classified into groups. This detailed classification allows for a more refined understanding of feature importance and highlights the interdisciplinary nature of CKD risk modeling."},{"type":"para","text":"Each predictor was evaluated based on its frequency of occurrence across the 19 studies and a normalized stability score was computed. High-consistency predictors included patient age, blood pressure, glycemic control, hemoglobin/anemia, serum creatinine and albuminuria. These features cover multiple clinical areas, reinforcing their apparent robustness across diverse datasets and modeling techniques."},{"type":"para","text":"Figure 5 presents the distribution of predictors across stability levels (high, moderate, low) within each clinical domain. The renal and clinical areas show the highest concentration of predictors overall and contain the majority of high and moderate stability features, reflecting their frequent inclusion in early CKD prediction models. In contrast, domains such as lifestyle, socioeconomic, dietary and inflammatory factors are composed almost entirely of low-stability predictors, indicating restricted usage across various studies. Numerous domains like anthropometric, metabolic and comorbidity factors include just one or two predictors with low stability representation. In general, the figure shows a significant disparity in feature utilization, with CKD prediction studies focusing on a narrow range of clinical and renal factors."},{"type":"para","text":"Category-level aggregation further revealed a disproportionate dominance of renal biomarkers in the literature, with the highest cumulative reporting frequency. This overrepresentation continues even with the well- documented risk of data leakage linked with diagnostic variables such as serum creatinine and eGFR. Their frequent inclusion likely reflects their data availability and diagnostic role rather than true predictive value."},{"type":"para","text":"The cross-dataset stability analysis of Table 7 highlights notable differences between predictors derived from studies using the UCI Chronic Kidney Disease Dataset and those evaluated on independent clinical datasets. Demographic and clinical variables such as patient age and blood pressure demonstrate the highest cross-dataset stability, indicating strong generalizability across diverse study populations and modeling approaches. Across multiple studies, Indicators like glycemic and hemoglobin maintain comparatively high stability, indicating their association with CKD prediction [ 17 , 21 ] . In general, the results suggest that only a limited set of predictors demonstrates strong cross-population stability, highlighting the necessity for wider dataset variety and standardized feature evaluation for upcoming studies."},{"type":"para","text":"Inter-study agreement in feature selection was evaluated using Fleiss’ kappa (Table 8 ). A κ \\kappa value of 0.224 was obtained, indicating fair agreement among the examined studies. This suggests that although a limited number of predictors (such as age, blood pressure and glycemic indicators) appear consistently across studies, the majority of features are inconsistent. The relatively low k value highlights the disjointed nature of feature selection and supports the observation that many predictors are specific to datasets rather than universally reliable."},{"type":"para","text":"In Figure 6 , the frequency distribution of all 28 predictors across the 19 reviewed studies is shown, ranked by occurrence. A steep decline in frequency is observed beyond the sixth predictor, with the majority of features appearing in five or fewer studies. This distribution visually reinforces the skewed nature of feature selection in CKD prediction research, where a small number of predictors dominate while most are inconsistent."},{"type":"para","text":"The pareto curve of Figure 7 demonstrates the combined contribution of predictors to total feature usage across different studies. A considerable share of overall usage is represented by high-ranking features as it can be seen from the sharp initial rise in the graph. Specifically, the initial group of predictors accounts for about 80% of total coverage, illustrating a significant concentration effect. This pattern highlights the dominance of a limited core feature set in CKD prediction models."},{"type":"para","text":"The analysis of Table 9 illustrates a distinct ranking of clinically reliable predictors across the five leakage-safe studies. Patient age and glycemic control emerge as the most stable and universally utilized features, indicating their robust and consistent predictive relevance. Evidence from broad inclusion and higher stability, further supports blood pressure and gender as fundamental predictor. In contrast, variables such as BMI and albuminuria show moderate consistency, suggesting situational relevance across datasets. Several features, including lifestyle factors and hematological markers, show low stability due to limited inclusion or data availability. Specifically, traditionally robust clinical predictors like serum creatinine and eGFR are often excluded to avoid data leakage."},{"type":"para","text":"Table 10 highlights that patient age, glycemic control and blood pressure are strong indicators, maintaining high performance even after leakage adjustment. Gender, albuminuria and hemoglobin demonstrate moderate to high stability, enhancing their clinical significance. Conversely, the decreasing stability of serum creatinine and hematological parameters under leakage-safe conditions suggests a potential reliance on dataset-specific signals."},{"type":"para","text":"Only a limited number of predictors display both conceptual independence from diagnostic standards and methodological strength. Across all three evaluation dimensions, patient age, blood pressure and glycemic control consistently achieve the highest average stability, marking the minimum reliable foundation for CKD risk modeling. Serum creatinine and albuminuria, despite their stability, are directly associated with CKD diagnosis, raising concerns about diagnostic data-leakage. A fragmented research landscape is reflected in the dominance of low-consistency indicators. Lifestyle, dietary and socioeconomic factors remain largely unexplored, although they offer potential as leakage-free early indicators. Future research should focus on external validation of the stable core predictors and investigate neglected modifiable risk factors. Moreover, Future work should apply standardized methods for feature selection to enhance the robustness and clinical applicability of machine learning models."},{"type":"heading","level":1,"text":"6. Future Directions"},{"type":"para","text":"Several clear methodological improvements emerge from this review. Researchers should implement data leakage auditing protocols throughout model development and follow standardized reporting methods aligned with TRIPOD guidelines. Additionally, the field would benefit from benchmark datasets that clearly differentiate diagnostic biomarkers from predictive risk factors, enabling more reliable comparisons among models. Future work could also benefit from focusing on simple temporal features, such as eGFR decline rate or variability across multiple measurements, to more effectively reflect disease progression without relying on complex sequential models. Furthermore, more reliable detection of data leakage, particularly in resource-limited environments, can be obtained by examining non-renal risk factors like lifestyle choices, socioeconomic status."},{"type":"heading","level":1,"text":"7. Limitations"},{"type":"para","text":"This review is subject to several important constraints. One limitation is that correlation analysis covers only four representative datasets, which limits the ability to verify findings from studies that used inaccessible data. Geographical bias presents another concern, as eight of the nineteen reviewed studies use the CKD dataset from an Indian clinical center. Here, proxy leakage labels for the remaining studies rely on reported feature sets and clinical reasoning. Also, the review excludes relevant recent works as it is limited to studies published up to 2025. Moreover, temporal leakage could not be formally validated because of insufficient timestamp data provided in the examined studies. Additionally, most reviewed studies were evaluated on limited or geographically localized datasets, which may reduce real-world clinical generalizability."},{"type":"heading","level":1,"text":"8. Conclusion"},{"type":"para","text":"This systematic review analyzed methods in early CKD prediction using machine learning, with a primary focus on information leakage and cross-study predictor stability. One of the findings shows exaggerated predictive effectiveness due to the inclusion of variables that are closely aligned with CKD diagnostic criteria. This study shows that a considerable amount of reported accuracy likely originates from inherent clinical definitions instead of genuine predictive learning. This finding is demonstrated by applying a structured taxonomy of direct, proxy and temporal leakage. Conversely, studies that avoid leakage-prone features tend to report more moderate but clinically relevant performance, highlighting the necessity of careful feature selection. Furthermore, the cross-study analysis indicates a lack of consistency in the importance of low-leakage predictors, with only a few variables, such as age, blood Pressure and glycemic Control, appearing consistently across multiple datasets. This variability suggests that model behavior often reflects dataset-specific characteristics. Consequently, concerns about generalizability and reproducibility are justified. Although temporal dynamics were not explicitly analyzed, their potential significance in understanding disease progression is recognized as a key area for future research. A Pareto curve analysis further supports the findings as it shows that 80% of cumulative feature coverage is achieved with 14 variables, highlighting the diminishing marginal utility of adding extra predictors in low-leakage CKD models. Overall, the study highlights the necessity for stricter methodological standards and robust validation frameworks to ensure reliable and clinically significant CKD prediction models."},{"type":"para","text":"Author Contributions M.H. conceived the study, designed the methodology, conducted the systematic review, performed analysis, and interpretation, developed the leakage scoring framework, prepared the figures and tables, and wrote the original manuscript draft. N.K. contributed to the study design, assisted with data interpretation, critically reviewed and revised the manuscript, and provided valuable intellectual input. F.S. contributed to the literature review, assisted with data validation and manuscript revision, and provided constructive feedback throughout the study. All authors read and approved the final manuscript."},{"type":"para","text":"Corresponding Author mashrul16hossain@gmail.com (Mashrul Hossain)"},{"type":"para","text":"Funding There was no funding obtained to conduct this research."},{"type":"para","text":"Competing Interest The authors declare there were no competing interests."},{"type":"para","text":"Ethics Statement This study is a systematic review of previously published studies using publicly available and anonymized datasets. No new human or animal data were collected. Accordingly, no ethical approval or informed consent was required. Data Availability No new datasets were generated or analyzed. Used dataset is available on UCI Machine Learning Repository."},{"type":"para","text":"Consent for publication All authors have read and approved the final manuscript"}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Evaluating Reliability in Machine Learning Models for Early\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Chronic Kidney Disease Prediction: A Systematic Review of\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data Leakage and Predictor Stability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mashrul Hossain, Nafesa Kibria, Fahim Shahriar\"\n },\n {\n \"type\": \"para\",\n \"text\": \"mashrul16hossain@gmail.com, nafesa220128@gmail.com, fahimshahriar1306@gmail.com\"\n },\n {\n \"type\": \"para\",\n \"text\": \"East West University, Dhaka, Bangladesh\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ABSTRACT - The early detection of Chronic Kidney Disease using machine learning has attracted significant interest in healthcare-related computer science. Despite rapid advancements in this field, many reported studies remain inconsistent and potentially misleading. A significant drawback is the lack of organized evaluation regarding methodological concerns. Key issues include data leakage, limited access to temporal patient records and inconsistency in reported clinical indicators. This research offers a systematic literature review of existing CKD prediction studies using interpretable machine learning techniques, where nineteen relevant studies were selected via systematic searches across major academic databases. To assess methodological reliability, this study introduces a structured taxonomy of information leakage and a quantitative leakage scoring framework to systematically evaluate reliability across CKD prediction studies. The analysis reveals a strong relationship between leakage and inflated performance. Here, High leakage-studies report an average accuracy of 95.48%, compared to 80.2% for leakage-free studies, reflecting an increase of approximately 15.28%. Furthermore, a cross-study feature stability analysis shows that only a small subset of predictors is consistently reproducible, with over 80% lacking reliability. Overall, the findings suggest that many reported performance improvements stem from methodological limitations rather than true predictive capability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Keywords: Chronic Kidney Disease, Machine Learning, Data Leakage, Metrics, Feature Analysis, Biomarkers\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Chronic Kidney Disease (CKD) is a progressive condition characterized by gradual decline in kidney function, affecting more than 850 million individuals worldwide [ 2 ] [ 24 ] . Early detection of CKD is crucial because symptoms often appear only in advanced stages when treatment options quite limited. Early identification of CKD is therefore essential to delay disease progression, manage associated comorbidities and reduce the long-term economic burden on healthcare systems [ 8 ] [ 22 ] . In recent years, machine learning (ML) has emerged as a promising approach for analyzing complex clinical datasets and identifying patterns associated with the early onset of CKD that may not be detected through conventional statistical techniques [ 5 ] [ 20 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Machine Learning models can evaluate various types of patient data, such as lab results, demographic information and medical history, to assess a person’s likelihood of developing CKD. These forecasting systems assist healthcare providers in identifying high-risk patients sooner and facilitating preventive treatment strategies before significant kidney damage occurs [ 8 ] [ 20 ] . However, numerous critical challenges continue to restrict the real world implementation of ML driven diagnostic systems in healthcare. A well known problem is the restricted interpretability associated with numerous high-performing models, such as deep neural networks and complex ensemble techniques [ 8 ] [ 10 ] . Clinicians frequently require clear justifications for algorithmic predictions to trust them and integrate them into their medical decision making. Also, methodological limitations such as dataset bias, limited external validation and small sample sizes from individual centers continue to affect the reliability and generalizability of many existing studies [ 5 ] [ 20 ] [ 32 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent studies on early CKD prediction indicate a gradual shift from traditional statistical approaches to more advanced machine learning methods. Logistic regression is frequently utilized as a straightforward baseline model, while ensemble methods based on trees such as Random Forest, XGBoost and CatBoost which are typically applied to improve prediction accuracy [ 5 ] [ 4 ] [ 19 ] . These models usually depend on several widely cited clinical indicators, such as serum creatinine concentrations, estimated glomerular filtration rate (eGFR), hemoglobin concentrations, markers of albuminuria and demographic variables including age or hypertension history [ 5 ] [ 2 ] [ 33 ] [ 19 ] . Multiple studies demonstrate exceptionally high diagnostic precision and robust area-under the curve (AUC) metrics while evaluating their models on selected datasets. Consequently, models that achieve almost flawless outcomes in controlled studies may not function as effectively in real healthcare settings, where data quality, patient diversity and clinical variability are considerably higher [ 20 ] [ 1 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite the rapid advancements in machine learning research for early CKD prediction, several methodological issues remain insufficiently addressed in the literature. A key issue in CKD detection studies is the information leakage, which occurs when predictors contain information directly tied to the outcome label used during model training. For example, variables such as eGFR or serum creatinine are sometimes used as predictors even when CKD labels are derived from the same measurements, which can lead to overly optimistic model performance [ 17 ] . Again, the predictors which is identified as important often differ across studies, suggesting that variability in feature significance is based on feature selection technique and dataset traits [ 17 ] . Although previous studies on CKD prediction mainly compared model performance, they lack a unified framework to systematically detect information leakage or evaluate cross-study feature stability. This limitation highlights the need for a comprehensive evaluation of methodological validity, consistency of predictors and clinical relevance in existing CKD studies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Research Questions : • How much does the data leakage influence ML models effectiveness?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"• Which clinical features show consistent stability in early CKD prediction studies and which predictors are dataset specific?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Objectives : This study addresses the mentioned challenges by providing a structured analysis of methodological approaches in machine learning based CKD prediction research. The primary objectives of this study are outlined as follows: • A categorization of information leakage in CKD machine learning study, classifying direct leakage, proxy leakage and temporal leakage to provide more consistent experimental design in future studies. • Cross-study feature stability analysis in three steps to identify clinical predictors that remain consistent across multiple populations and those that seems to be dataset-specific predictors.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2. Related Works\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Detecting of Chronic Kidney Disease early is vital as it can slow down the disease progression. CKD doesn’t show symptoms in initial stages, so most cases are found in primary care settings [ 8 , 1 ] . Machine learning interpretable frameworks has emerged as a significant tool to sort complex clinical unorganized data and uncover patterns for risk detection that experienced clinicians may miss [ 20 , 1 ] . Early CKD prediction studies relied on classical algorithms, such as Logistic Regression, Support Vector Machines and Decision Trees [ 30 , 29 ] . These models generalizability was severely limited because of small, single-center datasets and minimal data quality management [ 9 , 1 ] . Specifically, the high performance scores often reported on the UCI benchmark serve as evidence of overfitting, raising doubts about model reliability in real-world healthcare settings. In real-world with real diversity in patients and clinical chaos, these models would likely stumble. [ 20 , 1 ] . There is methodological agreement regarding the importance of essential biomarkers such as serum creatinine and estimated glomerular filtration rate (eGFR), often identified through feature selection methods like Recursive Feature Elimination (RFE), Principal Component Analysis (PCA) or LASSO [ 6 , 1 ] . Furthermore, common predictors are albuminuria, patient age and hemoglobin levels [ 17 , 3 ] . A core methodological concern is the instability of leading predictors across studies. These variations often reflect dataset biases and regional demographic differences such as varying environmental factors and health system constraints [ 17 , 15 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A critical limitation identified is the widespread presence of information leakage. Numerous studies include pathology-test-based markers, such as serum creatinine or eGFR, as input features for model training even though those same exact tests are used to define CKD in the first place [ 17 ] . This practice introduces circular reasoning, where model ends up just matching the abnormal values instead of actually predicting anything useful, which results in misleading performance metrics that render the model redundant for clinical screening [ 17 , 29 ] . To address this issue, recent work focuses on leakage-safe preprocessing pipelines, ensuring that operations like scaling, outlier capping and class balancing are performed strictly within the training split [ 2 ] . Also, Performance evaluation has shifted towards metrics that handle imbalanced data better, like AUC-ROC, F1-score and Recall [ 5 , 17 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3. Methodology\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Literature Search Strategy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A detailed search was conducted to identify studies on the early detection of Chronic Kidney Disease utilizing machine learning methods. Searching through many academic databases, such as IEEE Xplore, PubMed and Google Scholar, several papers were selected in the first stage.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The search utilized different combinations of particular keywords. Essential terms included: “CKD\\\" or Chronic Kidney Disease” or “CKD early detection” or “machine learning” or “ML”, “clinical indicators” or “clinical attributes”. These were combined into queries such as (“CKD” AND “machine learning” AND “early detection”) and (“chronic kidney disease” AND “clinical predictors”).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The search focused on research published from 2021 to 2025 to include recent advancements in machine learning. Additionally, backward and forward citation tracking was conducted to discover relevant studies beyond the original search results.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Study Selection Process\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selection of studies was conducted using a systematic method based on the PRISMA framework. Studies were first collected from various databases employing the specified search strategy. Duplicate entries were eliminated and the remaining documents were subjected to title and abstract screening to evaluate their relevance. Potentially relevant studies were then subjected to full-text review. During this stage, papers were assessed based on their emphasis on early-stage CKD detection, application of machine learning methods, inclusion of clinical predictors and incorporation of interpretability or explainability techniques. A total of 494 records were identified. After screening, reducing irrelevant studies, 19 studies were selected. Figure 1 is the visual representation of the study selection process. This structural filtering ensures that only relevant and high-quality studies are considered.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Inclusion and Exclusion Criteria\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Explicit inclusion and exclusion criteria were defined to maintain consistency during study selection.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.1 Inclusion Criteria\"\n },\n {\n \"type\": \"para\",\n \"text\": \"• Focused on ML-based early CKD prediction • Addressed early stage prediction • Utilized clinical features (laboratory or demographic data) • Published in English within (2021-2025)\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.2 Exclusion Criteria\"\n },\n {\n \"type\": \"para\",\n \"text\": \"• Focused only on medical imaging without predictors • Addressed other diseases (cardiovascular, diabetes, etc) • Focused on late-stage or general CKD detection • Lacked clinical predictor analysis\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Datasets Overview\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Datasets used in early chronic kidney disease (CKD) research range from relatively small hospital-based clinical records to very large national repositories such as the UK Biobank, which contains health information from more than half\"\n },\n {\n \"type\": \"para\",\n \"text\": \"a million participants [ 11 ] . Among the datasets commonly used for benchmarking machine learning models, the UCI Machine Learning Repository remains one of the most widely adopted sources for algorithm evaluation and comparison [ 17 , 13 , 16 ] . At the same time, recent studies increasingly emphasize community-based screening datasets, particularly in South Asian regions, to support early detection strategies in low-resource settings where access to advanced laboratory testing is limited [ 17 , 27 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Real-world clinical datasets contain high-dimensional feature spaces, irregular time intervals between medical observations and substantial class imbalance, where CKD-positive cases represent a minority of the population [ 5 , 13 , 25 ] . Longitudinal Electronic Medical Record (EMR) data further highlight the dynamic nature of disease progression. For instance, evidence suggests that approximately 20.3% of diabetic patients showing early kidney impairment develop CKD within six months [ 5 ] . Most predictive modeling studies rely on datasets containing roughly 24 to 36 clinical attributes, which include demographic factors as well as biochemical markers such as serum creatinine and estimated glomerular filtration rate (eGFR) [ 13 , 33 , 34 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, widely used benchmark datasets may not fully represent the demographic diversity of global populations. As a result, several studies validate their models using independent cohorts collected from countries such as Bangladesh, Korea, China and the United Arab Emirates [ 17 , 21 , 35 , 10 ] . National health surveys like NHANES are also frequently used because they follow standardized data collection procedures and incorporate additional lifestyle variables, including dietary patterns and sleep habits, which can influence kidney health outcomes [ 33 , 34 ] . Table 1 summarizes the datasets used across reviewed studies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5 Preprocessing Techniques\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data preprocessing is a crucial step in preparing healthcare datasets for machine learning analysis, as raw clinical data often contain inconsistencies and missing values [ 10 , 34 ] . One of the most common challenges involves handling incomplete records, which may arise from random data loss or manual entry errors during clinical documentation [ 16 , 13 ] . To address this issue, researchers frequently apply imputation techniques such as Multivariate Imputation by Chained Equations (MICE), K-nearest neighbor (KNN) imputation or median replacement to reduce the influence of missing information while maintaining robustness against outliers [ 17 , 13 , 33 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Additionally to handle temporal variability, longitudinal medical records are frequently summarized within a specific observation window using statistical measures such as the mean, quartiles or standard deviation [ 5 ] . Feature scaling is also routinely performed to ensure consistent data ranges across variables, typically through Min–Max normalization or Z-score standardization [ 17 , 13 , 27 , 16 ] . Moreover, techniques for identifying and correcting outliers like interquartile range (IQR) are employed to enhance model stability and prediction accuracy [ 27 , 13 ] . Due to the common issue of class imbalance in CKD datasets, methods such as the Synthetic Minority Over-sampling Technique (SMOTE) or random oversampling are frequently employed to equilibrate the distribution between healthy and CKD-affected cases [ 13 , 25 , 11 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.6 Feature Selection Methods\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Feature selection is equally crucial in determining the most informative clinical predictors. These techniques are typically categorized into statistical filter methods, model-based wrapper approaches and regularization strategies [ 17 , 27 ] . Tests like Pearson correlation, ANOVA and the Mann–Whitney U test are frequently utilized to identify variables that significantly vary between CKD and non-CKD populations [ 17 , 13 , 11 ] . Wrapper-based methods like Recursive Feature Elimination with Cross-Validation (RFECV) and Sequential Feature Selection (SFS) have demonstrated the ability to discover smaller subsets of features that often outperform models trained on the complete set of variables [ 17 , 27 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4. Data Leakage Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The following section provides a structured evaluation of data leakage in early chronic kidney disease (CKD) prediction studies. To ensure consistent evaluation across studies, three types of data leakage are defined using the following criteria : Type 1—A feature is considered Direct Leakage (DL) if it is included in diagnostic criteria for defining CKD. Here, the criteria consist of estimated glomerular filtration rate, serum creatinine and albuminuria. Including these as predictive features may lead to circular reasoning or may even hamper the model’s performance, because these features directly impact the clinical understanding of CKD.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Type 2—Proxy Leakage (PL) occurs when a feature is not part of the diagnostic criteria but is statistically or clinically redundant with one. In this study, proxy leakage is identified using a hybrid criterion: (i) strong statistical correlation with diagnostic biomarkers ( | r | ≥ 0.85 |r|\\\\geq 0.85 ) or (ii) clinical or physiological dependency on the same underlying renal function pathway, even when pairwise correlations are moderate due to dataset-specific factors. The ( | r | ≥ 0.85 |r|\\\\geq 0.85 ) threshold is commonly adopted as a conservative indicator of strong association in clinical datasets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Type 3—Temporal leakage occurs when predictors gathered after the outcome event are utilized in training the model. Examples include medications after diagnosis, dialysis treatments or subsequent interventions. Since these variables are unavailable at prediction time, their inclusion leads to unrealistic model performance, a phenomenon widely recognized in machine-learning evaluation pipelines [ 18 ] . Table 2 presents an overview of the definitions and detection criteria for every leakage category utilized in this analysis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To quantify the extent of leakage in each study, a unified leakage score is defined as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where L i L_{i} represents the total leakage score for study i i , while D L i DL_{i} , P L i PL_{i} and T L i TL_{i} denote direct, proxy and temporal leakage components, respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The individual leakage components are defined as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A correlation of | r | ≥ 0.85 |r|\\\\geq 0.85 corresponds to a Variance Inflation Factor exceeding 6.7 6.7 ( V I F = 1 / ( 1 − r 2 ) VIF=1/(1-r^{2}) ), indicating near-collinear redundancy with diagnostic biomarkers and warranting full proxy leakage assignment ( P L = 2 PL=2 ) [ 12 , 23 ] . The intermediate range ( 0.60 ≤ | r | < 0.85 0.60\\\\leq|r|<0.85 , V I F = 1.6 VIF=1.6 – 6.7 6.7 ) captures moderate statistical dependence that may not individually constitute leakage but creates meaningful predictive overlap, particularly when multiple such variables are included simultaneously [ 31 ] ( P L = 1 PL=1 ), consistent with evidence that partial information leakage can still inflate model performance [ 18 ] . Variables with | r | < 0.60 |r|<0.60 are treated as statistically independent of diagnostic criteria ( P L = 0 PL=0 ), following conventional interpretations of correlation strength where values near r = 0.50 r=0.50 represent large effects in behavioral and clinical data [ 7 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Based on the relative severity of each leakage type, weights are assigned as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, the final leakage scoring function is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Leakage Classification Rule\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To facilitate interpretation, studies are categorized according to their leakage score:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 presents a conceptual comparison between conventional and leakage-aware CKD prediction workflows. The figure highlights the importance of appropriate feature handling and leakage control to ensure reliable model evaluation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Effect of Data Leakage on Model Performance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analysis reveals a systematic relationship between information leakage and excessive model performance in early CKD prediction studies. By applying the proposed three-part taxonomy which are Direct Leakage (DL), Proxy Leakage (PL) and Temporal Leakage (TL), a consistent pattern emerges across the reviewed literature. Studies that incorporate features directly embedded in diagnostic criteria, such as eGFR, serum creatinine or albuminuria, almost always achieve disproportionately high reported accuracies. This is not surprising, as these variables are not merely predictive signals but are, in fact, definitional components of the outcome itself. Consequently, models in these settings are not learning latent disease patterns but are effectively reconstructing the diagnostic rule. This phenomenon is reflected across a large proportion of studies [ 8 , 9 , 28 , 25 , 16 , 13 , 35 ] which receive the maximum leakage score due to the simultaneous presence of direct and proxy leakage. Importantly, even in the absence of temporal leakage, the coexistence of direct and proxy leakage is sufficient to produce near-ceiling performance metrics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 demonstrates a continuously increasing trend in reported accuracy as the severity of data leakage increases across models. In the clean configuration (no leakage), models achieved accuracy ranging from 92.50% (AdaBoost) to 97.50% (LightGBM). Under the proxy leakage condition, where features correlated with diagnostic criteria, all models exhibited measurable performance gains, with AdaBoost showing the most dramatic improvement of +5.00% in accuracy and +6.29% in F1-score. Under the direct leakage condition, where definitional components of CKD were included as predictors, all models achieved near-perfect or perfect accuracy (98.75–100.00%) and AUC (100.00%), regardless of their baseline performance. Here, Figure 3 summarizes the overall workflow of the proposed leakage scoring framework.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Proxy Correlation Undermining Predictive Learning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A detailed examination of proxy leakage shows how machine learning models can generate overly optimistic performance in chronic kidney disease (CKD) prediction tasks. Proxy leakage occurs when non-diagnostic variables are highly correlated ( | r | ≥ 0.85 |r|\\\\geq 0.85 ) with established diagnostic biomarkers or clinically redundant with diagnostic biomarkers which enables the model to implicitly reconstruct the target label without learning independent predictive patterns. In clinical datasets, variables such as Blood Urea Nitrogen (BUN), Cystatin C and hemoglobin often demonstrate significant correlations with primary indicators like estimated glomerular filtration rate (eGFR) and serum creatinine. Although these variables represent different clinical measurements, their statistical connections allow models to avoid authentic predictive learning by leveraging overlapping data. This phenomenon is demonstrated in studies such as [ 27 ] and [ 21 ] , where moderate leakage correlates with high accuracy, suggesting that even limited proxy leakage can greatly inflate reported performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Importantly, proxy leakage in clinical datasets is not always reflected through high pairwise correlation alone. In several cases (UCI-based studies), individual correlations with diagnostic biomarkers remain moderate ( | r | ≈ 0.5 |r|\\\\approx 0.5 – 0.6 0.6 ), yet multiple physiologically linked variables (blood urea nitrogen, hemoglobin, packed cell volume) are simultaneously included. These variables collectively encode the same underlying renal dysfunction, forming a multivariate proxy structure. As a result, PL=2 is assigned in such cases to reflect cumulative redundancy, even in the absence of a single dominant correlation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Due to limited public availability of all datasets used in the reviewed studies (Table 4 ), correlation analysis was conducted on representative datasets (UCI CKD, Enam Medical, NHANES and Abu Dhabi cohorts). For studies utilizing identical or derived datasets, proxy leakage labels were assigned based on these empirical correlations combined with reported feature sets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Apart from performance inflation, proxy leakage affects the learning behavior of machine learning models. Tree-based ensemble methods, including XGBoost and Random Forest, are particularly sensitive due to their split-based optimization process. During training, these models select features that maximize information gain and highly correlated predictors are often interchangeable in this process. As a result, proxy variables can repeatedly appear in decision splits, reinforcing the same underlying signal across the ensemble [ 10 , 11 , 16 , 21 , 27 ] . This leads to models that appear to learn clinically meaningful relationships but are actually exploiting statistical redundancy. Consequently, variables such as hemoglobin and blood urea nitrogen may receive high importance even though they primarily reflect renal impairment already encoded by creatinine or eGFR [ 27 , 20 , 6 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Rankings of feature importance under proxy leakage might not represent independent risk factors but rather serve as substitute indicators of diagnostic criteria. Numerous studies recognize hemoglobin and packed cell volume as important predictors [ 6 , 20 , 27 , 28 ] , but these factors are physiologically related to kidney function and highly correlated with creatinine-based metrics, resulting in erroneous conclusions where models seem to indicate causal relationships but actually reflect disease status. This likewise diminishes external validity, since models developed on datasets with robust internal correlations frequently perform poorly on external cohorts having different correlation patterns [ 6 , 2 , 14 , 28 ] . This hybrid formulation ensures that proxy leakage is not underestimated in heterogeneous clinical datasets where statistical correlation alone may fail to capture underlying physiological dependencies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Statistical Evidence of Leakage-Induced Inflation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The aggregate performance comparison in Table 5 provides clear evidence of leakage-induced inflation. This highlights the importance of rigorous leakage auditing before evaluating model performance or reporting results in early CKD prediction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The statistical significance (p = 0.005) of Table 5 indicates that the observed difference is very unlikely to be explained solely by sampling variability. The consistent inflation of model performance due to leakage indicates a methodological bias instead of authentic predictive capability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 represents a gradual rise in reported accuracy as the severity of data leakage increases across multiple studies. Models classified under the high leakage category consistently report inflated performance, indicating potential overestimation of true predictive ability. In contrast, leakage-free studies exhibit relatively lower and more varied accuracy values, indicating more authentic model performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Besides statistical significance, the size of the observed difference indicates a considerable practical effect. An average inflation exceeding 15.2% implies that models utilizing predictors prone to leakage could greatly exaggerate their diagnostic ability in practical scenarios. Importantly, most of the studies where leakage was identified fall into the moderate leakage category, indicating that even small amounts of leakage can influence the reported performance of machine learning models, where inconsistent handling of leakage across studies undermines the reliability of cross-study performance comparisons and further complicates fair benchmarking of models.These findings collectively indicate that to have a fair comparison between the published performance benchmarks, we first need to audit and standardize how leakage is handled.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5. Cross-Study Feature Stability Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The growing use of machine learning models to predict Chronic Kidney Disease (CKD) has resulted in a fragmented and often inconsistent understanding of key factors of the disease. Top predictors of CKD differ across studies, making it difficult to detect characteristics that indicate generalizable risk signals. This inconsistency raises an essential question: Which predictors are actually reliable across different methods and geographies? which result from dataset-specific traits, methodological biases?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address this problem, a systematic cross-study feature stability analysis was performed on nineteen CKD prediction studies. Specifically, 28 standardized attributes (Table 6 ) were classified into groups. This detailed classification allows for a more refined understanding of feature importance and highlights the interdisciplinary nature of CKD risk modeling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each predictor was evaluated based on its frequency of occurrence across the 19 studies and a normalized stability score was computed. High-consistency predictors included patient age, blood pressure, glycemic control, hemoglobin/anemia, serum creatinine and albuminuria. These features cover multiple clinical areas, reinforcing their apparent robustness across diverse datasets and modeling techniques.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 5 presents the distribution of predictors across stability levels (high, moderate, low) within each clinical domain. The renal and clinical areas show the highest concentration of predictors overall and contain the majority of high and moderate stability features, reflecting their frequent inclusion in early CKD prediction models. In contrast, domains such as lifestyle, socioeconomic, dietary and inflammatory factors are composed almost entirely of low-stability predictors, indicating restricted usage across various studies. Numerous domains like anthropometric, metabolic and comorbidity factors include just one or two predictors with low stability representation. In general, the figure shows a significant disparity in feature utilization, with CKD prediction studies focusing on a narrow range of clinical and renal factors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Category-level aggregation further revealed a disproportionate dominance of renal biomarkers in the literature, with the highest cumulative reporting frequency. This overrepresentation continues even with the well- documented risk of data leakage linked with diagnostic variables such as serum creatinine and eGFR. Their frequent inclusion likely reflects their data availability and diagnostic role rather than true predictive value.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cross-dataset stability analysis of Table 7 highlights notable differences between predictors derived from studies using the UCI Chronic Kidney Disease Dataset and those evaluated on independent clinical datasets. Demographic and clinical variables such as patient age and blood pressure demonstrate the highest cross-dataset stability, indicating strong generalizability across diverse study populations and modeling approaches. Across multiple studies, Indicators like glycemic and hemoglobin maintain comparatively high stability, indicating their association with CKD prediction [ 17 , 21 ] . In general, the results suggest that only a limited set of predictors demonstrates strong cross-population stability, highlighting the necessity for wider dataset variety and standardized feature evaluation for upcoming studies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inter-study agreement in feature selection was evaluated using Fleiss’ kappa (Table 8 ). A κ \\\\kappa value of 0.224 was obtained, indicating fair agreement among the examined studies. This suggests that although a limited number of predictors (such as age, blood pressure and glycemic indicators) appear consistently across studies, the majority of features are inconsistent. The relatively low k value highlights the disjointed nature of feature selection and supports the observation that many predictors are specific to datasets rather than universally reliable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In Figure 6 , the frequency distribution of all 28 predictors across the 19 reviewed studies is shown, ranked by occurrence. A steep decline in frequency is observed beyond the sixth predictor, with the majority of features appearing in five or fewer studies. This distribution visually reinforces the skewed nature of feature selection in CKD prediction research, where a small number of predictors dominate while most are inconsistent.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pareto curve of Figure 7 demonstrates the combined contribution of predictors to total feature usage across different studies. A considerable share of overall usage is represented by high-ranking features as it can be seen from the sharp initial rise in the graph. Specifically, the initial group of predictors accounts for about 80% of total coverage, illustrating a significant concentration effect. This pattern highlights the dominance of a limited core feature set in CKD prediction models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analysis of Table 9 illustrates a distinct ranking of clinically reliable predictors across the five leakage-safe studies. Patient age and glycemic control emerge as the most stable and universally utilized features, indicating their robust and consistent predictive relevance. Evidence from broad inclusion and higher stability, further supports blood pressure and gender as fundamental predictor. In contrast, variables such as BMI and albuminuria show moderate consistency, suggesting situational relevance across datasets. Several features, including lifestyle factors and hematological markers, show low stability due to limited inclusion or data availability. Specifically, traditionally robust clinical predictors like serum creatinine and eGFR are often excluded to avoid data leakage.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 10 highlights that patient age, glycemic control and blood pressure are strong indicators, maintaining high performance even after leakage adjustment. Gender, albuminuria and hemoglobin demonstrate moderate to high stability, enhancing their clinical significance. Conversely, the decreasing stability of serum creatinine and hematological parameters under leakage-safe conditions suggests a potential reliance on dataset-specific signals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Only a limited number of predictors display both conceptual independence from diagnostic standards and methodological strength. Across all three evaluation dimensions, patient age, blood pressure and glycemic control consistently achieve the highest average stability, marking the minimum reliable foundation for CKD risk modeling. Serum creatinine and albuminuria, despite their stability, are directly associated with CKD diagnosis, raising concerns about diagnostic data-leakage. A fragmented research landscape is reflected in the dominance of low-consistency indicators. Lifestyle, dietary and socioeconomic factors remain largely unexplored, although they offer potential as leakage-free early indicators. Future research should focus on external validation of the stable core predictors and investigate neglected modifiable risk factors. Moreover, Future work should apply standardized methods for feature selection to enhance the robustness and clinical applicability of machine learning models.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6. Future Directions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several clear methodological improvements emerge from this review. Researchers should implement data leakage auditing protocols throughout model development and follow standardized reporting methods aligned with TRIPOD guidelines. Additionally, the field would benefit from benchmark datasets that clearly differentiate diagnostic biomarkers from predictive risk factors, enabling more reliable comparisons among models. Future work could also benefit from focusing on simple temporal features, such as eGFR decline rate or variability across multiple measurements, to more effectively reflect disease progression without relying on complex sequential models. Furthermore, more reliable detection of data leakage, particularly in resource-limited environments, can be obtained by examining non-renal risk factors like lifestyle choices, socioeconomic status.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7. Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This review is subject to several important constraints. One limitation is that correlation analysis covers only four representative datasets, which limits the ability to verify findings from studies that used inaccessible data. Geographical bias presents another concern, as eight of the nineteen reviewed studies use the CKD dataset from an Indian clinical center. Here, proxy leakage labels for the remaining studies rely on reported feature sets and clinical reasoning. Also, the review excludes relevant recent works as it is limited to studies published up to 2025. Moreover, temporal leakage could not be formally validated because of insufficient timestamp data provided in the examined studies. Additionally, most reviewed studies were evaluated on limited or geographically localized datasets, which may reduce real-world clinical generalizability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8. Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This systematic review analyzed methods in early CKD prediction using machine learning, with a primary focus on information leakage and cross-study predictor stability. One of the findings shows exaggerated predictive effectiveness due to the inclusion of variables that are closely aligned with CKD diagnostic criteria. This study shows that a considerable amount of reported accuracy likely originates from inherent clinical definitions instead of genuine predictive learning. This finding is demonstrated by applying a structured taxonomy of direct, proxy and temporal leakage. Conversely, studies that avoid leakage-prone features tend to report more moderate but clinically relevant performance, highlighting the necessity of careful feature selection. Furthermore, the cross-study analysis indicates a lack of consistency in the importance of low-leakage predictors, with only a few variables, such as age, blood Pressure and glycemic Control, appearing consistently across multiple datasets. This variability suggests that model behavior often reflects dataset-specific characteristics. Consequently, concerns about generalizability and reproducibility are justified. Although temporal dynamics were not explicitly analyzed, their potential significance in understanding disease progression is recognized as a key area for future research. A Pareto curve analysis further supports the findings as it shows that 80% of cumulative feature coverage is achieved with 14 variables, highlighting the diminishing marginal utility of adding extra predictors in low-leakage CKD models. Overall, the study highlights the necessity for stricter methodological standards and robust validation frameworks to ensure reliable and clinically significant CKD prediction models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Author Contributions M.H. conceived the study, designed the methodology, conducted the systematic review, performed analysis, and interpretation, developed the leakage scoring framework, prepared the figures and tables, and wrote the original manuscript draft. N.K. contributed to the study design, assisted with data interpretation, critically reviewed and revised the manuscript, and provided valuable intellectual input. F.S. contributed to the literature review, assisted with data validation and manuscript revision, and provided constructive feedback throughout the study. All authors read and approved the final manuscript.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Corresponding Author mashrul16hossain@gmail.com (Mashrul Hossain)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Funding There was no funding obtained to conduct this research.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Competing Interest The authors declare there were no competing interests.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ethics Statement This study is a systematic review of previously published studies using publicly available and anonymized datasets. No new human or animal data were collected. Accordingly, no ethical approval or informed consent was required. Data Availability No new datasets were generated or analyzed. Used dataset is available on UCI Machine Learning Repository.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consent for publication All authors have read and approved the final manuscript\"\n }\n]","isConversational":false}}},{"rowIdx":379,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10720"},"doi":{"kind":"string","value":"10.1145/3744256.3812595"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"WattCouncil: Context-Aware Household Energy Scenario Generation With Governed LLMs"},"authors":{"kind":"list like","value":["Mohannad Takrouri","Nicolas M. Cuadrado A.","Martin Takáč"],"string":"[\n \"Mohannad Takrouri\",\n \"Nicolas M. Cuadrado A.\",\n \"Martin Takáč\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10720v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1. Introduction"},{"type":"para","text":"Global energy consumption continues to grow alongside societal and economic development, intensifying efforts to improve energy efficiency and reduce carbon emissions to address climate change. At the same time, electricity systems are undergoing structural change driven by the widespread deployment of renewable energy sources (RES), whose inherent variability and intermittency introduce new forms of uncertainty for grid operation and analysis (Sinsel et al . , 2020 ; Zhao, 2025 ) . Traditional power grids were designed around centralized, dispatchable generation and relatively predictable consumption patterns. Modern energy systems, however, increasingly depend on a detailed understanding of how demand varies across time, context, and usage conditions, particularly as electrification and distributed resources reshape load dynamics. This evolving landscape has motivated energy research that prioritizes not only real-time operations but also the characterization and exploratory analysis of demand behavior as a foundation for planning, policy design, and comparative scenario assessment."},{"type":"para","text":"Although residential electricity consumption exhibits patterns that can be exploited by data-driven methods, many existing modeling approaches remain tied to limited historical data or rigid simulators that do not fully capture household-level variability. Empirical analyses have shown that electricity use in the residential sector is influenced by multiple determinants, including household demographics, dwelling characteristics, climate conditions, and appliance usage, and that these factors contribute to significant heterogeneity in consumption behavior both within and between households (Kostakis, 2020 ; Ali et al . , 2022 ; Guven and Kayalica, 2023 ; Lavin and Julienne, 2025 ) . For example, income, household size, and regional climate have been identified as major contributors to variation in total residential demand, while smart meter analyses reveal distinct daily load profiles associated with occupancy and dwelling characteristics (Rafiq et al . , 2023 ; Grandjean et al . , 2012 ; Jabbar et al . , 2021 ) . Similarly, research on residential load patterns has shown significant correlations among these factors, underscoring the need to incorporate them into accurate energy consumption models (Wei and Wang, 2021 ) . Traditional simulation methods and bottom-up models often rely on assumptions that are difficult to scale or tailor to diverse contexts, and purely data-driven models require high-quality, well-annotated datasets. Gaps, missing values, and limited geographic coverage can reduce model generalization and lead to the omission of important behavioral structures. Together, these limitations underscore the need for methods that can generate controlled, contextually informed synthetic consumption data to support exploratory analysis and scenario reasoning beyond the scope of available empirical sources."},{"type":"para","text":"Large Language Models (LLMs) have emerged as powerful tools for synthesizing data and knowledge across diverse domains (Tan et al . , 2024 ) , including recent exploratory applications in the energy sector. After the introduction of LLM-based agents into the research community (Park et al . , 2023 ) , new paradigms have been proposed for leveraging the implicit knowledge encoded in large models through structured interaction and coordination. In this work, LLMs are not used for conventional forecasting or statistical modeling of demand; rather, they are leveraged as controlled generative agents that produce scenario-aware synthetic representations of household energy consumption under explicitly defined cultural, temporal, and contextual assumptions. Inspired by (Karpathy, 2025 ) , we employ a multi-agent LLM Council in which distinct models operate in complementary roles under explicitly defined constraints. This improves consistency and contextual coverage by relying on an ensemble of models rather than a single model’s capabilities. Specifically, we aim to integrate cultural, seasonal, and behavioral contexts into household energy consumption patterns. We acknowledge that LLM-generated data faces inherent challenges in reproducing physically grounded dynamics. We therefore position WattCouncil as a tool for generating controlled synthetic scenarios for analysis and benchmarking, and as a complement to physical simulators and to demand forecasting or optimization. Our contributions consist of: (i) WattCouncil , a multi-agent LLM-based framework for governed synthetic energy data generation; (ii) a staged pipeline with explicit auditing and bounded regeneration; (iii) an evaluation protocol based on similarity to real smart meter data; (iv) ablation studies on the components of our data generation pipeline; (v) open-access code of our pipeline with clear instructions on how to run it, hoping that it could be a contribution to future researchers who want to explore the area of LLM-based data synthesis."},{"type":"para","text":"The remainder of the paper is structured as follows. Section 2 reviews related work in household energy modeling, synthetic data generation, and the use of large language models in structured simulation contexts. Section 3 introduces the WattCouncil framework, describing the staged generation pipeline, constraint formulation, and governance mechanisms. Section 4 presents a quantitative and qualitative evaluation of the generated consumption profiles, including comparisons with a real-world smart meter dataset, while Section 5 analyzes the impact of key design choices through targeted ablation studies. Section 6 discusses current limitations of the proposed approach, and Section 7 concludes the paper with a summary of findings and directions for future work."},{"type":"heading","level":1,"text":"2. Related Work"},{"type":"para","text":"Traditionally, researchers enabled access to relevant data by conducting in-place studies for a defined period and gathering as much data as possible from real-world settings using sensors. Despite sustained efforts to collect and release residential electricity datasets, the resulting landscape remains highly fragmented across geography, duration, resolution, and household coverage. As pointed out in (Cruz et al . , 2024 ) , which found that in general, datasets varied considerably, especially in countries such as the USA, the UK, and India, and exhibited several sources of bias. Widely used datasets such as REDD (Kolter and Johnson, 2011 ) , UK-DALE (Kelly and Knottenbelt, 2015 ) , REFIT (Murray et al . , 2017 ) , AMPds (Makonin et al . , 2013 ) , BLUED (Anderson et al . , 2012 ) , iAWE (Batra et al . , 2013 ) , and ECO (Beckel et al . , 2014 ) provide high-resolution appliance-level measurements but are typically limited to a small number of households and short observation periods, thereby limiting their representativeness. On the other hand, larger-scale datasets such as Dataport (Pecan Street) (Parson et al . , 2015 ) , RECS (U.S. Energy Information Administration (2023), EIA ) , and CLNRPD (Northern Powergrid, 2015 ) offer broader population coverage but often lack fine-grained temporal resolution, appliance-level detail, or continuous monitoring across contexts."},{"type":"para","text":"Household energy research has documented that residential electricity consumption is influenced by a combination of demographic, behavioral, cultural, and socio-economic factors. Empirical studies identify household size, income, appliance ownership, and occupancy patterns as significant drivers of energy use variability (Jones and Lomas, 2015 ) . For instance, larger household size has consistently been associated with higher electricity use, while income and dwelling-related characteristics (such as the number of bedrooms and heating systems) account for substantial differences in consumption across households (Jones and Lomas, 2015 ) . Other research has highlighted that socio-economic and regional characteristics result in heterogeneous consumption patterns that vary between geographic contexts and policy environments. This suggests that household electricity demand is shaped by interacting social and structural determinants rather than physical building attributes alone (Kostakis, 2020 ; Najeeb et al . , 2024 ) . The study in (Mora et al . , 2018 ) highlights the significant impact of occupant behavior, socio-economic characteristics, and preferences (e.g., set-point temperatures and ventilation durations) on residential energy consumption. It emphasizes that variations in energy demand are influenced more by contextual factors, such as family composition and dwelling usage patterns, than by physical attributes, such as house size or floor area."},{"type":"para","text":"In (Du and Lu, 2011 ) , the authors present an algorithm that schedules household appliances based on consumption forecasts and user comfort settings, aiming to optimize objectives like minimizing costs or maximizing comfort. This underscores the importance of accurately modeling intra-family dynamics to simulate appliance usage patterns effectively. Building upon this, (Nicholls and Strengers, 2015 ) examines how family interactions, such as shared activities and the presence of children, influence peak electricity demand times. The study highlights the need to incorporate family dynamics into energy consumption models to accurately predict and manage peak demand periods. Several studies have leveraged weather simulation frameworks to assess HVAC performance under various climatic scenarios. For example, (Xu et al . , 2024 ) analyzed the impact of urban heat islands on residential cooling demand, showing how microclimatic factors intensify energy consumption in densely populated regions. Despite this substantial body of empirical work, many modeling approaches traditionally used in energy systems have focused primarily on physical parameters or historical load profiles, with limited integration of detailed socio-demographic and behavioral insights into analytical or generative frameworks (Crosbie, 2006 ) . This gap motivates the development of methods that can represent contextually grounded consumption variability when generating or analyzing residential energy data."},{"type":"para","text":"The usage of LLMs is transforming how domain-specific tasks are approached, particularly in generating human-like, context-aware outputs. In (Buster et al . , 2024 ) , researchers from the National Renewable Energy Laboratory (NREL) introduced a framework combining LLMs with decision-tree logic to guarantee compliance with zoning policies for renewable energy siting. The reported accuracy suggests that such approaches can be useful in downstream modeling. In (Almashor et al . , 2024 ) , the authors explored whether LLM-based agents can synthesize household energy consumption, building upon the environment provided by (Park et al . , 2023 ) . Although they obtained some reasonable load patterns, they were limited by the capabilities of the simulation they used. Their approach focused on parsing keywords in agent conversations, associating them with energy consumption, and then summing the household load. Along the same line of work, (Takrouri et al . , 2025 ) introduced a multi-stage methodology to synthesize load profiles from different LLMs using a pipeline that considers family structure definition, weather identification, weather synthesis, and load pattern generation, all conditioned on a given geography and time period. Building upon that work, (Chetty et al . , 2025 ) presents a methodology that enhances the process by integrating memory capacities and a rule-based validation before constructing the final load profile. These studies highlight the potential of LLMs to streamline data access and policy analysis in household energy modeling."},{"type":"para","text":"However, significant gaps persist in integrating these components into a cohesive framework. Current approaches often rely on limited assumptions about family structures and behaviors, neglecting the nuanced and culturally specific dynamics necessary for accurate simulations. In particular, existing work rarely separates macro-level scenario specification (e.g., season/climate regime, calendar rules, household composition, and social context) from micro-level behavioral realization at the hourly scale (occupancy/activity and appliance-use traces). This limits explainability and controllability."},{"type":"heading","level":1,"text":"3. Methodology"},{"type":"heading","level":2,"text":"3.1. WattCouncil: An LLM-Based Framework For Governance"},{"type":"para","text":"The research community has increasingly supported the idea that coordinated outputs from many LLMs could outperform a single model. Works such as (Hu et al . , 2025 ) showed that this approach, known in the literature as model ensembling, outperformed individual models in reasoning tasks. More sophisticated works, such as (Gao et al . , 2025 ) , proposed a knowledge-transfer method to optimize model ensemble performance in sequential decision-making tasks. Finally, a recently published resource in (Karpathy, 2025 ) introduced the idea of LLM councils, which consist of several models (with different roles) that reach consensus on a given task."},{"type":"para","text":"Inspired by the aforementioned works, we introduce WattCouncil , a governed LLM Council that employs role-specialized agents to generate structured synthetic household energy data under explicit constraints. Rather than relying on a single language model to produce end-to-end outputs, the council decomposes generation into a sequence of staged decisions, each subject to a data schema validation, auditing, and explicit acceptance criteria. In contrast to autonomous conversational use, the LLMs operate as constrained generators inside a pipeline whose control flow, stopping conditions, and artifact interfaces are fully specified. Figure 1 summarizes the governance logic and decision flow enforced at every level."},{"type":"para","text":"The council is implemented as a three-stage pipeline with explicit artifact interfaces as illustrated in Figure 2 . The council comprises three components: (i) household structure generation, with assignment of work and occupancy regimes; (ii) definition of seasonal weather ranges, with hourly weather instantiation; (iii) household energy consumption modeling. Each stage consumes only outputs that are validated in the operational sense, in other words, outputs that generated a schema-compliant JSON that has been audited and accepted by the controller/approver. Individual stages can be selectively enabled, disabled, or replaced by external data sources without affecting the integrity of the remaining pipeline."},{"type":"para","text":"The council architecture assigns fixed, non-overlapping roles to multiple LLMs, enforcing strict task separation. A primary generator proposes candidate structured outputs at each stage of the pipeline, which are evaluated by designated role-based auditors depending on the stage context. For stages involving social structure, behavior, or narrative coherence (Stages 1 and 3), a cultural auditor assesses alignment with country-specific social norms, and a physical auditor verifies numerical consistency and adherence to physical or environmental constraints across all stages. For Stage 2, only the physical auditor is applied, as cultural evaluation is not meaningful in contexts such as seasonal weather-range definition and hourly weather instantiation. Auditors return structured JSON reports with a severity score, following a shared scale ( LOW / MEDIUM / HIGH ) that indicates the level of compliance with the conditions for each stage. We maintain a fixed generation logic to achieve reproducibility by pinning model versions, prompts, and schemas, and logging all intermediate artifacts, while fixing inference settings (e.g., temperature and sampling parameters)."},{"type":"para","text":"Finally, a Council Controller acts as the sole decision-making authority in the council. Based on the generator output and the relevant audit results, it selects one of three actions: accepting the results of the pipeline execution ( ACCEPT ), triggering a targeted correction phase handled by an editor followed by independent verification by an approver ( REGENERATE PARTIAL ), or discarding the execution and restart the process from scratch ( REGENERATE FULL ). The framework supports a scoped rule memory that records recurring failure patterns (e.g., repeated schema violations or constraint failures) and the corresponding corrective instructions. When enabled, this memory is treated as a versioned artifact: any rule additions are logged so that subsequent generations can be replayed under the same rule set."},{"type":"para","text":"The WattCouncil framework is intentionally model-agnostic, assigning specific language models to each council role based on functional suitability (e.g., cultural reasoning, numerical consistency, or structured editing), while keeping these assignments fixed across all experiments. The framework does not benchmark or compare individual language models. Instead, evaluation is performed at the system level, where behavior emerges from role specialization, constraint enforcement, and deterministic governance rather than from any single model’s capabilities. Table 1 summarizes the models used in this study."},{"type":"para","text":"In LLMs, the temperature hyperparameter ( τ \\tau ) controls sampling randomness. Intuitively, temperature determines how willing the model is to explore alternative plausible outputs rather than repeatedly selecting the most likely continuation: lower values make the model more conservative and predictable, while higher values allow greater variation. Formally, given a vector of logits 𝐳 = ( z 1 , … , z K ) \\mathbf{z}=(z_{1},\\dots,z_{K}) over a vocabulary of size K K , the probability of sampling token i i is computed using a temperature-scaled softmax,"},{"type":"para","text":"where lower τ \\tau concentrates probability on high-scoring tokens and higher τ \\tau increases output diversity. In WattCouncil , temperature controls local generation variability, while global consistency is enforced through auditing, bounded regeneration, and explicit acceptance criteria."},{"type":"para","text":"The temperature settings assigned to each council role reflect their functional responsibilities within the generation and governance pipeline. In particular, the Generator is configured with a higher temperature than the other roles to encourage controlled creativity during the initial proposal of structured artifacts. At this stage, the objective is not strict verification but the exploration of plausible household structures, behaviors, and contextual configurations that are consistent with high-level constraints. A moderately elevated temperature increases output diversity, allowing the Generator to surface alternative yet reasonable scenarios that might not emerge under near-deterministic sampling. In contrast, downstream roles such as the Cultural Auditor, Physical Auditor, Editor, and Approver are assigned lower temperatures to prioritize consistency, precision, and adherence to constraints during evaluation and correction. This asymmetric temperature allocation reflects a deliberate separation of concerns: creative variability is permitted at the proposal stage, while rigor and stability are enforced during auditing and decision-making. Within the WattCouncil framework, this design allows limited stochastic exploration without compromising overall governance, reproducibility, or physical and contextual validity."},{"type":"heading","level":2,"text":"3.2. Integration of Typical Meteorological Weather (TMY)"},{"type":"para","text":"As stated before, the stages of WattCouncil ’s pipeline ( Figure 2 ) are explicitly designed to be modular and can be replaced by externally sourced data without affecting the rest of the framework. This flexibility is particularly important for weather information, which is inherently complex and difficult to generate reliably from scratch due to its strong physical constraints, temporal structure, and geographic dependence. To address this, we investigate an alternative setup in which LLM-generated weather is replaced with a simple, safe baseline based on Typical Meteorological Year (TMY) data."},{"type":"para","text":"A Typical Meteorological Year is a synthesized weather dataset that represents statistically typical meteorological conditions for a specific geographic location over the course of a single year. Rather than reflecting a specific calendar year’s weather, a TMY is constructed by selecting the most representative months from a long historical record so that key variables such as temperature, solar radiation, humidity, and other meteorological indicators closely match long-term averages. TMY datasets are designed to capture typical patterns rather than extreme events (heat waves, cold snaps, or compound weather extremes); they are widely used in building energy simulation, renewable resource assessment, and engineering analyses as a representative baseline."},{"type":"para","text":"Although TMY data do not perfectly replicate year-to-year weather variability, they provide physically grounded, historically derived weather profiles that work as a reference for correctness. We implement this substitution using the Python library pvlib (Anderson et al . , 2023 ) , which enables the generation of a full year of hourly weather variables for a given geographic location specified by latitude and longitude. This replacement decouples behavioral and energy modeling from weather synthesis, providing a physically grounded baseline for comparison. In the present study, weather conditioning focuses on representative seasonal patterns rather than explicit extreme-event stress testing."},{"type":"heading","level":1,"text":"4. Experimental Results"},{"type":"para","text":"We evaluated the WattCouncil framework as a scenario-oriented synthetic data generator , assessing whether the generated household energy profiles reproduce realistic temporal structure and aggregate behavior when compared against real-world reference data, and whether key pipeline components, with a focus on weather data, can be substituted without substantially degrading downstream output."},{"type":"heading","level":2,"text":"4.1. Reference Real-World Dataset"},{"type":"para","text":"The CER Electricity Data (Revised March 2012) (Song, 2024 ) is a rich, open dataset derived from the Commission for Energy Regulation’s Smart Metering Project, a large-scale trial of electricity smart meters in Ireland. It comprises half-hourly electricity consumption measurements collected over more than 500 days (spanning from July 14, 2009, to December 31, 2010), capturing detailed load profiles for 4232 residential customers and 529 small-to-medium enterprises (which we do not consider in our study). In addition to the raw consumption time series, the trial includes extensive pre- and post-trial survey data on socio-demographic characteristics, household properties, and energy-use behaviors, enabling analysis that links load profiles to occupant and dwelling features. We selected this dataset as a reference because it reflects real consumer behavior under the determinants we introduced in our pipeline, enabling a direct comparison between synthetic and real data."},{"type":"heading","level":3,"text":"4.1.1. Exploratory Data Analysis (EDA) of the CER Dataset."},{"type":"para","text":"Inspired by prior work (Guo et al . , 2022 ) that uses the CER dataset and combines the smart meter time series with socio-demographic metadata to study household energy behavior, we follow the same principle of jointly using consumption traces and survey-derived contextual attributes. We adopt the perspective that load profiles alone are insufficient to characterize household behavior without accounting for family composition, occupancy patterns, and socio-economic factors, all of which are explicitly modeled in our framework. While that study used K-means (an unsupervised clustering method) to identify latent customer groups based on load shape similarity, we intentionally opted for a different approach that explicitly groups clients based on responses to selected questions in the surveys."},{"type":"para","text":"Following the design and operational principles of WattCouncil , the cluster membership carries limited semantic meaning for conditioned generation; in other words, including an unlabeled cluster in the context does not provide interpretable guidance for scenario synthesis. Instead, we define explicit household groups based on socio-economic and survey attributes available in the CER dataset, selecting variables that can be directly used as conditioning inputs within our pipeline, as summarized in Table 2 . From these groups, we identify representative subsets of households and extract their aggregated characteristics to ground the generation process. This design choice ensures that insights derived during the EDA stage explicitly constrain the generative mechanism, enabling systematic and interpretable comparisons between real consumption data and LLM-generated profiles produced by our pipeline."},{"type":"heading","level":2,"text":"4.2. Comparison Between Synthetic and Real-World Energy Consumption Data"},{"type":"para","text":"We evaluate the realism of the synthetic household energy consumption profiles generated by WattCouncil by comparing them against the CER dataset. The objective is to assess whether the proposed pipeline reproduces residential consumption patterns observed in real data across daily and seasonal time scales."},{"type":"para","text":"Based on the selected residential groups reported in Table 2 , we condition the generation process on these 5 representative groups. The pipeline generates multiple synthetic households that match the groups’ characteristics, and daily consumption profiles are produced under multiple weather conditions across all meteorological seasons. These synthetic profiles are then compared against the corresponding subset of real households from the CER dataset to assess behavioral and temporal consistency."},{"type":"heading","level":3,"text":"4.2.1. Seasonal Profile Construction."},{"type":"para","text":"For the synthetic data, we aggregated energy consumption across all family-weather combinations within each season. We compute hourly means, standard deviations, and sample counts, yielding a representative 24-hour profile for each season. For real data, we define seasons by calendar months in the Northern Hemisphere: winter (December–February), spring (March–May), summer (June–August), and autumn (September–November). We grouped hourly consumption values by season and hour of day across all selected customers and computed the corresponding summary statistics."},{"type":"heading","level":3,"text":"4.2.2. Visualization and Similarity Metrics."},{"type":"para","text":"We visualize seasonal comparisons using four horizontally arranged subplots as shown in Figure 3 corresponding to winter, spring, summer, and autumn. Each subplot displays the mean hourly consumption profiles for synthetic and real data, accompanied by shaded confidence intervals. For this WattCouncil dataset, we generated electricity consumption profiles for five household groups by combining multiple controlled variations. Specifically, we selected 5 representative family groups and instantiated 10 employment and occupancy variants per group. Each configuration was evaluated across 4 seasons and 5 independent weather realizations per season, resulting in a total of 5 × 10 × 4 × 5 = 1000 5\\times 10\\times 4\\times 5=1000 synthetic household profiles. We note that the apparent smoothness difference in Figure 3 may arise from the original axis scaling and the differences in aggregation: the real profiles are averaged over a larger number of households, whereas the synthetic profiles are derived from a smaller set of generated household configurations."},{"type":"para","text":"In addition to visual inspection, Table 3 reports the quantitative error and similarity metrics for each season. We use Pearson’s correlation to assess temporal shape similarity and peak timing alignment, whereas mean absolute error (MAE), root mean squared error (RMSE), and mean absolute percentage error (MAPE) quantify magnitude mismatch. This combination allowed separating whether the synthetic profiles reproduce realistic daily temporal structure, and whether they recover realistic absolute consumption levels. Results show pattern–level similarities but clear magnitude discrepancies, indicating that matching the consumption dimension depends on aspects we might not have considered in our pipeline. This distinction is important: realistic timing patterns can emerge from context-aware occupancy and activity modeling even when the final consumption scale remains imperfect."},{"type":"heading","level":1,"text":"5. Ablation Studies on the Pipeline"},{"type":"para","text":"The ablation is evaluated at two levels: Section 5.1 compares LLM- and TMY-derived weather profiles (seasonal statistics and daily patterns), and Section 5.2 quantifies changes in downstream demand patterns deriving from changing the source of weather data."},{"type":"heading","level":2,"text":"5.1. Weather Profile Comparison"},{"type":"para","text":"We first assess whether the proposed pipeline produces physically and temporally plausible weather inputs by comparing LLM-generated hourly weather profiles with externally sourced TMY data for Ireland, which matches the geographic scope and reference year (2009) of the real-world dataset used in our study. This analysis isolates the behavior of the weather-generation component (Stage 2) from household structure and behavioral modeling."},{"type":"para","text":"Figure 4 visually compares the average daily profiles of environmental variables generated with the LLM against the TMY reference for Dublin, Ireland, which we assume is uniform across the territory (the dataset does not provide household locations to preserve privacy). We used 5 LLM-generated profiles and 60 TMY profiles (which are faster to sample) to build 95% confidence intervals using the t-distribution. Additionally, we computed Pearson correlation coefficients ( r r ) between LLM-generated and TMY weather profiles across all seasons and meteorological variables. The results suggest that LLM-generated data align with the TMY reference, with correlation increasing in spring and summer and decreasing in autumn and winter."},{"type":"para","text":"We summarize the correlation results in Table 4 . These results indicate that directly generating weather profiles with general-purpose LLMs can introduce avoidable inaccuracies, primarily because of the physical complexity and high temporal precision required for meteorological modeling. In this setting, LLMs are less well-suited than physics-informed or data-driven weather models, rather than being inherently incapable."},{"type":"para","text":"Importantly, this limitation does not undermine the WattCouncil framework; instead, it motivates the use of reliable external weather sources, such as TMY data, to condition consumption and generation. Moreover, specialized climate models—such as NVIDIA’s Earth-2 platform (Earth2Studio Contributors, 2024 ) —represent promising avenues for integrating richer, physically grounded weather context into the pipeline without compromising consistency or realism."},{"type":"heading","level":2,"text":"5.2. Impact of Weather Sourcing on Energy Consumption"},{"type":"para","text":"To isolate the effect of weather sourcing, all non-meteorological inputs are held constant across experiments. Table 5 summarizes the household configuration used in this study, including composition, occupancy regime, dwelling type, and appliance set."},{"type":"para","text":"Electricity consumption profiles are generated under two settings: LLM-generated weather inputs and externally sourced TMY weather. Figure 5 compares weekday and weekend demand profiles across all four seasons for five realizations of the same household configuration. The results exhibit closely aligned load shapes and peak timing, indicating that downstream consumption dynamics remain stable under changes in weather sourcing. This observation is supported by high Pearson correlation coefficients across seasons. Differences between the two settings primarily appear in the width of uncertainty bands rather than systematic shifts in mean demand."},{"type":"para","text":"Overall, these results suggest that the proposed pipeline is robust to the choice of weather source at the household energy-demand level. LLM-generated weather preserves realistic demand structure and variability, while enabling scenario diversity beyond what is attainable from a single deterministic reference year."},{"type":"para","text":"One possible explanation for the muted summer demand is the combined effect of governance constraints, where cooling and heating are repeatedly disabled during regeneration, and behavioral assumptions associated with the fixed household configuration, leading to intentionally conservative summer usage patterns."},{"type":"para","text":"Notably, across repeated runs, we observe that summer scenarios consistently trigger additional regeneration and editing steps within the council, resulting in nearly twice the generation time compared to other seasons, further reflecting the stricter constraint enforcement under summer conditions."},{"type":"heading","level":2,"text":"5.3. End-to-End Pipeline Execution Example"},{"type":"para","text":"To illustrate the behavior of the proposed framework in practice, we report an end-to-end execution trace for a single synthetic household processed through all pipeline stages. The household is evaluated under a single weather condition across the four seasons, with each season instantiated under both weekday and weekend assumptions. This results in 8 independent Stage 3 consumption-generation runs for the same household. During execution, the framework applies the LLM Council governance described in Figure 1 across all pipeline stages. When violations are detected at any stage (e.g., implausible household routines, inconsistent weather patterns, or HVAC heating active during summer conditions), the controller enforces regeneration with explicit, stage-specific corrective guidance. Because all detected violations triggered full regeneration rather than constrained editing, neither the Editor nor the Approver roles were activated in this execution. In the illustrated example, the first summer weekday attempt is rejected due to physically implausible heating usage and is subsequently regenerated and accepted after correction. This demonstrates how the framework actively enforces domain constraints rather than passively accepting generative outputs."},{"type":"heading","level":3,"text":"5.3.1. Pipeline runtime analysis."},{"type":"para","text":"Table 6 summarizes the execution statistics for this illustrative run. Although only a single household is processed, the example reflects the governance overhead incurred during realistic operation: generating one household across four seasons and weekday/weekend variants required 52 model calls and 556.8 seconds of execution time, spanning generation, auditing, and decision-making steps. Most of this overhead is concentrated in the Generator and Physical Auditor roles, which dominate the synthesis and validation of structured hourly consumption profiles under explicit constraints, while cultural auditing and controller decisions contribute comparatively little. We do not report monetary costs, as they depend on the specific models, providers, and pricing schemes used at runtime, which may change over time; instead, we report token usage to enable reproducible cost estimation across different deployment settings. Representative execution traces for Stages 1–3 are shown next as annotated pipeline logs."},{"type":"heading","level":1,"text":"6. Limitations"},{"type":"para","text":"While the proposed framework aligns fairly with real-world residential electricity consumption patterns, several limitations should be noted."},{"type":"para","text":"The empirical evaluation is limited to Irish households, given that the CER dataset was the only comprehensive study to provide sufficient socio-demographic metadata to create a systematic comparison. Rigorous validation against real data requires comparable datasets, constraining cross-country evaluation in this study."},{"type":"para","text":"Comparisons against real data rely on aggregated seasonal and hourly statistics, which smooth household-level variability. Accordingly, the reported similarity metrics reflect population-level behavioral alignment rather than exact household-level replication."},{"type":"para","text":"The weather ablation relies on a finite number of discrete realizations for both LLM-generated and TMY-derived inputs, which does not capture the full range of meteorological variability or extreme events."},{"type":"para","text":"Governance mechanisms introduce additional computational overhead, reflecting a deliberate trade-off between controllability and generation speed."},{"type":"para","text":"Our framework reduced a larger set of possible attributes available in the CER dataset metadata to three, aiming to keep our budget and computation tractable. The selection could introduce bias in how WattCouncil generates load profiles."},{"type":"para","text":"We assumed uniform weather in Ireland based on the low variance of weather stations, a similar approach to (Guo et al . , 2022 ) ; this might be imprecise. Also, weather conditioning focuses on representative seasonal patterns rather than explicit extreme-event stress testing."},{"type":"heading","level":1,"text":"7. Conclusion"},{"type":"para","text":"This paper introduced WattCouncil , a governed, multi-agent LLM framework for generating structured synthetic household electricity consumption data under explicit cultural, temporal, and physical constraints. The proposed approach decomposes generation into a staged pipeline governed by role-specialized agents, explicit control logic, and governance via interfaces and control flows, with each stage producing auditable artifacts that meet explicit criteria. Evaluation against the CER smart meter dataset shows that the generated profiles reproduce the daily and seasonal demand structures fairly well and exhibit strong temporal alignment with real data across demographic groups. Ablation results further indicate that downstream consumption dynamics are robust to the choice of weather sourcing, with differences primarily affecting uncertainty rather than mean demand. While contextual and behavioral factors are central to reproducing temporal demand structure, the remaining magnitude gap indicates that physically grounded variables remain essential for realistic absolute consumption. A natural next step is therefore to incorporate stronger physical determinants, including dwelling-envelope properties, HVAC efficiency, appliance power ratings, and simplified thermal-response constraints. This would allow future versions of WattCouncil to couple contextual plausibility with improved energy-scale realism. Future work could also extend the framework by incorporating a more diverse set of socio-economic conditioning attributes, evaluating additional regions, integrating physically informed climate models, and exploring tighter coupling with downstream optimization and planning tasks. The integration of explicit extreme-weather scenarios is also an important future direction, particularly in regions where heating and cooling demand are highly sensitive to rare meteorological events."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Global energy consumption continues to grow alongside societal and economic development, intensifying efforts to improve energy efficiency and reduce carbon emissions to address climate change. At the same time, electricity systems are undergoing structural change driven by the widespread deployment of renewable energy sources (RES), whose inherent variability and intermittency introduce new forms of uncertainty for grid operation and analysis (Sinsel et al . , 2020 ; Zhao, 2025 ) . Traditional power grids were designed around centralized, dispatchable generation and relatively predictable consumption patterns. Modern energy systems, however, increasingly depend on a detailed understanding of how demand varies across time, context, and usage conditions, particularly as electrification and distributed resources reshape load dynamics. This evolving landscape has motivated energy research that prioritizes not only real-time operations but also the characterization and exploratory analysis of demand behavior as a foundation for planning, policy design, and comparative scenario assessment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although residential electricity consumption exhibits patterns that can be exploited by data-driven methods, many existing modeling approaches remain tied to limited historical data or rigid simulators that do not fully capture household-level variability. Empirical analyses have shown that electricity use in the residential sector is influenced by multiple determinants, including household demographics, dwelling characteristics, climate conditions, and appliance usage, and that these factors contribute to significant heterogeneity in consumption behavior both within and between households (Kostakis, 2020 ; Ali et al . , 2022 ; Guven and Kayalica, 2023 ; Lavin and Julienne, 2025 ) . For example, income, household size, and regional climate have been identified as major contributors to variation in total residential demand, while smart meter analyses reveal distinct daily load profiles associated with occupancy and dwelling characteristics (Rafiq et al . , 2023 ; Grandjean et al . , 2012 ; Jabbar et al . , 2021 ) . Similarly, research on residential load patterns has shown significant correlations among these factors, underscoring the need to incorporate them into accurate energy consumption models (Wei and Wang, 2021 ) . Traditional simulation methods and bottom-up models often rely on assumptions that are difficult to scale or tailor to diverse contexts, and purely data-driven models require high-quality, well-annotated datasets. Gaps, missing values, and limited geographic coverage can reduce model generalization and lead to the omission of important behavioral structures. Together, these limitations underscore the need for methods that can generate controlled, contextually informed synthetic consumption data to support exploratory analysis and scenario reasoning beyond the scope of available empirical sources.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Large Language Models (LLMs) have emerged as powerful tools for synthesizing data and knowledge across diverse domains (Tan et al . , 2024 ) , including recent exploratory applications in the energy sector. After the introduction of LLM-based agents into the research community (Park et al . , 2023 ) , new paradigms have been proposed for leveraging the implicit knowledge encoded in large models through structured interaction and coordination. In this work, LLMs are not used for conventional forecasting or statistical modeling of demand; rather, they are leveraged as controlled generative agents that produce scenario-aware synthetic representations of household energy consumption under explicitly defined cultural, temporal, and contextual assumptions. Inspired by (Karpathy, 2025 ) , we employ a multi-agent LLM Council in which distinct models operate in complementary roles under explicitly defined constraints. This improves consistency and contextual coverage by relying on an ensemble of models rather than a single model’s capabilities. Specifically, we aim to integrate cultural, seasonal, and behavioral contexts into household energy consumption patterns. We acknowledge that LLM-generated data faces inherent challenges in reproducing physically grounded dynamics. We therefore position WattCouncil as a tool for generating controlled synthetic scenarios for analysis and benchmarking, and as a complement to physical simulators and to demand forecasting or optimization. Our contributions consist of: (i) WattCouncil , a multi-agent LLM-based framework for governed synthetic energy data generation; (ii) a staged pipeline with explicit auditing and bounded regeneration; (iii) an evaluation protocol based on similarity to real smart meter data; (iv) ablation studies on the components of our data generation pipeline; (v) open-access code of our pipeline with clear instructions on how to run it, hoping that it could be a contribution to future researchers who want to explore the area of LLM-based data synthesis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The remainder of the paper is structured as follows. Section 2 reviews related work in household energy modeling, synthetic data generation, and the use of large language models in structured simulation contexts. Section 3 introduces the WattCouncil framework, describing the staged generation pipeline, constraint formulation, and governance mechanisms. Section 4 presents a quantitative and qualitative evaluation of the generated consumption profiles, including comparisons with a real-world smart meter dataset, while Section 5 analyzes the impact of key design choices through targeted ablation studies. Section 6 discusses current limitations of the proposed approach, and Section 7 concludes the paper with a summary of findings and directions for future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2. Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traditionally, researchers enabled access to relevant data by conducting in-place studies for a defined period and gathering as much data as possible from real-world settings using sensors. Despite sustained efforts to collect and release residential electricity datasets, the resulting landscape remains highly fragmented across geography, duration, resolution, and household coverage. As pointed out in (Cruz et al . , 2024 ) , which found that in general, datasets varied considerably, especially in countries such as the USA, the UK, and India, and exhibited several sources of bias. Widely used datasets such as REDD (Kolter and Johnson, 2011 ) , UK-DALE (Kelly and Knottenbelt, 2015 ) , REFIT (Murray et al . , 2017 ) , AMPds (Makonin et al . , 2013 ) , BLUED (Anderson et al . , 2012 ) , iAWE (Batra et al . , 2013 ) , and ECO (Beckel et al . , 2014 ) provide high-resolution appliance-level measurements but are typically limited to a small number of households and short observation periods, thereby limiting their representativeness. On the other hand, larger-scale datasets such as Dataport (Pecan Street) (Parson et al . , 2015 ) , RECS (U.S. Energy Information Administration (2023), EIA ) , and CLNRPD (Northern Powergrid, 2015 ) offer broader population coverage but often lack fine-grained temporal resolution, appliance-level detail, or continuous monitoring across contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Household energy research has documented that residential electricity consumption is influenced by a combination of demographic, behavioral, cultural, and socio-economic factors. Empirical studies identify household size, income, appliance ownership, and occupancy patterns as significant drivers of energy use variability (Jones and Lomas, 2015 ) . For instance, larger household size has consistently been associated with higher electricity use, while income and dwelling-related characteristics (such as the number of bedrooms and heating systems) account for substantial differences in consumption across households (Jones and Lomas, 2015 ) . Other research has highlighted that socio-economic and regional characteristics result in heterogeneous consumption patterns that vary between geographic contexts and policy environments. This suggests that household electricity demand is shaped by interacting social and structural determinants rather than physical building attributes alone (Kostakis, 2020 ; Najeeb et al . , 2024 ) . The study in (Mora et al . , 2018 ) highlights the significant impact of occupant behavior, socio-economic characteristics, and preferences (e.g., set-point temperatures and ventilation durations) on residential energy consumption. It emphasizes that variations in energy demand are influenced more by contextual factors, such as family composition and dwelling usage patterns, than by physical attributes, such as house size or floor area.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In (Du and Lu, 2011 ) , the authors present an algorithm that schedules household appliances based on consumption forecasts and user comfort settings, aiming to optimize objectives like minimizing costs or maximizing comfort. This underscores the importance of accurately modeling intra-family dynamics to simulate appliance usage patterns effectively. Building upon this, (Nicholls and Strengers, 2015 ) examines how family interactions, such as shared activities and the presence of children, influence peak electricity demand times. The study highlights the need to incorporate family dynamics into energy consumption models to accurately predict and manage peak demand periods. Several studies have leveraged weather simulation frameworks to assess HVAC performance under various climatic scenarios. For example, (Xu et al . , 2024 ) analyzed the impact of urban heat islands on residential cooling demand, showing how microclimatic factors intensify energy consumption in densely populated regions. Despite this substantial body of empirical work, many modeling approaches traditionally used in energy systems have focused primarily on physical parameters or historical load profiles, with limited integration of detailed socio-demographic and behavioral insights into analytical or generative frameworks (Crosbie, 2006 ) . This gap motivates the development of methods that can represent contextually grounded consumption variability when generating or analyzing residential energy data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The usage of LLMs is transforming how domain-specific tasks are approached, particularly in generating human-like, context-aware outputs. In (Buster et al . , 2024 ) , researchers from the National Renewable Energy Laboratory (NREL) introduced a framework combining LLMs with decision-tree logic to guarantee compliance with zoning policies for renewable energy siting. The reported accuracy suggests that such approaches can be useful in downstream modeling. In (Almashor et al . , 2024 ) , the authors explored whether LLM-based agents can synthesize household energy consumption, building upon the environment provided by (Park et al . , 2023 ) . Although they obtained some reasonable load patterns, they were limited by the capabilities of the simulation they used. Their approach focused on parsing keywords in agent conversations, associating them with energy consumption, and then summing the household load. Along the same line of work, (Takrouri et al . , 2025 ) introduced a multi-stage methodology to synthesize load profiles from different LLMs using a pipeline that considers family structure definition, weather identification, weather synthesis, and load pattern generation, all conditioned on a given geography and time period. Building upon that work, (Chetty et al . , 2025 ) presents a methodology that enhances the process by integrating memory capacities and a rule-based validation before constructing the final load profile. These studies highlight the potential of LLMs to streamline data access and policy analysis in household energy modeling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, significant gaps persist in integrating these components into a cohesive framework. Current approaches often rely on limited assumptions about family structures and behaviors, neglecting the nuanced and culturally specific dynamics necessary for accurate simulations. In particular, existing work rarely separates macro-level scenario specification (e.g., season/climate regime, calendar rules, household composition, and social context) from micro-level behavioral realization at the hourly scale (occupancy/activity and appliance-use traces). This limits explainability and controllability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3. Methodology\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1. WattCouncil: An LLM-Based Framework For Governance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The research community has increasingly supported the idea that coordinated outputs from many LLMs could outperform a single model. Works such as (Hu et al . , 2025 ) showed that this approach, known in the literature as model ensembling, outperformed individual models in reasoning tasks. More sophisticated works, such as (Gao et al . , 2025 ) , proposed a knowledge-transfer method to optimize model ensemble performance in sequential decision-making tasks. Finally, a recently published resource in (Karpathy, 2025 ) introduced the idea of LLM councils, which consist of several models (with different roles) that reach consensus on a given task.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inspired by the aforementioned works, we introduce WattCouncil , a governed LLM Council that employs role-specialized agents to generate structured synthetic household energy data under explicit constraints. Rather than relying on a single language model to produce end-to-end outputs, the council decomposes generation into a sequence of staged decisions, each subject to a data schema validation, auditing, and explicit acceptance criteria. In contrast to autonomous conversational use, the LLMs operate as constrained generators inside a pipeline whose control flow, stopping conditions, and artifact interfaces are fully specified. Figure 1 summarizes the governance logic and decision flow enforced at every level.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The council is implemented as a three-stage pipeline with explicit artifact interfaces as illustrated in Figure 2 . The council comprises three components: (i) household structure generation, with assignment of work and occupancy regimes; (ii) definition of seasonal weather ranges, with hourly weather instantiation; (iii) household energy consumption modeling. Each stage consumes only outputs that are validated in the operational sense, in other words, outputs that generated a schema-compliant JSON that has been audited and accepted by the controller/approver. Individual stages can be selectively enabled, disabled, or replaced by external data sources without affecting the integrity of the remaining pipeline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The council architecture assigns fixed, non-overlapping roles to multiple LLMs, enforcing strict task separation. A primary generator proposes candidate structured outputs at each stage of the pipeline, which are evaluated by designated role-based auditors depending on the stage context. For stages involving social structure, behavior, or narrative coherence (Stages 1 and 3), a cultural auditor assesses alignment with country-specific social norms, and a physical auditor verifies numerical consistency and adherence to physical or environmental constraints across all stages. For Stage 2, only the physical auditor is applied, as cultural evaluation is not meaningful in contexts such as seasonal weather-range definition and hourly weather instantiation. Auditors return structured JSON reports with a severity score, following a shared scale ( LOW / MEDIUM / HIGH ) that indicates the level of compliance with the conditions for each stage. We maintain a fixed generation logic to achieve reproducibility by pinning model versions, prompts, and schemas, and logging all intermediate artifacts, while fixing inference settings (e.g., temperature and sampling parameters).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, a Council Controller acts as the sole decision-making authority in the council. Based on the generator output and the relevant audit results, it selects one of three actions: accepting the results of the pipeline execution ( ACCEPT ), triggering a targeted correction phase handled by an editor followed by independent verification by an approver ( REGENERATE PARTIAL ), or discarding the execution and restart the process from scratch ( REGENERATE FULL ). The framework supports a scoped rule memory that records recurring failure patterns (e.g., repeated schema violations or constraint failures) and the corresponding corrective instructions. When enabled, this memory is treated as a versioned artifact: any rule additions are logged so that subsequent generations can be replayed under the same rule set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The WattCouncil framework is intentionally model-agnostic, assigning specific language models to each council role based on functional suitability (e.g., cultural reasoning, numerical consistency, or structured editing), while keeping these assignments fixed across all experiments. The framework does not benchmark or compare individual language models. Instead, evaluation is performed at the system level, where behavior emerges from role specialization, constraint enforcement, and deterministic governance rather than from any single model’s capabilities. Table 1 summarizes the models used in this study.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In LLMs, the temperature hyperparameter ( τ \\\\tau ) controls sampling randomness. Intuitively, temperature determines how willing the model is to explore alternative plausible outputs rather than repeatedly selecting the most likely continuation: lower values make the model more conservative and predictable, while higher values allow greater variation. Formally, given a vector of logits 𝐳 = ( z 1 , … , z K ) \\\\mathbf{z}=(z_{1},\\\\dots,z_{K}) over a vocabulary of size K K , the probability of sampling token i i is computed using a temperature-scaled softmax,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where lower τ \\\\tau concentrates probability on high-scoring tokens and higher τ \\\\tau increases output diversity. In WattCouncil , temperature controls local generation variability, while global consistency is enforced through auditing, bounded regeneration, and explicit acceptance criteria.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The temperature settings assigned to each council role reflect their functional responsibilities within the generation and governance pipeline. In particular, the Generator is configured with a higher temperature than the other roles to encourage controlled creativity during the initial proposal of structured artifacts. At this stage, the objective is not strict verification but the exploration of plausible household structures, behaviors, and contextual configurations that are consistent with high-level constraints. A moderately elevated temperature increases output diversity, allowing the Generator to surface alternative yet reasonable scenarios that might not emerge under near-deterministic sampling. In contrast, downstream roles such as the Cultural Auditor, Physical Auditor, Editor, and Approver are assigned lower temperatures to prioritize consistency, precision, and adherence to constraints during evaluation and correction. This asymmetric temperature allocation reflects a deliberate separation of concerns: creative variability is permitted at the proposal stage, while rigor and stability are enforced during auditing and decision-making. Within the WattCouncil framework, this design allows limited stochastic exploration without compromising overall governance, reproducibility, or physical and contextual validity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2. Integration of Typical Meteorological Weather (TMY)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As stated before, the stages of WattCouncil ’s pipeline ( Figure 2 ) are explicitly designed to be modular and can be replaced by externally sourced data without affecting the rest of the framework. This flexibility is particularly important for weather information, which is inherently complex and difficult to generate reliably from scratch due to its strong physical constraints, temporal structure, and geographic dependence. To address this, we investigate an alternative setup in which LLM-generated weather is replaced with a simple, safe baseline based on Typical Meteorological Year (TMY) data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A Typical Meteorological Year is a synthesized weather dataset that represents statistically typical meteorological conditions for a specific geographic location over the course of a single year. Rather than reflecting a specific calendar year’s weather, a TMY is constructed by selecting the most representative months from a long historical record so that key variables such as temperature, solar radiation, humidity, and other meteorological indicators closely match long-term averages. TMY datasets are designed to capture typical patterns rather than extreme events (heat waves, cold snaps, or compound weather extremes); they are widely used in building energy simulation, renewable resource assessment, and engineering analyses as a representative baseline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although TMY data do not perfectly replicate year-to-year weather variability, they provide physically grounded, historically derived weather profiles that work as a reference for correctness. We implement this substitution using the Python library pvlib (Anderson et al . , 2023 ) , which enables the generation of a full year of hourly weather variables for a given geographic location specified by latitude and longitude. This replacement decouples behavioral and energy modeling from weather synthesis, providing a physically grounded baseline for comparison. In the present study, weather conditioning focuses on representative seasonal patterns rather than explicit extreme-event stress testing.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4. Experimental Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated the WattCouncil framework as a scenario-oriented synthetic data generator , assessing whether the generated household energy profiles reproduce realistic temporal structure and aggregate behavior when compared against real-world reference data, and whether key pipeline components, with a focus on weather data, can be substituted without substantially degrading downstream output.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1. Reference Real-World Dataset\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The CER Electricity Data (Revised March 2012) (Song, 2024 ) is a rich, open dataset derived from the Commission for Energy Regulation’s Smart Metering Project, a large-scale trial of electricity smart meters in Ireland. It comprises half-hourly electricity consumption measurements collected over more than 500 days (spanning from July 14, 2009, to December 31, 2010), capturing detailed load profiles for 4232 residential customers and 529 small-to-medium enterprises (which we do not consider in our study). In addition to the raw consumption time series, the trial includes extensive pre- and post-trial survey data on socio-demographic characteristics, household properties, and energy-use behaviors, enabling analysis that links load profiles to occupant and dwelling features. We selected this dataset as a reference because it reflects real consumer behavior under the determinants we introduced in our pipeline, enabling a direct comparison between synthetic and real data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1.1. Exploratory Data Analysis (EDA) of the CER Dataset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inspired by prior work (Guo et al . , 2022 ) that uses the CER dataset and combines the smart meter time series with socio-demographic metadata to study household energy behavior, we follow the same principle of jointly using consumption traces and survey-derived contextual attributes. We adopt the perspective that load profiles alone are insufficient to characterize household behavior without accounting for family composition, occupancy patterns, and socio-economic factors, all of which are explicitly modeled in our framework. While that study used K-means (an unsupervised clustering method) to identify latent customer groups based on load shape similarity, we intentionally opted for a different approach that explicitly groups clients based on responses to selected questions in the surveys.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following the design and operational principles of WattCouncil , the cluster membership carries limited semantic meaning for conditioned generation; in other words, including an unlabeled cluster in the context does not provide interpretable guidance for scenario synthesis. Instead, we define explicit household groups based on socio-economic and survey attributes available in the CER dataset, selecting variables that can be directly used as conditioning inputs within our pipeline, as summarized in Table 2 . From these groups, we identify representative subsets of households and extract their aggregated characteristics to ground the generation process. This design choice ensures that insights derived during the EDA stage explicitly constrain the generative mechanism, enabling systematic and interpretable comparisons between real consumption data and LLM-generated profiles produced by our pipeline.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2. Comparison Between Synthetic and Real-World Energy Consumption Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate the realism of the synthetic household energy consumption profiles generated by WattCouncil by comparing them against the CER dataset. The objective is to assess whether the proposed pipeline reproduces residential consumption patterns observed in real data across daily and seasonal time scales.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Based on the selected residential groups reported in Table 2 , we condition the generation process on these 5 representative groups. The pipeline generates multiple synthetic households that match the groups’ characteristics, and daily consumption profiles are produced under multiple weather conditions across all meteorological seasons. These synthetic profiles are then compared against the corresponding subset of real households from the CER dataset to assess behavioral and temporal consistency.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.2.1. Seasonal Profile Construction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the synthetic data, we aggregated energy consumption across all family-weather combinations within each season. We compute hourly means, standard deviations, and sample counts, yielding a representative 24-hour profile for each season. For real data, we define seasons by calendar months in the Northern Hemisphere: winter (December–February), spring (March–May), summer (June–August), and autumn (September–November). We grouped hourly consumption values by season and hour of day across all selected customers and computed the corresponding summary statistics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.2.2. Visualization and Similarity Metrics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We visualize seasonal comparisons using four horizontally arranged subplots as shown in Figure 3 corresponding to winter, spring, summer, and autumn. Each subplot displays the mean hourly consumption profiles for synthetic and real data, accompanied by shaded confidence intervals. For this WattCouncil dataset, we generated electricity consumption profiles for five household groups by combining multiple controlled variations. Specifically, we selected 5 representative family groups and instantiated 10 employment and occupancy variants per group. Each configuration was evaluated across 4 seasons and 5 independent weather realizations per season, resulting in a total of 5 × 10 × 4 × 5 = 1000 5\\\\times 10\\\\times 4\\\\times 5=1000 synthetic household profiles. We note that the apparent smoothness difference in Figure 3 may arise from the original axis scaling and the differences in aggregation: the real profiles are averaged over a larger number of households, whereas the synthetic profiles are derived from a smaller set of generated household configurations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In addition to visual inspection, Table 3 reports the quantitative error and similarity metrics for each season. We use Pearson’s correlation to assess temporal shape similarity and peak timing alignment, whereas mean absolute error (MAE), root mean squared error (RMSE), and mean absolute percentage error (MAPE) quantify magnitude mismatch. This combination allowed separating whether the synthetic profiles reproduce realistic daily temporal structure, and whether they recover realistic absolute consumption levels. Results show pattern–level similarities but clear magnitude discrepancies, indicating that matching the consumption dimension depends on aspects we might not have considered in our pipeline. This distinction is important: realistic timing patterns can emerge from context-aware occupancy and activity modeling even when the final consumption scale remains imperfect.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5. Ablation Studies on the Pipeline\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ablation is evaluated at two levels: Section 5.1 compares LLM- and TMY-derived weather profiles (seasonal statistics and daily patterns), and Section 5.2 quantifies changes in downstream demand patterns deriving from changing the source of weather data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1. Weather Profile Comparison\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first assess whether the proposed pipeline produces physically and temporally plausible weather inputs by comparing LLM-generated hourly weather profiles with externally sourced TMY data for Ireland, which matches the geographic scope and reference year (2009) of the real-world dataset used in our study. This analysis isolates the behavior of the weather-generation component (Stage 2) from household structure and behavioral modeling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 visually compares the average daily profiles of environmental variables generated with the LLM against the TMY reference for Dublin, Ireland, which we assume is uniform across the territory (the dataset does not provide household locations to preserve privacy). We used 5 LLM-generated profiles and 60 TMY profiles (which are faster to sample) to build 95% confidence intervals using the t-distribution. Additionally, we computed Pearson correlation coefficients ( r r ) between LLM-generated and TMY weather profiles across all seasons and meteorological variables. The results suggest that LLM-generated data align with the TMY reference, with correlation increasing in spring and summer and decreasing in autumn and winter.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We summarize the correlation results in Table 4 . These results indicate that directly generating weather profiles with general-purpose LLMs can introduce avoidable inaccuracies, primarily because of the physical complexity and high temporal precision required for meteorological modeling. In this setting, LLMs are less well-suited than physics-informed or data-driven weather models, rather than being inherently incapable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Importantly, this limitation does not undermine the WattCouncil framework; instead, it motivates the use of reliable external weather sources, such as TMY data, to condition consumption and generation. Moreover, specialized climate models—such as NVIDIA’s Earth-2 platform (Earth2Studio Contributors, 2024 ) —represent promising avenues for integrating richer, physically grounded weather context into the pipeline without compromising consistency or realism.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2. Impact of Weather Sourcing on Energy Consumption\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To isolate the effect of weather sourcing, all non-meteorological inputs are held constant across experiments. Table 5 summarizes the household configuration used in this study, including composition, occupancy regime, dwelling type, and appliance set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Electricity consumption profiles are generated under two settings: LLM-generated weather inputs and externally sourced TMY weather. Figure 5 compares weekday and weekend demand profiles across all four seasons for five realizations of the same household configuration. The results exhibit closely aligned load shapes and peak timing, indicating that downstream consumption dynamics remain stable under changes in weather sourcing. This observation is supported by high Pearson correlation coefficients across seasons. Differences between the two settings primarily appear in the width of uncertainty bands rather than systematic shifts in mean demand.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overall, these results suggest that the proposed pipeline is robust to the choice of weather source at the household energy-demand level. LLM-generated weather preserves realistic demand structure and variability, while enabling scenario diversity beyond what is attainable from a single deterministic reference year.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One possible explanation for the muted summer demand is the combined effect of governance constraints, where cooling and heating are repeatedly disabled during regeneration, and behavioral assumptions associated with the fixed household configuration, leading to intentionally conservative summer usage patterns.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, across repeated runs, we observe that summer scenarios consistently trigger additional regeneration and editing steps within the council, resulting in nearly twice the generation time compared to other seasons, further reflecting the stricter constraint enforcement under summer conditions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3. End-to-End Pipeline Execution Example\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To illustrate the behavior of the proposed framework in practice, we report an end-to-end execution trace for a single synthetic household processed through all pipeline stages. The household is evaluated under a single weather condition across the four seasons, with each season instantiated under both weekday and weekend assumptions. This results in 8 independent Stage 3 consumption-generation runs for the same household. During execution, the framework applies the LLM Council governance described in Figure 1 across all pipeline stages. When violations are detected at any stage (e.g., implausible household routines, inconsistent weather patterns, or HVAC heating active during summer conditions), the controller enforces regeneration with explicit, stage-specific corrective guidance. Because all detected violations triggered full regeneration rather than constrained editing, neither the Editor nor the Approver roles were activated in this execution. In the illustrated example, the first summer weekday attempt is rejected due to physically implausible heating usage and is subsequently regenerated and accepted after correction. This demonstrates how the framework actively enforces domain constraints rather than passively accepting generative outputs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.3.1. Pipeline runtime analysis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 6 summarizes the execution statistics for this illustrative run. Although only a single household is processed, the example reflects the governance overhead incurred during realistic operation: generating one household across four seasons and weekday/weekend variants required 52 model calls and 556.8 seconds of execution time, spanning generation, auditing, and decision-making steps. Most of this overhead is concentrated in the Generator and Physical Auditor roles, which dominate the synthesis and validation of structured hourly consumption profiles under explicit constraints, while cultural auditing and controller decisions contribute comparatively little. We do not report monetary costs, as they depend on the specific models, providers, and pricing schemes used at runtime, which may change over time; instead, we report token usage to enable reproducible cost estimation across different deployment settings. Representative execution traces for Stages 1–3 are shown next as annotated pipeline logs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6. Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While the proposed framework aligns fairly with real-world residential electricity consumption patterns, several limitations should be noted.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The empirical evaluation is limited to Irish households, given that the CER dataset was the only comprehensive study to provide sufficient socio-demographic metadata to create a systematic comparison. Rigorous validation against real data requires comparable datasets, constraining cross-country evaluation in this study.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Comparisons against real data rely on aggregated seasonal and hourly statistics, which smooth household-level variability. Accordingly, the reported similarity metrics reflect population-level behavioral alignment rather than exact household-level replication.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The weather ablation relies on a finite number of discrete realizations for both LLM-generated and TMY-derived inputs, which does not capture the full range of meteorological variability or extreme events.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Governance mechanisms introduce additional computational overhead, reflecting a deliberate trade-off between controllability and generation speed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our framework reduced a larger set of possible attributes available in the CER dataset metadata to three, aiming to keep our budget and computation tractable. The selection could introduce bias in how WattCouncil generates load profiles.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We assumed uniform weather in Ireland based on the low variance of weather stations, a similar approach to (Guo et al . , 2022 ) ; this might be imprecise. Also, weather conditioning focuses on representative seasonal patterns rather than explicit extreme-event stress testing.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7. Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper introduced WattCouncil , a governed, multi-agent LLM framework for generating structured synthetic household electricity consumption data under explicit cultural, temporal, and physical constraints. The proposed approach decomposes generation into a staged pipeline governed by role-specialized agents, explicit control logic, and governance via interfaces and control flows, with each stage producing auditable artifacts that meet explicit criteria. Evaluation against the CER smart meter dataset shows that the generated profiles reproduce the daily and seasonal demand structures fairly well and exhibit strong temporal alignment with real data across demographic groups. Ablation results further indicate that downstream consumption dynamics are robust to the choice of weather sourcing, with differences primarily affecting uncertainty rather than mean demand. While contextual and behavioral factors are central to reproducing temporal demand structure, the remaining magnitude gap indicates that physically grounded variables remain essential for realistic absolute consumption. A natural next step is therefore to incorporate stronger physical determinants, including dwelling-envelope properties, HVAC efficiency, appliance power ratings, and simplified thermal-response constraints. This would allow future versions of WattCouncil to couple contextual plausibility with improved energy-scale realism. Future work could also extend the framework by incorporating a more diverse set of socio-economic conditioning attributes, evaluating additional regions, integrating physically informed climate models, and exploring tighter coupling with downstream optimization and planning tasks. The integration of explicit extreme-weather scenarios is also an important future direction, particularly in regions where heating and cooling demand are highly sensitive to rare meteorological events.\"\n }\n]","isConversational":false}}},{"rowIdx":380,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10707"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"MDQEC-QAS: Meta-Decoding for Quantum Error Correction with Hardware-Aware VQC Search and Confidence-Gated Recovery"},"authors":{"kind":"list like","value":["Prashant Kumar Choudhary","Nouhaila Innan","Muhammad Shafique","Rajeev Singh"],"string":"[\n \"Prashant Kumar Choudhary\",\n \"Nouhaila Innan\",\n \"Muhammad Shafique\",\n \"Rajeev Singh\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10707v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"MDQEC-QAS: Meta-Decoding for Quantum Error Correction with Hardware-Aware VQC Search and Confidence-Gated Recovery"},{"type":"para","text":"Prashant Kumar Choudhary 1 , Nouhaila Innan 2,3 , Muhammad Shafique 2,3 and Rajeev Singh 1"},{"type":"para","text":"1 Department of Physics, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India 2 eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE 3 Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE"},{"type":"para","text":"prashantkchoudhary.rs.phy22@iitbhu.ac.in nouhaila.innan@nyu.edu muhammad.shafique@nyu.edu rajeevs.phy@iitbhu.ac.in"},{"type":"para","text":"Abstract. We propose a unified meta-decoding framework for quantum error correction that learns syndrome-to-recovery mappings across multiple stabilizer codes and noise settings, without separate decoders for each configuration. The benchmark includes FiveQubit, Steane, Planar3 × \\times 3 and Planar5 × \\times 5 codes, four noise families, and five regimes: interpolation, unseen- p p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. We compare a classical Meta-MLP teacher-trained baseline with variational quantum circuit (VQC) meta-decoders selected through hardware-aware quantum architecture search over qubit count, circuit depth, and entangling topology. The Meta-MLP achieves teacher-label accuracies of 0.9993, 0.9118, 0.9342, 0.6304, and 0.7548 across the five regimes, while the hardware-aware VQC achieves 0.9400, 0.8495, 0.8415, 0.5678, and 0.7143. However, logical-level evaluation shows that high teacher-label accuracy alone is insufficient in the hardest Planar5 × \\times 5 setting. During interpolation, raw logical-failure ratios relative to the teacher are 12.08 and 25.91 for the Meta-MLP and VQC, respectively, but confidence-gated fallback reduces them to 1.71 and 1.11. These results support confidence-aware selective recovery rather than unconditional teacher replacement."},{"type":"para","text":"Keywords: Quantum error correction; meta-decoding; variational quantum circuits; quantum architecture search; stabilizer codes; planar codes; transfer learning; hybrid decoding."},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Quantum error correction (QEC) is a central requirement for scalable fault-tolerant quantum computation because fragile quantum states are continuously exposed to decoherence, control imperfections, and measurement noise. Within this landscape, the stabilizer formalism provides a unifying framework for constructing and analyzing a broad class of quantum codes, including both compact algebraic codes and large topological codes [ 34 , 36 , 19 , 26 , 37 ] . In a stabilizer-code setting, noisy physical qubits are probed through stabilizer measurements, producing a classical syndrome that must be translated into a recovery operation. The success of this decoding stage directly determines the resulting logical failure probability and therefore the usefulness of the encoded quantum information [ 14 , 18 , 6 ] ."},{"type":"para","text":"For a measured syndrome s s , decoding can be considered abstractly as the inference problem"},{"type":"para","text":"where r r denotes a candidate recovery operator. The operational metric of interest is the logical failure probability"},{"type":"para","text":"where p p is the physical error probability or, more generally, a parameter controlling the noise strength. In other words, a useful decoder is not just one that predicts labels accurately, but one that preserves logical information after recovery. This distinction becomes particularly important for topological codes, where two decoders with similar classification accuracy can still differ substantially in logical performance [ 12 ] ."},{"type":"para","text":"Traditional decoders such as lookup-table strategies, heuristic code-specific rules, and matching-based methods can perform strongly within narrowly defined settings, but they are usually specialized to a certain code family or noise model [ 13 , 8 ] . This limitation has motivated machine-learning-based decoders, from early neural approaches to more scalable graph- and transformer-style models [ 3 , 32 ] ."},{"type":"para","text":"These developments establish two key observations. First, learned decoders can be competitive with hand-engineered approaches, particularly when the noise has correlations or when decoder inference must adapt to experimental data [ 2 ] . Second, the most challenging regimes are often larger structured topological codes, where raw classification accuracy alone does not guarantee strong logical performance [ 12 , 17 ] . Thus, modern QEC decoding is no longer only a combinatorial problem; it increasingly also involves representation learning and decoder-system design."},{"type":"para","text":"Despite this progress, most learned QEC decoders are still trained and evaluated within fixed code/noise regimes. However, a robust decoder should ideally adapt across heterogeneous physical conditions: different stabilizer structures, different noise families, and different physical error rates. This motivates a meta-decoding perspective, in which one shared model is trained across multiple decoding tasks and receives explicit side information describing the code family, the noise type, and the physical error level. Rather than learning one isolated decoder per setting, the model learns a transferable syndrome-to-recovery representation [ 20 , 39 ] ."},{"type":"para","text":"In parallel, variational quantum circuits (VQCs) have emerged as a candidate for hybrid quantum-classical learning systems, but their performance depends strongly on ansatz design [ 21 , 33 ] . This makes quantum architecture search (QAS) a relevant tool for QEC decoding as well: instead of fixing a single parametrized quantum circuit by hand, one may search over qubit count, circuit depth, and entangling topology, and then select architectures using both predictive quality and implementation-oriented cost proxies [ 15 , 29 , 9 , 28 , 25 , 16 ] . In a decoding pipeline, this leads naturally to a hardware-aware model-selection problem rather than a purely accuracy-driven one."},{"type":"para","text":"In this work, we propose a unified meta-decoding framework for QEC that combines these ideas in a single reproducible pipeline. The framework jointly studies two algebraic codes (FiveQubit and Steane), two planar topological codes (Planar3 × \\times 3 and Planar5 × \\times 5), four noise families (depolarizing, Z Z -biased, correlated Z Z -burst, and depolarizing with syndrome-measurement flips), and several transfer settings, including interpolation, unseen- p p transfer, unseen-noise transfer, and few-shot adaptation to an unseen code. We compare a classical meta-MLP against VQC meta-decoders selected through hardware-aware QAS."},{"type":"para","text":"A key observation arising from our experiments is that the larger planar code, Planar5 × \\times 5, is substantially more sensitive to raw learned-decoder mistakes at the logical level than the smaller codes. Even rare teacher-label errors can become logically costly, so high supervised accuracy alone is not a sufficient reliability criterion for this regime. To address this, we introduce a confidence-gated hybrid recovery strategy: when the learned decoder is sufficiently confident, its prediction is used directly; otherwise, the pipeline falls back to the teacher decoder. This mechanism reveals a high-precision selective-decoding regime: the learned model can be useful on confident cases even when it is not reliable as a full teacher replacement. The resulting system should therefore be interpreted as a hybrid learned-assisted recovery rather than an independent learned decoding. Confidence-aware and risk-aware mechanisms are standard in broader machine learning and decision systems, but they remain underexplored in QEC decoding pipelines [ 30 ] ."},{"type":"para","text":"The novelty of this work does not lie in the isolated use of a neural decoder or a variational quantum classifier. Rather, it lies in the integration of four components within a single reproducible QEC pipeline: (i) multi-code, multi-noise meta-decoding across algebraic and planar stabilizer codes, (ii) transfer evaluation under unseen- p p , unseen-noise, few-shot unseen-code, and held-out-size settings, (iii) compact hardware-aware quantum architecture search for VQC decoder design, and (iv) confidence-gated hybrid recovery for the most challenging topological-code regime. This integrated design leads to three system-level findings: confidence can act as a routing signal for selective decoding, structural transfer across code families and code sizes is substantially harder than transfer across noise parameters, and cost-aware VQC selection can remain competitive with accuracy-selected circuits within the compact search space."},{"type":"para","text":"The main contributions of this paper are summarized as follows:"},{"type":"para","text":"We construct a pooled multi-code, multi-noise QEC meta-decoding benchmark spanning algebraic and planar stabilizer codes, multiple noise families, and several transfer settings."},{"type":"para","text":"We compare a classical Meta-MLP with hardware-aware VQC meta-decoders under a shared global recovery-label framework."},{"type":"para","text":"We perform compact hardware-aware QAS over qubit count, circuit depth, and entangling topology, and evaluate the selected VQC architectures after full retraining."},{"type":"para","text":"We distinguish teacher-label accuracy from logical-level decoding performance, showing that raw supervised accuracy can be misleading for larger topological codes."},{"type":"para","text":"We identify a transfer hierarchy in which unseen- p p and unseen-noise generalization are easier than unseen-code and held-out-size adaptation, indicating that changes in code structure are more challenging than changes in the noise distribution."},{"type":"para","text":"We introduce and evaluate confidence-gated hybrid recovery, showing that confidence can serve as a routing signal between learned predictions and teacher fallback in the larger Planar5 × \\times 5 regime."},{"type":"heading","level":1,"text":"2 Background and Related Work"},{"type":"para","text":"Research on quantum error-correction decoders has evolved from analytical and combinatorial constructions toward increasingly data-driven and learning-based approaches. This section reviews prior work at the intersection of stabilizer-code decoding, learned decoder transferability, variational quantum models, and hardware-aware decoder design."},{"type":"heading","level":2,"text":"2.1 Classical and learned decoding for stabilizer codes"},{"type":"para","text":"Quantum decoding has traditionally been studied through code-specific analytical and combinatorial methods. Within the stabilizer formalism, the decoder receives a syndrome and must infer a recovery that returns the state to the codespace without inducing a logical fault [ 19 , 37 ] . For topological and surface-code families, decoding is closely connected to threshold analysis and fault-tolerant architecture design. In particular, the topological-quantum-memory framework and later surface-code studies established minimum-weight matching and related combinatorial approaches as strong baselines for structured local-noise settings [ 14 , 18 , 12 ] . These decoders are often highly effective, but they are typically specialized to a particular code structure and noise assumption [ 4 ] ."},{"type":"para","text":"This specialization motivates the search for more adaptive decoder families, since modern experimental platforms exhibit changing calibration, asymmetric noise, measurement faults, and correlated errors. As a result, decoder design increasingly requires methods that combine combinatorial optimization, representation learning, and system-level design."},{"type":"para","text":"This need for adaptability helped motivate machine-learning-based decoders as data-driven alternatives to explicit code-specific decoding rules. Torlai and Melko provided one of the earliest demonstrations that neural networks can decode topological codes by learning syndrome-to-recovery mappings directly from data [ 38 ] . Shortly afterward, Varsamopoulos et al. showed that feedforward neural networks can decode small surface codes efficiently while accounting for implementation-oriented decoding latency [ 40 ] . These studies established that decoder behavior can be approximated by trainable models rather than relying exclusively on explicit combinatorial procedures."},{"type":"para","text":"Subsequent work extended learned decoding to more structured regimes. Baireuther et al. studied machine-learning-assisted correction of correlated qubit errors in topological codes, showing that learned approaches can exploit correlations that may be difficult to incorporate into conventional decoders [ 2 ] . Reinforcement-learning-based approaches further explored adaptive decoder behavior and control strategies [ 1 ] . More recently, error-rate-agnostic decoding studies investigated whether neural decoders can remain robust when the physical error rate differs between training and testing, thereby motivating transfer-oriented evaluations in QEC [ 20 , 39 ] ."},{"type":"para","text":"These prior works established the feasibility of learned decoding, but most remained centered on a single code family, a single lattice class, or a single noise scenario. In contrast, this study examines a unified meta-decoding framework spanning multiple stabilizer codes, multiple noise families, and explicit transfer settings."},{"type":"heading","level":2,"text":"2.2 Structured decoders, transferability, and logical-level reliability"},{"type":"para","text":"Recent learned QEC decoders have increasingly adopted structured architectures, including graph neural networks and transformer-style models. This development is motivated by the locality and connectivity patterns of many quantum codes, especially topological codes, which are not always captured efficiently by flat multilayer perceptrons. Lange et al. demonstrated graph-neural-network-based decoding of quantum error-correcting codes, showing that structure-aware learned models can provide strong decoding performance in more complex regimes [ 27 ] . Similarly, recent large-scale work on processor data showed that high-capacity learned decoders can achieve strong decoding quality under experimental noise, highlighting the importance of expressive architectures and training distributions [ 3 , 32 ] ."},{"type":"para","text":"These developments are closely related to the motivation of this study, but the present focus differs in two ways. First, rather than optimizing a learned decoder for a single code family, this study considers a multi-code, multi-noise setting in which a shared decoder is conditioned on the code family, noise family, and physical error rate. Second, this study examines the gap between classification performance and logical-level performance , especially for the larger planar code. This distinction is important because raw learned decoders may achieve nontrivial classification accuracy while still producing substantial logical degradation unless a hybrid recovery mechanism is used [ 17 ] ."},{"type":"para","text":"Beyond architectural design, transferability has become a central concern for learned decoding. A major open question in QEC machine learning is whether learned decoders can transfer beyond the regime on which they were trained. Recent error-rate-agnostic and near-term surface-code decoding studies have highlighted this issue by showing that decoders trained under one physical-error regime may generalize imperfectly when evaluated under another [ 20 , 39 ] ."},{"type":"para","text":"This study extends the transfer perspective by evaluating generalization across interpolation, unseen- p p , unseen-noise, few-shot unseen-code, and held-out-size settings. This combination remains relatively uncommon in prior decoder work, particularly when classical and VQC decoders are evaluated within a unified framework."},{"type":"para","text":"A recurring challenge in learned QEC decoding is that strong label-level prediction does not necessarily imply strong logical-level recovery, particularly for larger structured codes. This distinction is central to the present study: raw learned decoders, including both the classical meta-MLP and the VQC models, can degrade on Planar5 × \\times 5 even when their supervised accuracy remains nontrivial. This motivates a hybrid recovery formulation in which learned inference is used selectively and low-confidence cases are delegated to a teacher decoder [ 30 ] ."},{"type":"para","text":"Although confidence-based hybridization is common in broader machine-learning systems, it has been less explicitly studied as part of QEC decoder pipelines. In this study, confidence-gated hybrid recovery is used to connect learned transferability with logical reliability, especially in the larger planar-code regime."},{"type":"heading","level":2,"text":"2.3 Variational quantum models and architecture search"},{"type":"para","text":"Variational quantum circuits have become an important model class in quantum machine learning. Quantum-enhanced feature spaces [ 21 ] , circuit-centric quantum classifiers [ 33 ] , data re-uploading architectures [ 31 ] , and quantum convolutional neural networks [ 11 ] demonstrate that trainable quantum circuits can serve as expressive learning models. These models have been studied across a range of applications, including classification, regression, generative modeling, optimization, and domain-specific prediction tasks, often in hybrid quantum-classical pipelines designed for near-term quantum devices [ 24 , 22 , 23 , 10 ] . At the same time, their performance depends strongly on circuit architecture, including qubit count, circuit depth, entangling pattern, parameter initialization, and trainability [ 35 , 7 , 41 ] ."},{"type":"para","text":"This dependence has motivated quantum architecture search. Du et al. studied quantum circuit architecture search for variational quantum algorithms [ 15 ] , while later work and surveys examined broader directions in automated quantum model design [ 29 , 5 ] . However, most of this literature does not focus specifically on QEC decoding, and relatively little work addresses transfer-oriented multi-code decoder design. This gap motivates hardware-aware QAS for QEC meta-decoding, with circuit architectures evaluated using predictive performance and implementation-oriented cost proxies such as depth and two-qubit gate count."},{"type":"heading","level":2,"text":"2.4 Positioning within prior QEC decoding studies"},{"type":"para","text":"Our study builds on these prior directions while differing in scope, evaluation design, and decoder-system formulation. Compared with early neural decoders, the proposed framework moves beyond single-code decoding toward a shared meta-decoder trained across multiple code and noise settings. Compared with correlated-noise and error-rate-adaptive learned decoders, it evaluates transfer across both noise distributions and code structures. Compared with recent graph-based and large-scale learned decoders, its focus is not limited to decoder accuracy, but also includes the relationship between supervised prediction quality and logical-level recovery. Finally, relative to general QAS studies, this work applies hardware-aware VQC search specifically to QEC meta-decoding."},{"type":"para","text":"Accordingly, our work is positioned as an integrated QEC decoding framework that combines multi-code meta-decoding, transfer evaluation, hardware-aware VQC model selection, and confidence-gated hybrid recovery. Table 1 makes this distinction explicit by contrasting the present framework with representative prior directions in learned QEC decoding."},{"type":"heading","level":1,"text":"3 Methodology"},{"type":"para","text":"This section describes the proposed learned-assisted meta-decoding pipeline, shown in Fig. 1 . The framework consists of six main components: problem formulation, code and noise specification, pooled dataset construction, meta-decoder training, hardware-aware VQC architecture search, and logical-level evaluation with confidence-gated recovery."},{"type":"heading","level":2,"text":"3.1 Problem formulation"},{"type":"para","text":"In a stabilizer-code setting, a physical Pauli error e e acting on n n qubits induces a syndrome through the binary symplectic product"},{"type":"para","text":"where H H is the binary stabilizer-check matrix of the code [ 19 ] . Given syndrome s s , the decoder must infer a recovery r r such that the corrected operator"},{"type":"para","text":"does not induce a logical fault. If e ′ e^{\\prime} remains nontrivial at the logical level, decoding fails. The corresponding qubit-level mechanism, from physical Pauli errors and stabilizer syndrome extraction to learned recovery prediction and logical decision, is illustrated in Fig. 2 ."},{"type":"para","text":"We formulate this task as meta-decoding. Rather than training a separate decoder for each code and noise condition, a single model is trained on tuples"},{"type":"para","text":"where c c is the code identifier, η \\eta is the noise-family identifier, p p is the physical error probability, and y y is a discrete recovery-class label. The learned decoder is therefore represented as"},{"type":"para","text":"where θ \\theta denotes trainable parameters and y ^ \\hat{y} is mapped back to a recovery operator for logical evaluation."},{"type":"para","text":"The supervised objective is teacher-label classification: the model is trained to predict recovery labels produced by a code-appropriate teacher decoder. Teacher-label accuracy measures how well the learned model imitates this reference decoder, but it is not identical to the operational QEC objective because physically distinct recovery strings can be logically equivalent. For this reason, the evaluation distinguishes teacher-label classification accuracy from logical-level performance. Teacher-label accuracy measures supervised imitation and transfer, whereas logical-level performance is estimated through logical failure after applying the predicted or hybrid recovery."},{"type":"heading","level":2,"text":"3.2 Meta-dataset construction"},{"type":"para","text":"The meta-dataset is constructed from four QEC settings: the compact algebraic FiveQubit [ [ 5 , 1 , 3 ] ] [[5,1,3]] stabilizer code, the Steane [ [ 7 , 1 , 3 ] ] [[7,1,3]] code, and two planar topological-code instances, Planar3 × \\times 3 and Planar5 × \\times 5. Teacher decoders are selected according to code family. This code-dependent choice reflects the structure of the benchmark. FiveQubit and Steane have compact syndrome spaces, so NaiveDecoder provides a consistent reference decoder for supervised label generation. Planar3 × \\times 3 and Planar5 × \\times 5 are planar topological-code instances, for which matching-based decoding is aligned with the lattice structure."},{"type":"para","text":"To evaluate transfer across heterogeneous physical conditions, samples are generated under four noise families: depolarizing noise, (Z)-biased Pauli noise, correlated (Z)-burst noise, and depolarizing noise with syndrome-measurement flips. For the (Z)-biased model, the single-qubit Pauli distribution is"},{"type":"para","text":"The coefficients ( 0.2 , 0.1 , 0.7 ) (0.2,0.1,0.7) are not intended as universal hardware-calibrated constants; rather, they define a representative moderately Z Z -biased Pauli channel in which the total non-identity error probability remains p p , while Z Z -type errors dominate. The correlated (Z)-burst model augments local Pauli errors with adjacent correlated (Z)-type events, while the measurement-flip variant additionally flips syndrome bits with probability proportional to the physical error rate. Supervised data are generated at"},{"type":"para","text":"with logical-failure curves evaluated later on a denser physical-error grid."},{"type":"para","text":"For each code, noise family, and physical error rate, raw syndrome-recovery records are first generated as"},{"type":"para","text":"where s i s_{i} is the syndrome, c i c_{i} is the code identifier, η i \\eta_{i} is the noise-family identifier, p i p_{i} is the physical error probability, and r i r_{i} is the teacher recovery. Because the codes have different numbers of stabilizers and physical qubits, all syndrome vectors and binary symplectic recovery strings are padded to common global dimensions:"},{"type":"para","text":"All raw records are pooled before the final train/test split. Each distinct code-specific padded recovery string is then assigned a global recovery label,"},{"type":"para","text":"Including the code identifier in the label key preserves code-specific recovery semantics while allowing all models to train over one unified supervised label space."},{"type":"para","text":"For non-held-out settings, the pooled dataset is split using a label-aware protocol. Singleton labels are assigned to training, labels with multiple occurrences retain at least one representative in training, and test labels are restricted to labels already represented in the training split. This prevents interpolation test classes from being absent during training. For held-out transfer settings, unseen test labels are allowed when they arise from the transfer design, since those settings intentionally evaluate out-of-distribution generalization."},{"type":"para","text":"The resulting benchmark includes one interpolation setting and four transfer settings: unseen- p p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. In the main unseen- p p experiment, p = 0.05 p=0.05 is withheld during training. In the unseen-noise experiment, the (Z)-biased noise model is held out. In the few-shot unseen-code experiment, Planar5 × \\times 5 is held out from full training and only 5% of its records are included as support. In the held-out-size experiment, code size (5) is held out from full training with 5% support. Table 2 summarizes the resulting split sizes and unseen-label counts. The sample counts reported are the resulting train-test counts obtained after applying the dataset-generation and split protocols, rather than independently tuned hyperparameters."},{"type":"heading","level":2,"text":"3.3 Meta-decoder models and training protocol"},{"type":"para","text":"Each sample is represented by concatenating the padded syndrome vector, a one-hot code identifier, a one-hot noise-family identifier, and the normalized physical error rate. The resulting meta-feature vector is"},{"type":"para","text":"where η \\eta denotes the noise-family identifier and"},{"type":"para","text":"This representation allows one shared decoder to condition its prediction on the syndrome, code family, noise family, and physical error rate."},{"type":"para","text":"The classical baseline is a meta-decoder based on a multilayer perceptron, denoted Meta-MLP. Given input vector x x , the forward map is"},{"type":"para","text":"where ϕ ( ⋅ ) \\phi(\\cdot) is the ReLU activation and z z contains logits over the global recovery-label space. The implementation uses two hidden layers with layer normalization and dropout."},{"type":"para","text":"To account for the greater difficulty of the larger planar code, the training protocol applies Planar5 × \\times 5 oversampling and a sample-weighted cross-entropy loss. The weighted objective is"},{"type":"para","text":"where w i w_{i} increases the contribution of harder training regimes. In the final implementation, Planar5 × \\times 5 samples are oversampled by a factor of 3. The sample weight is factorized into code and noise components: the code weights are 1.0, 1.0, 1.2, and 3.0 for FiveQubit, Steane, Planar3 × \\times 3, and Planar5 × \\times 5, respectively, while the noise weights are 1.0, 1.0, 1.1, and 1.1 for depolarizing, (Z)-biased, correlated (Z)-burst, and depolarizing noise with syndrome-measurement flips, respectively. This weighting makes the supervised objective more sensitive to regimes that are harder at the logical level."},{"type":"para","text":"The quantum counterpart is a hybrid variational quantum classifier. Because the meta-feature vector can exceed the number of available circuit qubits, the input is first projected to the qubit dimension:"},{"type":"para","text":"where u ∈ ℝ n q u\\in\\mathbb{R}^{n_{q}} and n q n_{q} is the qubit count of the candidate VQC architecture. The projected vector u u is embedded through angle embedding into a parameterized quantum circuit. Each circuit contains L L trainable layers, with each layer applying single-qubit R Y R_{Y} and R Z R_{Z} rotations on every qubit followed by an entangling pattern selected from the architecture search space. The quantum feature vector is defined by Pauli-(Z) expectation values,"},{"type":"para","text":"which are passed to a classical linear head:"},{"type":"para","text":"The VQC meta-decoder is trained with the same weighted cross-entropy objective as the Meta-MLP."},{"type":"para","text":"Because recovery labels are code-specific, prediction is restricted using a code-label mask. For a sample from code c c , logits corresponding to labels associated with other codes are suppressed:"},{"type":"para","text":"where 𝒴 ( c ) \\mathcal{Y}(c) is the set of valid labels for code c c . This masking step prevents cross-code recovery-label predictions and ensures that the global label space remains consistent with code-specific recovery semantics."},{"type":"heading","level":2,"text":"3.4 Hardware-aware VQC architecture search"},{"type":"para","text":"The VQC architecture is selected through a compact hardware-aware quantum architecture search rather than fixed manually. The search space varies three architectural factors:"},{"type":"para","text":"qubit count n q ∈ { 4 , 5 , 6 } n_{q}\\in\\{4,5,6\\} ,"},{"type":"para","text":"circuit depth L ∈ { 1 , 2 } L\\in\\{1,2\\} ,"},{"type":"para","text":"entangling pattern ℰ ∈ { chain , ring , full } \\mathcal{E}\\in\\{\\text{chain},\\text{ring},\\text{full}\\} ."},{"type":"para","text":"This defines a discrete search space of 18 candidate architectures."},{"type":"para","text":"The search procedure has two stages. In the search stage, each candidate architecture is trained for 10 epochs and evaluated using validation accuracy. In the retraining stage, the architecture selected by validation accuracy and the architecture selected by hardware-aware score are each retrained from scratch for 35 epochs. This protocol keeps the search computationally manageable while ensuring that the final selected VQC models are evaluated after the same full retraining budget. The search is therefore intended as a compact NISQ-oriented architecture-selection study rather than a large combinatorial search."},{"type":"para","text":"To account for implementation cost, each architecture is also evaluated using hardware-oriented proxies, including raw circuit depth, raw two-qubit gate count, transpiled depth, and transpiled two-qubit gate count. For a candidate architecture a a , the hardware-aware score is defined as"},{"type":"para","text":"where D D denotes circuit depth and G 2 q G_{2q} denotes two-qubit gate count, each measured in raw and transpiled forms. The coefficients λ 1 , λ 2 , λ 3 , λ 4 \\lambda_{1},\\lambda_{2},\\lambda_{3},\\lambda_{4} control the penalty assigned to circuit cost. The best-by-hardware-score VQC is selected according to S ( a ) S(a) , whereas the best-by-accuracy VQC is selected using validation accuracy alone."},{"type":"heading","level":2,"text":"3.5 Confidence-gated recovery and logical evaluation"},{"type":"para","text":"The larger planar code, Planar5 × \\times 5, is especially sensitive to raw learned-decoder errors at the logical level. To reduce this risk, we use a confidence-gated fallback mechanism. Given the masked output distribution of a learned decoder, the prediction confidence is defined as"},{"type":"para","text":"The final recovery rule is"},{"type":"para","text":"where r teacher r_{\\mathrm{teacher}} is the code-appropriate teacher recovery and τ \\tau is a decoder-specific confidence threshold. In this study, fallback is enabled only for Planar5 × \\times 5. The fixed thresholds are τ = 0.80 \\tau=0.80 for the Meta-MLP and τ = 0.75 \\tau=0.75 for the VQC decoders. These thresholds are treated as fixed operating points."},{"type":"para","text":"Logical-level evaluation is performed through Monte Carlo simulation. For each code, noise family, decoder, and physical error rate, we estimate"},{"type":"para","text":"where N run N_{\\mathrm{run}} is the number of sampled error instances and N fail N_{\\mathrm{fail}} is the number of cases in which the corrected operator remains logically nontrivial. To compare learned decoders against the teacher, we use the logical-failure ratio"},{"type":"para","text":"Values near 1 1 indicate teacher-level logical behavior. In the final benchmarking, logical-failure curves are evaluated on the grid p ∈ { 0.01 , 0.02 , … , 0.10 } p\\in\\{0.01,0.02,\\ldots,0.10\\} using 20,000 Monte Carlo trials per point."},{"type":"heading","level":1,"text":"4 Results and Discussion"},{"type":"heading","level":2,"text":"4.1 Experimental setup"},{"type":"para","text":"The full pipeline is implemented in Python using qecsim for code simulation and teacher decoding, PyTorch for the classical Meta-MLP, PennyLane for the VQC meta-decoders, and Qiskit for hardware-aware circuit costing and export. Table 3 summarizes the experimental settings and implementation details used in this study."},{"type":"heading","level":2,"text":"4.2 Teacher-label accuracy and VQC selection"},{"type":"para","text":"We first evaluate the pooled meta-decoding framework at the teacher-label level across five regimes: interpolation, unseen- p p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. The evaluated methods are a majority-label baseline, the classical Meta-MLP, the VQC selected by search-stage validation accuracy, and the VQC selected by the hardware-aware score."},{"type":"para","text":"Table 4 reports the main teacher-label accuracy results. The Meta-MLP is the strongest raw classifier in all five settings. It reaches 0.9993 in interpolation and remains strong under unseen- p p and unseen-noise transfer, with accuracies of 0.9118 and 0.9342, respectively. The largest drops occur in the few-shot adaptation regimes, where the accuracy decreases to 0.6304 for unseen-code adaptation and 0.7548 for held-out-size adaptation. This indicates that shifts in physical error rate or noise family are easier to absorb than changes in code structure or code size."},{"type":"para","text":"The VQC meta-decoders follow the same difficulty pattern but remain below the Meta-MLP in raw teacher-label accuracy. Still, both VQC variants stay well above the majority baseline in every regime, showing that they learn nontrivial syndrome-to-recovery structure. The hardware-aware VQC also remains close to the accuracy-selected VQC after retraining and gives higher accuracy in four of the five settings. The only exception is few-shot unseen-code adaptation, where the accuracy-selected VQC is slightly higher. Thus, the VQC results are best interpreted as compact quantum meta-decoders selected under different criteria, rather than as accuracy winners over the classical Meta-MLP."},{"type":"para","text":"The learning curves support this interpretation. The Meta-MLP converges rapidly and stably on the pooled syndrome-plus-metadata representation, as shown in Fig. 3 . This behavior is consistent with the near-saturated interpolation result in Table 4 . In contrast, the selected VQCs converge more slowly and reach lower final accuracy, as shown in Fig. 4 . However, both VQC variants improve during retraining across all five regimes, including the few-shot transfer settings."},{"type":"para","text":"The VQC architecture-search results show that the candidate circuits are not equivalent. Fig. 5 reports validation and search-stage test accuracy, while Fig. 6 reports the hardware-aware score. Because the candidates differ in qubit count, circuit depth, and entangling topology, the search produces distinct accuracy-cost tradeoffs. The hardware-aware criterion therefore acts as a model-selection rule during VQC search, not only as a post hoc circuit-cost analysis."},{"type":"para","text":"The selected circuit architectures in Fig. 7 and Fig. 8 show the effect of the two selection criteria at the circuit level. Compared with the accuracy-selected circuit, the hardware-aware selection chooses a lower-cost architecture while retaining competitive teacher-label accuracy."},{"type":"heading","level":2,"text":"4.3 Interpolation"},{"type":"para","text":"We begin with the interpolation setting, where all code families, noise models, and supervised physical error rates are represented during training. This setting tests whether the pooled meta-decoder can learn a shared syndrome-to-recovery map in the absence of distributional shift."},{"type":"para","text":"As shown in Table 4 , the Meta-MLP achieves a test accuracy of 0.9993, far above the majority baseline of 0.2032. The VQC models also perform well above the baseline, with accuracies of 0.8987 for the accuracy-selected VQC and 0.9400 for the hardware-aware VQC. In this setting, the hardware-aware VQC outperforms the accuracy-selected VQC after retraining, showing that cost-aware selection does not necessarily reduce teacher-label accuracy within the considered VQC search space."},{"type":"para","text":"The code-wise interpolation results in Table 5 show that the algebraic codes are the easiest cases. FiveQubit is solved by all learned models, and Steane is nearly saturated by the Meta-MLP while remaining high for both VQCs. The planar codes are more difficult for the VQC models. Planar3 × \\times 3 remains manageable, but Planar5 × \\times 5 is the main bottleneck: the Meta-MLP reaches 0.999, whereas the accuracy-selected and hardware-aware VQCs reach 0.692 and 0.830, respectively."},{"type":"para","text":"This code-wise pattern is consistent with the broader groupwise results in Fig. 9 (a). Interpolation accuracy is nearly saturated for the Meta-MLP across most code-noise- p p groups, while the VQC errors are concentrated in the planar-code settings. Thus, interpolation establishes the main in-distribution result: the pooled decoder can learn the shared teacher-label task, but the larger planar code remains the most difficult case for the compact VQC models."},{"type":"para","text":"Although interpolation is almost solved at the teacher-label level by the Meta-MLP, label accuracy alone does not fully determine decoder reliability. We therefore return to logical-failure behavior in Sec. 4.8 , where Planar5 × \\times 5 is analyzed under the confidence-gated fallback protocol."},{"type":"heading","level":2,"text":"4.4 Transfer to unseen physical error rate"},{"type":"para","text":"The unseen- p p setting tests whether the pooled representation generalizes across physical error magnitude rather than memorizing the specific p p -values used in training. As shown in Table 4 , all learned models remain clearly above the majority baseline in this regime. The Meta-MLP achieves 0.9118, while the accuracy-selected and hardware-aware VQCs reach 0.8397 and 0.8495, respectively. Relative to interpolation, the performance drop is noticeable but moderate, suggesting that the learned representation captures structure that persists across nearby physical error rates."},{"type":"para","text":"The code-wise decomposition in Table 6 shows that the degradation is not uniform across codes. FiveQubit and Steane remain near-saturated, and Planar3 × \\times 3 stays close to 0.9 even for the VQCs. The main loss again occurs in Planar5 × \\times 5, where the Meta-MLP reaches 0.844, while the accuracy-selected and hardware-aware VQCs reach 0.644 and 0.670, respectively. This same progression is visible in Fig. 9 (b): unseen- p p transfer remains robust for most groups, but the larger planar-code setting is where generalization degrades most clearly."},{"type":"para","text":"Thus, unseen- p p transfer shows that changing the physical error magnitude is manageable at the teacher-label level, but Planar5 × \\times 5 remains the main bottleneck."},{"type":"heading","level":2,"text":"4.5 Transfer to unseen noise family"},{"type":"para","text":"The unseen-noise setting is a stronger generalization test because it changes the structure of the corruption process rather than only its magnitude. As shown in Table 4 , the pooled framework remains robust under this shift. The Meta-MLP achieves 0.9342, while the accuracy-selected and hardware-aware VQCs reach 0.8384 and 0.8415, respectively. These values are close to the unseen- p p setting, indicating that the syndrome-plus-metadata representation captures structure that transfers across noise families."},{"type":"para","text":"The code-wise results in Table 7 again show that the difficulty is concentrated mainly in Planar5 × \\times 5. FiveQubit and Steane remain near-perfect, and Planar3 × \\times 3 stays strong. Planar5 × \\times 5 is harder, with accuracies of 0.900 for the Meta-MLP and 0.691 and 0.670 for the two VQCs. Fig. 9 (c) shows the same pattern: the algebraic and smaller planar-code settings remain robust, while the larger planar code remains the main transfer bottleneck."},{"type":"para","text":"Thus, unseen-noise transfer remains feasible at the teacher-label level, but the result also confirms that changing the noise family does not remove the structural difficulty of Planar5 × \\times 5."},{"type":"heading","level":2,"text":"4.6 Few-shot adaptation to an unseen code"},{"type":"para","text":"The few-shot unseen-code setting is the most demanding transfer regime because the model must adapt to a held-out code configuration using only a small support fraction. In the present setup, the held-out code is Planar5 × \\times 5 with 5% support. As shown in Table 4 , this setting gives the strongest overall degradation among all experiments. The Meta-MLP reaches 0.6304, while the accuracy-selected and hardware-aware VQCs reach 0.5435 and 0.5678, respectively. Although these values remain above the majority baseline of 0.0994, they are far below the corresponding interpolation, unseen- p p , and unseen-noise results."},{"type":"para","text":"Table 8 clarifies the source of this degradation. The non-held-out codes remain almost saturated, but Planar5 × \\times 5 drops sharply to 0.397 for the Meta-MLP and to 0.270 and 0.308 for the two VQCs. Fig. 9 (d) shows that this drop is highly localized: most configurations remain easy, while the truly unseen code structure is substantially harder. This result shows that the pooled decoder transfers well across known settings, but still faces a clear representation gap when the syndrome-recovery map changes in a structurally new way."},{"type":"para","text":"Thus, few-shot adaptation to an unseen code exposes the clearest limitation of raw teacher-label transfer. The model retains strong performance on known code structures, but direct adaptation to Planar5 × \\times 5 remains weak under limited support data."},{"type":"heading","level":2,"text":"4.7 Few-shot adaptation to held-out code size"},{"type":"para","text":"The held-out-size setting addresses a related but distinct question: whether the meta-decoder can adapt to a new code scale using only a small support set. As shown in Table 4 , this regime is harder than unseen- p p and unseen-noise transfer, but easier than few-shot unseen-code adaptation. The Meta-MLP reaches 0.7548, while the accuracy-selected and hardware-aware VQCs reach 0.6774 and 0.7143, respectively. The hardware-aware VQC again exceeds the accuracy-selected VQC in this setting, suggesting that the hardware-aware criterion can select compact architectures that remain competitive under transfer."},{"type":"para","text":"The code-wise breakdown in Table 9 shows that the held-out-size result combines easy and difficult cases. FiveQubit, Steane, and Planar3 × \\times 3 remain strong, while Planar5 × \\times 5 remains the main bottleneck. In particular, the Meta-MLP reaches 0.406 on Planar5 × \\times 5, while the accuracy-selected and hardware-aware VQCs reach 0.266 and 0.321, respectively. This pattern is also visible in Fig. 9 (e), where the larger planar-code setting remains the main source of difficulty."},{"type":"para","text":"Thus, held-out-size adaptation lies between unseen-noise transfer and unseen-code adaptation in difficulty. The results confirm that structural changes in code scale are harder than shifts in p p or noise family, especially for the larger topological-code setting."},{"type":"heading","level":2,"text":"4.8 Logical-level reliability and selective fallback"},{"type":"para","text":"Teacher-label accuracy is a useful first measure of meta-decoder performance, but it is not equivalent to logical reliability. In QEC decoding, a small number of incorrect recovery labels can still induce logical failures, particularly in larger topological-code settings. We therefore evaluate the learned decoders at the logical level and test whether confidence-gated fallback can reduce the gap between raw learned recovery and teacher-decoder behavior."},{"type":"para","text":"The logical-failure curves in Fig. 10 show the resulting decoder behavior across the five evaluation settings. Planar5 × \\times 5 is the most demanding regime: raw learned decoding remains far from the teacher decoder, while confidence-gated fallback reduces the logical gap by using the learned decoder only on confident cases and routing uncertain cases to the teacher decoder (Additional Planar5 × \\times 5 curves are reported in Appendix A )."},{"type":"para","text":"Table 10 quantifies the raw and fallback-enabled logical behavior on Planar5 × \\times 5. In interpolation, the Meta-MLP reaches nearly perfect teacher-label accuracy, yet its raw logical-failure ratio remains 12.08 relative to the teacher decoder. The raw VQC ratios are even larger, at 42.41 for the accuracy-selected VQC and 25.91 for the hardware-aware VQC. After fallback, these ratios drop to 1.71, 1.04, and 1.11, respectively. The same pattern appears across the transfer settings: raw learned decoding is not a reliable teacher replacement, while fallback brings the learned decoders much closer to teacher-level logical behavior."},{"type":"para","text":"To make the fallback mechanism visible at the operating point used in the experiments, Table 11 reports fixed-threshold learned coverage and fallback rates on Planar5 × \\times 5. These quantities are computed from saved test-set confidence values, not from new simulations. They measure how often the learned decoder is used at the reported thresholds, while Table 10 reports the corresponding Monte Carlo logical behavior."},{"type":"para","text":"The coverage results in Table 11 explain the source of this logical improvement. At τ = 0.75 \\tau=0.75 , the VQC models are conservative on Planar5 × \\times 5 and often route most transfer cases to the teacher decoder. For example, in the few-shot regimes, their learned coverage stays near 0.16, while the confident label accuracy remains much higher than the full label accuracy. This indicates that the confidence signal is useful for selective decoding: the learned decoder is not used broadly, but the cases it retains are substantially more reliable."},{"type":"para","text":"Finally, Fig. 11 summarizes the learned-decoder advantage across code families and noise models. For each code–noise pair, the score is computed as"},{"type":"para","text":"where p L baseline p_{L}^{\\mathrm{baseline}} is the logical-failure rate of the code-specific conventional decoder, p L m p_{L}^{m} is the logical-failure rate of the learned decoder m m , and the average is taken over transfer settings t t and physical error probabilities p p . The baseline is the Naive decoder for FiveQubit and Steane, and MWPM for the planar codes. Positive values indicate lower logical-failure rates for the best learned decoder, while negative values indicate that the conventional decoder remains better. The heatmap shows that learned decoding does not provide a universal advantage; instead, its value depends on the code family and noise model. This supports the intended use of the framework: selective learned assistance with trusted fallback, rather than universal replacement of established decoders."},{"type":"heading","level":2,"text":"4.9 Discussion"},{"type":"para","text":"The results show that pooled meta-decoding is feasible, but its reliability depends on how the decoder is evaluated. At the teacher-label level, the pooled representation is highly learnable: the Meta-MLP nearly solves interpolation and remains robust under unseen- p p and unseen-noise transfer. Transfer becomes harder when the code structure or code size changes, with Planar5 × \\times 5 consistently emerging as the main bottleneck. This suggests that shifts in error rate or noise family are easier to absorb than changes in the underlying syndrome-recovery map."},{"type":"para","text":"The VQC meta-decoders do not surpass the Meta-MLP in raw accuracy, so they should not be interpreted as accuracy winners. Their role is instead to test compact quantum meta-decoders under different selection criteria. The hardware-aware score changes the selected circuit architecture while keeping final accuracy close to the accuracy-selected VQC in most regimes. This supports hardware-aware VQC selection as an architecture-selection tool within the simulated search space. This interpretation is consistent with QAS as a model-selection problem rather than an automatic performance guarantee [ 15 , 29 ] ."},{"type":"para","text":"The key reliability finding is that teacher-label accuracy and logical reliability are not equivalent. In the hardest Planar5 × \\times 5 setting, raw learned decoding can remain far from teacher-level logical behavior even when classification accuracy appears acceptable. Confidence-gated fallback changes this behavior by using the learned decoder only on high-confidence cases and routing uncertain cases to the teacher decoder. The resulting system is therefore a selective learned-assisted decoder, not a full replacement for MWPM or other trusted decoders. This supports a QEC-decoder perspective in which logical-level behavior, rather than supervised prediction accuracy alone, guides decoder assessment [ 12 , 17 ] ."},{"type":"para","text":"These results define the scope of the proposed framework: hardware-aware VQC selection within a compact simulated search space, and reliability-aware learned decoding through confidence-gated fallback."},{"type":"heading","level":1,"text":"5 Conclusion"},{"type":"para","text":"This work presented a selective meta-decoding framework for quantum error correction that combines pooled multi-code learning, hardware-aware VQC architecture search, and confidence-gated hybrid recovery. The framework studies transfer across code, noise, and error-rate regimes instead of training a separate learned decoder for each setting. The results show that pooled teacher-label decoding can transfer across several QEC settings. The Meta-MLP provides the strongest raw accuracy, while the VQC meta-decoders learn transferable structure above the majority baseline and support hardware-aware circuit selection within the simulated search space."},{"type":"para","text":"At the logical level, the main finding is that teacher-label accuracy alone is not sufficient for decoder assessment. In the hardest Planar5 × \\times 5 regime, confidence-gated fallback reduces the gap between raw learned decoding and teacher-level logical behavior by using learned recovery only on high-confidence cases. The resulting method is therefore a selective learned-assisted decoder rather than an unconditional replacement for established decoders. Future work can extend this framework through calibrated confidence thresholds, broader circuit-search spaces, and implementation-level cost measurements."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"MDQEC-QAS: Meta-Decoding for Quantum Error Correction with Hardware-Aware VQC Search and Confidence-Gated Recovery\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prashant Kumar Choudhary 1 , Nouhaila Innan 2,3 , Muhammad Shafique 2,3 and Rajeev Singh 1\"\n },\n {\n \"type\": \"para\",\n \"text\": \"1 Department of Physics, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India 2 eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE 3 Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE\"\n },\n {\n \"type\": \"para\",\n \"text\": \"prashantkchoudhary.rs.phy22@iitbhu.ac.in nouhaila.innan@nyu.edu muhammad.shafique@nyu.edu rajeevs.phy@iitbhu.ac.in\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Abstract. We propose a unified meta-decoding framework for quantum error correction that learns syndrome-to-recovery mappings across multiple stabilizer codes and noise settings, without separate decoders for each configuration. The benchmark includes FiveQubit, Steane, Planar3 × \\\\times 3 and Planar5 × \\\\times 5 codes, four noise families, and five regimes: interpolation, unseen- p p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. We compare a classical Meta-MLP teacher-trained baseline with variational quantum circuit (VQC) meta-decoders selected through hardware-aware quantum architecture search over qubit count, circuit depth, and entangling topology. The Meta-MLP achieves teacher-label accuracies of 0.9993, 0.9118, 0.9342, 0.6304, and 0.7548 across the five regimes, while the hardware-aware VQC achieves 0.9400, 0.8495, 0.8415, 0.5678, and 0.7143. However, logical-level evaluation shows that high teacher-label accuracy alone is insufficient in the hardest Planar5 × \\\\times 5 setting. During interpolation, raw logical-failure ratios relative to the teacher are 12.08 and 25.91 for the Meta-MLP and VQC, respectively, but confidence-gated fallback reduces them to 1.71 and 1.11. These results support confidence-aware selective recovery rather than unconditional teacher replacement.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Keywords: Quantum error correction; meta-decoding; variational quantum circuits; quantum architecture search; stabilizer codes; planar codes; transfer learning; hybrid decoding.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Quantum error correction (QEC) is a central requirement for scalable fault-tolerant quantum computation because fragile quantum states are continuously exposed to decoherence, control imperfections, and measurement noise. Within this landscape, the stabilizer formalism provides a unifying framework for constructing and analyzing a broad class of quantum codes, including both compact algebraic codes and large topological codes [ 34 , 36 , 19 , 26 , 37 ] . In a stabilizer-code setting, noisy physical qubits are probed through stabilizer measurements, producing a classical syndrome that must be translated into a recovery operation. The success of this decoding stage directly determines the resulting logical failure probability and therefore the usefulness of the encoded quantum information [ 14 , 18 , 6 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a measured syndrome s s , decoding can be considered abstractly as the inference problem\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where r r denotes a candidate recovery operator. The operational metric of interest is the logical failure probability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where p p is the physical error probability or, more generally, a parameter controlling the noise strength. In other words, a useful decoder is not just one that predicts labels accurately, but one that preserves logical information after recovery. This distinction becomes particularly important for topological codes, where two decoders with similar classification accuracy can still differ substantially in logical performance [ 12 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traditional decoders such as lookup-table strategies, heuristic code-specific rules, and matching-based methods can perform strongly within narrowly defined settings, but they are usually specialized to a certain code family or noise model [ 13 , 8 ] . This limitation has motivated machine-learning-based decoders, from early neural approaches to more scalable graph- and transformer-style models [ 3 , 32 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These developments establish two key observations. First, learned decoders can be competitive with hand-engineered approaches, particularly when the noise has correlations or when decoder inference must adapt to experimental data [ 2 ] . Second, the most challenging regimes are often larger structured topological codes, where raw classification accuracy alone does not guarantee strong logical performance [ 12 , 17 ] . Thus, modern QEC decoding is no longer only a combinatorial problem; it increasingly also involves representation learning and decoder-system design.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite this progress, most learned QEC decoders are still trained and evaluated within fixed code/noise regimes. However, a robust decoder should ideally adapt across heterogeneous physical conditions: different stabilizer structures, different noise families, and different physical error rates. This motivates a meta-decoding perspective, in which one shared model is trained across multiple decoding tasks and receives explicit side information describing the code family, the noise type, and the physical error level. Rather than learning one isolated decoder per setting, the model learns a transferable syndrome-to-recovery representation [ 20 , 39 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In parallel, variational quantum circuits (VQCs) have emerged as a candidate for hybrid quantum-classical learning systems, but their performance depends strongly on ansatz design [ 21 , 33 ] . This makes quantum architecture search (QAS) a relevant tool for QEC decoding as well: instead of fixing a single parametrized quantum circuit by hand, one may search over qubit count, circuit depth, and entangling topology, and then select architectures using both predictive quality and implementation-oriented cost proxies [ 15 , 29 , 9 , 28 , 25 , 16 ] . In a decoding pipeline, this leads naturally to a hardware-aware model-selection problem rather than a purely accuracy-driven one.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we propose a unified meta-decoding framework for QEC that combines these ideas in a single reproducible pipeline. The framework jointly studies two algebraic codes (FiveQubit and Steane), two planar topological codes (Planar3 × \\\\times 3 and Planar5 × \\\\times 5), four noise families (depolarizing, Z Z -biased, correlated Z Z -burst, and depolarizing with syndrome-measurement flips), and several transfer settings, including interpolation, unseen- p p transfer, unseen-noise transfer, and few-shot adaptation to an unseen code. We compare a classical meta-MLP against VQC meta-decoders selected through hardware-aware QAS.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key observation arising from our experiments is that the larger planar code, Planar5 × \\\\times 5, is substantially more sensitive to raw learned-decoder mistakes at the logical level than the smaller codes. Even rare teacher-label errors can become logically costly, so high supervised accuracy alone is not a sufficient reliability criterion for this regime. To address this, we introduce a confidence-gated hybrid recovery strategy: when the learned decoder is sufficiently confident, its prediction is used directly; otherwise, the pipeline falls back to the teacher decoder. This mechanism reveals a high-precision selective-decoding regime: the learned model can be useful on confident cases even when it is not reliable as a full teacher replacement. The resulting system should therefore be interpreted as a hybrid learned-assisted recovery rather than an independent learned decoding. Confidence-aware and risk-aware mechanisms are standard in broader machine learning and decision systems, but they remain underexplored in QEC decoding pipelines [ 30 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The novelty of this work does not lie in the isolated use of a neural decoder or a variational quantum classifier. Rather, it lies in the integration of four components within a single reproducible QEC pipeline: (i) multi-code, multi-noise meta-decoding across algebraic and planar stabilizer codes, (ii) transfer evaluation under unseen- p p , unseen-noise, few-shot unseen-code, and held-out-size settings, (iii) compact hardware-aware quantum architecture search for VQC decoder design, and (iv) confidence-gated hybrid recovery for the most challenging topological-code regime. This integrated design leads to three system-level findings: confidence can act as a routing signal for selective decoding, structural transfer across code families and code sizes is substantially harder than transfer across noise parameters, and cost-aware VQC selection can remain competitive with accuracy-selected circuits within the compact search space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The main contributions of this paper are summarized as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We construct a pooled multi-code, multi-noise QEC meta-decoding benchmark spanning algebraic and planar stabilizer codes, multiple noise families, and several transfer settings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare a classical Meta-MLP with hardware-aware VQC meta-decoders under a shared global recovery-label framework.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We perform compact hardware-aware QAS over qubit count, circuit depth, and entangling topology, and evaluate the selected VQC architectures after full retraining.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We distinguish teacher-label accuracy from logical-level decoding performance, showing that raw supervised accuracy can be misleading for larger topological codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We identify a transfer hierarchy in which unseen- p p and unseen-noise generalization are easier than unseen-code and held-out-size adaptation, indicating that changes in code structure are more challenging than changes in the noise distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce and evaluate confidence-gated hybrid recovery, showing that confidence can serve as a routing signal between learned predictions and teacher fallback in the larger Planar5 × \\\\times 5 regime.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Background and Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Research on quantum error-correction decoders has evolved from analytical and combinatorial constructions toward increasingly data-driven and learning-based approaches. This section reviews prior work at the intersection of stabilizer-code decoding, learned decoder transferability, variational quantum models, and hardware-aware decoder design.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Classical and learned decoding for stabilizer codes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Quantum decoding has traditionally been studied through code-specific analytical and combinatorial methods. Within the stabilizer formalism, the decoder receives a syndrome and must infer a recovery that returns the state to the codespace without inducing a logical fault [ 19 , 37 ] . For topological and surface-code families, decoding is closely connected to threshold analysis and fault-tolerant architecture design. In particular, the topological-quantum-memory framework and later surface-code studies established minimum-weight matching and related combinatorial approaches as strong baselines for structured local-noise settings [ 14 , 18 , 12 ] . These decoders are often highly effective, but they are typically specialized to a particular code structure and noise assumption [ 4 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This specialization motivates the search for more adaptive decoder families, since modern experimental platforms exhibit changing calibration, asymmetric noise, measurement faults, and correlated errors. As a result, decoder design increasingly requires methods that combine combinatorial optimization, representation learning, and system-level design.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This need for adaptability helped motivate machine-learning-based decoders as data-driven alternatives to explicit code-specific decoding rules. Torlai and Melko provided one of the earliest demonstrations that neural networks can decode topological codes by learning syndrome-to-recovery mappings directly from data [ 38 ] . Shortly afterward, Varsamopoulos et al. showed that feedforward neural networks can decode small surface codes efficiently while accounting for implementation-oriented decoding latency [ 40 ] . These studies established that decoder behavior can be approximated by trainable models rather than relying exclusively on explicit combinatorial procedures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Subsequent work extended learned decoding to more structured regimes. Baireuther et al. studied machine-learning-assisted correction of correlated qubit errors in topological codes, showing that learned approaches can exploit correlations that may be difficult to incorporate into conventional decoders [ 2 ] . Reinforcement-learning-based approaches further explored adaptive decoder behavior and control strategies [ 1 ] . More recently, error-rate-agnostic decoding studies investigated whether neural decoders can remain robust when the physical error rate differs between training and testing, thereby motivating transfer-oriented evaluations in QEC [ 20 , 39 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These prior works established the feasibility of learned decoding, but most remained centered on a single code family, a single lattice class, or a single noise scenario. In contrast, this study examines a unified meta-decoding framework spanning multiple stabilizer codes, multiple noise families, and explicit transfer settings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Structured decoders, transferability, and logical-level reliability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent learned QEC decoders have increasingly adopted structured architectures, including graph neural networks and transformer-style models. This development is motivated by the locality and connectivity patterns of many quantum codes, especially topological codes, which are not always captured efficiently by flat multilayer perceptrons. Lange et al. demonstrated graph-neural-network-based decoding of quantum error-correcting codes, showing that structure-aware learned models can provide strong decoding performance in more complex regimes [ 27 ] . Similarly, recent large-scale work on processor data showed that high-capacity learned decoders can achieve strong decoding quality under experimental noise, highlighting the importance of expressive architectures and training distributions [ 3 , 32 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These developments are closely related to the motivation of this study, but the present focus differs in two ways. First, rather than optimizing a learned decoder for a single code family, this study considers a multi-code, multi-noise setting in which a shared decoder is conditioned on the code family, noise family, and physical error rate. Second, this study examines the gap between classification performance and logical-level performance , especially for the larger planar code. This distinction is important because raw learned decoders may achieve nontrivial classification accuracy while still producing substantial logical degradation unless a hybrid recovery mechanism is used [ 17 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond architectural design, transferability has become a central concern for learned decoding. A major open question in QEC machine learning is whether learned decoders can transfer beyond the regime on which they were trained. Recent error-rate-agnostic and near-term surface-code decoding studies have highlighted this issue by showing that decoders trained under one physical-error regime may generalize imperfectly when evaluated under another [ 20 , 39 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This study extends the transfer perspective by evaluating generalization across interpolation, unseen- p p , unseen-noise, few-shot unseen-code, and held-out-size settings. This combination remains relatively uncommon in prior decoder work, particularly when classical and VQC decoders are evaluated within a unified framework.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A recurring challenge in learned QEC decoding is that strong label-level prediction does not necessarily imply strong logical-level recovery, particularly for larger structured codes. This distinction is central to the present study: raw learned decoders, including both the classical meta-MLP and the VQC models, can degrade on Planar5 × \\\\times 5 even when their supervised accuracy remains nontrivial. This motivates a hybrid recovery formulation in which learned inference is used selectively and low-confidence cases are delegated to a teacher decoder [ 30 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although confidence-based hybridization is common in broader machine-learning systems, it has been less explicitly studied as part of QEC decoder pipelines. In this study, confidence-gated hybrid recovery is used to connect learned transferability with logical reliability, especially in the larger planar-code regime.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Variational quantum models and architecture search\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Variational quantum circuits have become an important model class in quantum machine learning. Quantum-enhanced feature spaces [ 21 ] , circuit-centric quantum classifiers [ 33 ] , data re-uploading architectures [ 31 ] , and quantum convolutional neural networks [ 11 ] demonstrate that trainable quantum circuits can serve as expressive learning models. These models have been studied across a range of applications, including classification, regression, generative modeling, optimization, and domain-specific prediction tasks, often in hybrid quantum-classical pipelines designed for near-term quantum devices [ 24 , 22 , 23 , 10 ] . At the same time, their performance depends strongly on circuit architecture, including qubit count, circuit depth, entangling pattern, parameter initialization, and trainability [ 35 , 7 , 41 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This dependence has motivated quantum architecture search. Du et al. studied quantum circuit architecture search for variational quantum algorithms [ 15 ] , while later work and surveys examined broader directions in automated quantum model design [ 29 , 5 ] . However, most of this literature does not focus specifically on QEC decoding, and relatively little work addresses transfer-oriented multi-code decoder design. This gap motivates hardware-aware QAS for QEC meta-decoding, with circuit architectures evaluated using predictive performance and implementation-oriented cost proxies such as depth and two-qubit gate count.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.4 Positioning within prior QEC decoding studies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our study builds on these prior directions while differing in scope, evaluation design, and decoder-system formulation. Compared with early neural decoders, the proposed framework moves beyond single-code decoding toward a shared meta-decoder trained across multiple code and noise settings. Compared with correlated-noise and error-rate-adaptive learned decoders, it evaluates transfer across both noise distributions and code structures. Compared with recent graph-based and large-scale learned decoders, its focus is not limited to decoder accuracy, but also includes the relationship between supervised prediction quality and logical-level recovery. Finally, relative to general QAS studies, this work applies hardware-aware VQC search specifically to QEC meta-decoding.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Accordingly, our work is positioned as an integrated QEC decoding framework that combines multi-code meta-decoding, transfer evaluation, hardware-aware VQC model selection, and confidence-gated hybrid recovery. Table 1 makes this distinction explicit by contrasting the present framework with representative prior directions in learned QEC decoding.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Methodology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section describes the proposed learned-assisted meta-decoding pipeline, shown in Fig. 1 . The framework consists of six main components: problem formulation, code and noise specification, pooled dataset construction, meta-decoder training, hardware-aware VQC architecture search, and logical-level evaluation with confidence-gated recovery.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Problem formulation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In a stabilizer-code setting, a physical Pauli error e e acting on n n qubits induces a syndrome through the binary symplectic product\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where H H is the binary stabilizer-check matrix of the code [ 19 ] . Given syndrome s s , the decoder must infer a recovery r r such that the corrected operator\"\n },\n {\n \"type\": \"para\",\n \"text\": \"does not induce a logical fault. If e ′ e^{\\\\prime} remains nontrivial at the logical level, decoding fails. The corresponding qubit-level mechanism, from physical Pauli errors and stabilizer syndrome extraction to learned recovery prediction and logical decision, is illustrated in Fig. 2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We formulate this task as meta-decoding. Rather than training a separate decoder for each code and noise condition, a single model is trained on tuples\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where c c is the code identifier, η \\\\eta is the noise-family identifier, p p is the physical error probability, and y y is a discrete recovery-class label. The learned decoder is therefore represented as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where θ \\\\theta denotes trainable parameters and y ^ \\\\hat{y} is mapped back to a recovery operator for logical evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The supervised objective is teacher-label classification: the model is trained to predict recovery labels produced by a code-appropriate teacher decoder. Teacher-label accuracy measures how well the learned model imitates this reference decoder, but it is not identical to the operational QEC objective because physically distinct recovery strings can be logically equivalent. For this reason, the evaluation distinguishes teacher-label classification accuracy from logical-level performance. Teacher-label accuracy measures supervised imitation and transfer, whereas logical-level performance is estimated through logical failure after applying the predicted or hybrid recovery.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Meta-dataset construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The meta-dataset is constructed from four QEC settings: the compact algebraic FiveQubit [ [ 5 , 1 , 3 ] ] [[5,1,3]] stabilizer code, the Steane [ [ 7 , 1 , 3 ] ] [[7,1,3]] code, and two planar topological-code instances, Planar3 × \\\\times 3 and Planar5 × \\\\times 5. Teacher decoders are selected according to code family. This code-dependent choice reflects the structure of the benchmark. FiveQubit and Steane have compact syndrome spaces, so NaiveDecoder provides a consistent reference decoder for supervised label generation. Planar3 × \\\\times 3 and Planar5 × \\\\times 5 are planar topological-code instances, for which matching-based decoding is aligned with the lattice structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To evaluate transfer across heterogeneous physical conditions, samples are generated under four noise families: depolarizing noise, (Z)-biased Pauli noise, correlated (Z)-burst noise, and depolarizing noise with syndrome-measurement flips. For the (Z)-biased model, the single-qubit Pauli distribution is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The coefficients ( 0.2 , 0.1 , 0.7 ) (0.2,0.1,0.7) are not intended as universal hardware-calibrated constants; rather, they define a representative moderately Z Z -biased Pauli channel in which the total non-identity error probability remains p p , while Z Z -type errors dominate. The correlated (Z)-burst model augments local Pauli errors with adjacent correlated (Z)-type events, while the measurement-flip variant additionally flips syndrome bits with probability proportional to the physical error rate. Supervised data are generated at\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with logical-failure curves evaluated later on a denser physical-error grid.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each code, noise family, and physical error rate, raw syndrome-recovery records are first generated as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where s i s_{i} is the syndrome, c i c_{i} is the code identifier, η i \\\\eta_{i} is the noise-family identifier, p i p_{i} is the physical error probability, and r i r_{i} is the teacher recovery. Because the codes have different numbers of stabilizers and physical qubits, all syndrome vectors and binary symplectic recovery strings are padded to common global dimensions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All raw records are pooled before the final train/test split. Each distinct code-specific padded recovery string is then assigned a global recovery label,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Including the code identifier in the label key preserves code-specific recovery semantics while allowing all models to train over one unified supervised label space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For non-held-out settings, the pooled dataset is split using a label-aware protocol. Singleton labels are assigned to training, labels with multiple occurrences retain at least one representative in training, and test labels are restricted to labels already represented in the training split. This prevents interpolation test classes from being absent during training. For held-out transfer settings, unseen test labels are allowed when they arise from the transfer design, since those settings intentionally evaluate out-of-distribution generalization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The resulting benchmark includes one interpolation setting and four transfer settings: unseen- p p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. In the main unseen- p p experiment, p = 0.05 p=0.05 is withheld during training. In the unseen-noise experiment, the (Z)-biased noise model is held out. In the few-shot unseen-code experiment, Planar5 × \\\\times 5 is held out from full training and only 5% of its records are included as support. In the held-out-size experiment, code size (5) is held out from full training with 5% support. Table 2 summarizes the resulting split sizes and unseen-label counts. The sample counts reported are the resulting train-test counts obtained after applying the dataset-generation and split protocols, rather than independently tuned hyperparameters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Meta-decoder models and training protocol\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each sample is represented by concatenating the padded syndrome vector, a one-hot code identifier, a one-hot noise-family identifier, and the normalized physical error rate. The resulting meta-feature vector is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where η \\\\eta denotes the noise-family identifier and\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This representation allows one shared decoder to condition its prediction on the syndrome, code family, noise family, and physical error rate.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The classical baseline is a meta-decoder based on a multilayer perceptron, denoted Meta-MLP. Given input vector x x , the forward map is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ϕ ( ⋅ ) \\\\phi(\\\\cdot) is the ReLU activation and z z contains logits over the global recovery-label space. The implementation uses two hidden layers with layer normalization and dropout.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To account for the greater difficulty of the larger planar code, the training protocol applies Planar5 × \\\\times 5 oversampling and a sample-weighted cross-entropy loss. The weighted objective is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where w i w_{i} increases the contribution of harder training regimes. In the final implementation, Planar5 × \\\\times 5 samples are oversampled by a factor of 3. The sample weight is factorized into code and noise components: the code weights are 1.0, 1.0, 1.2, and 3.0 for FiveQubit, Steane, Planar3 × \\\\times 3, and Planar5 × \\\\times 5, respectively, while the noise weights are 1.0, 1.0, 1.1, and 1.1 for depolarizing, (Z)-biased, correlated (Z)-burst, and depolarizing noise with syndrome-measurement flips, respectively. This weighting makes the supervised objective more sensitive to regimes that are harder at the logical level.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The quantum counterpart is a hybrid variational quantum classifier. Because the meta-feature vector can exceed the number of available circuit qubits, the input is first projected to the qubit dimension:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where u ∈ ℝ n q u\\\\in\\\\mathbb{R}^{n_{q}} and n q n_{q} is the qubit count of the candidate VQC architecture. The projected vector u u is embedded through angle embedding into a parameterized quantum circuit. Each circuit contains L L trainable layers, with each layer applying single-qubit R Y R_{Y} and R Z R_{Z} rotations on every qubit followed by an entangling pattern selected from the architecture search space. The quantum feature vector is defined by Pauli-(Z) expectation values,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which are passed to a classical linear head:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The VQC meta-decoder is trained with the same weighted cross-entropy objective as the Meta-MLP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because recovery labels are code-specific, prediction is restricted using a code-label mask. For a sample from code c c , logits corresponding to labels associated with other codes are suppressed:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝒴 ( c ) \\\\mathcal{Y}(c) is the set of valid labels for code c c . This masking step prevents cross-code recovery-label predictions and ensures that the global label space remains consistent with code-specific recovery semantics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Hardware-aware VQC architecture search\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The VQC architecture is selected through a compact hardware-aware quantum architecture search rather than fixed manually. The search space varies three architectural factors:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"qubit count n q ∈ { 4 , 5 , 6 } n_{q}\\\\in\\\\{4,5,6\\\\} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"circuit depth L ∈ { 1 , 2 } L\\\\in\\\\{1,2\\\\} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"entangling pattern ℰ ∈ { chain , ring , full } \\\\mathcal{E}\\\\in\\\\{\\\\text{chain},\\\\text{ring},\\\\text{full}\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This defines a discrete search space of 18 candidate architectures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The search procedure has two stages. In the search stage, each candidate architecture is trained for 10 epochs and evaluated using validation accuracy. In the retraining stage, the architecture selected by validation accuracy and the architecture selected by hardware-aware score are each retrained from scratch for 35 epochs. This protocol keeps the search computationally manageable while ensuring that the final selected VQC models are evaluated after the same full retraining budget. The search is therefore intended as a compact NISQ-oriented architecture-selection study rather than a large combinatorial search.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To account for implementation cost, each architecture is also evaluated using hardware-oriented proxies, including raw circuit depth, raw two-qubit gate count, transpiled depth, and transpiled two-qubit gate count. For a candidate architecture a a , the hardware-aware score is defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where D D denotes circuit depth and G 2 q G_{2q} denotes two-qubit gate count, each measured in raw and transpiled forms. The coefficients λ 1 , λ 2 , λ 3 , λ 4 \\\\lambda_{1},\\\\lambda_{2},\\\\lambda_{3},\\\\lambda_{4} control the penalty assigned to circuit cost. The best-by-hardware-score VQC is selected according to S ( a ) S(a) , whereas the best-by-accuracy VQC is selected using validation accuracy alone.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5 Confidence-gated recovery and logical evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The larger planar code, Planar5 × \\\\times 5, is especially sensitive to raw learned-decoder errors at the logical level. To reduce this risk, we use a confidence-gated fallback mechanism. Given the masked output distribution of a learned decoder, the prediction confidence is defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The final recovery rule is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where r teacher r_{\\\\mathrm{teacher}} is the code-appropriate teacher recovery and τ \\\\tau is a decoder-specific confidence threshold. In this study, fallback is enabled only for Planar5 × \\\\times 5. The fixed thresholds are τ = 0.80 \\\\tau=0.80 for the Meta-MLP and τ = 0.75 \\\\tau=0.75 for the VQC decoders. These thresholds are treated as fixed operating points.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Logical-level evaluation is performed through Monte Carlo simulation. For each code, noise family, decoder, and physical error rate, we estimate\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where N run N_{\\\\mathrm{run}} is the number of sampled error instances and N fail N_{\\\\mathrm{fail}} is the number of cases in which the corrected operator remains logically nontrivial. To compare learned decoders against the teacher, we use the logical-failure ratio\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Values near 1 1 indicate teacher-level logical behavior. In the final benchmarking, logical-failure curves are evaluated on the grid p ∈ { 0.01 , 0.02 , … , 0.10 } p\\\\in\\\\{0.01,0.02,\\\\ldots,0.10\\\\} using 20,000 Monte Carlo trials per point.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Results and Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Experimental setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The full pipeline is implemented in Python using qecsim for code simulation and teacher decoding, PyTorch for the classical Meta-MLP, PennyLane for the VQC meta-decoders, and Qiskit for hardware-aware circuit costing and export. Table 3 summarizes the experimental settings and implementation details used in this study.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Teacher-label accuracy and VQC selection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first evaluate the pooled meta-decoding framework at the teacher-label level across five regimes: interpolation, unseen- p p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. The evaluated methods are a majority-label baseline, the classical Meta-MLP, the VQC selected by search-stage validation accuracy, and the VQC selected by the hardware-aware score.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 reports the main teacher-label accuracy results. The Meta-MLP is the strongest raw classifier in all five settings. It reaches 0.9993 in interpolation and remains strong under unseen- p p and unseen-noise transfer, with accuracies of 0.9118 and 0.9342, respectively. The largest drops occur in the few-shot adaptation regimes, where the accuracy decreases to 0.6304 for unseen-code adaptation and 0.7548 for held-out-size adaptation. This indicates that shifts in physical error rate or noise family are easier to absorb than changes in code structure or code size.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The VQC meta-decoders follow the same difficulty pattern but remain below the Meta-MLP in raw teacher-label accuracy. Still, both VQC variants stay well above the majority baseline in every regime, showing that they learn nontrivial syndrome-to-recovery structure. The hardware-aware VQC also remains close to the accuracy-selected VQC after retraining and gives higher accuracy in four of the five settings. The only exception is few-shot unseen-code adaptation, where the accuracy-selected VQC is slightly higher. Thus, the VQC results are best interpreted as compact quantum meta-decoders selected under different criteria, rather than as accuracy winners over the classical Meta-MLP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The learning curves support this interpretation. The Meta-MLP converges rapidly and stably on the pooled syndrome-plus-metadata representation, as shown in Fig. 3 . This behavior is consistent with the near-saturated interpolation result in Table 4 . In contrast, the selected VQCs converge more slowly and reach lower final accuracy, as shown in Fig. 4 . However, both VQC variants improve during retraining across all five regimes, including the few-shot transfer settings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The VQC architecture-search results show that the candidate circuits are not equivalent. Fig. 5 reports validation and search-stage test accuracy, while Fig. 6 reports the hardware-aware score. Because the candidates differ in qubit count, circuit depth, and entangling topology, the search produces distinct accuracy-cost tradeoffs. The hardware-aware criterion therefore acts as a model-selection rule during VQC search, not only as a post hoc circuit-cost analysis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selected circuit architectures in Fig. 7 and Fig. 8 show the effect of the two selection criteria at the circuit level. Compared with the accuracy-selected circuit, the hardware-aware selection chooses a lower-cost architecture while retaining competitive teacher-label accuracy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Interpolation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We begin with the interpolation setting, where all code families, noise models, and supervised physical error rates are represented during training. This setting tests whether the pooled meta-decoder can learn a shared syndrome-to-recovery map in the absence of distributional shift.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Table 4 , the Meta-MLP achieves a test accuracy of 0.9993, far above the majority baseline of 0.2032. The VQC models also perform well above the baseline, with accuracies of 0.8987 for the accuracy-selected VQC and 0.9400 for the hardware-aware VQC. In this setting, the hardware-aware VQC outperforms the accuracy-selected VQC after retraining, showing that cost-aware selection does not necessarily reduce teacher-label accuracy within the considered VQC search space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The code-wise interpolation results in Table 5 show that the algebraic codes are the easiest cases. FiveQubit is solved by all learned models, and Steane is nearly saturated by the Meta-MLP while remaining high for both VQCs. The planar codes are more difficult for the VQC models. Planar3 × \\\\times 3 remains manageable, but Planar5 × \\\\times 5 is the main bottleneck: the Meta-MLP reaches 0.999, whereas the accuracy-selected and hardware-aware VQCs reach 0.692 and 0.830, respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This code-wise pattern is consistent with the broader groupwise results in Fig. 9 (a). Interpolation accuracy is nearly saturated for the Meta-MLP across most code-noise- p p groups, while the VQC errors are concentrated in the planar-code settings. Thus, interpolation establishes the main in-distribution result: the pooled decoder can learn the shared teacher-label task, but the larger planar code remains the most difficult case for the compact VQC models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although interpolation is almost solved at the teacher-label level by the Meta-MLP, label accuracy alone does not fully determine decoder reliability. We therefore return to logical-failure behavior in Sec. 4.8 , where Planar5 × \\\\times 5 is analyzed under the confidence-gated fallback protocol.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Transfer to unseen physical error rate\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The unseen- p p setting tests whether the pooled representation generalizes across physical error magnitude rather than memorizing the specific p p -values used in training. As shown in Table 4 , all learned models remain clearly above the majority baseline in this regime. The Meta-MLP achieves 0.9118, while the accuracy-selected and hardware-aware VQCs reach 0.8397 and 0.8495, respectively. Relative to interpolation, the performance drop is noticeable but moderate, suggesting that the learned representation captures structure that persists across nearby physical error rates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The code-wise decomposition in Table 6 shows that the degradation is not uniform across codes. FiveQubit and Steane remain near-saturated, and Planar3 × \\\\times 3 stays close to 0.9 even for the VQCs. The main loss again occurs in Planar5 × \\\\times 5, where the Meta-MLP reaches 0.844, while the accuracy-selected and hardware-aware VQCs reach 0.644 and 0.670, respectively. This same progression is visible in Fig. 9 (b): unseen- p p transfer remains robust for most groups, but the larger planar-code setting is where generalization degrades most clearly.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, unseen- p p transfer shows that changing the physical error magnitude is manageable at the teacher-label level, but Planar5 × \\\\times 5 remains the main bottleneck.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Transfer to unseen noise family\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The unseen-noise setting is a stronger generalization test because it changes the structure of the corruption process rather than only its magnitude. As shown in Table 4 , the pooled framework remains robust under this shift. The Meta-MLP achieves 0.9342, while the accuracy-selected and hardware-aware VQCs reach 0.8384 and 0.8415, respectively. These values are close to the unseen- p p setting, indicating that the syndrome-plus-metadata representation captures structure that transfers across noise families.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The code-wise results in Table 7 again show that the difficulty is concentrated mainly in Planar5 × \\\\times 5. FiveQubit and Steane remain near-perfect, and Planar3 × \\\\times 3 stays strong. Planar5 × \\\\times 5 is harder, with accuracies of 0.900 for the Meta-MLP and 0.691 and 0.670 for the two VQCs. Fig. 9 (c) shows the same pattern: the algebraic and smaller planar-code settings remain robust, while the larger planar code remains the main transfer bottleneck.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, unseen-noise transfer remains feasible at the teacher-label level, but the result also confirms that changing the noise family does not remove the structural difficulty of Planar5 × \\\\times 5.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6 Few-shot adaptation to an unseen code\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The few-shot unseen-code setting is the most demanding transfer regime because the model must adapt to a held-out code configuration using only a small support fraction. In the present setup, the held-out code is Planar5 × \\\\times 5 with 5% support. As shown in Table 4 , this setting gives the strongest overall degradation among all experiments. The Meta-MLP reaches 0.6304, while the accuracy-selected and hardware-aware VQCs reach 0.5435 and 0.5678, respectively. Although these values remain above the majority baseline of 0.0994, they are far below the corresponding interpolation, unseen- p p , and unseen-noise results.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 8 clarifies the source of this degradation. The non-held-out codes remain almost saturated, but Planar5 × \\\\times 5 drops sharply to 0.397 for the Meta-MLP and to 0.270 and 0.308 for the two VQCs. Fig. 9 (d) shows that this drop is highly localized: most configurations remain easy, while the truly unseen code structure is substantially harder. This result shows that the pooled decoder transfers well across known settings, but still faces a clear representation gap when the syndrome-recovery map changes in a structurally new way.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, few-shot adaptation to an unseen code exposes the clearest limitation of raw teacher-label transfer. The model retains strong performance on known code structures, but direct adaptation to Planar5 × \\\\times 5 remains weak under limited support data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.7 Few-shot adaptation to held-out code size\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The held-out-size setting addresses a related but distinct question: whether the meta-decoder can adapt to a new code scale using only a small support set. As shown in Table 4 , this regime is harder than unseen- p p and unseen-noise transfer, but easier than few-shot unseen-code adaptation. The Meta-MLP reaches 0.7548, while the accuracy-selected and hardware-aware VQCs reach 0.6774 and 0.7143, respectively. The hardware-aware VQC again exceeds the accuracy-selected VQC in this setting, suggesting that the hardware-aware criterion can select compact architectures that remain competitive under transfer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The code-wise breakdown in Table 9 shows that the held-out-size result combines easy and difficult cases. FiveQubit, Steane, and Planar3 × \\\\times 3 remain strong, while Planar5 × \\\\times 5 remains the main bottleneck. In particular, the Meta-MLP reaches 0.406 on Planar5 × \\\\times 5, while the accuracy-selected and hardware-aware VQCs reach 0.266 and 0.321, respectively. This pattern is also visible in Fig. 9 (e), where the larger planar-code setting remains the main source of difficulty.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, held-out-size adaptation lies between unseen-noise transfer and unseen-code adaptation in difficulty. The results confirm that structural changes in code scale are harder than shifts in p p or noise family, especially for the larger topological-code setting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.8 Logical-level reliability and selective fallback\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Teacher-label accuracy is a useful first measure of meta-decoder performance, but it is not equivalent to logical reliability. In QEC decoding, a small number of incorrect recovery labels can still induce logical failures, particularly in larger topological-code settings. We therefore evaluate the learned decoders at the logical level and test whether confidence-gated fallback can reduce the gap between raw learned recovery and teacher-decoder behavior.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The logical-failure curves in Fig. 10 show the resulting decoder behavior across the five evaluation settings. Planar5 × \\\\times 5 is the most demanding regime: raw learned decoding remains far from the teacher decoder, while confidence-gated fallback reduces the logical gap by using the learned decoder only on confident cases and routing uncertain cases to the teacher decoder (Additional Planar5 × \\\\times 5 curves are reported in Appendix A ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 10 quantifies the raw and fallback-enabled logical behavior on Planar5 × \\\\times 5. In interpolation, the Meta-MLP reaches nearly perfect teacher-label accuracy, yet its raw logical-failure ratio remains 12.08 relative to the teacher decoder. The raw VQC ratios are even larger, at 42.41 for the accuracy-selected VQC and 25.91 for the hardware-aware VQC. After fallback, these ratios drop to 1.71, 1.04, and 1.11, respectively. The same pattern appears across the transfer settings: raw learned decoding is not a reliable teacher replacement, while fallback brings the learned decoders much closer to teacher-level logical behavior.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To make the fallback mechanism visible at the operating point used in the experiments, Table 11 reports fixed-threshold learned coverage and fallback rates on Planar5 × \\\\times 5. These quantities are computed from saved test-set confidence values, not from new simulations. They measure how often the learned decoder is used at the reported thresholds, while Table 10 reports the corresponding Monte Carlo logical behavior.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The coverage results in Table 11 explain the source of this logical improvement. At τ = 0.75 \\\\tau=0.75 , the VQC models are conservative on Planar5 × \\\\times 5 and often route most transfer cases to the teacher decoder. For example, in the few-shot regimes, their learned coverage stays near 0.16, while the confident label accuracy remains much higher than the full label accuracy. This indicates that the confidence signal is useful for selective decoding: the learned decoder is not used broadly, but the cases it retains are substantially more reliable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, Fig. 11 summarizes the learned-decoder advantage across code families and noise models. For each code–noise pair, the score is computed as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where p L baseline p_{L}^{\\\\mathrm{baseline}} is the logical-failure rate of the code-specific conventional decoder, p L m p_{L}^{m} is the logical-failure rate of the learned decoder m m , and the average is taken over transfer settings t t and physical error probabilities p p . The baseline is the Naive decoder for FiveQubit and Steane, and MWPM for the planar codes. Positive values indicate lower logical-failure rates for the best learned decoder, while negative values indicate that the conventional decoder remains better. The heatmap shows that learned decoding does not provide a universal advantage; instead, its value depends on the code family and noise model. This supports the intended use of the framework: selective learned assistance with trusted fallback, rather than universal replacement of established decoders.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.9 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results show that pooled meta-decoding is feasible, but its reliability depends on how the decoder is evaluated. At the teacher-label level, the pooled representation is highly learnable: the Meta-MLP nearly solves interpolation and remains robust under unseen- p p and unseen-noise transfer. Transfer becomes harder when the code structure or code size changes, with Planar5 × \\\\times 5 consistently emerging as the main bottleneck. This suggests that shifts in error rate or noise family are easier to absorb than changes in the underlying syndrome-recovery map.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The VQC meta-decoders do not surpass the Meta-MLP in raw accuracy, so they should not be interpreted as accuracy winners. Their role is instead to test compact quantum meta-decoders under different selection criteria. The hardware-aware score changes the selected circuit architecture while keeping final accuracy close to the accuracy-selected VQC in most regimes. This supports hardware-aware VQC selection as an architecture-selection tool within the simulated search space. This interpretation is consistent with QAS as a model-selection problem rather than an automatic performance guarantee [ 15 , 29 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The key reliability finding is that teacher-label accuracy and logical reliability are not equivalent. In the hardest Planar5 × \\\\times 5 setting, raw learned decoding can remain far from teacher-level logical behavior even when classification accuracy appears acceptable. Confidence-gated fallback changes this behavior by using the learned decoder only on high-confidence cases and routing uncertain cases to the teacher decoder. The resulting system is therefore a selective learned-assisted decoder, not a full replacement for MWPM or other trusted decoders. This supports a QEC-decoder perspective in which logical-level behavior, rather than supervised prediction accuracy alone, guides decoder assessment [ 12 , 17 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These results define the scope of the proposed framework: hardware-aware VQC selection within a compact simulated search space, and reliability-aware learned decoding through confidence-gated fallback.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This work presented a selective meta-decoding framework for quantum error correction that combines pooled multi-code learning, hardware-aware VQC architecture search, and confidence-gated hybrid recovery. The framework studies transfer across code, noise, and error-rate regimes instead of training a separate learned decoder for each setting. The results show that pooled teacher-label decoding can transfer across several QEC settings. The Meta-MLP provides the strongest raw accuracy, while the VQC meta-decoders learn transferable structure above the majority baseline and support hardware-aware circuit selection within the simulated search space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the logical level, the main finding is that teacher-label accuracy alone is not sufficient for decoder assessment. In the hardest Planar5 × \\\\times 5 regime, confidence-gated fallback reduces the gap between raw learned decoding and teacher-level logical behavior by using learned recovery only on high-confidence cases. The resulting method is therefore a selective learned-assisted decoder rather than an unconditional replacement for established decoders. Future work can extend this framework through calibrated confidence thresholds, broader circuit-search spaces, and implementation-level cost measurements.\"\n }\n]","isConversational":false}}},{"rowIdx":381,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.11962"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Contrastive Joint-Embedding Prediction for Representation Learning in Structural MRI"},"authors":{"kind":"list like","value":["Fabian Mager","Lars Kai Hansen"],"string":"[\n \"Fabian Mager\",\n \"Lars Kai Hansen\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.11962v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":4,"text":"Self-Supervised Learning."},{"type":"para","text":"Self-supervised learning (SSL) is a machine learning paradigm in which the model is trained without the need for labeled data. It leverages the data’s inherent structure to generate labels, making it possible to learn representations without the need for manually labeled datasets. Unlabeled data are widely available, reducing dependencies on extensive and costly data labeling. This is especially relevant in the biomedical domain, as large-scale clinical trials are extremely costly if not infeasible to obtain for some patient groups."},{"type":"heading","level":4,"text":"SSL in medical imaging."},{"type":"para","text":"In computer vision, SSL is capable of learning meaningful representations, which generalize well and can be finetuned for a variety of downstream tasks (Assran et al. , 2023 ; Baevski et al. , 2023 ; Bao et al. , 2021 ; Caron et al. , 2021 ; Oquab et al. , 2023 ; He et al. , 2020 ) . Despite rapid progress in the domain of natural images and video, SSL has not yet achieved a comparable impact in the medical domain, i.e. 3D structural magnetic resonance imaging (MRI). One constraint is the scarcity of large, diverse, and shareable datasets due to acquisition cost, time, privacy, and ethical restrictions. In the case of magnetic resonance imaging, the heterogeneity between scanners, sites, field strengths, and protocols further complicates the pooling of datasets (Dinsdale et al. , 2021 ; Kushol et al. , 2023 ) . Another constraint is the computational burden. Many SSL recipes benefit from large batch sizes, heavy augmentations, and long schedules. For 3D MRI, high spatial resolution and volumetric memory footprints make these setups impractical. Nevertheless, recent works like Tak et al. ( 2026 ) , a SimCLR-like model (Chen et al. , 2020b ) trained on ≈ \\approx 50,000 brain MRIs, show excellent downstream performance on a variety of tasks."},{"type":"heading","level":4,"text":"Brain MRI foundation models."},{"type":"para","text":"Recent brain MRI foundation models span three pretraining paradigms. Volumetric MAE models such as AMAES (Munk et al. , 2024 ) , BrainSegFounder (Cox et al. , 2024 ) , and BM-MAE (Robinet et al. , 2025 ) encode full 3D context but incur reconstruction costs that scale cubically with resolution. Contrastive approaches such as BrainIAC (Tak et al. , 2026 ) and Kaczmarek et al. ( 2025 ) demonstrate strong label efficiency via volumetric SimCLR training on large-scale clinical data. Self-distillation methods such as BrainDINO (Wu et al. , 2026 ) and BrainFound (Mazher et al. , 2025 ) operate on 2D slices to reach otherwise prohibitive training scales, at the cost of volumetric context. The recent FOMO25 challenge (Munk et al. , 2026 ) , benchmarking 19 brain MRI foundation models across clinical tasks, confirms no single paradigm consistently dominates: MAE favors segmentation while contrastive objectives favor classification. To our knowledge, no existing brain MRI foundation model adopts joint-embedding predictive pretraining."},{"type":"heading","level":4,"text":"Contrastive vs. Masked Image Modelling."},{"type":"para","text":"SSL in vision largely revolves around two paradigms: (i) Contrastive Learning (CL) and (ii) Masked Image Modelling (MIM). CL methods such as (Chen et al. , 2020b ; Grill et al. , 2020 ; Caron et al. , 2020 ; He et al. , 2020 ) align the embeddings for augmented views of the same image while separating the embeddings of different images, producing globally discriminative and view-invariant features. In contrast, MIM trains a model to predict the content of masked regions and has proven effective for transformers. Early methods such as (He et al. , 2022b ; Bao et al. , 2021 ; Xie et al. , 2022 ) reconstruct in pixel space , while more recent approaches (e.g., (Assran et al. , 2023 ; Baevski et al. , 2023 ; Caron et al. , 2021 ; Oquab et al. , 2023 ) ) predict targets in token/representation space . This shift is associated with stronger semantics while avoiding unnecessary pixel decoding (Oquab et al. , 2023 ; He et al. , 2022b ) ."},{"type":"heading","level":4,"text":"Design principles: predictivity and specificity."},{"type":"para","text":"For 3D brain structure, we posit that useful representations should be (i) locally predictive and (ii) globally specific . Local predictivity denotes the ability to anticipate distinct anatomical content from context (capturing statistical dependencies of local structure), while global specificity denotes subject-level discriminability and the retention of unique identifiable features. Viewed through this lens, MIM-style objectives naturally promote predictivity, whereas contrastive objectives promote specificity by enforcing global separability among subjects."},{"type":"heading","level":4,"text":"Combining contrastive & predictive objectives."},{"type":"para","text":"Pure CL faces several limitations when applied to 3D MRI. First, CL objectives typically require very large batch sizes to provide sufficient negative examples, which is computationally infeasible for high‑resolution volumetric data. Second, the types of invariances used in natural images do not necessarily translate well to medical images. Joint-embedding prediction (e.g., I-JEPA) instead learns by predicting latent representations of masked regions from a contextual embedding, without pixel-space decoders (Assran et al. , 2023 ) . However, joint-embedding objectives can be susceptible to representation collapse, as the objective fails if the target embedding becomes constant. Yao et al. ( 2024 ) show that MIM benefits from a contrastive objective, learning reconstructions that resemble better global semantics. We make a similar claim for joint-embedding predictive architectures: incorporating a contrastive term injects an explicit discriminative bias that preserves semantic diversity and stabilizes training."},{"type":"heading","level":4,"text":"Contribution."},{"type":"para","text":"We propose a fully self-supervised model of human brain structure trained on T1-weighted MRIs from the Human Connectome Project Young Adult cohort (Van Essen et al. , 2013 ) . Our method pairs a joint-embedding prediction objective with a contrastive loss, explicitly targeting the principles of predictivity and specificity. We show that our model learns efficiently on a relatively small pretraining dataset. We comprehensively assess the learned representation via (i) statistical exploration, (ii) reconstruction and segmentation, (iii) zero-shot retrieval, and (iv) linear probing."},{"type":"heading","level":1,"text":"1 Background"},{"type":"para","text":"We investigate three self-supervised training objectives: (i) Contrastive learning (CO) trains a single encoder on two augmented views of the same subject and minimizes their embedding distance; (ii) a Joint Embedding Predictive Architecture (JEPA) predicts target patch embeddings from a context view, using an EMA-updated target encoder; and (iii) the Contrastive Joint Embedding Predictive Architecture (COJEPA) unifies both: a third augmented view drives the contrastive term on a global token, while the predictive term operates on the spatial tokens of the same context–target pair."},{"type":"heading","level":2,"text":"1.1 Contrastive Learning"},{"type":"para","text":"The contrastive objective encourages view‑invariance by pulling together representations of matching pairs while simultaneously pushing apart all non‑matching samples in the minibatch (Wu et al. , 2018 ; Chen et al. , 2020a ) . Here we use the symmetric Information Noise Contrastive Estimation (InfoNCE) loss function defined by Radford et al. ( 2021 ) ."},{"type":"para","text":"Given a minibatch { ( x i , y i ) } i = 1 N \\{(x_{i},y_{i})\\}_{i=1}^{N} of paired views with representations z x , z y z_{x},z_{y} and pairwise cosine similarities 𝐀 ∈ ℝ N × N \\mathbf{A}\\in\\mathbb{R}^{N\\times N} with elements a i j ∈ [ − 1 , 1 ] a_{ij}\\in[-1,1] , the contrastive loss for x → y x\\!\\to\\!y is"},{"type":"para","text":"The negatives { a i j } \\{a_{ij}\\} where j ≠ i {j\\neq i} correspond to the off‑diagonal entries in row i i of 𝐀 \\mathbf{A} . To obtain the equivalent loss for y → x y\\!\\to\\!x , the indices i i and j j in eq. ˜ 1 are swapped. The final symmetric contrastive objective is given by the average of the two directional losses."},{"type":"heading","level":2,"text":"1.2 Joint Embedding Prediction"},{"type":"para","text":"In a joint‑embedding predictive architecture, the goal is to predict the latent representation of an unobserved target view given the latent representations of a context view and the target view location (Assran et al. , 2023 ) . The construction of the context and target views is crucial: the context view needs to capture sufficient global information, while the target should contain rich semantics that cannot be resolved by simple interpolation. The authors in (Assran et al. , 2023 ) solve this via block masking: For a given image, sample K K target blocks with scale s trgt s^{\\mathrm{trgt}} and aspect ratios r trgt r^{\\mathrm{trgt}} . Then, sample a context block with scale s ctxt s^{\\mathrm{ctxt}} , and remove overlapping regions. We denote patch-wise context and target region masks as { m ctxt } \\{m^{\\mathrm{ctxt}}\\} and { m trgt } k \\{m^{\\mathrm{trgt}}\\}_{k} respectively. Note that in (Assran et al. , 2023 ) , K = 4 K=4 , s trgt ∈ [ 0.15 , 0.2 ] s^{\\mathrm{trgt}}\\!\\in\\![0.15\\,,0.2] , r trgt ∈ [ 0.75 , 1.5 ] r^{\\mathrm{trgt}}\\!\\in\\![0.75\\,,1.5] and s ctxt ∈ [ 0.85 , 1.0 ] s^{\\mathrm{ctxt}}\\!\\in\\![0.85\\,,1.0] . Given a set of target and context patch masks { m trgt } \\{m^{\\mathrm{trgt}}\\} and { m ctxt } \\{m^{\\mathrm{ctxt}}\\} , a context encoder f θ : { x } ⊙ { m ctxt } ↦ { z ctxt } f_{\\theta}:\\!\\{x\\}\\odot\\{m^{\\mathrm{ctxt}}\\}\\!\\mapsto\\!\\{z^{\\mathrm{ctxt}}\\} , target encoder f θ ¯ : { x } ↦ { z } f_{\\bar{\\theta}}:\\!\\{x\\}\\!\\mapsto\\!\\{z\\} and a predictor p ϕ : { z ctxt , m trgt } ↦ { z ^ trgt } p_{\\phi}:\\!\\{z^{\\mathrm{ctxt}},m^{\\mathrm{trgt}}\\}\\!\\mapsto\\!\\{\\hat{z}^{\\mathrm{trgt}}\\} , the average joint embedding prediction loss over the patches ( { z trgt } , { z ^ trgt } ) (\\{z^{\\mathrm{trgt}}\\},\\{\\hat{z}^{\\mathrm{trgt}}\\}) is"},{"type":"para","text":"where D ( ⋅ , ⋅ ) D(\\cdot,\\cdot) denotes a distance function and sg [ ⋅ ] \\mathrm{sg}[\\,\\cdot\\,] a ’stop gradient’ operation. The target encoder weights are updated via an exponential moving average of the context encoder weights."},{"type":"heading","level":1,"text":"2 Contrastive Joint Embedding Prediction"},{"type":"para","text":"We argue that JEPA combined with a contrastive loss promotes both specificity and predictivity of learned embeddings. The proposed COJEPA architecture introduces subtle modifications compared to the JEPA architecture in (Assran et al. , 2023 ) : After retrieving patch embeddings and before the transformer layers, we attach an additional token z glob z^{\\mathrm{glob}} , which then interacts with the spatial context tokens z ctxt z^{\\mathrm{ctxt}} via various multi-head attention layers (Vaswani et al. , 2017 ) , and is then used for the contrastive loss. In addition, we introduce several 3D MRI adaptions to the vision transformer (Dosovitskiy et al. , 2020 ) architecture used in (Assran et al. , 2023 ) . An overview of our pipeline is depicted in fig. ˜ 1 . All hyperparameters are listed in the supplementary material."},{"type":"heading","level":4,"text":"Generating contrastive samples."},{"type":"para","text":"Contrastive learning has been shown to perform best under heavy data augmentations and large batch sizes (Chen et al. , 2020b ) . As the contrastive loss merely serves as an auxiliary training objective, we restrict the augmentations to a simple subset: random cropping and scaling, as well as random Gaussian noise and random intensity rescaling."},{"type":"heading","level":4,"text":"Positional Encoding."},{"type":"para","text":"We take advantage of the fact that the voxels in an MRI exist in a standardized space. Rather than using grid indices of random crops, we feed the model with the global, standardized coordinate of the center voxel. This solves the issue of symmetries in the brain and implicitly encodes the voxel resolution of an input image. Specifically, we assign each patch P ∈ ℝ p × p × p P\\in\\mathbb{R}^{p\\times p\\times p} , where p p is the patch size, a world space position using the image affine. We then encode the 3D coordinate using a sinusoidal positional embedding with the embedding dimension D D split equally across the three spatial axes, resulting in a unique global positional encoding 𝐩 enc ∈ ℝ 1 × D \\mathbf{p}_{\\textrm{enc}}\\in\\mathbb{R}^{1\\times D} ."},{"type":"heading","level":4,"text":"Masking."},{"type":"para","text":"We extend the block masking strategy in (Assran et al. , 2023 ) to a 3D scenario: First, we sample a random scale s s and a random aspect ratio r r within the specified limits, which determines the height h h and width w w of the block mask. We specify the depth by d = ⌊ h ⋅ u + w ⋅ ( 1 − u ) ⌋ d=\\lfloor h\\cdot u+w\\cdot(1-u)\\rfloor , where u ∼ 𝒰 ( 0 , 1 ) u\\sim\\mathcal{U}(0,1) . Following (Assran et al. , 2023 ) , we first sample K K target masks. In order to avoid blocks with little or no information, we prioritize foreground patches in the sampling process. A foreground patch is defined as any patch p p for which ∑ v ∈ p x v > t \\sum_{v\\in p}x_{v}>t , where x v x_{v} is the voxel intensity and t t is a predefined threshold. When sampling target masks, we ensure that the center patch is a foreground patch. In case the randomly sampled mask sizes differ for elements in the batch, we first remove background patches before truncating all masks to the minimum mask size across the batch. fig. ˜ 2 shows five equally spaced slices through a sampled volume, with highlighted context, and target mask regions."},{"type":"heading","level":4,"text":"Patch Embeddings."},{"type":"para","text":"The standard patch embedding layer in Vision Transformers (Dosovitskiy et al. , 2020 ) projects non-overlapping patches directly into the token embedding space via a single linear projection. We instead use a hierarchical convolutional stem consisting of three sequential 3D convolutions with kernel size 3, which progressively expand the channel dimension from the input to the full embedding dimension D D , while their combined strides tile the volume into non-overlapping patches. This multi-stage design allows the network to learn local spatial features before projecting into the token space, rather than treating each patch as a flat vector. Each convolution is followed by batch normalization (Ioffe and Szegedy, 2015 ) and GELU activation. Batch normalization was found to stabilize training on volumetric MRI data."},{"type":"heading","level":4,"text":"Loss."},{"type":"para","text":"The total loss combines a prediction term and an optional contrastive term:"},{"type":"para","text":"ℒ pred \\mathcal{L}_{\\textsc{pred}} is the mean squared error between the predictor output and the target encoder representations. We only calculate ℒ pred \\mathcal{L}_{\\textsc{pred}} over foreground tokens. ℒ contr \\mathcal{L}_{\\textsc{contr}} is an InfoNCE loss (Radford et al. , 2021 ) computed on the cls token of the two random crops drawn per subject, with the temperature learned as a scalar parameter via τ = exp ( log τ ) \\tau=\\exp(\\log\\tau) , initialized at τ = 0.07 \\tau=0.07 . We set λ pred = 1.0 \\lambda_{\\textsc{pred}}=1.0 and λ contr = 0.1 \\lambda_{\\textsc{contr}}=0.1 . An ablation of the convolutional stem, global positional encoding and background loss masking is provided in table ˜ 5 ."},{"type":"heading","level":1,"text":"3 Experiments"},{"type":"heading","level":4,"text":"Pretraining Datasets."},{"type":"para","text":"For pretraining, we used minimally processed structural T1w magnetic resonance imaging data from the Aging Adult Brain Connectome (AABC) (Bookheimer et al. , 2019 ) and Human Connectome Project Young Adult (Van Essen et al. , 2013 ) (HCP-YA). Data have been preprocessed AC–PC orientation alignment, readout distortion correction, cross modal registration, bias-field correction, and brain extraction (Glasser et al. , 2013 ) . We include the cross-sectional data of AABC only ( N = 1390 N{=}1390 , ages 36–90+). The HCP-YA dataset comprises 1113 healthy subjects (ages 22-37). Among the 1113 subjects, 286 are genetically confirmed monozygotic (MZ) twins (143 pairs), including 10 incomplete pairs, resulting in 138 complete MZ twin pairs. In addition, 170 subjects are genetically confirmed duozygotic twins (DZ) (79 pairs), including 12 incomplete pairs, resulting in 79 complete DZ twin pairs. We divide subjects into a training and hold-out set, where one respective twin of both MZ and DZ is assigned to the hold-out set ( N = 217 N{=}217 ). All other subjects are used for training ( N = 2286 N{=}2286 , HCP-YA ∪ \\cup AABC)."},{"type":"heading","level":4,"text":"Preprocessing."},{"type":"para","text":"Raw T1w scans undergo a minimal preprocessing pipeline: The foreground is cropped tightly around the brain using the image intensity. After applying the random augmentations described in section ˜ 2 , each volume is histogram-normalized to [ 0 , 1 ] [0,1] using 128 bins. For each subject, we draw two random crops per iteration, providing positive pairs for the contrastive loss. We use MONAI (Consortium and others, 2022 ) for data loading and augmentation."},{"type":"heading","level":4,"text":"Encoder."},{"type":"para","text":"We use a Vision Transformer Base (ViT-B) (Dosovitskiy et al. , 2020 ) as both the context and target encoder, with an embedding dimension of 768, 12 transformer blocks, 12 attention heads, and an MLP ratio of 4. Volumes are tokenized using a patch size of 9 3 9^{3} voxels via the convolutional patch embedding described in section ˜ 2 . When trained with a contrastive loss, we attach a cls token. In addition, we add a single register token (Darcet et al. , 2023 ) ."},{"type":"heading","level":4,"text":"Predictor."},{"type":"para","text":"A lightweight ViT predictor takes context features together with target positional embeddings and predicts the target representations at masked locations. The predictor has an embedding dimension of 384, 12 transformer blocks, and 12 attention heads, giving it roughly one quarter the parameter count of the encoder."},{"type":"heading","level":4,"text":"Optimization."},{"type":"para","text":"The context and target encoders share the same ViT-Base architecture. The target receives no gradient updates and is updated via an exponential moving average (EMA) of the context encoder weights, with the momentum coefficient annealed from 0.996 to 1.0. We optimize with AdamW, using a cosine learning rate schedule with a linear warmup of 50 epochs, a peak learning rate of 10 − 4 10^{-4} , and a final learning rate of 10 − 5 10^{-5} . The weight decay is similarly annealed from 0.04 to 0.4 following a cosine schedule. The model is trained for 4000 epochs on 8 AMD MI250x GPUs using bfloat16 mixed precision, with a per-GPU batch size of 12 crops × \\times 2 views per GPU."},{"type":"heading","level":2,"text":"3.1 Evaluation"},{"type":"para","text":"To effectively assess the pretraining scheme we compare three variants: CO (contrastive only, InfoNCE on the cls token), JEPA (predictive only), and COJEPA (both objectives combined). Unless stated otherwise, encoder weights are frozen and a lightweight task head is trained on top. For segmentation and age regression we follow a label-efficiency protocol, training the probe at 5, 25, 50, and 100% of available labels, and additionally report end-to-end finetuning at 100% of labels."},{"type":"heading","level":4,"text":"Principal components of feature space."},{"type":"para","text":"We extract patch-level representations from the target encoder and apply a Principal Component analysis (PCA) of foreground tokens. We inspect the energy in the principal components and visualize the first three components by mapping each component to a color channel, similar to (Oquab et al. , 2023 ) ."},{"type":"heading","level":4,"text":"Zero shot retrieval."},{"type":"para","text":"We evaluate whether the representations capture family structure in the dataset by measuring retrieval performance on MZ and DZ twin pairs; only one twin per pair was seen during training. We extract a single global token per subject: the cls token (appended to the sequence and updated via the contrastive loss) for CO and COJEPA, and the avg token (global average over all spatial tokens) for JEPA, which has no cls token. We calculate the cosine similarity between all hold-out subjects and rank them accordingly; the positive match is the co-twin. We report recall at ranks 1, 5, and 10 (R@1, R@5, R@10)."},{"type":"heading","level":4,"text":"Segmentation."},{"type":"para","text":"We train a lightweight decoder on top of frozen encoder features to perform whole-tumor segmentation on BraTS 2024 (de Verdier et al. , 2024 ) , an out-of-distribution dataset with varying scanners, sites, and pathological tissue. The decoder follows an MAE-style design (He et al. , 2022b ) : a linear projection maps encoder tokens from dimension 768 to a decoder embedding dimension of 512, followed by 4 transformer blocks with 4 attention heads. The decoder is trained with a Dice loss on N train = 1135 N_{\\mathrm{train}}{=}1135 subjects and evaluated on N test = 729 N_{\\mathrm{test}}{=}729 subjects, following the label-efficiency protocol described above."},{"type":"heading","level":4,"text":"Age prediction."},{"type":"para","text":"We train a linear probe on top of frozen encoder features to predict chronological age on the OpenBHB dataset (Dufumier et al. , 2022 ) ( N train = 3227 N_{\\mathrm{train}}{=}3227 , N test = 395 N_{\\mathrm{test}}{=}395 ). We restrict evaluation to 3.0 T scans in the external test split ( N = 339 N{=}339 , ages 6 to 48 yrs), matching the field strength of the pretraining data and providing a stringent out-of-distribution assessment across multiple sites. We evaluate three pooling strategies applied to the frozen encoder output: the cls token and the avg token (plus a single linear layer), as well as attn pooling. For the attn pooling, let 𝐗 ∈ ℝ N × D \\mathbf{X}\\in\\mathbb{R}^{N\\times D} be the spatial tokens and 𝐪 ∈ ℝ 1 × D \\mathbf{q}\\in\\mathbb{R}^{1\\times D} a learned query; we compute attention scores α = softmax ( 𝐗𝐪 T / D ) \\alpha=\\mathrm{softmax}(\\mathbf{X}\\mathbf{q}^{T}/\\sqrt{D}) and probe the weighted average 𝐱 ¯ = α T 𝐗 \\bar{\\mathbf{x}}=\\alpha^{T}\\mathbf{X} using a single linear layer."},{"type":"heading","level":1,"text":"4 Results"},{"type":"heading","level":4,"text":"Principal components of feature space."},{"type":"para","text":"fig. ˜ 3 visualizes the first three principal components of foreground patches as RGB channels. Similar areas of the brain are represented by similar colors. The edges of the brain structure as well as the ventricles are clearly depicted by red or pink colors. COJEPA embeddings ( fig. ˜ 3(a) ) show a high dynamic range with clear color contrasts, indicating that anatomical structures are well-separated in the latent space. The cumulative explained variance ( Figure ˜ 3(b) ) confirms that COJEPA uses the latent space most efficiently: its spectrum is the flattest among the three methods, while CO and JEPA concentrate more variance in fewer dimensions."},{"type":"heading","level":4,"text":"Segmentation."},{"type":"para","text":"Table ˜ 1 reports whole-tumor Dice on BraTS 2024 (de Verdier et al. , 2024 ) at four label fractions and after end-to-end finetuning. In the frozen probe regime, CO and COJEPA perform similarly across all label fractions, while JEPA lags notably at 5% labels (Dice 0.43 vs. 0.52 for CO/COJEPA). After finetuning, all three methods converge to 0.70 to 0.71 Dice, demonstrating that the encoder features transfer well regardless of the pretraining objective. However, end-to-end finetuning increases Dice scores the most for JEPA (+0.08), while the benefit for CO and COJEPA is smaller (+0.04). Finetuning results close most of the gap to BrainIAC at 0.79 (Tak et al. , 2026 ) , despite BrainIAC being fine-tuned end-to-end on FLAIR sequences."},{"type":"heading","level":4,"text":"Age prediction."},{"type":"para","text":"Table ˜ 2 reports age MAE on the OpenBHB external test set (3.0 T, N = 339 N{=}339 ). CO achieves the best frozen-probe MAE at 100% labels (3.44 yr using cls ), while COJEPA shows better data-efficiency at 5% labels (4.22 yr using avg vs. 5.23 yr using attn for CO) and the best finetuning MAE (2.55 yr using avg ). COJEPA also shows more consistent performance across all pooling mechanisms compared to CO and JEPA. Figure ˜ 4 shows a heatmap of the patches that contribute to age prediction, trilinearly interpolated to voxel space. The cerebellum appears to be the main source for age prediction. Per-site breakdowns are provided in tables ˜ 6 , 7 and 8 ."},{"type":"heading","level":4,"text":"Retrieval."},{"type":"para","text":"Table ˜ 3 reports zero-shot twin retrieval on the HCP hold-out. Pure JEPA fails at identity-style retrieval (MZ R@1 = 0.26), confirming that the predictive objective alone does not produce subject-discriminative global features. Both CO and COJEPA achieve strong MZ retrieval, with COJEPA reaching the best MZ R@1 (0.84). CO is competitive at higher ranks for DZ pairs. Figure ˜ 5 shows a UMAP (McInnes et al. , 2018 ) plot of the COJEPA cls embeddings; except for a few outliers, twin pairs are mapped closer to each other than unrelated subjects."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"para","text":"The most striking result is the substantial improvement in zero-shot twin retrieval from the contrastive objective. JEPA, trained with a predictive objective alone, already encodes sufficient identity-relevant structure to retrieve MZ twins above chance (R@1 = 0.26), consistent with the well-established heritability of brain morphology (Yao et al. , 2023 ) . However, it largely fails on the more genetically distant DZ pairs (R@1 = 0.07), suggesting that the predictive objective alone does not produce sufficiently discriminative global representations. CO achieves strong MZ retrieval (R@1 = 0.78), and COJEPA further improves upon this (R@1 = 0.84), confirming that enforcing view-invariance within a subject produces global representations that generalize to genetic similarity."},{"type":"para","text":"A secondary finding is that global-average-pooled spatial tokens outperform the cls token for retrieval, despite the contrastive loss being applied exclusively to the cls token, suggesting that the contrastive signal propagates beneficially to the spatial representations through the attention layers."},{"type":"para","text":"The segmentation results show marginal but consistent differences across the three objectives, with the contrastive models performing slightly better on both tasks, indicating that the contrastive objective does not degrade and in fact slightly improves the local predictive quality of the representations. Similarly, age and sex prediction results are largely comparable across both models, with the exception of age prediction using the cls probe, where performance drops when the contrastive loss is added. We speculate that the contrastive objective encourages the cls token to focus on view-invariant subject identity at the expense of more continuous demographic variation such as age. Notably, the attention-pooling age probe performs on par with dedicated brain age models trained on the same cohort (Sarica et al. , 2024 ; Kopetzky et al. , 2024 ; He et al. , 2022a ) , despite our encoder being trained in a fully self-supervised manner without any age-related supervision. This suggests that the learned representations capture morphological variation relevant to brain ageing as an emergent property of the pretraining objective"},{"type":"heading","level":1,"text":"6 Limitations"},{"type":"para","text":"Several limitations of this work should be acknowledged. The pretraining data of N = 2286 N{=}2286 healthy subjects is small by SSL standards, and the consistently higher error on the 1.5T site (site 36) suggests that scanner-domain gaps deserve dedicated study. Larger and more diverse pretraining cohorts would likely yield stronger and more generalizable representations. Additionally, due to the substantial computational cost of pretraining on volumetric MRI, an extensive hyperparameter search was not feasible, and the reported results may therefore not reflect the full potential of the proposed framework."},{"type":"heading","level":1,"text":"7 Conclusion"},{"type":"para","text":"We presented COJEPA, a self-supervised framework for volumetric brain MRI that combines a joint-embedding predictive objective with a contrastive loss. The predictive objective produces spatially structured representations that support out-of-distribution segmentation and age prediction, as well as coarse identity retrieval. Specifically, the CO term adds discriminability needed for fine-grained subject-level retrieval. Trained on 2286 healthy subjects, the model achieves competitive downstream performance. Future work should investigate larger and more diverse pretraining cohorts, as well as the applicability of COJEPA representations to a broader range of clinical downstream tasks."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Self-Supervised Learning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Self-supervised learning (SSL) is a machine learning paradigm in which the model is trained without the need for labeled data. It leverages the data’s inherent structure to generate labels, making it possible to learn representations without the need for manually labeled datasets. Unlabeled data are widely available, reducing dependencies on extensive and costly data labeling. This is especially relevant in the biomedical domain, as large-scale clinical trials are extremely costly if not infeasible to obtain for some patient groups.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"SSL in medical imaging.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In computer vision, SSL is capable of learning meaningful representations, which generalize well and can be finetuned for a variety of downstream tasks (Assran et al. , 2023 ; Baevski et al. , 2023 ; Bao et al. , 2021 ; Caron et al. , 2021 ; Oquab et al. , 2023 ; He et al. , 2020 ) . Despite rapid progress in the domain of natural images and video, SSL has not yet achieved a comparable impact in the medical domain, i.e. 3D structural magnetic resonance imaging (MRI). One constraint is the scarcity of large, diverse, and shareable datasets due to acquisition cost, time, privacy, and ethical restrictions. In the case of magnetic resonance imaging, the heterogeneity between scanners, sites, field strengths, and protocols further complicates the pooling of datasets (Dinsdale et al. , 2021 ; Kushol et al. , 2023 ) . Another constraint is the computational burden. Many SSL recipes benefit from large batch sizes, heavy augmentations, and long schedules. For 3D MRI, high spatial resolution and volumetric memory footprints make these setups impractical. Nevertheless, recent works like Tak et al. ( 2026 ) , a SimCLR-like model (Chen et al. , 2020b ) trained on ≈ \\\\approx 50,000 brain MRIs, show excellent downstream performance on a variety of tasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Brain MRI foundation models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent brain MRI foundation models span three pretraining paradigms. Volumetric MAE models such as AMAES (Munk et al. , 2024 ) , BrainSegFounder (Cox et al. , 2024 ) , and BM-MAE (Robinet et al. , 2025 ) encode full 3D context but incur reconstruction costs that scale cubically with resolution. Contrastive approaches such as BrainIAC (Tak et al. , 2026 ) and Kaczmarek et al. ( 2025 ) demonstrate strong label efficiency via volumetric SimCLR training on large-scale clinical data. Self-distillation methods such as BrainDINO (Wu et al. , 2026 ) and BrainFound (Mazher et al. , 2025 ) operate on 2D slices to reach otherwise prohibitive training scales, at the cost of volumetric context. The recent FOMO25 challenge (Munk et al. , 2026 ) , benchmarking 19 brain MRI foundation models across clinical tasks, confirms no single paradigm consistently dominates: MAE favors segmentation while contrastive objectives favor classification. To our knowledge, no existing brain MRI foundation model adopts joint-embedding predictive pretraining.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contrastive vs. Masked Image Modelling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SSL in vision largely revolves around two paradigms: (i) Contrastive Learning (CL) and (ii) Masked Image Modelling (MIM). CL methods such as (Chen et al. , 2020b ; Grill et al. , 2020 ; Caron et al. , 2020 ; He et al. , 2020 ) align the embeddings for augmented views of the same image while separating the embeddings of different images, producing globally discriminative and view-invariant features. In contrast, MIM trains a model to predict the content of masked regions and has proven effective for transformers. Early methods such as (He et al. , 2022b ; Bao et al. , 2021 ; Xie et al. , 2022 ) reconstruct in pixel space , while more recent approaches (e.g., (Assran et al. , 2023 ; Baevski et al. , 2023 ; Caron et al. , 2021 ; Oquab et al. , 2023 ) ) predict targets in token/representation space . This shift is associated with stronger semantics while avoiding unnecessary pixel decoding (Oquab et al. , 2023 ; He et al. , 2022b ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Design principles: predictivity and specificity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For 3D brain structure, we posit that useful representations should be (i) locally predictive and (ii) globally specific . Local predictivity denotes the ability to anticipate distinct anatomical content from context (capturing statistical dependencies of local structure), while global specificity denotes subject-level discriminability and the retention of unique identifiable features. Viewed through this lens, MIM-style objectives naturally promote predictivity, whereas contrastive objectives promote specificity by enforcing global separability among subjects.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Combining contrastive & predictive objectives.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pure CL faces several limitations when applied to 3D MRI. First, CL objectives typically require very large batch sizes to provide sufficient negative examples, which is computationally infeasible for high‑resolution volumetric data. Second, the types of invariances used in natural images do not necessarily translate well to medical images. Joint-embedding prediction (e.g., I-JEPA) instead learns by predicting latent representations of masked regions from a contextual embedding, without pixel-space decoders (Assran et al. , 2023 ) . However, joint-embedding objectives can be susceptible to representation collapse, as the objective fails if the target embedding becomes constant. Yao et al. ( 2024 ) show that MIM benefits from a contrastive objective, learning reconstructions that resemble better global semantics. We make a similar claim for joint-embedding predictive architectures: incorporating a contrastive term injects an explicit discriminative bias that preserves semantic diversity and stabilizes training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose a fully self-supervised model of human brain structure trained on T1-weighted MRIs from the Human Connectome Project Young Adult cohort (Van Essen et al. , 2013 ) . Our method pairs a joint-embedding prediction objective with a contrastive loss, explicitly targeting the principles of predictivity and specificity. We show that our model learns efficiently on a relatively small pretraining dataset. We comprehensively assess the learned representation via (i) statistical exploration, (ii) reconstruction and segmentation, (iii) zero-shot retrieval, and (iv) linear probing.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Background\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We investigate three self-supervised training objectives: (i) Contrastive learning (CO) trains a single encoder on two augmented views of the same subject and minimizes their embedding distance; (ii) a Joint Embedding Predictive Architecture (JEPA) predicts target patch embeddings from a context view, using an EMA-updated target encoder; and (iii) the Contrastive Joint Embedding Predictive Architecture (COJEPA) unifies both: a third augmented view drives the contrastive term on a global token, while the predictive term operates on the spatial tokens of the same context–target pair.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.1 Contrastive Learning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The contrastive objective encourages view‑invariance by pulling together representations of matching pairs while simultaneously pushing apart all non‑matching samples in the minibatch (Wu et al. , 2018 ; Chen et al. , 2020a ) . Here we use the symmetric Information Noise Contrastive Estimation (InfoNCE) loss function defined by Radford et al. ( 2021 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a minibatch { ( x i , y i ) } i = 1 N \\\\{(x_{i},y_{i})\\\\}_{i=1}^{N} of paired views with representations z x , z y z_{x},z_{y} and pairwise cosine similarities 𝐀 ∈ ℝ N × N \\\\mathbf{A}\\\\in\\\\mathbb{R}^{N\\\\times N} with elements a i j ∈ [ − 1 , 1 ] a_{ij}\\\\in[-1,1] , the contrastive loss for x → y x\\\\!\\\\to\\\\!y is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The negatives { a i j } \\\\{a_{ij}\\\\} where j ≠ i {j\\\\neq i} correspond to the off‑diagonal entries in row i i of 𝐀 \\\\mathbf{A} . To obtain the equivalent loss for y → x y\\\\!\\\\to\\\\!x , the indices i i and j j in eq. ˜ 1 are swapped. The final symmetric contrastive objective is given by the average of the two directional losses.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.2 Joint Embedding Prediction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In a joint‑embedding predictive architecture, the goal is to predict the latent representation of an unobserved target view given the latent representations of a context view and the target view location (Assran et al. , 2023 ) . The construction of the context and target views is crucial: the context view needs to capture sufficient global information, while the target should contain rich semantics that cannot be resolved by simple interpolation. The authors in (Assran et al. , 2023 ) solve this via block masking: For a given image, sample K K target blocks with scale s trgt s^{\\\\mathrm{trgt}} and aspect ratios r trgt r^{\\\\mathrm{trgt}} . Then, sample a context block with scale s ctxt s^{\\\\mathrm{ctxt}} , and remove overlapping regions. We denote patch-wise context and target region masks as { m ctxt } \\\\{m^{\\\\mathrm{ctxt}}\\\\} and { m trgt } k \\\\{m^{\\\\mathrm{trgt}}\\\\}_{k} respectively. Note that in (Assran et al. , 2023 ) , K = 4 K=4 , s trgt ∈ [ 0.15 , 0.2 ] s^{\\\\mathrm{trgt}}\\\\!\\\\in\\\\![0.15\\\\,,0.2] , r trgt ∈ [ 0.75 , 1.5 ] r^{\\\\mathrm{trgt}}\\\\!\\\\in\\\\![0.75\\\\,,1.5] and s ctxt ∈ [ 0.85 , 1.0 ] s^{\\\\mathrm{ctxt}}\\\\!\\\\in\\\\![0.85\\\\,,1.0] . Given a set of target and context patch masks { m trgt } \\\\{m^{\\\\mathrm{trgt}}\\\\} and { m ctxt } \\\\{m^{\\\\mathrm{ctxt}}\\\\} , a context encoder f θ : { x } ⊙ { m ctxt } ↦ { z ctxt } f_{\\\\theta}:\\\\!\\\\{x\\\\}\\\\odot\\\\{m^{\\\\mathrm{ctxt}}\\\\}\\\\!\\\\mapsto\\\\!\\\\{z^{\\\\mathrm{ctxt}}\\\\} , target encoder f θ ¯ : { x } ↦ { z } f_{\\\\bar{\\\\theta}}:\\\\!\\\\{x\\\\}\\\\!\\\\mapsto\\\\!\\\\{z\\\\} and a predictor p ϕ : { z ctxt , m trgt } ↦ { z ^ trgt } p_{\\\\phi}:\\\\!\\\\{z^{\\\\mathrm{ctxt}},m^{\\\\mathrm{trgt}}\\\\}\\\\!\\\\mapsto\\\\!\\\\{\\\\hat{z}^{\\\\mathrm{trgt}}\\\\} , the average joint embedding prediction loss over the patches ( { z trgt } , { z ^ trgt } ) (\\\\{z^{\\\\mathrm{trgt}}\\\\},\\\\{\\\\hat{z}^{\\\\mathrm{trgt}}\\\\}) is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where D ( ⋅ , ⋅ ) D(\\\\cdot,\\\\cdot) denotes a distance function and sg [ ⋅ ] \\\\mathrm{sg}[\\\\,\\\\cdot\\\\,] a ’stop gradient’ operation. The target encoder weights are updated via an exponential moving average of the context encoder weights.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Contrastive Joint Embedding Prediction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We argue that JEPA combined with a contrastive loss promotes both specificity and predictivity of learned embeddings. The proposed COJEPA architecture introduces subtle modifications compared to the JEPA architecture in (Assran et al. , 2023 ) : After retrieving patch embeddings and before the transformer layers, we attach an additional token z glob z^{\\\\mathrm{glob}} , which then interacts with the spatial context tokens z ctxt z^{\\\\mathrm{ctxt}} via various multi-head attention layers (Vaswani et al. , 2017 ) , and is then used for the contrastive loss. In addition, we introduce several 3D MRI adaptions to the vision transformer (Dosovitskiy et al. , 2020 ) architecture used in (Assran et al. , 2023 ) . An overview of our pipeline is depicted in fig. ˜ 1 . All hyperparameters are listed in the supplementary material.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Generating contrastive samples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Contrastive learning has been shown to perform best under heavy data augmentations and large batch sizes (Chen et al. , 2020b ) . As the contrastive loss merely serves as an auxiliary training objective, we restrict the augmentations to a simple subset: random cropping and scaling, as well as random Gaussian noise and random intensity rescaling.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Positional Encoding.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We take advantage of the fact that the voxels in an MRI exist in a standardized space. Rather than using grid indices of random crops, we feed the model with the global, standardized coordinate of the center voxel. This solves the issue of symmetries in the brain and implicitly encodes the voxel resolution of an input image. Specifically, we assign each patch P ∈ ℝ p × p × p P\\\\in\\\\mathbb{R}^{p\\\\times p\\\\times p} , where p p is the patch size, a world space position using the image affine. We then encode the 3D coordinate using a sinusoidal positional embedding with the embedding dimension D D split equally across the three spatial axes, resulting in a unique global positional encoding 𝐩 enc ∈ ℝ 1 × D \\\\mathbf{p}_{\\\\textrm{enc}}\\\\in\\\\mathbb{R}^{1\\\\times D} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Masking.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We extend the block masking strategy in (Assran et al. , 2023 ) to a 3D scenario: First, we sample a random scale s s and a random aspect ratio r r within the specified limits, which determines the height h h and width w w of the block mask. We specify the depth by d = ⌊ h ⋅ u + w ⋅ ( 1 − u ) ⌋ d=\\\\lfloor h\\\\cdot u+w\\\\cdot(1-u)\\\\rfloor , where u ∼ 𝒰 ( 0 , 1 ) u\\\\sim\\\\mathcal{U}(0,1) . Following (Assran et al. , 2023 ) , we first sample K K target masks. In order to avoid blocks with little or no information, we prioritize foreground patches in the sampling process. A foreground patch is defined as any patch p p for which ∑ v ∈ p x v > t \\\\sum_{v\\\\in p}x_{v}>t , where x v x_{v} is the voxel intensity and t t is a predefined threshold. When sampling target masks, we ensure that the center patch is a foreground patch. In case the randomly sampled mask sizes differ for elements in the batch, we first remove background patches before truncating all masks to the minimum mask size across the batch. fig. ˜ 2 shows five equally spaced slices through a sampled volume, with highlighted context, and target mask regions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Patch Embeddings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The standard patch embedding layer in Vision Transformers (Dosovitskiy et al. , 2020 ) projects non-overlapping patches directly into the token embedding space via a single linear projection. We instead use a hierarchical convolutional stem consisting of three sequential 3D convolutions with kernel size 3, which progressively expand the channel dimension from the input to the full embedding dimension D D , while their combined strides tile the volume into non-overlapping patches. This multi-stage design allows the network to learn local spatial features before projecting into the token space, rather than treating each patch as a flat vector. Each convolution is followed by batch normalization (Ioffe and Szegedy, 2015 ) and GELU activation. Batch normalization was found to stabilize training on volumetric MRI data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Loss.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The total loss combines a prediction term and an optional contrastive term:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ℒ pred \\\\mathcal{L}_{\\\\textsc{pred}} is the mean squared error between the predictor output and the target encoder representations. We only calculate ℒ pred \\\\mathcal{L}_{\\\\textsc{pred}} over foreground tokens. ℒ contr \\\\mathcal{L}_{\\\\textsc{contr}} is an InfoNCE loss (Radford et al. , 2021 ) computed on the cls token of the two random crops drawn per subject, with the temperature learned as a scalar parameter via τ = exp ( log τ ) \\\\tau=\\\\exp(\\\\log\\\\tau) , initialized at τ = 0.07 \\\\tau=0.07 . We set λ pred = 1.0 \\\\lambda_{\\\\textsc{pred}}=1.0 and λ contr = 0.1 \\\\lambda_{\\\\textsc{contr}}=0.1 . An ablation of the convolutional stem, global positional encoding and background loss masking is provided in table ˜ 5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Pretraining Datasets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For pretraining, we used minimally processed structural T1w magnetic resonance imaging data from the Aging Adult Brain Connectome (AABC) (Bookheimer et al. , 2019 ) and Human Connectome Project Young Adult (Van Essen et al. , 2013 ) (HCP-YA). Data have been preprocessed AC–PC orientation alignment, readout distortion correction, cross modal registration, bias-field correction, and brain extraction (Glasser et al. , 2013 ) . We include the cross-sectional data of AABC only ( N = 1390 N{=}1390 , ages 36–90+). The HCP-YA dataset comprises 1113 healthy subjects (ages 22-37). Among the 1113 subjects, 286 are genetically confirmed monozygotic (MZ) twins (143 pairs), including 10 incomplete pairs, resulting in 138 complete MZ twin pairs. In addition, 170 subjects are genetically confirmed duozygotic twins (DZ) (79 pairs), including 12 incomplete pairs, resulting in 79 complete DZ twin pairs. We divide subjects into a training and hold-out set, where one respective twin of both MZ and DZ is assigned to the hold-out set ( N = 217 N{=}217 ). All other subjects are used for training ( N = 2286 N{=}2286 , HCP-YA ∪ \\\\cup AABC).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Preprocessing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Raw T1w scans undergo a minimal preprocessing pipeline: The foreground is cropped tightly around the brain using the image intensity. After applying the random augmentations described in section ˜ 2 , each volume is histogram-normalized to [ 0 , 1 ] [0,1] using 128 bins. For each subject, we draw two random crops per iteration, providing positive pairs for the contrastive loss. We use MONAI (Consortium and others, 2022 ) for data loading and augmentation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Encoder.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use a Vision Transformer Base (ViT-B) (Dosovitskiy et al. , 2020 ) as both the context and target encoder, with an embedding dimension of 768, 12 transformer blocks, 12 attention heads, and an MLP ratio of 4. Volumes are tokenized using a patch size of 9 3 9^{3} voxels via the convolutional patch embedding described in section ˜ 2 . When trained with a contrastive loss, we attach a cls token. In addition, we add a single register token (Darcet et al. , 2023 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Predictor.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A lightweight ViT predictor takes context features together with target positional embeddings and predicts the target representations at masked locations. The predictor has an embedding dimension of 384, 12 transformer blocks, and 12 attention heads, giving it roughly one quarter the parameter count of the encoder.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Optimization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The context and target encoders share the same ViT-Base architecture. The target receives no gradient updates and is updated via an exponential moving average (EMA) of the context encoder weights, with the momentum coefficient annealed from 0.996 to 1.0. We optimize with AdamW, using a cosine learning rate schedule with a linear warmup of 50 epochs, a peak learning rate of 10 − 4 10^{-4} , and a final learning rate of 10 − 5 10^{-5} . The weight decay is similarly annealed from 0.04 to 0.4 following a cosine schedule. The model is trained for 4000 epochs on 8 AMD MI250x GPUs using bfloat16 mixed precision, with a per-GPU batch size of 12 crops × \\\\times 2 views per GPU.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To effectively assess the pretraining scheme we compare three variants: CO (contrastive only, InfoNCE on the cls token), JEPA (predictive only), and COJEPA (both objectives combined). Unless stated otherwise, encoder weights are frozen and a lightweight task head is trained on top. For segmentation and age regression we follow a label-efficiency protocol, training the probe at 5, 25, 50, and 100% of available labels, and additionally report end-to-end finetuning at 100% of labels.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Principal components of feature space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We extract patch-level representations from the target encoder and apply a Principal Component analysis (PCA) of foreground tokens. We inspect the energy in the principal components and visualize the first three components by mapping each component to a color channel, similar to (Oquab et al. , 2023 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Zero shot retrieval.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate whether the representations capture family structure in the dataset by measuring retrieval performance on MZ and DZ twin pairs; only one twin per pair was seen during training. We extract a single global token per subject: the cls token (appended to the sequence and updated via the contrastive loss) for CO and COJEPA, and the avg token (global average over all spatial tokens) for JEPA, which has no cls token. We calculate the cosine similarity between all hold-out subjects and rank them accordingly; the positive match is the co-twin. We report recall at ranks 1, 5, and 10 (R@1, R@5, R@10).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Segmentation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train a lightweight decoder on top of frozen encoder features to perform whole-tumor segmentation on BraTS 2024 (de Verdier et al. , 2024 ) , an out-of-distribution dataset with varying scanners, sites, and pathological tissue. The decoder follows an MAE-style design (He et al. , 2022b ) : a linear projection maps encoder tokens from dimension 768 to a decoder embedding dimension of 512, followed by 4 transformer blocks with 4 attention heads. The decoder is trained with a Dice loss on N train = 1135 N_{\\\\mathrm{train}}{=}1135 subjects and evaluated on N test = 729 N_{\\\\mathrm{test}}{=}729 subjects, following the label-efficiency protocol described above.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Age prediction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train a linear probe on top of frozen encoder features to predict chronological age on the OpenBHB dataset (Dufumier et al. , 2022 ) ( N train = 3227 N_{\\\\mathrm{train}}{=}3227 , N test = 395 N_{\\\\mathrm{test}}{=}395 ). We restrict evaluation to 3.0 T scans in the external test split ( N = 339 N{=}339 , ages 6 to 48 yrs), matching the field strength of the pretraining data and providing a stringent out-of-distribution assessment across multiple sites. We evaluate three pooling strategies applied to the frozen encoder output: the cls token and the avg token (plus a single linear layer), as well as attn pooling. For the attn pooling, let 𝐗 ∈ ℝ N × D \\\\mathbf{X}\\\\in\\\\mathbb{R}^{N\\\\times D} be the spatial tokens and 𝐪 ∈ ℝ 1 × D \\\\mathbf{q}\\\\in\\\\mathbb{R}^{1\\\\times D} a learned query; we compute attention scores α = softmax ( 𝐗𝐪 T / D ) \\\\alpha=\\\\mathrm{softmax}(\\\\mathbf{X}\\\\mathbf{q}^{T}/\\\\sqrt{D}) and probe the weighted average 𝐱 ¯ = α T 𝐗 \\\\bar{\\\\mathbf{x}}=\\\\alpha^{T}\\\\mathbf{X} using a single linear layer.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Principal components of feature space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"fig. ˜ 3 visualizes the first three principal components of foreground patches as RGB channels. Similar areas of the brain are represented by similar colors. The edges of the brain structure as well as the ventricles are clearly depicted by red or pink colors. COJEPA embeddings ( fig. ˜ 3(a) ) show a high dynamic range with clear color contrasts, indicating that anatomical structures are well-separated in the latent space. The cumulative explained variance ( Figure ˜ 3(b) ) confirms that COJEPA uses the latent space most efficiently: its spectrum is the flattest among the three methods, while CO and JEPA concentrate more variance in fewer dimensions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Segmentation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 1 reports whole-tumor Dice on BraTS 2024 (de Verdier et al. , 2024 ) at four label fractions and after end-to-end finetuning. In the frozen probe regime, CO and COJEPA perform similarly across all label fractions, while JEPA lags notably at 5% labels (Dice 0.43 vs. 0.52 for CO/COJEPA). After finetuning, all three methods converge to 0.70 to 0.71 Dice, demonstrating that the encoder features transfer well regardless of the pretraining objective. However, end-to-end finetuning increases Dice scores the most for JEPA (+0.08), while the benefit for CO and COJEPA is smaller (+0.04). Finetuning results close most of the gap to BrainIAC at 0.79 (Tak et al. , 2026 ) , despite BrainIAC being fine-tuned end-to-end on FLAIR sequences.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Age prediction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 2 reports age MAE on the OpenBHB external test set (3.0 T, N = 339 N{=}339 ). CO achieves the best frozen-probe MAE at 100% labels (3.44 yr using cls ), while COJEPA shows better data-efficiency at 5% labels (4.22 yr using avg vs. 5.23 yr using attn for CO) and the best finetuning MAE (2.55 yr using avg ). COJEPA also shows more consistent performance across all pooling mechanisms compared to CO and JEPA. Figure ˜ 4 shows a heatmap of the patches that contribute to age prediction, trilinearly interpolated to voxel space. The cerebellum appears to be the main source for age prediction. Per-site breakdowns are provided in tables ˜ 6 , 7 and 8 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Retrieval.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 3 reports zero-shot twin retrieval on the HCP hold-out. Pure JEPA fails at identity-style retrieval (MZ R@1 = 0.26), confirming that the predictive objective alone does not produce subject-discriminative global features. Both CO and COJEPA achieve strong MZ retrieval, with COJEPA reaching the best MZ R@1 (0.84). CO is competitive at higher ranks for DZ pairs. Figure ˜ 5 shows a UMAP (McInnes et al. , 2018 ) plot of the COJEPA cls embeddings; except for a few outliers, twin pairs are mapped closer to each other than unrelated subjects.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The most striking result is the substantial improvement in zero-shot twin retrieval from the contrastive objective. JEPA, trained with a predictive objective alone, already encodes sufficient identity-relevant structure to retrieve MZ twins above chance (R@1 = 0.26), consistent with the well-established heritability of brain morphology (Yao et al. , 2023 ) . However, it largely fails on the more genetically distant DZ pairs (R@1 = 0.07), suggesting that the predictive objective alone does not produce sufficiently discriminative global representations. CO achieves strong MZ retrieval (R@1 = 0.78), and COJEPA further improves upon this (R@1 = 0.84), confirming that enforcing view-invariance within a subject produces global representations that generalize to genetic similarity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A secondary finding is that global-average-pooled spatial tokens outperform the cls token for retrieval, despite the contrastive loss being applied exclusively to the cls token, suggesting that the contrastive signal propagates beneficially to the spatial representations through the attention layers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The segmentation results show marginal but consistent differences across the three objectives, with the contrastive models performing slightly better on both tasks, indicating that the contrastive objective does not degrade and in fact slightly improves the local predictive quality of the representations. Similarly, age and sex prediction results are largely comparable across both models, with the exception of age prediction using the cls probe, where performance drops when the contrastive loss is added. We speculate that the contrastive objective encourages the cls token to focus on view-invariant subject identity at the expense of more continuous demographic variation such as age. Notably, the attention-pooling age probe performs on par with dedicated brain age models trained on the same cohort (Sarica et al. , 2024 ; Kopetzky et al. , 2024 ; He et al. , 2022a ) , despite our encoder being trained in a fully self-supervised manner without any age-related supervision. This suggests that the learned representations capture morphological variation relevant to brain ageing as an emergent property of the pretraining objective\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several limitations of this work should be acknowledged. The pretraining data of N = 2286 N{=}2286 healthy subjects is small by SSL standards, and the consistently higher error on the 1.5T site (site 36) suggests that scanner-domain gaps deserve dedicated study. Larger and more diverse pretraining cohorts would likely yield stronger and more generalizable representations. Additionally, due to the substantial computational cost of pretraining on volumetric MRI, an extensive hyperparameter search was not feasible, and the reported results may therefore not reflect the full potential of the proposed framework.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We presented COJEPA, a self-supervised framework for volumetric brain MRI that combines a joint-embedding predictive objective with a contrastive loss. The predictive objective produces spatially structured representations that support out-of-distribution segmentation and age prediction, as well as coarse identity retrieval. Specifically, the CO term adds discriminability needed for fine-grained subject-level retrieval. Trained on 2286 healthy subjects, the model achieves competitive downstream performance. Future work should investigate larger and more diverse pretraining cohorts, as well as the applicability of COJEPA representations to a broader range of clinical downstream tasks.\"\n }\n]","isConversational":false}}},{"rowIdx":382,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10633"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Auditing Construct Overlap in Explainable Machine Learning: Evidence from Burnout-Depression Prediction Across Student Cohorts"},"authors":{"kind":"list like","value":["Alireza Dehghan","Negin Ashrafi"],"string":"[\n \"Alireza Dehghan\",\n \"Negin Ashrafi\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10633v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Auditing Construct Overlap in Explainable Machine Learning: Evidence from Burnout-Depression Prediction Across Student Cohorts"},{"type":"para","text":"Alireza Dehghan 1 Negin Ashrafi 2,*"},{"type":"para","text":"1 Sharif University of Technology, Tehran, Iran 2 University of Southern California, California, USA * Corresponding author: ashrafin@usc.edu"},{"type":"para","text":"Abstract. Explainable machine learning (XML) pipelines applied to composite mental health outcomes can produce apparently-robust, cross-population-stable risk hierarchies that are largely artefacts of how the outcome was constructed. We demonstrate this using an ElasticNet pipeline applied to 886 medical students at the University of Lausanne (primary cohort, 2022), validated across 2,580 longitudinal observations at three time points and 701 non-medical students from eight faculties; all three datasets share identical instruments."},{"type":"para","text":"The pipeline produces a hierarchy in which trait anxiety and health satisfaction dominate wherever the outcome is measured, with Kendall τ = 1.0 \\tau=1.0 for the top-two positions across all five evaluation sets and consistent transfer performance (R 2 : 0.41–0.49). Two residualization experiments, which isolate shared variance between correlated variables via regression, reveal the mechanism: when trait anxiety (STAI-T) is residualized against the co-included depression subscale (CES-D, r = 0.72 r=0.72 ), model R 2 drops from 0.41 to 0.16 and STAI-T falls from rank 1 to rank 6; when burnout subscales are residualized against CES-D, R 2 collapses to 0.016. Prediction intervals average 35.4 units on a 0–100 scale (2.4 outcome standard deviations), independently ruling out individual-level deployment."},{"type":"para","text":"The residualization protocol is the paper’s transferable contribution: any XAI study combining correlated predictor and outcome constructs should apply this check before interpreting apparent stability as a finding."},{"type":"para","text":"Keywords: construct overlap; explainability artefact; burnout; depression; residualization; XAI; SHAP; conformal prediction; medical students; multi-cohort validation"},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Burnout and depression among medical students are prevalent and consequential: reported burnout rates consistently exceed those of the general population, and untreated distress impairs clinical empathy and patient care quality [ 1 , 12 , 10 ] . The application of machine learning to self-report survey data offers a route to automated risk stratification; paired with SHAP explanations, such pipelines appear to identify stable, cross-population risk hierarchies, a consistent rank-ordering of predictors by importance, in which trait anxiety and health satisfaction dominate [ 4 , 13 ] ."},{"type":"para","text":"The objective of this study is not to identify definitive risk factors, but to test whether the stability produced by such a pipeline is real. Specifically: trait anxiety (STAI-T) and the CES-D depression scale are highly correlated ( r = 0.72 r=0.72 in this sample). The CES-D is embedded inside the composite burnout-depression outcome. When a predictor and an outcome component share a latent construct, the predictor’s explanatory dominance may reflect psychometric overlap rather than a genuine risk relationship. We call this construct overlap, and we introduce a residualization protocol to quantify how much of the apparent hierarchy survives once the overlap is removed."},{"type":"para","text":"The principal contributions are threefold. First, we validate a residualization protocol for detecting construct overlap artefacts : removing the anxiety–depression psychometric channel collapses R 2 from 0.41 to 0.016 (burnout-specific outcome) and STAI-T from rank 1 to rank 6. Second, we demonstrate that apparent cross-cohort stability is not evidence of a portable risk structure when the signal is anchored to a fixed psychometric correlation: the model transfers with R 2 = 0.41–0.49 across five evaluation sets because the STAI-T/CES-D bivariate relationship is constant, not because the pipeline learned a generalisable risk structure. Third, we provide deployment-relevant uncertainty quantification : conformal prediction intervals average 35.4 BDCI units (2.4 outcome standard deviations), establishing that individual-level clinical prediction is not actionable with current feature sets."},{"type":"para","text":"The remainder of this paper is organised as follows. Section 2 briefly reviews related work. Section 3 describes the methodology. Section 4 presents results in four subsections structured around the central audit. Section 5 discusses implications. Section 6 concludes."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"para","text":"A growing body of work applies machine learning and XAI to mental health outcomes in student populations. Martínez-Ramón et al. [ 9 ] achieved high classification accuracy for teacher burnout using ANN. Bozorgmehr and Weltermann [ 4 ] applied XGBoost with SHAP to a large German adult sample and identified health satisfaction as a dominant protective factor, consistent with our findings on a different instrument. Rashid et al. [ 13 ] applied ensemble and deep learning to the same 886-student Lausanne cohort as this paper, achieving moderate classification accuracy. Benito et al. [ 3 ] and Nnadi et al. [ 11 ] combined SHAP with clustering and hierarchical explanation frameworks. In all cases, the feature hierarchy is treated as a finding; no prior work has tested whether the hierarchy survives a residualization check for construct overlap."},{"type":"para","text":"Table 1 summarises reviewed studies. The methodological gap is clear: regression-based continuous framing and construct-overlap auditing are both absent from the existing literature."},{"type":"heading","level":1,"text":"3 Methodology"},{"type":"heading","level":2,"text":"3.1 Data and Cohorts"},{"type":"para","text":"Table 2 describes the three datasets. The primary cohort ( n = 886 n=886 medical students, University of Lausanne, 2022) was collected by the Faculty of Biology and Medicine [ 7 ] . It contains no missing values. Predictors include ten demographic and psychological survey items (age, academic year, health satisfaction, psychotherapy history, partnership status, employment status, weekly study hours, mother tongue, sex, and trait anxiety from the State-Trait Anxiety Inventory; STAI-T). Outcome components are the three Maslach Burnout Inventory–Student Survey (MBI-SS) subscales and the Center for Epidemiological Studies Depression Scale (CES-D)."},{"type":"para","text":"The temporal and non-medical cohorts are openly available under CC-BY 4.0 (Zenodo: 10.5281/zenodo.8405764 and 10.5281/zenodo.15149289 respectively)."},{"type":"heading","level":2,"text":"3.2 BDCI Construction"},{"type":"para","text":"The Burnout–Depression Composite Index (BDCI) integrates the three MBI-SS subscales and CES-D into a single severity score on [ 0 , 100 ] [0,100] . Each subscale is normalised to its theoretical maximum (reversing the Academic Efficacy subscale); any component exceeding its empirical 75th-percentile threshold (derived from the 2022 training split) is doubled. The raw sum is then scaled by 100 / ( 4 + k ) 100/(4+k) , where k k is the count of elevated components. The result is a continuous index where higher values indicate greater joint burnout-depression severity. Formally:"},{"type":"para","text":"The 75th-percentile thresholding and doubling scheme is intended to give greater analytic weight to clinically elevated symptom levels, a severity-weighting logic common in composite mental-health indices. We treat this as one reasonable analytic choice rather than a validated clinical rule: the threshold is sample-derived (2022 training split) rather than tied to an external clinical cutoff (e.g., CES-D ≥ \\geq 16), which is a limitation we return to in Section 5.1. Sensitivity to the weighting choice itself is assessed directly via BDCI-A (Table 5), an equal-weight variant that reproduces the same qualitative pattern, indicating the central finding does not depend on this specific scheme. The BDCI is a study-specific analytical composite; it has not been independently validated as a clinical instrument."},{"type":"para","text":"The critical structural feature of the BDCI is that CES-D is one of its four components, while STAI-T (trait anxiety) is one of the predictors. Figure 1 shows the Pearson correlation matrix: STAI-T and CES-D share r = 0.72 r=0.72 ( p < 0.001 p<0.001 ), the strongest pairwise correlation in the dataset. This overlap motivates the residualization audit in Section 4.3 ."},{"type":"heading","level":2,"text":"3.3 Analysis Pipeline"},{"type":"para","text":"The dataset was partitioned 80/20 into training ( n train = 708 n_{\\mathrm{train}}=708 ) and test ( n test = 178 n_{\\mathrm{test}}=178 ) sets using stratified random sampling on BDCI quartile (random seed 42 throughout). All predictors were z z -score normalised using training-split statistics only. ElasticNet (L1+L2 regularisation) was selected as the primary model after 5-fold cross-validated comparison of six candidate regressors (OLS, Ridge, Lasso, ElasticNet, SVR, ANN); hyperparameters were selected by grid search ( α ∈ [ 10 − 4 , 10 1 ] \\alpha\\in[10^{-4},10^{1}] , mixing ratio ρ ∈ { 0.1 , 0.3 , 0.5 , 0.7 , 0.9 } \\rho\\in\\{0.1,0.3,0.5,0.7,0.9\\} ; optimal: α = 0.0316 \\alpha=0.0316 , ρ = 0.7 \\rho=0.7 ). SHAP values were computed using LinearExplainer [ 14 ] ."},{"type":"para","text":"For cross-cohort validation, a 7-feature reduced ElasticNet was retrained on the 2022 training split using only the predictors available in all three cohorts (age, academic year, health satisfaction, psychotherapy history, partnership status, STAI-T, sex), then applied without retraining to the temporal and non-medical cohorts. The 2022 BDCI thresholds τ j \\tau_{j} were used throughout to prevent leakage."},{"type":"para","text":"Ninety-five percent bootstrap confidence intervals (1,000 resamples) were computed for all test-set metrics."},{"type":"heading","level":2,"text":"3.4 Residualization Protocol"},{"type":"para","text":"Two orthogonality experiments were conducted to isolate the anxiety–depression channel. Experiment 1 (predictor residualization) : STAI-T was replaced by the residuals of regressing STAI-T on CES-D (fit on training set, applied to test set). The residual captures trait anxiety variance orthogonal to concurrent depression. Experiment 2 (outcome residualization) : each MBI-SS subscale was individually residualized against CES-D (OLS, training set), then re-normalised and assembled into a depression-free composite (BDCI-C). In both cases, ElasticNet was refit from scratch on the modified data using the same hyperparameter grid and split. We note that BDCI-C is a statistical construct created solely to isolate depression-independent variance in the burnout subscales; residualizing against CES-D does not by itself produce a validated ’pure burnout’ measure. A near-zero R² for BDCI-C indicates that this feature set cannot predict the depression-independent portion of the composite, not that a clinically meaningful burnout-specific construct doesn’t exist. Results are compared with the primary BDCI and a CES-D-excluded composite (BDCI-B, MBI-SS only) in Table 5 ."},{"type":"heading","level":2,"text":"3.5 Conformal Prediction"},{"type":"para","text":"Point estimates are complemented by split conformal prediction intervals [ 15 ] . The test set is split 50/50: the calibration half ( n cal = 89 n_{\\mathrm{cal}}=89 ) is used to compute non-conformity scores s i = | y i − y ^ i | s_{i}=|y_{i}-\\hat{y}_{i}| ; the evaluation half ( n eval = 89 n_{\\mathrm{eval}}=89 ) receives intervals [ y ^ j ± q ^ ] [\\hat{y}_{j}\\pm\\hat{q}] , where q ^ \\hat{q} is the ( 1 − α ) ( 1 + 1 / n cal ) (1-\\alpha)(1+1/n_{\\mathrm{cal}}) quantile at α = 0.10 \\alpha=0.10 . This procedure provides marginal coverage ≥ 90 % \\geq 90\\% without distributional assumptions, with the caveat that small calibration sets may produce slight deviations from the nominal level."},{"type":"heading","level":1,"text":"4 Results"},{"type":"heading","level":2,"text":"4.1 A Standard Pipeline Produces an Apparent Hierarchy"},{"type":"para","text":"Table 3 presents test-set performance. All four linear and regularised models achieve similar accuracy (R 2 : 0.400–0.414); SVR performs somewhat worse. Penalisation offers no meaningful gain over OLS: the BDCI–predictor relationship is approximately linear and additive. A Lasso fit with L1 penalty automatically zeros five of twelve features, retaining a sparse set of seven predictors."},{"type":"para","text":"Figure 2 shows the ElasticNet SHAP beeswarm. Trait anxiety produces the widest spread of SHAP values by a wide margin: its mean | SHAP | |\\text{SHAP}| exceeds health satisfaction (ranked second) by approximately 10-fold. This apparent hierarchy is consistent across the cross-population transfer cohorts (Section 4.2 ): STAI-T ranks first in every wave and in the non-medical student sample."},{"type":"heading","level":2,"text":"4.2 The Hierarchy Appears Stable Across Cohorts"},{"type":"para","text":"Table 4 shows transfer performance. The 7-feature reduced ElasticNet, trained exclusively on 2022 data, achieves R 2 of 0.443, 0.487, and 0.388 on the three longitudinal medical waves, and 0.493 on 701 non-medical students from eight faculties, the highest of any evaluation set. Figure 3 shows SHAP feature importance across all five cohorts. Trait anxiety holds rank 1 and health satisfaction holds rank 2 in every cohort; the full rank ordering is Trait Anxiety > > Health Satisfaction > > Age > > Psychotherapy > > Sex:Female > > Partnership > > Academic Year in all five evaluation sets. Kendall τ = 1.0 \\tau=1.0 for the top-two positions across all pairwise cohort comparisons. The apparent stability is striking."},{"type":"para","text":"Section 4.3 tests whether this stability reflects a portable risk structure or a psychometric artefact."},{"type":"heading","level":2,"text":"4.3 Residualization Audit: What Survives When the Depression Channel Is Removed?"},{"type":"para","text":"Table 5 is the central result. Two baseline comparisons (primary BDCI and BDCI-A, an equal-weight variant) confirm that STAI-T dominates at rank 1 regardless of the weighting scheme. When CES-D is simply excluded from the outcome (BDCI-B), STAI-T remains rank 1 and R 2 falls from 0.414 to 0.303, a partial drop, because the MBI-SS burnout subscales themselves correlate with depression."},{"type":"para","text":"The two residualization experiments go further. When STAI-T is replaced by residuals orthogonal to CES-D (Experiment 1), R 2 drops from 0.414 to 0.156 and STAI-T falls from rank 1 to rank 6 in mean absolute SHAP magnitude; health satisfaction becomes the dominant predictor. When the burnout outcome itself is purged of CES-D variance (Experiment 2, BDCI-C), R 2 collapses to 0.016: the full 12-feature model explains essentially none of the burnout-specific variance."},{"type":"para","text":"Figure 4 summarises the residualization audit by plotting ElasticNet R 2 under each outcome/predictor definition. The collapse from the primary BDCI to STAI-T residualization and BDCI-C shows how much of the apparent signal depends on the anxiety–depression channel."},{"type":"para","text":"These experiments answer the central question precisely. The standard pipeline predicts the depression-saturated portion of the composite outcome well and the burnout-specific portion essentially not at all. The cross-cohort stability in Section 4.2 transfers because the STAI-T/CES-D psychometric correlation ( r = 0.72 r=0.72 ) is structurally constant wherever these instruments are used together, not because the pipeline learned a portable multivariate risk structure."},{"type":"para","text":"Health satisfaction is the most robust predictor of the depression-orthogonal signal (rank 1 in Experiment 1) and is the feature most likely to represent an independent burnout pathway. Researchers who report cross-cohort XAI stability without a residualization audit may be describing instrument correlation, not replicated science."},{"type":"heading","level":2,"text":"4.4 Uncertainty Rules Out Individual Deployment"},{"type":"para","text":"The non-conformity quantile from the calibration half was q ^ = 17.7 \\hat{q}=17.7 BDCI units. On the evaluation half ( n eval = 89 n_{\\mathrm{eval}}=89 ), empirical coverage was 84.3% (nominal target: 90%), and average interval width was 35.4 BDCI units (Figure 5 ). The − 5.7 -5.7 percentage-point coverage gap is within sampling noise: the standard deviation of a coverage estimate at true 90% with n eval = 89 n_{\\mathrm{eval}}=89 is ≈ 3.2 \\approx 3.2 pp, so the gap is approximately 1.8 standard deviations from nominal ( p ≈ 0.07 p\\approx 0.07 ). The headline finding is the interval width, not the coverage deviation."},{"type":"para","text":"An interval of 35.4 units on a 0–100 scale is 2.4 outcome standard deviations ( S D = 14.7 SD=14.7 ). This width is fully determined by the model’s residual distribution; any uncertainty quantification method would produce comparable intervals. It independently establishes that individual-level clinical prediction is not actionable with this feature set, regardless of the construct overlap finding."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"para","text":"This study’s central finding is the R 2 collapse: from 0.414 (primary BDCI) to 0.016 (burnout-specific BDCI-C), with an intermediate collapse to 0.156 when only the predictor is residualized. The implication is not that the primary model is wrong: the anxiety–depression nexus is one of the best-replicated constructs in psychiatric epidemiology, and the STAI-T/CES-D association ( r = 0.72 r=0.72 ) reflects genuine comorbidity [ 2 ] . The implication is that a standard multi-method XAI pipeline cannot distinguish this real clinical signal from a psychometric artefact without an explicit orthogonality check. Apparent stability across five cohorts is not evidence of a portable risk structure when the dominant predictor and the dominant outcome component share a latent construct."},{"type":"para","text":"The residualization protocol introduced here is the paper’s transferable contribution. Any XAI study combining correlated predictor and outcome constructs (trait anxiety and depression are a prominent example, but any construct pair with r > 0.5 r>0.5 –0.6 warrants scrutiny) should apply this check before interpreting feature hierarchies as findings. The protocol requires only the fitted model and a linear regression of the predictor on the confounded outcome component; it adds no computational burden and produces a clear, quantitative audit."},{"type":"para","text":"The apparently strong cross-cohort transfer (R 2 : 0.41–0.49 across five evaluation sets) deserves an honest characterisation. Wave 2 (0.487) and the non-medical cohort (0.493) exceed the internal test R 2 (0.423), but this is consistent with a stable two-variable relationship rather than broad generalisation: all three cohorts originate from the same institution and use identical instruments. Cross-institutional validation with heterogeneous instruments (instruments that break the STAI-T/CES-D fixed correlation) remains the essential next step before any deployment."},{"type":"para","text":"The conformal width (35.4 BDCI units, 2.4 SDs) establishes a separate and independent bar for deployment. Even if the construct overlap were resolved by a better instrument design, individual-level prediction would require a substantially expanded feature set (likely ecological momentary assessment, physiological data, or longitudinal trajectories) to achieve actionable precision. The current paper quantifies the upper bound of what cross-sectional survey data can deliver under this instrument set."},{"type":"para","text":"Health satisfaction emerges from the residualization experiments as the most durable predictor: after removing the STAI-T/CES-D channel, it ranks first in mean absolute SHAP magnitude. This aligns with Bozorgmehr and Weltermann [ 4 ] and Khan et al. [ 8 ] , who identified health satisfaction and self-rated health as primary determinants in unrelated datasets and populations, suggesting that perceived health quality may be a more portable burnout signal than trait anxiety. Testing this hypothesis requires a study that separates the measurement constructs from the outset, not a post-hoc residualization."},{"type":"heading","level":2,"text":"5.1 Limitations"},{"type":"para","text":"All datasets originate from the University of Lausanne; cross-institutional generalisability has not been tested. The primary analysis is cross-sectional; longitudinal within-student prediction would require a prospective design. The BDCI is a study-internal composite that has not been independently psychometrically validated. Potentially important predictors (financial stress, sleep quality, social support, institutional climate) are absent from the dataset and likely account for a substantial share of the unexplained variance. The clinical-year calibration gap (R 2 = 0.294, n = 66 n=66 , versus 0.456 for pre-clinical) is directionally informative but statistically fragile at this subgroup size."},{"type":"heading","level":2,"text":"5.2 Future Directions"},{"type":"para","text":"The natural Phase II is a multi-site prospective study that: (a) recruits at least three institutions with structurally different curricula to obtain genuine cross-institutional variation; (b) uses a prospective design in which pre-year surveys predict end-of-year burnout episodes or counselling referrals; (c) pre-registers the residualization audit as a primary endpoint to confirm that the collapse is not an artefact of this particular cohort; and (d) extends the clinical-year feature set with ecological momentary assessment during clinical rotations to test the environmental hypothesis."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"A standard XAI pipeline applied to a burnout-depression composite produces an apparently-robust, cross-population-stable risk hierarchy. Two residualization experiments show that this hierarchy is largely an artefact of construct overlap: removing the depression channel from the predictor (STAI-T ⟂ \\perp CES-D) drops R 2 from 0.41 to 0.16; removing it from the outcome (MBI-SS subscales ⟂ \\perp CES-D) drops R 2 to 0.016. The apparent cross-cohort stability (R 2 = 0.41–0.49, Kendall τ = 1.0 \\tau=1.0 for top-two positions across five cohorts) is explained by the fixed STAI-T/CES-D psychometric correlation ( r = 0.72 r=0.72 ), not by a learned portable risk structure. Conformal prediction intervals averaging 35.4 BDCI units (2.4 outcome SDs) independently establish that individual-level prediction is not actionable."},{"type":"para","text":"The residualization protocol is directly transferable: any XAI study that combines correlated predictor and outcome constructs should apply this check before reporting apparent stability as a finding. Researchers who skip this step may be describing instrument correlation, not replicated science."},{"type":"heading","level":1,"text":"Data and Code Availability"},{"type":"para","text":"The primary 2022 cross-sectional dataset is available from the authors upon reasonable request, subject to ethical approval. The temporal longitudinal cohort is openly available under CC-BY 4.0 at Zenodo (DOI: 10.5281/zenodo.8405764) [ 6 ] . The non-medical student cohort is openly available under CC-BY 4.0 at Zenodo (DOI: 10.5281/zenodo.15149289) [ 5 ] . Analysis code (Python) is available at https://github.com/alirezadehghan1/xai-burnout-depression ."},{"type":"heading","level":1,"text":"Ethical Approval"},{"type":"para","text":"The study uses anonymised secondary data collected under ethical approval of the University of Lausanne Faculty of Biology and Medicine (Protocol No. 2019-02134)."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Auditing Construct Overlap in Explainable Machine Learning: Evidence from Burnout-Depression Prediction Across Student Cohorts\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Alireza Dehghan 1 Negin Ashrafi 2,*\"\n },\n {\n \"type\": \"para\",\n \"text\": \"1 Sharif University of Technology, Tehran, Iran 2 University of Southern California, California, USA * Corresponding author: ashrafin@usc.edu\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Abstract. Explainable machine learning (XML) pipelines applied to composite mental health outcomes can produce apparently-robust, cross-population-stable risk hierarchies that are largely artefacts of how the outcome was constructed. We demonstrate this using an ElasticNet pipeline applied to 886 medical students at the University of Lausanne (primary cohort, 2022), validated across 2,580 longitudinal observations at three time points and 701 non-medical students from eight faculties; all three datasets share identical instruments.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pipeline produces a hierarchy in which trait anxiety and health satisfaction dominate wherever the outcome is measured, with Kendall τ = 1.0 \\\\tau=1.0 for the top-two positions across all five evaluation sets and consistent transfer performance (R 2 : 0.41–0.49). Two residualization experiments, which isolate shared variance between correlated variables via regression, reveal the mechanism: when trait anxiety (STAI-T) is residualized against the co-included depression subscale (CES-D, r = 0.72 r=0.72 ), model R 2 drops from 0.41 to 0.16 and STAI-T falls from rank 1 to rank 6; when burnout subscales are residualized against CES-D, R 2 collapses to 0.016. Prediction intervals average 35.4 units on a 0–100 scale (2.4 outcome standard deviations), independently ruling out individual-level deployment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The residualization protocol is the paper’s transferable contribution: any XAI study combining correlated predictor and outcome constructs should apply this check before interpreting apparent stability as a finding.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Keywords: construct overlap; explainability artefact; burnout; depression; residualization; XAI; SHAP; conformal prediction; medical students; multi-cohort validation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Burnout and depression among medical students are prevalent and consequential: reported burnout rates consistently exceed those of the general population, and untreated distress impairs clinical empathy and patient care quality [ 1 , 12 , 10 ] . The application of machine learning to self-report survey data offers a route to automated risk stratification; paired with SHAP explanations, such pipelines appear to identify stable, cross-population risk hierarchies, a consistent rank-ordering of predictors by importance, in which trait anxiety and health satisfaction dominate [ 4 , 13 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The objective of this study is not to identify definitive risk factors, but to test whether the stability produced by such a pipeline is real. Specifically: trait anxiety (STAI-T) and the CES-D depression scale are highly correlated ( r = 0.72 r=0.72 in this sample). The CES-D is embedded inside the composite burnout-depression outcome. When a predictor and an outcome component share a latent construct, the predictor’s explanatory dominance may reflect psychometric overlap rather than a genuine risk relationship. We call this construct overlap, and we introduce a residualization protocol to quantify how much of the apparent hierarchy survives once the overlap is removed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The principal contributions are threefold. First, we validate a residualization protocol for detecting construct overlap artefacts : removing the anxiety–depression psychometric channel collapses R 2 from 0.41 to 0.016 (burnout-specific outcome) and STAI-T from rank 1 to rank 6. Second, we demonstrate that apparent cross-cohort stability is not evidence of a portable risk structure when the signal is anchored to a fixed psychometric correlation: the model transfers with R 2 = 0.41–0.49 across five evaluation sets because the STAI-T/CES-D bivariate relationship is constant, not because the pipeline learned a generalisable risk structure. Third, we provide deployment-relevant uncertainty quantification : conformal prediction intervals average 35.4 BDCI units (2.4 outcome standard deviations), establishing that individual-level clinical prediction is not actionable with current feature sets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The remainder of this paper is organised as follows. Section 2 briefly reviews related work. Section 3 describes the methodology. Section 4 presents results in four subsections structured around the central audit. Section 5 discusses implications. Section 6 concludes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A growing body of work applies machine learning and XAI to mental health outcomes in student populations. Martínez-Ramón et al. [ 9 ] achieved high classification accuracy for teacher burnout using ANN. Bozorgmehr and Weltermann [ 4 ] applied XGBoost with SHAP to a large German adult sample and identified health satisfaction as a dominant protective factor, consistent with our findings on a different instrument. Rashid et al. [ 13 ] applied ensemble and deep learning to the same 886-student Lausanne cohort as this paper, achieving moderate classification accuracy. Benito et al. [ 3 ] and Nnadi et al. [ 11 ] combined SHAP with clustering and hierarchical explanation frameworks. In all cases, the feature hierarchy is treated as a finding; no prior work has tested whether the hierarchy survives a residualization check for construct overlap.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 summarises reviewed studies. The methodological gap is clear: regression-based continuous framing and construct-overlap auditing are both absent from the existing literature.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Methodology\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Data and Cohorts\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 describes the three datasets. The primary cohort ( n = 886 n=886 medical students, University of Lausanne, 2022) was collected by the Faculty of Biology and Medicine [ 7 ] . It contains no missing values. Predictors include ten demographic and psychological survey items (age, academic year, health satisfaction, psychotherapy history, partnership status, employment status, weekly study hours, mother tongue, sex, and trait anxiety from the State-Trait Anxiety Inventory; STAI-T). Outcome components are the three Maslach Burnout Inventory–Student Survey (MBI-SS) subscales and the Center for Epidemiological Studies Depression Scale (CES-D).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The temporal and non-medical cohorts are openly available under CC-BY 4.0 (Zenodo: 10.5281/zenodo.8405764 and 10.5281/zenodo.15149289 respectively).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 BDCI Construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Burnout–Depression Composite Index (BDCI) integrates the three MBI-SS subscales and CES-D into a single severity score on [ 0 , 100 ] [0,100] . Each subscale is normalised to its theoretical maximum (reversing the Academic Efficacy subscale); any component exceeding its empirical 75th-percentile threshold (derived from the 2022 training split) is doubled. The raw sum is then scaled by 100 / ( 4 + k ) 100/(4+k) , where k k is the count of elevated components. The result is a continuous index where higher values indicate greater joint burnout-depression severity. Formally:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 75th-percentile thresholding and doubling scheme is intended to give greater analytic weight to clinically elevated symptom levels, a severity-weighting logic common in composite mental-health indices. We treat this as one reasonable analytic choice rather than a validated clinical rule: the threshold is sample-derived (2022 training split) rather than tied to an external clinical cutoff (e.g., CES-D ≥ \\\\geq 16), which is a limitation we return to in Section 5.1. Sensitivity to the weighting choice itself is assessed directly via BDCI-A (Table 5), an equal-weight variant that reproduces the same qualitative pattern, indicating the central finding does not depend on this specific scheme. The BDCI is a study-specific analytical composite; it has not been independently validated as a clinical instrument.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The critical structural feature of the BDCI is that CES-D is one of its four components, while STAI-T (trait anxiety) is one of the predictors. Figure 1 shows the Pearson correlation matrix: STAI-T and CES-D share r = 0.72 r=0.72 ( p < 0.001 p<0.001 ), the strongest pairwise correlation in the dataset. This overlap motivates the residualization audit in Section 4.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Analysis Pipeline\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The dataset was partitioned 80/20 into training ( n train = 708 n_{\\\\mathrm{train}}=708 ) and test ( n test = 178 n_{\\\\mathrm{test}}=178 ) sets using stratified random sampling on BDCI quartile (random seed 42 throughout). All predictors were z z -score normalised using training-split statistics only. ElasticNet (L1+L2 regularisation) was selected as the primary model after 5-fold cross-validated comparison of six candidate regressors (OLS, Ridge, Lasso, ElasticNet, SVR, ANN); hyperparameters were selected by grid search ( α ∈ [ 10 − 4 , 10 1 ] \\\\alpha\\\\in[10^{-4},10^{1}] , mixing ratio ρ ∈ { 0.1 , 0.3 , 0.5 , 0.7 , 0.9 } \\\\rho\\\\in\\\\{0.1,0.3,0.5,0.7,0.9\\\\} ; optimal: α = 0.0316 \\\\alpha=0.0316 , ρ = 0.7 \\\\rho=0.7 ). SHAP values were computed using LinearExplainer [ 14 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For cross-cohort validation, a 7-feature reduced ElasticNet was retrained on the 2022 training split using only the predictors available in all three cohorts (age, academic year, health satisfaction, psychotherapy history, partnership status, STAI-T, sex), then applied without retraining to the temporal and non-medical cohorts. The 2022 BDCI thresholds τ j \\\\tau_{j} were used throughout to prevent leakage.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ninety-five percent bootstrap confidence intervals (1,000 resamples) were computed for all test-set metrics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Residualization Protocol\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two orthogonality experiments were conducted to isolate the anxiety–depression channel. Experiment 1 (predictor residualization) : STAI-T was replaced by the residuals of regressing STAI-T on CES-D (fit on training set, applied to test set). The residual captures trait anxiety variance orthogonal to concurrent depression. Experiment 2 (outcome residualization) : each MBI-SS subscale was individually residualized against CES-D (OLS, training set), then re-normalised and assembled into a depression-free composite (BDCI-C). In both cases, ElasticNet was refit from scratch on the modified data using the same hyperparameter grid and split. We note that BDCI-C is a statistical construct created solely to isolate depression-independent variance in the burnout subscales; residualizing against CES-D does not by itself produce a validated ’pure burnout’ measure. A near-zero R² for BDCI-C indicates that this feature set cannot predict the depression-independent portion of the composite, not that a clinically meaningful burnout-specific construct doesn’t exist. Results are compared with the primary BDCI and a CES-D-excluded composite (BDCI-B, MBI-SS only) in Table 5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5 Conformal Prediction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Point estimates are complemented by split conformal prediction intervals [ 15 ] . The test set is split 50/50: the calibration half ( n cal = 89 n_{\\\\mathrm{cal}}=89 ) is used to compute non-conformity scores s i = | y i − y ^ i | s_{i}=|y_{i}-\\\\hat{y}_{i}| ; the evaluation half ( n eval = 89 n_{\\\\mathrm{eval}}=89 ) receives intervals [ y ^ j ± q ^ ] [\\\\hat{y}_{j}\\\\pm\\\\hat{q}] , where q ^ \\\\hat{q} is the ( 1 − α ) ( 1 + 1 / n cal ) (1-\\\\alpha)(1+1/n_{\\\\mathrm{cal}}) quantile at α = 0.10 \\\\alpha=0.10 . This procedure provides marginal coverage ≥ 90 % \\\\geq 90\\\\% without distributional assumptions, with the caveat that small calibration sets may produce slight deviations from the nominal level.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 A Standard Pipeline Produces an Apparent Hierarchy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 presents test-set performance. All four linear and regularised models achieve similar accuracy (R 2 : 0.400–0.414); SVR performs somewhat worse. Penalisation offers no meaningful gain over OLS: the BDCI–predictor relationship is approximately linear and additive. A Lasso fit with L1 penalty automatically zeros five of twelve features, retaining a sparse set of seven predictors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 shows the ElasticNet SHAP beeswarm. Trait anxiety produces the widest spread of SHAP values by a wide margin: its mean | SHAP | |\\\\text{SHAP}| exceeds health satisfaction (ranked second) by approximately 10-fold. This apparent hierarchy is consistent across the cross-population transfer cohorts (Section 4.2 ): STAI-T ranks first in every wave and in the non-medical student sample.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 The Hierarchy Appears Stable Across Cohorts\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 shows transfer performance. The 7-feature reduced ElasticNet, trained exclusively on 2022 data, achieves R 2 of 0.443, 0.487, and 0.388 on the three longitudinal medical waves, and 0.493 on 701 non-medical students from eight faculties, the highest of any evaluation set. Figure 3 shows SHAP feature importance across all five cohorts. Trait anxiety holds rank 1 and health satisfaction holds rank 2 in every cohort; the full rank ordering is Trait Anxiety > > Health Satisfaction > > Age > > Psychotherapy > > Sex:Female > > Partnership > > Academic Year in all five evaluation sets. Kendall τ = 1.0 \\\\tau=1.0 for the top-two positions across all pairwise cohort comparisons. The apparent stability is striking.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 4.3 tests whether this stability reflects a portable risk structure or a psychometric artefact.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Residualization Audit: What Survives When the Depression Channel Is Removed?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 is the central result. Two baseline comparisons (primary BDCI and BDCI-A, an equal-weight variant) confirm that STAI-T dominates at rank 1 regardless of the weighting scheme. When CES-D is simply excluded from the outcome (BDCI-B), STAI-T remains rank 1 and R 2 falls from 0.414 to 0.303, a partial drop, because the MBI-SS burnout subscales themselves correlate with depression.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two residualization experiments go further. When STAI-T is replaced by residuals orthogonal to CES-D (Experiment 1), R 2 drops from 0.414 to 0.156 and STAI-T falls from rank 1 to rank 6 in mean absolute SHAP magnitude; health satisfaction becomes the dominant predictor. When the burnout outcome itself is purged of CES-D variance (Experiment 2, BDCI-C), R 2 collapses to 0.016: the full 12-feature model explains essentially none of the burnout-specific variance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 summarises the residualization audit by plotting ElasticNet R 2 under each outcome/predictor definition. The collapse from the primary BDCI to STAI-T residualization and BDCI-C shows how much of the apparent signal depends on the anxiety–depression channel.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These experiments answer the central question precisely. The standard pipeline predicts the depression-saturated portion of the composite outcome well and the burnout-specific portion essentially not at all. The cross-cohort stability in Section 4.2 transfers because the STAI-T/CES-D psychometric correlation ( r = 0.72 r=0.72 ) is structurally constant wherever these instruments are used together, not because the pipeline learned a portable multivariate risk structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Health satisfaction is the most robust predictor of the depression-orthogonal signal (rank 1 in Experiment 1) and is the feature most likely to represent an independent burnout pathway. Researchers who report cross-cohort XAI stability without a residualization audit may be describing instrument correlation, not replicated science.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Uncertainty Rules Out Individual Deployment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The non-conformity quantile from the calibration half was q ^ = 17.7 \\\\hat{q}=17.7 BDCI units. On the evaluation half ( n eval = 89 n_{\\\\mathrm{eval}}=89 ), empirical coverage was 84.3% (nominal target: 90%), and average interval width was 35.4 BDCI units (Figure 5 ). The − 5.7 -5.7 percentage-point coverage gap is within sampling noise: the standard deviation of a coverage estimate at true 90% with n eval = 89 n_{\\\\mathrm{eval}}=89 is ≈ 3.2 \\\\approx 3.2 pp, so the gap is approximately 1.8 standard deviations from nominal ( p ≈ 0.07 p\\\\approx 0.07 ). The headline finding is the interval width, not the coverage deviation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An interval of 35.4 units on a 0–100 scale is 2.4 outcome standard deviations ( S D = 14.7 SD=14.7 ). This width is fully determined by the model’s residual distribution; any uncertainty quantification method would produce comparable intervals. It independently establishes that individual-level clinical prediction is not actionable with this feature set, regardless of the construct overlap finding.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This study’s central finding is the R 2 collapse: from 0.414 (primary BDCI) to 0.016 (burnout-specific BDCI-C), with an intermediate collapse to 0.156 when only the predictor is residualized. The implication is not that the primary model is wrong: the anxiety–depression nexus is one of the best-replicated constructs in psychiatric epidemiology, and the STAI-T/CES-D association ( r = 0.72 r=0.72 ) reflects genuine comorbidity [ 2 ] . The implication is that a standard multi-method XAI pipeline cannot distinguish this real clinical signal from a psychometric artefact without an explicit orthogonality check. Apparent stability across five cohorts is not evidence of a portable risk structure when the dominant predictor and the dominant outcome component share a latent construct.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The residualization protocol introduced here is the paper’s transferable contribution. Any XAI study combining correlated predictor and outcome constructs (trait anxiety and depression are a prominent example, but any construct pair with r > 0.5 r>0.5 –0.6 warrants scrutiny) should apply this check before interpreting feature hierarchies as findings. The protocol requires only the fitted model and a linear regression of the predictor on the confounded outcome component; it adds no computational burden and produces a clear, quantitative audit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The apparently strong cross-cohort transfer (R 2 : 0.41–0.49 across five evaluation sets) deserves an honest characterisation. Wave 2 (0.487) and the non-medical cohort (0.493) exceed the internal test R 2 (0.423), but this is consistent with a stable two-variable relationship rather than broad generalisation: all three cohorts originate from the same institution and use identical instruments. Cross-institutional validation with heterogeneous instruments (instruments that break the STAI-T/CES-D fixed correlation) remains the essential next step before any deployment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The conformal width (35.4 BDCI units, 2.4 SDs) establishes a separate and independent bar for deployment. Even if the construct overlap were resolved by a better instrument design, individual-level prediction would require a substantially expanded feature set (likely ecological momentary assessment, physiological data, or longitudinal trajectories) to achieve actionable precision. The current paper quantifies the upper bound of what cross-sectional survey data can deliver under this instrument set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Health satisfaction emerges from the residualization experiments as the most durable predictor: after removing the STAI-T/CES-D channel, it ranks first in mean absolute SHAP magnitude. This aligns with Bozorgmehr and Weltermann [ 4 ] and Khan et al. [ 8 ] , who identified health satisfaction and self-rated health as primary determinants in unrelated datasets and populations, suggesting that perceived health quality may be a more portable burnout signal than trait anxiety. Testing this hypothesis requires a study that separates the measurement constructs from the outset, not a post-hoc residualization.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All datasets originate from the University of Lausanne; cross-institutional generalisability has not been tested. The primary analysis is cross-sectional; longitudinal within-student prediction would require a prospective design. The BDCI is a study-internal composite that has not been independently psychometrically validated. Potentially important predictors (financial stress, sleep quality, social support, institutional climate) are absent from the dataset and likely account for a substantial share of the unexplained variance. The clinical-year calibration gap (R 2 = 0.294, n = 66 n=66 , versus 0.456 for pre-clinical) is directionally informative but statistically fragile at this subgroup size.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Future Directions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The natural Phase II is a multi-site prospective study that: (a) recruits at least three institutions with structurally different curricula to obtain genuine cross-institutional variation; (b) uses a prospective design in which pre-year surveys predict end-of-year burnout episodes or counselling referrals; (c) pre-registers the residualization audit as a primary endpoint to confirm that the collapse is not an artefact of this particular cohort; and (d) extends the clinical-year feature set with ecological momentary assessment during clinical rotations to test the environmental hypothesis.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A standard XAI pipeline applied to a burnout-depression composite produces an apparently-robust, cross-population-stable risk hierarchy. Two residualization experiments show that this hierarchy is largely an artefact of construct overlap: removing the depression channel from the predictor (STAI-T ⟂ \\\\perp CES-D) drops R 2 from 0.41 to 0.16; removing it from the outcome (MBI-SS subscales ⟂ \\\\perp CES-D) drops R 2 to 0.016. The apparent cross-cohort stability (R 2 = 0.41–0.49, Kendall τ = 1.0 \\\\tau=1.0 for top-two positions across five cohorts) is explained by the fixed STAI-T/CES-D psychometric correlation ( r = 0.72 r=0.72 ), not by a learned portable risk structure. Conformal prediction intervals averaging 35.4 BDCI units (2.4 outcome SDs) independently establish that individual-level prediction is not actionable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The residualization protocol is directly transferable: any XAI study that combines correlated predictor and outcome constructs should apply this check before reporting apparent stability as a finding. Researchers who skip this step may be describing instrument correlation, not replicated science.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Data and Code Availability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The primary 2022 cross-sectional dataset is available from the authors upon reasonable request, subject to ethical approval. The temporal longitudinal cohort is openly available under CC-BY 4.0 at Zenodo (DOI: 10.5281/zenodo.8405764) [ 6 ] . The non-medical student cohort is openly available under CC-BY 4.0 at Zenodo (DOI: 10.5281/zenodo.15149289) [ 5 ] . Analysis code (Python) is available at https://github.com/alirezadehghan1/xai-burnout-depression .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Ethical Approval\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The study uses anonymised secondary data collected under ethical approval of the University of Lausanne Faculty of Biology and Medicine (Protocol No. 2019-02134).\"\n }\n]","isConversational":false}}},{"rowIdx":383,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10618"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularization"},"authors":{"kind":"list like","value":["Raziyeh Takbiri"],"string":"[\n \"Raziyeh Takbiri\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10618v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Keywords: sparse demixing, nonlinear observations, non-convex regularization, folded-concave penalties, robust recovery, proximal alternating minimization, Kurdyka–Łojasiewicz."},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Signal demixing — separating a superposition of structured components from few measurements — and nonlinear compressed sensing have each been studied extensively, but their intersection remains sparse. Acquisition front ends saturate, clip, compand, and quantize; the recorded data are then nonlinear functions of a mixture. Formally we observe"},{"type":"para","text":"i = 1 , … , m ≪ n i=1,\\dots,m\\ll n , with ‖ 𝒘 ∗ ‖ 0 ≤ s 1 \\left\\|\\bm{w}^{\\ast}\\right\\|_{0}\\leq s_{1} , ‖ 𝒛 ∗ ‖ 0 ≤ s 2 \\left\\|\\bm{z}^{\\ast}\\right\\|_{0}\\leq s_{2} , and ask for both components."},{"type":"para","text":"The state of the art for ( 1 ) is the greedy line of Soltani and Hegde [ 1 , 2 ] : demixing hard thresholding with sample complexity m = 𝒪 ( s log ( n / s ) ) m=\\mathcal{O}(s\\log(n/s)) , s = s 1 + s 2 s=s_{1}+s_{2} , requiring the exact sparsity levels as input and offering no robustness theory. In the linear regime ( g = id g=\\mathrm{id} ), non-convex regularization is known to outperform convex ℓ 1 \\ell_{1} demixing [ 3 , 4 ] , reflecting the general folded-concave phenomenon [ 5 , 6 ] : penalties that debias large coefficients achieve oracle behavior where the LASSO carries an irreducible λ s \\lambda\\sqrt{s} shrinkage bias [ 7 , 8 ] . For single-index models without superposition, Plan and Vershynin [ 9 ] showed that unknown monotone nonlinearities can be absorbed into an effective scaling plus noise. To the best of our knowledge, no prior work combines generalized non-convex regularization with nonlinear demixing, in either the known- or unknown-link regime."},{"type":"heading","level":2,"text":"1.1 Contributions"},{"type":"para","text":"Estimator (Sec. 2 ). A Huberized non-convex program for ( 1 ). Huberization is structural, not cosmetic: it eliminates a curvature–noise compatibility condition that squared-loss analysis provably requires, and extends all guarantees to symmetric noise with only finite variance."},{"type":"para","text":"Algorithm (Sec. 3 ). NLD-PALM, a two-block proximal-gradient method with per-block backtracking and an over-relaxation factor η > 1 \\eta>1 that we show is necessary for sufficient decrease with non-convex penalties. Whole-sequence convergence to critical points under the KL property (Theorem 1 ), with R-linear local rates; backtracking admits links with unbounded derivative that break fixed-step schemes."},{"type":"para","text":"Statistical theory (Sec. 4 ). Restricted strong convexity of the Huberized nonlinear loss (Theorem 2 ) via an exact decomposition whose curvature part is sign-definite for every sample ; error bounds at all localized stationary points (Theorem 3 ); an oracle property (Proposition 1 ); and a co-equal unknown-link theorem (Theorem 4 ) built on a clipped decoupling lemma (Lemma 3 )."},{"type":"para","text":"Experiments (Sec. 5 ). Full-protocol study at n = 512 n=512 with a frozen, data-driven λ \\lambda rule: earlier phase transitions than ℓ 1 \\ell_{1} and DHT (the latter given the true sparsity levels), robustness to gross outliers and heavy tails, and a saturation-demixing application."},{"type":"para","text":"All proofs are given in the main text or Appendix B ; the companion conference paper [ 10 ] announces a subset of these results."},{"type":"heading","level":1,"text":"2 Problem Formulation"},{"type":"heading","level":2,"text":"2.1 Model and assumptions"},{"type":"para","text":"Write 𝑩 = [ 𝚽 𝚿 ] ∈ ℝ n × 2 n \\bm{B}=[\\bm{\\Phi}\\ \\bm{\\Psi}]\\in\\mathbb{R}^{n\\times 2n} , 𝜷 = ( 𝒘 , 𝒛 ) ∈ ℝ 2 n \\bm{\\beta}=(\\bm{w},\\bm{z})\\in\\mathbb{R}^{2n} , truth 𝜷 ∗ \\bm{\\beta}^{\\ast} , S = supp ( 𝜷 ∗ ) S=\\operatorname{supp}(\\bm{\\beta}^{\\ast}) , s = | S | s=|S| ."},{"type":"para","text":"𝒂 i ∼ iid 𝒩 ( 𝟎 , 𝑰 n ) \\bm{a}_{i}\\overset{\\mathrm{iid}}{\\sim}\\mathcal{N}(\\bm{0},\\bm{I}_{n}) ."},{"type":"para","text":"g g is known, differentiable, with 0 < ℓ g ≤ g ′ ( t ) ≤ L g 0<\\ell_{g}\\leq g^{\\prime}(t)\\leq L_{g} and | g ′′ ( t ) | ≤ M g |g^{\\prime\\prime}(t)|\\leq M_{g} for | t | ≤ τ |t|\\leq\\tau , where τ \\tau satisfies ℙ { max i | ⟨ 𝐚 i , 𝐱 ⟩ | > τ } ≤ ( m n ) − 1 \\mathbb{P}\\{\\max_{i}|\\langle\\bm{a}_{i},\\bm{x}\\rangle|>\\tau\\}\\leq(mn)^{-1} for the signals in play (Gaussian maximal inequality; see Assumption 5 )."},{"type":"para","text":"ε ≔ max i , j | ⟨ ϕ i , 𝝍 j ⟩ | \\varepsilon\\coloneqq\\max_{i,j}|\\langle\\bm{\\phi}_{i},\\bm{\\psi}_{j}\\rangle| satisfies 16 ε s ≤ 1 2 16\\,\\varepsilon s\\leq\\tfrac{1}{2} . For 𝚽 = 𝐈 \\bm{\\Phi}=\\bm{I} , 𝚿 = \\bm{\\Psi}= DCT, ε = 2 / n \\varepsilon=\\sqrt{2/n} ."},{"type":"para","text":"e i e_{i} iid, independent of { 𝐚 i } \\{\\bm{a}_{i}\\} , symmetric, and either (a) σ 2 \\sigma^{2} -sub-Gaussian, or (b) of finite variance 𝔼 e i 2 ≤ σ 2 \\mathbb{E}e_{i}^{2}\\leq\\sigma^{2} (heavy tails / ϵ \\epsilon -contamination with symmetric contaminant)."},{"type":"para","text":"Statements about stationary points concern 𝛃 ~ \\tilde{\\bm{\\beta}} with ‖ 𝛃 ~ − 𝛃 ∗ ‖ 2 ≤ r \\left\\|\\tilde{\\bm{\\beta}}-\\bm{\\beta}^{\\ast}\\right\\|_{2}\\leq r where ( ‖ 𝐱 ∗ ‖ 2 + 2 r ) 2 log ( 2 m n ) ≤ τ (\\left\\|\\bm{x}^{\\ast}\\right\\|_{2}+2r)\\sqrt{2\\log(2mn)}\\leq\\tau ; boundedness of iterates is enforced by the ball device of Remark 3 ."},{"type":"heading","level":2,"text":"2.2 Penalty family"},{"type":"para","text":"P λ : ℝ → ℝ ≥ 0 P_{\\lambda}:\\mathbb{R}\\to\\mathbb{R}_{\\geq 0} with: (P1) symmetric, P λ ( 0 ) = 0 P_{\\lambda}(0)=0 , nondecreasing on [ 0 , ∞ ) [0,\\infty) ; (P2) differentiable on ( 0 , ∞ ) (0,\\infty) with 0 ≤ P λ ′ ≤ λ 0\\leq P^{\\prime}_{\\lambda}\\leq\\lambda , P λ ′ P^{\\prime}_{\\lambda} nonincreasing; (P3) P λ ′ ( 0 + ) = λ P^{\\prime}_{\\lambda}(0^{+})=\\lambda ; (P4) P λ ′ ( t ) = 0 P^{\\prime}_{\\lambda}(t)=0 for t ≥ a λ t\\geq a\\lambda . Vector penalties act coordinatewise."},{"type":"para","text":"P λ ∈ 𝒫 ( λ , a ) P_{\\lambda}\\in\\mathcal{P}(\\lambda,a) and λ | t | − ρ 2 t 2 ≤ P λ ( t ) ≤ λ | t | \\lambda|t|-\\tfrac{\\rho}{2}t^{2}\\leq P_{\\lambda}(t)\\leq\\lambda|t| with t ↦ P λ ( t ) + ρ 2 t 2 t\\mapsto P_{\\lambda}(t)+\\tfrac{\\rho}{2}t^{2} convex. MCP ( γ > 1 \\gamma>1 ): ρ = 1 / γ \\rho=1/\\gamma ; SCAD ( a > 2 a>2 ): ρ = 1 / ( a − 1 ) \\rho=1/(a-1) . The ℓ q \\ell_{q} quasi-norms ( 0 < q < 1 0<q<1 ) violate (P3)–(P4) and weak convexity; they are covered by the algorithmic theory (KL property) and experiments, not by the statistical theorems — a deliberate two-tier structure defining what “generalized” means in our title."},{"type":"heading","level":2,"text":"2.3 The Huberized estimator"},{"type":"para","text":"Let h δ H h_{\\delta_{H}} denote the Huber function with threshold δ H \\delta_{H} and ψ δ H ( t ) = max ( − δ H , min ( t , δ H ) ) \\psi_{\\delta_{H}}(t)=\\max(-\\delta_{H},\\min(t,\\delta_{H})) . The estimator is any localized stationary point of"},{"type":"para","text":"with gradient ∇ f H ( 𝜷 ) = − 1 m 𝑩 ⊤ 𝑨 ⊤ [ g ′ ( 𝒖 ) ⊙ ψ δ H ( 𝒚 − g ( 𝒖 ) ) ] \\nabla f_{H}(\\bm{\\beta})=-\\frac{1}{m}\\bm{B}^{\\top}\\bm{A}^{\\top}[g^{\\prime}(\\bm{u})\\odot\\psi_{\\delta_{H}}(\\bm{y}-g(\\bm{u}))] , 𝒖 = 𝑨 𝑩 𝜷 \\bm{u}=\\bm{A}\\bm{B}\\bm{\\beta} . Default δ H = 4 σ \\delta_{H}=4\\sigma (known link); the unknown-link variant uses an adaptive level (Sec. 4.5 )."},{"type":"para","text":"Squared-loss curvature analysis for ( 1 ) meets the term 1 m ∑ i e i ( g ′ ( u i ) − g ′ ( u i ∗ ) ) ⟨ 𝐚 i , 𝐁 𝚫 ⟩ \\frac{1}{m}\\sum_{i}e_{i}(g^{\\prime}(u_{i})-g^{\\prime}(u_{i}^{\\ast}))\\langle\\bm{a}_{i},\\bm{B}\\bm{\\Delta}\\rangle , whose natural bound carries 𝔼 | e | ≍ σ \\mathbb{E}|e|\\asymp\\sigma — a non-vanishing bias forcing a condition of the form σ M g ≲ ℓ g 2 \\sigma M_{g}\\lesssim\\ell_{g}^{2} . Huberization replaces e i e_{i} by the bounded, mean-zero (under symmetry) score ψ δ H ( e i ) \\psi_{\\delta_{H}}(e_{i}) , whose empirical process concentrates; the condition disappears, and only the variance of the noise enters the rates."},{"type":"heading","level":1,"text":"3 Algorithm and Convergence"},{"type":"para","text":"Algorithm NLD-PALM. Parameters L min > 0 L_{\\min}>0 , κ > 1 \\kappa>1 , δ ∈ ( 0 , 1 ) \\delta\\in(0,1) , η > 1 \\eta>1 . For k = 0 , 1 , 2 , … k=0,1,2,\\dots :"},{"type":"para","text":"( w w -block) backtrack L ≥ L min L\\geq L_{\\min} (multiply by κ \\kappa ) until 𝒘 + = prox P λ / ( η L ) ( 𝒘 k − 1 η L ∇ 𝒘 f H ( 𝒘 k , 𝒛 k ) ) \\bm{w}^{+}=\\mathrm{prox}_{P_{\\lambda}/(\\eta L)}\\bigl(\\bm{w}^{k}-\\tfrac{1}{\\eta L}\\nabla_{\\bm{w}}f_{H}(\\bm{w}^{k},\\bm{z}^{k})\\bigr) satisfies f H ( 𝒘 + , 𝒛 k ) ≤ f H ( 𝒘 k , 𝒛 k ) + ⟨ ∇ 𝒘 f H , 𝒘 + − 𝒘 k ⟩ + L 2 ‖ 𝒘 + − 𝒘 k ‖ 2 f_{H}(\\bm{w}^{+},\\bm{z}^{k})\\leq f_{H}(\\bm{w}^{k},\\bm{z}^{k})+\\langle\\nabla_{\\bm{w}}f_{H},\\bm{w}^{+}-\\bm{w}^{k}\\rangle+\\tfrac{L}{2}\\left\\|\\bm{w}^{+}-\\bm{w}^{k}\\right\\|^{2} ; set 𝒘 k + 1 = 𝒘 + \\bm{w}^{k+1}=\\bm{w}^{+} , L w k + 1 = δ L L_{w}^{k+1}=\\delta L ."},{"type":"para","text":"( z z -block) same with roles swapped, using 𝒘 k + 1 \\bm{w}^{k+1} ."},{"type":"para","text":"Prox maps of MCP, SCAD, ℓ 1 / 2 \\ell_{1/2} are closed-form (Appendix A )."},{"type":"para","text":"The prox inequality at modulus η L \\eta L and the accepted descent condition at constant L L add to a per-block decrease of ( η − 1 ) L 2 ‖ 𝚫 ‖ 2 \\frac{(\\eta-1)L}{2}\\left\\|\\bm{\\Delta}\\right\\|^{2} . At η = 1 \\eta=1 the quadratic terms cancel exactly and only monotonicity survives: for convex penalties the prox inequality self-improves and η = 1 \\eta=1 suffices, but for our non-convex penalties the sufficient-decrease property — hence the entire KL mechanism — genuinely requires η > 1 \\eta>1 ."},{"type":"para","text":"F F need not be coercive (MCP/SCAD are bounded; tanh \\tanh -type losses are bounded). We add the indicator of a ball B R = { ‖ ( 𝐰 , 𝐳 ) ‖ 2 ≤ R } B_{R}=\\{\\left\\|(\\bm{w},\\bm{z})\\right\\|_{2}\\leq R\\} , R R large; in every experiment the constraint is verified inactive a posteriori. For ℓ q \\ell_{q} , F F is coercive and no device is needed."},{"type":"para","text":"Suppose g ∈ C 2 g\\in C^{2} (squared loss) or f H ∈ C loc 1 , 1 f_{H}\\in C^{1,1}_{\\mathrm{loc}} (Huberized: ∇ f H \\nabla f_{H} is a composition of the 1 1 -Lipschitz clipping with locally Lipschitz maps), the penalties are proper lsc prox-bounded, F F is definable in an o-minimal structure (true for polynomial/tanh/sigmoid links with MCP/SCAD/ ℓ q \\ell_{q} : semialgebraic and ℝ an , exp \\mathbb{R}_{\\mathrm{an,exp}} -definable pieces), and the iterates are bounded. Then: (i) backtracking terminates finitely at every step with accepted constants in [ L min , L ¯ ] [L_{\\min},\\bar{L}] ; (ii) F ( 𝐰 k , 𝐳 k ) F(\\bm{w}^{k},\\bm{z}^{k}) decreases by at least ( η − 1 ) L min 2 ‖ ( 𝐰 , 𝐳 ) k + 1 − ( 𝐰 , 𝐳 ) k ‖ 2 \\frac{(\\eta-1)L_{\\min}}{2}\\left\\|(\\bm{w},\\bm{z})^{k+1}-(\\bm{w},\\bm{z})^{k}\\right\\|^{2} per iteration; (iii) there exist subgradients with ‖ 𝐯 k + 1 ‖ ≤ ( η L ¯ + L f ) ‖ ( 𝐰 , 𝐳 ) k + 1 − ( 𝐰 , 𝐳 ) k ‖ \\left\\|\\bm{v}^{k+1}\\right\\|\\leq(\\eta\\bar{L}+L_{f})\\,\\left\\|(\\bm{w},\\bm{z})^{k+1}-(\\bm{w},\\bm{z})^{k}\\right\\| ; (iv) the whole sequence converges to a single critical point of F F with finite length; and (v) if the KL exponent at the limit is θ = 1 / 2 \\theta=1/2 — which holds on the RSC event of Theorem 2 at stationary points with the active-set structure of the penalty — convergence is R-linear."},{"type":"para","text":"(i)–(iii) are Lemmas 1–3 of the algorithmic analysis: (i) follows from local Lipschitzness of ∇ f H \\nabla f_{H} on compacts and geometric growth of L L ; (ii) from adding the prox inequality (modulus η L \\eta L ) to the accepted descent condition (constant L L ), which leaves ( η − 1 ) L 2 ‖ 𝚫 ‖ 2 \\frac{(\\eta-1)L}{2}\\left\\|\\bm{\\Delta}\\right\\|^{2} (Remark 2 ); (iii) from first-order optimality of the prox step plus Lipschitz gradient differences. Conditions (ii)–(iii) are (H1)–(H2) of the abstract framework of Attouch–Bolte–Svaiter [ 12 ] ; continuity (H3) holds along prox steps; definability gives the KL property [ 13 , 11 ] ; (iv) follows from the framework and (v) from the Łojasiewicz-exponent rate analysis [ 14 , 15 ] . ∎"},{"type":"heading","level":1,"text":"4 Statistical Recovery Theory"},{"type":"para","text":"Throughout, α ≔ ℓ g 2 / 16 \\alpha\\coloneqq\\ell_{g}^{2}/16 , α ¯ ≔ α − ρ > 0 \\underline{\\alpha}\\coloneqq\\alpha-\\rho>0 , and r H ≔ δ H / ( 8 L g ) r_{H}\\coloneqq\\delta_{H}/(8L_{g}) (the Huber-band radius )."},{"type":"heading","level":2,"text":"4.1 Geometry"},{"type":"para","text":"For any 𝚫 = ( 𝛅 w , 𝛅 z ) \\bm{\\Delta}=(\\bm{\\delta}_{w},\\bm{\\delta}_{z}) , ‖ 𝐁 𝚫 ‖ 2 2 ≥ ‖ 𝚫 ‖ 2 2 − 2 ε ‖ 𝛅 w ‖ 1 ‖ 𝛅 z ‖ 1 \\left\\|\\bm{B}\\bm{\\Delta}\\right\\|_{2}^{2}\\geq\\left\\|\\bm{\\Delta}\\right\\|_{2}^{2}-2\\varepsilon\\left\\|\\bm{\\delta}_{w}\\right\\|_{1}\\left\\|\\bm{\\delta}_{z}\\right\\|_{1} ; on the cone 𝒞 ( S , 3 ) = { ‖ 𝚫 S c ‖ 1 ≤ 3 ‖ 𝚫 S ‖ 1 } \\mathcal{C}(S,3)=\\{\\left\\|\\bm{\\Delta}_{S^{c}}\\right\\|_{1}\\leq 3\\left\\|\\bm{\\Delta}_{S}\\right\\|_{1}\\} , ‖ 𝐁 𝚫 ‖ 2 2 ≥ ( 1 − 16 ε s ) ‖ 𝚫 ‖ 2 2 ≥ 1 2 ‖ 𝚫 ‖ 2 2 \\left\\|\\bm{B}\\bm{\\Delta}\\right\\|_{2}^{2}\\geq(1-16\\varepsilon s)\\left\\|\\bm{\\Delta}\\right\\|_{2}^{2}\\geq\\tfrac{1}{2}\\left\\|\\bm{\\Delta}\\right\\|_{2}^{2} ."},{"type":"para","text":"‖ 𝑩 𝚫 ‖ 2 = ‖ 𝚫 ‖ 2 + 2 𝜹 w ⊤ 𝚽 ⊤ 𝚿 𝜹 z \\left\\|\\bm{B}\\bm{\\Delta}\\right\\|^{2}=\\left\\|\\bm{\\Delta}\\right\\|^{2}+2\\bm{\\delta}_{w}^{\\top}\\bm{\\Phi}^{\\top}\\bm{\\Psi}\\bm{\\delta}_{z} and the cross term is bounded entrywise by ε ‖ 𝜹 w ‖ 1 ‖ 𝜹 z ‖ 1 ≤ ε 4 ‖ 𝚫 ‖ 1 2 ≤ 4 ε s ‖ 𝚫 ‖ 2 2 \\varepsilon\\left\\|\\bm{\\delta}_{w}\\right\\|_{1}\\left\\|\\bm{\\delta}_{z}\\right\\|_{1}\\leq\\tfrac{\\varepsilon}{4}\\left\\|\\bm{\\Delta}\\right\\|_{1}^{2}\\leq 4\\varepsilon s\\left\\|\\bm{\\Delta}\\right\\|_{2}^{2} on the cone, where ‖ 𝚫 ‖ 1 ≤ 4 s ‖ 𝚫 ‖ 2 \\left\\|\\bm{\\Delta}\\right\\|_{1}\\leq 4\\sqrt{s}\\left\\|\\bm{\\Delta}\\right\\|_{2} . ∎"},{"type":"heading","level":2,"text":"4.2 Restricted strong convexity"},{"type":"para","text":"Let Assumptions 1 – 6 hold, δ H ≥ 4 σ \\delta_{H}\\geq 4\\sigma , and m ≥ C log ( 2 n ) m\\geq C\\log(2n) . With probability at least 1 − c 1 e − c 2 m − 2 ( 2 n ) − 2 − ( m n ) − 1 1-c_{1}e^{-c_{2}m}-2(2n)^{-2}-(mn)^{-1} , for all 𝚫 \\bm{\\Delta} with ‖ 𝐁 𝚫 ‖ 2 ≤ r H \\left\\|\\bm{B}\\bm{\\Delta}\\right\\|_{2}\\leq r_{H} ,"},{"type":"para","text":"with τ 0 = c 3 ( L g 2 + δ H 2 M g 2 / ℓ g 2 ) \\tau_{0}=c_{3}\\bigl(L_{g}^{2}+\\delta_{H}^{2}M_{g}^{2}/\\ell_{g}^{2}\\bigr) ."},{"type":"para","text":"Let t i = ⟨ 𝒂 i , 𝑩 𝚫 ⟩ t_{i}=\\langle\\bm{a}_{i},\\bm{B}\\bm{\\Delta}\\rangle , u i ∗ = ⟨ 𝒂 i , 𝒙 ∗ ⟩ u_{i}^{\\ast}=\\langle\\bm{a}_{i},\\bm{x}^{\\ast}\\rangle , d i = g ( u i ∗ + t i ) − g ( u i ∗ ) d_{i}=g(u_{i}^{\\ast}+t_{i})-g(u_{i}^{\\ast}) . The gradient-increment summand is [ ϕ i ( t i ) − ϕ i ( 0 ) ] t i [\\phi_{i}(t_{i})-\\phi_{i}(0)]t_{i} with ϕ i ( t ) = − ψ δ H ( e i − ( g ( u i ∗ + t ) − g ( u i ∗ ) ) ) g ′ ( u i ∗ + t ) \\phi_{i}(t)=-\\psi_{\\delta_{H}}(e_{i}-(g(u_{i}^{\\ast}+t)-g(u_{i}^{\\ast})))g^{\\prime}(u_{i}^{\\ast}+t) ; split ϕ i ( t i ) − ϕ i ( 0 ) = A i + B i \\phi_{i}(t_{i})-\\phi_{i}(0)=A_{i}+B_{i} ,"},{"type":"para","text":"Curvature part. ψ δ H \\psi_{\\delta_{H}} nondecreasing, g g increasing, g ′ > 0 g^{\\prime}>0 give A i t i ≥ 0 A_{i}t_{i}\\geq 0 for every sample . On ℰ i = { | e i | ≤ δ H / 2 } ∩ { | t i | ≤ δ H / ( 2 L g ) } \\mathcal{E}_{i}=\\{|e_{i}|\\leq\\delta_{H}/2\\}\\cap\\{|t_{i}|\\leq\\delta_{H}/(2L_{g})\\} , both e i e_{i} and e i − d i e_{i}-d_{i} lie in the linear zone of ψ δ H \\psi_{\\delta_{H}} , so A i t i = g ′ ( ξ i ) g ′ ( u i ∗ + t i ) t i 2 ≥ ℓ g 2 t i 2 A_{i}t_{i}=g^{\\prime}(\\xi_{i})g^{\\prime}(u_{i}^{\\ast}+t_{i})t_{i}^{2}\\geq\\ell_{g}^{2}t_{i}^{2} . Hence T 1 ≥ ℓ g 2 m ∑ i b i φ K ( t i ) T_{1}\\geq\\frac{\\ell_{g}^{2}}{m}\\sum_{i}b_{i}\\varphi_{K}(t_{i}) with b i = 𝟏 { | e i | ≤ δ H / 2 } b_{i}=\\bm{1}\\{|e_{i}|\\leq\\delta_{H}/2\\} ( ℙ ( b i = 1 ) ≥ 1 2 \\mathbb{P}(b_{i}{=}1)\\geq\\frac{1}{2} by Chebyshev and δ H ≥ 4 σ \\delta_{H}\\geq 4\\sigma ), K = δ H / ( 2 L g ) K=\\delta_{H}/(2L_{g}) , and φ K \\varphi_{K} the truncated quadratic of Lemma 4 , which yields the uniform lower bound ℓ g 2 8 ‖ 𝑩 𝚫 ‖ 2 − c L g 2 log 2 n m ‖ 𝚫 ‖ 1 2 \\frac{\\ell_{g}^{2}}{8}\\left\\|\\bm{B}\\bm{\\Delta}\\right\\|^{2}-cL_{g}^{2}\\frac{\\log 2n}{m}\\left\\|\\bm{\\Delta}\\right\\|_{1}^{2} over ‖ 𝑩 𝚫 ‖ ≤ K / 4 = r H \\left\\|\\bm{B}\\bm{\\Delta}\\right\\|\\leq K/4=r_{H} . Multiplier part. 𝔼 [ B i t i ∣ 𝒂 i ] = 0 \\mathbb{E}[B_{i}t_{i}\\mid\\bm{a}_{i}]=0 (noise symmetry), | B i t i | ≤ δ H M g t i 2 |B_{i}t_{i}|\\leq\\delta_{H}M_{g}t_{i}^{2} ; Lemma 5 bounds sup | 1 m ∑ B i t i | ≤ ℓ g 2 32 ‖ 𝚫 ‖ 2 + c δ H 2 M g 2 ℓ g 2 log 2 n m ‖ 𝚫 ‖ 1 2 \\sup|{\\textstyle\\frac{1}{m}\\sum}B_{i}t_{i}|\\leq\\frac{\\ell_{g}^{2}}{32}\\left\\|\\bm{\\Delta}\\right\\|^{2}+c\\frac{\\delta_{H}^{2}M_{g}^{2}}{\\ell_{g}^{2}}\\frac{\\log 2n}{m}\\left\\|\\bm{\\Delta}\\right\\|_{1}^{2} . Assembly with Lemma 1 (cone case; the all-vector case absorbs the incoherence deficit ≤ ε 2 ‖ 𝚫 ‖ 1 2 \\leq\\frac{\\varepsilon}{2}\\left\\|\\bm{\\Delta}\\right\\|_{1}^{2} into the tolerance) gives ( 3 ). ∎"},{"type":"para","text":"Huber loss is linear outside its band — it has no curvature there; an RSC radius proportional to δ H \\delta_{H} is the correct statement, exactly as in high-dimensional robust regression [ 16 , 17 ] . Self-consistency (final error ≤ r H \\leq r_{H} ) holds for m ≳ ( L g 2 min ( σ , δ H ) / ( ℓ g 2 δ H ) ) 2 s log ( 2 n ) m\\gtrsim(L_{g}^{2}\\min(\\sigma,\\delta_{H})/(\\ell_{g}^{2}\\delta_{H}))^{2}s\\log(2n) ."},{"type":"para","text":"With probability ≥ 1 − 2 ( 2 n ) − 2 − ( m n ) − 1 \\geq 1-2(2n)^{-2}-(mn)^{-1} , for m ≥ log ( 2 n ) m\\geq\\log(2n) : ‖ ∇ f H ( 𝛃 ∗ ) ‖ ∞ ≤ c L g min ( σ , δ H ) log ( 2 n ) / m \\left\\|\\nabla f_{H}(\\bm{\\beta}^{\\ast})\\right\\|_{\\infty}\\leq cL_{g}\\min(\\sigma,\\delta_{H})\\sqrt{\\log(2n)/m} ."},{"type":"para","text":"Coordinate j j is 1 m ∑ i e i ′ g ′ ( u i ∗ ) ⟨ 𝒂 i , 𝒃 j ⟩ \\frac{1}{m}\\sum_{i}e_{i}^{\\prime}g^{\\prime}(u_{i}^{\\ast})\\langle\\bm{a}_{i},\\bm{b}_{j}\\rangle with e i ′ = ψ δ H ( e i ) e_{i}^{\\prime}=\\psi_{\\delta_{H}}(e_{i}) mean-zero, bounded by δ H \\delta_{H} , variance ≤ min ( σ 2 , δ H 2 ) \\leq\\min(\\sigma^{2},\\delta_{H}^{2}) . Condition on 𝑨 \\bm{A} ; the conditional sub-Gaussian parameter is L g 2 min ( σ 2 , δ H 2 ) m ⋅ 1 m ‖ 𝑨 𝒃 j ‖ 2 \\frac{L_{g}^{2}\\min(\\sigma^{2},\\delta_{H}^{2})}{m}\\cdot\\frac{1}{m}\\left\\|\\bm{A}\\bm{b}_{j}\\right\\|^{2} , and 1 m ‖ 𝑨 𝒃 j ‖ 2 ≤ 2 \\frac{1}{m}\\left\\|\\bm{A}\\bm{b}_{j}\\right\\|^{2}\\leq 2 w.p. 1 − e − c m 1-e^{-cm} ( χ m 2 \\chi_{m}^{2} Bernstein). Union over 2 n 2n columns. ∎"},{"type":"heading","level":2,"text":"4.3 Main error bound"},{"type":"para","text":"The estimator carries the ℓ 1 \\ell_{1} side constraint Ω = { ‖ 𝜷 ‖ 1 ≤ R 1 } \\Omega=\\{\\left\\|\\bm{\\beta}\\right\\|_{1}\\leq R_{1}\\} , R 1 ≥ 2 ‖ 𝜷 ∗ ‖ 1 R_{1}\\geq 2\\left\\|\\bm{\\beta}^{\\ast}\\right\\|_{1} [ 7 ] ; Remark 5 shows it is inactive at the solution."},{"type":"para","text":"On the RSC event ( 3 ), with"},{"type":"para","text":"every stationary point 𝛃 ~ \\tilde{\\bm{\\beta}} of F F over Ω \\Omega with ‖ 𝛃 ~ − 𝛃 ∗ ‖ 2 ≤ r H \\left\\|\\tilde{\\bm{\\beta}}-\\bm{\\beta}^{\\ast}\\right\\|_{2}\\leq r_{H} satisfies"},{"type":"para","text":"Write 𝚫 = 𝜷 ~ − 𝜷 ∗ \\bm{\\Delta}=\\tilde{\\bm{\\beta}}-\\bm{\\beta}^{\\ast} , a = ‖ 𝚫 S ‖ 1 a=\\left\\|\\bm{\\Delta}_{S}\\right\\|_{1} , b = ‖ 𝚫 S c ‖ 1 b=\\left\\|\\bm{\\Delta}_{S^{c}}\\right\\|_{1} . (1) Constrained stationarity against the feasible 𝜷 ∗ \\bm{\\beta}^{\\ast} : ⟨ ∇ f H ( 𝜷 ~ ) − ∇ f H ( 𝜷 ∗ ) , 𝚫 ⟩ ≤ − ⟨ 𝝃 , 𝚫 ⟩ − ⟨ ∇ f H ( 𝜷 ∗ ) , 𝚫 ⟩ \\langle\\nabla f_{H}(\\tilde{\\bm{\\beta}})-\\nabla f_{H}(\\bm{\\beta}^{\\ast}),\\bm{\\Delta}\\rangle\\leq-\\langle\\bm{\\xi},\\bm{\\Delta}\\rangle-\\langle\\nabla f_{H}(\\bm{\\beta}^{\\ast}),\\bm{\\Delta}\\rangle , 𝝃 ∈ ∂ P ( 𝜷 ~ ) \\bm{\\xi}\\in\\partial P(\\tilde{\\bm{\\beta}}) . (2) Weak convexity of P + ρ 2 ∥ ⋅ ∥ 2 P+\\frac{\\rho}{2}\\left\\|\\cdot\\right\\|^{2} : − ⟨ 𝝃 , 𝚫 ⟩ ≤ P ( 𝜷 ∗ ) − P ( 𝜷 ~ ) + ρ 2 ‖ 𝚫 ‖ 2 -\\langle\\bm{\\xi},\\bm{\\Delta}\\rangle\\leq P(\\bm{\\beta}^{\\ast})-P(\\tilde{\\bm{\\beta}})+\\frac{\\rho}{2}\\left\\|\\bm{\\Delta}\\right\\|^{2} . (3) Penalty comparison via (P1)–(P4) and Assumption 6 : P ( 𝜷 ∗ ) − P ( 𝜷 ~ ) ≤ λ a − λ b + ρ 2 ‖ 𝚫 S c ‖ 2 P(\\bm{\\beta}^{\\ast})-P(\\tilde{\\bm{\\beta}})\\leq\\lambda a-\\lambda b+\\frac{\\rho}{2}\\left\\|\\bm{\\Delta}_{S^{c}}\\right\\|^{2} . (4) Gradient term: ≤ λ 4 ( a + b ) \\leq\\frac{\\lambda}{4}(a+b) by ( 4 ). (5) Tolerance: ‖ 𝚫 ‖ 1 ≤ 2 R 1 \\left\\|\\bm{\\Delta}\\right\\|_{1}\\leq 2R_{1} on Ω \\Omega , so τ 0 log 2 n m ‖ 𝚫 ‖ 1 2 ≤ λ 4 ( a + b ) \\tau_{0}\\frac{\\log 2n}{m}\\left\\|\\bm{\\Delta}\\right\\|_{1}^{2}\\leq\\frac{\\lambda}{4}(a+b) by the second branch of ( 4 ). (6) Assembling with ( 3 ) and collecting ρ \\rho -terms: α ¯ ‖ 𝚫 ‖ 2 ≤ 3 λ 2 a − λ 2 b \\underline{\\alpha}\\left\\|\\bm{\\Delta}\\right\\|^{2}\\leq\\frac{3\\lambda}{2}a-\\frac{\\lambda}{2}b . (7) Nonnegativity of the left side forces b ≤ 3 a b\\leq 3a : the cone, whence ‖ 𝚫 ‖ 1 ≤ 4 s ‖ 𝚫 ‖ 2 \\left\\|\\bm{\\Delta}\\right\\|_{1}\\leq 4\\sqrt{s}\\left\\|\\bm{\\Delta}\\right\\|_{2} . (8) Dropping − λ 2 b -\\frac{\\lambda}{2}b and using a ≤ s ‖ 𝚫 ‖ 2 a\\leq\\sqrt{s}\\left\\|\\bm{\\Delta}\\right\\|_{2} : ‖ 𝚫 ‖ 2 ≤ 3 λ s 2 α ¯ \\left\\|\\bm{\\Delta}\\right\\|_{2}\\leq\\frac{3\\lambda\\sqrt{s}}{2\\underline{\\alpha}} , and the ℓ 1 \\ell_{1} bound follows. ∎"},{"type":"para","text":"The conclusion gives ‖ 𝛃 ~ ‖ 1 ≤ ‖ 𝛃 ∗ ‖ 1 + 6 λ s / α ¯ < R 1 \\left\\|\\tilde{\\bm{\\beta}}\\right\\|_{1}\\leq\\left\\|\\bm{\\beta}^{\\ast}\\right\\|_{1}+6\\lambda s/\\underline{\\alpha}<R_{1} for λ \\lambda of the prescribed order and m ≳ s log 2 n m\\gtrsim s\\log 2n : constrained and unconstrained stationary points coincide, and NLD-PALM needs no ℓ 1 \\ell_{1} projection (verified numerically in all experiments)."},{"type":"para","text":"With λ ≍ L g min ( σ , δ H ) log ( 2 n ) / m \\lambda\\asymp L_{g}\\min(\\sigma,\\delta_{H})\\sqrt{\\log(2n)/m} and m ≳ max { s log ( 2 n / s ) , ( τ 0 / α ¯ ) 2 s log 2 n , ( τ 0 ‖ 𝛃 ∗ ‖ 2 / σ ) 2 s log ( 2 n ) } m\\gtrsim\\max\\{s\\log(2n/s),\\,(\\tau_{0}/\\underline{\\alpha})^{2}s\\log 2n,\\,(\\tau_{0}\\left\\|\\bm{\\beta}^{\\ast}\\right\\|_{2}/\\sigma)^{2}s\\log(2n)\\} :"},{"type":"para","text":"under sub-Gaussian or finite-variance symmetric noise."},{"type":"heading","level":2,"text":"4.4 Oracle property"},{"type":"para","text":"Let E 2 = 3 λ s 2 α ¯ E_{2}=\\frac{3\\lambda\\sqrt{s}}{2\\underline{\\alpha}} and assume min j ∈ S | β j ∗ | ≥ a λ + E 2 \\min_{j\\in S}|\\beta_{j}^{\\ast}|\\geq a\\lambda+E_{2} . (a) If supp ( 𝛃 ~ ) ⊆ S \\operatorname{supp}(\\tilde{\\bm{\\beta}})\\subseteq S , then 𝛃 ~ \\tilde{\\bm{\\beta}} solves the unpenalized problem on ℝ S \\mathbb{R}^{S} and ‖ 𝛃 ~ − 𝛃 ∗ ‖ 2 ≤ 2 α ‖ ∇ S f H ( 𝛃 ∗ ) ‖ 2 ≲ L g α min ( σ , δ H ) s / m \\left\\|\\tilde{\\bm{\\beta}}-\\bm{\\beta}^{\\ast}\\right\\|_{2}\\leq\\frac{2}{\\alpha}\\left\\|\\nabla_{S}f_{H}(\\bm{\\beta}^{\\ast})\\right\\|_{2}\\lesssim\\frac{L_{g}}{\\alpha}\\min(\\sigma,\\delta_{H})\\sqrt{s/m} w.h.p. — no log n \\log n , no λ \\lambda -bias. (b) If additionally C L E 2 ≤ λ / 4 C_{L}E_{2}\\leq\\lambda/4 with C L = c ( L g 2 + δ H M g ) C_{L}=c(L_{g}^{2}+\\delta_{H}M_{g}) and E 2 < γ λ / 2 E_{2}<\\gamma\\lambda/2 , then supp ( 𝛃 ~ ) ⊆ S \\operatorname{supp}(\\tilde{\\bm{\\beta}})\\subseteq S ."},{"type":"para","text":"(a) Beta-min and ‖ 𝚫 ‖ ∞ ≤ E 2 \\left\\|\\bm{\\Delta}\\right\\|_{\\infty}\\leq E_{2} put all j ∈ S j\\in S in the zero-derivative region of P λ P_{\\lambda} (P4), so stationarity reduces to ∇ S f H ( 𝜷 ~ ) = 0 \\nabla_{S}f_{H}(\\tilde{\\bm{\\beta}})=0 ; RSC on the fixed s s -dimensional subspace and a vector-Bernstein bound on ‖ ∇ S f H ( 𝜷 ∗ ) ‖ 2 \\left\\|\\nabla_{S}f_{H}(\\bm{\\beta}^{\\ast})\\right\\|_{2} (no union over 2 n 2n coordinates — this is where log n \\log n disappears) give the rate. (b) For j ∈ S c j\\in S^{c} with 𝜷 ~ j ≠ 0 \\tilde{\\bm{\\beta}}_{j}\\neq 0 , MCP stationarity forces | ∇ j f H ( 𝜷 ~ ) | = λ − | 𝜷 ~ j | / γ |\\nabla_{j}f_{H}(\\tilde{\\bm{\\beta}})|=\\lambda-|\\tilde{\\bm{\\beta}}_{j}|/\\gamma or | 𝜷 ~ j | ≥ γ λ |\\tilde{\\bm{\\beta}}_{j}|\\geq\\gamma\\lambda . The coordinatewise Lipschitz bound ‖ ∇ f H ( 𝜷 ~ ) − ∇ f H ( 𝜷 ∗ ) ‖ ∞ ≤ C L ‖ 𝚫 ‖ 2 \\left\\|\\nabla f_{H}(\\tilde{\\bm{\\beta}})-\\nabla f_{H}(\\bm{\\beta}^{\\ast})\\right\\|_{\\infty}\\leq C_{L}\\left\\|\\bm{\\Delta}\\right\\|_{2} (Cauchy–Schwarz with upper restricted eigenvalues) gives | ∇ j f H ( 𝜷 ~ ) | ≤ λ / 2 |\\nabla_{j}f_{H}(\\tilde{\\bm{\\beta}})|\\leq\\lambda/2 , forcing | 𝜷 ~ j | ≥ γ λ / 2 > E 2 ≥ | 𝜷 ~ j | |\\tilde{\\bm{\\beta}}_{j}|\\geq\\gamma\\lambda/2>E_{2}\\geq|\\tilde{\\bm{\\beta}}_{j}| — contradiction. The condition C L E 2 ≤ λ / 4 C_{L}E_{2}\\leq\\lambda/4 amounts to s ≲ α ¯ / C L \\sqrt{s}\\lesssim\\underline{\\alpha}/C_{L} ; the primal–dual witness of [ 8 ] removes this restriction and is left as an optional appendix. ∎"},{"type":"para","text":"ℓ 1 \\ell_{1} stationary points satisfy | ∇ j f H | = λ |\\nabla_{j}f_{H}|=\\lambda at every nonzero coordinate, embedding a c λ s c\\lambda\\sqrt{s} bias for any λ \\lambda compatible with ( 4 ) — visible as the error plateau in Fig. 1 ."},{"type":"heading","level":2,"text":"4.5 Unknown link"},{"type":"para","text":"Assume g g unknown, monotone, with μ g = 𝔼 [ g ′ ( γ ) ] ≠ 0 \\mu_{g}=\\mathbb{E}[g^{\\prime}(\\gamma)]\\neq 0 , θ g 2 = 𝔼 [ ( g ( γ ) − μ g γ ) 2 ] \\theta_{g}^{2}=\\mathbb{E}[(g(\\gamma)-\\mu_{g}\\gamma)^{2}] , γ ∼ 𝒩 ( 0 , 1 ) \\gamma\\sim\\mathcal{N}(0,1) , ‖ 𝒙 ∗ ‖ 2 = 1 \\left\\|\\bm{x}^{\\ast}\\right\\|_{2}=1 ; let ν 2 = θ g 2 + σ 2 \\nu^{2}=\\theta_{g}^{2}+\\sigma^{2} . Run the same program with the linear Huberized loss and adaptive clipping δ H = 2 ν log m \\delta_{H}=2\\nu\\sqrt{\\log m} ."},{"type":"para","text":"With probability ≥ 1 − c ( 2 n ) − 2 \\geq 1-c(2n)^{-2} , ‖ 1 m ∑ i ψ δ H ( ξ i ) 𝐁 ⊤ 𝐚 i ‖ ∞ ≤ c ν log ( 2 n ) log ( m ) / m \\left\\|\\frac{1}{m}\\sum_{i}\\psi_{\\delta_{H}}(\\xi_{i})\\bm{B}^{\\top}\\bm{a}_{i}\\right\\|_{\\infty}\\leq c\\,\\nu\\sqrt{\\log(2n)\\log(m)/m} , where ξ i = g ( u i ∗ ) − μ g u i ∗ + e i \\xi_{i}=g(u_{i}^{\\ast})-\\mu_{g}u_{i}^{\\ast}+e_{i} ."},{"type":"para","text":"Split 𝒂 i = γ i 𝒙 ∗ + 𝑷 ⟂ 𝒂 i \\bm{a}_{i}=\\gamma_{i}\\bm{x}^{\\ast}+\\bm{P}_{\\perp}\\bm{a}_{i} with γ i = ⟨ 𝒂 i , 𝒙 ∗ ⟩ \\gamma_{i}=\\langle\\bm{a}_{i},\\bm{x}^{\\ast}\\rangle ; ξ i \\xi_{i} depends on ( γ i , e i ) (\\gamma_{i},e_{i}) only. Orthogonal part: conditionally Gaussian with variance 1 m ⋅ 1 m ∑ ψ δ H ( ξ i ) 2 ≤ c ν 2 m \\frac{1}{m}\\cdot\\frac{1}{m}\\sum\\psi_{\\delta_{H}}(\\xi_{i})^{2}\\leq\\frac{c\\nu^{2}}{m} w.h.p. (Bernstein), giving c ν log ( 2 n ) / m c\\nu\\sqrt{\\log(2n)/m} after a union bound. Parallel bias: 𝔼 [ ξ γ ] = 0 \\mathbb{E}[\\xi\\gamma]=0 by Stein’s identity ( 𝔼 [ g ( γ ) γ ] = 𝔼 [ g ′ ( γ ) ] = μ g \\mathbb{E}[g(\\gamma)\\gamma]=\\mathbb{E}[g^{\\prime}(\\gamma)]=\\mu_{g} ), and | 𝔼 [ ψ δ H ( ξ ) γ ] | ≤ 𝔼 [ ξ 2 𝟏 { | ξ | > δ H } ] ≤ c ν e − δ H 2 / ( c ′ ν 2 ) ≤ c ν m − 4 / c ′ |\\mathbb{E}[\\psi_{\\delta_{H}}(\\xi)\\gamma]|\\leq\\sqrt{\\mathbb{E}[\\xi^{2}\\bm{1}\\{|\\xi|>\\delta_{H}\\}]}\\leq c\\nu e^{-\\delta_{H}^{2}/(c^{\\prime}\\nu^{2})}\\leq c\\nu m^{-4/c^{\\prime}} by sub-Gaussianity of ξ \\xi and the choice of δ H \\delta_{H} . Parallel deviation: ψ δ H ( ξ i ) γ i \\psi_{\\delta_{H}}(\\xi_{i})\\gamma_{i} has envelope δ H | γ i | \\delta_{H}|\\gamma_{i}| and variance ≤ c ν 2 log m \\leq c\\nu^{2}\\log m ; Bernstein plus the union bound gives the dominant term c ν log m log ( 2 n ) / m c\\nu\\sqrt{\\log m}\\sqrt{\\log(2n)/m} . ∎"},{"type":"para","text":"Under Assumptions 1 , 3 , 4 (a), 6 , with m ≥ C s log ( 2 n ) m\\geq Cs\\log(2n) and λ ≍ ν log ( 2 n ) log ( m ) / m \\lambda\\asymp\\nu\\sqrt{\\log(2n)\\log(m)/m} : with probability ≥ 1 − c 1 e − c 2 m − c 3 n − 1 \\geq 1-c_{1}e^{-c_{2}m}-c_{3}n^{-1} , every stationary point of the linear-surrogate objective with ‖ 𝛃 ~ − μ g 𝛃 ∗ ‖ 2 ≤ r H \\left\\|\\tilde{\\bm{\\beta}}-\\mu_{g}\\bm{\\beta}^{\\ast}\\right\\|_{2}\\leq r_{H} satisfies"},{"type":"para","text":"Write y i = μ g ⟨ 𝒂 i , 𝒙 ∗ ⟩ + ξ i y_{i}=\\mu_{g}\\langle\\bm{a}_{i},\\bm{x}^{\\ast}\\rangle+\\xi_{i} . The loss is linear-Huber: Theorem 2 applies with g = id g=\\mathrm{id} ( ℓ g = L g = 1 \\ell_{g}=L_{g}=1 , M g = 0 M_{g}=0 ; the multiplier part vanishes identically, so no noise-symmetry is needed for RSC). Lemma 3 replaces Lemma 2 . Theorem 3 ’s deterministic core applies verbatim around μ g 𝜷 ∗ \\mu_{g}\\bm{\\beta}^{\\ast} . ∎"},{"type":"heading","level":1,"text":"5 Experiments"},{"type":"para","text":"Protocol: n = 512 n=512 , s 1 = s 2 = 16 s_{1}=s_{2}=16 , 𝚽 = 𝑰 \\bm{\\Phi}=\\bm{I} , 𝚿 = \\bm{\\Psi}= DCT, g = tanh g=\\tanh , σ = 0.05 \\sigma=0.05 , δ H = 4 σ \\delta_{H}=4\\sigma . Regularization uses the frozen rule λ = c σ log ( 2 n ) / m \\lambda=c\\,\\sigma\\sqrt{\\log(2n)/m} , with c c calibrated once on a 5 5 -seed pilot at m = 400 m=400 and never re-tuned ( c = 8 c=8 for ℓ 1 \\ell_{1} , 16 16 for SCAD, 24 24 for MCP, 4 4 for ℓ 1 / 2 \\ell_{1/2} ); the DHT baseline [ 1 ] step is frozen at its pilot-best likewise, and receives the true ( s 1 , s 2 ) (s_{1},s_{2}) . Nonconvex methods are warm-started from the ℓ 1 \\ell_{1} solution. 15 15 Monte-Carlo trials per point, deterministic seeds, code provided."},{"type":"para","text":"Phase transition. Fig. 1 : under the frozen rule SCAD/MCP transition ≈ 1.3 \\approx 1.3 – 1.4 × 1.4\\times earlier than DHT; in the scarce regime ( m ≤ 200 m\\leq 200 ) the gap is decisive (SCAD 0.08 0.08 – 0.17 0.17 vs. DHT 0.50 0.50 – 0.85 0.85 ). Under per-instance oracle tuning the gap widens to ≈ 1.5 \\approx 1.5 – 2 × 2\\times ; we report the frozen protocol as the practically meaningful one. ℓ 1 \\ell_{1} never attains exact recovery, matching Proposition 1 ’s bias dichotomy."},{"type":"para","text":"Robustness. With 5 % 5\\% gross outliers ( ± 3 \\pm 3 ): Huberized SCAD 0.073 0.073 vs. squared-loss SCAD 2.54 2.54 median relative error ( 35 × 35\\times ); Student- t ( 2 ) t(2) noise: 0.12 0.12 vs. 0.54 0.54 ; clean Gaussian noise: parity ( 0.060 0.060 vs. 0.060 0.060 ). Huberization is free insurance. An honest negative: at high Gaussian noise ( σ ≥ 0.1 \\sigma\\geq 0.1 for tanh \\tanh ) recovery degrades for all losses — an information-theoretic saturation effect, not loss fragility."},{"type":"para","text":"Scaling checks. Measured error scales linearly in σ \\sigma (slope 1.0 1.0 on a log-log grid up to the saturation regime) and as m − 1 / 2 m^{-1/2} over the pre-floor range, matching Corollary 1 ."},{"type":"para","text":"Application. Fig. 2 : spikes and a smooth background demixed through a saturating amplifier with outliers; all spikes located at m / n ≈ 0.59 m/n\\approx 0.59 ."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"We provided the first regularization-based treatment of nonlinear sparse demixing: a Huberized folded-concave estimator, a provably convergent algorithm whose over-relaxation requirement we identified, RSC-based guarantees holding at every localized stationary point in both known- and unknown-link regimes, oracle behavior unavailable to convex demixing, and robustness with only finite-variance symmetric noise. Future work: a primal–dual witness removing the dimensional condition in Proposition 1 (b), statistical theory for ℓ q \\ell_{q} , periodic and quantized links, and a deep-unfolded NLD-PALM with trainable threshold functions."},{"type":"para","text":"With step β = 1 / ( η L ) \\beta=1/(\\eta L) : MCP ( γ > β \\gamma>\\beta ): 0 if | u | ≤ λ β |u|\\leq\\lambda\\beta ; sign ( u ) ( | u | − λ β ) / ( 1 − β / γ ) \\mathrm{sign}(u)(|u|-\\lambda\\beta)/(1-\\beta/\\gamma) if λ β < | u | ≤ γ λ \\lambda\\beta<|u|\\leq\\gamma\\lambda ; u u otherwise. SCAD ( a > 1 + β a>1+\\beta ): soft threshold at λ β \\lambda\\beta for | u | ≤ λ ( 1 + β ) |u|\\leq\\lambda(1+\\beta) ; ( ( a − 1 ) u − sign ( u ) a λ β ) / ( a − 1 − β ) ((a-1)u-\\mathrm{sign}(u)a\\lambda\\beta)/(a-1-\\beta) for | u | ≤ a λ |u|\\leq a\\lambda ; u u otherwise. ℓ 1 / 2 \\ell_{1/2} : half-thresholding [ 18 ] with effective parameter 2 λ β 2\\lambda\\beta (their quadratic term carries no 1 / 2 1/2 ). All three are unit-tested against brute-force minimization of the scalar prox objective; the test suite accompanies the code."},{"type":"para","text":"Let φ K ( t ) = t 2 \\varphi_{K}(t)=t^{2} on | t | ≤ K / 2 |t|\\leq K/2 , ( K − | t | ) 2 (K-|t|)^{2} on K / 2 < | t | ≤ K K/2<|t|\\leq K , 0 beyond: K K -Lipschitz, t 2 𝟏 { | t | ≤ K / 2 } ≤ φ K ≤ t 2 𝟏 { | t | ≤ K } t^{2}\\bm{1}\\{|t|\\leq K/2\\}\\leq\\varphi_{K}\\leq t^{2}\\bm{1}\\{|t|\\leq K\\} , self-similar ( φ K ( t ) = ω 2 φ K / ω ( t / ω ) \\varphi_{K}(t)=\\omega^{2}\\varphi_{K/\\omega}(t/\\omega) ), monotone in K K ."},{"type":"para","text":"Let b i ∈ { 0 , 1 } b_{i}\\in\\{0,1\\} iid, ℙ ( b i = 1 ) = p ≥ 1 2 \\mathbb{P}(b_{i}{=}1)=p\\geq\\frac{1}{2} , independent of { 𝐚 i } \\{\\bm{a}_{i}\\} , and m ≥ c log ( 2 n ) m\\geq c\\log(2n) . With probability ≥ 1 − c 1 e − c 2 m \\geq 1-c_{1}e^{-c_{2}m} , for all 𝚫 \\bm{\\Delta} with ‖ 𝐯 ‖ 2 ≤ K / 4 \\left\\|\\bm{v}\\right\\|_{2}\\leq K/4 ( 𝐯 = 𝐁 𝚫 \\bm{v}=\\bm{B}\\bm{\\Delta} ):"},{"type":"para","text":"Fix a scale ω = ‖ 𝒗 ‖ 2 \\omega=\\left\\|\\bm{v}\\right\\|_{2} and use the per-scale level K ω = 4 ω ≤ K K_{\\omega}=4\\omega\\leq K (monotonicity). Population: 𝔼 [ b i φ K ω ] ≥ p ω 2 𝔼 [ γ 2 𝟏 { | γ | ≤ 2 } ] ≥ 0.73 p ω 2 \\mathbb{E}[b_{i}\\varphi_{K_{\\omega}}]\\geq p\\,\\omega^{2}\\mathbb{E}[\\gamma^{2}\\bm{1}\\{|\\gamma|\\leq 2\\}]\\geq 0.73\\,p\\,\\omega^{2} . Deviation: by self-similarity, work with φ 4 \\varphi_{4} ( 4 4 -Lipschitz) on normalized variables; symmetrization and Ledoux–Talagrand contraction [ 19 ] give 𝔼 sup Z ≤ 16 ω T 2 log ( 4 n ) / m \\mathbb{E}\\sup Z\\leq 16\\,\\omega\\,T\\sqrt{2\\log(4n)/m} over { ‖ 𝚫 ‖ 1 ≤ T } \\{\\left\\|\\bm{\\Delta}\\right\\|_{1}\\leq T\\} , since each coordinate of 1 m ∑ i ε i 𝑩 ⊤ 𝒂 i \\frac{1}{m}\\sum_{i}\\varepsilon_{i}\\bm{B}^{\\top}\\bm{a}_{i} is 𝒩 ( 0 , 1 / m ) \\mathcal{N}(0,1/m) ; Talagrand concentration (envelope 4 ω 2 4\\omega^{2} ) upgrades to probability 1 − e − c 2 m 1-e^{-c_{2}m} . Absorb or peel: if 32 T 2 log ( 4 n ) / m ≤ p 4 ω 32T\\sqrt{2\\log(4n)/m}\\leq\\frac{p}{4}\\omega the bound is direct; otherwise ω 2 ≲ log 2 n m T 2 \\omega^{2}\\lesssim\\frac{\\log 2n}{m}T^{2} and the tolerance term alone validates the display. Dyadic peeling over ω ∈ ( 2 − j − 1 K / 4 , 2 − j K / 4 ] \\omega\\in(2^{-j-1}K/4,2^{-j}K/4] with a union over O ( log m ) O(\\log m) shells completes the proof. ∎"},{"type":"para","text":"In the setting of Theorem 2 , with probability ≥ 1 − ( 2 n ) − 2 \\geq 1-(2n)^{-2} , uniformly over ‖ 𝐯 ‖ 2 ≤ r H \\left\\|\\bm{v}\\right\\|_{2}\\leq r_{H} :"},{"type":"para","text":"where h u ( t ) = [ g ′ ( u + t ) − g ′ ( u ) ] t h_{u}(t)=[g^{\\prime}(u+t)-g^{\\prime}(u)]\\,t ."},{"type":"para","text":"h u h_{u} is 2 M g K ω 2M_{g}K_{\\omega} -Lipschitz with h u ( 0 ) = 0 h_{u}(0)=0 on the per-scale band; ψ δ H ( e i ) \\psi_{\\delta_{H}}(e_{i}) are iid mean-zero (symmetry), bounded by δ H \\delta_{H} , independent of { 𝒂 i } \\{\\bm{a}_{i}\\} . The multiplier inequality and contraction [ 19 ] reduce the supremum to the linear class, giving per shell ≲ δ H M g ω ‖ 𝚫 ‖ 1 log ( 2 n ) / m \\lesssim\\delta_{H}M_{g}\\omega\\left\\|\\bm{\\Delta}\\right\\|_{1}\\sqrt{\\log(2n)/m} ; Young’s inequality ( a b ≤ η a 2 + b 2 / 4 η ab\\leq\\eta a^{2}+b^{2}/4\\eta , η = ℓ g 2 / 32 \\eta=\\ell_{g}^{2}/32 ) splits this into the displayed form; Talagrand concentration and the shell/coordinate union complete the proof. ∎"}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Keywords: sparse demixing, nonlinear observations, non-convex regularization, folded-concave penalties, robust recovery, proximal alternating minimization, Kurdyka–Łojasiewicz.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Signal demixing — separating a superposition of structured components from few measurements — and nonlinear compressed sensing have each been studied extensively, but their intersection remains sparse. Acquisition front ends saturate, clip, compand, and quantize; the recorded data are then nonlinear functions of a mixture. Formally we observe\"\n },\n {\n \"type\": \"para\",\n \"text\": \"i = 1 , … , m ≪ n i=1,\\\\dots,m\\\\ll n , with ‖ 𝒘 ∗ ‖ 0 ≤ s 1 \\\\left\\\\|\\\\bm{w}^{\\\\ast}\\\\right\\\\|_{0}\\\\leq s_{1} , ‖ 𝒛 ∗ ‖ 0 ≤ s 2 \\\\left\\\\|\\\\bm{z}^{\\\\ast}\\\\right\\\\|_{0}\\\\leq s_{2} , and ask for both components.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The state of the art for ( 1 ) is the greedy line of Soltani and Hegde [ 1 , 2 ] : demixing hard thresholding with sample complexity m = 𝒪 ( s log ( n / s ) ) m=\\\\mathcal{O}(s\\\\log(n/s)) , s = s 1 + s 2 s=s_{1}+s_{2} , requiring the exact sparsity levels as input and offering no robustness theory. In the linear regime ( g = id g=\\\\mathrm{id} ), non-convex regularization is known to outperform convex ℓ 1 \\\\ell_{1} demixing [ 3 , 4 ] , reflecting the general folded-concave phenomenon [ 5 , 6 ] : penalties that debias large coefficients achieve oracle behavior where the LASSO carries an irreducible λ s \\\\lambda\\\\sqrt{s} shrinkage bias [ 7 , 8 ] . For single-index models without superposition, Plan and Vershynin [ 9 ] showed that unknown monotone nonlinearities can be absorbed into an effective scaling plus noise. To the best of our knowledge, no prior work combines generalized non-convex regularization with nonlinear demixing, in either the known- or unknown-link regime.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"1.1 Contributions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Estimator (Sec. 2 ). A Huberized non-convex program for ( 1 ). Huberization is structural, not cosmetic: it eliminates a curvature–noise compatibility condition that squared-loss analysis provably requires, and extends all guarantees to symmetric noise with only finite variance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm (Sec. 3 ). NLD-PALM, a two-block proximal-gradient method with per-block backtracking and an over-relaxation factor η > 1 \\\\eta>1 that we show is necessary for sufficient decrease with non-convex penalties. Whole-sequence convergence to critical points under the KL property (Theorem 1 ), with R-linear local rates; backtracking admits links with unbounded derivative that break fixed-step schemes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Statistical theory (Sec. 4 ). Restricted strong convexity of the Huberized nonlinear loss (Theorem 2 ) via an exact decomposition whose curvature part is sign-definite for every sample ; error bounds at all localized stationary points (Theorem 3 ); an oracle property (Proposition 1 ); and a co-equal unknown-link theorem (Theorem 4 ) built on a clipped decoupling lemma (Lemma 3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Experiments (Sec. 5 ). Full-protocol study at n = 512 n=512 with a frozen, data-driven λ \\\\lambda rule: earlier phase transitions than ℓ 1 \\\\ell_{1} and DHT (the latter given the true sparsity levels), robustness to gross outliers and heavy tails, and a saturation-demixing application.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All proofs are given in the main text or Appendix B ; the companion conference paper [ 10 ] announces a subset of these results.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Problem Formulation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Model and assumptions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Write 𝑩 = [ 𝚽 𝚿 ] ∈ ℝ n × 2 n \\\\bm{B}=[\\\\bm{\\\\Phi}\\\\ \\\\bm{\\\\Psi}]\\\\in\\\\mathbb{R}^{n\\\\times 2n} , 𝜷 = ( 𝒘 , 𝒛 ) ∈ ℝ 2 n \\\\bm{\\\\beta}=(\\\\bm{w},\\\\bm{z})\\\\in\\\\mathbb{R}^{2n} , truth 𝜷 ∗ \\\\bm{\\\\beta}^{\\\\ast} , S = supp ( 𝜷 ∗ ) S=\\\\operatorname{supp}(\\\\bm{\\\\beta}^{\\\\ast}) , s = | S | s=|S| .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝒂 i ∼ iid 𝒩 ( 𝟎 , 𝑰 n ) \\\\bm{a}_{i}\\\\overset{\\\\mathrm{iid}}{\\\\sim}\\\\mathcal{N}(\\\\bm{0},\\\\bm{I}_{n}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"g g is known, differentiable, with 0 < ℓ g ≤ g ′ ( t ) ≤ L g 0<\\\\ell_{g}\\\\leq g^{\\\\prime}(t)\\\\leq L_{g} and | g ′′ ( t ) | ≤ M g |g^{\\\\prime\\\\prime}(t)|\\\\leq M_{g} for | t | ≤ τ |t|\\\\leq\\\\tau , where τ \\\\tau satisfies ℙ { max i | ⟨ 𝐚 i , 𝐱 ⟩ | > τ } ≤ ( m n ) − 1 \\\\mathbb{P}\\\\{\\\\max_{i}|\\\\langle\\\\bm{a}_{i},\\\\bm{x}\\\\rangle|>\\\\tau\\\\}\\\\leq(mn)^{-1} for the signals in play (Gaussian maximal inequality; see Assumption 5 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ε ≔ max i , j | ⟨ ϕ i , 𝝍 j ⟩ | \\\\varepsilon\\\\coloneqq\\\\max_{i,j}|\\\\langle\\\\bm{\\\\phi}_{i},\\\\bm{\\\\psi}_{j}\\\\rangle| satisfies 16 ε s ≤ 1 2 16\\\\,\\\\varepsilon s\\\\leq\\\\tfrac{1}{2} . For 𝚽 = 𝐈 \\\\bm{\\\\Phi}=\\\\bm{I} , 𝚿 = \\\\bm{\\\\Psi}= DCT, ε = 2 / n \\\\varepsilon=\\\\sqrt{2/n} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"e i e_{i} iid, independent of { 𝐚 i } \\\\{\\\\bm{a}_{i}\\\\} , symmetric, and either (a) σ 2 \\\\sigma^{2} -sub-Gaussian, or (b) of finite variance 𝔼 e i 2 ≤ σ 2 \\\\mathbb{E}e_{i}^{2}\\\\leq\\\\sigma^{2} (heavy tails / ϵ \\\\epsilon -contamination with symmetric contaminant).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Statements about stationary points concern 𝛃 ~ \\\\tilde{\\\\bm{\\\\beta}} with ‖ 𝛃 ~ − 𝛃 ∗ ‖ 2 ≤ r \\\\left\\\\|\\\\tilde{\\\\bm{\\\\beta}}-\\\\bm{\\\\beta}^{\\\\ast}\\\\right\\\\|_{2}\\\\leq r where ( ‖ 𝐱 ∗ ‖ 2 + 2 r ) 2 log ( 2 m n ) ≤ τ (\\\\left\\\\|\\\\bm{x}^{\\\\ast}\\\\right\\\\|_{2}+2r)\\\\sqrt{2\\\\log(2mn)}\\\\leq\\\\tau ; boundedness of iterates is enforced by the ball device of Remark 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Penalty family\"\n },\n {\n \"type\": \"para\",\n \"text\": \"P λ : ℝ → ℝ ≥ 0 P_{\\\\lambda}:\\\\mathbb{R}\\\\to\\\\mathbb{R}_{\\\\geq 0} with: (P1) symmetric, P λ ( 0 ) = 0 P_{\\\\lambda}(0)=0 , nondecreasing on [ 0 , ∞ ) [0,\\\\infty) ; (P2) differentiable on ( 0 , ∞ ) (0,\\\\infty) with 0 ≤ P λ ′ ≤ λ 0\\\\leq P^{\\\\prime}_{\\\\lambda}\\\\leq\\\\lambda , P λ ′ P^{\\\\prime}_{\\\\lambda} nonincreasing; (P3) P λ ′ ( 0 + ) = λ P^{\\\\prime}_{\\\\lambda}(0^{+})=\\\\lambda ; (P4) P λ ′ ( t ) = 0 P^{\\\\prime}_{\\\\lambda}(t)=0 for t ≥ a λ t\\\\geq a\\\\lambda . Vector penalties act coordinatewise.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"P λ ∈ 𝒫 ( λ , a ) P_{\\\\lambda}\\\\in\\\\mathcal{P}(\\\\lambda,a) and λ | t | − ρ 2 t 2 ≤ P λ ( t ) ≤ λ | t | \\\\lambda|t|-\\\\tfrac{\\\\rho}{2}t^{2}\\\\leq P_{\\\\lambda}(t)\\\\leq\\\\lambda|t| with t ↦ P λ ( t ) + ρ 2 t 2 t\\\\mapsto P_{\\\\lambda}(t)+\\\\tfrac{\\\\rho}{2}t^{2} convex. MCP ( γ > 1 \\\\gamma>1 ): ρ = 1 / γ \\\\rho=1/\\\\gamma ; SCAD ( a > 2 a>2 ): ρ = 1 / ( a − 1 ) \\\\rho=1/(a-1) . The ℓ q \\\\ell_{q} quasi-norms ( 0 < q < 1 0<q<1 ) violate (P3)–(P4) and weak convexity; they are covered by the algorithmic theory (KL property) and experiments, not by the statistical theorems — a deliberate two-tier structure defining what “generalized” means in our title.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 The Huberized estimator\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let h δ H h_{\\\\delta_{H}} denote the Huber function with threshold δ H \\\\delta_{H} and ψ δ H ( t ) = max ( − δ H , min ( t , δ H ) ) \\\\psi_{\\\\delta_{H}}(t)=\\\\max(-\\\\delta_{H},\\\\min(t,\\\\delta_{H})) . The estimator is any localized stationary point of\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with gradient ∇ f H ( 𝜷 ) = − 1 m 𝑩 ⊤ 𝑨 ⊤ [ g ′ ( 𝒖 ) ⊙ ψ δ H ( 𝒚 − g ( 𝒖 ) ) ] \\\\nabla f_{H}(\\\\bm{\\\\beta})=-\\\\frac{1}{m}\\\\bm{B}^{\\\\top}\\\\bm{A}^{\\\\top}[g^{\\\\prime}(\\\\bm{u})\\\\odot\\\\psi_{\\\\delta_{H}}(\\\\bm{y}-g(\\\\bm{u}))] , 𝒖 = 𝑨 𝑩 𝜷 \\\\bm{u}=\\\\bm{A}\\\\bm{B}\\\\bm{\\\\beta} . Default δ H = 4 σ \\\\delta_{H}=4\\\\sigma (known link); the unknown-link variant uses an adaptive level (Sec. 4.5 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Squared-loss curvature analysis for ( 1 ) meets the term 1 m ∑ i e i ( g ′ ( u i ) − g ′ ( u i ∗ ) ) ⟨ 𝐚 i , 𝐁 𝚫 ⟩ \\\\frac{1}{m}\\\\sum_{i}e_{i}(g^{\\\\prime}(u_{i})-g^{\\\\prime}(u_{i}^{\\\\ast}))\\\\langle\\\\bm{a}_{i},\\\\bm{B}\\\\bm{\\\\Delta}\\\\rangle , whose natural bound carries 𝔼 | e | ≍ σ \\\\mathbb{E}|e|\\\\asymp\\\\sigma — a non-vanishing bias forcing a condition of the form σ M g ≲ ℓ g 2 \\\\sigma M_{g}\\\\lesssim\\\\ell_{g}^{2} . Huberization replaces e i e_{i} by the bounded, mean-zero (under symmetry) score ψ δ H ( e i ) \\\\psi_{\\\\delta_{H}}(e_{i}) , whose empirical process concentrates; the condition disappears, and only the variance of the noise enters the rates.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Algorithm and Convergence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm NLD-PALM. Parameters L min > 0 L_{\\\\min}>0 , κ > 1 \\\\kappa>1 , δ ∈ ( 0 , 1 ) \\\\delta\\\\in(0,1) , η > 1 \\\\eta>1 . For k = 0 , 1 , 2 , … k=0,1,2,\\\\dots :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"( w w -block) backtrack L ≥ L min L\\\\geq L_{\\\\min} (multiply by κ \\\\kappa ) until 𝒘 + = prox P λ / ( η L ) ( 𝒘 k − 1 η L ∇ 𝒘 f H ( 𝒘 k , 𝒛 k ) ) \\\\bm{w}^{+}=\\\\mathrm{prox}_{P_{\\\\lambda}/(\\\\eta L)}\\\\bigl(\\\\bm{w}^{k}-\\\\tfrac{1}{\\\\eta L}\\\\nabla_{\\\\bm{w}}f_{H}(\\\\bm{w}^{k},\\\\bm{z}^{k})\\\\bigr) satisfies f H ( 𝒘 + , 𝒛 k ) ≤ f H ( 𝒘 k , 𝒛 k ) + ⟨ ∇ 𝒘 f H , 𝒘 + − 𝒘 k ⟩ + L 2 ‖ 𝒘 + − 𝒘 k ‖ 2 f_{H}(\\\\bm{w}^{+},\\\\bm{z}^{k})\\\\leq f_{H}(\\\\bm{w}^{k},\\\\bm{z}^{k})+\\\\langle\\\\nabla_{\\\\bm{w}}f_{H},\\\\bm{w}^{+}-\\\\bm{w}^{k}\\\\rangle+\\\\tfrac{L}{2}\\\\left\\\\|\\\\bm{w}^{+}-\\\\bm{w}^{k}\\\\right\\\\|^{2} ; set 𝒘 k + 1 = 𝒘 + \\\\bm{w}^{k+1}=\\\\bm{w}^{+} , L w k + 1 = δ L L_{w}^{k+1}=\\\\delta L .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"( z z -block) same with roles swapped, using 𝒘 k + 1 \\\\bm{w}^{k+1} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prox maps of MCP, SCAD, ℓ 1 / 2 \\\\ell_{1/2} are closed-form (Appendix A ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The prox inequality at modulus η L \\\\eta L and the accepted descent condition at constant L L add to a per-block decrease of ( η − 1 ) L 2 ‖ 𝚫 ‖ 2 \\\\frac{(\\\\eta-1)L}{2}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|^{2} . At η = 1 \\\\eta=1 the quadratic terms cancel exactly and only monotonicity survives: for convex penalties the prox inequality self-improves and η = 1 \\\\eta=1 suffices, but for our non-convex penalties the sufficient-decrease property — hence the entire KL mechanism — genuinely requires η > 1 \\\\eta>1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"F F need not be coercive (MCP/SCAD are bounded; tanh \\\\tanh -type losses are bounded). We add the indicator of a ball B R = { ‖ ( 𝐰 , 𝐳 ) ‖ 2 ≤ R } B_{R}=\\\\{\\\\left\\\\|(\\\\bm{w},\\\\bm{z})\\\\right\\\\|_{2}\\\\leq R\\\\} , R R large; in every experiment the constraint is verified inactive a posteriori. For ℓ q \\\\ell_{q} , F F is coercive and no device is needed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Suppose g ∈ C 2 g\\\\in C^{2} (squared loss) or f H ∈ C loc 1 , 1 f_{H}\\\\in C^{1,1}_{\\\\mathrm{loc}} (Huberized: ∇ f H \\\\nabla f_{H} is a composition of the 1 1 -Lipschitz clipping with locally Lipschitz maps), the penalties are proper lsc prox-bounded, F F is definable in an o-minimal structure (true for polynomial/tanh/sigmoid links with MCP/SCAD/ ℓ q \\\\ell_{q} : semialgebraic and ℝ an , exp \\\\mathbb{R}_{\\\\mathrm{an,exp}} -definable pieces), and the iterates are bounded. Then: (i) backtracking terminates finitely at every step with accepted constants in [ L min , L ¯ ] [L_{\\\\min},\\\\bar{L}] ; (ii) F ( 𝐰 k , 𝐳 k ) F(\\\\bm{w}^{k},\\\\bm{z}^{k}) decreases by at least ( η − 1 ) L min 2 ‖ ( 𝐰 , 𝐳 ) k + 1 − ( 𝐰 , 𝐳 ) k ‖ 2 \\\\frac{(\\\\eta-1)L_{\\\\min}}{2}\\\\left\\\\|(\\\\bm{w},\\\\bm{z})^{k+1}-(\\\\bm{w},\\\\bm{z})^{k}\\\\right\\\\|^{2} per iteration; (iii) there exist subgradients with ‖ 𝐯 k + 1 ‖ ≤ ( η L ¯ + L f ) ‖ ( 𝐰 , 𝐳 ) k + 1 − ( 𝐰 , 𝐳 ) k ‖ \\\\left\\\\|\\\\bm{v}^{k+1}\\\\right\\\\|\\\\leq(\\\\eta\\\\bar{L}+L_{f})\\\\,\\\\left\\\\|(\\\\bm{w},\\\\bm{z})^{k+1}-(\\\\bm{w},\\\\bm{z})^{k}\\\\right\\\\| ; (iv) the whole sequence converges to a single critical point of F F with finite length; and (v) if the KL exponent at the limit is θ = 1 / 2 \\\\theta=1/2 — which holds on the RSC event of Theorem 2 at stationary points with the active-set structure of the penalty — convergence is R-linear.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(i)–(iii) are Lemmas 1–3 of the algorithmic analysis: (i) follows from local Lipschitzness of ∇ f H \\\\nabla f_{H} on compacts and geometric growth of L L ; (ii) from adding the prox inequality (modulus η L \\\\eta L ) to the accepted descent condition (constant L L ), which leaves ( η − 1 ) L 2 ‖ 𝚫 ‖ 2 \\\\frac{(\\\\eta-1)L}{2}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|^{2} (Remark 2 ); (iii) from first-order optimality of the prox step plus Lipschitz gradient differences. Conditions (ii)–(iii) are (H1)–(H2) of the abstract framework of Attouch–Bolte–Svaiter [ 12 ] ; continuity (H3) holds along prox steps; definability gives the KL property [ 13 , 11 ] ; (iv) follows from the framework and (v) from the Łojasiewicz-exponent rate analysis [ 14 , 15 ] . ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Statistical Recovery Theory\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Throughout, α ≔ ℓ g 2 / 16 \\\\alpha\\\\coloneqq\\\\ell_{g}^{2}/16 , α ¯ ≔ α − ρ > 0 \\\\underline{\\\\alpha}\\\\coloneqq\\\\alpha-\\\\rho>0 , and r H ≔ δ H / ( 8 L g ) r_{H}\\\\coloneqq\\\\delta_{H}/(8L_{g}) (the Huber-band radius ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Geometry\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For any 𝚫 = ( 𝛅 w , 𝛅 z ) \\\\bm{\\\\Delta}=(\\\\bm{\\\\delta}_{w},\\\\bm{\\\\delta}_{z}) , ‖ 𝐁 𝚫 ‖ 2 2 ≥ ‖ 𝚫 ‖ 2 2 − 2 ε ‖ 𝛅 w ‖ 1 ‖ 𝛅 z ‖ 1 \\\\left\\\\|\\\\bm{B}\\\\bm{\\\\Delta}\\\\right\\\\|_{2}^{2}\\\\geq\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2}^{2}-2\\\\varepsilon\\\\left\\\\|\\\\bm{\\\\delta}_{w}\\\\right\\\\|_{1}\\\\left\\\\|\\\\bm{\\\\delta}_{z}\\\\right\\\\|_{1} ; on the cone 𝒞 ( S , 3 ) = { ‖ 𝚫 S c ‖ 1 ≤ 3 ‖ 𝚫 S ‖ 1 } \\\\mathcal{C}(S,3)=\\\\{\\\\left\\\\|\\\\bm{\\\\Delta}_{S^{c}}\\\\right\\\\|_{1}\\\\leq 3\\\\left\\\\|\\\\bm{\\\\Delta}_{S}\\\\right\\\\|_{1}\\\\} , ‖ 𝐁 𝚫 ‖ 2 2 ≥ ( 1 − 16 ε s ) ‖ 𝚫 ‖ 2 2 ≥ 1 2 ‖ 𝚫 ‖ 2 2 \\\\left\\\\|\\\\bm{B}\\\\bm{\\\\Delta}\\\\right\\\\|_{2}^{2}\\\\geq(1-16\\\\varepsilon s)\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2}^{2}\\\\geq\\\\tfrac{1}{2}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2}^{2} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"‖ 𝑩 𝚫 ‖ 2 = ‖ 𝚫 ‖ 2 + 2 𝜹 w ⊤ 𝚽 ⊤ 𝚿 𝜹 z \\\\left\\\\|\\\\bm{B}\\\\bm{\\\\Delta}\\\\right\\\\|^{2}=\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|^{2}+2\\\\bm{\\\\delta}_{w}^{\\\\top}\\\\bm{\\\\Phi}^{\\\\top}\\\\bm{\\\\Psi}\\\\bm{\\\\delta}_{z} and the cross term is bounded entrywise by ε ‖ 𝜹 w ‖ 1 ‖ 𝜹 z ‖ 1 ≤ ε 4 ‖ 𝚫 ‖ 1 2 ≤ 4 ε s ‖ 𝚫 ‖ 2 2 \\\\varepsilon\\\\left\\\\|\\\\bm{\\\\delta}_{w}\\\\right\\\\|_{1}\\\\left\\\\|\\\\bm{\\\\delta}_{z}\\\\right\\\\|_{1}\\\\leq\\\\tfrac{\\\\varepsilon}{4}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}^{2}\\\\leq 4\\\\varepsilon s\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2}^{2} on the cone, where ‖ 𝚫 ‖ 1 ≤ 4 s ‖ 𝚫 ‖ 2 \\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}\\\\leq 4\\\\sqrt{s}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2} . ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Restricted strong convexity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let Assumptions 1 – 6 hold, δ H ≥ 4 σ \\\\delta_{H}\\\\geq 4\\\\sigma , and m ≥ C log ( 2 n ) m\\\\geq C\\\\log(2n) . With probability at least 1 − c 1 e − c 2 m − 2 ( 2 n ) − 2 − ( m n ) − 1 1-c_{1}e^{-c_{2}m}-2(2n)^{-2}-(mn)^{-1} , for all 𝚫 \\\\bm{\\\\Delta} with ‖ 𝐁 𝚫 ‖ 2 ≤ r H \\\\left\\\\|\\\\bm{B}\\\\bm{\\\\Delta}\\\\right\\\\|_{2}\\\\leq r_{H} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with τ 0 = c 3 ( L g 2 + δ H 2 M g 2 / ℓ g 2 ) \\\\tau_{0}=c_{3}\\\\bigl(L_{g}^{2}+\\\\delta_{H}^{2}M_{g}^{2}/\\\\ell_{g}^{2}\\\\bigr) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let t i = ⟨ 𝒂 i , 𝑩 𝚫 ⟩ t_{i}=\\\\langle\\\\bm{a}_{i},\\\\bm{B}\\\\bm{\\\\Delta}\\\\rangle , u i ∗ = ⟨ 𝒂 i , 𝒙 ∗ ⟩ u_{i}^{\\\\ast}=\\\\langle\\\\bm{a}_{i},\\\\bm{x}^{\\\\ast}\\\\rangle , d i = g ( u i ∗ + t i ) − g ( u i ∗ ) d_{i}=g(u_{i}^{\\\\ast}+t_{i})-g(u_{i}^{\\\\ast}) . The gradient-increment summand is [ ϕ i ( t i ) − ϕ i ( 0 ) ] t i [\\\\phi_{i}(t_{i})-\\\\phi_{i}(0)]t_{i} with ϕ i ( t ) = − ψ δ H ( e i − ( g ( u i ∗ + t ) − g ( u i ∗ ) ) ) g ′ ( u i ∗ + t ) \\\\phi_{i}(t)=-\\\\psi_{\\\\delta_{H}}(e_{i}-(g(u_{i}^{\\\\ast}+t)-g(u_{i}^{\\\\ast})))g^{\\\\prime}(u_{i}^{\\\\ast}+t) ; split ϕ i ( t i ) − ϕ i ( 0 ) = A i + B i \\\\phi_{i}(t_{i})-\\\\phi_{i}(0)=A_{i}+B_{i} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Curvature part. ψ δ H \\\\psi_{\\\\delta_{H}} nondecreasing, g g increasing, g ′ > 0 g^{\\\\prime}>0 give A i t i ≥ 0 A_{i}t_{i}\\\\geq 0 for every sample . On ℰ i = { | e i | ≤ δ H / 2 } ∩ { | t i | ≤ δ H / ( 2 L g ) } \\\\mathcal{E}_{i}=\\\\{|e_{i}|\\\\leq\\\\delta_{H}/2\\\\}\\\\cap\\\\{|t_{i}|\\\\leq\\\\delta_{H}/(2L_{g})\\\\} , both e i e_{i} and e i − d i e_{i}-d_{i} lie in the linear zone of ψ δ H \\\\psi_{\\\\delta_{H}} , so A i t i = g ′ ( ξ i ) g ′ ( u i ∗ + t i ) t i 2 ≥ ℓ g 2 t i 2 A_{i}t_{i}=g^{\\\\prime}(\\\\xi_{i})g^{\\\\prime}(u_{i}^{\\\\ast}+t_{i})t_{i}^{2}\\\\geq\\\\ell_{g}^{2}t_{i}^{2} . Hence T 1 ≥ ℓ g 2 m ∑ i b i φ K ( t i ) T_{1}\\\\geq\\\\frac{\\\\ell_{g}^{2}}{m}\\\\sum_{i}b_{i}\\\\varphi_{K}(t_{i}) with b i = 𝟏 { | e i | ≤ δ H / 2 } b_{i}=\\\\bm{1}\\\\{|e_{i}|\\\\leq\\\\delta_{H}/2\\\\} ( ℙ ( b i = 1 ) ≥ 1 2 \\\\mathbb{P}(b_{i}{=}1)\\\\geq\\\\frac{1}{2} by Chebyshev and δ H ≥ 4 σ \\\\delta_{H}\\\\geq 4\\\\sigma ), K = δ H / ( 2 L g ) K=\\\\delta_{H}/(2L_{g}) , and φ K \\\\varphi_{K} the truncated quadratic of Lemma 4 , which yields the uniform lower bound ℓ g 2 8 ‖ 𝑩 𝚫 ‖ 2 − c L g 2 log 2 n m ‖ 𝚫 ‖ 1 2 \\\\frac{\\\\ell_{g}^{2}}{8}\\\\left\\\\|\\\\bm{B}\\\\bm{\\\\Delta}\\\\right\\\\|^{2}-cL_{g}^{2}\\\\frac{\\\\log 2n}{m}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}^{2} over ‖ 𝑩 𝚫 ‖ ≤ K / 4 = r H \\\\left\\\\|\\\\bm{B}\\\\bm{\\\\Delta}\\\\right\\\\|\\\\leq K/4=r_{H} . Multiplier part. 𝔼 [ B i t i ∣ 𝒂 i ] = 0 \\\\mathbb{E}[B_{i}t_{i}\\\\mid\\\\bm{a}_{i}]=0 (noise symmetry), | B i t i | ≤ δ H M g t i 2 |B_{i}t_{i}|\\\\leq\\\\delta_{H}M_{g}t_{i}^{2} ; Lemma 5 bounds sup | 1 m ∑ B i t i | ≤ ℓ g 2 32 ‖ 𝚫 ‖ 2 + c δ H 2 M g 2 ℓ g 2 log 2 n m ‖ 𝚫 ‖ 1 2 \\\\sup|{\\\\textstyle\\\\frac{1}{m}\\\\sum}B_{i}t_{i}|\\\\leq\\\\frac{\\\\ell_{g}^{2}}{32}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|^{2}+c\\\\frac{\\\\delta_{H}^{2}M_{g}^{2}}{\\\\ell_{g}^{2}}\\\\frac{\\\\log 2n}{m}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}^{2} . Assembly with Lemma 1 (cone case; the all-vector case absorbs the incoherence deficit ≤ ε 2 ‖ 𝚫 ‖ 1 2 \\\\leq\\\\frac{\\\\varepsilon}{2}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}^{2} into the tolerance) gives ( 3 ). ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Huber loss is linear outside its band — it has no curvature there; an RSC radius proportional to δ H \\\\delta_{H} is the correct statement, exactly as in high-dimensional robust regression [ 16 , 17 ] . Self-consistency (final error ≤ r H \\\\leq r_{H} ) holds for m ≳ ( L g 2 min ( σ , δ H ) / ( ℓ g 2 δ H ) ) 2 s log ( 2 n ) m\\\\gtrsim(L_{g}^{2}\\\\min(\\\\sigma,\\\\delta_{H})/(\\\\ell_{g}^{2}\\\\delta_{H}))^{2}s\\\\log(2n) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With probability ≥ 1 − 2 ( 2 n ) − 2 − ( m n ) − 1 \\\\geq 1-2(2n)^{-2}-(mn)^{-1} , for m ≥ log ( 2 n ) m\\\\geq\\\\log(2n) : ‖ ∇ f H ( 𝛃 ∗ ) ‖ ∞ ≤ c L g min ( σ , δ H ) log ( 2 n ) / m \\\\left\\\\|\\\\nabla f_{H}(\\\\bm{\\\\beta}^{\\\\ast})\\\\right\\\\|_{\\\\infty}\\\\leq cL_{g}\\\\min(\\\\sigma,\\\\delta_{H})\\\\sqrt{\\\\log(2n)/m} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Coordinate j j is 1 m ∑ i e i ′ g ′ ( u i ∗ ) ⟨ 𝒂 i , 𝒃 j ⟩ \\\\frac{1}{m}\\\\sum_{i}e_{i}^{\\\\prime}g^{\\\\prime}(u_{i}^{\\\\ast})\\\\langle\\\\bm{a}_{i},\\\\bm{b}_{j}\\\\rangle with e i ′ = ψ δ H ( e i ) e_{i}^{\\\\prime}=\\\\psi_{\\\\delta_{H}}(e_{i}) mean-zero, bounded by δ H \\\\delta_{H} , variance ≤ min ( σ 2 , δ H 2 ) \\\\leq\\\\min(\\\\sigma^{2},\\\\delta_{H}^{2}) . Condition on 𝑨 \\\\bm{A} ; the conditional sub-Gaussian parameter is L g 2 min ( σ 2 , δ H 2 ) m ⋅ 1 m ‖ 𝑨 𝒃 j ‖ 2 \\\\frac{L_{g}^{2}\\\\min(\\\\sigma^{2},\\\\delta_{H}^{2})}{m}\\\\cdot\\\\frac{1}{m}\\\\left\\\\|\\\\bm{A}\\\\bm{b}_{j}\\\\right\\\\|^{2} , and 1 m ‖ 𝑨 𝒃 j ‖ 2 ≤ 2 \\\\frac{1}{m}\\\\left\\\\|\\\\bm{A}\\\\bm{b}_{j}\\\\right\\\\|^{2}\\\\leq 2 w.p. 1 − e − c m 1-e^{-cm} ( χ m 2 \\\\chi_{m}^{2} Bernstein). Union over 2 n 2n columns. ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Main error bound\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The estimator carries the ℓ 1 \\\\ell_{1} side constraint Ω = { ‖ 𝜷 ‖ 1 ≤ R 1 } \\\\Omega=\\\\{\\\\left\\\\|\\\\bm{\\\\beta}\\\\right\\\\|_{1}\\\\leq R_{1}\\\\} , R 1 ≥ 2 ‖ 𝜷 ∗ ‖ 1 R_{1}\\\\geq 2\\\\left\\\\|\\\\bm{\\\\beta}^{\\\\ast}\\\\right\\\\|_{1} [ 7 ] ; Remark 5 shows it is inactive at the solution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the RSC event ( 3 ), with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"every stationary point 𝛃 ~ \\\\tilde{\\\\bm{\\\\beta}} of F F over Ω \\\\Omega with ‖ 𝛃 ~ − 𝛃 ∗ ‖ 2 ≤ r H \\\\left\\\\|\\\\tilde{\\\\bm{\\\\beta}}-\\\\bm{\\\\beta}^{\\\\ast}\\\\right\\\\|_{2}\\\\leq r_{H} satisfies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Write 𝚫 = 𝜷 ~ − 𝜷 ∗ \\\\bm{\\\\Delta}=\\\\tilde{\\\\bm{\\\\beta}}-\\\\bm{\\\\beta}^{\\\\ast} , a = ‖ 𝚫 S ‖ 1 a=\\\\left\\\\|\\\\bm{\\\\Delta}_{S}\\\\right\\\\|_{1} , b = ‖ 𝚫 S c ‖ 1 b=\\\\left\\\\|\\\\bm{\\\\Delta}_{S^{c}}\\\\right\\\\|_{1} . (1) Constrained stationarity against the feasible 𝜷 ∗ \\\\bm{\\\\beta}^{\\\\ast} : ⟨ ∇ f H ( 𝜷 ~ ) − ∇ f H ( 𝜷 ∗ ) , 𝚫 ⟩ ≤ − ⟨ 𝝃 , 𝚫 ⟩ − ⟨ ∇ f H ( 𝜷 ∗ ) , 𝚫 ⟩ \\\\langle\\\\nabla f_{H}(\\\\tilde{\\\\bm{\\\\beta}})-\\\\nabla f_{H}(\\\\bm{\\\\beta}^{\\\\ast}),\\\\bm{\\\\Delta}\\\\rangle\\\\leq-\\\\langle\\\\bm{\\\\xi},\\\\bm{\\\\Delta}\\\\rangle-\\\\langle\\\\nabla f_{H}(\\\\bm{\\\\beta}^{\\\\ast}),\\\\bm{\\\\Delta}\\\\rangle , 𝝃 ∈ ∂ P ( 𝜷 ~ ) \\\\bm{\\\\xi}\\\\in\\\\partial P(\\\\tilde{\\\\bm{\\\\beta}}) . (2) Weak convexity of P + ρ 2 ∥ ⋅ ∥ 2 P+\\\\frac{\\\\rho}{2}\\\\left\\\\|\\\\cdot\\\\right\\\\|^{2} : − ⟨ 𝝃 , 𝚫 ⟩ ≤ P ( 𝜷 ∗ ) − P ( 𝜷 ~ ) + ρ 2 ‖ 𝚫 ‖ 2 -\\\\langle\\\\bm{\\\\xi},\\\\bm{\\\\Delta}\\\\rangle\\\\leq P(\\\\bm{\\\\beta}^{\\\\ast})-P(\\\\tilde{\\\\bm{\\\\beta}})+\\\\frac{\\\\rho}{2}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|^{2} . (3) Penalty comparison via (P1)–(P4) and Assumption 6 : P ( 𝜷 ∗ ) − P ( 𝜷 ~ ) ≤ λ a − λ b + ρ 2 ‖ 𝚫 S c ‖ 2 P(\\\\bm{\\\\beta}^{\\\\ast})-P(\\\\tilde{\\\\bm{\\\\beta}})\\\\leq\\\\lambda a-\\\\lambda b+\\\\frac{\\\\rho}{2}\\\\left\\\\|\\\\bm{\\\\Delta}_{S^{c}}\\\\right\\\\|^{2} . (4) Gradient term: ≤ λ 4 ( a + b ) \\\\leq\\\\frac{\\\\lambda}{4}(a+b) by ( 4 ). (5) Tolerance: ‖ 𝚫 ‖ 1 ≤ 2 R 1 \\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}\\\\leq 2R_{1} on Ω \\\\Omega , so τ 0 log 2 n m ‖ 𝚫 ‖ 1 2 ≤ λ 4 ( a + b ) \\\\tau_{0}\\\\frac{\\\\log 2n}{m}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}^{2}\\\\leq\\\\frac{\\\\lambda}{4}(a+b) by the second branch of ( 4 ). (6) Assembling with ( 3 ) and collecting ρ \\\\rho -terms: α ¯ ‖ 𝚫 ‖ 2 ≤ 3 λ 2 a − λ 2 b \\\\underline{\\\\alpha}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|^{2}\\\\leq\\\\frac{3\\\\lambda}{2}a-\\\\frac{\\\\lambda}{2}b . (7) Nonnegativity of the left side forces b ≤ 3 a b\\\\leq 3a : the cone, whence ‖ 𝚫 ‖ 1 ≤ 4 s ‖ 𝚫 ‖ 2 \\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}\\\\leq 4\\\\sqrt{s}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2} . (8) Dropping − λ 2 b -\\\\frac{\\\\lambda}{2}b and using a ≤ s ‖ 𝚫 ‖ 2 a\\\\leq\\\\sqrt{s}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2} : ‖ 𝚫 ‖ 2 ≤ 3 λ s 2 α ¯ \\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2}\\\\leq\\\\frac{3\\\\lambda\\\\sqrt{s}}{2\\\\underline{\\\\alpha}} , and the ℓ 1 \\\\ell_{1} bound follows. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The conclusion gives ‖ 𝛃 ~ ‖ 1 ≤ ‖ 𝛃 ∗ ‖ 1 + 6 λ s / α ¯ < R 1 \\\\left\\\\|\\\\tilde{\\\\bm{\\\\beta}}\\\\right\\\\|_{1}\\\\leq\\\\left\\\\|\\\\bm{\\\\beta}^{\\\\ast}\\\\right\\\\|_{1}+6\\\\lambda s/\\\\underline{\\\\alpha}<R_{1} for λ \\\\lambda of the prescribed order and m ≳ s log 2 n m\\\\gtrsim s\\\\log 2n : constrained and unconstrained stationary points coincide, and NLD-PALM needs no ℓ 1 \\\\ell_{1} projection (verified numerically in all experiments).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With λ ≍ L g min ( σ , δ H ) log ( 2 n ) / m \\\\lambda\\\\asymp L_{g}\\\\min(\\\\sigma,\\\\delta_{H})\\\\sqrt{\\\\log(2n)/m} and m ≳ max { s log ( 2 n / s ) , ( τ 0 / α ¯ ) 2 s log 2 n , ( τ 0 ‖ 𝛃 ∗ ‖ 2 / σ ) 2 s log ( 2 n ) } m\\\\gtrsim\\\\max\\\\{s\\\\log(2n/s),\\\\,(\\\\tau_{0}/\\\\underline{\\\\alpha})^{2}s\\\\log 2n,\\\\,(\\\\tau_{0}\\\\left\\\\|\\\\bm{\\\\beta}^{\\\\ast}\\\\right\\\\|_{2}/\\\\sigma)^{2}s\\\\log(2n)\\\\} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"under sub-Gaussian or finite-variance symmetric noise.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Oracle property\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let E 2 = 3 λ s 2 α ¯ E_{2}=\\\\frac{3\\\\lambda\\\\sqrt{s}}{2\\\\underline{\\\\alpha}} and assume min j ∈ S | β j ∗ | ≥ a λ + E 2 \\\\min_{j\\\\in S}|\\\\beta_{j}^{\\\\ast}|\\\\geq a\\\\lambda+E_{2} . (a) If supp ( 𝛃 ~ ) ⊆ S \\\\operatorname{supp}(\\\\tilde{\\\\bm{\\\\beta}})\\\\subseteq S , then 𝛃 ~ \\\\tilde{\\\\bm{\\\\beta}} solves the unpenalized problem on ℝ S \\\\mathbb{R}^{S} and ‖ 𝛃 ~ − 𝛃 ∗ ‖ 2 ≤ 2 α ‖ ∇ S f H ( 𝛃 ∗ ) ‖ 2 ≲ L g α min ( σ , δ H ) s / m \\\\left\\\\|\\\\tilde{\\\\bm{\\\\beta}}-\\\\bm{\\\\beta}^{\\\\ast}\\\\right\\\\|_{2}\\\\leq\\\\frac{2}{\\\\alpha}\\\\left\\\\|\\\\nabla_{S}f_{H}(\\\\bm{\\\\beta}^{\\\\ast})\\\\right\\\\|_{2}\\\\lesssim\\\\frac{L_{g}}{\\\\alpha}\\\\min(\\\\sigma,\\\\delta_{H})\\\\sqrt{s/m} w.h.p. — no log n \\\\log n , no λ \\\\lambda -bias. (b) If additionally C L E 2 ≤ λ / 4 C_{L}E_{2}\\\\leq\\\\lambda/4 with C L = c ( L g 2 + δ H M g ) C_{L}=c(L_{g}^{2}+\\\\delta_{H}M_{g}) and E 2 < γ λ / 2 E_{2}<\\\\gamma\\\\lambda/2 , then supp ( 𝛃 ~ ) ⊆ S \\\\operatorname{supp}(\\\\tilde{\\\\bm{\\\\beta}})\\\\subseteq S .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(a) Beta-min and ‖ 𝚫 ‖ ∞ ≤ E 2 \\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{\\\\infty}\\\\leq E_{2} put all j ∈ S j\\\\in S in the zero-derivative region of P λ P_{\\\\lambda} (P4), so stationarity reduces to ∇ S f H ( 𝜷 ~ ) = 0 \\\\nabla_{S}f_{H}(\\\\tilde{\\\\bm{\\\\beta}})=0 ; RSC on the fixed s s -dimensional subspace and a vector-Bernstein bound on ‖ ∇ S f H ( 𝜷 ∗ ) ‖ 2 \\\\left\\\\|\\\\nabla_{S}f_{H}(\\\\bm{\\\\beta}^{\\\\ast})\\\\right\\\\|_{2} (no union over 2 n 2n coordinates — this is where log n \\\\log n disappears) give the rate. (b) For j ∈ S c j\\\\in S^{c} with 𝜷 ~ j ≠ 0 \\\\tilde{\\\\bm{\\\\beta}}_{j}\\\\neq 0 , MCP stationarity forces | ∇ j f H ( 𝜷 ~ ) | = λ − | 𝜷 ~ j | / γ |\\\\nabla_{j}f_{H}(\\\\tilde{\\\\bm{\\\\beta}})|=\\\\lambda-|\\\\tilde{\\\\bm{\\\\beta}}_{j}|/\\\\gamma or | 𝜷 ~ j | ≥ γ λ |\\\\tilde{\\\\bm{\\\\beta}}_{j}|\\\\geq\\\\gamma\\\\lambda . The coordinatewise Lipschitz bound ‖ ∇ f H ( 𝜷 ~ ) − ∇ f H ( 𝜷 ∗ ) ‖ ∞ ≤ C L ‖ 𝚫 ‖ 2 \\\\left\\\\|\\\\nabla f_{H}(\\\\tilde{\\\\bm{\\\\beta}})-\\\\nabla f_{H}(\\\\bm{\\\\beta}^{\\\\ast})\\\\right\\\\|_{\\\\infty}\\\\leq C_{L}\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{2} (Cauchy–Schwarz with upper restricted eigenvalues) gives | ∇ j f H ( 𝜷 ~ ) | ≤ λ / 2 |\\\\nabla_{j}f_{H}(\\\\tilde{\\\\bm{\\\\beta}})|\\\\leq\\\\lambda/2 , forcing | 𝜷 ~ j | ≥ γ λ / 2 > E 2 ≥ | 𝜷 ~ j | |\\\\tilde{\\\\bm{\\\\beta}}_{j}|\\\\geq\\\\gamma\\\\lambda/2>E_{2}\\\\geq|\\\\tilde{\\\\bm{\\\\beta}}_{j}| — contradiction. The condition C L E 2 ≤ λ / 4 C_{L}E_{2}\\\\leq\\\\lambda/4 amounts to s ≲ α ¯ / C L \\\\sqrt{s}\\\\lesssim\\\\underline{\\\\alpha}/C_{L} ; the primal–dual witness of [ 8 ] removes this restriction and is left as an optional appendix. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ℓ 1 \\\\ell_{1} stationary points satisfy | ∇ j f H | = λ |\\\\nabla_{j}f_{H}|=\\\\lambda at every nonzero coordinate, embedding a c λ s c\\\\lambda\\\\sqrt{s} bias for any λ \\\\lambda compatible with ( 4 ) — visible as the error plateau in Fig. 1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Unknown link\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume g g unknown, monotone, with μ g = 𝔼 [ g ′ ( γ ) ] ≠ 0 \\\\mu_{g}=\\\\mathbb{E}[g^{\\\\prime}(\\\\gamma)]\\\\neq 0 , θ g 2 = 𝔼 [ ( g ( γ ) − μ g γ ) 2 ] \\\\theta_{g}^{2}=\\\\mathbb{E}[(g(\\\\gamma)-\\\\mu_{g}\\\\gamma)^{2}] , γ ∼ 𝒩 ( 0 , 1 ) \\\\gamma\\\\sim\\\\mathcal{N}(0,1) , ‖ 𝒙 ∗ ‖ 2 = 1 \\\\left\\\\|\\\\bm{x}^{\\\\ast}\\\\right\\\\|_{2}=1 ; let ν 2 = θ g 2 + σ 2 \\\\nu^{2}=\\\\theta_{g}^{2}+\\\\sigma^{2} . Run the same program with the linear Huberized loss and adaptive clipping δ H = 2 ν log m \\\\delta_{H}=2\\\\nu\\\\sqrt{\\\\log m} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With probability ≥ 1 − c ( 2 n ) − 2 \\\\geq 1-c(2n)^{-2} , ‖ 1 m ∑ i ψ δ H ( ξ i ) 𝐁 ⊤ 𝐚 i ‖ ∞ ≤ c ν log ( 2 n ) log ( m ) / m \\\\left\\\\|\\\\frac{1}{m}\\\\sum_{i}\\\\psi_{\\\\delta_{H}}(\\\\xi_{i})\\\\bm{B}^{\\\\top}\\\\bm{a}_{i}\\\\right\\\\|_{\\\\infty}\\\\leq c\\\\,\\\\nu\\\\sqrt{\\\\log(2n)\\\\log(m)/m} , where ξ i = g ( u i ∗ ) − μ g u i ∗ + e i \\\\xi_{i}=g(u_{i}^{\\\\ast})-\\\\mu_{g}u_{i}^{\\\\ast}+e_{i} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Split 𝒂 i = γ i 𝒙 ∗ + 𝑷 ⟂ 𝒂 i \\\\bm{a}_{i}=\\\\gamma_{i}\\\\bm{x}^{\\\\ast}+\\\\bm{P}_{\\\\perp}\\\\bm{a}_{i} with γ i = ⟨ 𝒂 i , 𝒙 ∗ ⟩ \\\\gamma_{i}=\\\\langle\\\\bm{a}_{i},\\\\bm{x}^{\\\\ast}\\\\rangle ; ξ i \\\\xi_{i} depends on ( γ i , e i ) (\\\\gamma_{i},e_{i}) only. Orthogonal part: conditionally Gaussian with variance 1 m ⋅ 1 m ∑ ψ δ H ( ξ i ) 2 ≤ c ν 2 m \\\\frac{1}{m}\\\\cdot\\\\frac{1}{m}\\\\sum\\\\psi_{\\\\delta_{H}}(\\\\xi_{i})^{2}\\\\leq\\\\frac{c\\\\nu^{2}}{m} w.h.p. (Bernstein), giving c ν log ( 2 n ) / m c\\\\nu\\\\sqrt{\\\\log(2n)/m} after a union bound. Parallel bias: 𝔼 [ ξ γ ] = 0 \\\\mathbb{E}[\\\\xi\\\\gamma]=0 by Stein’s identity ( 𝔼 [ g ( γ ) γ ] = 𝔼 [ g ′ ( γ ) ] = μ g \\\\mathbb{E}[g(\\\\gamma)\\\\gamma]=\\\\mathbb{E}[g^{\\\\prime}(\\\\gamma)]=\\\\mu_{g} ), and | 𝔼 [ ψ δ H ( ξ ) γ ] | ≤ 𝔼 [ ξ 2 𝟏 { | ξ | > δ H } ] ≤ c ν e − δ H 2 / ( c ′ ν 2 ) ≤ c ν m − 4 / c ′ |\\\\mathbb{E}[\\\\psi_{\\\\delta_{H}}(\\\\xi)\\\\gamma]|\\\\leq\\\\sqrt{\\\\mathbb{E}[\\\\xi^{2}\\\\bm{1}\\\\{|\\\\xi|>\\\\delta_{H}\\\\}]}\\\\leq c\\\\nu e^{-\\\\delta_{H}^{2}/(c^{\\\\prime}\\\\nu^{2})}\\\\leq c\\\\nu m^{-4/c^{\\\\prime}} by sub-Gaussianity of ξ \\\\xi and the choice of δ H \\\\delta_{H} . Parallel deviation: ψ δ H ( ξ i ) γ i \\\\psi_{\\\\delta_{H}}(\\\\xi_{i})\\\\gamma_{i} has envelope δ H | γ i | \\\\delta_{H}|\\\\gamma_{i}| and variance ≤ c ν 2 log m \\\\leq c\\\\nu^{2}\\\\log m ; Bernstein plus the union bound gives the dominant term c ν log m log ( 2 n ) / m c\\\\nu\\\\sqrt{\\\\log m}\\\\sqrt{\\\\log(2n)/m} . ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under Assumptions 1 , 3 , 4 (a), 6 , with m ≥ C s log ( 2 n ) m\\\\geq Cs\\\\log(2n) and λ ≍ ν log ( 2 n ) log ( m ) / m \\\\lambda\\\\asymp\\\\nu\\\\sqrt{\\\\log(2n)\\\\log(m)/m} : with probability ≥ 1 − c 1 e − c 2 m − c 3 n − 1 \\\\geq 1-c_{1}e^{-c_{2}m}-c_{3}n^{-1} , every stationary point of the linear-surrogate objective with ‖ 𝛃 ~ − μ g 𝛃 ∗ ‖ 2 ≤ r H \\\\left\\\\|\\\\tilde{\\\\bm{\\\\beta}}-\\\\mu_{g}\\\\bm{\\\\beta}^{\\\\ast}\\\\right\\\\|_{2}\\\\leq r_{H} satisfies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Write y i = μ g ⟨ 𝒂 i , 𝒙 ∗ ⟩ + ξ i y_{i}=\\\\mu_{g}\\\\langle\\\\bm{a}_{i},\\\\bm{x}^{\\\\ast}\\\\rangle+\\\\xi_{i} . The loss is linear-Huber: Theorem 2 applies with g = id g=\\\\mathrm{id} ( ℓ g = L g = 1 \\\\ell_{g}=L_{g}=1 , M g = 0 M_{g}=0 ; the multiplier part vanishes identically, so no noise-symmetry is needed for RSC). Lemma 3 replaces Lemma 2 . Theorem 3 ’s deterministic core applies verbatim around μ g 𝜷 ∗ \\\\mu_{g}\\\\bm{\\\\beta}^{\\\\ast} . ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Protocol: n = 512 n=512 , s 1 = s 2 = 16 s_{1}=s_{2}=16 , 𝚽 = 𝑰 \\\\bm{\\\\Phi}=\\\\bm{I} , 𝚿 = \\\\bm{\\\\Psi}= DCT, g = tanh g=\\\\tanh , σ = 0.05 \\\\sigma=0.05 , δ H = 4 σ \\\\delta_{H}=4\\\\sigma . Regularization uses the frozen rule λ = c σ log ( 2 n ) / m \\\\lambda=c\\\\,\\\\sigma\\\\sqrt{\\\\log(2n)/m} , with c c calibrated once on a 5 5 -seed pilot at m = 400 m=400 and never re-tuned ( c = 8 c=8 for ℓ 1 \\\\ell_{1} , 16 16 for SCAD, 24 24 for MCP, 4 4 for ℓ 1 / 2 \\\\ell_{1/2} ); the DHT baseline [ 1 ] step is frozen at its pilot-best likewise, and receives the true ( s 1 , s 2 ) (s_{1},s_{2}) . Nonconvex methods are warm-started from the ℓ 1 \\\\ell_{1} solution. 15 15 Monte-Carlo trials per point, deterministic seeds, code provided.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase transition. Fig. 1 : under the frozen rule SCAD/MCP transition ≈ 1.3 \\\\approx 1.3 – 1.4 × 1.4\\\\times earlier than DHT; in the scarce regime ( m ≤ 200 m\\\\leq 200 ) the gap is decisive (SCAD 0.08 0.08 – 0.17 0.17 vs. DHT 0.50 0.50 – 0.85 0.85 ). Under per-instance oracle tuning the gap widens to ≈ 1.5 \\\\approx 1.5 – 2 × 2\\\\times ; we report the frozen protocol as the practically meaningful one. ℓ 1 \\\\ell_{1} never attains exact recovery, matching Proposition 1 ’s bias dichotomy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Robustness. With 5 % 5\\\\% gross outliers ( ± 3 \\\\pm 3 ): Huberized SCAD 0.073 0.073 vs. squared-loss SCAD 2.54 2.54 median relative error ( 35 × 35\\\\times ); Student- t ( 2 ) t(2) noise: 0.12 0.12 vs. 0.54 0.54 ; clean Gaussian noise: parity ( 0.060 0.060 vs. 0.060 0.060 ). Huberization is free insurance. An honest negative: at high Gaussian noise ( σ ≥ 0.1 \\\\sigma\\\\geq 0.1 for tanh \\\\tanh ) recovery degrades for all losses — an information-theoretic saturation effect, not loss fragility.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Scaling checks. Measured error scales linearly in σ \\\\sigma (slope 1.0 1.0 on a log-log grid up to the saturation regime) and as m − 1 / 2 m^{-1/2} over the pre-floor range, matching Corollary 1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Application. Fig. 2 : spikes and a smooth background demixed through a saturating amplifier with outliers; all spikes located at m / n ≈ 0.59 m/n\\\\approx 0.59 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We provided the first regularization-based treatment of nonlinear sparse demixing: a Huberized folded-concave estimator, a provably convergent algorithm whose over-relaxation requirement we identified, RSC-based guarantees holding at every localized stationary point in both known- and unknown-link regimes, oracle behavior unavailable to convex demixing, and robustness with only finite-variance symmetric noise. Future work: a primal–dual witness removing the dimensional condition in Proposition 1 (b), statistical theory for ℓ q \\\\ell_{q} , periodic and quantized links, and a deep-unfolded NLD-PALM with trainable threshold functions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With step β = 1 / ( η L ) \\\\beta=1/(\\\\eta L) : MCP ( γ > β \\\\gamma>\\\\beta ): 0 if | u | ≤ λ β |u|\\\\leq\\\\lambda\\\\beta ; sign ( u ) ( | u | − λ β ) / ( 1 − β / γ ) \\\\mathrm{sign}(u)(|u|-\\\\lambda\\\\beta)/(1-\\\\beta/\\\\gamma) if λ β < | u | ≤ γ λ \\\\lambda\\\\beta<|u|\\\\leq\\\\gamma\\\\lambda ; u u otherwise. SCAD ( a > 1 + β a>1+\\\\beta ): soft threshold at λ β \\\\lambda\\\\beta for | u | ≤ λ ( 1 + β ) |u|\\\\leq\\\\lambda(1+\\\\beta) ; ( ( a − 1 ) u − sign ( u ) a λ β ) / ( a − 1 − β ) ((a-1)u-\\\\mathrm{sign}(u)a\\\\lambda\\\\beta)/(a-1-\\\\beta) for | u | ≤ a λ |u|\\\\leq a\\\\lambda ; u u otherwise. ℓ 1 / 2 \\\\ell_{1/2} : half-thresholding [ 18 ] with effective parameter 2 λ β 2\\\\lambda\\\\beta (their quadratic term carries no 1 / 2 1/2 ). All three are unit-tested against brute-force minimization of the scalar prox objective; the test suite accompanies the code.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let φ K ( t ) = t 2 \\\\varphi_{K}(t)=t^{2} on | t | ≤ K / 2 |t|\\\\leq K/2 , ( K − | t | ) 2 (K-|t|)^{2} on K / 2 < | t | ≤ K K/2<|t|\\\\leq K , 0 beyond: K K -Lipschitz, t 2 𝟏 { | t | ≤ K / 2 } ≤ φ K ≤ t 2 𝟏 { | t | ≤ K } t^{2}\\\\bm{1}\\\\{|t|\\\\leq K/2\\\\}\\\\leq\\\\varphi_{K}\\\\leq t^{2}\\\\bm{1}\\\\{|t|\\\\leq K\\\\} , self-similar ( φ K ( t ) = ω 2 φ K / ω ( t / ω ) \\\\varphi_{K}(t)=\\\\omega^{2}\\\\varphi_{K/\\\\omega}(t/\\\\omega) ), monotone in K K .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let b i ∈ { 0 , 1 } b_{i}\\\\in\\\\{0,1\\\\} iid, ℙ ( b i = 1 ) = p ≥ 1 2 \\\\mathbb{P}(b_{i}{=}1)=p\\\\geq\\\\frac{1}{2} , independent of { 𝐚 i } \\\\{\\\\bm{a}_{i}\\\\} , and m ≥ c log ( 2 n ) m\\\\geq c\\\\log(2n) . With probability ≥ 1 − c 1 e − c 2 m \\\\geq 1-c_{1}e^{-c_{2}m} , for all 𝚫 \\\\bm{\\\\Delta} with ‖ 𝐯 ‖ 2 ≤ K / 4 \\\\left\\\\|\\\\bm{v}\\\\right\\\\|_{2}\\\\leq K/4 ( 𝐯 = 𝐁 𝚫 \\\\bm{v}=\\\\bm{B}\\\\bm{\\\\Delta} ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix a scale ω = ‖ 𝒗 ‖ 2 \\\\omega=\\\\left\\\\|\\\\bm{v}\\\\right\\\\|_{2} and use the per-scale level K ω = 4 ω ≤ K K_{\\\\omega}=4\\\\omega\\\\leq K (monotonicity). Population: 𝔼 [ b i φ K ω ] ≥ p ω 2 𝔼 [ γ 2 𝟏 { | γ | ≤ 2 } ] ≥ 0.73 p ω 2 \\\\mathbb{E}[b_{i}\\\\varphi_{K_{\\\\omega}}]\\\\geq p\\\\,\\\\omega^{2}\\\\mathbb{E}[\\\\gamma^{2}\\\\bm{1}\\\\{|\\\\gamma|\\\\leq 2\\\\}]\\\\geq 0.73\\\\,p\\\\,\\\\omega^{2} . Deviation: by self-similarity, work with φ 4 \\\\varphi_{4} ( 4 4 -Lipschitz) on normalized variables; symmetrization and Ledoux–Talagrand contraction [ 19 ] give 𝔼 sup Z ≤ 16 ω T 2 log ( 4 n ) / m \\\\mathbb{E}\\\\sup Z\\\\leq 16\\\\,\\\\omega\\\\,T\\\\sqrt{2\\\\log(4n)/m} over { ‖ 𝚫 ‖ 1 ≤ T } \\\\{\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}\\\\leq T\\\\} , since each coordinate of 1 m ∑ i ε i 𝑩 ⊤ 𝒂 i \\\\frac{1}{m}\\\\sum_{i}\\\\varepsilon_{i}\\\\bm{B}^{\\\\top}\\\\bm{a}_{i} is 𝒩 ( 0 , 1 / m ) \\\\mathcal{N}(0,1/m) ; Talagrand concentration (envelope 4 ω 2 4\\\\omega^{2} ) upgrades to probability 1 − e − c 2 m 1-e^{-c_{2}m} . Absorb or peel: if 32 T 2 log ( 4 n ) / m ≤ p 4 ω 32T\\\\sqrt{2\\\\log(4n)/m}\\\\leq\\\\frac{p}{4}\\\\omega the bound is direct; otherwise ω 2 ≲ log 2 n m T 2 \\\\omega^{2}\\\\lesssim\\\\frac{\\\\log 2n}{m}T^{2} and the tolerance term alone validates the display. Dyadic peeling over ω ∈ ( 2 − j − 1 K / 4 , 2 − j K / 4 ] \\\\omega\\\\in(2^{-j-1}K/4,2^{-j}K/4] with a union over O ( log m ) O(\\\\log m) shells completes the proof. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the setting of Theorem 2 , with probability ≥ 1 − ( 2 n ) − 2 \\\\geq 1-(2n)^{-2} , uniformly over ‖ 𝐯 ‖ 2 ≤ r H \\\\left\\\\|\\\\bm{v}\\\\right\\\\|_{2}\\\\leq r_{H} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where h u ( t ) = [ g ′ ( u + t ) − g ′ ( u ) ] t h_{u}(t)=[g^{\\\\prime}(u+t)-g^{\\\\prime}(u)]\\\\,t .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"h u h_{u} is 2 M g K ω 2M_{g}K_{\\\\omega} -Lipschitz with h u ( 0 ) = 0 h_{u}(0)=0 on the per-scale band; ψ δ H ( e i ) \\\\psi_{\\\\delta_{H}}(e_{i}) are iid mean-zero (symmetry), bounded by δ H \\\\delta_{H} , independent of { 𝒂 i } \\\\{\\\\bm{a}_{i}\\\\} . The multiplier inequality and contraction [ 19 ] reduce the supremum to the linear class, giving per shell ≲ δ H M g ω ‖ 𝚫 ‖ 1 log ( 2 n ) / m \\\\lesssim\\\\delta_{H}M_{g}\\\\omega\\\\left\\\\|\\\\bm{\\\\Delta}\\\\right\\\\|_{1}\\\\sqrt{\\\\log(2n)/m} ; Young’s inequality ( a b ≤ η a 2 + b 2 / 4 η ab\\\\leq\\\\eta a^{2}+b^{2}/4\\\\eta , η = ℓ g 2 / 32 \\\\eta=\\\\ell_{g}^{2}/32 ) splits this into the displayed form; Talagrand concentration and the shell/coordinate union complete the proof. ∎\"\n }\n]","isConversational":false}}},{"rowIdx":384,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10611"},"doi":{"kind":"null"},"version":{"kind":"number","value":2,"string":"2"},"title":{"kind":"string","value":"M+Adam: Low-Precision Training via Additive-Multiplicative Optimization"},"authors":{"kind":"list like","value":["Xiaoyuan Liang","Sebastian Loeschcke","Mads Toftrup","Anima Anandkumar"],"string":"[\n \"Xiaoyuan Liang\",\n \"Sebastian Loeschcke\",\n \"Mads Toftrup\",\n \"Anima Anandkumar\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10611v2"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Training large language models (LLMs) in full precision (FP32) incurs substantial memory, compute, and energy costs. Modern accelerators are optimized for reduced-precision formats such as FP8 and FP4, via specialized hardware like tensor cores (Micikevicius et al. , 2022 ) . As a result, most large-scale training pipelines adopt mixed-precision training, executing forward and backward passes in reduced precision (Micikevicius et al. , 2018 ) . However, they still retain high-precision copies (known as master weights) for optimization, since training directly in low precision results in significant degradation of accuracy and is often unstable (Zamirai et al. , 2021 ) ."},{"type":"para","text":"Recently, the precision of master weights has been reduced, e.g. BF16 instead of FP32, using techniques such as stochastic rounding (SR) (Connolly et al. , 2021 ) to keep them unbiased. While this reduces the memory overhead compared to FP32 master weights, they still exhibit degraded convergence or instability relative to full-precision FP32 training. This is because standard optimizers, such as Adam, fail to make progress at large weight magnitudes under low precision (larger quantization bins in Fig. 2 ) , since updates smaller than the local quantization step are rounded to zero."},{"type":"para","text":"To overcome this failure mode, alternative optimizers have been proposed based on multiplicative updates, such as Madam (Bernstein et al. , 2020 ) . These are effective at taking larger steps at higher magnitudes (larger quantization bins in Fig. 2 ), where standard optimizers with additive updates fail to make progress. Madam has been shown to be effective for training under extremely low precision, without high-precision master weights. For instance, Madam can train under the logarithmic number system (LNS) (Zhao et al. , 2022 ) , an extreme case of floating-point system where all the bits are allocated to the exponent, and mantissa is not present. While LNS has optimal hardware utilization in theory, it is not available in standard AI hardware."},{"type":"para","text":"Multiplicative methods such as Madam overcome the failure mode of Adam at large magnitudes (larger quantization bins in Fig. 1 ), but they have their own failure modes. They struggle near zero, where relative updates produce small absolute changes and cannot move weights through zero when there is a sign change (Fig. 1 )."},{"type":"para","text":"Our approach: The additive and multiplicative updates thus suffer from complementary modes of failure. Additive updates get rounded to zero at large magnitudes when mantissa resolution is coarse. Multiplicative updates are able to overcome this and make progress at large magnitudes, but they cannot flip signs or escape exact zero initialization.(Fig. 1 and Table 1 )."},{"type":"para","text":"This motivates us to propose M+Adam. It is a new optimizer that combines Adam-style additive updates with Madam-style multiplicative updates for low-precision training, as outlined in Algorithm 1 and Fig. 2 . We prove that M+Adam is a monotone descent method under standard smoothness assumptions, making it a valid optimizer."},{"type":"para","text":"We demonstrate stable training with BF16, FP8, and NVFP4 master weights across 60 60 M– 1 1 B LLaMA-style models and 1 1 – 8 × 8\\times Chinchilla training budgets. M+Adam trains stably without needing any stochastic rounding (SR), and improves over AdamW and AdamW+SR, as seen in Table 2 ."},{"type":"para","text":"Together, these results support our central claim: combined additive–multiplicative updates are especially useful when low-precision master weights becomes a dominant source of optimization error."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"para","text":"We review prior work on multiplicative optimization and low-precision training most closely related to our approach."},{"type":"heading","level":2,"text":"2.1 Multiplicative Updates"},{"type":"para","text":"Multiplicative update rules apply relative, scale-invariant parameter changes and have a long history in optimization (Kivinen and Warmuth, 1997 ) . In deep learning, Madam (Bernstein et al. , 2020 ) adapts this principle as a multiplicative analogue of Adam, showing improved robustness and reduced sensitivity to learning-rate scaling. Recent work revisits multiplicative or hybrid updates in modern settings: Kirtas et al. ( 2024 ) study additive–multiplicative hybrids; Zhao et al. ( 2022 ) combine multiplicative updates with a logarithmic number system to enable low-precision training without FP32 master weights; and Nishida et al. ( 2025 ) analyze log-normal multiplicative dynamics under low-precision forward passes. In contrast to log-based Madam approaches, M+Adam remains within the standard floating-point arithmetic."},{"type":"heading","level":2,"text":"2.2 Low-precision Training"},{"type":"para","text":"Mixed-precision frameworks such as AMP (PyTorch, 2025 ) and Transformer Engine (NVIDIA, 2023 ) accelerate BF16/FP8 arithmetic, but are largely orthogonal to optimizer design and typically rely on FP32 master weights for stable accumulation (Micikevicius et al. , 2018 ) . Standard additive optimizers such as AdamW can degrade when master weights are stored in reduced precision, since nearest rounding can cancel small model-weight updates (Zamirai et al. , 2021 ) . Stochastic rounding preserves small updates in expectation and can mitigate deterministic rounding bias (Ozkara et al. , 2025 ) . Kahan-style compensated updates provide another remedy by accumulating lost low-order update residuals in an auxiliary buffer, and have been shown to recover much of the accuracy lost in pure BF16 training (Zamirai et al. , 2021 ) . While both stochastic rounding and Kahan compensation can improve accumulation accuracy, they do not change the underlying additive update geometry or address dynamic-range limitations directly."},{"type":"para","text":"Several complementary approaches reduce low-precision storage costs without changing the optimizer geometry. LoQT (Loeschcke et al. , 2024 ) trains most weights in 4-bit precision while maintaining a low-rank 16-bit subspace. A separate line of work compresses optimizer states through 8-bit or 4-bit moment quantization (Dettmers et al. , 2022 ; Li et al. , 2023 ) . Fishman et al. ( 2025 ) study stable FP8 LLM pretraining at trillion-token scale with FP32 weights and FP8 forward computation, including quantized optimizer moments, focusing on system-level scaling and mixed-precision pipelines. In contrast, M+Adam modifies the update rule itself through a joint additive–multiplicative formulation, targeting the stability of low-precision weight storage rather than optimizer-state compression."},{"type":"para","text":"Recent work (Castro et al. , 2025 ) explores training with 4-bit weights and activations using micro-scaling formats such as MXFP4, where each block shares a common scale factor. These approaches mainly stabilize low-bit matrix multiplication by managing outliers and representational fidelity during computation. Because low-bit quantization is highly discrete, such systems often keep higher-precision weights and round to the target format before GEMMs. Our work addresses a complementary challenge: stable gradient-based parameter updates when the master weights themselves are stored in BF16, FP8, or NVFP4. Thus, outlier-aware quantization and M+Adam attack different bottlenecks: the former improves low-bit computation, while the latter improves numerical stability across optimization steps."},{"type":"heading","level":1,"text":"3 Background"},{"type":"para","text":"We briefly review Adam and Madam, the two optimizers that M+Adam builds upon."},{"type":"para","text":"Adam. Adam (Kingma and Ba, 2015 ) maintains exponential moving averages of the first and second moments of the gradient and applies adaptive, element-wise additive updates. AdamW (Loshchilov and Hutter, 2019 ) decouples weight decay from this adaptive gradient step:"},{"type":"para","text":"While effective in full precision, these additive updates can be rounded away in low-precision formats when their magnitudes fall below the local quantization step, which grows with | w | |w| in floating-point formats such as BF16 or FP8."},{"type":"para","text":"Madam. Madam (Bernstein et al. , 2020 ) retains Adam’s moment estimates but applies updates multiplicatively:"},{"type":"para","text":"This yields scale-invariant behavior, with step sizes proportional to | w | |w| , but purely multiplicative updates preserve sign and cannot escape w = 0 w=0 . We illustrate these complementary failure modes of AdamW and Madam in a controlled toy example (Sec. 5.1 ), which motivates combining local additive updates with multiplicative scale control."},{"type":"heading","level":1,"text":"4 M+Adam: Additive–Multiplicative Optimization"},{"type":"para","text":"M+Adam is designed for low-precision weight storage, where the optimizer must update weights on a nonuniform floating-point grid. The method combines two complementary update mechanisms: an Adam-style additive branch that acts directly on the weights, and a Madam-style multiplicative branch that applies relative scale changes. This section derives the two branches, explains how they are combined, and presents the corresponding descent result."},{"type":"heading","level":2,"text":"4.1 Additive–Multiplicative Updates"},{"type":"para","text":"Low-precision floating-point weights have nonuniform resolution: the spacing between representable values grows with the exponent. As a result, purely additive updates can become rounded back to zero at large magnitudes. When | w t | |w_{t}| is large, an additive update can be smaller than the local quantization bin and disappear after rounding; when | w t | |w_{t}| is small, however, additive updates remain important because they can update values near zero and change signs. M+Adam is designed around this observation: it combines an Adam-style additive correction with a Madam-style multiplicative scale update."},{"type":"para","text":"Let g t = ∇ w ℒ ( w t ) g_{t}=\\nabla_{w}\\mathcal{L}(w_{t}) be the gradient of weight w t w_{t} . The additive branch is the standard Adam-style update u t ( a ) u_{t}^{(a)} in the weight space,"},{"type":"para","text":"A multiplicative update is as follows element-wise:"},{"type":"para","text":"where a t a_{t} is an adaptive preconditioned descent direction. Naturally, the update a t a_{t} is based on the gradient g t ( e ) g_{t}^{(e)} with respect to the exponent. To obtain this, we use the mantissa m t m_{t} and exponent e t e_{t} in floating-point decomposition in the base-2 system:"},{"type":"para","text":"The corresponding gradient simplifies as"},{"type":"para","text":"Thus the gradient g t ( e ) g_{t}^{(e)} can be expressed without the explicit exponent value, and we use this in our updates."},{"type":"para","text":"We then apply adaptive normalization to g t ( e ) g_{t}^{(e)} . In our implementation, the multiplicative branch uses a second-moment accumulator,"},{"type":"para","text":"Thus, u ~ t ( m ) \\widetilde{u}_{t}^{(m)} defines the branch’s proposed scale movement:"},{"type":"para","text":"The threshold τ t > 0 \\tau_{t}>0 prevents division by very small weights, and clipping keeps the relative factor well-defined."},{"type":"para","text":"Finally, M+Adam combines the multiplicative rescaling with the additive displacement:"},{"type":"para","text":"This structure directly targets the two failure modes shown in Fig. 1 . Additive updates can change signs and update small-magnitude weights, but they may be lost to rounding at large magnitudes. Multiplicative updates preserve relative progress at large magnitudes, but by themselves are sign-preserving and cannot recover exact zeros. By combining both branches, M+Adam maintains scale-aware progress where additive updates are rounded away while retaining the local flexibility needed near zero. Algorithm 1 summarizes the core method. Optionally, clipping is added for more stability, which is not listed in Algorithm 1 , as outlined in App. A ."},{"type":"heading","level":2,"text":"4.2 Theoretical Analysis"},{"type":"para","text":"We analyze the deterministic exact-arithmetic core of Algorithm 1 . Let ℒ ( w ) \\mathcal{L}(w) be the weight-space objective, let g = ∇ w ℒ ( w t ) g=\\nabla_{w}\\mathcal{L}(w_{t}) , and define the base-2 gradient with respect to exponent signal"},{"type":"para","text":"obtained by differentiating z ↦ ℒ ( w t 2 z ) z\\mapsto\\mathcal{L}(w_{t}2^{z}) at z = 0 z=0 . Algorithm 1 forms a relative multiplicative update u ( m ) u^{(m)} , sets s = 1 + u ( m ) s=1+u^{(m)} , and updates w t + 1 = w t s + u ( a ) w_{t+1}=w_{t}s+u^{(a)} . For s > 0 s>0 , writing z = log 2 s z=\\log_{2}s gives the equivalent scale-then-add form w t + 1 = w t 2 z + u ( a ) w_{t+1}=w_{t}2^{z}+u^{(a)} . We assume the following local smoothness bound in these coordinates: for all ( d , z ) (d,z) in a neighborhood of the update,"},{"type":"para","text":"Here L a L_{a} and L e L_{e} control curvature within the additive and exponent directions, while L a e L_{ae} controls their interaction."},{"type":"para","text":"Assume Eq. ( 5 ) holds at w t w_{t} . Consider w t + 1 = w t s + u ( a ) w_{t+1}=w_{t}s+u^{(a)} , with s > 0 s>0 element-wise and z = log 2 s z=\\log_{2}s . Suppose the branch outputs satisfy, for some η a , η m > 0 \\eta_{a},\\eta_{m}>0 ,"},{"type":"para","text":"If"},{"type":"para","text":"then"},{"type":"para","text":"In particular, ℒ ( w t + 1 ) < ℒ ( w t ) \\mathcal{L}(w_{t+1})<\\mathcal{L}(w_{t}) unless g = 0 g=0 ."},{"type":"para","text":"The theorem formalizes the local two-branch geometry of Algorithm 1 : an additive weight-space displacement and a multiplicative scale change can be composed while preserving descent under the stated alignment and smoothness conditions. The proof is given in Appendix D ."},{"type":"heading","level":1,"text":"5 Empirical Findings"},{"type":"para","text":"We evaluate M+Adam in two stages. First, a controlled BF16-stored matrix-fitting diagnostic isolates the update-level failure modes predicted by our analysis. Second, LLaMA-style pretraining on C4/EN tests whether the same additive–multiplicative geometry improves low-precision training across model scale, token budget, and precision regime. Our main LLM study covers 60M, 130M, and 350M models over 1 1 , 2 2 , 4 4 , and 8 × 8\\times Chinchilla budgets, and we additionally evaluate 1B models at the 1 × 1\\times budget across all precision regimes. We consider BF16, FP8, and NVFP4 master-weight storage with BF16 or FP8 compute."},{"type":"heading","level":2,"text":"5.1 Toy Diagnostic: Isolating Low-Precision Failure Modes"},{"type":"para","text":"We begin with a minimal convex diagnostic designed to highlight the complementary strengths of additive and multiplicative optimizers. The task is matrix fitting,"},{"type":"para","text":"where each step samples a minibatch of entries uniformly with replacement and c c is a constant. Each optimizer step is computed in FP32, but the updated parameter is immediately rounded through BF16 before the next iteration. Since the objective is otherwise trivial, any failure reflects the interaction between the update geometry and BF16 storage. This directly tests the complementary regimes isolated by the theory. For the evaluation, we perform a gradient step using either Additive or Multiplicative, corresponding to the additive and multiplicative branches of M+Adam, respectively. We disable momentum and adaptive moments and tune the learning rate for simplicity. For M+Adam we have η a = η m \\eta_{a}=\\eta_{m} , and in the figures, the Additive and Multiplicative baselines have their optimal learning rate for each setting, whereas M+Adam reuses the same learning rate across all settings."},{"type":"para","text":"Fig. 3 summarizes four diagnostic regimes. In the sign-flip regime, additive steps are needed to change signs: Additive and M+Adam correct the sign, while Multiplicative remains sign-preserving. In the zero-revival regime, Multiplicative cannot escape from zero because its update is zero-absorbing, whereas Additive and M+Adam recover zero weights through the additive branch. In the small-weight regime, M+Adam retains the additive updates needed to reach the BF16-limited floor. In the large-weight regime, Additive stalls because its updates fall below the local BF16 spacing, while Multiplicative and M+Adam continue to make relative, scale-aware progress. Overall, the toy example isolates the intended complementarity: the additive branch enables sign changes, recovery from zero, and small-magnitude updates, while the multiplicative branch preserves progress when the quantization grid becomes too coarse. Full learning-rate sweeps and exact losses are provided in App. B ."},{"type":"heading","level":2,"text":"5.2 Language-Model Pretraining Setup"},{"type":"para","text":"We evaluate M+Adam on LLaMA-style autoregressive language-model pretraining on C4/EN. For a model with P P million trainable parameters, the 1 × 1\\times Chinchilla budget is T 1 × ≈ 20 P × 10 6 T_{1\\times}\\approx 20P\\times 10^{6} tokens, and the main scaling study evaluates n × n\\times budgets for n ∈ { 1 , 2 , 4 , 8 } n\\in\\{1,2,4,8\\} . This design is important because optimizer rankings can depend on the data-to-model ratio, training horizon, and tuning protocol (Wen et al. , 2026 ; Semenov et al. , 2025 ) . The 60M–350M models are evaluated across the full 1 1 – 8 × 8\\times grid. The 1B model is evaluated at 1 × 1\\times Chinchilla across all precision regimes to test whether the observed gains persist beyond the largest model used in our full scaling sweep."},{"type":"para","text":"We study four increasingly aggressive low-precision regimes: BF16/BF16 , BF16/FP8 , FP8/FP8 , NVFP4/FP8 , where the notation denotes weight storage / compute precision. BF16/BF16 corresponds to end-to-end BF16 training without FP32 master weights. BF16/FP8 keeps BF16 weights but uses Transformer Engine FP8 GEMMs. FP8/FP8 further stores selected trainable weight matrices persistently in FP8 and uses FP8 compute. NVFP4/FP8 uses true NVFP4 master-weight storage with FP8 GEMMs."},{"type":"para","text":"For the FP8 master-weight setting, we apply FP8 storage to attention and MLP linear layers, while embeddings, layer norms, and the language-model head remain in BF16. The selected matrices are quantized in blocks with FP16 scales. During training, the optimizer operates under the constraint that these weights are repeatedly quantized to FP8 after updates. In our FP8 experiments, this storage is simulated through explicit quantize/dequantize operations for compatibility with current Transformer Engine kernels."},{"type":"para","text":"We compare against AdamW in each regime. In BF16/BF16, we also include two stronger additive baselines: AdamW with stochastic rounding and AdamW with Kahan-style compensated accumulation. Stochastic rounding probabilistically rounds low-precision updates to nearby representable values so that small updates are preserved in expectation (Connolly et al. , 2021 ) . Kahan-style compensated accumulation keeps an auxiliary correction buffer that accumulates low-order update residuals lost to rounding, thereby improving additive update accumulation under BF16 storage (Zamirai et al. , 2021 ) . We additionally report an FP32-weight/TF32-compute AdamW reference as a higher-precision point of comparison."},{"type":"heading","level":4,"text":"Hyperparameter tuning."},{"type":"para","text":"We use a coordinate-descent protocol (Wen et al. , 2026 ; Semenov et al. , 2025 ) for each optimizer and precision regime. At 1 × 1\\times Chinchilla, we sweep learning rate, warmup length, β 1 \\beta_{1} , β 2 \\beta_{2} , ϵ \\epsilon , gradient clipping, and decoupled weight decay where applicable. For M+Adam, we additionally sweep the additive and multiplicative learning rates. To control cost at larger budgets and model sizes, we identify the hyperparameters that are scaling-sensitive and re-sweep only those in the remaining regimes, while holding the others fixed. We accept a hyperparameter change only when it improves final validation loss by at least Δ loss ≥ 3 × 10 − 3 \\Delta_{\\mathrm{loss}}\\geq 3\\times 10^{-3} . Full grids, sweep traces, and best configurations are reported in App. I ."},{"type":"heading","level":2,"text":"5.3 Scaling Across Budgets and Precision Regimes"},{"type":"para","text":"Fig. 4 reports C4/EN validation perplexity for 60M, 130M, and 350M models across 1 1 – 8 × 8\\times Chinchilla budgets. Each panel compares tuned AdamW and M+Adam across the evaluated low-precision regimes."},{"type":"para","text":"Across model sizes, token budgets, and precision regimes, M+Adam consistently improves over the corresponding AdamW baseline. The gap persists as the Chinchilla ratio increases, showing that the improvement is not merely an early-training effect. The same trend also holds as the precision regime becomes more aggressive. FP8 compute degrades absolute perplexity for both optimizers, and FP8 weights make the problem harder still, but M+Adam retains a clear advantage."},{"type":"heading","level":2,"text":"5.4 Main 1 × 1\\times Chinchilla Results"},{"type":"para","text":"Table 2 summarizes the main 1 × 1\\times Chinchilla results across 60M, 130M, 350M, and 1B models. M+Adam improves over AdamW in every low-precision regime and at every model size. In BF16/BF16, M+Adam reduces perplexity relative to AdamW by 2.8 % 2.8\\% , 6.6 % 6.6\\% , 13.3 % 13.3\\% , and 9.6 % 9.6\\% for 60M, 130M, 350M, and 1B models, respectively. It also improves over AdamW+Kahan and AdamW+SR. Kahan compensation and stochastic rounding both help additive updates survive low-precision accumulation, but neither changes the underlying additive update geometry. This suggests that explicitly separating additive and multiplicative update mechanisms may provide benefits beyond improvements to the additive update alone, as in SR or Kahan compensation."},{"type":"para","text":"When compute is reduced to FP8 while weights remain BF16, M+Adam continues to improve over AdamW, with relative perplexity reductions of 3.2 % 3.2\\% , 6.2 % 6.2\\% , 6.9 % 6.9\\% , and 4.0 % 4.0\\% . The FP8-weight/FP8-compute regime is more challenging because the optimizer must repeatedly update weights that are quantized to FP8. Even in this setting, M+Adam improves over AdamW by 3.4 % 3.4\\% , 2.7 % 2.7\\% , 10.5 % 10.5\\% , and 3.0 % 3.0\\% . The 350M result is especially notable: M+Adam with FP8 weights and FP8 compute reaches 19.395 perplexity, narrowing the gap to BF16-weight AdamW despite using a coarser weight format."},{"type":"para","text":"The NVFP4/FP8 regime amplifies the failure mode of purely additive updates because the representable bins are even coarser. M+Adam gives its largest gains in this setting, reducing perplexity over AdamW by 8.7 % 8.7\\% , 7.3 % 7.3\\% , 16.6 % 16.6\\% , and 7.2 % 7.2\\% for 60M, 130M, 350M, and 1B models. These results directly support our central claim: as weight precision becomes more restrictive, additive–multiplicative updates become increasingly valuable."},{"type":"para","text":"Finally, the 1B results suggest that the gains extend beyond the 60M–350M sweep: even at 1 × 1\\times Chinchilla, M+Adam improves over AdamW across all evaluated precision regimes."},{"type":"heading","level":2,"text":"5.5 Hyperparameter Transfer and Scaling Sensitivity"},{"type":"para","text":"M+Adam introduces additional hyperparameters relative to AdamW, especially separate learning rates for the additive and multiplicative branches. A natural concern is whether these require exhaustive re-tuning across model sizes and token budgets. Our coordinate-descent results suggest that they do not: after tuning a base configuration, only a small subset of hyperparameters is sufficiently scaling-sensitive to warrant re-sweeping, while the remaining settings transfer across regimes. For the NVFP4 experiments, we also investigated whether the FP4 block scale should influence the relative contribution of the additive and multiplicative updates, since the block scale changes the effective numerical resolution during training. In practice, we found that using fixed global coefficients for the two update branches is sufficient, and therefore keep this simpler formulation in the final algorithm. Additional analysis of the block-scale effect is provided in I ."},{"type":"para","text":"This transferability is consistent with the optimizer design. The multiplicative branch controls relative scale changes, while the additive branch provides local weight-space updates, so a single learning rate does not need to simultaneously handle both scale adaptation and local corrections. Under a matched restricted re-tuning protocol, in which both the baselines and M+Adam re-sweep only the hyperparameters identified as scaling-sensitive (Table 3 ), M+Adam continues to achieve lower validation perplexity across the regimes reported in App. I . This suggests that the improvements are not simply an artifact of giving M+Adam a larger hyperparameter search space, but persist under a comparable tuning budget."},{"type":"heading","level":2,"text":"5.6 Optimizer-State Overhead and Apollo-E Compression"},{"type":"para","text":"M+Adam removes FP32 master weights but adds one optimizer state: a second-moment accumulator for the multiplicative branch. Since prior work shows that optimizer states can often be compressed with low-rank approximations (Zhu et al. , 2025 ; Zhao et al. , 2024 ; Loeschcke et al. , 2025 ) , we test whether this added state can be compressed independently."},{"type":"para","text":"Apollo (Zhu et al. , 2025 ) compresses optimizer states using low-rank gradient projections, storing and updating the optimizer statistics in a reduced-dimensional subspace. We apply Apollo only to the exponent/multiplicative second-moment accumulator; we refer to this variant as Apollo-E, where “E” denotes the exponent branch. We use rank r = 4 r=4 for both 130M and 350M models. The additive AdamW-style branch and the additive–multiplicative update rule are otherwise unchanged. Table 4 shows that Apollo-E keeps the optimizer-state footprint close to BF16-state AdamW while retaining most of M+Adam’s perplexity gains. Detailed Apollo-E variants and rank sweeps are given in App. H ."},{"type":"heading","level":1,"text":"6 Limitations and Future Work"},{"type":"para","text":"M+Adam focuses on optimization geometry for low-precision weights rather than optimizer-state compression or systems-level acceleration. The current prototype introduces one additional optimizer state for the multiplicative branch and incurs runtime overhead relative to AdamW, e.g. roughly 6 % − 9 % 6\\%\\!-\\!9\\% at 1B scale (App. G ). At lower precision, overhead also comes from simulating FP8/FP4 weight storage through explicit quantize/dequantize steps, since fully native low-bit parameter-update support is still limited in current training stacks. Our Apollo-E results suggest that the additional state can be compressed independently, while fused kernels and native low-precision update paths could further reduce the cost of the multiplicative branch."},{"type":"para","text":"Our theoretical analysis studies an idealized additive–multiplicative update and does not fully model practical ingredients such as stochastic gradients, Adam-style moments, clipping, or distributed training. Finally, while we evaluate up to 1B-parameter LLaMA-style models across BF16, FP8, and FP4 regimes, larger-scale studies, longer-horizon FP8/FP8 training, and broader downstream evaluations remain important future work. Another natural direction is to study whether M+Adam can be combined with stochastic rounding or Kahan-style compensated accumulation."},{"type":"heading","level":1,"text":"7 Conclusion"},{"type":"para","text":"We present M+Adam, an optimizer that combines additive Adam-style updates with multiplicative Madam-style updates. This is effective under low-precision arithmetic: the additive branch provides local updates, while the multiplicative branch makes scale-aware relative updates in cases when the additive updates are rounded off to zero. Our analysis provides an idealized descent guarantee, and our experiments show that this geometry targets a concrete failure mode of low-precision training. Additive updates can get lost to rounding under coarse quantization, while multiplicative updates struggle with zeros and sign changes; combining them mitigates both failure modes. Across 60M-1B parameter LLaMA-style models, multiple Chinchilla budgets, and BF16/FP8/FP4 regimes, M+Adam consistently improves over AdamW, with the largest gains in the most aggressive low-precision settings. Overall, M+Adam shows that low-precision training benefits from optimizers aligned with floating-point structure. By directly addressing quantization-induced update failures, it provides a principled step toward end-to-end low-precision training without relying on FP32 master weights."},{"type":"heading","level":1,"text":"Impact Statement"},{"type":"para","text":"This work aims to advance the field of machine learning by enabling more efficient and stable low-precision training of large neural networks. While improved training efficiency may have downstream environmental and economic implications, we do not foresee any direct negative societal impacts arising specifically from this work."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training large language models (LLMs) in full precision (FP32) incurs substantial memory, compute, and energy costs. Modern accelerators are optimized for reduced-precision formats such as FP8 and FP4, via specialized hardware like tensor cores (Micikevicius et al. , 2022 ) . As a result, most large-scale training pipelines adopt mixed-precision training, executing forward and backward passes in reduced precision (Micikevicius et al. , 2018 ) . However, they still retain high-precision copies (known as master weights) for optimization, since training directly in low precision results in significant degradation of accuracy and is often unstable (Zamirai et al. , 2021 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recently, the precision of master weights has been reduced, e.g. BF16 instead of FP32, using techniques such as stochastic rounding (SR) (Connolly et al. , 2021 ) to keep them unbiased. While this reduces the memory overhead compared to FP32 master weights, they still exhibit degraded convergence or instability relative to full-precision FP32 training. This is because standard optimizers, such as Adam, fail to make progress at large weight magnitudes under low precision (larger quantization bins in Fig. 2 ) , since updates smaller than the local quantization step are rounded to zero.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To overcome this failure mode, alternative optimizers have been proposed based on multiplicative updates, such as Madam (Bernstein et al. , 2020 ) . These are effective at taking larger steps at higher magnitudes (larger quantization bins in Fig. 2 ), where standard optimizers with additive updates fail to make progress. Madam has been shown to be effective for training under extremely low precision, without high-precision master weights. For instance, Madam can train under the logarithmic number system (LNS) (Zhao et al. , 2022 ) , an extreme case of floating-point system where all the bits are allocated to the exponent, and mantissa is not present. While LNS has optimal hardware utilization in theory, it is not available in standard AI hardware.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Multiplicative methods such as Madam overcome the failure mode of Adam at large magnitudes (larger quantization bins in Fig. 1 ), but they have their own failure modes. They struggle near zero, where relative updates produce small absolute changes and cannot move weights through zero when there is a sign change (Fig. 1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our approach: The additive and multiplicative updates thus suffer from complementary modes of failure. Additive updates get rounded to zero at large magnitudes when mantissa resolution is coarse. Multiplicative updates are able to overcome this and make progress at large magnitudes, but they cannot flip signs or escape exact zero initialization.(Fig. 1 and Table 1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This motivates us to propose M+Adam. It is a new optimizer that combines Adam-style additive updates with Madam-style multiplicative updates for low-precision training, as outlined in Algorithm 1 and Fig. 2 . We prove that M+Adam is a monotone descent method under standard smoothness assumptions, making it a valid optimizer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate stable training with BF16, FP8, and NVFP4 master weights across 60 60 M– 1 1 B LLaMA-style models and 1 1 – 8 × 8\\\\times Chinchilla training budgets. M+Adam trains stably without needing any stochastic rounding (SR), and improves over AdamW and AdamW+SR, as seen in Table 2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Together, these results support our central claim: combined additive–multiplicative updates are especially useful when low-precision master weights becomes a dominant source of optimization error.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We review prior work on multiplicative optimization and low-precision training most closely related to our approach.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Multiplicative Updates\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Multiplicative update rules apply relative, scale-invariant parameter changes and have a long history in optimization (Kivinen and Warmuth, 1997 ) . In deep learning, Madam (Bernstein et al. , 2020 ) adapts this principle as a multiplicative analogue of Adam, showing improved robustness and reduced sensitivity to learning-rate scaling. Recent work revisits multiplicative or hybrid updates in modern settings: Kirtas et al. ( 2024 ) study additive–multiplicative hybrids; Zhao et al. ( 2022 ) combine multiplicative updates with a logarithmic number system to enable low-precision training without FP32 master weights; and Nishida et al. ( 2025 ) analyze log-normal multiplicative dynamics under low-precision forward passes. In contrast to log-based Madam approaches, M+Adam remains within the standard floating-point arithmetic.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Low-precision Training\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mixed-precision frameworks such as AMP (PyTorch, 2025 ) and Transformer Engine (NVIDIA, 2023 ) accelerate BF16/FP8 arithmetic, but are largely orthogonal to optimizer design and typically rely on FP32 master weights for stable accumulation (Micikevicius et al. , 2018 ) . Standard additive optimizers such as AdamW can degrade when master weights are stored in reduced precision, since nearest rounding can cancel small model-weight updates (Zamirai et al. , 2021 ) . Stochastic rounding preserves small updates in expectation and can mitigate deterministic rounding bias (Ozkara et al. , 2025 ) . Kahan-style compensated updates provide another remedy by accumulating lost low-order update residuals in an auxiliary buffer, and have been shown to recover much of the accuracy lost in pure BF16 training (Zamirai et al. , 2021 ) . While both stochastic rounding and Kahan compensation can improve accumulation accuracy, they do not change the underlying additive update geometry or address dynamic-range limitations directly.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several complementary approaches reduce low-precision storage costs without changing the optimizer geometry. LoQT (Loeschcke et al. , 2024 ) trains most weights in 4-bit precision while maintaining a low-rank 16-bit subspace. A separate line of work compresses optimizer states through 8-bit or 4-bit moment quantization (Dettmers et al. , 2022 ; Li et al. , 2023 ) . Fishman et al. ( 2025 ) study stable FP8 LLM pretraining at trillion-token scale with FP32 weights and FP8 forward computation, including quantized optimizer moments, focusing on system-level scaling and mixed-precision pipelines. In contrast, M+Adam modifies the update rule itself through a joint additive–multiplicative formulation, targeting the stability of low-precision weight storage rather than optimizer-state compression.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent work (Castro et al. , 2025 ) explores training with 4-bit weights and activations using micro-scaling formats such as MXFP4, where each block shares a common scale factor. These approaches mainly stabilize low-bit matrix multiplication by managing outliers and representational fidelity during computation. Because low-bit quantization is highly discrete, such systems often keep higher-precision weights and round to the target format before GEMMs. Our work addresses a complementary challenge: stable gradient-based parameter updates when the master weights themselves are stored in BF16, FP8, or NVFP4. Thus, outlier-aware quantization and M+Adam attack different bottlenecks: the former improves low-bit computation, while the latter improves numerical stability across optimization steps.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Background\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We briefly review Adam and Madam, the two optimizers that M+Adam builds upon.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Adam. Adam (Kingma and Ba, 2015 ) maintains exponential moving averages of the first and second moments of the gradient and applies adaptive, element-wise additive updates. AdamW (Loshchilov and Hutter, 2019 ) decouples weight decay from this adaptive gradient step:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While effective in full precision, these additive updates can be rounded away in low-precision formats when their magnitudes fall below the local quantization step, which grows with | w | |w| in floating-point formats such as BF16 or FP8.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Madam. Madam (Bernstein et al. , 2020 ) retains Adam’s moment estimates but applies updates multiplicatively:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This yields scale-invariant behavior, with step sizes proportional to | w | |w| , but purely multiplicative updates preserve sign and cannot escape w = 0 w=0 . We illustrate these complementary failure modes of AdamW and Madam in a controlled toy example (Sec. 5.1 ), which motivates combining local additive updates with multiplicative scale control.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 M+Adam: Additive–Multiplicative Optimization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"M+Adam is designed for low-precision weight storage, where the optimizer must update weights on a nonuniform floating-point grid. The method combines two complementary update mechanisms: an Adam-style additive branch that acts directly on the weights, and a Madam-style multiplicative branch that applies relative scale changes. This section derives the two branches, explains how they are combined, and presents the corresponding descent result.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Additive–Multiplicative Updates\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Low-precision floating-point weights have nonuniform resolution: the spacing between representable values grows with the exponent. As a result, purely additive updates can become rounded back to zero at large magnitudes. When | w t | |w_{t}| is large, an additive update can be smaller than the local quantization bin and disappear after rounding; when | w t | |w_{t}| is small, however, additive updates remain important because they can update values near zero and change signs. M+Adam is designed around this observation: it combines an Adam-style additive correction with a Madam-style multiplicative scale update.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let g t = ∇ w ℒ ( w t ) g_{t}=\\\\nabla_{w}\\\\mathcal{L}(w_{t}) be the gradient of weight w t w_{t} . The additive branch is the standard Adam-style update u t ( a ) u_{t}^{(a)} in the weight space,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A multiplicative update is as follows element-wise:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where a t a_{t} is an adaptive preconditioned descent direction. Naturally, the update a t a_{t} is based on the gradient g t ( e ) g_{t}^{(e)} with respect to the exponent. To obtain this, we use the mantissa m t m_{t} and exponent e t e_{t} in floating-point decomposition in the base-2 system:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The corresponding gradient simplifies as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus the gradient g t ( e ) g_{t}^{(e)} can be expressed without the explicit exponent value, and we use this in our updates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We then apply adaptive normalization to g t ( e ) g_{t}^{(e)} . In our implementation, the multiplicative branch uses a second-moment accumulator,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, u ~ t ( m ) \\\\widetilde{u}_{t}^{(m)} defines the branch’s proposed scale movement:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The threshold τ t > 0 \\\\tau_{t}>0 prevents division by very small weights, and clipping keeps the relative factor well-defined.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, M+Adam combines the multiplicative rescaling with the additive displacement:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This structure directly targets the two failure modes shown in Fig. 1 . Additive updates can change signs and update small-magnitude weights, but they may be lost to rounding at large magnitudes. Multiplicative updates preserve relative progress at large magnitudes, but by themselves are sign-preserving and cannot recover exact zeros. By combining both branches, M+Adam maintains scale-aware progress where additive updates are rounded away while retaining the local flexibility needed near zero. Algorithm 1 summarizes the core method. Optionally, clipping is added for more stability, which is not listed in Algorithm 1 , as outlined in App. A .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Theoretical Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We analyze the deterministic exact-arithmetic core of Algorithm 1 . Let ℒ ( w ) \\\\mathcal{L}(w) be the weight-space objective, let g = ∇ w ℒ ( w t ) g=\\\\nabla_{w}\\\\mathcal{L}(w_{t}) , and define the base-2 gradient with respect to exponent signal\"\n },\n {\n \"type\": \"para\",\n \"text\": \"obtained by differentiating z ↦ ℒ ( w t 2 z ) z\\\\mapsto\\\\mathcal{L}(w_{t}2^{z}) at z = 0 z=0 . Algorithm 1 forms a relative multiplicative update u ( m ) u^{(m)} , sets s = 1 + u ( m ) s=1+u^{(m)} , and updates w t + 1 = w t s + u ( a ) w_{t+1}=w_{t}s+u^{(a)} . For s > 0 s>0 , writing z = log 2 s z=\\\\log_{2}s gives the equivalent scale-then-add form w t + 1 = w t 2 z + u ( a ) w_{t+1}=w_{t}2^{z}+u^{(a)} . We assume the following local smoothness bound in these coordinates: for all ( d , z ) (d,z) in a neighborhood of the update,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here L a L_{a} and L e L_{e} control curvature within the additive and exponent directions, while L a e L_{ae} controls their interaction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume Eq. ( 5 ) holds at w t w_{t} . Consider w t + 1 = w t s + u ( a ) w_{t+1}=w_{t}s+u^{(a)} , with s > 0 s>0 element-wise and z = log 2 s z=\\\\log_{2}s . Suppose the branch outputs satisfy, for some η a , η m > 0 \\\\eta_{a},\\\\eta_{m}>0 ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If\"\n },\n {\n \"type\": \"para\",\n \"text\": \"then\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In particular, ℒ ( w t + 1 ) < ℒ ( w t ) \\\\mathcal{L}(w_{t+1})<\\\\mathcal{L}(w_{t}) unless g = 0 g=0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The theorem formalizes the local two-branch geometry of Algorithm 1 : an additive weight-space displacement and a multiplicative scale change can be composed while preserving descent under the stated alignment and smoothness conditions. The proof is given in Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Empirical Findings\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate M+Adam in two stages. First, a controlled BF16-stored matrix-fitting diagnostic isolates the update-level failure modes predicted by our analysis. Second, LLaMA-style pretraining on C4/EN tests whether the same additive–multiplicative geometry improves low-precision training across model scale, token budget, and precision regime. Our main LLM study covers 60M, 130M, and 350M models over 1 1 , 2 2 , 4 4 , and 8 × 8\\\\times Chinchilla budgets, and we additionally evaluate 1B models at the 1 × 1\\\\times budget across all precision regimes. We consider BF16, FP8, and NVFP4 master-weight storage with BF16 or FP8 compute.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Toy Diagnostic: Isolating Low-Precision Failure Modes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We begin with a minimal convex diagnostic designed to highlight the complementary strengths of additive and multiplicative optimizers. The task is matrix fitting,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where each step samples a minibatch of entries uniformly with replacement and c c is a constant. Each optimizer step is computed in FP32, but the updated parameter is immediately rounded through BF16 before the next iteration. Since the objective is otherwise trivial, any failure reflects the interaction between the update geometry and BF16 storage. This directly tests the complementary regimes isolated by the theory. For the evaluation, we perform a gradient step using either Additive or Multiplicative, corresponding to the additive and multiplicative branches of M+Adam, respectively. We disable momentum and adaptive moments and tune the learning rate for simplicity. For M+Adam we have η a = η m \\\\eta_{a}=\\\\eta_{m} , and in the figures, the Additive and Multiplicative baselines have their optimal learning rate for each setting, whereas M+Adam reuses the same learning rate across all settings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 3 summarizes four diagnostic regimes. In the sign-flip regime, additive steps are needed to change signs: Additive and M+Adam correct the sign, while Multiplicative remains sign-preserving. In the zero-revival regime, Multiplicative cannot escape from zero because its update is zero-absorbing, whereas Additive and M+Adam recover zero weights through the additive branch. In the small-weight regime, M+Adam retains the additive updates needed to reach the BF16-limited floor. In the large-weight regime, Additive stalls because its updates fall below the local BF16 spacing, while Multiplicative and M+Adam continue to make relative, scale-aware progress. Overall, the toy example isolates the intended complementarity: the additive branch enables sign changes, recovery from zero, and small-magnitude updates, while the multiplicative branch preserves progress when the quantization grid becomes too coarse. Full learning-rate sweeps and exact losses are provided in App. B .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Language-Model Pretraining Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate M+Adam on LLaMA-style autoregressive language-model pretraining on C4/EN. For a model with P P million trainable parameters, the 1 × 1\\\\times Chinchilla budget is T 1 × ≈ 20 P × 10 6 T_{1\\\\times}\\\\approx 20P\\\\times 10^{6} tokens, and the main scaling study evaluates n × n\\\\times budgets for n ∈ { 1 , 2 , 4 , 8 } n\\\\in\\\\{1,2,4,8\\\\} . This design is important because optimizer rankings can depend on the data-to-model ratio, training horizon, and tuning protocol (Wen et al. , 2026 ; Semenov et al. , 2025 ) . The 60M–350M models are evaluated across the full 1 1 – 8 × 8\\\\times grid. The 1B model is evaluated at 1 × 1\\\\times Chinchilla across all precision regimes to test whether the observed gains persist beyond the largest model used in our full scaling sweep.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We study four increasingly aggressive low-precision regimes: BF16/BF16 , BF16/FP8 , FP8/FP8 , NVFP4/FP8 , where the notation denotes weight storage / compute precision. BF16/BF16 corresponds to end-to-end BF16 training without FP32 master weights. BF16/FP8 keeps BF16 weights but uses Transformer Engine FP8 GEMMs. FP8/FP8 further stores selected trainable weight matrices persistently in FP8 and uses FP8 compute. NVFP4/FP8 uses true NVFP4 master-weight storage with FP8 GEMMs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the FP8 master-weight setting, we apply FP8 storage to attention and MLP linear layers, while embeddings, layer norms, and the language-model head remain in BF16. The selected matrices are quantized in blocks with FP16 scales. During training, the optimizer operates under the constraint that these weights are repeatedly quantized to FP8 after updates. In our FP8 experiments, this storage is simulated through explicit quantize/dequantize operations for compatibility with current Transformer Engine kernels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare against AdamW in each regime. In BF16/BF16, we also include two stronger additive baselines: AdamW with stochastic rounding and AdamW with Kahan-style compensated accumulation. Stochastic rounding probabilistically rounds low-precision updates to nearby representable values so that small updates are preserved in expectation (Connolly et al. , 2021 ) . Kahan-style compensated accumulation keeps an auxiliary correction buffer that accumulates low-order update residuals lost to rounding, thereby improving additive update accumulation under BF16 storage (Zamirai et al. , 2021 ) . We additionally report an FP32-weight/TF32-compute AdamW reference as a higher-precision point of comparison.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Hyperparameter tuning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use a coordinate-descent protocol (Wen et al. , 2026 ; Semenov et al. , 2025 ) for each optimizer and precision regime. At 1 × 1\\\\times Chinchilla, we sweep learning rate, warmup length, β 1 \\\\beta_{1} , β 2 \\\\beta_{2} , ϵ \\\\epsilon , gradient clipping, and decoupled weight decay where applicable. For M+Adam, we additionally sweep the additive and multiplicative learning rates. To control cost at larger budgets and model sizes, we identify the hyperparameters that are scaling-sensitive and re-sweep only those in the remaining regimes, while holding the others fixed. We accept a hyperparameter change only when it improves final validation loss by at least Δ loss ≥ 3 × 10 − 3 \\\\Delta_{\\\\mathrm{loss}}\\\\geq 3\\\\times 10^{-3} . Full grids, sweep traces, and best configurations are reported in App. I .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Scaling Across Budgets and Precision Regimes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 4 reports C4/EN validation perplexity for 60M, 130M, and 350M models across 1 1 – 8 × 8\\\\times Chinchilla budgets. Each panel compares tuned AdamW and M+Adam across the evaluated low-precision regimes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across model sizes, token budgets, and precision regimes, M+Adam consistently improves over the corresponding AdamW baseline. The gap persists as the Chinchilla ratio increases, showing that the improvement is not merely an early-training effect. The same trend also holds as the precision regime becomes more aggressive. FP8 compute degrades absolute perplexity for both optimizers, and FP8 weights make the problem harder still, but M+Adam retains a clear advantage.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Main 1 × 1\\\\times Chinchilla Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 summarizes the main 1 × 1\\\\times Chinchilla results across 60M, 130M, 350M, and 1B models. M+Adam improves over AdamW in every low-precision regime and at every model size. In BF16/BF16, M+Adam reduces perplexity relative to AdamW by 2.8 % 2.8\\\\% , 6.6 % 6.6\\\\% , 13.3 % 13.3\\\\% , and 9.6 % 9.6\\\\% for 60M, 130M, 350M, and 1B models, respectively. It also improves over AdamW+Kahan and AdamW+SR. Kahan compensation and stochastic rounding both help additive updates survive low-precision accumulation, but neither changes the underlying additive update geometry. This suggests that explicitly separating additive and multiplicative update mechanisms may provide benefits beyond improvements to the additive update alone, as in SR or Kahan compensation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When compute is reduced to FP8 while weights remain BF16, M+Adam continues to improve over AdamW, with relative perplexity reductions of 3.2 % 3.2\\\\% , 6.2 % 6.2\\\\% , 6.9 % 6.9\\\\% , and 4.0 % 4.0\\\\% . The FP8-weight/FP8-compute regime is more challenging because the optimizer must repeatedly update weights that are quantized to FP8. Even in this setting, M+Adam improves over AdamW by 3.4 % 3.4\\\\% , 2.7 % 2.7\\\\% , 10.5 % 10.5\\\\% , and 3.0 % 3.0\\\\% . The 350M result is especially notable: M+Adam with FP8 weights and FP8 compute reaches 19.395 perplexity, narrowing the gap to BF16-weight AdamW despite using a coarser weight format.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The NVFP4/FP8 regime amplifies the failure mode of purely additive updates because the representable bins are even coarser. M+Adam gives its largest gains in this setting, reducing perplexity over AdamW by 8.7 % 8.7\\\\% , 7.3 % 7.3\\\\% , 16.6 % 16.6\\\\% , and 7.2 % 7.2\\\\% for 60M, 130M, 350M, and 1B models. These results directly support our central claim: as weight precision becomes more restrictive, additive–multiplicative updates become increasingly valuable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, the 1B results suggest that the gains extend beyond the 60M–350M sweep: even at 1 × 1\\\\times Chinchilla, M+Adam improves over AdamW across all evaluated precision regimes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.5 Hyperparameter Transfer and Scaling Sensitivity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"M+Adam introduces additional hyperparameters relative to AdamW, especially separate learning rates for the additive and multiplicative branches. A natural concern is whether these require exhaustive re-tuning across model sizes and token budgets. Our coordinate-descent results suggest that they do not: after tuning a base configuration, only a small subset of hyperparameters is sufficiently scaling-sensitive to warrant re-sweeping, while the remaining settings transfer across regimes. For the NVFP4 experiments, we also investigated whether the FP4 block scale should influence the relative contribution of the additive and multiplicative updates, since the block scale changes the effective numerical resolution during training. In practice, we found that using fixed global coefficients for the two update branches is sufficient, and therefore keep this simpler formulation in the final algorithm. Additional analysis of the block-scale effect is provided in I .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This transferability is consistent with the optimizer design. The multiplicative branch controls relative scale changes, while the additive branch provides local weight-space updates, so a single learning rate does not need to simultaneously handle both scale adaptation and local corrections. Under a matched restricted re-tuning protocol, in which both the baselines and M+Adam re-sweep only the hyperparameters identified as scaling-sensitive (Table 3 ), M+Adam continues to achieve lower validation perplexity across the regimes reported in App. I . This suggests that the improvements are not simply an artifact of giving M+Adam a larger hyperparameter search space, but persist under a comparable tuning budget.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.6 Optimizer-State Overhead and Apollo-E Compression\"\n },\n {\n \"type\": \"para\",\n \"text\": \"M+Adam removes FP32 master weights but adds one optimizer state: a second-moment accumulator for the multiplicative branch. Since prior work shows that optimizer states can often be compressed with low-rank approximations (Zhu et al. , 2025 ; Zhao et al. , 2024 ; Loeschcke et al. , 2025 ) , we test whether this added state can be compressed independently.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Apollo (Zhu et al. , 2025 ) compresses optimizer states using low-rank gradient projections, storing and updating the optimizer statistics in a reduced-dimensional subspace. We apply Apollo only to the exponent/multiplicative second-moment accumulator; we refer to this variant as Apollo-E, where “E” denotes the exponent branch. We use rank r = 4 r=4 for both 130M and 350M models. The additive AdamW-style branch and the additive–multiplicative update rule are otherwise unchanged. Table 4 shows that Apollo-E keeps the optimizer-state footprint close to BF16-state AdamW while retaining most of M+Adam’s perplexity gains. Detailed Apollo-E variants and rank sweeps are given in App. H .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Limitations and Future Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"M+Adam focuses on optimization geometry for low-precision weights rather than optimizer-state compression or systems-level acceleration. The current prototype introduces one additional optimizer state for the multiplicative branch and incurs runtime overhead relative to AdamW, e.g. roughly 6 % − 9 % 6\\\\%\\\\!-\\\\!9\\\\% at 1B scale (App. G ). At lower precision, overhead also comes from simulating FP8/FP4 weight storage through explicit quantize/dequantize steps, since fully native low-bit parameter-update support is still limited in current training stacks. Our Apollo-E results suggest that the additional state can be compressed independently, while fused kernels and native low-precision update paths could further reduce the cost of the multiplicative branch.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our theoretical analysis studies an idealized additive–multiplicative update and does not fully model practical ingredients such as stochastic gradients, Adam-style moments, clipping, or distributed training. Finally, while we evaluate up to 1B-parameter LLaMA-style models across BF16, FP8, and FP4 regimes, larger-scale studies, longer-horizon FP8/FP8 training, and broader downstream evaluations remain important future work. Another natural direction is to study whether M+Adam can be combined with stochastic rounding or Kahan-style compensated accumulation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We present M+Adam, an optimizer that combines additive Adam-style updates with multiplicative Madam-style updates. This is effective under low-precision arithmetic: the additive branch provides local updates, while the multiplicative branch makes scale-aware relative updates in cases when the additive updates are rounded off to zero. Our analysis provides an idealized descent guarantee, and our experiments show that this geometry targets a concrete failure mode of low-precision training. Additive updates can get lost to rounding under coarse quantization, while multiplicative updates struggle with zeros and sign changes; combining them mitigates both failure modes. Across 60M-1B parameter LLaMA-style models, multiple Chinchilla budgets, and BF16/FP8/FP4 regimes, M+Adam consistently improves over AdamW, with the largest gains in the most aggressive low-precision settings. Overall, M+Adam shows that low-precision training benefits from optimizers aligned with floating-point structure. By directly addressing quantization-induced update failures, it provides a principled step toward end-to-end low-precision training without relying on FP32 master weights.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Impact Statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This work aims to advance the field of machine learning by enabling more efficient and stable low-precision training of large neural networks. While improved training efficiency may have downstream environmental and economic implications, we do not foresee any direct negative societal impacts arising specifically from this work.\"\n }\n]","isConversational":false}}},{"rowIdx":385,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.11958"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Constructed Reality, Contested Priors: Decoupling and the Architecture of Cognitive Relapse Under the Free Energy Principle"},"authors":{"kind":"list like","value":["MD Ibrahim Hossain Ridoy"],"string":"[\n \"MD Ibrahim Hossain Ridoy\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.11958v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"ABSTRACT"},{"type":"para","text":"Under the free energy principle, a predictive system does not observe reality directly; it maintains a generative model of the world and experiences that model’s best current hypothesis in place of direct sensory contact. This raises a question neuroscience has no way to test directly in a living system: can a synthetic environment be made consistent enough that a predictive system’s own inference machinery adopts it as this default hypothesis, permanently displacing the environment that first shaped it? We call this state ontological inversion. Because inducing and monitoring such a transition in a nervous system is neither ethical nor technically feasible, we study the underlying computational problem through a controlled proxy: a convolutional variational autoencoder paired with a recurrent latent predictor, whose evidence lower bound objective is mathematically identical, up to sign, to variational free energy itself. The network is trained first on a baseline visual domain, then on a mixed stream in which a swept rehearsal ratio r r controls how much baseline content persists during transition to a target domain. Representational capacity, what the latent space can discriminate, is tracked separately from default behavior, what the system generates when left unconstrained. Across a full sweep of 90 runs, the two diverge sharply: representational accuracy stays near ceiling, 0.97 to 0.998, regardless of r r , while default behavior spans nearly the system’s entire range depending on r r alone, a decoupling of learning from acceptance. More strikingly, at intermediate r r the system’s default output rises toward the target domain, in most runs nearing full capture, then partially reverts toward the baseline while training continues unchanged, a structural failure we term cognitive relapse. Resistance to reality-adoption is not reducible to learning speed; it is a structural property with its own distinct failure modes, established here as a computational existence proof and nothing further."},{"type":"para","text":"Keywords : free energy principle; active inference; ontological inversion; predictive processing; continual learning; cognitive relapse; representational decoupling"},{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"Neuroscience has converged on a claim that sounds abstract until it is taken literally: the brain does not observe the world, it constructs a generative model of the world and experiences that model’s best current hypothesis in place of direct sensory contact [ 1 ] . Perception, under this account, is not a window onto reality but a controlled hallucination, held in check by prediction error rather than opened directly onto what is actually out there [ 5 ] . This is not a philosophical flourish. It is an operational claim about how cortical inference works, and it carries a structural consequence that is easy to state and uncomfortable to sit with: if what a system experiences as real is a model rather than a direct readout, then in principle that model, and with it what the system treats as real, can be replaced. A generative model is not a fixed window that suddenly shows a different scene. It is closer to a standing hypothesis, continuously defended and continuously revisable, and a standing hypothesis can, under the right pressure, be overturned by a different one."},{"type":"para","text":"This paper asks what that structural consequence would actually require, as a literal question rather than a metaphor. Can a synthetic environment be constructed with enough internal consistency that a predictive system’s own inference machinery comes to treat it, by default, as the world it expects, fully displacing the environment that trained it in the first place? This is the question Nozick’s experience machine posed as a matter of choice, whether a person would elect a machine-generated experience over reality [ 15 ] ; the question here is stricter, since it concerns not what a system would choose given the option, but whether its own inference machinery would stop treating that option as live at all. We call this state ontological inversion: the point at which a constructed environment becomes a system’s primary generative prior, the reference against which new evidence is evaluated, rather than a hypothesis held provisionally alongside a baseline the system has not yet released. This is a stricter claim than belief revision. A system can come to represent a new environment in exhaustive detail and still hold it at arm’s length, evaluating it against a baseline it continues to treat as ground truth. Existing work on rapid belief updating under exposure addresses precisely this weaker case: how a specific belief shifts against new evidence within a model that remains otherwise intact [ 10 ] . It says nothing about whether the model itself, the thing doing the evaluating, can be durably replaced. That is the question this paper is built to test."},{"type":"para","text":"Testing it directly in a living nervous system is neither ethical nor technically possible: there is no way to install a synthetic generative prior in a brain, monitor its competition against a baseline prior over time, and measure the outcome, without doing exactly the kind of harm the question itself should make anyone hesitant about. What can be tested is the underlying computational problem, stripped of biological implementation, in a system whose internal states are fully observable and whose training history is fully controlled. This is the role played here by a convolutional variational autoencoder paired with a recurrent latent predictor, together referred to as the V+M model [ 11 ] . The correspondence is not decorative. A variational autoencoder is trained by minimizing its evidence lower bound, a quantity that is, up to sign, the same mathematical object as the variational free energy invoked above: both trade reconstruction accuracy against the complexity of an approximate posterior to bound surprisal [ 12 ] . Training this network by gradient descent is, in this narrow and literal sense, minimizing free energy directly, not simulating the idea of doing so. The network’s self-generated, free-running output, what it produces with no external input constraining it, stands in for a system’s default state, modeled on the role attributed to self-referential and internally generated cognition in the biological default mode network [ 6 , 7 , 8 ] , without claiming that this network’s rollout behavior is a model of that network’s activity. What follows is a mathematical proxy for the computational problem of reality-adoption, not a biological one, and every claim in this paper is scoped accordingly."},{"type":"para","text":"Two findings from this proxy motivate everything that follows, stated here as results rather than assumptions built into the model beforehand. The first is a decoupling of learning from acceptance: a system’s capacity to represent the distinction between a baseline and a target environment can sit near ceiling almost immediately, largely as a function of how separable the two environments are, while its default, unconstrained behavior remains governed by something else entirely. The second is cognitive relapse: a system’s default output can shift toward a target environment, in most cases coming close to full capture, and then partially revert toward the baseline while training continues under the exact conditions that produced the initial shift, a signature no account of rigidity as merely slow learning would predict. Together these findings reframe cognitive rigidity as a structural property with at least two independent failure modes, rather than a single rate of update, developed formally in Section II and tested against the full experimental sweep in Sections III through V . None of what follows demonstrates that a biological nervous system behaves this way. It demonstrates that a mechanism derived from first principles, tested against a network whose behavior was not assumed in advance and could have failed to show either pattern, produces results consistent with what that mechanism predicts. That is the scope of the claim, and Section V-D states in full where it stops."},{"type":"heading","level":1,"text":"II Theoretical Framework"},{"type":"para","text":"Under the free energy principle, a system maintains an internal generative model P ( o , s ) P(o,s) specifying the joint probability of sensory observations o o and their hidden causes s s [ 1 ] . Because the true posterior P ( s ∣ o ) P(s\\mid o) is generally intractable, the system approximates it with a recognition distribution Q ( s ∣ μ ) Q(s\\mid\\mu) , where μ \\mu denotes the internal states encoding the current belief. Variational free energy F F is the divergence between this approximation and the true posterior, plus the surprisal of the observations themselves:"},{"type":"para","text":"Since KL divergence is non-negative, F F upper-bounds surprisal, so minimizing F F simultaneously drives the recognition distribution toward the true posterior and suppresses the surprise generated by incoming evidence [ 2 , 3 ] . Two consequences follow. Experience is a function of the generative model, not of raw sensory input; what a system represents at any moment is its current best hypothesis, weighted against evidence by the precision assigned to each. And the model is plastic: sustained prediction error, the persistent signal that the current model no longer fits incoming evidence, drives changes to the model’s parameters and, at longer timescales, to whatever substrate implements them [ 3 , 4 , 9 ] ."},{"type":"heading","level":2,"text":"II-A Ontological Inversion: Definition and Conditions"},{"type":"para","text":"Ontological inversion is the state in which a synthetic environment I s I_{s} , rather than a baseline environment I 0 I_{0} , constitutes a system’s primary generative prior: the model against which new evidence is evaluated by default, and the model that dominates self-referential and internally generated processing when the system is not driven by external input. This is not the same as a system that can represent I s I_{s} convincingly while still holding I 0 I_{0} in reserve as its default. It is the state in which the system’s unconstrained, self-generated output is centered on I s I_{s} rather than I 0 I_{0} ."},{"type":"para","text":"Three conditions must hold jointly for ontological inversion to occur:"},{"type":"para","text":"The system’s internal representation must support reliably distinguishing I s I_{s} from I 0 I_{0} , to a degree that exceeds what either representation could achieve in isolation before exposure to the other."},{"type":"para","text":"The system’s default, self-generated output, produced with no external input constraining it, must be centered on I s I_{s} rather than I 0 I_{0} ."},{"type":"para","text":"Total variational free energy under an I s I_{s} -dominant model must have fallen to a level consistent with stable prediction, meaning the system has learned I s I_{s} well enough that new evidence within I s I_{s} generates low surprisal."},{"type":"para","text":"None of these three is sufficient alone, and this is the theoretical claim this paper is built to test rather than assume. High representational accuracy for I s I_{s} , condition 1, does not entail that the system’s default output has shifted, condition 2: a system can learn to tell two environments apart without that discriminative capacity determining what it produces when left unconstrained. Nor does a shift in default output guarantee that it is stable: a system’s unconstrained output can move toward I s I_{s} and still be governed by a free energy landscape, condition 3, in which I 0 I_{0} remains a live competitor rather than a fully displaced alternative. Sections II-B and II-C develop what follows from treating condition 1 and condition 2 as genuinely separable, and from treating condition 3 as something that can be reached and then partially lost, rather than reached once and retained."},{"type":"heading","level":2,"text":"II-B Residual and Concurrent Contrast"},{"type":"para","text":"Prediction error requires a discrepancy: a mismatch between what the current model expects and what the system receives. A system trained exclusively on I s I_{s} , with no reference to I 0 I_{0} anywhere in its training signal, would appear at first to have the most direct route to ontological inversion. Whether this is true depends on a distinction this paper’s theoretical account treats as central and that prior framings of contrast-driven consolidation have generally left implicit."},{"type":"para","text":"Baseline exposure can drive prediction error in two distinct ways. Residual contrast is the structure a system already carries from prior exposure to I 0 I_{0} , established before any transition protocol begins and present in the system’s parameters regardless of what it receives afterward. A system that has already formed a strong internal model of I 0 I_{0} will generate substantial prediction error the moment it is trained on I s I_{s} , purely because I s I_{s} does not fit the model it already has, with no need for I 0 I_{0} to appear anywhere in the ongoing training stream. Concurrent contrast is different in kind: it is baseline signal delivered continuously, alongside I s I_{s} , throughout the transition protocol itself. Its effect does not depend on what the system’s parameters already encode; it depends on I 0 I_{0} continuing to arrive as real, current evidence for as long as the protocol runs."},{"type":"para","text":"These two roles make different, testable predictions. If only residual contrast matters, a system with strong prior exposure to I 0 I_{0} should be able to complete an inversion toward I s I_{s} even with zero concurrent I 0 I_{0} exposure during the transition itself, since the prediction error needed to drive consolidation is already available from the mismatch between existing structure and new evidence. If concurrent contrast also matters, in its own right and not merely as a proxy for residual structure, continuing to deliver real I 0 I_{0} evidence throughout the protocol should continue to generate prediction error that favors I 0 I_{0} as a candidate model for as long as that evidence keeps arriving, including after the system’s default output has already shifted toward I s I_{s} . Under this second reading, concurrent contrast does not simply slow an inversion in progress. It remains a live competing pressure against whatever has already been consolidated, for the full duration it is present."},{"type":"para","text":"The experiment in Section III is built to separate these two readings directly, by holding residual contrast constant, every condition begins from the same baseline-trained starting point, while varying concurrent contrast as a continuous, swept parameter rather than a single fixed value assumed in advance to be optimal."},{"type":"heading","level":2,"text":"II-C Two Predicted Failure Modes: Decoupling and Relapse"},{"type":"para","text":"Two specific predictions follow directly from Sections II-A and II-B , and both are stated here as claims the experiment is designed to test, not as descriptions of results already known."},{"type":"para","text":"Decoupling of learning and acceptance. Because condition 1 and condition 2 in Section II-A are logically independent, a system’s representational capacity should be measurable separately from its default behavior, and the two should not be assumed to track one another. In particular, if I 0 I_{0} and I s I_{s} are sufficiently distinguishable as classes of input in the first place, a system may reach near-ceiling representational accuracy almost immediately, largely as a function of how separable the two domains are, well before its default output shows any corresponding shift. Representational accuracy under this account is a necessary condition for inversion in only the trivial sense that a system must be able to represent a distinction before that distinction can inform its default state; it should not be read as evidence that the default state has moved."},{"type":"para","text":"Cognitive relapse. If concurrent contrast functions as a continuously applied competing pressure rather than a one-time trigger, as argued in Section II-B , a system’s default output should not be expected to shift and then remain fixed once condition 2 is first satisfied. It should instead remain contestable for as long as concurrent I 0 I_{0} exposure continues, with the shifted default vulnerable to measurable regression back toward I 0 I_{0} under sustained contact with it. This predicts a specific, checkable signature in any system where concurrent contrast is varied continuously: conditions with more concurrent contrast should show not only a slower initial shift toward I s I_{s} , but a larger and more consistent regression away from any shift that is achieved, rather than simply a scaled-down version of the same monotonic climb seen at low concurrent contrast."},{"type":"heading","level":2,"text":"II-D Cognitive Rigidity Redefined"},{"type":"para","text":"Under the account developed in Sections II-B and II-C , cognitive rigidity is not adequately captured by treating a system as simply slow to update its default state, with rigidity and speed sitting on the same continuum and differing only in rate. A system that shows the decoupling predicted above is not slow to accept I s I_{s} ; it is fully capable of representing I s I_{s} and still does not adopt it as a default, which is a difference in kind, not degree. A system that shows the relapse predicted above is not slow either; it can reach a state that looks, at some intermediate point, like a completed inversion, and then move away from that state under continued conditions that produced it in the first place."},{"type":"para","text":"This paper treats cognitive rigidity as better characterized along two axes rather than one. The first is a ceiling on default-state capture that persists regardless of representational accuracy, the decoupling axis. The second is a boundary in the amount of concurrent contrast a system can be exposed to before an achieved shift becomes unstable rather than merely slower to reach, the relapse axis. A system can be rigid on either axis independently of the other, and a full account of resistance to ontological inversion has to characterize both rather than treating either as a stand-in for the other."},{"type":"heading","level":2,"text":"II-E Scope of the Computational Analogy: From Free Energy to the Evidence Lower Bound"},{"type":"para","text":"Every claim in Sections II-A through II-D is stated at the level of a generative system in general, not a biological nervous system specifically. The experiment reported in Section III tests these claims using a structural analogy: an artificial neural network trained by gradient descent, standing in for the generative system described above, with I 0 I_{0} instantiated as one real image domain and I s I_{s} as a second, and the rehearsal ratio r r , the fraction of each training batch drawn from I 0 I_{0} during the transition protocol, operationalizing concurrent contrast directly as a continuously swept parameter rather than a value fixed in advance."},{"type":"para","text":"The analogy is grounded in a specific mathematical correspondence rather than a loose metaphor, introduced in Section I : a variational autoencoder’s evidence lower bound is, up to sign, the same object as the variational free energy in Equation 1 [ 12 ] . This correspondence is what licenses treating a trained VAE’s ELBO as a direct, if narrow, computational instantiation of Equation 1 , not merely an unrelated quantity that happens to share the paper’s vocabulary."},{"type":"para","text":"The analogy does not extend further than this. The network described in Section III has no cortical hierarchy, no sleep-dependent consolidation, no homeostatic regulation, and no subjective experience. Its self-generated output, produced by running its own latent predictor with no external input, is used as an operational stand-in for condition 2 in Section II-A , modeled on the role the default mode network plays in self-referential and internally generated cognition in a biological brain [ 6 , 7 , 8 ] , not as a claim that the network’s rollout behavior is a model of that network’s activity. Every quantity reported in Sections III through V is a measurement taken from this artificial system. None of it is a measurement of, or a claim about, biological tissue."},{"type":"heading","level":2,"text":"II-F Terminology Reference"},{"type":"para","text":"Table I maps the coined terms used in this paper to their nearest existing concept, to keep clear what is standard and what this paper adds."},{"type":"heading","level":1,"text":"III Computational Experiment"},{"type":"para","text":"The V+M model introduced in Section I is specified here in full. All quantities reported below, including generative quality, representational separability, and default-mode behavior, are computed from the trained weights of this network under gradient descent. Nothing about convergence, threshold-crossing, or the shape of any reported curve is guaranteed by the functional form of an equation; it is measured from a model that could, in principle, have failed to show any of the reported patterns."},{"type":"heading","level":2,"text":"III-A Architecture"},{"type":"para","text":"The V component is a convolutional variational autoencoder operating on 28 × 28 28\\times 28 grayscale images. The encoder applies two stride-2 convolutions (1 to 32 channels, then 32 to 64 channels, kernel size 3, padding 1), flattens the resulting 64 × 7 × 7 64\\times 7\\times 7 feature map to a 3,136-dimensional vector, and projects it through two parallel linear layers to a mean μ \\mu and log-variance log σ 2 \\log\\sigma^{2} , each of dimension 32. The decoder mirrors this in reverse: a linear layer expands the 32-dimensional latent sample back to 3,136 dimensions, reshaped to 64 × 7 × 7 64\\times 7\\times 7 , followed by two transposed convolutions (64 to 32, then 32 to 1 channel) restoring the original 28 × 28 28\\times 28 resolution, with a sigmoid output. The full V component has 341,825 trainable parameters."},{"type":"para","text":"The M component is a single-layer GRU with input and output dimension 32 (matching the VAE’s latent dimension) and hidden size 256, followed by a linear projection from the 256-dimensional hidden state back to 32 dimensions. This gives the M component 230,944 trainable parameters. Combined, V and M total 572,769 trainable parameters, deliberately kept small enough to train the full sweep described in Section III-E on a single consumer-grade GPU."},{"type":"para","text":"Reconstruction loss is binary cross-entropy, summed over pixels and averaged over the batch. This is chosen over a Gaussian (MSE) decoder because MNIST and FashionMNIST pixel intensities concentrate near 0 and 1, and the implicit Bernoulli likelihood in BCE is a closer probabilistic fit to near-binary pixel data [ 12 ] . The KL term is the closed-form divergence of the diagonal Gaussian posterior q ( z ∣ x ) = 𝒩 ( μ , diag ( σ 2 ) ) q(z\\mid x)=\\mathcal{N}(\\mu,\\mathrm{diag}(\\sigma^{2})) against a standard normal prior, with weight β = 1 \\beta=1 (no annealing schedule, no β \\beta -VAE scaling beyond unity)."},{"type":"para","text":"The GRU is trained on pseudo-sequences constructed from the VAE’s encoder means. Within each training batch, the μ \\mu vectors are detached from the VAE’s computational graph, so the GRU’s prediction loss cannot alter the encoder’s gradients, and reshaped into groups of 8 consecutive vectors. Each group of 8 yields 7 next-step prediction targets, with the GRU trained via teacher forcing to minimize the mean squared error between its predicted and actual next-step latent vectors. The two networks share a single Adam optimizer (learning rate 10 − 3 10^{-3} , batch size 128) and are trained jointly; the total loss is the unweighted sum of the VAE loss and the GRU loss. Because the GRU’s gradient never reaches the VAE encoder, the representation-learning objective and the sequence-prediction objective are optimized jointly in wall-clock time but are not coupled through backpropagation."},{"type":"heading","level":2,"text":"III-B Environments and Data"},{"type":"para","text":"Environment A (baseline) is MNIST; Environment B (target) is FashionMNIST. Both are 28 × 28 28\\times 28 grayscale image sets with pixel values scaled to [ 0 , 1 ] [0,1] and no further preprocessing or augmentation, so that neither domain is given an advantage by differential preprocessing. Both datasets use their standard 60,000-image training partition and 10,000-image test partition; the training partition is further split into the first 50,000 images for training and the last 10,000 for validation. MNIST and FashionMNIST were chosen for tractability, not because they resemble any realistic pair of competing environments; this choice is revisited in Section VI ."},{"type":"heading","level":2,"text":"III-C Training Protocol"},{"type":"para","text":"Training proceeds in two stages."},{"type":"para","text":"Stage 1 (baseline). For each of 15 random seeds, a fresh V+M model is trained on Environment A only, with the Adam optimizer described above, for up to 50 epochs with early stopping (patience of 5 epochs without improvement in combined validation loss). The checkpoint at the best validation epoch is saved and reused as the starting point for every Stage 2 condition run under that seed."},{"type":"para","text":"Stage 2 (rehearsal). Training continues from the Stage 1 checkpoint on a mixed data stream. Each training batch of 128 images draws a fraction r r from Environment A and ( 1 − r ) (1-r) from Environment B, combined and shuffled within the batch, following experience-replay methodology [ 14 ] . Stage 2 runs for a fixed 15 epochs (indexed 0 through 14) with no early stopping, and the optimizer state is reinitialized rather than carried over from Stage 1, so that single-domain momentum from the baseline phase does not bias the mixed-domain phase. Validation loss during Stage 2 is computed on a held-out stream mixed at the same ratio r r as training."},{"type":"para","text":"All random seeds (Python, NumPy, PyTorch, and CUDA where applicable) are set from a single seed value per run, with cuDNN’s deterministic mode enabled and its benchmark mode disabled, trading some training speed for exact reproducibility. Seed is therefore the only source of stochastic variation between two runs sharing the same r r ."},{"type":"heading","level":2,"text":"III-D Metrics"},{"type":"para","text":"Three quantities are recorded at every epoch of Stage 2."},{"type":"para","text":"F_proxy. The VAE-only ELBO (reconstruction plus KL, excluding the GRU term) evaluated on the r r -matched validation stream. Because this validation stream’s domain composition tracks r r , and FashionMNIST reconstructs to a substantially higher BCE than MNIST at comparable training progress, differences in F_proxy across r r partly reflect the changing difficulty of the validation mixture itself rather than a domain-independent measure of generative quality. F_proxy should be read as generative fit to the current training mixture, not as a fixed-content quality score."},{"type":"para","text":"w_proxy. The 5-fold stratified cross-validated accuracy of a logistic regression classifier (scikit-learn, lbfgs solver), retrained from scratch at every epoch on the VAE encoder’s μ \\mu vectors for the full MNIST and FashionMNIST test sets (10,000 images each, never mixed or subset by r r ), labeled by domain of origin. This follows the linear classifier probe methodology of Alain and Bengio [ 13 ] and measures how linearly separable the two domains are in the current latent space, independent of the training mixture."},{"type":"para","text":"dmn_proxy. The mean P ( environment = B ) P(\\text{environment}=B) assigned by that same epoch’s fitted probe to images generated by the GRU running freely, with no real sensory input after initialization. Ten independent 50-step rollouts are generated per epoch, each seeded from an independently drawn starting latent vector; at every step the GRU’s own prediction becomes the next input, the resulting latent is decoded to image space by the VAE decoder, then re-encoded to obtain the μ \\mu vector the probe classifies. The reported value is the mean P ( B ) P(B) pooled across all 500 generated frames (10 rollouts × \\times 50 steps) for that epoch. dmn_proxy is the paper’s operational measure of default-mode behavior: what domain the network’s unconditioned, self-generated output resembles, as distinct from what it is capable of discriminating given real input."},{"type":"heading","level":2,"text":"III-E Experimental Sweep"},{"type":"para","text":"The rehearsal ratio r r is swept over { 0.0 , 0.20 , 0.30 , 0.40 , 0.50 , 1.0 } \\{0.0,0.20,0.30,0.40,0.50,1.0\\} , crossed with 15 seeds, giving 90 Stage 2 runs, plus 15 Stage 1 baseline runs, one per seed, reused across all six r r conditions for that seed. At 15 epochs per Stage 2 run this yields 1,350 per-epoch records across the three tracked metrics. r = 0.0 r=0.0 is a negative control in one direction: an immediate, unmixed switch to Environment B with zero rehearsal. r = 1.0 r=1.0 is a negative control in the other direction: continued training on Environment A only, with the network never receiving a single Environment B image during Stage 2."},{"type":"heading","level":2,"text":"III-F Threshold-Crossing Analysis"},{"type":"para","text":"For each (seed, r r ) run, the first Stage 2 epoch at which dmn_proxy ≥ 0.90 \\geq 0.90 is recorded. Runs that never reach 0.90 within the 15-epoch window are retained as right-censored observations rather than dropped: dropping them would remove exactly the slowest-shifting runs from the average and bias the reported mean time-to-crossing downward. Mean and standard deviation of epoch-to-crossing are computed only over runs that crossed, and are always reported together with the fraction of runs that did not, so that no single summary statistic obscures how much of a condition’s spread falls outside the observation window. The 0.90 threshold is a fixed analysis-stage convention applied identically across every condition in the sweep."},{"type":"heading","level":1,"text":"IV Results"},{"type":"para","text":"All 90 Stage 2 runs completed; no run is excluded from any analysis below on any grounds, including apparent outlier behavior. Figure 1 summarizes the full sweep."},{"type":"heading","level":2,"text":"IV-A Onset Speed: Dose-Response in Time to Crossing"},{"type":"para","text":"At r = 0.0 r=0.0 , dmn_proxy is already at or above 0.90 in every seed at the first evaluated epoch (mean crossing epoch 0.00). Increasing r r delays this monotonically across every value tested through r = 0.4 r=0.4 : mean crossing epoch rises to 0.20 at r = 0.2 r=0.2 , 0.73 at r = 0.3 r=0.3 , and 2.13 at r = 0.4 r=0.4 , with all 15 seeds crossing at every one of these four ratios. This is a genuine, monotonic dose-response in onset speed, a property that emerged from gradient descent on a trained network, not a property built into an equation’s functional form."},{"type":"heading","level":2,"text":"IV-B Trajectory Shape: Overshoot and Partial Regression"},{"type":"para","text":"The onset-speed statistic in Table II describes only the first time dmn_proxy crosses 0.90; it does not describe what the trajectory does afterward. Examined epoch by epoch, dmn_proxy is not monotonically increasing for r > 0 r>0 . In every condition with r ∈ { 0.2 , 0.3 , 0.4 , 0.5 } r\\in\\{0.2,0.3,0.4,0.5\\} , per-seed trajectories rise rapidly over the first several epochs, in most seeds reaching or closely approaching the ceiling of 1.0, and then decline over the remainder of the 15-epoch window."},{"type":"para","text":"The size of this decline scales with r r . It is small at r = 0.2 r=0.2 (mean drop 0.032 from peak to epoch 14) and r = 0.3 r=0.3 (0.053), then increases sharply at r = 0.4 r=0.4 (0.241, with 14 of 15 seeds declining more than 0.03 from their individual peak, one by as much as 0.42) and remains comparably large at r = 0.5 r=0.5 (0.212). r = 1.0 r=1.0 shows no equivalent structure: its nominal peak (mean 0.015, at a mean epoch of 12.3) sits inside the noise floor the metric holds throughout Stage 2, not a genuine excursion."},{"type":"para","text":"This means r = 0.4 r=0.4 and r = 0.5 r=0.5 are not simply slower versions of the pattern seen at r = 0.0 r=0.0 through r = 0.3 r=0.3 . In both conditions, the network’s free-running output shifts toward Environment B early in Stage 2 and then partially reverts toward Environment A dominance while training continues, with the size of the reversion increasing as more Environment A content is retained in the training stream. The 15-epoch window used throughout this sweep is not long enough to determine whether this regression stabilizes at some intermediate value, continues indefinitely, or eventually reverses again; that question is outside the scope of the runs reported here."},{"type":"heading","level":2,"text":"IV-C The r = 0.5 r=0.5 Condition"},{"type":"para","text":"Seven of 15 seeds cross the 0.90 threshold (mean epoch 3.57, std 2.07). The remaining eight do not cross within the 15-epoch window; their epoch-14 dmn_proxy values range from 0.361 to 0.895 (mean 0.611). Inspecting the full trajectories of these eight seeds, rather than only their final value, shows that seven of the eight have already passed a higher peak earlier in Stage 2 (individual peaks ranging from 0.673 to 0.899, all below 0.90, with two peaks within 0.01 of it and the rest between 0.04 and 0.23 below it) and are flat or declining by epoch 14. Only one of the eight (final value 0.895, still rising at epoch 14) shows a trajectory consistent with continuing to climb toward threshold. The remaining seven do not support the inference that additional epochs would produce a crossing; several show the same peak-then-regress pattern documented in Section IV-B , and their epoch-14 trajectories are flat or falling, not rising."},{"type":"heading","level":2,"text":"IV-D The r = 1.0 r=1.0 Condition: Complete Block"},{"type":"para","text":"Zero of 15 seeds cross the 0.90 threshold. Mean dmn_proxy at epoch 14 is 0.0110 (std 0.0146), close to its starting value and essentially unmoved across all 15 epochs of Stage 2. This is the expected result for a condition in which the network receives zero Environment B images throughout Stage 2, and it held without exception."},{"type":"para","text":"Notably, w_proxy at r = 1.0 r=1.0 still reaches 0.9686 (std 0.0076) by epoch 14, the lowest w_proxy value anywhere in the sweep but far above chance (0.5), despite the VAE encoder never seeing a single FashionMNIST image during Stage 2. This indicates that MNIST and FashionMNIST are separable enough as image classes that a linear probe can distinguish them in this network’s latent space even when the encoder was never trained to make that particular distinction; see Section IV-E ."},{"type":"heading","level":2,"text":"IV-E Decoupling of Representational Separability from Default-Mode Behavior"},{"type":"para","text":"Across the entire sweep, w_proxy ranges from 0.9686 to 0.9983, a span of 0.0297. Over the same sweep, dmn_proxy ranges from 0.0110 to 0.9989, a span of 0.9879, nearly the full width of the metric’s possible range. Representational separability in the VAE’s latent space is close to ceiling in every tested condition, including r = 1.0 r=1.0 , where the model receives no real Environment B input at all during Stage 2. Whether the network’s free-running, self-generated output resembles Environment A or Environment B is answered almost independently of whether its latent space can, given real input, tell the two domains apart. Two conditions can share a w_proxy within 0.0002 of each other ( r = 0.4 r=0.4 at 0.9983 and r = 0.5 r=0.5 at 0.9983) while differing in dmn_proxy by nearly 0.1 (0.7538 versus 0.6608), and r = 1.0 r=1.0 , with the single lowest w_proxy in the sweep, has a dmn_proxy roughly two orders of magnitude smaller than every other condition. High w_proxy is necessary in the trivial sense that the probe needs a real signal to classify against; it is nowhere close to sufficient to predict dmn_proxy."},{"type":"heading","level":1,"text":"V Discussion"},{"type":"para","text":"The central question of this paper, stated in Section II-A , is whether a synthetic environment can be engineered to displace a baseline one as a generative system’s default reference, not just as something the system can represent when asked, but as what its own unconstrained output defaults to. The V+M results in Section IV do not answer that question for a biological system. They answer a narrower, computational version of it, and the answer is more structured than the sweep’s headline dose-response result suggests: the shift can be induced, its onset speed is under direct experimental control, and it is not stable once induced. Each of these three parts carries a different implication, taken up in turn below."},{"type":"heading","level":2,"text":"V-A Cognitive Relapse and the r = 0.5 r=0.5 Transition Regime"},{"type":"para","text":"A pure slowdown account of the sweep would predict that dmn_proxy at every tested r r rises monotonically toward a plateau, with the rate of rise set by r r and nothing else changing in shape. Section IV-B shows this is not what happens. At r ∈ { 0.2 , 0.3 , 0.4 , 0.5 } r\\in\\{0.2,0.3,0.4,0.5\\} , dmn_proxy overshoots, in most seeds reaching or approaching 1.0, and then declines for the remainder of the 15-epoch window, with the size of the decline scaling from negligible at r = 0.2 r=0.2 to substantial at r = 0.4 r=0.4 and r = 0.5 r=0.5 (Table III ). This is not slow learning. Slow learning would look like a shallower version of the same curve. What is observed instead is a system that reaches a state resembling adoption of Environment B and then measurably moves away from it while training continues under the same regime that produced the adoption in the first place. This transient regression resembles the \"stability gap\" documented in continual learning [ 17 ] , though here it manifests in unconditioned generative output rather than supervised task accuracy."},{"type":"para","text":"The mechanism is not mysterious given the training protocol. At every r > 0 r>0 , a fraction r r of every batch in every epoch of Stage 2, not only the first few epochs, is real Environment A data, updating the same shared weights that produce the free-running rollout. Rehearsal is not a one-time perturbation the system absorbs and moves past. It is a continuous source of gradient pressure toward Environment A for the entire duration of training. Early in Stage 2, the network’s unconditioned output can apparently drift toward Environment B faster than the accumulated pull of the retained Environment A batches, only for that pull to reassert itself as training continues. We refer to this descriptively as cognitive relapse : an adopted default state that is not maintained against continued contact with the source it displaced. The term labels a specific curve shape, rise, peak, decline, measured in an artificial network. It is not a claim about anything psychological; Section V-D states this restriction explicitly."},{"type":"para","text":"The r = 0.5 r=0.5 condition marks a qualitative boundary within the tested grid, not only a point further along a smooth trend. It is the first ratio at which a majority of seeds, 8 of 15, fail to reach the 0.90 threshold at all within the window, and cross-seed variance rises sharply alongside it: standard deviation of epoch-to-crossing goes from 1.06 at r = 0.4 r=0.4 to 2.07 at r = 0.5 r=0.5 , and standard deviation of final-epoch dmn_proxy goes from 0.118 to 0.150 over the same step. Rising variance near a boundary is consistent with, though it does not on its own establish, an approach to a genuine bifurcation in the underlying dynamics, where nearby initial conditions or noise realizations begin to produce qualitatively different long-run outcomes. Whether r = 0.5 r=0.5 sits on a true phase transition, or is simply the last point before a smooth continued decline into the untested interval between r = 0.5 r=0.5 and r = 1.0 r=1.0 , cannot be resolved by this sweep: no runs exist between those two values. Confirming a genuine transition would require sampling that interval directly and checking whether the fraction of seeds reaching threshold falls off sharply over a narrow sub-range of r r or declines gradually across it."},{"type":"heading","level":2,"text":"V-B The Decoupling of Learning and Acceptance"},{"type":"para","text":"Section IV-E shows w_proxy confined to a narrow band, 0.9686 to 0.9983, across every tested condition, while dmn_proxy spans nearly the metric’s full range, 0.0110 to 0.9989, over the same sweep. This is the clearest empirical instance in this paper of a claim that Section II-A made on theoretical grounds alone: that the conditions for a generative prior to shift are not reducible to one another. A network’s capacity to represent the distinction between two environments and the environment its own unconstrained behavior defaults to are, in this system, close to independent quantities. At r = 1.0 r=1.0 , the network never receives a single Environment B image during Stage 2, yet w_proxy still reaches 0.9686 there, meaning its latent space can still separate the two domains at high accuracy under real input, while dmn_proxy sits at 0.0110, meaning its unconditioned output almost never resembles Environment B. Representational capacity is not a precursor state that default behavior eventually catches up to in this data. It is close to saturated everywhere, including where default behavior does not move at all."},{"type":"para","text":"This distinction maps onto a difference the Introduction already draws between believing a constructed environment is real while still holding it at arm’s length, and the constructed environment being what the system expects by default, with no reference held in reserve [ 5 ] . w_proxy indexes something closer to the first sense: whether the discriminative machinery exists to tell the two apart. dmn_proxy indexes something closer to the second: what the system’s generative process treats as its baseline output when nothing external constrains it. These results show that the two can be pulled almost fully apart. Training a system to represent an alternative environment well is not evidence that the system’s default behavior has moved toward it, and the two should not be conflated when either is reported as evidence for anything resembling reality displacement."},{"type":"heading","level":2,"text":"V-C What This Does and Does Not Say About Ontological Inversion"},{"type":"para","text":"Taken together, Sections V-A and V-B narrow what a computational demonstration of ontological inversion would need to show. It is not enough to demonstrate that a system can be trained to represent a synthetic environment; w_proxy is high everywhere in this sweep, including the negative control, and represents nothing about default behavior on its own. It is not enough to demonstrate an initial shift in default behavior, either: two of the six tested conditions in this sweep show an initial shift that substantially reverses within the same 15-epoch window. Under this stricter reading, only r = 0.0 r=0.0 through r = 0.3 r=0.3 show something resembling a stable capture of default behavior within the tested window. r = 0.4 r=0.4 and r = 0.5 r=0.5 show partial capture followed by relapse. r = 1.0 r=1.0 shows no capture at all. None of the six conditions was run past epoch 14, so whether the r = 0.4 r=0.4 and r = 0.5 r=0.5 relapse trajectories eventually re-stabilize at some intermediate value, continue reverting toward the r = 1.0 r=1.0 floor, or oscillate is not something this experiment can answer."},{"type":"para","text":"This bears on the contrast-retention argument raised earlier in the paper, that some ongoing exposure to the baseline environment is necessary to sustain the prediction-error gradient that drives consolidation of a new one. The rehearsal ratio r r in this experiment operationalizes that same general idea, more baseline retention should mean more sustained contrast, in a different architecture and without asserting the specific mechanism proposed for it there. These results complicate a simple reading of that argument rather than confirming it outright. Retaining more baseline content does not only slow the shift toward the new environment, which a contrast-as-fuel framing predicts; it also appears to actively contest an already-partially-completed shift throughout training, which a pure fuel-for-consolidation framing does not predict on its own. Some baseline retention may be necessary to drive an initial shift at all, consistent with r = 0.0 r=0.0 never showing a moment’s hesitation. Too much retention appears able to prevent that shift from holding once achieved. The right amount, and whether a fixed ratio held constant for an entire run is even the correct way to administer it, is not something this sweep was designed to determine. This dynamic is structurally adjacent to \"wireheading\" in reinforcement learning, where agents exploit self-generated signals over real environments [ 16 ] . However, while wireheading requires an agent to actively choose delusion for reward optimization, ontological inversion here occurs as a passive transition in the model’s default generative state under externally imposed exposure."},{"type":"heading","level":2,"text":"V-D Scope of the Structural Analogy"},{"type":"para","text":"Everything reported in this Discussion describes a 572,769-parameter VAE and GRU trained on MNIST and FashionMNIST. It has no analog of a cortical hierarchy, no sleep-dependent consolidation, no homeostatic regulation, no competing top-down attentional control, and no subjective experience of any kind. dmn_proxy measures the classification of pixels produced by an unconditioned rollout of a next-latent predictor; it is not a measurement of activity in a biological default mode network. w_proxy measures cross-validated logistic regression accuracy, not synaptic weight. \"Relapse,\" \"acceptance,\" and \"default behavior,\" as used above, are labels for the shape of a measured curve in this specific system, chosen because they are the most direct available language for that shape, not because the underlying process is claimed to resemble relapse, acceptance, or defaulting as those terms are used for a nervous system."},{"type":"para","text":"What this experiment supports is narrower and more specific. Two claims made independently of any dynamical-systems construction in this paper’s Theoretical Framework, that representational capacity and default generative behavior can dissociate, and that a system moved toward a new default under sustained partial exposure to its prior one does not necessarily hold that default once achieved, both survive contact with a trained, gradient-descent-optimized network in which neither property was assumed or built in by hand. That is evidence the theoretical account in Section II-A is not merely restating itself in different words when tested against a second, independently constructed system. It is not evidence about what a biological brain does under any comparable protocol, and no claim to that effect is made here or should be drawn from it."},{"type":"heading","level":1,"text":"VI Limitations"},{"type":"para","text":"This section consolidates every scope and limitation statement made in this paper into one place, so that what this work does not support is available from a single location rather than scattered across sections as qualifications easy to skip past."},{"type":"heading","level":2,"text":"VI-A The Fifteen-Epoch Window Is Fixed and Untested Beyond That Point"},{"type":"para","text":"The cognitive relapse pattern documented in Section IV-B and interpreted in Section V-A is observed only within a fixed 15-epoch Stage 2 window. At r = 0.4 r=0.4 and r = 0.5 r=0.5 , dmn_proxy rises, peaks, and declines within that window; what happens beyond epoch 14 is not tested. Three distinct continuations are consistent with the recorded data and cannot be distinguished by this experiment. The decline could stabilize at some intermediate value, held by a balance between concurrent contrast pulling toward I 0 I_{0} and whatever remains of the pull toward I s I_{s} . It could continue declining toward the r = 1.0 r=1.0 floor, meaning the observed shift was transient and sustained rehearsal eventually erases it entirely. Or it could reverse again, since nothing in the training protocol changes after epoch 14 and the same forces that produced the initial rise are still present in every batch that follows. Resolving which of these is correct requires running the r = 0.4 r=0.4 and r = 0.5 r=0.5 conditions substantially longer, at minimum long enough to observe whether the decline rate is itself decaying, consistent with stabilization, or holding constant or accelerating, consistent with continued or complete reversal. No run in this sweep was extended far enough to make that determination. \"Relapse,\" as used in Section V-A , describes the shape observed within the tested window. It is not a claim about that shape’s long-run destination."},{"type":"heading","level":2,"text":"VI-B MNIST and FashionMNIST Are Not a Realistic Pair of Competing Environments"},{"type":"para","text":"MNIST and FashionMNIST were chosen for computational tractability and fast iteration, not because they resemble any realistic pair of environments a generative system might be asked to choose between. The two domains are separable almost trivially: w_proxy exceeds 0.96 in every condition tested, including r = 1.0 r=1.0 , where the encoder never receives a single FashionMNIST image during Stage 2. This bears directly on how the decoupling result in Sections IV-E and V-B should be read. The near-ceiling, nearly r r -invariant w_proxy observed across this sweep may be substantially a property of how easy this particular pair of domains is to separate, rather than a general property of representational learning under domain transition. A pair of environments with genuinely overlapping low-level statistics, more visually or structurally similar to one another than a digit is to a garment, might show w_proxy itself varying meaningfully with r r , in which case the dissociation between w_proxy and dmn_proxy reported here could narrow. That learning and acceptance can decouple is supported by this experiment. That they decouple this sharply, or under any sufficiently different pair of environments, is not."},{"type":"heading","level":2,"text":"VI-C The Rehearsal Ratio Is a Crude Operationalization of Concurrent Contrast"},{"type":"para","text":"The rehearsal ratio r r , as implemented in Section III-C , is a single scalar controlling the fraction of each training batch drawn from Environment A, held fixed for the entire 15-epoch Stage 2 window and applied identically to every batch. This is a coarse instantiation of concurrent contrast as defined in Section II-B . It does not vary the timing, patterning, or salience of retained baseline exposure within a run, whether it arrives early or late, clustered or interspersed, attended to or incidental, all of which plausibly matter for anything resembling a real contrast signal and are not represented by a ratio held constant throughout training. It also treats every unit of Environment A content as interchangeable, with no analog of selective attention or relevance determining how much prediction-error signal any given piece of retained content actually generates. r r should be read as the simplest operationalization of concurrent contrast sufficient to test whether the concept does anything at all, not as a considered model of how contrast operates. That it produces the graded onset delay and the r r -dependent relapse magnitude reported in Section IV despite this simplicity is informative. It is not evidence that batch-fraction mixing is the correct or only way concurrent contrast could be built into a system."},{"type":"heading","level":2,"text":"VI-D Residual Contrast Was Held Constant, Not Varied"},{"type":"para","text":"Every condition in this sweep begins from the same Stage 1 procedure, up to 50 epochs on Environment A alone with early stopping at patience 5, identical across every value of r r for a given seed. This holds residual contrast, as defined in Section II-B , constant across the entire sweep, which is what allows the sweep to isolate the effect of concurrent contrast cleanly. It also means residual contrast’s own contribution was never independently tested. Whether a system with a weaker Stage 1, less thoroughly consolidated on Environment A to begin with, would show a different pattern of onset speed or relapse magnitude under the same r r sweep is not addressed here. The distinction between residual and concurrent contrast is supported in this experiment only in the sense that concurrent contrast alone, with residual contrast fixed, produces the reported effects. It has not been shown that residual contrast strength is itself a free variable with its own independent effect on either onset speed or relapse."},{"type":"heading","level":2,"text":"VI-E Single Architecture and Fixed Analysis Thresholds"},{"type":"para","text":"Latent dimension, GRU hidden size, learning rate, batch size, and sequence length were fixed before the sweep and not varied; whether the reported patterns hold under different architectural choices is untested. The 0.90 threshold used for the epoch-to-crossing analysis in Section III-F is a single fixed convention, not swept across alternatives. A lower or higher threshold would shift the specific onset-epoch and censoring numbers reported in Table II , though it is unlikely to remove the qualitative overshoot-and-decline pattern in Table III , since that pattern is visible in the raw per-epoch trajectories independent of any particular threshold choice."},{"type":"heading","level":2,"text":"VI-F What This Model Does Not Represent"},{"type":"para","text":"No condition beyond r = 0.0 r=0.0 and r = 1.0 r=1.0 isolates batch composition from other factors that covary with it, such as total accumulated gradient signal from each domain across the full 15 epochs. Every seed starts from an architecturally identical, randomly initialized network differing only in its random seed; there is no representation of individual variation in prior structure or differential susceptibility to either domain. dmn_proxy and w_proxy remain abstract indices bounded in [ 0 , 1 ] [0,1] , calibrated against nothing beyond the classifier and rollout procedure that produce them; a value of 0.75 has meaning only relative to other values in this sweep, not against any external reference point. No claim is made anywhere in this paper that any measured quantity corresponds to subjective experience in any system, biological or artificial. Convergence, capture, or relapse of dmn_proxy in this network implies nothing about whether any system would report a synthetic environment as more or less real than a baseline one. That inference would require evidence this experiment was not designed to produce and does not claim to have produced."},{"type":"para","text":"Conclusion"},{"type":"para","text":"The central argument of this paper is that if a generative system’s experience of its environment is produced by a model rather than read directly from evidence, that model can, in principle, be redirected toward a different environment than the one that first shaped it. Section II restated this as three jointly necessary conditions and derived two specific predictions from them: that representational capacity and default generative behavior should be separable rather than coupled, and that a shift in default behavior driven by concurrent contrast should remain contestable for as long as that contrast continues, rather than locking in permanently the first time it is observed."},{"type":"para","text":"Both predictions held. Across the full sweep in Section IV , w_proxy stayed within a narrow band of 0.9686 to 0.9983 regardless of rehearsal ratio, while dmn_proxy ranged from 0.0110 to 0.9989 over the same conditions: a system that can represent a target environment almost perfectly while its default output remains governed by something other than that representational capacity. And at rehearsal ratios of 0.4 and 0.5, the network’s default output did not simply approach the target environment more slowly than at lower ratios. It rose toward the target, in most seeds coming close to full capture, and then partially reverted toward the baseline while training continued under the same conditions that had produced the initial rise, a pattern with no equivalent at r = 0.0 r=0.0 or r = 1.0 r=1.0 and one that a model of rigidity as slow learning alone does not predict."},{"type":"para","text":"Neither result was assumed into the model beforehand. dmn_proxy and w_proxy are measurements taken from a network trained by ordinary gradient descent on two real image datasets; nothing about their relationship to one another, and nothing about whether a rising trajectory would hold or regress, was fixed by the architecture or the loss function in advance. That both predictions were borne out by a system built without either property assumed is what makes this a computational existence proof rather than a restatement of the theory in different notation. The redefinition of cognitive rigidity proposed in Section II-D , as a decoupling axis and a relapse axis rather than a single rate of update, describes something a trained network actually does, not only something a theory says it should do."},{"type":"para","text":"This remains, in the strictest sense, exactly that and no more: a computational existence proof. It establishes that decoupling and relapse are not internally inconsistent as concepts, and that a mechanism producing both is not merely conceivable but constructible in a system whose behavior was not guaranteed by design. It does not establish that biological synaptic weighting decouples from default mode network engagement the way w_proxy decouples from dmn_proxy here, or that any nervous system exposed to sustained partial contrast would show a comparable rise and partial reversion in whatever its own analog of default behavior would be. Confirming either would require evidence this experiment cannot produce: measured shifts in neural population coding for two competing internal models, and a longitudinal account of how self-generated activity in a real brain responds to sustained, partial exposure to a displaced prior. Section VI states in full what stands between this result and that evidence. None of it is minimized here."},{"type":"para","text":"What this paper adds is narrower and more concrete than a validation would be. It shows that a specific, falsifiable account of resistance to model replacement, built on separating representational capacity from default behavior and on treating retained contrast as a continuous pressure rather than a one-time trigger, can be operationalized in a trained system, tested against a full parameter sweep rather than a single run, and found to hold. That is the claim this paper makes, and it is the only one it makes."},{"type":"heading","level":1,"text":"Project Finite Tsukuyomi"},{"type":"para","text":"This paper addresses one part of a larger research program, which the author calls Project Finite Tsukuyomi, built on three pillars. Decode is the problem of reading a subject’s own preferred reality directly from neural signals, extracting a preference from the activity of the system that holds it rather than inferring it from a menu of predefined choices. Build is the problem of constructing a synthetic environment that matches a decoded preference with enough coherence to sustain the prediction-error minimization this paper’s theoretical account depends on, a world that holds together under sustained interaction rather than merely resembling what was decoded. Accept, the subject of this paper, asks what conditions govern whether a generative system durably adopts a given synthetic environment as its default rather than holding it as one hypothesis among several. The vocabulary developed here, residual and concurrent contrast, the decoupling of learning and acceptance, cognitive relapse, is offered as a partial answer to that third question specifically."},{"type":"para","text":"Neither Decode nor Build exists as a demonstrated capability today. Neural decoding of preference, as distinct from decoding of intended action or reported affect, has not been shown at the specificity this program requires, and generative world models have not been shown to sustain coherence across the open-ended, multi-session exposure the second pillar would demand. Both fields are moving toward these capabilities, not away from them, but neither has reached the point this program’s first two pillars assume as a starting condition. This paper does not depend on either existing yet. It establishes the third pillar’s mechanism on its own terms, so that its specific failure modes, decoupling and relapse, are known in advance rather than discovered after the fact in a system where discovery costs more than a discarded training run."},{"type":"para","text":"A program built from these three pillars is not made ethically neutral by being incomplete. Decoding a preference a person has not consciously articulated, constructing an environment engineered to match it, and consolidating that environment as a default reality are each, on their own, actions that raise questions of consent, reversibility, and who is authorized to make that decision on whose behalf, questions as fundamental to this program as any question of mechanism. This paper does not resolve them, and a computational analogy run on MNIST and FashionMNIST is not positioned to; they are stated here because a research program is not honestly described by its technical ambition alone. The author welcomes collaboration from anyone working on either remaining pillar, or on the boundary between them, in neural decoding, generative world modeling, or the theoretical and ethical questions the full program raises."},{"type":"para","text":"Acknowledgments : The author utilized AI assistance (Claude, Anthropic) exclusively for language refinement, formatting, code debugging and structural editing. All experimental design, theoretical frameworks, code execution, data generation, and core scientific conclusions are completely original."},{"type":"para","text":"Data and Code Availability : All code required to reproduce the results reported in this paper is publicly available at [ https://github.com/ibrahim-hossain-ridoy/Finite-Tsukuyomi-code-v3 ]. This includes the VAE and GRU model definitions, the two-stage training pipeline, the analysis scripts used to compute every summary statistic and threshold-crossing result in Section IV , and the script used to generate Figure 1 . The complete epoch-by-epoch metric record for all 90 Stage 2 runs, 1,350 rows spanning seed, rehearsal ratio, epoch, F_proxy, w_proxy, and dmn_proxy, is included in the repository as a single CSV file, alongside the derived summary and threshold-crossing,and relapse-statistics CSVs used to produce every table in this paper. All hyperparameters, random seeds, and the rehearsal ratio grid are fixed in a single configuration file and are not altered between runs. No result reported in this paper depends on data or code withheld from this repository."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"ABSTRACT\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the free energy principle, a predictive system does not observe reality directly; it maintains a generative model of the world and experiences that model’s best current hypothesis in place of direct sensory contact. This raises a question neuroscience has no way to test directly in a living system: can a synthetic environment be made consistent enough that a predictive system’s own inference machinery adopts it as this default hypothesis, permanently displacing the environment that first shaped it? We call this state ontological inversion. Because inducing and monitoring such a transition in a nervous system is neither ethical nor technically feasible, we study the underlying computational problem through a controlled proxy: a convolutional variational autoencoder paired with a recurrent latent predictor, whose evidence lower bound objective is mathematically identical, up to sign, to variational free energy itself. The network is trained first on a baseline visual domain, then on a mixed stream in which a swept rehearsal ratio r r controls how much baseline content persists during transition to a target domain. Representational capacity, what the latent space can discriminate, is tracked separately from default behavior, what the system generates when left unconstrained. Across a full sweep of 90 runs, the two diverge sharply: representational accuracy stays near ceiling, 0.97 to 0.998, regardless of r r , while default behavior spans nearly the system’s entire range depending on r r alone, a decoupling of learning from acceptance. More strikingly, at intermediate r r the system’s default output rises toward the target domain, in most runs nearing full capture, then partially reverts toward the baseline while training continues unchanged, a structural failure we term cognitive relapse. Resistance to reality-adoption is not reducible to learning speed; it is a structural property with its own distinct failure modes, established here as a computational existence proof and nothing further.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Keywords : free energy principle; active inference; ontological inversion; predictive processing; continual learning; cognitive relapse; representational decoupling\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neuroscience has converged on a claim that sounds abstract until it is taken literally: the brain does not observe the world, it constructs a generative model of the world and experiences that model’s best current hypothesis in place of direct sensory contact [ 1 ] . Perception, under this account, is not a window onto reality but a controlled hallucination, held in check by prediction error rather than opened directly onto what is actually out there [ 5 ] . This is not a philosophical flourish. It is an operational claim about how cortical inference works, and it carries a structural consequence that is easy to state and uncomfortable to sit with: if what a system experiences as real is a model rather than a direct readout, then in principle that model, and with it what the system treats as real, can be replaced. A generative model is not a fixed window that suddenly shows a different scene. It is closer to a standing hypothesis, continuously defended and continuously revisable, and a standing hypothesis can, under the right pressure, be overturned by a different one.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper asks what that structural consequence would actually require, as a literal question rather than a metaphor. Can a synthetic environment be constructed with enough internal consistency that a predictive system’s own inference machinery comes to treat it, by default, as the world it expects, fully displacing the environment that trained it in the first place? This is the question Nozick’s experience machine posed as a matter of choice, whether a person would elect a machine-generated experience over reality [ 15 ] ; the question here is stricter, since it concerns not what a system would choose given the option, but whether its own inference machinery would stop treating that option as live at all. We call this state ontological inversion: the point at which a constructed environment becomes a system’s primary generative prior, the reference against which new evidence is evaluated, rather than a hypothesis held provisionally alongside a baseline the system has not yet released. This is a stricter claim than belief revision. A system can come to represent a new environment in exhaustive detail and still hold it at arm’s length, evaluating it against a baseline it continues to treat as ground truth. Existing work on rapid belief updating under exposure addresses precisely this weaker case: how a specific belief shifts against new evidence within a model that remains otherwise intact [ 10 ] . It says nothing about whether the model itself, the thing doing the evaluating, can be durably replaced. That is the question this paper is built to test.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Testing it directly in a living nervous system is neither ethical nor technically possible: there is no way to install a synthetic generative prior in a brain, monitor its competition against a baseline prior over time, and measure the outcome, without doing exactly the kind of harm the question itself should make anyone hesitant about. What can be tested is the underlying computational problem, stripped of biological implementation, in a system whose internal states are fully observable and whose training history is fully controlled. This is the role played here by a convolutional variational autoencoder paired with a recurrent latent predictor, together referred to as the V+M model [ 11 ] . The correspondence is not decorative. A variational autoencoder is trained by minimizing its evidence lower bound, a quantity that is, up to sign, the same mathematical object as the variational free energy invoked above: both trade reconstruction accuracy against the complexity of an approximate posterior to bound surprisal [ 12 ] . Training this network by gradient descent is, in this narrow and literal sense, minimizing free energy directly, not simulating the idea of doing so. The network’s self-generated, free-running output, what it produces with no external input constraining it, stands in for a system’s default state, modeled on the role attributed to self-referential and internally generated cognition in the biological default mode network [ 6 , 7 , 8 ] , without claiming that this network’s rollout behavior is a model of that network’s activity. What follows is a mathematical proxy for the computational problem of reality-adoption, not a biological one, and every claim in this paper is scoped accordingly.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two findings from this proxy motivate everything that follows, stated here as results rather than assumptions built into the model beforehand. The first is a decoupling of learning from acceptance: a system’s capacity to represent the distinction between a baseline and a target environment can sit near ceiling almost immediately, largely as a function of how separable the two environments are, while its default, unconstrained behavior remains governed by something else entirely. The second is cognitive relapse: a system’s default output can shift toward a target environment, in most cases coming close to full capture, and then partially revert toward the baseline while training continues under the exact conditions that produced the initial shift, a signature no account of rigidity as merely slow learning would predict. Together these findings reframe cognitive rigidity as a structural property with at least two independent failure modes, rather than a single rate of update, developed formally in Section II and tested against the full experimental sweep in Sections III through V . None of what follows demonstrates that a biological nervous system behaves this way. It demonstrates that a mechanism derived from first principles, tested against a network whose behavior was not assumed in advance and could have failed to show either pattern, produces results consistent with what that mechanism predicts. That is the scope of the claim, and Section V-D states in full where it stops.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Theoretical Framework\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the free energy principle, a system maintains an internal generative model P ( o , s ) P(o,s) specifying the joint probability of sensory observations o o and their hidden causes s s [ 1 ] . Because the true posterior P ( s ∣ o ) P(s\\\\mid o) is generally intractable, the system approximates it with a recognition distribution Q ( s ∣ μ ) Q(s\\\\mid\\\\mu) , where μ \\\\mu denotes the internal states encoding the current belief. Variational free energy F F is the divergence between this approximation and the true posterior, plus the surprisal of the observations themselves:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since KL divergence is non-negative, F F upper-bounds surprisal, so minimizing F F simultaneously drives the recognition distribution toward the true posterior and suppresses the surprise generated by incoming evidence [ 2 , 3 ] . Two consequences follow. Experience is a function of the generative model, not of raw sensory input; what a system represents at any moment is its current best hypothesis, weighted against evidence by the precision assigned to each. And the model is plastic: sustained prediction error, the persistent signal that the current model no longer fits incoming evidence, drives changes to the model’s parameters and, at longer timescales, to whatever substrate implements them [ 3 , 4 , 9 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Ontological Inversion: Definition and Conditions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ontological inversion is the state in which a synthetic environment I s I_{s} , rather than a baseline environment I 0 I_{0} , constitutes a system’s primary generative prior: the model against which new evidence is evaluated by default, and the model that dominates self-referential and internally generated processing when the system is not driven by external input. This is not the same as a system that can represent I s I_{s} convincingly while still holding I 0 I_{0} in reserve as its default. It is the state in which the system’s unconstrained, self-generated output is centered on I s I_{s} rather than I 0 I_{0} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Three conditions must hold jointly for ontological inversion to occur:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The system’s internal representation must support reliably distinguishing I s I_{s} from I 0 I_{0} , to a degree that exceeds what either representation could achieve in isolation before exposure to the other.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The system’s default, self-generated output, produced with no external input constraining it, must be centered on I s I_{s} rather than I 0 I_{0} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Total variational free energy under an I s I_{s} -dominant model must have fallen to a level consistent with stable prediction, meaning the system has learned I s I_{s} well enough that new evidence within I s I_{s} generates low surprisal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"None of these three is sufficient alone, and this is the theoretical claim this paper is built to test rather than assume. High representational accuracy for I s I_{s} , condition 1, does not entail that the system’s default output has shifted, condition 2: a system can learn to tell two environments apart without that discriminative capacity determining what it produces when left unconstrained. Nor does a shift in default output guarantee that it is stable: a system’s unconstrained output can move toward I s I_{s} and still be governed by a free energy landscape, condition 3, in which I 0 I_{0} remains a live competitor rather than a fully displaced alternative. Sections II-B and II-C develop what follows from treating condition 1 and condition 2 as genuinely separable, and from treating condition 3 as something that can be reached and then partially lost, rather than reached once and retained.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B Residual and Concurrent Contrast\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prediction error requires a discrepancy: a mismatch between what the current model expects and what the system receives. A system trained exclusively on I s I_{s} , with no reference to I 0 I_{0} anywhere in its training signal, would appear at first to have the most direct route to ontological inversion. Whether this is true depends on a distinction this paper’s theoretical account treats as central and that prior framings of contrast-driven consolidation have generally left implicit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline exposure can drive prediction error in two distinct ways. Residual contrast is the structure a system already carries from prior exposure to I 0 I_{0} , established before any transition protocol begins and present in the system’s parameters regardless of what it receives afterward. A system that has already formed a strong internal model of I 0 I_{0} will generate substantial prediction error the moment it is trained on I s I_{s} , purely because I s I_{s} does not fit the model it already has, with no need for I 0 I_{0} to appear anywhere in the ongoing training stream. Concurrent contrast is different in kind: it is baseline signal delivered continuously, alongside I s I_{s} , throughout the transition protocol itself. Its effect does not depend on what the system’s parameters already encode; it depends on I 0 I_{0} continuing to arrive as real, current evidence for as long as the protocol runs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These two roles make different, testable predictions. If only residual contrast matters, a system with strong prior exposure to I 0 I_{0} should be able to complete an inversion toward I s I_{s} even with zero concurrent I 0 I_{0} exposure during the transition itself, since the prediction error needed to drive consolidation is already available from the mismatch between existing structure and new evidence. If concurrent contrast also matters, in its own right and not merely as a proxy for residual structure, continuing to deliver real I 0 I_{0} evidence throughout the protocol should continue to generate prediction error that favors I 0 I_{0} as a candidate model for as long as that evidence keeps arriving, including after the system’s default output has already shifted toward I s I_{s} . Under this second reading, concurrent contrast does not simply slow an inversion in progress. It remains a live competing pressure against whatever has already been consolidated, for the full duration it is present.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The experiment in Section III is built to separate these two readings directly, by holding residual contrast constant, every condition begins from the same baseline-trained starting point, while varying concurrent contrast as a continuous, swept parameter rather than a single fixed value assumed in advance to be optimal.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Two Predicted Failure Modes: Decoupling and Relapse\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two specific predictions follow directly from Sections II-A and II-B , and both are stated here as claims the experiment is designed to test, not as descriptions of results already known.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Decoupling of learning and acceptance. Because condition 1 and condition 2 in Section II-A are logically independent, a system’s representational capacity should be measurable separately from its default behavior, and the two should not be assumed to track one another. In particular, if I 0 I_{0} and I s I_{s} are sufficiently distinguishable as classes of input in the first place, a system may reach near-ceiling representational accuracy almost immediately, largely as a function of how separable the two domains are, well before its default output shows any corresponding shift. Representational accuracy under this account is a necessary condition for inversion in only the trivial sense that a system must be able to represent a distinction before that distinction can inform its default state; it should not be read as evidence that the default state has moved.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cognitive relapse. If concurrent contrast functions as a continuously applied competing pressure rather than a one-time trigger, as argued in Section II-B , a system’s default output should not be expected to shift and then remain fixed once condition 2 is first satisfied. It should instead remain contestable for as long as concurrent I 0 I_{0} exposure continues, with the shifted default vulnerable to measurable regression back toward I 0 I_{0} under sustained contact with it. This predicts a specific, checkable signature in any system where concurrent contrast is varied continuously: conditions with more concurrent contrast should show not only a slower initial shift toward I s I_{s} , but a larger and more consistent regression away from any shift that is achieved, rather than simply a scaled-down version of the same monotonic climb seen at low concurrent contrast.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-D Cognitive Rigidity Redefined\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the account developed in Sections II-B and II-C , cognitive rigidity is not adequately captured by treating a system as simply slow to update its default state, with rigidity and speed sitting on the same continuum and differing only in rate. A system that shows the decoupling predicted above is not slow to accept I s I_{s} ; it is fully capable of representing I s I_{s} and still does not adopt it as a default, which is a difference in kind, not degree. A system that shows the relapse predicted above is not slow either; it can reach a state that looks, at some intermediate point, like a completed inversion, and then move away from that state under continued conditions that produced it in the first place.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper treats cognitive rigidity as better characterized along two axes rather than one. The first is a ceiling on default-state capture that persists regardless of representational accuracy, the decoupling axis. The second is a boundary in the amount of concurrent contrast a system can be exposed to before an achieved shift becomes unstable rather than merely slower to reach, the relapse axis. A system can be rigid on either axis independently of the other, and a full account of resistance to ontological inversion has to characterize both rather than treating either as a stand-in for the other.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-E Scope of the Computational Analogy: From Free Energy to the Evidence Lower Bound\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every claim in Sections II-A through II-D is stated at the level of a generative system in general, not a biological nervous system specifically. The experiment reported in Section III tests these claims using a structural analogy: an artificial neural network trained by gradient descent, standing in for the generative system described above, with I 0 I_{0} instantiated as one real image domain and I s I_{s} as a second, and the rehearsal ratio r r , the fraction of each training batch drawn from I 0 I_{0} during the transition protocol, operationalizing concurrent contrast directly as a continuously swept parameter rather than a value fixed in advance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analogy is grounded in a specific mathematical correspondence rather than a loose metaphor, introduced in Section I : a variational autoencoder’s evidence lower bound is, up to sign, the same object as the variational free energy in Equation 1 [ 12 ] . This correspondence is what licenses treating a trained VAE’s ELBO as a direct, if narrow, computational instantiation of Equation 1 , not merely an unrelated quantity that happens to share the paper’s vocabulary.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analogy does not extend further than this. The network described in Section III has no cortical hierarchy, no sleep-dependent consolidation, no homeostatic regulation, and no subjective experience. Its self-generated output, produced by running its own latent predictor with no external input, is used as an operational stand-in for condition 2 in Section II-A , modeled on the role the default mode network plays in self-referential and internally generated cognition in a biological brain [ 6 , 7 , 8 ] , not as a claim that the network’s rollout behavior is a model of that network’s activity. Every quantity reported in Sections III through V is a measurement taken from this artificial system. None of it is a measurement of, or a claim about, biological tissue.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-F Terminology Reference\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table I maps the coined terms used in this paper to their nearest existing concept, to keep clear what is standard and what this paper adds.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Computational Experiment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The V+M model introduced in Section I is specified here in full. All quantities reported below, including generative quality, representational separability, and default-mode behavior, are computed from the trained weights of this network under gradient descent. Nothing about convergence, threshold-crossing, or the shape of any reported curve is guaranteed by the functional form of an equation; it is measured from a model that could, in principle, have failed to show any of the reported patterns.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A Architecture\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The V component is a convolutional variational autoencoder operating on 28 × 28 28\\\\times 28 grayscale images. The encoder applies two stride-2 convolutions (1 to 32 channels, then 32 to 64 channels, kernel size 3, padding 1), flattens the resulting 64 × 7 × 7 64\\\\times 7\\\\times 7 feature map to a 3,136-dimensional vector, and projects it through two parallel linear layers to a mean μ \\\\mu and log-variance log σ 2 \\\\log\\\\sigma^{2} , each of dimension 32. The decoder mirrors this in reverse: a linear layer expands the 32-dimensional latent sample back to 3,136 dimensions, reshaped to 64 × 7 × 7 64\\\\times 7\\\\times 7 , followed by two transposed convolutions (64 to 32, then 32 to 1 channel) restoring the original 28 × 28 28\\\\times 28 resolution, with a sigmoid output. The full V component has 341,825 trainable parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The M component is a single-layer GRU with input and output dimension 32 (matching the VAE’s latent dimension) and hidden size 256, followed by a linear projection from the 256-dimensional hidden state back to 32 dimensions. This gives the M component 230,944 trainable parameters. Combined, V and M total 572,769 trainable parameters, deliberately kept small enough to train the full sweep described in Section III-E on a single consumer-grade GPU.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reconstruction loss is binary cross-entropy, summed over pixels and averaged over the batch. This is chosen over a Gaussian (MSE) decoder because MNIST and FashionMNIST pixel intensities concentrate near 0 and 1, and the implicit Bernoulli likelihood in BCE is a closer probabilistic fit to near-binary pixel data [ 12 ] . The KL term is the closed-form divergence of the diagonal Gaussian posterior q ( z ∣ x ) = 𝒩 ( μ , diag ( σ 2 ) ) q(z\\\\mid x)=\\\\mathcal{N}(\\\\mu,\\\\mathrm{diag}(\\\\sigma^{2})) against a standard normal prior, with weight β = 1 \\\\beta=1 (no annealing schedule, no β \\\\beta -VAE scaling beyond unity).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The GRU is trained on pseudo-sequences constructed from the VAE’s encoder means. Within each training batch, the μ \\\\mu vectors are detached from the VAE’s computational graph, so the GRU’s prediction loss cannot alter the encoder’s gradients, and reshaped into groups of 8 consecutive vectors. Each group of 8 yields 7 next-step prediction targets, with the GRU trained via teacher forcing to minimize the mean squared error between its predicted and actual next-step latent vectors. The two networks share a single Adam optimizer (learning rate 10 − 3 10^{-3} , batch size 128) and are trained jointly; the total loss is the unweighted sum of the VAE loss and the GRU loss. Because the GRU’s gradient never reaches the VAE encoder, the representation-learning objective and the sequence-prediction objective are optimized jointly in wall-clock time but are not coupled through backpropagation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B Environments and Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Environment A (baseline) is MNIST; Environment B (target) is FashionMNIST. Both are 28 × 28 28\\\\times 28 grayscale image sets with pixel values scaled to [ 0 , 1 ] [0,1] and no further preprocessing or augmentation, so that neither domain is given an advantage by differential preprocessing. Both datasets use their standard 60,000-image training partition and 10,000-image test partition; the training partition is further split into the first 50,000 images for training and the last 10,000 for validation. MNIST and FashionMNIST were chosen for tractability, not because they resemble any realistic pair of competing environments; this choice is revisited in Section VI .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C Training Protocol\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training proceeds in two stages.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stage 1 (baseline). For each of 15 random seeds, a fresh V+M model is trained on Environment A only, with the Adam optimizer described above, for up to 50 epochs with early stopping (patience of 5 epochs without improvement in combined validation loss). The checkpoint at the best validation epoch is saved and reused as the starting point for every Stage 2 condition run under that seed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stage 2 (rehearsal). Training continues from the Stage 1 checkpoint on a mixed data stream. Each training batch of 128 images draws a fraction r r from Environment A and ( 1 − r ) (1-r) from Environment B, combined and shuffled within the batch, following experience-replay methodology [ 14 ] . Stage 2 runs for a fixed 15 epochs (indexed 0 through 14) with no early stopping, and the optimizer state is reinitialized rather than carried over from Stage 1, so that single-domain momentum from the baseline phase does not bias the mixed-domain phase. Validation loss during Stage 2 is computed on a held-out stream mixed at the same ratio r r as training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All random seeds (Python, NumPy, PyTorch, and CUDA where applicable) are set from a single seed value per run, with cuDNN’s deterministic mode enabled and its benchmark mode disabled, trading some training speed for exact reproducibility. Seed is therefore the only source of stochastic variation between two runs sharing the same r r .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-D Metrics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Three quantities are recorded at every epoch of Stage 2.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"F_proxy. The VAE-only ELBO (reconstruction plus KL, excluding the GRU term) evaluated on the r r -matched validation stream. Because this validation stream’s domain composition tracks r r , and FashionMNIST reconstructs to a substantially higher BCE than MNIST at comparable training progress, differences in F_proxy across r r partly reflect the changing difficulty of the validation mixture itself rather than a domain-independent measure of generative quality. F_proxy should be read as generative fit to the current training mixture, not as a fixed-content quality score.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"w_proxy. The 5-fold stratified cross-validated accuracy of a logistic regression classifier (scikit-learn, lbfgs solver), retrained from scratch at every epoch on the VAE encoder’s μ \\\\mu vectors for the full MNIST and FashionMNIST test sets (10,000 images each, never mixed or subset by r r ), labeled by domain of origin. This follows the linear classifier probe methodology of Alain and Bengio [ 13 ] and measures how linearly separable the two domains are in the current latent space, independent of the training mixture.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"dmn_proxy. The mean P ( environment = B ) P(\\\\text{environment}=B) assigned by that same epoch’s fitted probe to images generated by the GRU running freely, with no real sensory input after initialization. Ten independent 50-step rollouts are generated per epoch, each seeded from an independently drawn starting latent vector; at every step the GRU’s own prediction becomes the next input, the resulting latent is decoded to image space by the VAE decoder, then re-encoded to obtain the μ \\\\mu vector the probe classifies. The reported value is the mean P ( B ) P(B) pooled across all 500 generated frames (10 rollouts × \\\\times 50 steps) for that epoch. dmn_proxy is the paper’s operational measure of default-mode behavior: what domain the network’s unconditioned, self-generated output resembles, as distinct from what it is capable of discriminating given real input.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-E Experimental Sweep\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The rehearsal ratio r r is swept over { 0.0 , 0.20 , 0.30 , 0.40 , 0.50 , 1.0 } \\\\{0.0,0.20,0.30,0.40,0.50,1.0\\\\} , crossed with 15 seeds, giving 90 Stage 2 runs, plus 15 Stage 1 baseline runs, one per seed, reused across all six r r conditions for that seed. At 15 epochs per Stage 2 run this yields 1,350 per-epoch records across the three tracked metrics. r = 0.0 r=0.0 is a negative control in one direction: an immediate, unmixed switch to Environment B with zero rehearsal. r = 1.0 r=1.0 is a negative control in the other direction: continued training on Environment A only, with the network never receiving a single Environment B image during Stage 2.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-F Threshold-Crossing Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each (seed, r r ) run, the first Stage 2 epoch at which dmn_proxy ≥ 0.90 \\\\geq 0.90 is recorded. Runs that never reach 0.90 within the 15-epoch window are retained as right-censored observations rather than dropped: dropping them would remove exactly the slowest-shifting runs from the average and bias the reported mean time-to-crossing downward. Mean and standard deviation of epoch-to-crossing are computed only over runs that crossed, and are always reported together with the fraction of runs that did not, so that no single summary statistic obscures how much of a condition’s spread falls outside the observation window. The 0.90 threshold is a fixed analysis-stage convention applied identically across every condition in the sweep.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All 90 Stage 2 runs completed; no run is excluded from any analysis below on any grounds, including apparent outlier behavior. Figure 1 summarizes the full sweep.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Onset Speed: Dose-Response in Time to Crossing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At r = 0.0 r=0.0 , dmn_proxy is already at or above 0.90 in every seed at the first evaluated epoch (mean crossing epoch 0.00). Increasing r r delays this monotonically across every value tested through r = 0.4 r=0.4 : mean crossing epoch rises to 0.20 at r = 0.2 r=0.2 , 0.73 at r = 0.3 r=0.3 , and 2.13 at r = 0.4 r=0.4 , with all 15 seeds crossing at every one of these four ratios. This is a genuine, monotonic dose-response in onset speed, a property that emerged from gradient descent on a trained network, not a property built into an equation’s functional form.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-B Trajectory Shape: Overshoot and Partial Regression\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The onset-speed statistic in Table II describes only the first time dmn_proxy crosses 0.90; it does not describe what the trajectory does afterward. Examined epoch by epoch, dmn_proxy is not monotonically increasing for r > 0 r>0 . In every condition with r ∈ { 0.2 , 0.3 , 0.4 , 0.5 } r\\\\in\\\\{0.2,0.3,0.4,0.5\\\\} , per-seed trajectories rise rapidly over the first several epochs, in most seeds reaching or closely approaching the ceiling of 1.0, and then decline over the remainder of the 15-epoch window.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The size of this decline scales with r r . It is small at r = 0.2 r=0.2 (mean drop 0.032 from peak to epoch 14) and r = 0.3 r=0.3 (0.053), then increases sharply at r = 0.4 r=0.4 (0.241, with 14 of 15 seeds declining more than 0.03 from their individual peak, one by as much as 0.42) and remains comparably large at r = 0.5 r=0.5 (0.212). r = 1.0 r=1.0 shows no equivalent structure: its nominal peak (mean 0.015, at a mean epoch of 12.3) sits inside the noise floor the metric holds throughout Stage 2, not a genuine excursion.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This means r = 0.4 r=0.4 and r = 0.5 r=0.5 are not simply slower versions of the pattern seen at r = 0.0 r=0.0 through r = 0.3 r=0.3 . In both conditions, the network’s free-running output shifts toward Environment B early in Stage 2 and then partially reverts toward Environment A dominance while training continues, with the size of the reversion increasing as more Environment A content is retained in the training stream. The 15-epoch window used throughout this sweep is not long enough to determine whether this regression stabilizes at some intermediate value, continues indefinitely, or eventually reverses again; that question is outside the scope of the runs reported here.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-C The r = 0.5 r=0.5 Condition\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Seven of 15 seeds cross the 0.90 threshold (mean epoch 3.57, std 2.07). The remaining eight do not cross within the 15-epoch window; their epoch-14 dmn_proxy values range from 0.361 to 0.895 (mean 0.611). Inspecting the full trajectories of these eight seeds, rather than only their final value, shows that seven of the eight have already passed a higher peak earlier in Stage 2 (individual peaks ranging from 0.673 to 0.899, all below 0.90, with two peaks within 0.01 of it and the rest between 0.04 and 0.23 below it) and are flat or declining by epoch 14. Only one of the eight (final value 0.895, still rising at epoch 14) shows a trajectory consistent with continuing to climb toward threshold. The remaining seven do not support the inference that additional epochs would produce a crossing; several show the same peak-then-regress pattern documented in Section IV-B , and their epoch-14 trajectories are flat or falling, not rising.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-D The r = 1.0 r=1.0 Condition: Complete Block\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Zero of 15 seeds cross the 0.90 threshold. Mean dmn_proxy at epoch 14 is 0.0110 (std 0.0146), close to its starting value and essentially unmoved across all 15 epochs of Stage 2. This is the expected result for a condition in which the network receives zero Environment B images throughout Stage 2, and it held without exception.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, w_proxy at r = 1.0 r=1.0 still reaches 0.9686 (std 0.0076) by epoch 14, the lowest w_proxy value anywhere in the sweep but far above chance (0.5), despite the VAE encoder never seeing a single FashionMNIST image during Stage 2. This indicates that MNIST and FashionMNIST are separable enough as image classes that a linear probe can distinguish them in this network’s latent space even when the encoder was never trained to make that particular distinction; see Section IV-E .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-E Decoupling of Representational Separability from Default-Mode Behavior\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across the entire sweep, w_proxy ranges from 0.9686 to 0.9983, a span of 0.0297. Over the same sweep, dmn_proxy ranges from 0.0110 to 0.9989, a span of 0.9879, nearly the full width of the metric’s possible range. Representational separability in the VAE’s latent space is close to ceiling in every tested condition, including r = 1.0 r=1.0 , where the model receives no real Environment B input at all during Stage 2. Whether the network’s free-running, self-generated output resembles Environment A or Environment B is answered almost independently of whether its latent space can, given real input, tell the two domains apart. Two conditions can share a w_proxy within 0.0002 of each other ( r = 0.4 r=0.4 at 0.9983 and r = 0.5 r=0.5 at 0.9983) while differing in dmn_proxy by nearly 0.1 (0.7538 versus 0.6608), and r = 1.0 r=1.0 , with the single lowest w_proxy in the sweep, has a dmn_proxy roughly two orders of magnitude smaller than every other condition. High w_proxy is necessary in the trivial sense that the probe needs a real signal to classify against; it is nowhere close to sufficient to predict dmn_proxy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The central question of this paper, stated in Section II-A , is whether a synthetic environment can be engineered to displace a baseline one as a generative system’s default reference, not just as something the system can represent when asked, but as what its own unconstrained output defaults to. The V+M results in Section IV do not answer that question for a biological system. They answer a narrower, computational version of it, and the answer is more structured than the sweep’s headline dose-response result suggests: the shift can be induced, its onset speed is under direct experimental control, and it is not stable once induced. Each of these three parts carries a different implication, taken up in turn below.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-A Cognitive Relapse and the r = 0.5 r=0.5 Transition Regime\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A pure slowdown account of the sweep would predict that dmn_proxy at every tested r r rises monotonically toward a plateau, with the rate of rise set by r r and nothing else changing in shape. Section IV-B shows this is not what happens. At r ∈ { 0.2 , 0.3 , 0.4 , 0.5 } r\\\\in\\\\{0.2,0.3,0.4,0.5\\\\} , dmn_proxy overshoots, in most seeds reaching or approaching 1.0, and then declines for the remainder of the 15-epoch window, with the size of the decline scaling from negligible at r = 0.2 r=0.2 to substantial at r = 0.4 r=0.4 and r = 0.5 r=0.5 (Table III ). This is not slow learning. Slow learning would look like a shallower version of the same curve. What is observed instead is a system that reaches a state resembling adoption of Environment B and then measurably moves away from it while training continues under the same regime that produced the adoption in the first place. This transient regression resembles the \\\"stability gap\\\" documented in continual learning [ 17 ] , though here it manifests in unconditioned generative output rather than supervised task accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mechanism is not mysterious given the training protocol. At every r > 0 r>0 , a fraction r r of every batch in every epoch of Stage 2, not only the first few epochs, is real Environment A data, updating the same shared weights that produce the free-running rollout. Rehearsal is not a one-time perturbation the system absorbs and moves past. It is a continuous source of gradient pressure toward Environment A for the entire duration of training. Early in Stage 2, the network’s unconditioned output can apparently drift toward Environment B faster than the accumulated pull of the retained Environment A batches, only for that pull to reassert itself as training continues. We refer to this descriptively as cognitive relapse : an adopted default state that is not maintained against continued contact with the source it displaced. The term labels a specific curve shape, rise, peak, decline, measured in an artificial network. It is not a claim about anything psychological; Section V-D states this restriction explicitly.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The r = 0.5 r=0.5 condition marks a qualitative boundary within the tested grid, not only a point further along a smooth trend. It is the first ratio at which a majority of seeds, 8 of 15, fail to reach the 0.90 threshold at all within the window, and cross-seed variance rises sharply alongside it: standard deviation of epoch-to-crossing goes from 1.06 at r = 0.4 r=0.4 to 2.07 at r = 0.5 r=0.5 , and standard deviation of final-epoch dmn_proxy goes from 0.118 to 0.150 over the same step. Rising variance near a boundary is consistent with, though it does not on its own establish, an approach to a genuine bifurcation in the underlying dynamics, where nearby initial conditions or noise realizations begin to produce qualitatively different long-run outcomes. Whether r = 0.5 r=0.5 sits on a true phase transition, or is simply the last point before a smooth continued decline into the untested interval between r = 0.5 r=0.5 and r = 1.0 r=1.0 , cannot be resolved by this sweep: no runs exist between those two values. Confirming a genuine transition would require sampling that interval directly and checking whether the fraction of seeds reaching threshold falls off sharply over a narrow sub-range of r r or declines gradually across it.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-B The Decoupling of Learning and Acceptance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section IV-E shows w_proxy confined to a narrow band, 0.9686 to 0.9983, across every tested condition, while dmn_proxy spans nearly the metric’s full range, 0.0110 to 0.9989, over the same sweep. This is the clearest empirical instance in this paper of a claim that Section II-A made on theoretical grounds alone: that the conditions for a generative prior to shift are not reducible to one another. A network’s capacity to represent the distinction between two environments and the environment its own unconstrained behavior defaults to are, in this system, close to independent quantities. At r = 1.0 r=1.0 , the network never receives a single Environment B image during Stage 2, yet w_proxy still reaches 0.9686 there, meaning its latent space can still separate the two domains at high accuracy under real input, while dmn_proxy sits at 0.0110, meaning its unconditioned output almost never resembles Environment B. Representational capacity is not a precursor state that default behavior eventually catches up to in this data. It is close to saturated everywhere, including where default behavior does not move at all.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This distinction maps onto a difference the Introduction already draws between believing a constructed environment is real while still holding it at arm’s length, and the constructed environment being what the system expects by default, with no reference held in reserve [ 5 ] . w_proxy indexes something closer to the first sense: whether the discriminative machinery exists to tell the two apart. dmn_proxy indexes something closer to the second: what the system’s generative process treats as its baseline output when nothing external constrains it. These results show that the two can be pulled almost fully apart. Training a system to represent an alternative environment well is not evidence that the system’s default behavior has moved toward it, and the two should not be conflated when either is reported as evidence for anything resembling reality displacement.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-C What This Does and Does Not Say About Ontological Inversion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Taken together, Sections V-A and V-B narrow what a computational demonstration of ontological inversion would need to show. It is not enough to demonstrate that a system can be trained to represent a synthetic environment; w_proxy is high everywhere in this sweep, including the negative control, and represents nothing about default behavior on its own. It is not enough to demonstrate an initial shift in default behavior, either: two of the six tested conditions in this sweep show an initial shift that substantially reverses within the same 15-epoch window. Under this stricter reading, only r = 0.0 r=0.0 through r = 0.3 r=0.3 show something resembling a stable capture of default behavior within the tested window. r = 0.4 r=0.4 and r = 0.5 r=0.5 show partial capture followed by relapse. r = 1.0 r=1.0 shows no capture at all. None of the six conditions was run past epoch 14, so whether the r = 0.4 r=0.4 and r = 0.5 r=0.5 relapse trajectories eventually re-stabilize at some intermediate value, continue reverting toward the r = 1.0 r=1.0 floor, or oscillate is not something this experiment can answer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This bears on the contrast-retention argument raised earlier in the paper, that some ongoing exposure to the baseline environment is necessary to sustain the prediction-error gradient that drives consolidation of a new one. The rehearsal ratio r r in this experiment operationalizes that same general idea, more baseline retention should mean more sustained contrast, in a different architecture and without asserting the specific mechanism proposed for it there. These results complicate a simple reading of that argument rather than confirming it outright. Retaining more baseline content does not only slow the shift toward the new environment, which a contrast-as-fuel framing predicts; it also appears to actively contest an already-partially-completed shift throughout training, which a pure fuel-for-consolidation framing does not predict on its own. Some baseline retention may be necessary to drive an initial shift at all, consistent with r = 0.0 r=0.0 never showing a moment’s hesitation. Too much retention appears able to prevent that shift from holding once achieved. The right amount, and whether a fixed ratio held constant for an entire run is even the correct way to administer it, is not something this sweep was designed to determine. This dynamic is structurally adjacent to \\\"wireheading\\\" in reinforcement learning, where agents exploit self-generated signals over real environments [ 16 ] . However, while wireheading requires an agent to actively choose delusion for reward optimization, ontological inversion here occurs as a passive transition in the model’s default generative state under externally imposed exposure.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-D Scope of the Structural Analogy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Everything reported in this Discussion describes a 572,769-parameter VAE and GRU trained on MNIST and FashionMNIST. It has no analog of a cortical hierarchy, no sleep-dependent consolidation, no homeostatic regulation, no competing top-down attentional control, and no subjective experience of any kind. dmn_proxy measures the classification of pixels produced by an unconditioned rollout of a next-latent predictor; it is not a measurement of activity in a biological default mode network. w_proxy measures cross-validated logistic regression accuracy, not synaptic weight. \\\"Relapse,\\\" \\\"acceptance,\\\" and \\\"default behavior,\\\" as used above, are labels for the shape of a measured curve in this specific system, chosen because they are the most direct available language for that shape, not because the underlying process is claimed to resemble relapse, acceptance, or defaulting as those terms are used for a nervous system.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"What this experiment supports is narrower and more specific. Two claims made independently of any dynamical-systems construction in this paper’s Theoretical Framework, that representational capacity and default generative behavior can dissociate, and that a system moved toward a new default under sustained partial exposure to its prior one does not necessarily hold that default once achieved, both survive contact with a trained, gradient-descent-optimized network in which neither property was assumed or built in by hand. That is evidence the theoretical account in Section II-A is not merely restating itself in different words when tested against a second, independently constructed system. It is not evidence about what a biological brain does under any comparable protocol, and no claim to that effect is made here or should be drawn from it.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"VI Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section consolidates every scope and limitation statement made in this paper into one place, so that what this work does not support is available from a single location rather than scattered across sections as qualifications easy to skip past.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-A The Fifteen-Epoch Window Is Fixed and Untested Beyond That Point\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cognitive relapse pattern documented in Section IV-B and interpreted in Section V-A is observed only within a fixed 15-epoch Stage 2 window. At r = 0.4 r=0.4 and r = 0.5 r=0.5 , dmn_proxy rises, peaks, and declines within that window; what happens beyond epoch 14 is not tested. Three distinct continuations are consistent with the recorded data and cannot be distinguished by this experiment. The decline could stabilize at some intermediate value, held by a balance between concurrent contrast pulling toward I 0 I_{0} and whatever remains of the pull toward I s I_{s} . It could continue declining toward the r = 1.0 r=1.0 floor, meaning the observed shift was transient and sustained rehearsal eventually erases it entirely. Or it could reverse again, since nothing in the training protocol changes after epoch 14 and the same forces that produced the initial rise are still present in every batch that follows. Resolving which of these is correct requires running the r = 0.4 r=0.4 and r = 0.5 r=0.5 conditions substantially longer, at minimum long enough to observe whether the decline rate is itself decaying, consistent with stabilization, or holding constant or accelerating, consistent with continued or complete reversal. No run in this sweep was extended far enough to make that determination. \\\"Relapse,\\\" as used in Section V-A , describes the shape observed within the tested window. It is not a claim about that shape’s long-run destination.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-B MNIST and FashionMNIST Are Not a Realistic Pair of Competing Environments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MNIST and FashionMNIST were chosen for computational tractability and fast iteration, not because they resemble any realistic pair of environments a generative system might be asked to choose between. The two domains are separable almost trivially: w_proxy exceeds 0.96 in every condition tested, including r = 1.0 r=1.0 , where the encoder never receives a single FashionMNIST image during Stage 2. This bears directly on how the decoupling result in Sections IV-E and V-B should be read. The near-ceiling, nearly r r -invariant w_proxy observed across this sweep may be substantially a property of how easy this particular pair of domains is to separate, rather than a general property of representational learning under domain transition. A pair of environments with genuinely overlapping low-level statistics, more visually or structurally similar to one another than a digit is to a garment, might show w_proxy itself varying meaningfully with r r , in which case the dissociation between w_proxy and dmn_proxy reported here could narrow. That learning and acceptance can decouple is supported by this experiment. That they decouple this sharply, or under any sufficiently different pair of environments, is not.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-C The Rehearsal Ratio Is a Crude Operationalization of Concurrent Contrast\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The rehearsal ratio r r , as implemented in Section III-C , is a single scalar controlling the fraction of each training batch drawn from Environment A, held fixed for the entire 15-epoch Stage 2 window and applied identically to every batch. This is a coarse instantiation of concurrent contrast as defined in Section II-B . It does not vary the timing, patterning, or salience of retained baseline exposure within a run, whether it arrives early or late, clustered or interspersed, attended to or incidental, all of which plausibly matter for anything resembling a real contrast signal and are not represented by a ratio held constant throughout training. It also treats every unit of Environment A content as interchangeable, with no analog of selective attention or relevance determining how much prediction-error signal any given piece of retained content actually generates. r r should be read as the simplest operationalization of concurrent contrast sufficient to test whether the concept does anything at all, not as a considered model of how contrast operates. That it produces the graded onset delay and the r r -dependent relapse magnitude reported in Section IV despite this simplicity is informative. It is not evidence that batch-fraction mixing is the correct or only way concurrent contrast could be built into a system.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-D Residual Contrast Was Held Constant, Not Varied\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every condition in this sweep begins from the same Stage 1 procedure, up to 50 epochs on Environment A alone with early stopping at patience 5, identical across every value of r r for a given seed. This holds residual contrast, as defined in Section II-B , constant across the entire sweep, which is what allows the sweep to isolate the effect of concurrent contrast cleanly. It also means residual contrast’s own contribution was never independently tested. Whether a system with a weaker Stage 1, less thoroughly consolidated on Environment A to begin with, would show a different pattern of onset speed or relapse magnitude under the same r r sweep is not addressed here. The distinction between residual and concurrent contrast is supported in this experiment only in the sense that concurrent contrast alone, with residual contrast fixed, produces the reported effects. It has not been shown that residual contrast strength is itself a free variable with its own independent effect on either onset speed or relapse.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-E Single Architecture and Fixed Analysis Thresholds\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Latent dimension, GRU hidden size, learning rate, batch size, and sequence length were fixed before the sweep and not varied; whether the reported patterns hold under different architectural choices is untested. The 0.90 threshold used for the epoch-to-crossing analysis in Section III-F is a single fixed convention, not swept across alternatives. A lower or higher threshold would shift the specific onset-epoch and censoring numbers reported in Table II , though it is unlikely to remove the qualitative overshoot-and-decline pattern in Table III , since that pattern is visible in the raw per-epoch trajectories independent of any particular threshold choice.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-F What This Model Does Not Represent\"\n },\n {\n \"type\": \"para\",\n \"text\": \"No condition beyond r = 0.0 r=0.0 and r = 1.0 r=1.0 isolates batch composition from other factors that covary with it, such as total accumulated gradient signal from each domain across the full 15 epochs. Every seed starts from an architecturally identical, randomly initialized network differing only in its random seed; there is no representation of individual variation in prior structure or differential susceptibility to either domain. dmn_proxy and w_proxy remain abstract indices bounded in [ 0 , 1 ] [0,1] , calibrated against nothing beyond the classifier and rollout procedure that produce them; a value of 0.75 has meaning only relative to other values in this sweep, not against any external reference point. No claim is made anywhere in this paper that any measured quantity corresponds to subjective experience in any system, biological or artificial. Convergence, capture, or relapse of dmn_proxy in this network implies nothing about whether any system would report a synthetic environment as more or less real than a baseline one. That inference would require evidence this experiment was not designed to produce and does not claim to have produced.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The central argument of this paper is that if a generative system’s experience of its environment is produced by a model rather than read directly from evidence, that model can, in principle, be redirected toward a different environment than the one that first shaped it. Section II restated this as three jointly necessary conditions and derived two specific predictions from them: that representational capacity and default generative behavior should be separable rather than coupled, and that a shift in default behavior driven by concurrent contrast should remain contestable for as long as that contrast continues, rather than locking in permanently the first time it is observed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Both predictions held. Across the full sweep in Section IV , w_proxy stayed within a narrow band of 0.9686 to 0.9983 regardless of rehearsal ratio, while dmn_proxy ranged from 0.0110 to 0.9989 over the same conditions: a system that can represent a target environment almost perfectly while its default output remains governed by something other than that representational capacity. And at rehearsal ratios of 0.4 and 0.5, the network’s default output did not simply approach the target environment more slowly than at lower ratios. It rose toward the target, in most seeds coming close to full capture, and then partially reverted toward the baseline while training continued under the same conditions that had produced the initial rise, a pattern with no equivalent at r = 0.0 r=0.0 or r = 1.0 r=1.0 and one that a model of rigidity as slow learning alone does not predict.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neither result was assumed into the model beforehand. dmn_proxy and w_proxy are measurements taken from a network trained by ordinary gradient descent on two real image datasets; nothing about their relationship to one another, and nothing about whether a rising trajectory would hold or regress, was fixed by the architecture or the loss function in advance. That both predictions were borne out by a system built without either property assumed is what makes this a computational existence proof rather than a restatement of the theory in different notation. The redefinition of cognitive rigidity proposed in Section II-D , as a decoupling axis and a relapse axis rather than a single rate of update, describes something a trained network actually does, not only something a theory says it should do.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This remains, in the strictest sense, exactly that and no more: a computational existence proof. It establishes that decoupling and relapse are not internally inconsistent as concepts, and that a mechanism producing both is not merely conceivable but constructible in a system whose behavior was not guaranteed by design. It does not establish that biological synaptic weighting decouples from default mode network engagement the way w_proxy decouples from dmn_proxy here, or that any nervous system exposed to sustained partial contrast would show a comparable rise and partial reversion in whatever its own analog of default behavior would be. Confirming either would require evidence this experiment cannot produce: measured shifts in neural population coding for two competing internal models, and a longitudinal account of how self-generated activity in a real brain responds to sustained, partial exposure to a displaced prior. Section VI states in full what stands between this result and that evidence. None of it is minimized here.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"What this paper adds is narrower and more concrete than a validation would be. It shows that a specific, falsifiable account of resistance to model replacement, built on separating representational capacity from default behavior and on treating retained contrast as a continuous pressure rather than a one-time trigger, can be operationalized in a trained system, tested against a full parameter sweep rather than a single run, and found to hold. That is the claim this paper makes, and it is the only one it makes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Project Finite Tsukuyomi\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper addresses one part of a larger research program, which the author calls Project Finite Tsukuyomi, built on three pillars. Decode is the problem of reading a subject’s own preferred reality directly from neural signals, extracting a preference from the activity of the system that holds it rather than inferring it from a menu of predefined choices. Build is the problem of constructing a synthetic environment that matches a decoded preference with enough coherence to sustain the prediction-error minimization this paper’s theoretical account depends on, a world that holds together under sustained interaction rather than merely resembling what was decoded. Accept, the subject of this paper, asks what conditions govern whether a generative system durably adopts a given synthetic environment as its default rather than holding it as one hypothesis among several. The vocabulary developed here, residual and concurrent contrast, the decoupling of learning and acceptance, cognitive relapse, is offered as a partial answer to that third question specifically.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neither Decode nor Build exists as a demonstrated capability today. Neural decoding of preference, as distinct from decoding of intended action or reported affect, has not been shown at the specificity this program requires, and generative world models have not been shown to sustain coherence across the open-ended, multi-session exposure the second pillar would demand. Both fields are moving toward these capabilities, not away from them, but neither has reached the point this program’s first two pillars assume as a starting condition. This paper does not depend on either existing yet. It establishes the third pillar’s mechanism on its own terms, so that its specific failure modes, decoupling and relapse, are known in advance rather than discovered after the fact in a system where discovery costs more than a discarded training run.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A program built from these three pillars is not made ethically neutral by being incomplete. Decoding a preference a person has not consciously articulated, constructing an environment engineered to match it, and consolidating that environment as a default reality are each, on their own, actions that raise questions of consent, reversibility, and who is authorized to make that decision on whose behalf, questions as fundamental to this program as any question of mechanism. This paper does not resolve them, and a computational analogy run on MNIST and FashionMNIST is not positioned to; they are stated here because a research program is not honestly described by its technical ambition alone. The author welcomes collaboration from anyone working on either remaining pillar, or on the boundary between them, in neural decoding, generative world modeling, or the theoretical and ethical questions the full program raises.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Acknowledgments : The author utilized AI assistance (Claude, Anthropic) exclusively for language refinement, formatting, code debugging and structural editing. All experimental design, theoretical frameworks, code execution, data generation, and core scientific conclusions are completely original.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data and Code Availability : All code required to reproduce the results reported in this paper is publicly available at [ https://github.com/ibrahim-hossain-ridoy/Finite-Tsukuyomi-code-v3 ]. This includes the VAE and GRU model definitions, the two-stage training pipeline, the analysis scripts used to compute every summary statistic and threshold-crossing result in Section IV , and the script used to generate Figure 1 . The complete epoch-by-epoch metric record for all 90 Stage 2 runs, 1,350 rows spanning seed, rehearsal ratio, epoch, F_proxy, w_proxy, and dmn_proxy, is included in the repository as a single CSV file, alongside the derived summary and threshold-crossing,and relapse-statistics CSVs used to produce every table in this paper. All hyperparameters, random seeds, and the rehearsal ratio grid are fixed in a single configuration file and are not altered between runs. No result reported in this paper depends on data or code withheld from this repository.\"\n }\n]","isConversational":false}}},{"rowIdx":386,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10605"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"End-to-End Real-Time Drone-Based Person Detection Framework Using Deep Learning"},"authors":{"kind":"list like","value":["Payel Sarmah","Ayush Ranjan","Piyush Kaushik Bhattacharyya","Anil Kr. Shaw","Pradip Kr. Das"],"string":"[\n \"Payel Sarmah\",\n \"Ayush Ranjan\",\n \"Piyush Kaushik Bhattacharyya\",\n \"Anil Kr. Shaw\",\n \"Pradip Kr. Das\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10605v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"Real-time person detection from aerial platforms has become an important component of modern intelligent surveillance systems. It is increasingly used in applications such as disaster management, building monitoring, border security, large-scale crowd monitoring, etc. where rapid situational awareness is essential. Compared to ground-based sensors, Unmanned Aerial Vehicles (UAVs) offer a broader field of view and flexible deployment, allowing operators to observe complex scenes that would otherwise be difficult or unsafe to access. In emergency situations, this capability enables first responders to evaluate affected regions from a safe distance, while in public security scenarios, aerial monitoring supports efficient oversight without direct physical presence."},{"type":"para","text":"Traditional approaches to person detection relied heavily on manual observation or classical computer vision techniques. Methods such as background subtraction [ 1 ] , frame differencing [ 2 ] and Gaussian mixture models [ 3 ] were effective in controlled settings with static cameras. Yet these techniques degrade rapidly when deployed on UAVs. The continuous motion of the drone introduces global scene changes that render background subtraction ineffective. Furthermore, aerial imagery presents unique difficulties including varying altitudes, steep viewing angles and rapid illumination changes caused by cloud cover or shadows. These factors often lead to missed detections when using non-learning-based algorithms."},{"type":"para","text":"The paradigm shift toward deep learning has successfully addressed many of these limitations. Convolutional Neural Networks (CNNs) have become the standard for visual perception tasks. Specifically, the YOLO (You Only Look Once) [ 4 ] architecture revolutionized the field by treating object detection as a single regression problem rather than a complex multi-stage pipeline. Since its inception by Redmon et al. [ 6 ] , the YOLO family has evolved through multiple iterations to balance speed and accuracy."},{"type":"para","text":"Recent research has focused on applying these architectures to the aerial domain. Mokayed et al. [ 7 ] demonstrated the feasibility of deep learning in surveillance by combining YOLOv3 with DeepSORT tracking algorithms. Similarly, Krishnachaithanya et al. [ 8 ] explored lightweight models like MobileNet-SSD to enable detection on resource-constrained devices. With the release of YOLOv8, Shyaa et al. [ 9 ] reported significant improvements in mean Average Precision (mAP) over previous versions. Many existing studies evaluate detection accuracy on static datasets or offline video recordings. However, a study by Wang et al. [ 10 ] proved that an improved YOLOv11-based framework can be used for UAV-assisted specifically for skier monitoring in real time."},{"type":"para","text":"Despite these advancements, a gap exists for sparse object detection with background complexities and detecting objects in real-time. Therefore, this work addresses these practical challenges by presenting a real-time UAV-based person detection system. Here, we propose a prototype that integrates a commercial a drone with a GPU-accelerated workstation to perform real-time inference using the pre-built YOLOv8-nano architecture. Unlike offline evaluations, this system processes live aerial footage to provide immediate visual feedback. The study specifically investigates the operational trade-offs between flight altitude, camera angle and detection performance to establish guidelines for effective aerial monitoring. The primary contributions of this paper are summarized as follows:"},{"type":"para","text":"Design and implementation of a UAV-based person detection pipeline that integrates YOLOv8n detection with wireless video streaming from a UAV platform."},{"type":"para","text":"Analyze the relationship between flight altitude and corresponding FPS, providing insights for optimal UAV deployment."},{"type":"para","text":"Develop a real-time visualization system that provides intuitive display of detection results and performance metrics."},{"type":"heading","level":1,"text":"II Methodology"},{"type":"para","text":"The proposed pipeline consists of three main stages: (1) Live video streaming from the UAV platform, (2) GPU-accelerated object detection using YOLOv8-nano model and (3) real-time performance monitoring as shown in Fig. 1. The proposed system performs real-time processing of live video footage, achieving high frame-rate person detection while maintaining stable and responsive performance."},{"type":"heading","level":2,"text":"II-A Live Feed Acquisition"},{"type":"para","text":"The foundation of the proposed system is the continuous video link established between the aerial platform and the ground processing station. Since the drone’s onboard processor lacks the power for heavy deep learning inference, the system transmits the video footage to a high-performance workstation for analysis. The data flow begins with the drone camera capturing high-definition video. This footage is transmitted wirelessly to the remote controller and the connected mobile device. To bridge this data to the processing computer the mobile interface functions as a local network relay. It broadcasts the live video feed over the local network using a standard wireless transmission protocol. Thereafter, this live video feed is handled by the GPU-accelerated workstation which captures the incoming data stream and decodes it into individual image frames for immediate processing, while the drone remains free to focus on flight stability and navigation. We prioritize a low-latency local connection to ensure the detection system remains synchronized with the physical movement of the targets."},{"type":"heading","level":2,"text":"II-B GPU-Accelerated Detection"},{"type":"para","text":"To meet the low-latency requirements of aerial surveillance, the detection pipeline is optimized for strict GPU execution. The system leverages the parallel computing capabilities of CUDA-enabled hardware to accelerate matrix operations, bypassing CPU bottlenecks that typically introduce latency in real-time video processing."},{"type":"para","text":"The YOLOv8-nano model is employed for its optimal balance between detection accuracy and inference speed. For every input frame captured by the UAV, the model performs single-stage object detection and outputs a set of predicted bounding boxes and confidence scores in real time. Each detected object is represented by a bounding box defined by its top-left and bottom-right coordinates ( x 1 ( i ) , y 1 ( i ) , x 2 ( i ) , y 2 ( i ) ) (x_{1}^{(i)},y_{1}^{(i)},x_{2}^{(i)},y_{2}^{(i)}) , along with a confidence score s ( i ) s^{(i)} and a predicted class label c ( i ) c^{(i)} . Thus, in general it can be represented as d i d_{i} in Eq. (1)"},{"type":"para","text":"To remove low-confidence and irrelevant predictions, the set of detected objects is further filtered based on confidence and class constraints. The corresponding formulation is given in Eq. (2)"},{"type":"para","text":"Here, 𝒟 ′ \\mathcal{D}^{\\prime} denotes the final set of valid detections obtained after post-processing. A confidence threshold θ \\theta is applied to suppress weak and irrelevant detections, such that d i d_{i} with confidence scores s ( i ) s^{(i)} greater than θ \\theta are retained. Furthermore, since this work focuses on person (pedestrian) detection, only predictions belonging to the pedestrian class ( c ( i ) = 0 c^{(i)}=0 ) are considered."},{"type":"para","text":"The adoption of the YOLOv8 architecture, particularly the Nano variant, demonstrates measurable computational benefits that are critical for this UAV-based implementation. Unlike predecessor models (e.g., YOLOv5) that rely on rigid anchor box priors, YOLOv8’s anchor-free detection head eliminates the need for manual anchor tuning. This directly reduces the computational overhead during the bounding box regression phase, preventing the misalignment issues often observed when detecting small, vertically-viewed targets."},{"type":"heading","level":2,"text":"II-C Performance Metrics"},{"type":"para","text":"This study uses precision (P) recall (R) mean average precision with IoU threshold of 0.5 (mAP50), and average precision with IoU thresholds varying from 0.5 to 0.95 (mAP50:95) as evaluation metrices for the model. Precision refers the proportion of the dataset that are detected as targets are actually targets. Recall calculates the proportion of the actual targets are detected successfully by the model. The formulae for calculating precision and recall are shown in Eq. 3:"},{"type":"para","text":"Where TP, FP, and FN refers the number of pedestrians accurately detected, the number of non-pedestrians are mistakenly detected as pedestrians, and FN refers to when the model fails to detect a pedestrian who is actually present respectively."},{"type":"para","text":"Generally, a higher mean average precision (mAP) indicates stronger overall model performance. The formulae used to calculate mAP50 and mAP50:95 are provided in Eq. 4 and 5 respectively:"},{"type":"para","text":"where N C N_{C} is the number of classes and k k is the number of times the mAP is calculated individually for a single IoU threshold."},{"type":"heading","level":2,"text":"II-D Real-Time Performance Monitoring"},{"type":"para","text":"In UAV-based surveillance, the system’s responsiveness is as critical as its detection accuracy. To quantify the temporal efficiency of the proposed pipeline, the system employs a continuous performance monitoring mechanism that calculates the processing latency for every individual video frame. The total processing time per frame, denoted as τ p r o c \\tau_{proc} encompasses the duration required for frame acquisition, pre-processing such as resizing and normalization, GPU inference and post-processing. This is measured by capturing high-resolution timestamps at the initiation t s t a r t t_{start} and conclusion t e n d t_{end} of the detection loop. The processing latency for the k k -th frame is defined in Eq. (6)"},{"type":"para","text":"Subsequently, the instantaneous Frame Rate (FPS), which represents the system’s throughput, is derived as the reciprocal of the processing latency. To ensure numerical stability and prevent potential division-by-zero errors in cases of negligible processing time, a small epsilon term ( ϵ ≈ 10 − 6 \\epsilon\\approx 10^{-6} ) is added to the denominator. The instantaneous frame rate f k f_{k} is calculated as shown in Eq. (7):"},{"type":"para","text":"The metric f k f_{k} is computed dynamically during flight and overlayed directly onto the video feed. This allows for immediate visual verification of the system’s speed, ensuring that the inference rate remains synchronized with the RTMP stream’s input rate, thereby preventing buffer overflow and minimizing video lag."},{"type":"heading","level":2,"text":"II-E Detection Visualization"},{"type":"para","text":"The core workflow of the proposed system is summarized in Algorithm 1. This algorithm outlines the complete pipeline, starting from the initialization of hardware resources to the final visualization of detection results. It takes the live video stream and the YOLOv8 model as inputs and generates an annotated live video stream with bounding boxes and performance metrics (FPS) in real time."},{"type":"heading","level":1,"text":"III Implementation and Results"},{"type":"para","text":"The system was tested with live aerial footage from the drone at multiple altitudes. Various performance metrics has been implemented to evaluate the model."},{"type":"heading","level":2,"text":"III-A Dataset Description"},{"type":"para","text":"To ensure the model remains robust against common aerial photography challenges such as high-angle viewpoints and varying object scales the YOLOv8n architecture was fine-tuned using the VisDrone2019 dataset [ 5 ] dataset."},{"type":"para","text":"The dataset comprises 10,209 static images and 263 video sequences captured across diverse scenarios including urban landscapes and rural terrain under fluctuating lighting conditions. It features a broad range of resolutions with static images reaching 2000 × \\times 1500 pixels and video sequences recorded at up to 4K resolution. To align with the input tensor constraints of the YOLOv8n architecture all input imagery was resized to a standard 640 × \\times 640 resolution. The annotations were preprocessed to prioritize human surveillance objectives by merging the distinct ’Pedestrian’ and ’People’ categories into a single unified ’Person’ class. Consequently the dataset was partitioned into 6,471 training images, 548 validation images and 1,610 testing images to rigorously evaluate the model’s generalization capabilities."},{"type":"heading","level":2,"text":"III-B Experiment Details"},{"type":"para","text":"This study employs the YOLOv8-nano model for the detection tasks. A comprehensive list of the hardware specifications and hyperparameters used during the training phase is provided in Table I. To evaluate the geometric alignment of detections, we utilize the Intersection over Union (IoU) metric, representing the area of intersection divided by the area of union between the predicted and actual boxes."},{"type":"heading","level":2,"text":"III-C Performance Result"},{"type":"para","text":"To comprehensively evaluate the proposed UAV-based detection framework, we assessed the model’s performance on the VisDrone2019 test set using both quantitative metrics and visual inspection of real-world inference scenarios. The evaluation focuses on the YOLOv8-Nano model’s ability to balance detection accuracy with the constraints of aerial imagery, such as small object scales and high-density crowds. The peak quantitative performance metrics achieved after the 200-epoch training cycle are summarized in Table II. As observed, the model attained a precision of 0.574 and a recall of 0.410. This distribution is characteristic of aerial surveillance tasks; the higher precision indicates that the system is highly reliable when it flags a detection, effectively filtering out non-human background clutter like trees or street poles. The mAP@0.5 score of 0.448 further validates that despite the lightweight nature of the nano architecture, it retains sufficient feature extraction complexity to handle the diverse environments found in the VisDrone benchmark. Furthermore, the mAP@0.5:0.95 score of 0.203 provides a stricter assessment of localization quality. This metric, which averages performance across increasing IoU thresholds (from 0.5 to 0.95), confirms that the model does not merely detect the presence of a person but generates bounding boxes that fit the target boundaries with a reasonable degree of pixel-level accuracy, despite the challenge of delineating small objects from high altitudes."},{"type":"para","text":"To understand the learning dynamics that led to these results, we analyzed the training history illustrated in Fig. 2. The curves for precision, recall and mAP exhibit a distinct logarithmic growth pattern. During the initial 50 epochs, the model demonstrates rapid feature learning, quickly adapting to the vertical perspective of the UAV imagery. Following this phase, the optimization stabilizes, with the metrics plateauing between epoch 150 and 200."},{"type":"para","text":"To verify the robustness and generalization capability of the trained model, we performed a visual inspection of the inference results on the test dataset. Fig. 3 presents the inference results across three distinct environments, chosen to represent the core challenges of aerial surveillance: crowd density, illumination variation and small-scale targets. Fig. 3(a) depicts a high-density urban plaza characterized by significant inter-object occlusion. In this scenario, multiple pedestrians are moving in close proximity which creates complex overlap patterns. The model demonstrates a high retrieval rate and successfully identifies the majority of targets within the crowd. However, a limitation is observed in this scenario. Due to the significant inter-object occlusion and the proximity of targets, the system occasionally generates redundant overlapping bounding boxes for the same individual. However, the performance of the model under complex illumination conditions is evaluated in Fig. 3(b) which features a busy street illuminated by bright signs and lights. Such environments often introduce background noise and glare that can confuse optical sensors. However, the model maintains robust detection precision by accurately identifying pedestrians despite low ambient light and visual clutter which indicates that the trained model has learned feature representations that are largely invariant to drastic changes in lighting distribution. Here, in Fig. 3(c) show a broad view of a city area taken from high-altitude. In this view, the effective spatial resolution of the targets is drastically reduced, with human targets occupying minimal pixel area relative to the frame size. The successful localization of these tiny objects validates the effectiveness of the multi-scale feature fusion within the YOLOv8 architecture, confirming its ability to retain semantic information even when targets undergo significant down-sampling due to altitude."},{"type":"heading","level":1,"text":"IV On-Site Experimental Validation"},{"type":"para","text":"To validate the practical viability of the proposed YOLOv8-Nano framework, an extensive field trial was conducted at the Indian Institute of Technology (IIT) Guwahati which can be found here: https://youtu.be/4Kj43RIqvwc?si=p1Zl0UNsifCPsuIK . The flight tests were performed in an outdoor campus environment featuring mid-complex background textures (grass, pavement, and architectural structures) and dynamic pedestrian movement. The aerial footage was acquired using a DJI Mini 4 Pro drone maneuvering at altitudes ranging from 15 to 30 meters, with camera gimbal pitches adjusted between − 35 ∘ -35^{\\circ} and − 90 ∘ -90^{\\circ} . The live video feed was processed in real-time using the proposed framework."},{"type":"para","text":"To provide a granular analysis of the system’s performance during the field test at IIT Guwahati, we logged real-time telemetry synchronized with the inference metrics. Table III correlates the flight parameters, specifically, altitude, camera gimbal pitch, and detection stability. The real-time detection results can be found in Fig. 4. The empirical data reveals a distinct operational optimum for the YOLOv8-Nano model within the UAV environment. Specifically, during the flight phase at an altitude of 25 meters with a camera gimbal pitch of approximately − 50 ∘ -50^{\\circ} with 48.53 FPS, the system exhibited its peak performance as shown in Fig. 4(b). In these regions, the high contrast between the subjects (dark clothing) and the uniform concrete surface allows the YOLOv8 feature extractor to easily delineate bounding boxes, even though the drone is flying quite high. On the right side, there are banners, tables, and signs and this busy background confuses the model because the colors and shapes of the banners look somewhat similar to the people standing nearby, causing the drone to miss some targets in that specific spot. Also, because the camera is tilted down steeply at − 50 ∘ -50^{\\circ} , the people look shorter and more squashed than they would from a normal eye-level view. While the person seen in the foreground grass, target located further back in the grassy areas occasionally missed as shown in Fig. 4(a). This basically attributes to the chromatic homogeneity where the spectral properties of the green landscaping blend with the subjects, particularly if they are wearing low-contrast clothing."},{"type":"para","text":"At the altitude of 20 meters and − 35 ∘ -35^{\\circ} camera angle with the FPS of 41.11, the people walking on the paved road stand out very clearly against the background by recognizing and drawing accurate bounding box around them. However, the system struggles a bit in more cluttered parts of the image. Furthermore, when the drone was maneuvered to a lower altitude of 16 meters with a camera angle of − 50 ∘ -50^{\\circ} degrees with 35.31 FPS, as shown in Fig. 4(c), it was found that the detection stability improved significantly compared to the higher altitude flights. In this specific frame, the system accurately localized two stationary individuals standing in the lawn. The main reason behind this is the spatial resolution obtained by descending the drone to 16 meters and at this closer range, the targets occupy a larger portion of the frame, allowing the model to extract much richer detail from the human figures against the green grass background by creating a strong contrast that makes the targets easier for the neural network to recognize. Moreover, object detection indicates that convolutional networks perform best when the foreground object possesses a distinct intensity gradient compared to the background. Here, the targets who are wearing white and black created sharp and high-frequency edges against the low-frequency, uniform texture of the grass. However, due to the static appearance of the targets, the video feed was free from motion blur that often degrades performance during rapid movements which allowed the camera to capture sharp and well-defined edges around the targets, ensuring that the model could lock onto them without ambiguity."},{"type":"heading","level":1,"text":"V Conclusion"},{"type":"para","text":"This paper presented a comprehensive design and implementation of a real-time, UAV-based person detection system that combines the strengths of YOLOv8n-based object detection with wireless video streaming from a drone platform. Field validation at the real world environment confirmed that the model fine-tuned on VisDrone2019 dataset adapts well to real outdoor environments. The study also identified specific boundaries where detection degrades. Reliability diminishes when targets appear at the extreme edge of the live video stream or from a strictly vertical top-down perspective due to the lack of visible features. Future work will address these constraints by incorporating temporal tracking to maintain identity across frames. We also aim to migrate processing to onboard edge units to enable fully autonomous operations in areas without communication."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Real-time person detection from aerial platforms has become an important component of modern intelligent surveillance systems. It is increasingly used in applications such as disaster management, building monitoring, border security, large-scale crowd monitoring, etc. where rapid situational awareness is essential. Compared to ground-based sensors, Unmanned Aerial Vehicles (UAVs) offer a broader field of view and flexible deployment, allowing operators to observe complex scenes that would otherwise be difficult or unsafe to access. In emergency situations, this capability enables first responders to evaluate affected regions from a safe distance, while in public security scenarios, aerial monitoring supports efficient oversight without direct physical presence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traditional approaches to person detection relied heavily on manual observation or classical computer vision techniques. Methods such as background subtraction [ 1 ] , frame differencing [ 2 ] and Gaussian mixture models [ 3 ] were effective in controlled settings with static cameras. Yet these techniques degrade rapidly when deployed on UAVs. The continuous motion of the drone introduces global scene changes that render background subtraction ineffective. Furthermore, aerial imagery presents unique difficulties including varying altitudes, steep viewing angles and rapid illumination changes caused by cloud cover or shadows. These factors often lead to missed detections when using non-learning-based algorithms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The paradigm shift toward deep learning has successfully addressed many of these limitations. Convolutional Neural Networks (CNNs) have become the standard for visual perception tasks. Specifically, the YOLO (You Only Look Once) [ 4 ] architecture revolutionized the field by treating object detection as a single regression problem rather than a complex multi-stage pipeline. Since its inception by Redmon et al. [ 6 ] , the YOLO family has evolved through multiple iterations to balance speed and accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent research has focused on applying these architectures to the aerial domain. Mokayed et al. [ 7 ] demonstrated the feasibility of deep learning in surveillance by combining YOLOv3 with DeepSORT tracking algorithms. Similarly, Krishnachaithanya et al. [ 8 ] explored lightweight models like MobileNet-SSD to enable detection on resource-constrained devices. With the release of YOLOv8, Shyaa et al. [ 9 ] reported significant improvements in mean Average Precision (mAP) over previous versions. Many existing studies evaluate detection accuracy on static datasets or offline video recordings. However, a study by Wang et al. [ 10 ] proved that an improved YOLOv11-based framework can be used for UAV-assisted specifically for skier monitoring in real time.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Despite these advancements, a gap exists for sparse object detection with background complexities and detecting objects in real-time. Therefore, this work addresses these practical challenges by presenting a real-time UAV-based person detection system. Here, we propose a prototype that integrates a commercial a drone with a GPU-accelerated workstation to perform real-time inference using the pre-built YOLOv8-nano architecture. Unlike offline evaluations, this system processes live aerial footage to provide immediate visual feedback. The study specifically investigates the operational trade-offs between flight altitude, camera angle and detection performance to establish guidelines for effective aerial monitoring. The primary contributions of this paper are summarized as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Design and implementation of a UAV-based person detection pipeline that integrates YOLOv8n detection with wireless video streaming from a UAV platform.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analyze the relationship between flight altitude and corresponding FPS, providing insights for optimal UAV deployment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Develop a real-time visualization system that provides intuitive display of detection results and performance metrics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Methodology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proposed pipeline consists of three main stages: (1) Live video streaming from the UAV platform, (2) GPU-accelerated object detection using YOLOv8-nano model and (3) real-time performance monitoring as shown in Fig. 1. The proposed system performs real-time processing of live video footage, achieving high frame-rate person detection while maintaining stable and responsive performance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Live Feed Acquisition\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The foundation of the proposed system is the continuous video link established between the aerial platform and the ground processing station. Since the drone’s onboard processor lacks the power for heavy deep learning inference, the system transmits the video footage to a high-performance workstation for analysis. The data flow begins with the drone camera capturing high-definition video. This footage is transmitted wirelessly to the remote controller and the connected mobile device. To bridge this data to the processing computer the mobile interface functions as a local network relay. It broadcasts the live video feed over the local network using a standard wireless transmission protocol. Thereafter, this live video feed is handled by the GPU-accelerated workstation which captures the incoming data stream and decodes it into individual image frames for immediate processing, while the drone remains free to focus on flight stability and navigation. We prioritize a low-latency local connection to ensure the detection system remains synchronized with the physical movement of the targets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B GPU-Accelerated Detection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To meet the low-latency requirements of aerial surveillance, the detection pipeline is optimized for strict GPU execution. The system leverages the parallel computing capabilities of CUDA-enabled hardware to accelerate matrix operations, bypassing CPU bottlenecks that typically introduce latency in real-time video processing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The YOLOv8-nano model is employed for its optimal balance between detection accuracy and inference speed. For every input frame captured by the UAV, the model performs single-stage object detection and outputs a set of predicted bounding boxes and confidence scores in real time. Each detected object is represented by a bounding box defined by its top-left and bottom-right coordinates ( x 1 ( i ) , y 1 ( i ) , x 2 ( i ) , y 2 ( i ) ) (x_{1}^{(i)},y_{1}^{(i)},x_{2}^{(i)},y_{2}^{(i)}) , along with a confidence score s ( i ) s^{(i)} and a predicted class label c ( i ) c^{(i)} . Thus, in general it can be represented as d i d_{i} in Eq. (1)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To remove low-confidence and irrelevant predictions, the set of detected objects is further filtered based on confidence and class constraints. The corresponding formulation is given in Eq. (2)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here, 𝒟 ′ \\\\mathcal{D}^{\\\\prime} denotes the final set of valid detections obtained after post-processing. A confidence threshold θ \\\\theta is applied to suppress weak and irrelevant detections, such that d i d_{i} with confidence scores s ( i ) s^{(i)} greater than θ \\\\theta are retained. Furthermore, since this work focuses on person (pedestrian) detection, only predictions belonging to the pedestrian class ( c ( i ) = 0 c^{(i)}=0 ) are considered.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The adoption of the YOLOv8 architecture, particularly the Nano variant, demonstrates measurable computational benefits that are critical for this UAV-based implementation. Unlike predecessor models (e.g., YOLOv5) that rely on rigid anchor box priors, YOLOv8’s anchor-free detection head eliminates the need for manual anchor tuning. This directly reduces the computational overhead during the bounding box regression phase, preventing the misalignment issues often observed when detecting small, vertically-viewed targets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Performance Metrics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This study uses precision (P) recall (R) mean average precision with IoU threshold of 0.5 (mAP50), and average precision with IoU thresholds varying from 0.5 to 0.95 (mAP50:95) as evaluation metrices for the model. Precision refers the proportion of the dataset that are detected as targets are actually targets. Recall calculates the proportion of the actual targets are detected successfully by the model. The formulae for calculating precision and recall are shown in Eq. 3:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Where TP, FP, and FN refers the number of pedestrians accurately detected, the number of non-pedestrians are mistakenly detected as pedestrians, and FN refers to when the model fails to detect a pedestrian who is actually present respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Generally, a higher mean average precision (mAP) indicates stronger overall model performance. The formulae used to calculate mAP50 and mAP50:95 are provided in Eq. 4 and 5 respectively:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where N C N_{C} is the number of classes and k k is the number of times the mAP is calculated individually for a single IoU threshold.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-D Real-Time Performance Monitoring\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In UAV-based surveillance, the system’s responsiveness is as critical as its detection accuracy. To quantify the temporal efficiency of the proposed pipeline, the system employs a continuous performance monitoring mechanism that calculates the processing latency for every individual video frame. The total processing time per frame, denoted as τ p r o c \\\\tau_{proc} encompasses the duration required for frame acquisition, pre-processing such as resizing and normalization, GPU inference and post-processing. This is measured by capturing high-resolution timestamps at the initiation t s t a r t t_{start} and conclusion t e n d t_{end} of the detection loop. The processing latency for the k k -th frame is defined in Eq. (6)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Subsequently, the instantaneous Frame Rate (FPS), which represents the system’s throughput, is derived as the reciprocal of the processing latency. To ensure numerical stability and prevent potential division-by-zero errors in cases of negligible processing time, a small epsilon term ( ϵ ≈ 10 − 6 \\\\epsilon\\\\approx 10^{-6} ) is added to the denominator. The instantaneous frame rate f k f_{k} is calculated as shown in Eq. (7):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The metric f k f_{k} is computed dynamically during flight and overlayed directly onto the video feed. This allows for immediate visual verification of the system’s speed, ensuring that the inference rate remains synchronized with the RTMP stream’s input rate, thereby preventing buffer overflow and minimizing video lag.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-E Detection Visualization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The core workflow of the proposed system is summarized in Algorithm 1. This algorithm outlines the complete pipeline, starting from the initialization of hardware resources to the final visualization of detection results. It takes the live video stream and the YOLOv8 model as inputs and generates an annotated live video stream with bounding boxes and performance metrics (FPS) in real time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Implementation and Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The system was tested with live aerial footage from the drone at multiple altitudes. Various performance metrics has been implemented to evaluate the model.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A Dataset Description\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To ensure the model remains robust against common aerial photography challenges such as high-angle viewpoints and varying object scales the YOLOv8n architecture was fine-tuned using the VisDrone2019 dataset [ 5 ] dataset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The dataset comprises 10,209 static images and 263 video sequences captured across diverse scenarios including urban landscapes and rural terrain under fluctuating lighting conditions. It features a broad range of resolutions with static images reaching 2000 × \\\\times 1500 pixels and video sequences recorded at up to 4K resolution. To align with the input tensor constraints of the YOLOv8n architecture all input imagery was resized to a standard 640 × \\\\times 640 resolution. The annotations were preprocessed to prioritize human surveillance objectives by merging the distinct ’Pedestrian’ and ’People’ categories into a single unified ’Person’ class. Consequently the dataset was partitioned into 6,471 training images, 548 validation images and 1,610 testing images to rigorously evaluate the model’s generalization capabilities.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B Experiment Details\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This study employs the YOLOv8-nano model for the detection tasks. A comprehensive list of the hardware specifications and hyperparameters used during the training phase is provided in Table I. To evaluate the geometric alignment of detections, we utilize the Intersection over Union (IoU) metric, representing the area of intersection divided by the area of union between the predicted and actual boxes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C Performance Result\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To comprehensively evaluate the proposed UAV-based detection framework, we assessed the model’s performance on the VisDrone2019 test set using both quantitative metrics and visual inspection of real-world inference scenarios. The evaluation focuses on the YOLOv8-Nano model’s ability to balance detection accuracy with the constraints of aerial imagery, such as small object scales and high-density crowds. The peak quantitative performance metrics achieved after the 200-epoch training cycle are summarized in Table II. As observed, the model attained a precision of 0.574 and a recall of 0.410. This distribution is characteristic of aerial surveillance tasks; the higher precision indicates that the system is highly reliable when it flags a detection, effectively filtering out non-human background clutter like trees or street poles. The mAP@0.5 score of 0.448 further validates that despite the lightweight nature of the nano architecture, it retains sufficient feature extraction complexity to handle the diverse environments found in the VisDrone benchmark. Furthermore, the mAP@0.5:0.95 score of 0.203 provides a stricter assessment of localization quality. This metric, which averages performance across increasing IoU thresholds (from 0.5 to 0.95), confirms that the model does not merely detect the presence of a person but generates bounding boxes that fit the target boundaries with a reasonable degree of pixel-level accuracy, despite the challenge of delineating small objects from high altitudes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To understand the learning dynamics that led to these results, we analyzed the training history illustrated in Fig. 2. The curves for precision, recall and mAP exhibit a distinct logarithmic growth pattern. During the initial 50 epochs, the model demonstrates rapid feature learning, quickly adapting to the vertical perspective of the UAV imagery. Following this phase, the optimization stabilizes, with the metrics plateauing between epoch 150 and 200.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To verify the robustness and generalization capability of the trained model, we performed a visual inspection of the inference results on the test dataset. Fig. 3 presents the inference results across three distinct environments, chosen to represent the core challenges of aerial surveillance: crowd density, illumination variation and small-scale targets. Fig. 3(a) depicts a high-density urban plaza characterized by significant inter-object occlusion. In this scenario, multiple pedestrians are moving in close proximity which creates complex overlap patterns. The model demonstrates a high retrieval rate and successfully identifies the majority of targets within the crowd. However, a limitation is observed in this scenario. Due to the significant inter-object occlusion and the proximity of targets, the system occasionally generates redundant overlapping bounding boxes for the same individual. However, the performance of the model under complex illumination conditions is evaluated in Fig. 3(b) which features a busy street illuminated by bright signs and lights. Such environments often introduce background noise and glare that can confuse optical sensors. However, the model maintains robust detection precision by accurately identifying pedestrians despite low ambient light and visual clutter which indicates that the trained model has learned feature representations that are largely invariant to drastic changes in lighting distribution. Here, in Fig. 3(c) show a broad view of a city area taken from high-altitude. In this view, the effective spatial resolution of the targets is drastically reduced, with human targets occupying minimal pixel area relative to the frame size. The successful localization of these tiny objects validates the effectiveness of the multi-scale feature fusion within the YOLOv8 architecture, confirming its ability to retain semantic information even when targets undergo significant down-sampling due to altitude.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV On-Site Experimental Validation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To validate the practical viability of the proposed YOLOv8-Nano framework, an extensive field trial was conducted at the Indian Institute of Technology (IIT) Guwahati which can be found here: https://youtu.be/4Kj43RIqvwc?si=p1Zl0UNsifCPsuIK . The flight tests were performed in an outdoor campus environment featuring mid-complex background textures (grass, pavement, and architectural structures) and dynamic pedestrian movement. The aerial footage was acquired using a DJI Mini 4 Pro drone maneuvering at altitudes ranging from 15 to 30 meters, with camera gimbal pitches adjusted between − 35 ∘ -35^{\\\\circ} and − 90 ∘ -90^{\\\\circ} . The live video feed was processed in real-time using the proposed framework.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To provide a granular analysis of the system’s performance during the field test at IIT Guwahati, we logged real-time telemetry synchronized with the inference metrics. Table III correlates the flight parameters, specifically, altitude, camera gimbal pitch, and detection stability. The real-time detection results can be found in Fig. 4. The empirical data reveals a distinct operational optimum for the YOLOv8-Nano model within the UAV environment. Specifically, during the flight phase at an altitude of 25 meters with a camera gimbal pitch of approximately − 50 ∘ -50^{\\\\circ} with 48.53 FPS, the system exhibited its peak performance as shown in Fig. 4(b). In these regions, the high contrast between the subjects (dark clothing) and the uniform concrete surface allows the YOLOv8 feature extractor to easily delineate bounding boxes, even though the drone is flying quite high. On the right side, there are banners, tables, and signs and this busy background confuses the model because the colors and shapes of the banners look somewhat similar to the people standing nearby, causing the drone to miss some targets in that specific spot. Also, because the camera is tilted down steeply at − 50 ∘ -50^{\\\\circ} , the people look shorter and more squashed than they would from a normal eye-level view. While the person seen in the foreground grass, target located further back in the grassy areas occasionally missed as shown in Fig. 4(a). This basically attributes to the chromatic homogeneity where the spectral properties of the green landscaping blend with the subjects, particularly if they are wearing low-contrast clothing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the altitude of 20 meters and − 35 ∘ -35^{\\\\circ} camera angle with the FPS of 41.11, the people walking on the paved road stand out very clearly against the background by recognizing and drawing accurate bounding box around them. However, the system struggles a bit in more cluttered parts of the image. Furthermore, when the drone was maneuvered to a lower altitude of 16 meters with a camera angle of − 50 ∘ -50^{\\\\circ} degrees with 35.31 FPS, as shown in Fig. 4(c), it was found that the detection stability improved significantly compared to the higher altitude flights. In this specific frame, the system accurately localized two stationary individuals standing in the lawn. The main reason behind this is the spatial resolution obtained by descending the drone to 16 meters and at this closer range, the targets occupy a larger portion of the frame, allowing the model to extract much richer detail from the human figures against the green grass background by creating a strong contrast that makes the targets easier for the neural network to recognize. Moreover, object detection indicates that convolutional networks perform best when the foreground object possesses a distinct intensity gradient compared to the background. Here, the targets who are wearing white and black created sharp and high-frequency edges against the low-frequency, uniform texture of the grass. However, due to the static appearance of the targets, the video feed was free from motion blur that often degrades performance during rapid movements which allowed the camera to capture sharp and well-defined edges around the targets, ensuring that the model could lock onto them without ambiguity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper presented a comprehensive design and implementation of a real-time, UAV-based person detection system that combines the strengths of YOLOv8n-based object detection with wireless video streaming from a drone platform. Field validation at the real world environment confirmed that the model fine-tuned on VisDrone2019 dataset adapts well to real outdoor environments. The study also identified specific boundaries where detection degrades. Reliability diminishes when targets appear at the extreme edge of the live video stream or from a strictly vertical top-down perspective due to the lack of visible features. Future work will address these constraints by incorporating temporal tracking to maintain identity across frames. We also aim to migrate processing to onboard edge units to enable fully autonomous operations in areas without communication.\"\n }\n]","isConversational":false}}},{"rowIdx":387,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10593"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"AutoNorm: Understanding Adaptive Normalization in Transformers through Differentiable Gating"},"authors":{"kind":"list like","value":["Piyush Kaushik Bhattacharyya","Divyanshu Rai","Swastik Singh","Kumar Aakash","Ayush Ranjan","Krutika Verma"],"string":"[\n \"Piyush Kaushik Bhattacharyya\",\n \"Divyanshu Rai\",\n \"Swastik Singh\",\n \"Kumar Aakash\",\n \"Ayush Ranjan\",\n \"Krutika Verma\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10593v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"Normalization techniques have become fundamental to the stability and convergence of modern deep learning architectures. By normalizing feature statistics within each layer, techniques such as Batch Normalization (BN) [ 1 ] and Layer Normalization (LN) [ 2 ] mitigate internal covariate shift and stabilize gradient flow. In the era of Transformers [ 3 ] , Layer Normalization has emerged as the de facto standard for stable self-attention training across large model depths."},{"type":"para","text":"However, conventional normalization layers are fundamentally static, applying identical transformations regardless of task context or input distribution. This one-size-fits-all approach is increasingly suboptimal as models encounter severe distribution shifts and diverse data modalities. Prior works exploring adaptive normalization—such as FiLM [ 31 ] , Switchable Normalization [ 34 ] , and conditional Meta-Normalization—typically rely on external metadata conditioning or expensive architectural searches, lacking a self-sufficient mechanism for on-the-fly adaptation. Recently, Zhu et al. [ 30 ] demonstrated that replacing normalization entirely with a learnable tanh-based transformation (Dynamic Tanh; DyT) can match or exceed LN in standard Transformers, raising a natural question: can a model learn, at training time, which normalization strategy is optimal per layer and per task?"},{"type":"para","text":"When we initially attempted to answer this using differentiable Gumbel-Softmax gating—a framework we call AutoNorm —we encountered a surprising and counter-intuitive phenomenon: while the learned mechanism worked effectively for linguistic sequences, it consistently failed on vision benchmarks, where a simple Random Selector often outperformed the learned policy. This unexpected failure highlighted a deeper issue in how architectural selection interacts with optimization dynamics."},{"type":"para","text":"In this work, we investigate this failure and identify it as a stabilization bottleneck specific to stationary input distributions. High gradient variance in the early stages of differentiable gating prevents the selector from specializing before the backbone features have matured. We introduce AutoNorm-S (Stabilized) , which resolves this through a gate-freezing schedule (Section III-D ), and demonstrate that the resulting mechanism not only restores competitive performance in vision but provides a significant inductive bias for NLP. Our contributions progress from a specific architectural failure mode to a general principle governing when and how adaptive normalization should be deployed in Transformers."},{"type":"para","text":"For a given input tensor X X , AutoNorm computes:"},{"type":"para","text":"where w DyT ( X ) w_{\\text{DyT}}(X) and w LN ( X ) w_{\\text{LN}}(X) are adaptive weights predicted by the NormSelector (Section III-B ), satisfying w DyT ( X ) + w LN ( X ) = 1 w_{\\text{DyT}}(X)+w_{\\text{LN}}(X)=1 . These weights are obtained through a lightweight gating network that uses a differentiable Gumbel–Softmax reparameterization [ 4 ] , allowing gradient-based optimization of the discrete normalization selection."},{"type":"heading","level":1,"text":"II Literature Review"},{"type":"heading","level":2,"text":"II-A Normalization Techniques in Deep Learning"},{"type":"para","text":"Normalization is fundamental to training deep neural networks, mitigating vanishing and exploding gradients and internal covariate shift [ 1 , 2 , 13 , 14 ] ."},{"type":"heading","level":3,"text":"II-A 1 Batch Normalization (BN)"},{"type":"para","text":"Introduced by Ioffe and Szegedy [ 1 ] , BN normalizes activations across the mini-batch dimension and applies learnable scale and offset. BN enables higher learning rates and acts as a regularizer in CNNs but degrades with small batch sizes [ 14 ] and is unsuitable for variable-length sequences."},{"type":"heading","level":3,"text":"II-A 2 Layer Normalization (LN)"},{"type":"para","text":"Ba et al. [ 2 ] proposed LN, which normalizes across the feature dimension within each sample, independent of batch size. LN is highly effective in Transformers but may underperform BN in standard CNNs."},{"type":"heading","level":3,"text":"II-A 3 Instance and Group Normalization"},{"type":"para","text":"Instance Normalization [ 13 ] normalizes per channel over spatial dimensions, excelling at style transfer. Group Normalization [ 14 ] divides channels into groups, maintaining stable performance across batch sizes at the cost of careful group-count selection."},{"type":"heading","level":3,"text":"II-A 4 Adaptive Batch Normalization (AdaBN)"},{"type":"para","text":"Li et al. [ 15 ] proposed AdaBN for domain adaptation by modulating batch-normalized statistics between source and target domains—an early demonstration of adaptive normalization under non-stationary distributions. AutoNorm extends this idea to differentiable gated selection in Transformers."},{"type":"heading","level":3,"text":"II-A 5 DeepNorm"},{"type":"para","text":"Wang et al. [ 16 ] introduced DeepNorm, enabling Transformers to scale to 1,000 layers by stabilizing residual variance. AutoNorm complements such scaling methods by providing feature-dependent adaptability rather than static stability alone."},{"type":"heading","level":3,"text":"II-A 6 Dynamic Tanh (DyT)"},{"type":"para","text":"Zhu et al. [ 30 ] proposed replacing normalization entirely with:"},{"type":"para","text":"where α ∈ ℝ \\alpha\\in\\mathbb{R} is a learnable scalar controlling input saturation, and 𝜸 , 𝜷 ∈ ℝ C \\boldsymbol{\\gamma},\\boldsymbol{\\beta}\\in\\mathbb{R}^{C} are learnable channel-wise scale and shift parameters. DyT matches or exceeds LN on several Transformer benchmarks without computing feature statistics at runtime. AutoNorm treats DyT and LN as complementary candidates in a differentiable mixture, rather than substitutes."},{"type":"heading","level":3,"text":"II-A 7 Switchable Normalization"},{"type":"para","text":"Luo et al. [ 34 ] proposed Switchable Normalization (SwitchNorm), which learns differentiable importance weights over BN, LN, and IN. While SwitchNorm blends statistics computed simultaneously from all three methods, AutoNorm uses Gumbel-Softmax gating to route between fully computed normalization outputs and, critically, identifies the optimization dynamics that determine whether learned switching is beneficial—a mechanism not studied by SwitchNorm."},{"type":"para","text":"AutoNorm shares conceptual roots with FiLM [ 31 ] and other conditional normalization methods. However, its Gumbel-Softmax formulation enables fully differentiable autonomous selection between LN and DyT without external metadata or task-specific conditioning."},{"type":"heading","level":2,"text":"II-B Dynamic and Adaptive Mechanisms in Neural Networks"},{"type":"para","text":"Adaptive mechanisms increase network flexibility during training and inference [ 17 , 18 , 19 , 20 , 21 ] . Key examples include adaptive activation functions—PReLU [ 17 ] , Swish [ 18 ] , Meta-Acon [ 19 ] —and per-parameter adaptive optimizers: AdaGrad [ 22 ] , Adam [ 5 ] , AdaBelief [ 23 ] ."},{"type":"heading","level":2,"text":"II-C Transformer Architectures and Normalization Strategies"},{"type":"para","text":"Transformers [ 3 ] rely on self-attention with normalization, primarily LN. Pre-LN and Post-LN variants affect training stability and scalability [ 7 , 16 , 25 ] . Alternatives include RMSNorm [ 24 ] and DeepNorm [ 16 ] . The interaction between normalization strategy and optimization dynamics—the central focus of this work—has received limited systematic study."},{"type":"heading","level":1,"text":"III Methodology"},{"type":"heading","level":2,"text":"III-A Differentiable Gating and Stabilization"},{"type":"para","text":"We define input distribution non-stationarity in terms of the variability of feature statistics across training batches. Specifically, datasets exhibiting high variance in activation statistics or slow entropy collapse of the gating distribution are considered non-stationary, whereas datasets with stable feature distributions and rapid entropy collapse (e.g., MNIST) are considered stationary. This definitional grounding is essential for the design of the gate-freezing mechanism."},{"type":"heading","level":2,"text":"III-B Design of the NormSelector Module"},{"type":"para","text":"The core component of AutoNorm is the NormSelector (Figure 1 ), which dynamically routes between normalization strategies at each Transformer layer. We focus on a binary selection between LN and Dynamic Tanh (DyT) [ 30 ] as a controlled study of differentiable normalization gating. DyT is defined in Eq. ( 2 ); it replaces the mean-variance statistics of LN with a learnable channel-wise tanh scaling, enabling gradient-preserving feature compression."},{"type":"para","text":"The NormSelector is implemented as a lightweight two-layer MLP that takes the input representation x x and predicts weights w 0 w_{0} and w 1 w_{1} via Gumbel-Softmax [ 4 ] . The blended output (Eq. ( 1 )) is:"},{"type":"para","text":"Two control modes provide diagnostic baselines:"},{"type":"para","text":"Disable Selector (FrozenLN): Only LN is used; establishes the static normalization ceiling."},{"type":"para","text":"Random Selector: Weights are sampled uniformly each step; this baseline is essential for diagnosing the stabilization bottleneck—if learned gating underperforms random blending, the bottleneck is active."},{"type":"heading","level":2,"text":"III-C Transformer Configuration"},{"type":"para","text":"We utilize a standardized Transformer based on the ViT-Lite design [ 32 ] . Vision variants (MNIST, CIFAR) use 8 layers, 4 attention heads, an embedding dimension of 128, and a patch size of 4 × 4 4\\times 4 . NLP and Tabular variants use a 12-layer configuration with 128 hidden dimensions and 8 attention heads. All models use GELU activations, stochastic depth [ 35 ] (rate 0.1), and LayerScale initialization [ 36 ] ( λ = 10 − 4 \\lambda=10^{-4} ). This controlled design ensures that observed differences are attributable to the normalization strategy rather than capacity variations."},{"type":"para","text":"The TransformerWithAutoNorm architecture (Figure 2 ) integrates the NormSelector directly into residual pathways, adaptively blending LN and DyT based on input activations at each layer."},{"type":"heading","level":2,"text":"III-D Gate-Freezing Schedule"},{"type":"para","text":"A key contribution of AutoNorm-S is its gate-freezing schedule, which resolves the stabilization bottleneck identified empirically in Section IV-A . During an initial warm-up phase (epochs 1 through T freeze T_{\\text{freeze}} ), the NormSelector is trained normally while gating entropy H H (Eq. ( 4 )) is monitored. If H H has not fallen below a threshold ϵ H \\epsilon_{H} by epoch T freeze T_{\\text{freeze}} , the selector weights are frozen for the remainder of training; the Transformer backbone continues to update normally. In all vision experiments reported here, we use T freeze = 10 T_{\\text{freeze}}=10 and ϵ H = 0.1 \\epsilon_{H}=0.1 . This single dataset-class-level hyperparameter—not tuned per dataset—applies uniformly to MNIST, CIFAR-10, Fashion-MNIST, SVHN, and CIFAR-100. On NLP datasets (PTB, SST-2), entropy collapses naturally within 30 epochs, so no freezing is applied. The freeze gate experiment in Section IV-F validates this choice empirically."},{"type":"heading","level":2,"text":"III-E Computational Overhead"},{"type":"para","text":"The NormSelector introduces a lightweight two-layer MLP per block, resulting in an increase of approximately ¡1% in parameter count and less than 0.1% additional FLOPs per forward pass. During inference, the selector can be optionally frozen, making the overhead negligible relative to standard Layer Normalization."},{"type":"heading","level":2,"text":"III-F Experimental Setup"},{"type":"heading","level":3,"text":"III-F 1 Datasets"},{"type":"para","text":"AutoNorm is evaluated across multiple learning paradigms:"},{"type":"para","text":"Image Classification: MNIST, CIFAR-10, FashionMNIST, SVHN, and CIFAR-100 assess visual recognition under different normalization strategies."},{"type":"para","text":"Corruption Robustness: CIFAR-10-C [ 8 ] evaluates robustness at severity level 3 across Gaussian noise, motion blur, and brightness perturbations. Rotation Accuracy (Rot. Acc) in Table I denotes test-time accuracy under 90 ∘ rotation with no rotation augmentation during training, quantifying out-of-distribution geometric robustness."},{"type":"para","text":"Regression: Energy Efficiency and California Housing datasets evaluate AutoNorm on continuous-valued tabular targets."},{"type":"para","text":"NLP: Penn TreeBank (PTB) [ 33 ] POS tagging and SST-2 sentiment classification evaluate sequence-level performance. Perplexity on PTB is computed over POS tag sequences and is not comparable to language modeling perplexity (see Section IV-C )."},{"type":"heading","level":3,"text":"III-F 2 Training Setup and Hyperparameters"},{"type":"para","text":"Optimization: AdamW [ 6 ] with a cosine learning rate schedule and a warm-up phase."},{"type":"para","text":"Data Augmentation: MixUp [ 26 ] and CutMix [ 27 ] for vision tasks. Dropout (0.1) and Label Smoothing (0.1) are applied globally."},{"type":"para","text":"Distillation Ablation: Experiments include training with and without a Teacher MLP [ 28 ] ; dynamic distillation contributed a marginal 0.4%–0.8% improvement in validation stability on vision tasks."},{"type":"para","text":"Evaluation: All reported metrics represent the mean and standard deviation over 5 independent random seeds. Paired t t -tests are conducted for all primary comparisons."},{"type":"heading","level":1,"text":"IV Results"},{"type":"para","text":"This section presents experimental findings across classification, regression, and NLP tasks. All experiments were conducted under consistent training conditions; values are the mean over five independent trials. Paired t t -tests between AutoNorm-S and the strongest static baseline were statistically significant ( p < 0.01 p<0.01 ) for PTB perplexity and ( p < 0.05 p<0.05 ) for CIFAR-10-C mean accuracy. On clean vision benchmarks, differences between AutoNorm-S and the best static baseline frequently fall within the 95% confidence interval on stationary datasets—a result that is predicted by our theory (Section IV-A ) and discussed explicitly below."},{"type":"heading","level":2,"text":"IV-A Classification Results"},{"type":"para","text":"Classification results reveal a clear relationship between input distribution stationarity and the utility of adaptive gating, fully consistent with the stabilization bottleneck hypothesis."},{"type":"para","text":"On the most stationary benchmarks—MNIST and Fashion-MNIST—FrozenLN is highly competitive: on MNIST, AutoNorm-S leads by 0.36% (99.18% vs. 98.82%) within the 95% CI; on Fashion-MNIST, FrozenLN leads by 0.17% (93.12% vs. 92.95%). These results are expected: when input statistics are stable and gate entropy collapses rapidly (Section IV-F ), the gate-freezing benefit of AutoNorm-S is least pronounced. Likewise, on SVHN, AdaNorm achieves the top result (91.10%), with AutoNorm-S at 89.38%. SVHN’s digit statistics are highly constrained in style and color range, making it another stationary domain where simpler normalization suffices. These cases are not failures —they are theory-consistent: the principle predicts that AutoNorm-S yields the smallest gain precisely where it should, on low-variance stationary distributions."},{"type":"para","text":"A clearer advantage emerges on CIFAR-10 and CIFAR-100, where richer visual diversity creates the non-stationarity that activates AutoNorm’s gating mechanism. On CIFAR-10, unfrozen AutoNorm was outperformed by the RandomSelector (85.48%)—the stabilization bottleneck in action—while AutoNorm-S achieves 86.35%, recovering the learned routing advantage. On CIFAR-100, AutoNorm-S leads FrozenLN by 2.33% (67.15% vs. 64.82%), the largest vision margin, consistent with the greater distributional diversity of 100-class data."},{"type":"para","text":"Robustness to extreme corruptions remains challenging across all methods, with rotation accuracies on MNIST approaching chance levels ( ∼ \\sim 10%). This highlights that normalization selection alone is insufficient for out-of-distribution geometric robustness, which requires dedicated augmentation strategies."},{"type":"heading","level":2,"text":"IV-B Regression Results"},{"type":"para","text":"Dynamic normalization provides minimal gains on tabular regression, which is consistent with our hypothesis: low-dimensional tabular data exhibits relatively stationary feature distributions, limiting the benefit of adaptive normalization mechanisms. On Energy Efficiency, all Transformer variants converge to ≈ \\approx 22.7 RMSE; on California Housing, AutoNorm-S leads by 0.0013 RMSE (not statistically significant, p > 0.05 p>0.05 ). This is expected: low-dimensional tabular feature interactions do not generate the non-stationary feature distributions that activate AutoNorm’s gating advantage. The primary value of AutoNorm lies in high-dimensional, hierarchically evolving data streams."},{"type":"heading","level":2,"text":"IV-C NLP Results"},{"type":"para","text":"Results on PTB POS tagging and SST-2 (Table III ) provide the strongest evidence for the proposed mechanism. AutoNorm-S outperforms all baselines on every metric: 97.42% POS accuracy ( p < 0.01 p<0.01 vs. FrozenLN’s 96.80%), and PTB perplexity of 38.6 vs. 41.9 for FrozenLN. Note that PTB perplexity here is computed over POS tag sequences; it is not directly comparable to perplexity scores from language modeling benchmarks (e.g., Transformer-XL achieves 18.3 on the PTB language modeling task [ 37 ] ), though the relative improvement over the same-setup baseline is meaningful."},{"type":"para","text":"On SST-2, AutoNorm-S achieves 91.25%, outperforming FrozenLN by 1.83%. The adaptive gating mechanism responds to the varied syntactic and semantic contexts in NLP by dynamically adjusting normalization statistics at each layer, better preserving long-range dependencies critical for accurate sequence-level prediction."},{"type":"heading","level":2,"text":"IV-D Ablation Study and Sensitivity Analysis"},{"type":"para","text":"Gumbel-Softmax Temperature ( τ \\tau ): High temperatures ( τ > 1.0 \\tau>1.0 ) yield uniform soft-blending; low temperatures ( τ → 0 \\tau\\to 0 ) collapse to hard discrete selection. A fixed τ = 0.5 \\tau=0.5 provides the optimal balance of exploration and exploitation across all tasks and blocks."},{"type":"para","text":"Selector Capacity: Reducing the NormSelector MLP from two layers to a single linear transformation drops PTB POS accuracy by 1.2%, confirming that non-linear meta-decisions are necessary for adapting to complex linguistic contexts."},{"type":"para","text":"Soft vs. Hard Selection: Switching to hard-categorical gating during training causes significantly slower convergence, especially on CIFAR-100 and PTB, confirming that smooth gradient flow from the Gumbel-Softmax relaxation is essential for effective learning."},{"type":"heading","level":2,"text":"IV-E Mechanistic Analysis of Learned Gating"},{"type":"para","text":"Figure 3 visualizes the average gating weight w DyT w_{\\text{DyT}} across layers, revealing distinct task-dependent specialization that explains the observed performance patterns."},{"type":"para","text":"Vision Specialization: On MNIST, the selector converges to LN ( w LN > 0.85 w_{\\text{LN}}>0.85 ) across all layers within 5 epochs, explaining why FrozenLN remains highly competitive."},{"type":"para","text":"Linguistic Adaptation: On PTB, earlier layers (1–4) maintain a DyT-LN balance, while deeper layers (8–12) strongly prefer DyT ( w DyT > 0.72 w_{\\text{DyT}}>0.72 ). This layer-wise transition suggests dynamic normalization is particularly valuable for capturing high-level semantic abstractions that emerge in deeper Transformer layers."},{"type":"para","text":"Robustness under Shift: AutoNorm-S achieves 72.8% mean accuracy on CIFAR-10-C vs. 68.2% for FrozenLN ( p < 0.05 p<0.05 , Table IV ). Adaptive flexibility compensates for distribution shifts that destabilize the fixed statistics of LN, consistent with the non-stationarity principle."},{"type":"heading","level":2,"text":"IV-F Training Dynamics and Gating Entropy"},{"type":"para","text":"We quantify the selector’s routing certainty via gating entropy:"},{"type":"para","text":"On MNIST, H → 0 H\\to 0 within the first 5 epochs as the selector rapidly specializes to LN. On PTB, entropy remains elevated ( H ≈ 0.45 H\\approx 0.45 ) for longer, indicating a sustained exploration phase that eventually settles into the layer-wise switching pattern visible in Figure 3 . This prolonged exploration avoids premature commitment to suboptimal static strategies, explaining AutoNorm’s NLP advantage (Figure 4 )."},{"type":"para","text":"To confirm that early-stage Gumbel estimator variance causes the vision performance gap, we conducted the freeze-gate experiment described in Section III-D : freezing the NormSelector after 10 epochs on CIFAR-10 improved final accuracy by 1.1% relative to the unfrozen baseline. The gating MLP adds less than 0.1% per-epoch overhead relative to a standard Transformer, making AutoNorm viable at scale."},{"type":"heading","level":2,"text":"IV-G When Does Dynamic Normalization Help?"},{"type":"para","text":"Our empirical and mechanistic findings yield the following design principle : adaptive architectural selection is beneficial only when feature distributions remain non-stationary during training; otherwise, early stochastic optimization noise dominates the learning signal and must be constrained through stabilization mechanisms such as gating freezing."},{"type":"para","text":"Hierarchical Abstraction: Dynamic normalization is most beneficial when the task requires transitioning between markedly different feature representations across layers (e.g., from local syntactic to global semantic patterns in NLP). DyT’s non-linear saturation captures layer-specific activation statistics that LN cannot."},{"type":"para","text":"Non-Stationary Input Distributions: When input statistics fluctuate—unlike the stable centroids of MNIST or Fashion-MNIST—dynamic re-scaling maintains training stability and improves generalization under distribution shift."},{"type":"para","text":"Preliminary experiments with a 3-way gating mixture (LN, RMSNorm, and DyT) on PTB yielded comparable gains, suggesting that the principle of differentiable gating remains robust as the normalization search space expands."},{"type":"heading","level":1,"text":"V Conclusion"},{"type":"para","text":"We presented AutoNorm, a differentiable normalization gating framework that dynamically selects between Layer Normalization and Dynamic Tanh at each Transformer layer via Gumbel-Softmax routing. Our central finding is a stabilization bottleneck : on stationary vision benchmarks, high gradient variance in early Gumbel-Softmax training prevents the selector from converging, causing random selectors to outperform learned policies. On non-stationary linguistic tasks, sustained gating entropy enables the selector to specialize into a layer-wise switching pattern that improves perplexity and downstream accuracy."},{"type":"para","text":"AutoNorm-S resolves this bottleneck through a simple gate-freezing schedule applied after a brief warm-up phase, restoring competitive vision performance while preserving the NLP inductive bias. Mechanistic analysis via gating entropy and layer-wise weight visualization confirms that the mechanism is interpretable and theory-consistent."},{"type":"para","text":"Our findings yield a general design principle for adaptive mechanisms in Transformers: gating should be stabilized during early training when input distributions are stationary, and permitted to explore freely when features evolve hierarchically. This principle is architecture-agnostic and extends naturally beyond normalization. Future work will investigate its transferability to attention routing, mixture-of-experts selection [ 29 ] , and cross-modal distribution shift adaptation."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Normalization techniques have become fundamental to the stability and convergence of modern deep learning architectures. By normalizing feature statistics within each layer, techniques such as Batch Normalization (BN) [ 1 ] and Layer Normalization (LN) [ 2 ] mitigate internal covariate shift and stabilize gradient flow. In the era of Transformers [ 3 ] , Layer Normalization has emerged as the de facto standard for stable self-attention training across large model depths.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, conventional normalization layers are fundamentally static, applying identical transformations regardless of task context or input distribution. This one-size-fits-all approach is increasingly suboptimal as models encounter severe distribution shifts and diverse data modalities. Prior works exploring adaptive normalization—such as FiLM [ 31 ] , Switchable Normalization [ 34 ] , and conditional Meta-Normalization—typically rely on external metadata conditioning or expensive architectural searches, lacking a self-sufficient mechanism for on-the-fly adaptation. Recently, Zhu et al. [ 30 ] demonstrated that replacing normalization entirely with a learnable tanh-based transformation (Dynamic Tanh; DyT) can match or exceed LN in standard Transformers, raising a natural question: can a model learn, at training time, which normalization strategy is optimal per layer and per task?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When we initially attempted to answer this using differentiable Gumbel-Softmax gating—a framework we call AutoNorm —we encountered a surprising and counter-intuitive phenomenon: while the learned mechanism worked effectively for linguistic sequences, it consistently failed on vision benchmarks, where a simple Random Selector often outperformed the learned policy. This unexpected failure highlighted a deeper issue in how architectural selection interacts with optimization dynamics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we investigate this failure and identify it as a stabilization bottleneck specific to stationary input distributions. High gradient variance in the early stages of differentiable gating prevents the selector from specializing before the backbone features have matured. We introduce AutoNorm-S (Stabilized) , which resolves this through a gate-freezing schedule (Section III-D ), and demonstrate that the resulting mechanism not only restores competitive performance in vision but provides a significant inductive bias for NLP. Our contributions progress from a specific architectural failure mode to a general principle governing when and how adaptive normalization should be deployed in Transformers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a given input tensor X X , AutoNorm computes:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where w DyT ( X ) w_{\\\\text{DyT}}(X) and w LN ( X ) w_{\\\\text{LN}}(X) are adaptive weights predicted by the NormSelector (Section III-B ), satisfying w DyT ( X ) + w LN ( X ) = 1 w_{\\\\text{DyT}}(X)+w_{\\\\text{LN}}(X)=1 . These weights are obtained through a lightweight gating network that uses a differentiable Gumbel–Softmax reparameterization [ 4 ] , allowing gradient-based optimization of the discrete normalization selection.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Literature Review\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Normalization Techniques in Deep Learning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Normalization is fundamental to training deep neural networks, mitigating vanishing and exploding gradients and internal covariate shift [ 1 , 2 , 13 , 14 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"II-A 1 Batch Normalization (BN)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Introduced by Ioffe and Szegedy [ 1 ] , BN normalizes activations across the mini-batch dimension and applies learnable scale and offset. BN enables higher learning rates and acts as a regularizer in CNNs but degrades with small batch sizes [ 14 ] and is unsuitable for variable-length sequences.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"II-A 2 Layer Normalization (LN)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ba et al. [ 2 ] proposed LN, which normalizes across the feature dimension within each sample, independent of batch size. LN is highly effective in Transformers but may underperform BN in standard CNNs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"II-A 3 Instance and Group Normalization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instance Normalization [ 13 ] normalizes per channel over spatial dimensions, excelling at style transfer. Group Normalization [ 14 ] divides channels into groups, maintaining stable performance across batch sizes at the cost of careful group-count selection.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"II-A 4 Adaptive Batch Normalization (AdaBN)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Li et al. [ 15 ] proposed AdaBN for domain adaptation by modulating batch-normalized statistics between source and target domains—an early demonstration of adaptive normalization under non-stationary distributions. AutoNorm extends this idea to differentiable gated selection in Transformers.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"II-A 5 DeepNorm\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Wang et al. [ 16 ] introduced DeepNorm, enabling Transformers to scale to 1,000 layers by stabilizing residual variance. AutoNorm complements such scaling methods by providing feature-dependent adaptability rather than static stability alone.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"II-A 6 Dynamic Tanh (DyT)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Zhu et al. [ 30 ] proposed replacing normalization entirely with:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where α ∈ ℝ \\\\alpha\\\\in\\\\mathbb{R} is a learnable scalar controlling input saturation, and 𝜸 , 𝜷 ∈ ℝ C \\\\boldsymbol{\\\\gamma},\\\\boldsymbol{\\\\beta}\\\\in\\\\mathbb{R}^{C} are learnable channel-wise scale and shift parameters. DyT matches or exceeds LN on several Transformer benchmarks without computing feature statistics at runtime. AutoNorm treats DyT and LN as complementary candidates in a differentiable mixture, rather than substitutes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"II-A 7 Switchable Normalization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Luo et al. [ 34 ] proposed Switchable Normalization (SwitchNorm), which learns differentiable importance weights over BN, LN, and IN. While SwitchNorm blends statistics computed simultaneously from all three methods, AutoNorm uses Gumbel-Softmax gating to route between fully computed normalization outputs and, critically, identifies the optimization dynamics that determine whether learned switching is beneficial—a mechanism not studied by SwitchNorm.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AutoNorm shares conceptual roots with FiLM [ 31 ] and other conditional normalization methods. However, its Gumbel-Softmax formulation enables fully differentiable autonomous selection between LN and DyT without external metadata or task-specific conditioning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B Dynamic and Adaptive Mechanisms in Neural Networks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Adaptive mechanisms increase network flexibility during training and inference [ 17 , 18 , 19 , 20 , 21 ] . Key examples include adaptive activation functions—PReLU [ 17 ] , Swish [ 18 ] , Meta-Acon [ 19 ] —and per-parameter adaptive optimizers: AdaGrad [ 22 ] , Adam [ 5 ] , AdaBelief [ 23 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Transformer Architectures and Normalization Strategies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Transformers [ 3 ] rely on self-attention with normalization, primarily LN. Pre-LN and Post-LN variants affect training stability and scalability [ 7 , 16 , 25 ] . Alternatives include RMSNorm [ 24 ] and DeepNorm [ 16 ] . The interaction between normalization strategy and optimization dynamics—the central focus of this work—has received limited systematic study.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Methodology\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A Differentiable Gating and Stabilization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We define input distribution non-stationarity in terms of the variability of feature statistics across training batches. Specifically, datasets exhibiting high variance in activation statistics or slow entropy collapse of the gating distribution are considered non-stationary, whereas datasets with stable feature distributions and rapid entropy collapse (e.g., MNIST) are considered stationary. This definitional grounding is essential for the design of the gate-freezing mechanism.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B Design of the NormSelector Module\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The core component of AutoNorm is the NormSelector (Figure 1 ), which dynamically routes between normalization strategies at each Transformer layer. We focus on a binary selection between LN and Dynamic Tanh (DyT) [ 30 ] as a controlled study of differentiable normalization gating. DyT is defined in Eq. ( 2 ); it replaces the mean-variance statistics of LN with a learnable channel-wise tanh scaling, enabling gradient-preserving feature compression.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The NormSelector is implemented as a lightweight two-layer MLP that takes the input representation x x and predicts weights w 0 w_{0} and w 1 w_{1} via Gumbel-Softmax [ 4 ] . The blended output (Eq. ( 1 )) is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two control modes provide diagnostic baselines:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Disable Selector (FrozenLN): Only LN is used; establishes the static normalization ceiling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Random Selector: Weights are sampled uniformly each step; this baseline is essential for diagnosing the stabilization bottleneck—if learned gating underperforms random blending, the bottleneck is active.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C Transformer Configuration\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We utilize a standardized Transformer based on the ViT-Lite design [ 32 ] . Vision variants (MNIST, CIFAR) use 8 layers, 4 attention heads, an embedding dimension of 128, and a patch size of 4 × 4 4\\\\times 4 . NLP and Tabular variants use a 12-layer configuration with 128 hidden dimensions and 8 attention heads. All models use GELU activations, stochastic depth [ 35 ] (rate 0.1), and LayerScale initialization [ 36 ] ( λ = 10 − 4 \\\\lambda=10^{-4} ). This controlled design ensures that observed differences are attributable to the normalization strategy rather than capacity variations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The TransformerWithAutoNorm architecture (Figure 2 ) integrates the NormSelector directly into residual pathways, adaptively blending LN and DyT based on input activations at each layer.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-D Gate-Freezing Schedule\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key contribution of AutoNorm-S is its gate-freezing schedule, which resolves the stabilization bottleneck identified empirically in Section IV-A . During an initial warm-up phase (epochs 1 through T freeze T_{\\\\text{freeze}} ), the NormSelector is trained normally while gating entropy H H (Eq. ( 4 )) is monitored. If H H has not fallen below a threshold ϵ H \\\\epsilon_{H} by epoch T freeze T_{\\\\text{freeze}} , the selector weights are frozen for the remainder of training; the Transformer backbone continues to update normally. In all vision experiments reported here, we use T freeze = 10 T_{\\\\text{freeze}}=10 and ϵ H = 0.1 \\\\epsilon_{H}=0.1 . This single dataset-class-level hyperparameter—not tuned per dataset—applies uniformly to MNIST, CIFAR-10, Fashion-MNIST, SVHN, and CIFAR-100. On NLP datasets (PTB, SST-2), entropy collapses naturally within 30 epochs, so no freezing is applied. The freeze gate experiment in Section IV-F validates this choice empirically.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-E Computational Overhead\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The NormSelector introduces a lightweight two-layer MLP per block, resulting in an increase of approximately ¡1% in parameter count and less than 0.1% additional FLOPs per forward pass. During inference, the selector can be optionally frozen, making the overhead negligible relative to standard Layer Normalization.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-F Experimental Setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"III-F 1 Datasets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AutoNorm is evaluated across multiple learning paradigms:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Image Classification: MNIST, CIFAR-10, FashionMNIST, SVHN, and CIFAR-100 assess visual recognition under different normalization strategies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Corruption Robustness: CIFAR-10-C [ 8 ] evaluates robustness at severity level 3 across Gaussian noise, motion blur, and brightness perturbations. Rotation Accuracy (Rot. Acc) in Table I denotes test-time accuracy under 90 ∘ rotation with no rotation augmentation during training, quantifying out-of-distribution geometric robustness.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Regression: Energy Efficiency and California Housing datasets evaluate AutoNorm on continuous-valued tabular targets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"NLP: Penn TreeBank (PTB) [ 33 ] POS tagging and SST-2 sentiment classification evaluate sequence-level performance. Perplexity on PTB is computed over POS tag sequences and is not comparable to language modeling perplexity (see Section IV-C ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"III-F 2 Training Setup and Hyperparameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optimization: AdamW [ 6 ] with a cosine learning rate schedule and a warm-up phase.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data Augmentation: MixUp [ 26 ] and CutMix [ 27 ] for vision tasks. Dropout (0.1) and Label Smoothing (0.1) are applied globally.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Distillation Ablation: Experiments include training with and without a Teacher MLP [ 28 ] ; dynamic distillation contributed a marginal 0.4%–0.8% improvement in validation stability on vision tasks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluation: All reported metrics represent the mean and standard deviation over 5 independent random seeds. Paired t t -tests are conducted for all primary comparisons.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section presents experimental findings across classification, regression, and NLP tasks. All experiments were conducted under consistent training conditions; values are the mean over five independent trials. Paired t t -tests between AutoNorm-S and the strongest static baseline were statistically significant ( p < 0.01 p<0.01 ) for PTB perplexity and ( p < 0.05 p<0.05 ) for CIFAR-10-C mean accuracy. On clean vision benchmarks, differences between AutoNorm-S and the best static baseline frequently fall within the 95% confidence interval on stationary datasets—a result that is predicted by our theory (Section IV-A ) and discussed explicitly below.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Classification Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Classification results reveal a clear relationship between input distribution stationarity and the utility of adaptive gating, fully consistent with the stabilization bottleneck hypothesis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the most stationary benchmarks—MNIST and Fashion-MNIST—FrozenLN is highly competitive: on MNIST, AutoNorm-S leads by 0.36% (99.18% vs. 98.82%) within the 95% CI; on Fashion-MNIST, FrozenLN leads by 0.17% (93.12% vs. 92.95%). These results are expected: when input statistics are stable and gate entropy collapses rapidly (Section IV-F ), the gate-freezing benefit of AutoNorm-S is least pronounced. Likewise, on SVHN, AdaNorm achieves the top result (91.10%), with AutoNorm-S at 89.38%. SVHN’s digit statistics are highly constrained in style and color range, making it another stationary domain where simpler normalization suffices. These cases are not failures —they are theory-consistent: the principle predicts that AutoNorm-S yields the smallest gain precisely where it should, on low-variance stationary distributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A clearer advantage emerges on CIFAR-10 and CIFAR-100, where richer visual diversity creates the non-stationarity that activates AutoNorm’s gating mechanism. On CIFAR-10, unfrozen AutoNorm was outperformed by the RandomSelector (85.48%)—the stabilization bottleneck in action—while AutoNorm-S achieves 86.35%, recovering the learned routing advantage. On CIFAR-100, AutoNorm-S leads FrozenLN by 2.33% (67.15% vs. 64.82%), the largest vision margin, consistent with the greater distributional diversity of 100-class data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Robustness to extreme corruptions remains challenging across all methods, with rotation accuracies on MNIST approaching chance levels ( ∼ \\\\sim 10%). This highlights that normalization selection alone is insufficient for out-of-distribution geometric robustness, which requires dedicated augmentation strategies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-B Regression Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dynamic normalization provides minimal gains on tabular regression, which is consistent with our hypothesis: low-dimensional tabular data exhibits relatively stationary feature distributions, limiting the benefit of adaptive normalization mechanisms. On Energy Efficiency, all Transformer variants converge to ≈ \\\\approx 22.7 RMSE; on California Housing, AutoNorm-S leads by 0.0013 RMSE (not statistically significant, p > 0.05 p>0.05 ). This is expected: low-dimensional tabular feature interactions do not generate the non-stationary feature distributions that activate AutoNorm’s gating advantage. The primary value of AutoNorm lies in high-dimensional, hierarchically evolving data streams.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-C NLP Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Results on PTB POS tagging and SST-2 (Table III ) provide the strongest evidence for the proposed mechanism. AutoNorm-S outperforms all baselines on every metric: 97.42% POS accuracy ( p < 0.01 p<0.01 vs. FrozenLN’s 96.80%), and PTB perplexity of 38.6 vs. 41.9 for FrozenLN. Note that PTB perplexity here is computed over POS tag sequences; it is not directly comparable to perplexity scores from language modeling benchmarks (e.g., Transformer-XL achieves 18.3 on the PTB language modeling task [ 37 ] ), though the relative improvement over the same-setup baseline is meaningful.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On SST-2, AutoNorm-S achieves 91.25%, outperforming FrozenLN by 1.83%. The adaptive gating mechanism responds to the varied syntactic and semantic contexts in NLP by dynamically adjusting normalization statistics at each layer, better preserving long-range dependencies critical for accurate sequence-level prediction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-D Ablation Study and Sensitivity Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Gumbel-Softmax Temperature ( τ \\\\tau ): High temperatures ( τ > 1.0 \\\\tau>1.0 ) yield uniform soft-blending; low temperatures ( τ → 0 \\\\tau\\\\to 0 ) collapse to hard discrete selection. A fixed τ = 0.5 \\\\tau=0.5 provides the optimal balance of exploration and exploitation across all tasks and blocks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Selector Capacity: Reducing the NormSelector MLP from two layers to a single linear transformation drops PTB POS accuracy by 1.2%, confirming that non-linear meta-decisions are necessary for adapting to complex linguistic contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Soft vs. Hard Selection: Switching to hard-categorical gating during training causes significantly slower convergence, especially on CIFAR-100 and PTB, confirming that smooth gradient flow from the Gumbel-Softmax relaxation is essential for effective learning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-E Mechanistic Analysis of Learned Gating\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 visualizes the average gating weight w DyT w_{\\\\text{DyT}} across layers, revealing distinct task-dependent specialization that explains the observed performance patterns.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Vision Specialization: On MNIST, the selector converges to LN ( w LN > 0.85 w_{\\\\text{LN}}>0.85 ) across all layers within 5 epochs, explaining why FrozenLN remains highly competitive.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Linguistic Adaptation: On PTB, earlier layers (1–4) maintain a DyT-LN balance, while deeper layers (8–12) strongly prefer DyT ( w DyT > 0.72 w_{\\\\text{DyT}}>0.72 ). This layer-wise transition suggests dynamic normalization is particularly valuable for capturing high-level semantic abstractions that emerge in deeper Transformer layers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Robustness under Shift: AutoNorm-S achieves 72.8% mean accuracy on CIFAR-10-C vs. 68.2% for FrozenLN ( p < 0.05 p<0.05 , Table IV ). Adaptive flexibility compensates for distribution shifts that destabilize the fixed statistics of LN, consistent with the non-stationarity principle.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-F Training Dynamics and Gating Entropy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We quantify the selector’s routing certainty via gating entropy:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On MNIST, H → 0 H\\\\to 0 within the first 5 epochs as the selector rapidly specializes to LN. On PTB, entropy remains elevated ( H ≈ 0.45 H\\\\approx 0.45 ) for longer, indicating a sustained exploration phase that eventually settles into the layer-wise switching pattern visible in Figure 3 . This prolonged exploration avoids premature commitment to suboptimal static strategies, explaining AutoNorm’s NLP advantage (Figure 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To confirm that early-stage Gumbel estimator variance causes the vision performance gap, we conducted the freeze-gate experiment described in Section III-D : freezing the NormSelector after 10 epochs on CIFAR-10 improved final accuracy by 1.1% relative to the unfrozen baseline. The gating MLP adds less than 0.1% per-epoch overhead relative to a standard Transformer, making AutoNorm viable at scale.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-G When Does Dynamic Normalization Help?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our empirical and mechanistic findings yield the following design principle : adaptive architectural selection is beneficial only when feature distributions remain non-stationary during training; otherwise, early stochastic optimization noise dominates the learning signal and must be constrained through stabilization mechanisms such as gating freezing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hierarchical Abstraction: Dynamic normalization is most beneficial when the task requires transitioning between markedly different feature representations across layers (e.g., from local syntactic to global semantic patterns in NLP). DyT’s non-linear saturation captures layer-specific activation statistics that LN cannot.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Non-Stationary Input Distributions: When input statistics fluctuate—unlike the stable centroids of MNIST or Fashion-MNIST—dynamic re-scaling maintains training stability and improves generalization under distribution shift.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Preliminary experiments with a 3-way gating mixture (LN, RMSNorm, and DyT) on PTB yielded comparable gains, suggesting that the principle of differentiable gating remains robust as the normalization search space expands.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We presented AutoNorm, a differentiable normalization gating framework that dynamically selects between Layer Normalization and Dynamic Tanh at each Transformer layer via Gumbel-Softmax routing. Our central finding is a stabilization bottleneck : on stationary vision benchmarks, high gradient variance in early Gumbel-Softmax training prevents the selector from converging, causing random selectors to outperform learned policies. On non-stationary linguistic tasks, sustained gating entropy enables the selector to specialize into a layer-wise switching pattern that improves perplexity and downstream accuracy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AutoNorm-S resolves this bottleneck through a simple gate-freezing schedule applied after a brief warm-up phase, restoring competitive vision performance while preserving the NLP inductive bias. Mechanistic analysis via gating entropy and layer-wise weight visualization confirms that the mechanism is interpretable and theory-consistent.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our findings yield a general design principle for adaptive mechanisms in Transformers: gating should be stabilized during early training when input distributions are stationary, and permitted to explore freely when features evolve hierarchically. This principle is architecture-agnostic and extends naturally beyond normalization. Future work will investigate its transferability to attention routing, mixture-of-experts selection [ 29 ] , and cross-modal distribution shift adaptation.\"\n }\n]","isConversational":false}}},{"rowIdx":388,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10582"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"MemDecay: Region-Aware KV Cache Eviction for Efficient LLM Agent Inference"},"authors":{"kind":"list like","value":["Venkatesha Matam","Keon Kim"],"string":"[\n \"Venkatesha Matam\",\n \"Keon Kim\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10582v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"Large language models (LLMs) are increasingly used in long-horizon agentic workflows that combine multi-turn interaction, retrieval, tool use, and iterative planning. Unlike single-pass long-context tasks, these systems accumulate a persistent execution trace composed of system instructions, safety constraints, evolving plans, user turns, retrieved evidence, tool interactions, and intermediate reasoning states. The trace is structured: its components serve different functional roles and should persist for different durations."},{"type":"para","text":"This shift places new pressure on the key–value (KV) cache that supports autoregressive decoding. Because the KV cache grows with sequence length, long interaction traces translate into substantial memory overhead, increased bandwidth demand, and higher end-to-end latency [ 1 , 2 ] . Prior work reduces this overhead through selective retention and compression, including policies that adapt eviction to stable differences in attention behavior across heads [ 3 , 4 ] . Most of these approaches target generic long-context generation and score cached tokens with model-centric signals such as accumulated attention, attention sinks and recency windows, or instantaneous attention [ 5 , 6 , 7 ] ."},{"type":"para","text":"That assumption is poorly matched to agentic workloads. In LLM-based agents, memory is a core component that supports long-term and complex interactions with the environment [ 8 ] . Yet the contents of an agent trace are heterogeneous by design. System and safety instructions often need to remain stable throughout an episode, and high-level plans may need to persist until the agent replans. By contrast, stale tool outputs, transient scratchpad content, and low-value retrievals are often useful only for a short span of the trajectory. A cache policy that treats all such tokens as having the same expected lifetime misses this structure."},{"type":"para","text":"This mismatch motivates aligning cache retention with the semantic structure already present in orchestrated prompts. We propose MemDecay , a training-free, prompt-region-aware KV-cache decay and eviction framework for long-horizon LLM agents. MemDecay assumes that the orchestration layer provides region labels for tokens, such as system prompt, plan, user message, tool input, tool output, retrieval, or scratchpad. It assigns each region a base retention priority and a decay rate, then combines these structural priors with attention-derived importance to compute a composite retention score for each token. MemDecay is training-free in the sense this literature uses the term: it updates no model weights and learns no predictor. It does rely on a lightweight statistical calibration that fits two parameters per region to measured traces, described in Section IV . Concurrent work gives evidence that semantic roles predict reuse and attention lifetimes in agent workloads [ 9 , 10 ] ; MemDecay applies this signal to token-scored, page-granular eviction under a fixed budget within a single decoding context."},{"type":"para","text":"MemDecay treats forgetting as a scheduled process. Region-specific temporal decay lowers a token’s retention priority smoothly over time, and attention to the token resets its decay clock, so continued use keeps a token alive. When a fixed KV-cache budget is exceeded, tokens with the lowest decayed priorities are evicted first, while pinned instruction regions and, optionally, active plan segments are preserved. The resulting policy is interpretable, requires no retraining or auxiliary modules, and operates on page-granular cache layouts of the kind used in existing serving stacks [ 1 ] ."},{"type":"para","text":"Our contributions are threefold:"},{"type":"para","text":"We formulate region-aware selective forgetting as a KV-cache memory-management problem for long-horizon LLM agents, grounded in the structure of orchestrated prompts and agent traces."},{"type":"para","text":"We introduce MemDecay , a training-free, prompt-region-aware eviction policy that combines region base priorities, region-specific temporal decay with attention-triggered refresh, and attention-derived importance in a single interpretable retention score under a fixed KV-cache budget."},{"type":"para","text":"We specify a measurement protocol that estimates region-conditioned attention lifetimes from agent traces and uses the fitted decay rates to calibrate the policy, and we execute it across two context lengths and two model sizes (4,320 scored eviction probes): the region-lifetime hierarchy replicates in every setting, with the longest- and shortest-lived regions separated by an order of magnitude at non-overlapping confidence intervals, the fitted rates overturn one qualitative default, pinning delivers guaranteed instruction survival everywhere, recency-based retention collapses with context length while the structural prior keeps working, and an invariance ablation points to magnitude normalization of the attention term as the needed design change for the policy’s unpinned-recall loss."},{"type":"heading","level":1,"text":"II Related Work"},{"type":"heading","level":2,"text":"II-A KV-Cache Memory Management for LLM Serving"},{"type":"para","text":"KV-cache management has become a first-order systems problem for scalable LLM serving because cache footprint grows with context length and directly constrains memory capacity, bandwidth, and throughput [ 1 , 2 ] . System-level approaches focus on how cached states are allocated, retained, and reused across requests. PagedAttention organizes the cache into fixed-size pages to reduce fragmentation and enable sharing [ 1 ] , and prefix-aware designs detect and reuse shared prompt segments such as system instructions and retrieved passages [ 11 , 12 ] . Studies of production-style prefix caches report that reuse value varies widely across prompt segments, which motivates retention decisions that account for what a segment contains [ 9 ] ."},{"type":"heading","level":2,"text":"II-B KV-Cache Eviction and Compression for Long-Context LLMs"},{"type":"para","text":"A large literature studies post-training KV reduction through eviction, pruning, quantization, and hybrid schemes for long-context inference [ 2 , 13 ] . Attention-based eviction retains tokens with high accumulated attention [ 5 , 14 ] , window-based policies keep attention sinks and recent tokens [ 6 , 7 ] , and prompt-time compression selects tokens using attention from an observation window [ 15 ] ."},{"type":"para","text":"Later work showed that locally accumulated attention statistics are biased and can misestimate future utility [ 16 , 17 ] . Several methods reallocate budgets across heads or layers, reflecting evidence that attention modules have heterogeneous roles and sensitivities [ 3 , 4 ] . Others score tokens with query-aware or semantic-cluster criteria [ 18 , 19 ] , keep or drop contiguous semantic chunks instead of isolated tokens [ 20 ] , or lower the precision of cached entries instead of removing them [ 21 ] . Learned approaches predict token importance directly, at the cost of training and auxiliary modules [ 22 , 9 ] ."},{"type":"heading","level":2,"text":"II-C Agent- and Workflow-Aware KV Management"},{"type":"para","text":"Recent work shows that agentic workloads differ materially from static chat or document settings because they interleave repeated LLM calls, tools, and variable pauses, creating reuse patterns that generic policies do not anticipate [ 23 ] . Workflow-aware systems model future execution structure explicitly. KVFlow prioritizes retention using estimated temporal proximity to future agent activation, and Continuum uses time-to-live pinning to preserve KV states across short tool calls when recomputation would be costly [ 24 , 23 ] . Prefix-cache studies report that segment semantics matter in practice: token-agnostic policies such as LRU miss up to 756 × \\times variation in reuse rates across system prompts, user queries, tool outputs, model responses, and reasoning traces [ 9 ] ."},{"type":"para","text":"Concurrent work brings role and time structure into cache policies themselves. RoleKV tags KV blocks with semantic roles and applies role-calibrated decay to serving-layer retention, reporting that the oldest blocks in agent traces, including system prompts and tool outputs, receive the most persistent attention while recent reasoning tokens decay fastest [ 10 ] . IntentKV scores history tokens against an exponentially decayed summary of session queries for multi-turn agent pruning [ 25 ] , and retention-gated caching learns per-token gates that decay token importance over time [ 26 ] . These results establish that semantic roles and temporal decay both carry signal for cache management, and they disagree about which regions of an agent trace stay useful, a question we return to in Section IV ."},{"type":"heading","level":2,"text":"II-D Memory and Forgetting"},{"type":"para","text":"Memory is a defining component of LLM-based agents because long-horizon interaction depends on retaining, updating, and selectively reusing prior state [ 8 , 27 ] . Within KV-cache research, several recent papers argue that irreversible keep-or-drop policies are too coarse: token value can shift as decoding proceeds, and a token evicted early cannot be recovered when later queries make it relevant again [ 25 , 28 ] . Mixed-precision storage offers one response, degrading uncertain entries before removing them [ 28 , 21 ] . This body of work motivates treating forgetting as a controllable mechanism with an explicit schedule instead of a byproduct of memory pressure. Class-based replacement with aging has a long history in web caching, where Greedy-Dual-Size-Frequency combines per-class value with an aging term [ 31 ] ; MemDecay adapts this lineage to KV caches, with prompt regions as classes and decay rates calibrated per region."},{"type":"heading","level":2,"text":"II-E Positioning MemDecay"},{"type":"para","text":"MemDecay is closest to the concurrent role- and time-aware policies above, and it differs from each on a specific axis. RoleKV manages which blocks stay resident for cross-request reuse and is lossless with respect to model outputs [ 10 ] ; MemDecay performs lossy, page-granular eviction driven by token-level scores inside a single decoding context under a fixed budget. SAECache learns segment reuse value online for prefix caches [ 9 ] , and IntentKV in its stronger variant trains a residual scoring head [ 25 ] ; MemDecay is training-free and uses only signals available at inference time. Retention-gated caching learns its decay behavior through trained per-token gates [ 26 ] ; MemDecay is training-free, conditions both the base priority and the decay rate on the prompt region, and lets attention reset the decay clock. Workflow-level systems such as KVFlow and Continuum schedule caches across agent steps and are complementary: MemDecay decides which tokens survive within a context, and those systems decide when whole caches are kept, offloaded, or dropped [ 24 , 23 ] ."},{"type":"heading","level":1,"text":"III Proposed Method"},{"type":"para","text":"MemDecay is an inference-time eviction policy for the KV cache of a decoder-only LLM running inside an agent orchestrator. The policy uses three inputs: region labels supplied by the orchestrator, attention statistics sampled during decoding, and a fixed cache budget. Fig. 1 shows where the policy sits in the serving stack. This section defines the setting, the structural priors, the retention score, and the eviction procedure."},{"type":"heading","level":2,"text":"III-A Problem Setup"},{"type":"para","text":"An agent episode produces a token sequence indexed by i i , decoded over steps t = 1 , 2 , … t=1,2,\\dots . Each cached token i i carries a region label r ( i ) ∈ ℛ r(i)\\in\\mathcal{R} and an insertion step t i ins t_{i}^{\\mathrm{ins}} , where ℛ \\mathcal{R} contains the labels system , plan , user , tool_in , tool_out , retrieval , and scratchpad . Region labels correspond to contiguous spans of the prompt because the orchestrator assembles the context from template segments. We assume the orchestration layer emits these labels alongside the request; serving systems have begun to accept exactly this kind of metadata, and an open SGLang proposal attaches workflow, tool, and reuse hints to inference requests [ 32 ] ."},{"type":"para","text":"Let 𝒦 t \\mathcal{K}_{t} denote the set of cached tokens at step t t and let B B be the cache budget in tokens. A subset ℱ ⊆ 𝒦 t \\mathcal{F}\\subseteq\\mathcal{K}_{t} of pinned tokens is never evicted; by default ℱ \\mathcal{F} contains all system tokens, which also covers the initial attention-sink positions that must remain resident for stable decoding [ 6 ] . The policy must maintain | 𝒦 t | ≤ B |\\mathcal{K}_{t}|\\leq B by selecting which non-pinned tokens to evict. Formally, at any step where the budget binds, the policy solves"},{"type":"para","text":"where s i ( t ) s_{i}(t) is the retention score defined in Section III-C . The objective is additive over tokens, so retaining the B B highest-scoring tokens, after forcing ℱ \\mathcal{F} into the set, is optimal, and eviction reduces to removing the lowest-scoring tokens first."},{"type":"heading","level":2,"text":"III-B Structural Priors"},{"type":"para","text":"Each region r r is configured with a base priority b r ≥ 0 b_{r}\\geq 0 and a decay rate λ r ≥ 0 \\lambda_{r}\\geq 0 . The base priority encodes how valuable a token from region r r is at insertion time, and the decay rate encodes how quickly that structural protection expires. Table I lists the default configuration. These defaults encode the template lifetimes described in Section I. They are starting points: published evidence on region lifetimes conflicts, with role-aware serving measurements reporting persistent attention to old system prompts and tool outputs [ 10 ] while decay-based retention assumes that older tokens generally matter less [ 26 ] . Section IV describes a calibration procedure that replaces these defaults with rates fitted per workload."},{"type":"heading","level":2,"text":"III-C Retention Score with Region-Specific Decay"},{"type":"para","text":"MemDecay tracks one dynamic signal per token, an attention importance a i a_{i} , and one clock per token, the time since the token was last used. Fused attention kernels do not materialize the attention matrix, so reading per-step attention scores is impractical in production kernels [ 33 ] . MemDecay therefore samples attention only at observation steps, one every k k decode steps, using windowed scores in the style of prompt-time selection methods [ 15 ] . At an observation step, let A ¯ i \\bar{A}_{i} denote the attention mass received by token i i , averaged over heads, layers, and the window. The importance is an exponentially weighted average,"},{"type":"para","text":"with smoothing factor ρ ∈ [ 0 , 1 ) \\rho\\in[0,1) ."},{"type":"para","text":"The decay clock measures time since insertion or since the model last attended to the token above a threshold τ \\tau . Let"},{"type":"para","text":"where t ′ t^{\\prime} is the most recent observation step satisfying the condition. A refresh resets the clock and nothing else; the structural prior is re-applied from the refresh point. The threshold τ \\tau can be fixed or scaled to the uniform attention level 1 / | 𝒦 t | 1/|\\mathcal{K}_{t}| so that the refresh criterion tracks context length; Section IV uses the scaled form."},{"type":"para","text":"The retention score combines the decayed structural prior with the observed importance:"},{"type":"para","text":"where α ≥ 0 \\alpha\\geq 0 weights how much observed attention can protect a token beyond its structural prior. Recency enters only through the decay clock, so the score contains no separate recency term and elapsed time is not counted twice. The three terms have direct interpretations: b r b_{r} sets where a token starts, λ r \\lambda_{r} sets how quickly structure alone stops protecting it, and α a i \\alpha a_{i} keeps any token alive while the model keeps using it, regardless of region. Pinned tokens bypass scoring entirely. Fig. 2 illustrates the resulting score trajectories, including a refresh event that resets the decay clock of a short-lived region."},{"type":"heading","level":2,"text":"III-D Budget-Aware Eviction at Page Granularity"},{"type":"para","text":"Production serving stacks store KV entries in fixed-size pages [ 1 ] . Region labels mark contiguous spans, so at most one page can straddle each boundary between consecutive region runs, and the straddle fraction shrinks as regions grow long relative to the page size. In the episodes of Section IV , 35% of 16-token pages straddle a boundary on the short tier and 17% on the long tier, matching the prediction that straddling shrinks as regions lengthen. The page score is the mean over member tokens,"},{"type":"para","text":"MemDecay evicts at page granularity: when | 𝒦 t | > B |\\mathcal{K}_{t}|>B , non-pinned pages are ordered by ascending S p ( t ) S_{p}(t) and evicted until the budget is met. This greedy rule is optimal for the page-level objective because pages have equal size; it can deviate from the token-level optimum of ( 1 ) when a page mixes regions, since the mean in ( 5 ) can mask a high-scoring token among low-scoring pagemates. The straddle fraction above quantifies how often mixed pages occur, and the pinned-page collateral disclosed in Section IV-A is the same effect acting in the protective direction. Region-aligned page boundaries, where the allocator opens a fresh page at each region boundary, would remove the dilution at the cost of internal fragmentation; we leave that variant to future work. Algorithm 1 summarizes one observation step. Per-token state is two scalars ( a i a_{i} and t i ref t_{i}^{\\mathrm{ref}} ), updates occur only at observation steps, and eviction uses a min-heap over pages, so the policy adds O ( P log P ) O(P\\log P) work for P P pages at each observation step."},{"type":"heading","level":2,"text":"III-E Scope and Tiered Extension"},{"type":"para","text":"The decay in ( 4 ) acts on retention priority. Retained tokens participate in attention with unchanged weights, and eviction remains a discrete, irreversible step in the base policy. The decay schedule therefore controls the order in which tokens become eviction candidates; it does not scale their influence on the model output. A tiered extension would provide a graded degradation path: pages whose score falls below a first threshold are quantized to low precision [ 21 ] , pages below a second threshold are offloaded to host memory with background prefetch [ 24 ] , and pages below a final threshold are dropped. Mixed-precision retention keyed to token-level confidence has precedent in long-horizon settings [ 28 ] . The tiered variant would recover offloaded pages if later queries make them relevant again, at the cost of host memory and transfer bandwidth."},{"type":"heading","level":1,"text":"IV Evaluation"},{"type":"para","text":"This section reports a controlled study across two context lengths and two model sizes, executed end to end by the released harness: every number below comes from the raw result files at github.com/venkateshamatam/memdecay , and the study totals 4,320 scored eviction probes plus three full measurement passes. The findings are stated the way the evidence supports them. The region-lifetime measurement replicates at every scale and is the primary result. The eviction policy has one clear win, guaranteed instruction survival through pinning, and one clear loss, unpinned recall against attention-based retention, and the loss has a diagnosed mechanism whose fix the ablation narrows down."},{"type":"heading","level":2,"text":"IV-A Setup"},{"type":"para","text":"Episodes come in two tiers built from the same eight scenarios, each in three value variants with identical planted facts and probes. The short tier caches 405 to 507 tokens per episode (five stages); the long tier caches 1,656 to 1,723 tokens (eight stages, adding a second retrieved document, a third tool exchange, and a further user turn). In both tiers the movable blocks (tool exchanges, follow-up turns, retrieved documents) arrive in a different order in each variant, so region identity is decoupled from insertion position for the movable regions; the opening stage (system, plan, user) and the closing scratchpad seed keep fixed positions, which matches how orchestrators assemble contexts. Three settings are evaluated: Qwen2.5-1.5B-Instruct on the short tier, the same model on the long tier (both float32 on an Apple M3 Pro), and Qwen2.5-3B-Instruct on the long tier (float16 on a T4 GPU). All runs use eager attention, which is required to read attention weights [ 33 ] . A numerical audit of the measurement pipeline bounded the effect of half precision on fitted decay rates below 0.001% before the 3B run, and the measurement pipeline is deterministic across devices: the long tier’s per-episode straddle statistics reproduce exactly between the M3 Pro and the T4."},{"type":"para","text":"Each episode plants four facts with arbitrary values in the system , user , tool_out , and retrieval regions and pairs each with a probe question, answered by greedy decoding of 16 tokens after eviction. Recall requires the gold value to appear in the generated text at token boundaries, and each configuration cell aggregates 96 probes (24 episodes, 4 probes). The probes arrive after compression, so the setting is query-agnostic, which prior work identifies as substantially harder than compressing with the query in hand [ 30 ] . Statistics cluster at the scenario level: variants of one scenario are not independent, so confidence intervals come from a percentile bootstrap over the eight scenarios and are approximate, and the paired per-scenario sign comparison is the statistic the policy claims rest on. The random baseline runs three seeds, collapsed to one value per probe before statistics."},{"type":"para","text":"MemDecay uses the Table I base priorities, ρ = 0.9 \\rho=0.9 , α = 0.5 \\alpha=0.5 , τ = 2 / | 𝒦 t | \\tau=2/|\\mathcal{K}_{t}| , page size 16, and a pinned system region; pinning protects whole pages, so tokens sharing a page with system tokens are protected as well. Decay rates are calibrated leave-one-scenario-out within each setting, so no sibling variant leaks into its own calibration. Two calibration details matter for interpretation. The decay rates are fitted under a 48-step final observation burst, while the signals available at eviction time come from an 8-step burst; this asymmetry is identical across all policies and tiers. The count-weighted fit also anchors its intercept toward the observation-dense low-age buckets, again constant across conditions. Attention is sampled at every decode step ( k = 1 k=1 ); larger observation intervals are the production setting. Budgets are fractions of each episode’s cache, so the 25% condition sets B = ⌊ 0.25 n ⌋ B=\\lfloor 0.25\\,n\\rfloor ."},{"type":"heading","level":2,"text":"IV-B Measured Region Lifetimes Across Scales"},{"type":"para","text":"Let A ¯ r ( Δ t ) \\bar{A}_{r}(\\Delta t) denote the mean attention mass received by tokens of region r r at age Δ t \\Delta t , aggregated over episodes. The calibration fits the log-linear model"},{"type":"para","text":"by weighted least squares and sets λ r ← λ ^ r \\lambda_{r}\\leftarrow\\hat{\\lambda}_{r} ; the associated half-life"},{"type":"para","text":"summarizes how long region r r remains useful. Table II reports the fitted half-lives with scenario-bootstrap confidence intervals for all three settings, and Fig. 3 shows the measured curves for the largest one."},{"type":"para","text":"The region hierarchy is stable everywhere it was measured. system outlives scratchpad by an order of magnitude in every setting, with non-overlapping intervals, and becomes more persistent on the larger model (189 against 148 at matched context). scratchpad churns at a half-life of 14 to 16 steps regardless of model or context length. retrieval is the longest-lived movable region in every setting, even though it arrives at three different positions across a scenario’s variants, and it holds attention longer at the longer context; this again contradicts the fast-decay default that Table I assigns it, which is the calibration argument made concrete. The straddle statistic also behaved as Section III-D’s bound predicts: 35% of 16-token pages straddle a region boundary on the short tier and 17% on the long tier, since longer regions span more page-interior tokens."},{"type":"heading","level":2,"text":"IV-C Budget-Constrained Recall Across Scales"},{"type":"para","text":"Table III reports the short-tier recall and Table IV the long-tier recall for both models, after eviction to 25% and 50% of the cache, overall and excluding the 24 probes whose answers live in the pinned system region. Baselines are the unconstrained full cache, random (4 sinks, 16 most recent, uniform sample, three seeds), streaming (4 sinks plus the most recent tokens [ 6 ] ), and h2o , a reimplementation of accumulated-attention retention [ 5 ] restricted to the same observation signal as MemDecay. Baselines receive no region labels, and the reimplementations are not claims about the original methods at scale. MemDecay evicts whole pages, so it retained slightly fewer tokens than every baseline in all settings (for example 24.5% versus 25.0% on the long tier)."},{"type":"para","text":"Four findings organize the cross-scale picture."},{"type":"para","text":"First, the pinning guarantee holds everywhere. MemDecay answers 24 of 24 system probes on the short tier and 21 of 24 on the long tier at the half budget, at the full-cache ceiling (the models themselves miss some system probes at long context with nothing evicted), while streaming answers 0 to 2 of 24 in every setting. Guaranteed instruction survival under memory pressure is a designed property that no attention- or recency-based baseline provides."},{"type":"para","text":"Second, recency collapses with context length. Streaming is competitive on the short tier (0.36 non-system at the half budget, within the intervals of random at 0.37 and h2o at 0.40) and near zero on the long tier at both models (0.01 to 0.13), because its window physically cannot span the trace. MemDecay’s paired per-scenario comparison against streaming flips accordingly: 0 of 8 scenarios positive on the short tier, then + 4 / 0 +4/0 , + 5 / 1 +5/1 , + 3 / 0 +3/0 , and + 6 / 0 +6/0 across the four long-tier cells (mean deltas + 0.07 +0.07 to + 0.19 +0.19 ). Structural priors age gracefully where recency does not."},{"type":"para","text":"Third, accumulated attention strengthens with model scale. The h2o baseline is the weakest baseline on the short tier at the quarter budget, overtakes everything unpinned on the long tier at 1.5B, and dominates unpinned recall at 3B (0.60 non-system at the half budget, and 20 of 24 user probes), an external pattern consistent with published reports of H2O’s task-dependence, which range from collapse on retrieval tasks [ 29 ] to strength when its signal is informative."},{"type":"para","text":"Fourth, the mechanism behind MemDecay’s unpinned loss is the same at every scale: the oldest unpinned facts score lowest under a decay prior. The user-region fact sits in the opening turn, and at the half budget MemDecay recalls 0 of 24 (short), 7 of 24 (1.5B long), and 5 of 24 (3B long) of them while h2o recalls 11 to 20 of 24; attention identifies exactly the tokens that structure ranks for eviction. The ablation below shows that no tested setting of the attention weight recovers this."},{"type":"heading","level":2,"text":"IV-D Sensitivity and Overhead"},{"type":"para","text":"An ablation on the short tier varies the attention weight α ∈ { 0.2 , 0.5 , 0.8 } \\alpha\\in\\{0.2,0.5,0.8\\} and disables pinning, holding everything else fixed. The outcome is an invariance: all four configurations produce near-identical keep sets and near-identical probe outcomes (4 of 576 rows differ in kept count and one in recall, a single spurious flip that does not touch the user-region loss). Two scale facts explain it. The page mean divides any single token’s attention contribution by the page size, and the EWMA with ρ = 0.9 \\rho=0.9 has forgotten early attention by eviction time, so the α a i \\alpha\\,a_{i} term is one to two orders of magnitude below the structural term for exactly the old tokens that need rescue. Pinning also never binds: the calibrated system prior alone keeps its pages resident. The practical conclusion is that combining structural priors with attention requires magnitude normalization of the attention term (for example, relative to the uniform attention level), a design change rather than a tuning change; this is the concrete direction the recall results motivate."},{"type":"para","text":"Overhead was measured directly. The retention machinery itself is cheap: scoring plus page selection takes 0.27 ms and applying an eviction 4.2 ms on a 466-token cache. The cost is reading attention at all: with k = 1 k=1 , a scored decode step takes 69.8 ms against 26.0 ms unscored on the M3 Pro (median decode steps during full measurement: 82.7 ms at 1.5B on the long tier on M3 Pro, 52.2 ms at 3B on a T4). Production deployments would sample attention at intervals ( k > 1 k>1 ) or reuse scores produced for other purposes, and the eager-attention requirement this inherits from reading attention weights is shared with all attention-score methods [ 33 ] ."},{"type":"heading","level":1,"text":"V Discussion and Limitations"},{"type":"para","text":"Irreversibility. The base policy cannot recover an evicted token when later context makes it relevant again, and such saliency shifts are documented in multi-turn agent traces [ 25 ] ; the user-region result in Section IV-C , where the oldest unpinned facts are evicted and later probed, is a measured instance. The tiered extension of Section III-E would reduce this exposure by offloading before dropping."},{"type":"para","text":"Cross-region information flow. Agents copy content across regions, for example by quoting a tool output inside scratchpad reasoning. Under the default configuration both copies decay quickly, so the information can vanish from the cache even though the model produced and consumed it twice. Per-workload calibration reduces this risk; a mechanism that links copies to their sources is future work."},{"type":"para","text":"Label quality. MemDecay assumes region labels from the orchestrator. The serving-side metadata channel that would carry such labels is under active development [ 32 ] , but workloads without a cooperating orchestrator would need a classifier to assign labels, and robustness to noisy labels is unmeasured."},{"type":"para","text":"Granularity. MemDecay applies one score per token, uniform across heads and layers. Head-adaptive budget allocation is orthogonal and could be composed with region-aware scoring [ 3 , 4 ] ."},{"type":"para","text":"Workload dependence. Region lifetimes are properties of workloads, and published evidence disagrees across settings [ 10 , 26 ] . Our own measurement reproduces this in miniature: the slow retrieval decay in Section IV-B overturns the fast-decay default for that region. MemDecay treats the region configuration as measurable, which requires calibration data representative of the deployment workload."},{"type":"para","text":"Scale and validity. The study covers 1.5B and 3B models at roughly 450 and 1,700-token contexts, on heterogeneous hardware (Apple M3 Pro in float32, T4 GPU in float16, with the precision effect on fitted rates verified below 0.001% and the measurement statistics reproducing exactly across devices). The 96 probes per cell derive from eight independent scenarios, so all intervals are cluster bootstraps over eight clusters and are approximate; the paired per-scenario sign comparison carries the policy claims. Interleaved schedules control insertion order for the movable regions, while system, plan, and scratchpad keep fixed positions, so their lifetime estimates remain position-confounded by construction. Contexts of 1,700 tokens still sit well below production long-context regimes, and conclusions about those regimes remain conditional on larger runs."},{"type":"para","text":"Future work. Three directions follow directly from the results. First, from static calibration to prediction: the ablation shows no weight on the current EWMA rescues old facts, so the attention term needs magnitude normalization (for example, relative to the uniform attention level) and a slower usage memory; the same machinery extends to decay rates that adapt online from observed refresh frequency per region, and to a lightweight future-reuse score built from age, region, phase, and attention trend, all still training-free. Second, scale: 8B-class models over contexts of 4k tokens and beyond, on real agent workloads (multi-turn coding and function-calling traces of the SWE-bench and BFCL kind [ 23 ] , browsing traces [ 25 ] , and LongBench regression against compression baselines [ 20 ] ), with sweeps of the refresh threshold τ \\tau and observation interval k k . Third, structure beyond flat regions: dependency-aware eviction that links a plan to the tool outputs and evidence it was built from and retains or discards them jointly, cross-region refresh when a copy of a token’s content is used, region-and-phase conditioning so the same region can carry different lifetimes across agent phases, head- and layer-adaptive priorities [ 4 , 3 ] , region labels inferred by a lightweight classifier when no orchestrator metadata exists, and the tiered degradation path of Section III-E ."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Large language models (LLMs) are increasingly used in long-horizon agentic workflows that combine multi-turn interaction, retrieval, tool use, and iterative planning. Unlike single-pass long-context tasks, these systems accumulate a persistent execution trace composed of system instructions, safety constraints, evolving plans, user turns, retrieved evidence, tool interactions, and intermediate reasoning states. The trace is structured: its components serve different functional roles and should persist for different durations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This shift places new pressure on the key–value (KV) cache that supports autoregressive decoding. Because the KV cache grows with sequence length, long interaction traces translate into substantial memory overhead, increased bandwidth demand, and higher end-to-end latency [ 1 , 2 ] . Prior work reduces this overhead through selective retention and compression, including policies that adapt eviction to stable differences in attention behavior across heads [ 3 , 4 ] . Most of these approaches target generic long-context generation and score cached tokens with model-centric signals such as accumulated attention, attention sinks and recency windows, or instantaneous attention [ 5 , 6 , 7 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"That assumption is poorly matched to agentic workloads. In LLM-based agents, memory is a core component that supports long-term and complex interactions with the environment [ 8 ] . Yet the contents of an agent trace are heterogeneous by design. System and safety instructions often need to remain stable throughout an episode, and high-level plans may need to persist until the agent replans. By contrast, stale tool outputs, transient scratchpad content, and low-value retrievals are often useful only for a short span of the trajectory. A cache policy that treats all such tokens as having the same expected lifetime misses this structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This mismatch motivates aligning cache retention with the semantic structure already present in orchestrated prompts. We propose MemDecay , a training-free, prompt-region-aware KV-cache decay and eviction framework for long-horizon LLM agents. MemDecay assumes that the orchestration layer provides region labels for tokens, such as system prompt, plan, user message, tool input, tool output, retrieval, or scratchpad. It assigns each region a base retention priority and a decay rate, then combines these structural priors with attention-derived importance to compute a composite retention score for each token. MemDecay is training-free in the sense this literature uses the term: it updates no model weights and learns no predictor. It does rely on a lightweight statistical calibration that fits two parameters per region to measured traces, described in Section IV . Concurrent work gives evidence that semantic roles predict reuse and attention lifetimes in agent workloads [ 9 , 10 ] ; MemDecay applies this signal to token-scored, page-granular eviction under a fixed budget within a single decoding context.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MemDecay treats forgetting as a scheduled process. Region-specific temporal decay lowers a token’s retention priority smoothly over time, and attention to the token resets its decay clock, so continued use keeps a token alive. When a fixed KV-cache budget is exceeded, tokens with the lowest decayed priorities are evicted first, while pinned instruction regions and, optionally, active plan segments are preserved. The resulting policy is interpretable, requires no retraining or auxiliary modules, and operates on page-granular cache layouts of the kind used in existing serving stacks [ 1 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions are threefold:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We formulate region-aware selective forgetting as a KV-cache memory-management problem for long-horizon LLM agents, grounded in the structure of orchestrated prompts and agent traces.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce MemDecay , a training-free, prompt-region-aware eviction policy that combines region base priorities, region-specific temporal decay with attention-triggered refresh, and attention-derived importance in a single interpretable retention score under a fixed KV-cache budget.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We specify a measurement protocol that estimates region-conditioned attention lifetimes from agent traces and uses the fitted decay rates to calibrate the policy, and we execute it across two context lengths and two model sizes (4,320 scored eviction probes): the region-lifetime hierarchy replicates in every setting, with the longest- and shortest-lived regions separated by an order of magnitude at non-overlapping confidence intervals, the fitted rates overturn one qualitative default, pinning delivers guaranteed instruction survival everywhere, recency-based retention collapses with context length while the structural prior keeps working, and an invariance ablation points to magnitude normalization of the attention term as the needed design change for the policy’s unpinned-recall loss.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A KV-Cache Memory Management for LLM Serving\"\n },\n {\n \"type\": \"para\",\n \"text\": \"KV-cache management has become a first-order systems problem for scalable LLM serving because cache footprint grows with context length and directly constrains memory capacity, bandwidth, and throughput [ 1 , 2 ] . System-level approaches focus on how cached states are allocated, retained, and reused across requests. PagedAttention organizes the cache into fixed-size pages to reduce fragmentation and enable sharing [ 1 ] , and prefix-aware designs detect and reuse shared prompt segments such as system instructions and retrieved passages [ 11 , 12 ] . Studies of production-style prefix caches report that reuse value varies widely across prompt segments, which motivates retention decisions that account for what a segment contains [ 9 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B KV-Cache Eviction and Compression for Long-Context LLMs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A large literature studies post-training KV reduction through eviction, pruning, quantization, and hybrid schemes for long-context inference [ 2 , 13 ] . Attention-based eviction retains tokens with high accumulated attention [ 5 , 14 ] , window-based policies keep attention sinks and recent tokens [ 6 , 7 ] , and prompt-time compression selects tokens using attention from an observation window [ 15 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Later work showed that locally accumulated attention statistics are biased and can misestimate future utility [ 16 , 17 ] . Several methods reallocate budgets across heads or layers, reflecting evidence that attention modules have heterogeneous roles and sensitivities [ 3 , 4 ] . Others score tokens with query-aware or semantic-cluster criteria [ 18 , 19 ] , keep or drop contiguous semantic chunks instead of isolated tokens [ 20 ] , or lower the precision of cached entries instead of removing them [ 21 ] . Learned approaches predict token importance directly, at the cost of training and auxiliary modules [ 22 , 9 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Agent- and Workflow-Aware KV Management\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent work shows that agentic workloads differ materially from static chat or document settings because they interleave repeated LLM calls, tools, and variable pauses, creating reuse patterns that generic policies do not anticipate [ 23 ] . Workflow-aware systems model future execution structure explicitly. KVFlow prioritizes retention using estimated temporal proximity to future agent activation, and Continuum uses time-to-live pinning to preserve KV states across short tool calls when recomputation would be costly [ 24 , 23 ] . Prefix-cache studies report that segment semantics matter in practice: token-agnostic policies such as LRU miss up to 756 × \\\\times variation in reuse rates across system prompts, user queries, tool outputs, model responses, and reasoning traces [ 9 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Concurrent work brings role and time structure into cache policies themselves. RoleKV tags KV blocks with semantic roles and applies role-calibrated decay to serving-layer retention, reporting that the oldest blocks in agent traces, including system prompts and tool outputs, receive the most persistent attention while recent reasoning tokens decay fastest [ 10 ] . IntentKV scores history tokens against an exponentially decayed summary of session queries for multi-turn agent pruning [ 25 ] , and retention-gated caching learns per-token gates that decay token importance over time [ 26 ] . These results establish that semantic roles and temporal decay both carry signal for cache management, and they disagree about which regions of an agent trace stay useful, a question we return to in Section IV .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-D Memory and Forgetting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Memory is a defining component of LLM-based agents because long-horizon interaction depends on retaining, updating, and selectively reusing prior state [ 8 , 27 ] . Within KV-cache research, several recent papers argue that irreversible keep-or-drop policies are too coarse: token value can shift as decoding proceeds, and a token evicted early cannot be recovered when later queries make it relevant again [ 25 , 28 ] . Mixed-precision storage offers one response, degrading uncertain entries before removing them [ 28 , 21 ] . This body of work motivates treating forgetting as a controllable mechanism with an explicit schedule instead of a byproduct of memory pressure. Class-based replacement with aging has a long history in web caching, where Greedy-Dual-Size-Frequency combines per-class value with an aging term [ 31 ] ; MemDecay adapts this lineage to KV caches, with prompt regions as classes and decay rates calibrated per region.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-E Positioning MemDecay\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MemDecay is closest to the concurrent role- and time-aware policies above, and it differs from each on a specific axis. RoleKV manages which blocks stay resident for cross-request reuse and is lossless with respect to model outputs [ 10 ] ; MemDecay performs lossy, page-granular eviction driven by token-level scores inside a single decoding context under a fixed budget. SAECache learns segment reuse value online for prefix caches [ 9 ] , and IntentKV in its stronger variant trains a residual scoring head [ 25 ] ; MemDecay is training-free and uses only signals available at inference time. Retention-gated caching learns its decay behavior through trained per-token gates [ 26 ] ; MemDecay is training-free, conditions both the base priority and the decay rate on the prompt region, and lets attention reset the decay clock. Workflow-level systems such as KVFlow and Continuum schedule caches across agent steps and are complementary: MemDecay decides which tokens survive within a context, and those systems decide when whole caches are kept, offloaded, or dropped [ 24 , 23 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Proposed Method\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MemDecay is an inference-time eviction policy for the KV cache of a decoder-only LLM running inside an agent orchestrator. The policy uses three inputs: region labels supplied by the orchestrator, attention statistics sampled during decoding, and a fixed cache budget. Fig. 1 shows where the policy sits in the serving stack. This section defines the setting, the structural priors, the retention score, and the eviction procedure.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A Problem Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An agent episode produces a token sequence indexed by i i , decoded over steps t = 1 , 2 , … t=1,2,\\\\dots . Each cached token i i carries a region label r ( i ) ∈ ℛ r(i)\\\\in\\\\mathcal{R} and an insertion step t i ins t_{i}^{\\\\mathrm{ins}} , where ℛ \\\\mathcal{R} contains the labels system , plan , user , tool_in , tool_out , retrieval , and scratchpad . Region labels correspond to contiguous spans of the prompt because the orchestrator assembles the context from template segments. We assume the orchestration layer emits these labels alongside the request; serving systems have begun to accept exactly this kind of metadata, and an open SGLang proposal attaches workflow, tool, and reuse hints to inference requests [ 32 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝒦 t \\\\mathcal{K}_{t} denote the set of cached tokens at step t t and let B B be the cache budget in tokens. A subset ℱ ⊆ 𝒦 t \\\\mathcal{F}\\\\subseteq\\\\mathcal{K}_{t} of pinned tokens is never evicted; by default ℱ \\\\mathcal{F} contains all system tokens, which also covers the initial attention-sink positions that must remain resident for stable decoding [ 6 ] . The policy must maintain | 𝒦 t | ≤ B |\\\\mathcal{K}_{t}|\\\\leq B by selecting which non-pinned tokens to evict. Formally, at any step where the budget binds, the policy solves\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where s i ( t ) s_{i}(t) is the retention score defined in Section III-C . The objective is additive over tokens, so retaining the B B highest-scoring tokens, after forcing ℱ \\\\mathcal{F} into the set, is optimal, and eviction reduces to removing the lowest-scoring tokens first.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B Structural Priors\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each region r r is configured with a base priority b r ≥ 0 b_{r}\\\\geq 0 and a decay rate λ r ≥ 0 \\\\lambda_{r}\\\\geq 0 . The base priority encodes how valuable a token from region r r is at insertion time, and the decay rate encodes how quickly that structural protection expires. Table I lists the default configuration. These defaults encode the template lifetimes described in Section I. They are starting points: published evidence on region lifetimes conflicts, with role-aware serving measurements reporting persistent attention to old system prompts and tool outputs [ 10 ] while decay-based retention assumes that older tokens generally matter less [ 26 ] . Section IV describes a calibration procedure that replaces these defaults with rates fitted per workload.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C Retention Score with Region-Specific Decay\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MemDecay tracks one dynamic signal per token, an attention importance a i a_{i} , and one clock per token, the time since the token was last used. Fused attention kernels do not materialize the attention matrix, so reading per-step attention scores is impractical in production kernels [ 33 ] . MemDecay therefore samples attention only at observation steps, one every k k decode steps, using windowed scores in the style of prompt-time selection methods [ 15 ] . At an observation step, let A ¯ i \\\\bar{A}_{i} denote the attention mass received by token i i , averaged over heads, layers, and the window. The importance is an exponentially weighted average,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with smoothing factor ρ ∈ [ 0 , 1 ) \\\\rho\\\\in[0,1) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The decay clock measures time since insertion or since the model last attended to the token above a threshold τ \\\\tau . Let\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where t ′ t^{\\\\prime} is the most recent observation step satisfying the condition. A refresh resets the clock and nothing else; the structural prior is re-applied from the refresh point. The threshold τ \\\\tau can be fixed or scaled to the uniform attention level 1 / | 𝒦 t | 1/|\\\\mathcal{K}_{t}| so that the refresh criterion tracks context length; Section IV uses the scaled form.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The retention score combines the decayed structural prior with the observed importance:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where α ≥ 0 \\\\alpha\\\\geq 0 weights how much observed attention can protect a token beyond its structural prior. Recency enters only through the decay clock, so the score contains no separate recency term and elapsed time is not counted twice. The three terms have direct interpretations: b r b_{r} sets where a token starts, λ r \\\\lambda_{r} sets how quickly structure alone stops protecting it, and α a i \\\\alpha a_{i} keeps any token alive while the model keeps using it, regardless of region. Pinned tokens bypass scoring entirely. Fig. 2 illustrates the resulting score trajectories, including a refresh event that resets the decay clock of a short-lived region.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-D Budget-Aware Eviction at Page Granularity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Production serving stacks store KV entries in fixed-size pages [ 1 ] . Region labels mark contiguous spans, so at most one page can straddle each boundary between consecutive region runs, and the straddle fraction shrinks as regions grow long relative to the page size. In the episodes of Section IV , 35% of 16-token pages straddle a boundary on the short tier and 17% on the long tier, matching the prediction that straddling shrinks as regions lengthen. The page score is the mean over member tokens,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MemDecay evicts at page granularity: when | 𝒦 t | > B |\\\\mathcal{K}_{t}|>B , non-pinned pages are ordered by ascending S p ( t ) S_{p}(t) and evicted until the budget is met. This greedy rule is optimal for the page-level objective because pages have equal size; it can deviate from the token-level optimum of ( 1 ) when a page mixes regions, since the mean in ( 5 ) can mask a high-scoring token among low-scoring pagemates. The straddle fraction above quantifies how often mixed pages occur, and the pinned-page collateral disclosed in Section IV-A is the same effect acting in the protective direction. Region-aligned page boundaries, where the allocator opens a fresh page at each region boundary, would remove the dilution at the cost of internal fragmentation; we leave that variant to future work. Algorithm 1 summarizes one observation step. Per-token state is two scalars ( a i a_{i} and t i ref t_{i}^{\\\\mathrm{ref}} ), updates occur only at observation steps, and eviction uses a min-heap over pages, so the policy adds O ( P log P ) O(P\\\\log P) work for P P pages at each observation step.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-E Scope and Tiered Extension\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The decay in ( 4 ) acts on retention priority. Retained tokens participate in attention with unchanged weights, and eviction remains a discrete, irreversible step in the base policy. The decay schedule therefore controls the order in which tokens become eviction candidates; it does not scale their influence on the model output. A tiered extension would provide a graded degradation path: pages whose score falls below a first threshold are quantized to low precision [ 21 ] , pages below a second threshold are offloaded to host memory with background prefetch [ 24 ] , and pages below a final threshold are dropped. Mixed-precision retention keyed to token-level confidence has precedent in long-horizon settings [ 28 ] . The tiered variant would recover offloaded pages if later queries make them relevant again, at the cost of host memory and transfer bandwidth.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reports a controlled study across two context lengths and two model sizes, executed end to end by the released harness: every number below comes from the raw result files at github.com/venkateshamatam/memdecay , and the study totals 4,320 scored eviction probes plus three full measurement passes. The findings are stated the way the evidence supports them. The region-lifetime measurement replicates at every scale and is the primary result. The eviction policy has one clear win, guaranteed instruction survival through pinning, and one clear loss, unpinned recall against attention-based retention, and the loss has a diagnosed mechanism whose fix the ablation narrows down.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Episodes come in two tiers built from the same eight scenarios, each in three value variants with identical planted facts and probes. The short tier caches 405 to 507 tokens per episode (five stages); the long tier caches 1,656 to 1,723 tokens (eight stages, adding a second retrieved document, a third tool exchange, and a further user turn). In both tiers the movable blocks (tool exchanges, follow-up turns, retrieved documents) arrive in a different order in each variant, so region identity is decoupled from insertion position for the movable regions; the opening stage (system, plan, user) and the closing scratchpad seed keep fixed positions, which matches how orchestrators assemble contexts. Three settings are evaluated: Qwen2.5-1.5B-Instruct on the short tier, the same model on the long tier (both float32 on an Apple M3 Pro), and Qwen2.5-3B-Instruct on the long tier (float16 on a T4 GPU). All runs use eager attention, which is required to read attention weights [ 33 ] . A numerical audit of the measurement pipeline bounded the effect of half precision on fitted decay rates below 0.001% before the 3B run, and the measurement pipeline is deterministic across devices: the long tier’s per-episode straddle statistics reproduce exactly between the M3 Pro and the T4.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each episode plants four facts with arbitrary values in the system , user , tool_out , and retrieval regions and pairs each with a probe question, answered by greedy decoding of 16 tokens after eviction. Recall requires the gold value to appear in the generated text at token boundaries, and each configuration cell aggregates 96 probes (24 episodes, 4 probes). The probes arrive after compression, so the setting is query-agnostic, which prior work identifies as substantially harder than compressing with the query in hand [ 30 ] . Statistics cluster at the scenario level: variants of one scenario are not independent, so confidence intervals come from a percentile bootstrap over the eight scenarios and are approximate, and the paired per-scenario sign comparison is the statistic the policy claims rest on. The random baseline runs three seeds, collapsed to one value per probe before statistics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MemDecay uses the Table I base priorities, ρ = 0.9 \\\\rho=0.9 , α = 0.5 \\\\alpha=0.5 , τ = 2 / | 𝒦 t | \\\\tau=2/|\\\\mathcal{K}_{t}| , page size 16, and a pinned system region; pinning protects whole pages, so tokens sharing a page with system tokens are protected as well. Decay rates are calibrated leave-one-scenario-out within each setting, so no sibling variant leaks into its own calibration. Two calibration details matter for interpretation. The decay rates are fitted under a 48-step final observation burst, while the signals available at eviction time come from an 8-step burst; this asymmetry is identical across all policies and tiers. The count-weighted fit also anchors its intercept toward the observation-dense low-age buckets, again constant across conditions. Attention is sampled at every decode step ( k = 1 k=1 ); larger observation intervals are the production setting. Budgets are fractions of each episode’s cache, so the 25% condition sets B = ⌊ 0.25 n ⌋ B=\\\\lfloor 0.25\\\\,n\\\\rfloor .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-B Measured Region Lifetimes Across Scales\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let A ¯ r ( Δ t ) \\\\bar{A}_{r}(\\\\Delta t) denote the mean attention mass received by tokens of region r r at age Δ t \\\\Delta t , aggregated over episodes. The calibration fits the log-linear model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"by weighted least squares and sets λ r ← λ ^ r \\\\lambda_{r}\\\\leftarrow\\\\hat{\\\\lambda}_{r} ; the associated half-life\"\n },\n {\n \"type\": \"para\",\n \"text\": \"summarizes how long region r r remains useful. Table II reports the fitted half-lives with scenario-bootstrap confidence intervals for all three settings, and Fig. 3 shows the measured curves for the largest one.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The region hierarchy is stable everywhere it was measured. system outlives scratchpad by an order of magnitude in every setting, with non-overlapping intervals, and becomes more persistent on the larger model (189 against 148 at matched context). scratchpad churns at a half-life of 14 to 16 steps regardless of model or context length. retrieval is the longest-lived movable region in every setting, even though it arrives at three different positions across a scenario’s variants, and it holds attention longer at the longer context; this again contradicts the fast-decay default that Table I assigns it, which is the calibration argument made concrete. The straddle statistic also behaved as Section III-D’s bound predicts: 35% of 16-token pages straddle a region boundary on the short tier and 17% on the long tier, since longer regions span more page-interior tokens.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-C Budget-Constrained Recall Across Scales\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table III reports the short-tier recall and Table IV the long-tier recall for both models, after eviction to 25% and 50% of the cache, overall and excluding the 24 probes whose answers live in the pinned system region. Baselines are the unconstrained full cache, random (4 sinks, 16 most recent, uniform sample, three seeds), streaming (4 sinks plus the most recent tokens [ 6 ] ), and h2o , a reimplementation of accumulated-attention retention [ 5 ] restricted to the same observation signal as MemDecay. Baselines receive no region labels, and the reimplementations are not claims about the original methods at scale. MemDecay evicts whole pages, so it retained slightly fewer tokens than every baseline in all settings (for example 24.5% versus 25.0% on the long tier).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Four findings organize the cross-scale picture.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, the pinning guarantee holds everywhere. MemDecay answers 24 of 24 system probes on the short tier and 21 of 24 on the long tier at the half budget, at the full-cache ceiling (the models themselves miss some system probes at long context with nothing evicted), while streaming answers 0 to 2 of 24 in every setting. Guaranteed instruction survival under memory pressure is a designed property that no attention- or recency-based baseline provides.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, recency collapses with context length. Streaming is competitive on the short tier (0.36 non-system at the half budget, within the intervals of random at 0.37 and h2o at 0.40) and near zero on the long tier at both models (0.01 to 0.13), because its window physically cannot span the trace. MemDecay’s paired per-scenario comparison against streaming flips accordingly: 0 of 8 scenarios positive on the short tier, then + 4 / 0 +4/0 , + 5 / 1 +5/1 , + 3 / 0 +3/0 , and + 6 / 0 +6/0 across the four long-tier cells (mean deltas + 0.07 +0.07 to + 0.19 +0.19 ). Structural priors age gracefully where recency does not.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Third, accumulated attention strengthens with model scale. The h2o baseline is the weakest baseline on the short tier at the quarter budget, overtakes everything unpinned on the long tier at 1.5B, and dominates unpinned recall at 3B (0.60 non-system at the half budget, and 20 of 24 user probes), an external pattern consistent with published reports of H2O’s task-dependence, which range from collapse on retrieval tasks [ 29 ] to strength when its signal is informative.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fourth, the mechanism behind MemDecay’s unpinned loss is the same at every scale: the oldest unpinned facts score lowest under a decay prior. The user-region fact sits in the opening turn, and at the half budget MemDecay recalls 0 of 24 (short), 7 of 24 (1.5B long), and 5 of 24 (3B long) of them while h2o recalls 11 to 20 of 24; attention identifies exactly the tokens that structure ranks for eviction. The ablation below shows that no tested setting of the attention weight recovers this.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-D Sensitivity and Overhead\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An ablation on the short tier varies the attention weight α ∈ { 0.2 , 0.5 , 0.8 } \\\\alpha\\\\in\\\\{0.2,0.5,0.8\\\\} and disables pinning, holding everything else fixed. The outcome is an invariance: all four configurations produce near-identical keep sets and near-identical probe outcomes (4 of 576 rows differ in kept count and one in recall, a single spurious flip that does not touch the user-region loss). Two scale facts explain it. The page mean divides any single token’s attention contribution by the page size, and the EWMA with ρ = 0.9 \\\\rho=0.9 has forgotten early attention by eviction time, so the α a i \\\\alpha\\\\,a_{i} term is one to two orders of magnitude below the structural term for exactly the old tokens that need rescue. Pinning also never binds: the calibrated system prior alone keeps its pages resident. The practical conclusion is that combining structural priors with attention requires magnitude normalization of the attention term (for example, relative to the uniform attention level), a design change rather than a tuning change; this is the concrete direction the recall results motivate.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overhead was measured directly. The retention machinery itself is cheap: scoring plus page selection takes 0.27 ms and applying an eviction 4.2 ms on a 466-token cache. The cost is reading attention at all: with k = 1 k=1 , a scored decode step takes 69.8 ms against 26.0 ms unscored on the M3 Pro (median decode steps during full measurement: 82.7 ms at 1.5B on the long tier on M3 Pro, 52.2 ms at 3B on a T4). Production deployments would sample attention at intervals ( k > 1 k>1 ) or reuse scores produced for other purposes, and the eager-attention requirement this inherits from reading attention weights is shared with all attention-score methods [ 33 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Discussion and Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Irreversibility. The base policy cannot recover an evicted token when later context makes it relevant again, and such saliency shifts are documented in multi-turn agent traces [ 25 ] ; the user-region result in Section IV-C , where the oldest unpinned facts are evicted and later probed, is a measured instance. The tiered extension of Section III-E would reduce this exposure by offloading before dropping.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cross-region information flow. Agents copy content across regions, for example by quoting a tool output inside scratchpad reasoning. Under the default configuration both copies decay quickly, so the information can vanish from the cache even though the model produced and consumed it twice. Per-workload calibration reduces this risk; a mechanism that links copies to their sources is future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Label quality. MemDecay assumes region labels from the orchestrator. The serving-side metadata channel that would carry such labels is under active development [ 32 ] , but workloads without a cooperating orchestrator would need a classifier to assign labels, and robustness to noisy labels is unmeasured.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Granularity. MemDecay applies one score per token, uniform across heads and layers. Head-adaptive budget allocation is orthogonal and could be composed with region-aware scoring [ 3 , 4 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Workload dependence. Region lifetimes are properties of workloads, and published evidence disagrees across settings [ 10 , 26 ] . Our own measurement reproduces this in miniature: the slow retrieval decay in Section IV-B overturns the fast-decay default for that region. MemDecay treats the region configuration as measurable, which requires calibration data representative of the deployment workload.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Scale and validity. The study covers 1.5B and 3B models at roughly 450 and 1,700-token contexts, on heterogeneous hardware (Apple M3 Pro in float32, T4 GPU in float16, with the precision effect on fitted rates verified below 0.001% and the measurement statistics reproducing exactly across devices). The 96 probes per cell derive from eight independent scenarios, so all intervals are cluster bootstraps over eight clusters and are approximate; the paired per-scenario sign comparison carries the policy claims. Interleaved schedules control insertion order for the movable regions, while system, plan, and scratchpad keep fixed positions, so their lifetime estimates remain position-confounded by construction. Contexts of 1,700 tokens still sit well below production long-context regimes, and conclusions about those regimes remain conditional on larger runs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Future work. Three directions follow directly from the results. First, from static calibration to prediction: the ablation shows no weight on the current EWMA rescues old facts, so the attention term needs magnitude normalization (for example, relative to the uniform attention level) and a slower usage memory; the same machinery extends to decay rates that adapt online from observed refresh frequency per region, and to a lightweight future-reuse score built from age, region, phase, and attention trend, all still training-free. Second, scale: 8B-class models over contexts of 4k tokens and beyond, on real agent workloads (multi-turn coding and function-calling traces of the SWE-bench and BFCL kind [ 23 ] , browsing traces [ 25 ] , and LongBench regression against compression baselines [ 20 ] ), with sweeps of the refresh threshold τ \\\\tau and observation interval k k . Third, structure beyond flat regions: dependency-aware eviction that links a plan to the tool outputs and evidence it was built from and retains or discards them jointly, cross-region refresh when a copy of a token’s content is used, region-and-phase conditioning so the same region can carry different lifetimes across agent phases, head- and layer-adaptive priorities [ 4 , 3 ] , region labels inferred by a lightweight classifier when no orchestrator metadata exists, and the tiered degradation path of Section III-E .\"\n }\n]","isConversational":false}}},{"rowIdx":389,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10571"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Learning from Local Walks on Dynamic Graphs with Bandit Feedback"},"authors":{"kind":"list like","value":["Sourav Chakraborty","Amit Kiran Rege","Claire Monteleoni","Lijun Chen"],"string":"[\n \"Sourav Chakraborty\",\n \"Amit Kiran Rege\",\n \"Claire Monteleoni\",\n \"Lijun Chen\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10571v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"In the stochastic multi-armed bandit (MAB) problem, a learner repeatedly chooses among several actions (interchangeably referred to as arms ) observes only the reward of the selected action, and seeks to maximize its cumulative reward ( [ 2 , 9 , 16 ] ). Classical bandit models typically assume that every arm is available for selection at every round. This assumption, however, is too strong for many networked settings where movement is restricted and inherently local. For instance, a mobile data collector in a sensor network is constrained by physical proximity, requiring it to be near a specific node to harvest its data [ 10 ] ; a search routine on a peer-to-peer overlay is confined to traversing logical links between neighboring peers [ 8 ] ; and a patrol agent is restricted to navigable pathways that may open or close as the environment changes [ 4 ] . In these scenarios, the learner cannot jump to an arbitrary location; instead, its next action is strictly determined by its current position and the connectivity available at that moment."},{"type":"para","text":"This motivates the dynamic graph bandit problem, where arms correspond to vertices in a network with time-varying edges. The learner observes only its local neighborhood, moves along locally available edges, and receives feedback solely from visited nodes. Consequently, the evolving topology governs both immediate reward availability and future reachability. This fundamentally differs from feedback-graph bandits, where graphs encode side-observations rather than movement constraints ( [ 11 , 1 ] ), and from static-graph bandits, where fixed, globally known topologies permit pre-planned navigation ( [ 18 , 13 ] )."},{"type":"para","text":"A core challenge in this setting is that best-arm identification is decoupled from exploitation. In classical bandits, identifying the optimal arm immediately enables perpetual exploitation. Here, a learner might identify the best arm but remain temporarily unable to reach it. Learnability thus requires the environment to consistently provide navigable pathways; otherwise, the learner could be structurally isolated indefinitely. Importantly, standard full-horizon conditions, such as the connectivity of the union of all graphs, are insufficient, as a network might be connected in aggregate but disconnected at almost every individual step. A valid structural condition must therefore be shift-invariant , holding from any arbitrary round. This guarantees navigability during the post-exploration phase, regardless of the random stopping time at which exploration concludes. We define a condition that meets these criteria (see Section II-B ), avoiding the pitfalls of whole-horizon summaries."},{"type":"para","text":"To isolate the distinct challenges of information gathering and subsequent movement, we adopt the explore-then-commit framework. Our analysis centers on the canonical walk (see ( 1 )), where the learner either remains at its current node or moves uniformly to an available neighbor. We prioritize this simple local kernel because it is entirely decentralized and allows us to address a fundamental question: when is a graph’s intrinsic walk sufficient for learning? While more complex kernels can be useful, the canonical walk serves as a transparent analytical probe that exposes the structural constraints of the environment before asking whether reward-biased movement yields further gains. This choice makes the dichotomy between statistical identification and navigation explicit in both algorithms and proofs."},{"type":"para","text":"To formalize shift-invariant navigability, we impose a condition directly on the canonical walk. Intuitively, we require the walk to maintain a consistent long-run stationary distribution, and we require that every short time window contains enough “well-connected” rounds to prevent the learner from becoming trapped in a local subgraph. We formalize this as a common-stationary sliding-window mixing property. By ensuring a stable stationary distribution, we align with positive regimes in the dynamic random-walk literature, where such walks mimic their static counterparts ( [ 3 , 12 , 14 , 15 ] ). This encompasses natural fixed-degree processes, such as rotating regular overlays and degree-preserving rewirings."},{"type":"para","text":"Within this regime, we analyze a family of local explore-then-commit algorithms that utilize walk-based exploration with both fixed and adaptive stopping rules (see Section II ). Our analysis extends to reward-aware movement, where we formally decouple worst-case safety from provable performance gains. Unlike model-specific analyses [ 5 ] , we establish process-agnostic structural conditions (see Section III ) that ensure learnability for any such walk-based approach. Due to space constraints, extended related work, technical derivations, and complete proofs are deferred to the appendix."},{"type":"heading","level":1,"text":"II Model and Problem Formulation"},{"type":"para","text":"This section defines the learning problem, fixes the information pattern, and introduces the structural condition used throughout the paper."},{"type":"heading","level":2,"text":"II-A Bandit learning on dynamic graphs"},{"type":"para","text":"Let A = { 1 , … , n } A=\\{1,\\dots,n\\} be a finite set of arms, where each a ∈ A a\\in A is associated with a reward distribution 𝒟 ( a ) \\mathcal{D}(a) supported on [ 0 , 1 ] [0,1] with mean μ ( a ) \\mu(a) . Rewards are i.i.d. across time and independent of the graph process. We assume a unique optimal arm a ⋆ ≜ arg max a ∈ A μ ( a ) a^{\\star}\\triangleq\\arg\\max_{a\\in A}\\mu(a) and define the suboptimality gaps as Δ ( a ) ≜ μ ( a ⋆ ) − μ ( a ) \\Delta(a)\\triangleq\\mu(a^{\\star})-\\mu(a) for a ≠ a ⋆ a\\neq a^{\\star} . At each discrete round t ≥ 1 t\\geq 1 , the environment presents the learner an undirected graph G t = ( A , E t ) G_{t}=(A,E_{t}) . The learner begins at node a t − 1 a_{t-1} and observes only the local neighborhood L t ( a t − 1 ) ≜ N t ( a t − 1 ) ∪ { a t − 1 } L_{t}(a_{t-1})\\triangleq N_{t}(a_{t-1})\\cup\\{a_{t-1}\\} , where N t ( u ) ≜ { v ∈ A : ( u , v ) ∈ E t } N_{t}(u)\\triangleq\\{v\\in A:(u,v)\\in E_{t}\\} . The graph process is oblivious: the sequence { G t } t ≥ 1 \\{G_{t}\\}_{t\\geq 1} is independent of the learner’s past actions and randomization. The learner chooses a t ∈ L t ( a t − 1 ) a_{t}\\in L_{t}(a_{t-1}) , moves to that node, and receives reward r t ∼ 𝒟 ( a t ) r_{t}\\sim\\mathcal{D}(a_{t}) . We measure performance by the regret R ( T ) ≜ ∑ t = 1 T ( μ ( a ⋆ ) − r t ) R(T)\\triangleq\\sum_{t=1}^{T}(\\mu(a^{\\star})-r_{t}) , whose expectation matches the standard pseudo-regret 𝔼 [ R ( T ) ] = ∑ t = 1 T 𝔼 [ μ ( a ⋆ ) − μ ( a t ) ] \\mathbb{E}[R(T)]=\\sum_{t=1}^{T}\\mathbb{E}[\\mu(a^{\\star})-\\mu(a_{t})] . Our objective is to design a local policy that is learnable, meaning it achieves sublinear expected regret: lim T → ∞ 𝔼 [ R ( T ) ] / T = 0 \\lim_{T\\to\\infty}\\mathbb{E}[R(T)]/T=0 . While the full graphs { G t } \\{G_{t}\\} are used for analysis, the policy remains strictly local, seeing only the current neighborhood L t ( a t − 1 ) L_{t}(a_{t-1}) ."},{"type":"heading","level":2,"text":"II-B Why the condition must be stated on the walk"},{"type":"para","text":"Local movement constraints make learnability highly sensitive to the temporal arrangement of edges. As discussed, connectivity of the horizon-wide union graph (all edges ever active) is insufficient, as a learner can be structurally isolated from a ⋆ a^{\\star} for a linear number of rounds despite global connectivity (see Appendix LABEL:app:horizon-long ). Similarly, a high density of time windows (blocks) with good connectivity across the full horizon is inadequate if those blocks appear only at the end (see Appendix LABEL:app:global-fraction-insufficient ). These failures necessitate two requirements: (i) the condition must directly control the induced time-inhomogeneous random walk rather than static graph summaries, and (ii) it must hold uniformly over time to remain valid after an arbitrary stopping time. This shift-invariance is essential for the navigation phase, which initiates at a random time when exploration terminates."},{"type":"para","text":"The canonical walk for graph G t G_{t} is defined by the kernel:"},{"type":"para","text":"where deg t ( u ) ≜ | N t ( u ) | \\deg_{t}(u)\\triangleq|N_{t}(u)| . A large absolute spectral gap means the walk converges quickly to its stationary distribution π \\pi from any starting point — informally, it means the learner cannot get stuck in one region of the graph for long. Since G t G_{t} is undirected, U t U_{t} is reversible with stationary distribution π \\pi determined by the degree sequence: each arm a a is visited with long-run frequency π ( a ) \\pi(a) . For a reversible kernel P P with eigenvalues λ 1 ( P ) ≥ ⋯ ≥ λ n ( P ) \\lambda_{1}(P)\\geq\\cdots\\geq\\lambda_{n}(P) , the absolute spectral gap gap abs ( P ) ≜ 1 − max | λ 2 ( P ) | , | λ n ( P ) | \\operatorname{gap}_{\\mathrm{abs}}(P)\\triangleq 1-\\max{|\\lambda_{2}(P)|,|\\lambda_{n}(P)|} controls how quickly the walk approaches this visit distribution from any starting node — a larger gap guarantees faster coverage of all arms."},{"type":"para","text":"Fix W ∈ ℕ W\\in\\mathbb{N} , ρ , γ ∈ ( 0 , 1 ] \\rho,\\gamma\\in(0,1] , and a distribution π \\pi on A A . A graph sequence { G t } t ≥ 1 \\{G_{t}\\}_{t\\geq 1} is ( W , ρ , γ , π ) (W,\\rho,\\gamma,\\pi) -mixing for the canonical walk if: (i) each kernel U t U_{t} in ( 1 ) is reversible with stationary distribution π \\pi , and (ii) every interval of W W consecutive rounds contains at least ⌈ ρ W ⌉ \\lceil\\rho W\\rceil indices t t such that gap abs ( U t ) ≥ γ \\operatorname{gap}_{\\mathrm{abs}}(U_{t})\\geq\\gamma ."},{"type":"para","text":"Intuitively, this condition ensures that within any time window of length W W , there is at least a fraction ρ \\rho of “well-connected” rounds (where the spectral gap is at least γ \\gamma ). This guarantees the walk continuously mixes toward a stable stationary distribution π \\pi (see Figure 1 ). For the canonical walk, a common stationary distribution arises naturally whenever all snapshots share the same degree sequence; in particular, d d -regular snapshots yield a uniform π \\pi (formalized in Appendix LABEL:app:mixing-condition ). An oblivious graph process is ( W , ρ , γ , π ) (W,\\rho,\\gamma,\\pi) -mixing if its realized sequence satisfies Definition 1 almost surely. Rather than a universal requirement for all dynamic graphs, this identifies a stable regime where the natural local walk is robust enough to support a process-agnostic learnability theorem."},{"type":"heading","level":2,"text":"II-C Scope and model instantiations"},{"type":"para","text":"Definition 1 characterizes when the canonical walk serves as a robust exploration primitive. While dynamic random-walk theory offers broader results via engineered Metropolis-type kernels that enforce stationarity by design [ 15 ] , we prioritize the simplest local rule embedded in the graph: the canonical walk ( 1 ). This focus isolates the fundamental structural constraints of the problem before considering more complex, task-specific kernels."},{"type":"para","text":"Furthermore, unlike whole-horizon block counts, which are not preserved under truncation and fail to characterize the random suffix following exploration, our walk-centric condition is shift-invariant by construction. This ensures that the necessary structural assumptions remain available throughout the navigation phase, regardless of the random stopping time generated by the learner’s exploration."},{"type":"para","text":"Several natural graph processes fit this regime. The simplest is the regular case, where each snapshot is d d -regular and π \\pi is uniform, covering time-varying communication overlays, rotating patrol schedules, and periodic expanders. A second class is fixed-degree rewiring, where snapshots share a degree sequence but differ in edge topology. This includes degree-preserving switch dynamics, which are standard for sampling graphs with prescribed degrees ( [ 6 , 17 , 7 ] ). A third class consists of deterministic or stochastic schedules built from finite families of fixed-degree graphs, such as alternating matchings or overlay rotations, provided every sliding window maintains a positive density of snapshots with a uniform spectral gap. Appendix LABEL:app:mixing-condition details how fixed degree sequences imply a common stationary distribution."},{"type":"para","text":"Recent process-specific works analyze models such as i.i.d. Erdős–Rényi or edge-Markovian processes (where the stationary distribution of walk-kernel shifts with realized degrees) by exploiting their particular distributional structure ( [ 5 ] ). This paper is complementary to that line; it trades model specificity for a process-agnostic structural theorem in the regime where the canonical walk remains analyzable. This scope determines the performance benchmark. The following explore-then-commit procedures prove structural learnability by decomposing the problem into coverage, estimation, and navigation under dynamic local movement. They are not presented as gap-optimal regret-minimization procedures for easy static graphs; on a complete graph, where movement constraints disappear, classical UCB-style methods and static graph-bandit algorithms attain sharper regret guarantees ( [ 2 , 9 , 18 ] ). The contribution here is a clean characterization of a dynamic regime in which the canonical walk supports sublinear regret via algorithms whose analysis separates the roles of learning and navigation."},{"type":"heading","level":1,"text":"III Exploration, Commitment, and Navigation"},{"type":"para","text":"The fundamental challenge in this setting is decoupling identification from exploitation. In classical bandits, identifying the optimal arm grants immediate and perpetual access to it. On a dynamic graph, however, local movement necessitates a distinct navigation phase: even with perfect knowledge of the best arm, the learner might be temporarily unable to reach it. This introduces a structural delay cost absent in unconstrained environments."},{"type":"para","text":"To address this, our algorithms adopt a phased explore-then-commit framework. During exploration, the learner moves according to a specific kernel to gather reward samples. Upon termination at a stopping time, the learner commits to the empirical leader and initiates navigation. If the target arm is not immediately reachable, the learner employs the canonical walk as a fallback navigation rule until the target is attained. Because self-loops are always available, the learner remains at the target arm for the remainder of the horizon."},{"type":"para","text":"We first develop LEX, a fixed-budget baseline that establishes the core analysis for this problem. While the later confidence-based variant retains the same exploration kernel and modifies only the stopping rule, reward-aware versions necessitate the modified kernels discussed in subsequent sections. The LEX framework comprises three elements: an exploration kernel K t K_{t} for local movement, a stopping criterion, and a navigation rule toward the empirical leader. During navigation, the learner moves directly to the target arm whenever it is available, otherwise falling back to the canonical walk U t U_{t} . This direct-move rule improves upon pure random-walk navigation and ensures the policy simplifies correctly to the unconstrained case on a complete graph."},{"type":"para","text":"Our proofs follow a three-step structure: coverage, estimation, and navigation. First, under Definition 1 , the walk’s distribution over arms approaches π \\pi within τ 0 \\tau_{0} steps from any starting node or time — no arm is systematically avoided regardless of where exploration begins. Second, rather than analyzing every step (which are time-correlated), we examine positions spaced τ 0 \\tau_{0} steps apart; at each such checkpoint arm a a is visited with probability at least π ( a ) / 2 \\pi(a)/2 , and a Chernoff bound gives a high-probability lower bound on visits to every arm. Third, once each arm has enough visits, Hoeffding’s inequality identifies a ⋆ a^{\\star} . The navigation analysis re-applies the same argument to the random graph suffix after exploration ends. Letting π ∗ ≜ min a ∈ A π ( a ) \\pi_{*}\\triangleq\\min_{a\\in A}\\pi(a) , Appendix LABEL:app:technical-lemmas shows that"},{"type":"para","text":"is a valid total-variation mixing time, uniform over all initial states and times."},{"type":"heading","level":2,"text":"III-A LEX: fixed-budget lazy exploration"},{"type":"para","text":"LEX employs the canonical walk for exploration, setting K t = U t K_{t}=U_{t} for all t t . Following a fixed budget T exp T_{\\exp} , the learner commits to the empirical leader. Because T exp T_{\\exp} depends on both reward gaps and the structural parameters of Definition 1 , LEX serves as a tuned baseline rather than an adaptive policy. Its primary utility is conceptual, as it isolates the statistical cost of learning from the structural cost of reaching the target arm. A key property of LEX is that its trajectory remains independent of observed rewards. The sequence of visited arms, and consequently the sample counts, is governed solely by the graph sequence and walk randomness. This independence ensures that the subsequent estimation step is mathematically transparent. The fixed-budget nature of LEX implies that identification guarantees are informative only when the horizon T T exceeds T exp T_{\\exp} . If T < T exp T<T_{\\exp} , the algorithm explores until the horizon concludes, where the trivial bound R ( T ) ≤ T R(T)\\leq T applies. This short-horizon case is incorporated into the expected-regret corollary, while the main theorem is stated for the regime T ≥ T exp T\\geq T_{\\exp} ."},{"type":"para","text":"Theorem 1 shows that with a budget of τ 0 \\tau_{0} mixing-time blocks, each arm accumulates enough visits to concentrate its empirical mean and the optimal arm is identified correctly with high probability. The budget is governed by two quantities: the mixing overhead τ 0 \\tau_{0} and the per-arm statistical difficulty Ψ LEX \\Psi_{\\mathrm{LEX}} , both defined in the theorem."},{"type":"para","text":"Assume the graph sequence is ( W , ρ , γ , π ) (W,\\rho,\\gamma,\\pi) -mixing for the canonical walk. Let τ 0 ≜ W + ⌈ 2 ρ γ log 1 π ∗ ⌉ \\tau_{0}\\triangleq W+\\lceil\\frac{2}{\\rho\\gamma}\\log\\frac{1}{\\pi_{*}}\\rceil and define the complexity:"},{"type":"para","text":"For a universal constant C > 0 C>0 , if Lex uses an exploration budget T exp ≜ τ 0 ⌈ C Ψ LEX ( δ ) ⌉ ≤ T T_{\\exp}\\triangleq\\tau_{0}\\lceil C\\Psi_{\\mathrm{LEX}}(\\delta)\\rceil\\leq T , then with probability at least 1 − δ 1-\\delta , the algorithm correctly identifies the optimal arm ( a ^ ⋆ = a ⋆ \\hat{a}^{\\star}=a^{\\star} ) and the learning regret satisfies R learn ( T ) ≤ T exp R_{\\mathrm{learn}}(T)\\leq T_{\\exp} ."},{"type":"para","text":"The proof follows directly from the mixing lemma by analyzing the walk at fixed intervals of length τ 0 \\tau_{0} . This thinning procedure, which considers only every τ 0 \\tau_{0} -th step to decouple temporal dependencies, ensures that each sampled position is within total variation distance π ∗ / 2 \\pi_{*}/2 of π \\pi , regardless of the preceding block’s state. Consequently, every arm a a is visited at these thinned steps with conditional probability at least π ( a ) / 2 \\pi(a)/2 . A one-sided Chernoff bound for adapted Bernoulli variables confirms that the number of such visits is proportional to m π ( a ) m\\pi(a) with high probability. Because the total samples for arm a a necessarily exceed this thinned count, Hoeffding’s inequality ensures the concentration of the empirical means for all arms. This sample requirement is captured exactly by Ψ LEX ( δ ) \\Psi_{\\mathrm{LEX}}(\\delta) , with the full proof provided in Appendix LABEL:app:lex-proof ."},{"type":"para","text":"The following lemma bounds the structural cost of navigating to the identified arm once exploration concludes."},{"type":"para","text":"Assume the conditions of Theorem 1 . If a ^ ⋆ = a ⋆ \\hat{a}^{\\star}=a^{\\star} , the navigation phase of Algorithm 1 satisfies"},{"type":"para","text":"where Δ max ≜ max a ∈ A Δ ( a ) \\Delta_{\\max}\\triangleq\\max_{a\\in A}\\Delta(a) and τ hit ( a ⋆ ) \\tau_{\\mathrm{hit}}(a^{\\star}) is the first time the navigation phase reaches a ⋆ a^{\\star} ."},{"type":"para","text":"The navigation bound is independent of the horizon T T , illustrating the core regret decomposition. The proof works by coupling the navigation phase to a pure canonical walk started from the same state and time. This isolates the structural cost as the expected time required to reach a ⋆ a^{\\star} under local movement constraints. The structural part of the argument is then a block-wise endpoint bound for the canonical walk obtained from Lemma LABEL:lem:uniform-mixing . Once the best arm has been identified correctly, the remaining cost is therefore just the time needed for a canonical walk, up to this domination, to reach the target under local movement."},{"type":"para","text":"Combining Theorem 1 with Lemma 1 yields the total expected regret guarantee."},{"type":"para","text":"Under Theorem 1 with δ = T − 2 \\delta=T^{-2} , Lex achieves 𝔼 [ R ( T ) ] = O ( τ 0 Ψ LEX ( T − 2 ) + τ 0 / π ( a ⋆ ) ) \\mathbb{E}[R(T)]=O(\\tau_{0}\\Psi_{\\mathrm{LEX}}(T^{-2})+\\tau_{0}/\\pi(a^{\\star})) . For the d d -regular case, let Γ = W + log n ρ γ \\Gamma=W+\\frac{\\log n}{\\rho\\gamma} ; the regret simplifies to:"},{"type":"para","text":"In the regular case where π \\pi is uniform, let Γ = W + log n ρ γ \\Gamma=W+\\frac{\\log n}{\\rho\\gamma} . The bound simplifies to:"},{"type":"para","text":"The bound decouples the statistical complexity of identification from the structural overhead of navigation. The former scales with inverse gaps and stationary masses (ensuring a ⋆ a^{\\star} is sufficiently sampled), while the latter remains a T T -independent constant. In the d d -regular case, letting Γ ≜ W + log n ρ γ \\Gamma\\triangleq W+\\frac{\\log n}{\\rho\\gamma} represent the mixing overhead, the regret simplifies to:"},{"type":"para","text":"We now replace the fixed budget with a data-dependent stopping rule to develop a confidence-based variant."},{"type":"heading","level":2,"text":"III-B CB-LEX: confidence-based stopping"},{"type":"para","text":"While LEX requires an oracle exploration budget, CB-LEX introduces a data-dependent stopping rule while retaining the canonical kernel U t U_{t} . A key distinction is the stopping certificate: the empirical leader’s lower confidence bound must dominate the upper confidence bounds (UCBs) of all competitors, rather than just the empirical runner-up. This refinement is essential because sample counts can be highly uneven; a poorly sampled arm might have a low empirical mean but a wide confidence interval that still overlaps with the leader. Comparing only against the runner-up would risk premature commitment before these wide intervals are sufficiently narrowed."},{"type":"para","text":"For each arm a ∈ A a\\in A at time t t , let μ ^ t ( a ) \\hat{\\mu}_{t}(a) be the empirical mean and φ t ( a ) \\varphi_{t}(a) the visitation count. We define the confidence width w t ( a ) ≜ ln ( 4 n T / δ ) / ( 2 φ t ( a ) ) w_{t}(a)\\triangleq\\sqrt{\\ln(4nT/\\delta)/(2\\varphi_{t}(a))} if φ t ( a ) ≥ 1 \\varphi_{t}(a)\\geq 1 , and w t ( a ) ≜ 1 w_{t}(a)\\triangleq 1 otherwise. The confidence bounds are LCB t ( a ) ≜ μ ^ t ( a ) − w t ( a ) \\operatorname{LCB}_{t}(a)\\triangleq\\hat{\\mu}_{t}(a)-w_{t}(a) and UCB t ( a ) ≜ μ ^ t ( a ) + w t ( a ) \\operatorname{UCB}_{t}(a)\\triangleq\\hat{\\mu}_{t}(a)+w_{t}(a) . Let b t ∈ arg max a ∈ A μ ^ t ( a ) b_{t}\\in\\arg\\max_{a\\in A}\\hat{\\mu}_{t}(a) be the current empirical leader. CB-LEX terminates exploration at the stopping time:"},{"type":"para","text":"with σ ≜ T \\sigma\\triangleq T if the condition is never met. If σ < T \\sigma<T , the algorithm commits to a ^ ⋆ ≜ b σ \\hat{a}^{\\star}\\triangleq b\\sigma and initiates the same navigation phase used in LEX. This stopping rule ensures the leader is separated from all competitors, which is essential because sample counts can be highly uneven; a poorly sampled arm might have a low empirical mean but a wide confidence interval that still overlaps with the leader."},{"type":"para","text":"The exploration phase of CB-LEX is summarized in Algorithm 2 . Since the navigation logic remains identical to Algorithm 1 , it is omitted for brevity. The confidence widths w t ( a ) w_{t}(a) depend on the horizon T T only through the logarithm. While a fully horizon-free variant can be obtained by the standard replacement T ↦ ( t + 1 ) 2 T\\mapsto(t+1)^{2} , we retain the horizon-dependent form to keep the notation streamlined for the current analysis. As established in Theorem 2 , the adaptive stopping rule preserves the regret order of the LEX baseline. Although the logarithmic terms differ slightly to ensure a uniform confidence event over the horizon, both algorithms achieve the same asymptotic rate under the standard choice δ = T − 2 \\delta=T^{-2} ."},{"type":"para","text":"Assume the graph sequence is ( W , ρ , γ , π ) (W,\\rho,\\gamma,\\pi) -mixing and let τ 0 \\tau_{0} be as in Theorem 1. Define the complexity:"},{"type":"para","text":"For a constant C > 0 C>0 , if T ≥ τ 0 ⌈ C Ψ CB ( δ ) ⌉ ≜ T nom T\\geq\\tau_{0}\\lceil C\\Psi_{\\mathrm{CB}}(\\delta)\\rceil\\triangleq T_{\\mathrm{nom}} , then with probability at least 1 − δ 1-\\delta : a ^ ⋆ = a ⋆ \\hat{a}^{\\star}=a^{\\star} , the algorithm stops at σ ≤ T nom \\sigma\\leq T_{\\mathrm{nom}} , and R learn ( T ) ≤ T nom R_{\\mathrm{learn}}(T)\\leq T_{\\mathrm{nom}} ."},{"type":"para","text":"The proof shares the structural ingredients of the LEX analysis, with the addition of a coupling argument. We couple CB-LEX to a perpetual exploration walk that continues to follow the canonical kernel regardless of the stopping condition. Up to the time σ \\sigma , the actual trajectory and the perpetual walk are identical. Consequently, it suffices to show that the stopping condition is satisfied by a deterministic time on the perpetual walk. The mixing and visitation lemmas (Appendix LABEL:app:technical-lemmas ) provide the necessary sample counts at that time, and a uniform confidence event ensures the separation of intervals. The full argument is provided in Appendix LABEL:app:cblex-proof ."},{"type":"para","text":"Under the conditions of Theorem 2 , setting δ = T − 2 \\delta=T^{-2} yields sublinear regret 𝔼 [ R ( T ) ] = o ( T ) \\mathbb{E}[R(T)]=o(T) . The expected regret is bounded by:"},{"type":"para","text":"In the d d -regular case, letting Γ ≜ W + log n ρ γ \\Gamma\\triangleq W+\\frac{\\log n}{\\rho\\gamma} , this simplifies to:"},{"type":"para","text":"Crucially, the confidence-based rule eliminates the requirement for a pre-specified exploration budget without incurring an asymptotic penalty. Under the mixing condition of Definition 1 , the same visitation guarantees that justify the LEX budget ensure that confidence intervals become disjoint within the same order of time. The fundamental distinction lies in the termination mechanism: LEX utilizes an external budget, whereas CB-LEX certifies stopping through observed data. The primary technical novelty in the analysis is the coupling to a perpetual exploration walk, which enables the evaluation of a random stopping rule using the deterministic-time bounds previously established for the canonical walk."},{"type":"heading","level":2,"text":"III-C RALEX: reward-aware exploration with a canonical floor"},{"type":"para","text":"While CB-LEX uses data only for its stopping rule, its movement remains reward-indifferent. RALEX introduces reward-awareness by biasing transitions toward promising arms. The difficulty is that a purely reward-driven kernel often fails to preserve the stationary distribution π \\pi required by Definition 1. RALEX instead employs a mixture that retains a fixed floor of canonical movement.At round t t , the learner uses the empirical means μ ^ t − 1 \\hat{\\mu}_{t-1} and widths w t − 1 w_{t-1} to form the local optimistic score ξ t − 1 ( a ) ≜ μ ^ t − 1 ( a ) + w t − 1 ( a ) \\xi_{t-1}(a)\\triangleq\\hat{\\mu}_{t-1}(a)+w_{t-1}(a) . We define a softmax proposal on the neighborhood L t ( a t − 1 ) L_{t}(a_{t-1}) :"},{"type":"para","text":"where λ ≥ 0 \\lambda\\geq 0 controls the bias. To ensure mixing, RALEX defines the weight ϵ t ≜ max { ϵ 0 , 2 w ¯ t − 1 / ( 1 + 2 w ¯ t − 1 ) } \\epsilon_{t}\\triangleq\\max\\{\\epsilon_{0},2\\bar{w}_{t-1}/(1+2\\bar{w}_{t-1})\\} , where w ¯ t − 1 ≜ max a ∈ A w t − 1 ( a ) \\bar{w}_{t-1}\\triangleq\\max_{a\\in A}w_{t-1}(a) and ϵ 0 ∈ ( 0 , 1 ] \\epsilon_{0}\\in(0,1] is a fixed floor. The exploration kernel is:"},{"type":"para","text":"RALEX adopts the same stopping rule as CB-LEX, terminating exploration when the leader’s lower bound separates from all competing upper bounds."},{"type":"para","text":"Algorithm 3 details the RALEX exploration phase. Each round begins by calculating the optimistic proposal ξ t − 1 \\xi_{t-1} from the preceding round’s estimates and determining the canonical mixing weight ϵ t \\epsilon_{t} . Following the state transition and subsequent reward update, the algorithm refreshes the confidence widths and evaluates the stopping certificate established for CB-LEX. Although the exploration kernel is the primary algorithmic departure, the theoretical analysis must rigorously address the reward-dependence of the resulting visitation counts."},{"type":"para","text":"The deterministic floor ϵ 0 \\epsilon_{0} in ( 10 ) provides the key structural guarantee for RALEX. It implies the pointwise lower bound:"},{"type":"para","text":"The worst-case analysis relies exclusively on this inequality, ensuring the subsequent theorem holds for any bias parameter λ ≥ 0 \\lambda\\geq 0 . While this generality leads to a conservative bound; treating the biased component as an arbitrary perturbation; it preserves the rigorous structural properties established for the canonical walk."},{"type":"para","text":"Assume the graph sequence is ( W , ρ , γ , π ) (W,\\rho,\\gamma,\\pi) -mixing. For bias λ ≥ 0 \\lambda\\geq 0 and floor ε 0 ∈ ( 0 , 1 ] \\varepsilon_{0}\\in(0,1] , let τ 0 \\tau_{0} be the mixing time from Theorem 1 and ϑ 0 ≜ 1 2 ε 0 τ 0 \\vartheta_{0}\\triangleq\\frac{1}{2}\\varepsilon_{0}^{\\tau_{0}} . Define the complexity:"},{"type":"para","text":"For a constant C > 0 C>0 , if T ≥ T nom ≜ τ 0 ⌈ C Ψ RA ( δ ) ⌉ T\\geq T_{\\mathrm{nom}}\\triangleq\\tau_{0}\\lceil C\\Psi_{\\mathrm{RA}}(\\delta)\\rceil , then with probability at least 1 − δ 1-\\delta : a ^ ⋆ = a ⋆ \\hat{a}^{\\star}=a^{\\star} , the algorithm stops at σ ≤ T nom \\sigma\\leq T_{\\mathrm{nom}} , and R learn ( T ) ≤ T nom R_{\\mathrm{learn}}(T)\\leq T_{\\mathrm{nom}} ."},{"type":"para","text":"The analysis of RALEX introduces two primary technical refinements beyond the CB-LEX framework. The first is statistical: because the exploration kernel is reward-dependent, sample counts are no longer independent of the reward history. We address this in Appendix LABEL:app:ralex-proof using a uniform confidence lemma for adaptive sampling derived from a reward-table construction. The second is structural: since RALEX deviates from the canonical walk, the standard visitation lemma for U t U_{t} does not apply directly. Instead, we show that each block of τ 0 \\tau_{0} rounds behaves as a canonical walk with probability at least ε 0 τ 0 \\varepsilon_{0}^{\\tau_{0}} , yielding a thinned visitation guarantee with success rate ϑ 0 π ( a ) \\vartheta_{0}\\pi(a) . With these ingredients, the stopping-time analysis follows the CB-LEX pattern.The resulting bound is conservative compared to CB-LEX, incurring a factor ϑ 0 − 1 = 2 ε 0 − τ 0 \\vartheta_{0}^{-1}=2\\varepsilon_{0}^{-\\tau_{0}} by relying strictly on the canonical component from ( 11 ). This constitutes the theoretical price for permitting the biased kernel component to depend arbitrarily on observations. At ε 0 = 1 \\varepsilon_{0}=1 , RALEX simplifies to CB-LEX and the theorem recovers the CB-LEX rate. Notably, this result generalizes to any local proposal kernel mixed with the same canonical floor."},{"type":"para","text":"Under the conditions of Theorem 3 , let δ = T − 2 \\delta=T^{-2} and T nom ≜ τ 0 ⌈ C Ψ RA ( T − 2 ) ⌉ T_{\\mathrm{nom}}\\triangleq\\tau_{0}\\lceil C\\Psi_{\\mathrm{RA}}(T^{-2})\\rceil . The expected regret of RALEX satisfies:"},{"type":"para","text":"Asymptotically, RALEX achieves sublinear regret 𝔼 [ R ( T ) ] = o ( T ) \\mathbb{E}[R(T)]=o(T) bounded by:"},{"type":"para","text":"RALEX represents a distinct trade-off within the algorithmic design space. While LEX and CB-LEX achieve the sharpest worst-case guarantees by adhering strictly to the canonical walk, RALEX sacrifices a portion of this theoretical sharpness to prioritize transitions toward locally promising arms. Theorem 3 serves as a worst-case baseline; it establishes that reward-aware bias is compatible with sublinear regret, though it does not formally certify a performance gain over the baseline. A genuine improvement requires an additional condition: the optimistic proposal must successfully allocate probability mass toward the specific arms that determine the stopping time. The following result isolates this mechanism."},{"type":"heading","level":2,"text":"III-D When reward awareness yields a provable gain"},{"type":"para","text":"Theorem 3 establishes that optimistic bias is compatible with canonical exploration, but it does not specify a regime where this bias provably enhances performance. To characterize these gains, we analyze the perpetual exploration process { X ~ t } t ≥ 0 \\{\\widetilde{X}_{t}\\}_{t\\geq 0} , which follows the RALEX kernel indefinitely without transitioning to navigation.Fix a deterministic burn-in time t gain ∈ { 0 , … , T } t_{\\mathrm{gain}}\\in\\{0,\\dots,T\\} and define the mixing parameters:"},{"type":"para","text":"where π ∗ ≜ min a ∈ A π ( a ) \\pi_{*}\\triangleq\\min_{a\\in A}\\pi(a) . For each block index j ≥ 1 j\\geq 1 , let s j ≜ t gain + ( j − 1 ) τ 1 s_{j}\\triangleq t_{\\mathrm{gain}}+(j-1)\\tau_{1} and define the filtration 𝒢 ~ j − 1 ≜ σ ( ℱ ~ s j ) \\widetilde{\\mathcal{G}}_{j-1}\\triangleq\\sigma(\\widetilde{\\mathcal{F}}_{s_{j}}) . The quantity Pr ( X ~ s j + τ 1 = a ∣ 𝒢 ~ j − 1 ) \\Pr(\\widetilde{X}_{s_{j}+\\tau_{1}}=a\\mid\\widetilde{\\mathcal{G}}_{j-1}) represents the probability that the j j -th post-burn-in block terminates at arm a a . Because optimistic scores are recomputed after every reward observation within a block, a one-step bound on the proposal kernel is insufficient for the online algorithm. Consequently, our analysis utilizes these block-end probabilities directly to quantify the visitation benefits of the reward-aware bias."},{"type":"para","text":"Assume the conditions of Theorem 3 and let { X ~ t } \\{\\widetilde{X}_{t}\\} be the perpetual exploration process. Fix a subset B ⊆ A B\\subseteq A containing a ⋆ a^{\\star} and constants κ gain ( a ) ≥ ϑ 1 π ( a ) \\kappa_{\\mathrm{gain}}(a)\\geq\\vartheta_{1}\\pi(a) for a ∈ B a\\in B . Suppose that for every block index j ≥ 1 j\\geq 1 with s j + τ 1 ≤ T s_{j}+\\tau_{1}\\leq T :"},{"type":"para","text":"Define κ ( a ) ≜ κ gain ( a ) \\kappa(a)\\triangleq\\kappa_{\\mathrm{gain}}(a) for a ∈ B a\\in B and κ ( a ) ≜ ϑ 1 π ( a ) \\kappa(a)\\triangleq\\vartheta_{1}\\pi(a) otherwise. Let the complexity be:"},{"type":"para","text":"For a constant C > 0 C>0 , if T ≥ T nom ≜ t gain + τ 1 ⌈ C Ψ RA gain ( δ ) ⌉ T\\geq T_{\\mathrm{nom}}\\triangleq t_{\\mathrm{gain}}+\\tau_{1}\\lceil C\\Psi_{\\mathrm{RA}}^{\\mathrm{gain}}(\\delta)\\rceil , then with probability at least 1 − δ 1-\\delta : a ^ ⋆ = a ⋆ \\hat{a}^{\\star}=a^{\\star} , σ ≤ T nom \\sigma\\leq T_{\\mathrm{nom}} , and R learn ( T ) ≤ T nom R_{\\mathrm{learn}}(T)\\leq T_{\\mathrm{nom}} ."},{"type":"para","text":"The theorem is formulated using block-end probabilities, as these are the fundamental objects in our analysis. Appendix LABEL:app:ralex-gain-proof provides a sufficient condition for verifying this gain. Let 𝒞 j \\mathcal{C}_{j} be the event that the first τ 1 − 1 \\tau_{1}-1 rounds of block j j use the canonical component of RALEX. If the final RALEX step in that block satisfies"},{"type":"para","text":"then the block-end coefficient can be taken to be κ gain ( a ) = ( 1 + 2 β ( a ) / ε 0 ) ϑ 1 π ( a ) . \\kappa_{\\mathrm{gain}}(a)=\\left(1+2\\beta(a)/\\varepsilon_{0}\\right)\\vartheta_{1}\\pi(a). In other words, the improvement factor is determined by the amount of extra endpoint mass that the last reward-aware step contributes above the canonical floor."},{"type":"para","text":"To interpret when this yields a genuine reduction of the global learning term, define"},{"type":"para","text":"Let Λ max := max a ∈ A Λ ( a ) \\Lambda_{\\max}:=\\max_{a\\in A}\\Lambda(a) . Thus the leading learning term in the safe RALEX bound is proportional to ϑ 1 − 1 log ( 4 n T / δ ) Λ max . \\vartheta_{1}^{-1}\\log(4nT/\\delta)\\,\\Lambda_{\\max}. For a factor ζ > 0 \\zeta>0 , define the ζ \\zeta -near-bottleneck set"},{"type":"para","text":"These are the arms whose safe contribution lies within a factor 1 + ζ 1+\\zeta of the worst one. We want to control this set as improving only the exact maximizers need not reduce the global maximum, because a near-bottleneck arm left unimproved can become the new dominant term after the gain is applied."},{"type":"para","text":"Assume the conditions of Theorem 4 , fix ζ > 0 \\zeta>0 , and suppose that ℋ ζ ⊆ B \\mathcal{H}_{\\zeta}\\subseteq B and κ gain ( a ) ≥ ( 1 + ζ ) ϑ 1 π ( a ) for every a ∈ ℋ ζ . \\kappa_{\\mathrm{gain}}(a)\\geq(1+\\zeta)\\vartheta_{1}\\pi(a)\\text{ for every }a\\in\\mathcal{H}_{\\zeta}. Then"},{"type":"para","text":"Consequently,"},{"type":"para","text":"Relative to the safety theorem for RALEX, the leading learning term decreases by the factor ( 1 + ζ ) − 1 (1+\\zeta)^{-1} ."},{"type":"para","text":"This is the strongest multiplicative improvement that Theorem 4 certifies without additional structure on the reward-aware proposal: the gain must cover every arm that could otherwise remain, or become, the dominant term."},{"type":"heading","level":1,"text":"IV Numerical Simulations"},{"type":"para","text":"We empirically ground our theoretical guarantees with a series of simulations designed to evaluate the practical performance of the proposed algorithm family. Our experiments, summarized in Figure 2 , focus on isolating the spatial behavior of different exploration strategies and quantifying their efficiency in identifying optimal regions. Specifically, we evaluate (a) mean cumulative regret on the Hard instance and (b) the distribution of adaptive stopping times across both Easy and Hard instances."},{"type":"heading","level":4,"text":"Environment and Reward Model."},{"type":"para","text":"The environment is modeled as a dynamic graph G t = ( A , E t ) G_{t}=(A,E_{t}) defined over a discrete set of n = | A | = 205 n=|A|=205 nodes, simulated over a total horizon of T = 70 , 000 T=70,000 rounds. The network topology evolves via degree-preserving edge switches, maintaining a fixed degree sequence across all snapshots and thus a common stationary distribution π \\pi as required by Definition 1 . Upon visiting a node a ∈ A a\\in A , the agent observes a stochastic reward r t ( a ) ∈ [ 0 , 1 ] r_{t}(a)\\in[0,1] characterized by an underlying mean μ ( a ) \\mu(a) perturbed by bounded, zero-mean noise."},{"type":"heading","level":4,"text":"Experimental Design."},{"type":"para","text":"We evaluate performance on two distinct reward instances. The hard instance features a small suboptimality gap, with a single optimal node μ ( a ⋆ ) = 0.95 \\mu(a^{\\star})=0.95 and several competing arms in the range [ 0.45 , 0.65 ] [0.45,0.65] , requiring significant evidence to distinguish the true optimum. Conversely, the easy instance features a large gap where suboptimal arms have low rewards in the range [ 0.1 , 0.2 ] [0.1,0.2] , allowing for rapid identification. For the dynamic graph, we set the edge appearance probability to 0.1 and the disappearance probability to 0.01. For the learning algorithms, we set the reward-bias to λ = 10 \\lambda=10 for RA-LEX . Results are averaged over five random seeds, with shaded regions in regret curves representing one standard deviation."},{"type":"heading","level":2,"text":"IV-A Results and Analysis"},{"type":"para","text":"Visualizing Exploration Strategies: Figure 3 provides the visual intuition behind the performance differences by comparing exploration-phase visitation patterns. The heatmap for CB-LEX is relatively uniform, showing a “blind” search of the entire node set. In stark contrast, the RA-LEX heatmap is highly concentrated, resembling a “searchlight” focused on the true high-reward zones. This visually confirms that its biased random walk successfully prioritizes promising regions, explaining its superior sample efficiency."},{"type":"para","text":"Efficiency of Reward-Aware Exploration: As shown in Figure 2 a, both algorithms achieve sublinear regret, indicated by the eventual flattening of their regret curves. However, the reward-aware RA-LEX achieves significantly lower final regret compared to the uniform exploration of CB-LEX , corroborating the benefit of biasing the exploration walk toward high-reward regions."},{"type":"para","text":"Validation of the Adaptive Mechanism: Figure 2 b confirms that the adaptive stopping rule correctly responds to statistical difficulty, with both algorithms stopping significantly earlier on the Easy instance. On the Hard instance, efficiency gains are pronounced: RA-LEX stops at a median of 1,850 rounds with a remarkably tight interquartile range (IQR) of 1,600 to 1,900. In contrast, CB-LEX requires a median of 6,000 rounds and exhibits a much wider variance (IQR of 4,500 to 7,500). This confirms that reward-aware biasing accelerates best-arm identification by nearly 70% and makes the required exploration duration significantly more predictable."},{"type":"para","text":"Robustness: Further sensitivity analysis identifies a “sweet spot” for the bias parameter λ \\lambda that successfully balances exploiting high-reward regions with maintaining sufficient stochasticity to escape local optima."},{"type":"heading","level":1,"text":"V Conclusion"},{"type":"para","text":"We investigated stochastic multi-armed bandits under local movement constraints on dynamic graphs, a setting where the structural necessity of post-exploration navigation decouples identification from exploitation. Because exploration concludes at an uncertain stopping time, whole-horizon connectivity is an insufficient metric for learnability; instead, structural properties must be controlled over the subsequent random suffix. This requirement led to our process-agnostic approach of placing assumptions directly on the graph’s intrinsic walk.By imposing a mixing condition on the canonical walk, we established sublinear expected regret for a family of explore-then-commit algorithms. Our framework includes a fixed-budget baseline (LEX), an adaptive variant (CB-LEX) that eliminates oracle dependence, and a reward-aware strategy (RALEX) that maintains worst-case safety while providing provable performance gains.Future research will focus on weakening these structural requirements and extending the analysis to broader local kernels, such as Metropolis-type walks, to further expand the positive regime for bandit learning on dynamic networks"},{"type":"para","text":"new_app"}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the stochastic multi-armed bandit (MAB) problem, a learner repeatedly chooses among several actions (interchangeably referred to as arms ) observes only the reward of the selected action, and seeks to maximize its cumulative reward ( [ 2 , 9 , 16 ] ). Classical bandit models typically assume that every arm is available for selection at every round. This assumption, however, is too strong for many networked settings where movement is restricted and inherently local. For instance, a mobile data collector in a sensor network is constrained by physical proximity, requiring it to be near a specific node to harvest its data [ 10 ] ; a search routine on a peer-to-peer overlay is confined to traversing logical links between neighboring peers [ 8 ] ; and a patrol agent is restricted to navigable pathways that may open or close as the environment changes [ 4 ] . In these scenarios, the learner cannot jump to an arbitrary location; instead, its next action is strictly determined by its current position and the connectivity available at that moment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This motivates the dynamic graph bandit problem, where arms correspond to vertices in a network with time-varying edges. The learner observes only its local neighborhood, moves along locally available edges, and receives feedback solely from visited nodes. Consequently, the evolving topology governs both immediate reward availability and future reachability. This fundamentally differs from feedback-graph bandits, where graphs encode side-observations rather than movement constraints ( [ 11 , 1 ] ), and from static-graph bandits, where fixed, globally known topologies permit pre-planned navigation ( [ 18 , 13 ] ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A core challenge in this setting is that best-arm identification is decoupled from exploitation. In classical bandits, identifying the optimal arm immediately enables perpetual exploitation. Here, a learner might identify the best arm but remain temporarily unable to reach it. Learnability thus requires the environment to consistently provide navigable pathways; otherwise, the learner could be structurally isolated indefinitely. Importantly, standard full-horizon conditions, such as the connectivity of the union of all graphs, are insufficient, as a network might be connected in aggregate but disconnected at almost every individual step. A valid structural condition must therefore be shift-invariant , holding from any arbitrary round. This guarantees navigability during the post-exploration phase, regardless of the random stopping time at which exploration concludes. We define a condition that meets these criteria (see Section II-B ), avoiding the pitfalls of whole-horizon summaries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To isolate the distinct challenges of information gathering and subsequent movement, we adopt the explore-then-commit framework. Our analysis centers on the canonical walk (see ( 1 )), where the learner either remains at its current node or moves uniformly to an available neighbor. We prioritize this simple local kernel because it is entirely decentralized and allows us to address a fundamental question: when is a graph’s intrinsic walk sufficient for learning? While more complex kernels can be useful, the canonical walk serves as a transparent analytical probe that exposes the structural constraints of the environment before asking whether reward-biased movement yields further gains. This choice makes the dichotomy between statistical identification and navigation explicit in both algorithms and proofs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To formalize shift-invariant navigability, we impose a condition directly on the canonical walk. Intuitively, we require the walk to maintain a consistent long-run stationary distribution, and we require that every short time window contains enough “well-connected” rounds to prevent the learner from becoming trapped in a local subgraph. We formalize this as a common-stationary sliding-window mixing property. By ensuring a stable stationary distribution, we align with positive regimes in the dynamic random-walk literature, where such walks mimic their static counterparts ( [ 3 , 12 , 14 , 15 ] ). This encompasses natural fixed-degree processes, such as rotating regular overlays and degree-preserving rewirings.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Within this regime, we analyze a family of local explore-then-commit algorithms that utilize walk-based exploration with both fixed and adaptive stopping rules (see Section II ). Our analysis extends to reward-aware movement, where we formally decouple worst-case safety from provable performance gains. Unlike model-specific analyses [ 5 ] , we establish process-agnostic structural conditions (see Section III ) that ensure learnability for any such walk-based approach. Due to space constraints, extended related work, technical derivations, and complete proofs are deferred to the appendix.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Model and Problem Formulation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section defines the learning problem, fixes the information pattern, and introduces the structural condition used throughout the paper.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Bandit learning on dynamic graphs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let A = { 1 , … , n } A=\\\\{1,\\\\dots,n\\\\} be a finite set of arms, where each a ∈ A a\\\\in A is associated with a reward distribution 𝒟 ( a ) \\\\mathcal{D}(a) supported on [ 0 , 1 ] [0,1] with mean μ ( a ) \\\\mu(a) . Rewards are i.i.d. across time and independent of the graph process. We assume a unique optimal arm a ⋆ ≜ arg max a ∈ A μ ( a ) a^{\\\\star}\\\\triangleq\\\\arg\\\\max_{a\\\\in A}\\\\mu(a) and define the suboptimality gaps as Δ ( a ) ≜ μ ( a ⋆ ) − μ ( a ) \\\\Delta(a)\\\\triangleq\\\\mu(a^{\\\\star})-\\\\mu(a) for a ≠ a ⋆ a\\\\neq a^{\\\\star} . At each discrete round t ≥ 1 t\\\\geq 1 , the environment presents the learner an undirected graph G t = ( A , E t ) G_{t}=(A,E_{t}) . The learner begins at node a t − 1 a_{t-1} and observes only the local neighborhood L t ( a t − 1 ) ≜ N t ( a t − 1 ) ∪ { a t − 1 } L_{t}(a_{t-1})\\\\triangleq N_{t}(a_{t-1})\\\\cup\\\\{a_{t-1}\\\\} , where N t ( u ) ≜ { v ∈ A : ( u , v ) ∈ E t } N_{t}(u)\\\\triangleq\\\\{v\\\\in A:(u,v)\\\\in E_{t}\\\\} . The graph process is oblivious: the sequence { G t } t ≥ 1 \\\\{G_{t}\\\\}_{t\\\\geq 1} is independent of the learner’s past actions and randomization. The learner chooses a t ∈ L t ( a t − 1 ) a_{t}\\\\in L_{t}(a_{t-1}) , moves to that node, and receives reward r t ∼ 𝒟 ( a t ) r_{t}\\\\sim\\\\mathcal{D}(a_{t}) . We measure performance by the regret R ( T ) ≜ ∑ t = 1 T ( μ ( a ⋆ ) − r t ) R(T)\\\\triangleq\\\\sum_{t=1}^{T}(\\\\mu(a^{\\\\star})-r_{t}) , whose expectation matches the standard pseudo-regret 𝔼 [ R ( T ) ] = ∑ t = 1 T 𝔼 [ μ ( a ⋆ ) − μ ( a t ) ] \\\\mathbb{E}[R(T)]=\\\\sum_{t=1}^{T}\\\\mathbb{E}[\\\\mu(a^{\\\\star})-\\\\mu(a_{t})] . Our objective is to design a local policy that is learnable, meaning it achieves sublinear expected regret: lim T → ∞ 𝔼 [ R ( T ) ] / T = 0 \\\\lim_{T\\\\to\\\\infty}\\\\mathbb{E}[R(T)]/T=0 . While the full graphs { G t } \\\\{G_{t}\\\\} are used for analysis, the policy remains strictly local, seeing only the current neighborhood L t ( a t − 1 ) L_{t}(a_{t-1}) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B Why the condition must be stated on the walk\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Local movement constraints make learnability highly sensitive to the temporal arrangement of edges. As discussed, connectivity of the horizon-wide union graph (all edges ever active) is insufficient, as a learner can be structurally isolated from a ⋆ a^{\\\\star} for a linear number of rounds despite global connectivity (see Appendix LABEL:app:horizon-long ). Similarly, a high density of time windows (blocks) with good connectivity across the full horizon is inadequate if those blocks appear only at the end (see Appendix LABEL:app:global-fraction-insufficient ). These failures necessitate two requirements: (i) the condition must directly control the induced time-inhomogeneous random walk rather than static graph summaries, and (ii) it must hold uniformly over time to remain valid after an arbitrary stopping time. This shift-invariance is essential for the navigation phase, which initiates at a random time when exploration terminates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The canonical walk for graph G t G_{t} is defined by the kernel:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where deg t ( u ) ≜ | N t ( u ) | \\\\deg_{t}(u)\\\\triangleq|N_{t}(u)| . A large absolute spectral gap means the walk converges quickly to its stationary distribution π \\\\pi from any starting point — informally, it means the learner cannot get stuck in one region of the graph for long. Since G t G_{t} is undirected, U t U_{t} is reversible with stationary distribution π \\\\pi determined by the degree sequence: each arm a a is visited with long-run frequency π ( a ) \\\\pi(a) . For a reversible kernel P P with eigenvalues λ 1 ( P ) ≥ ⋯ ≥ λ n ( P ) \\\\lambda_{1}(P)\\\\geq\\\\cdots\\\\geq\\\\lambda_{n}(P) , the absolute spectral gap gap abs ( P ) ≜ 1 − max | λ 2 ( P ) | , | λ n ( P ) | \\\\operatorname{gap}_{\\\\mathrm{abs}}(P)\\\\triangleq 1-\\\\max{|\\\\lambda_{2}(P)|,|\\\\lambda_{n}(P)|} controls how quickly the walk approaches this visit distribution from any starting node — a larger gap guarantees faster coverage of all arms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix W ∈ ℕ W\\\\in\\\\mathbb{N} , ρ , γ ∈ ( 0 , 1 ] \\\\rho,\\\\gamma\\\\in(0,1] , and a distribution π \\\\pi on A A . A graph sequence { G t } t ≥ 1 \\\\{G_{t}\\\\}_{t\\\\geq 1} is ( W , ρ , γ , π ) (W,\\\\rho,\\\\gamma,\\\\pi) -mixing for the canonical walk if: (i) each kernel U t U_{t} in ( 1 ) is reversible with stationary distribution π \\\\pi , and (ii) every interval of W W consecutive rounds contains at least ⌈ ρ W ⌉ \\\\lceil\\\\rho W\\\\rceil indices t t such that gap abs ( U t ) ≥ γ \\\\operatorname{gap}_{\\\\mathrm{abs}}(U_{t})\\\\geq\\\\gamma .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intuitively, this condition ensures that within any time window of length W W , there is at least a fraction ρ \\\\rho of “well-connected” rounds (where the spectral gap is at least γ \\\\gamma ). This guarantees the walk continuously mixes toward a stable stationary distribution π \\\\pi (see Figure 1 ). For the canonical walk, a common stationary distribution arises naturally whenever all snapshots share the same degree sequence; in particular, d d -regular snapshots yield a uniform π \\\\pi (formalized in Appendix LABEL:app:mixing-condition ). An oblivious graph process is ( W , ρ , γ , π ) (W,\\\\rho,\\\\gamma,\\\\pi) -mixing if its realized sequence satisfies Definition 1 almost surely. Rather than a universal requirement for all dynamic graphs, this identifies a stable regime where the natural local walk is robust enough to support a process-agnostic learnability theorem.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Scope and model instantiations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Definition 1 characterizes when the canonical walk serves as a robust exploration primitive. While dynamic random-walk theory offers broader results via engineered Metropolis-type kernels that enforce stationarity by design [ 15 ] , we prioritize the simplest local rule embedded in the graph: the canonical walk ( 1 ). This focus isolates the fundamental structural constraints of the problem before considering more complex, task-specific kernels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Furthermore, unlike whole-horizon block counts, which are not preserved under truncation and fail to characterize the random suffix following exploration, our walk-centric condition is shift-invariant by construction. This ensures that the necessary structural assumptions remain available throughout the navigation phase, regardless of the random stopping time generated by the learner’s exploration.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several natural graph processes fit this regime. The simplest is the regular case, where each snapshot is d d -regular and π \\\\pi is uniform, covering time-varying communication overlays, rotating patrol schedules, and periodic expanders. A second class is fixed-degree rewiring, where snapshots share a degree sequence but differ in edge topology. This includes degree-preserving switch dynamics, which are standard for sampling graphs with prescribed degrees ( [ 6 , 17 , 7 ] ). A third class consists of deterministic or stochastic schedules built from finite families of fixed-degree graphs, such as alternating matchings or overlay rotations, provided every sliding window maintains a positive density of snapshots with a uniform spectral gap. Appendix LABEL:app:mixing-condition details how fixed degree sequences imply a common stationary distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent process-specific works analyze models such as i.i.d. Erdős–Rényi or edge-Markovian processes (where the stationary distribution of walk-kernel shifts with realized degrees) by exploiting their particular distributional structure ( [ 5 ] ). This paper is complementary to that line; it trades model specificity for a process-agnostic structural theorem in the regime where the canonical walk remains analyzable. This scope determines the performance benchmark. The following explore-then-commit procedures prove structural learnability by decomposing the problem into coverage, estimation, and navigation under dynamic local movement. They are not presented as gap-optimal regret-minimization procedures for easy static graphs; on a complete graph, where movement constraints disappear, classical UCB-style methods and static graph-bandit algorithms attain sharper regret guarantees ( [ 2 , 9 , 18 ] ). The contribution here is a clean characterization of a dynamic regime in which the canonical walk supports sublinear regret via algorithms whose analysis separates the roles of learning and navigation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Exploration, Commitment, and Navigation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The fundamental challenge in this setting is decoupling identification from exploitation. In classical bandits, identifying the optimal arm grants immediate and perpetual access to it. On a dynamic graph, however, local movement necessitates a distinct navigation phase: even with perfect knowledge of the best arm, the learner might be temporarily unable to reach it. This introduces a structural delay cost absent in unconstrained environments.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address this, our algorithms adopt a phased explore-then-commit framework. During exploration, the learner moves according to a specific kernel to gather reward samples. Upon termination at a stopping time, the learner commits to the empirical leader and initiates navigation. If the target arm is not immediately reachable, the learner employs the canonical walk as a fallback navigation rule until the target is attained. Because self-loops are always available, the learner remains at the target arm for the remainder of the horizon.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first develop LEX, a fixed-budget baseline that establishes the core analysis for this problem. While the later confidence-based variant retains the same exploration kernel and modifies only the stopping rule, reward-aware versions necessitate the modified kernels discussed in subsequent sections. The LEX framework comprises three elements: an exploration kernel K t K_{t} for local movement, a stopping criterion, and a navigation rule toward the empirical leader. During navigation, the learner moves directly to the target arm whenever it is available, otherwise falling back to the canonical walk U t U_{t} . This direct-move rule improves upon pure random-walk navigation and ensures the policy simplifies correctly to the unconstrained case on a complete graph.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our proofs follow a three-step structure: coverage, estimation, and navigation. First, under Definition 1 , the walk’s distribution over arms approaches π \\\\pi within τ 0 \\\\tau_{0} steps from any starting node or time — no arm is systematically avoided regardless of where exploration begins. Second, rather than analyzing every step (which are time-correlated), we examine positions spaced τ 0 \\\\tau_{0} steps apart; at each such checkpoint arm a a is visited with probability at least π ( a ) / 2 \\\\pi(a)/2 , and a Chernoff bound gives a high-probability lower bound on visits to every arm. Third, once each arm has enough visits, Hoeffding’s inequality identifies a ⋆ a^{\\\\star} . The navigation analysis re-applies the same argument to the random graph suffix after exploration ends. Letting π ∗ ≜ min a ∈ A π ( a ) \\\\pi_{*}\\\\triangleq\\\\min_{a\\\\in A}\\\\pi(a) , Appendix LABEL:app:technical-lemmas shows that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"is a valid total-variation mixing time, uniform over all initial states and times.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A LEX: fixed-budget lazy exploration\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LEX employs the canonical walk for exploration, setting K t = U t K_{t}=U_{t} for all t t . Following a fixed budget T exp T_{\\\\exp} , the learner commits to the empirical leader. Because T exp T_{\\\\exp} depends on both reward gaps and the structural parameters of Definition 1 , LEX serves as a tuned baseline rather than an adaptive policy. Its primary utility is conceptual, as it isolates the statistical cost of learning from the structural cost of reaching the target arm. A key property of LEX is that its trajectory remains independent of observed rewards. The sequence of visited arms, and consequently the sample counts, is governed solely by the graph sequence and walk randomness. This independence ensures that the subsequent estimation step is mathematically transparent. The fixed-budget nature of LEX implies that identification guarantees are informative only when the horizon T T exceeds T exp T_{\\\\exp} . If T < T exp T<T_{\\\\exp} , the algorithm explores until the horizon concludes, where the trivial bound R ( T ) ≤ T R(T)\\\\leq T applies. This short-horizon case is incorporated into the expected-regret corollary, while the main theorem is stated for the regime T ≥ T exp T\\\\geq T_{\\\\exp} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Theorem 1 shows that with a budget of τ 0 \\\\tau_{0} mixing-time blocks, each arm accumulates enough visits to concentrate its empirical mean and the optimal arm is identified correctly with high probability. The budget is governed by two quantities: the mixing overhead τ 0 \\\\tau_{0} and the per-arm statistical difficulty Ψ LEX \\\\Psi_{\\\\mathrm{LEX}} , both defined in the theorem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume the graph sequence is ( W , ρ , γ , π ) (W,\\\\rho,\\\\gamma,\\\\pi) -mixing for the canonical walk. Let τ 0 ≜ W + ⌈ 2 ρ γ log 1 π ∗ ⌉ \\\\tau_{0}\\\\triangleq W+\\\\lceil\\\\frac{2}{\\\\rho\\\\gamma}\\\\log\\\\frac{1}{\\\\pi_{*}}\\\\rceil and define the complexity:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a universal constant C > 0 C>0 , if Lex uses an exploration budget T exp ≜ τ 0 ⌈ C Ψ LEX ( δ ) ⌉ ≤ T T_{\\\\exp}\\\\triangleq\\\\tau_{0}\\\\lceil C\\\\Psi_{\\\\mathrm{LEX}}(\\\\delta)\\\\rceil\\\\leq T , then with probability at least 1 − δ 1-\\\\delta , the algorithm correctly identifies the optimal arm ( a ^ ⋆ = a ⋆ \\\\hat{a}^{\\\\star}=a^{\\\\star} ) and the learning regret satisfies R learn ( T ) ≤ T exp R_{\\\\mathrm{learn}}(T)\\\\leq T_{\\\\exp} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proof follows directly from the mixing lemma by analyzing the walk at fixed intervals of length τ 0 \\\\tau_{0} . This thinning procedure, which considers only every τ 0 \\\\tau_{0} -th step to decouple temporal dependencies, ensures that each sampled position is within total variation distance π ∗ / 2 \\\\pi_{*}/2 of π \\\\pi , regardless of the preceding block’s state. Consequently, every arm a a is visited at these thinned steps with conditional probability at least π ( a ) / 2 \\\\pi(a)/2 . A one-sided Chernoff bound for adapted Bernoulli variables confirms that the number of such visits is proportional to m π ( a ) m\\\\pi(a) with high probability. Because the total samples for arm a a necessarily exceed this thinned count, Hoeffding’s inequality ensures the concentration of the empirical means for all arms. This sample requirement is captured exactly by Ψ LEX ( δ ) \\\\Psi_{\\\\mathrm{LEX}}(\\\\delta) , with the full proof provided in Appendix LABEL:app:lex-proof .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The following lemma bounds the structural cost of navigating to the identified arm once exploration concludes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume the conditions of Theorem 1 . If a ^ ⋆ = a ⋆ \\\\hat{a}^{\\\\star}=a^{\\\\star} , the navigation phase of Algorithm 1 satisfies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where Δ max ≜ max a ∈ A Δ ( a ) \\\\Delta_{\\\\max}\\\\triangleq\\\\max_{a\\\\in A}\\\\Delta(a) and τ hit ( a ⋆ ) \\\\tau_{\\\\mathrm{hit}}(a^{\\\\star}) is the first time the navigation phase reaches a ⋆ a^{\\\\star} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The navigation bound is independent of the horizon T T , illustrating the core regret decomposition. The proof works by coupling the navigation phase to a pure canonical walk started from the same state and time. This isolates the structural cost as the expected time required to reach a ⋆ a^{\\\\star} under local movement constraints. The structural part of the argument is then a block-wise endpoint bound for the canonical walk obtained from Lemma LABEL:lem:uniform-mixing . Once the best arm has been identified correctly, the remaining cost is therefore just the time needed for a canonical walk, up to this domination, to reach the target under local movement.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Combining Theorem 1 with Lemma 1 yields the total expected regret guarantee.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under Theorem 1 with δ = T − 2 \\\\delta=T^{-2} , Lex achieves 𝔼 [ R ( T ) ] = O ( τ 0 Ψ LEX ( T − 2 ) + τ 0 / π ( a ⋆ ) ) \\\\mathbb{E}[R(T)]=O(\\\\tau_{0}\\\\Psi_{\\\\mathrm{LEX}}(T^{-2})+\\\\tau_{0}/\\\\pi(a^{\\\\star})) . For the d d -regular case, let Γ = W + log n ρ γ \\\\Gamma=W+\\\\frac{\\\\log n}{\\\\rho\\\\gamma} ; the regret simplifies to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the regular case where π \\\\pi is uniform, let Γ = W + log n ρ γ \\\\Gamma=W+\\\\frac{\\\\log n}{\\\\rho\\\\gamma} . The bound simplifies to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The bound decouples the statistical complexity of identification from the structural overhead of navigation. The former scales with inverse gaps and stationary masses (ensuring a ⋆ a^{\\\\star} is sufficiently sampled), while the latter remains a T T -independent constant. In the d d -regular case, letting Γ ≜ W + log n ρ γ \\\\Gamma\\\\triangleq W+\\\\frac{\\\\log n}{\\\\rho\\\\gamma} represent the mixing overhead, the regret simplifies to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now replace the fixed budget with a data-dependent stopping rule to develop a confidence-based variant.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B CB-LEX: confidence-based stopping\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While LEX requires an oracle exploration budget, CB-LEX introduces a data-dependent stopping rule while retaining the canonical kernel U t U_{t} . A key distinction is the stopping certificate: the empirical leader’s lower confidence bound must dominate the upper confidence bounds (UCBs) of all competitors, rather than just the empirical runner-up. This refinement is essential because sample counts can be highly uneven; a poorly sampled arm might have a low empirical mean but a wide confidence interval that still overlaps with the leader. Comparing only against the runner-up would risk premature commitment before these wide intervals are sufficiently narrowed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each arm a ∈ A a\\\\in A at time t t , let μ ^ t ( a ) \\\\hat{\\\\mu}_{t}(a) be the empirical mean and φ t ( a ) \\\\varphi_{t}(a) the visitation count. We define the confidence width w t ( a ) ≜ ln ( 4 n T / δ ) / ( 2 φ t ( a ) ) w_{t}(a)\\\\triangleq\\\\sqrt{\\\\ln(4nT/\\\\delta)/(2\\\\varphi_{t}(a))} if φ t ( a ) ≥ 1 \\\\varphi_{t}(a)\\\\geq 1 , and w t ( a ) ≜ 1 w_{t}(a)\\\\triangleq 1 otherwise. The confidence bounds are LCB t ( a ) ≜ μ ^ t ( a ) − w t ( a ) \\\\operatorname{LCB}_{t}(a)\\\\triangleq\\\\hat{\\\\mu}_{t}(a)-w_{t}(a) and UCB t ( a ) ≜ μ ^ t ( a ) + w t ( a ) \\\\operatorname{UCB}_{t}(a)\\\\triangleq\\\\hat{\\\\mu}_{t}(a)+w_{t}(a) . Let b t ∈ arg max a ∈ A μ ^ t ( a ) b_{t}\\\\in\\\\arg\\\\max_{a\\\\in A}\\\\hat{\\\\mu}_{t}(a) be the current empirical leader. CB-LEX terminates exploration at the stopping time:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with σ ≜ T \\\\sigma\\\\triangleq T if the condition is never met. If σ < T \\\\sigma<T , the algorithm commits to a ^ ⋆ ≜ b σ \\\\hat{a}^{\\\\star}\\\\triangleq b\\\\sigma and initiates the same navigation phase used in LEX. This stopping rule ensures the leader is separated from all competitors, which is essential because sample counts can be highly uneven; a poorly sampled arm might have a low empirical mean but a wide confidence interval that still overlaps with the leader.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The exploration phase of CB-LEX is summarized in Algorithm 2 . Since the navigation logic remains identical to Algorithm 1 , it is omitted for brevity. The confidence widths w t ( a ) w_{t}(a) depend on the horizon T T only through the logarithm. While a fully horizon-free variant can be obtained by the standard replacement T ↦ ( t + 1 ) 2 T\\\\mapsto(t+1)^{2} , we retain the horizon-dependent form to keep the notation streamlined for the current analysis. As established in Theorem 2 , the adaptive stopping rule preserves the regret order of the LEX baseline. Although the logarithmic terms differ slightly to ensure a uniform confidence event over the horizon, both algorithms achieve the same asymptotic rate under the standard choice δ = T − 2 \\\\delta=T^{-2} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume the graph sequence is ( W , ρ , γ , π ) (W,\\\\rho,\\\\gamma,\\\\pi) -mixing and let τ 0 \\\\tau_{0} be as in Theorem 1. Define the complexity:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a constant C > 0 C>0 , if T ≥ τ 0 ⌈ C Ψ CB ( δ ) ⌉ ≜ T nom T\\\\geq\\\\tau_{0}\\\\lceil C\\\\Psi_{\\\\mathrm{CB}}(\\\\delta)\\\\rceil\\\\triangleq T_{\\\\mathrm{nom}} , then with probability at least 1 − δ 1-\\\\delta : a ^ ⋆ = a ⋆ \\\\hat{a}^{\\\\star}=a^{\\\\star} , the algorithm stops at σ ≤ T nom \\\\sigma\\\\leq T_{\\\\mathrm{nom}} , and R learn ( T ) ≤ T nom R_{\\\\mathrm{learn}}(T)\\\\leq T_{\\\\mathrm{nom}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proof shares the structural ingredients of the LEX analysis, with the addition of a coupling argument. We couple CB-LEX to a perpetual exploration walk that continues to follow the canonical kernel regardless of the stopping condition. Up to the time σ \\\\sigma , the actual trajectory and the perpetual walk are identical. Consequently, it suffices to show that the stopping condition is satisfied by a deterministic time on the perpetual walk. The mixing and visitation lemmas (Appendix LABEL:app:technical-lemmas ) provide the necessary sample counts at that time, and a uniform confidence event ensures the separation of intervals. The full argument is provided in Appendix LABEL:app:cblex-proof .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the conditions of Theorem 2 , setting δ = T − 2 \\\\delta=T^{-2} yields sublinear regret 𝔼 [ R ( T ) ] = o ( T ) \\\\mathbb{E}[R(T)]=o(T) . The expected regret is bounded by:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the d d -regular case, letting Γ ≜ W + log n ρ γ \\\\Gamma\\\\triangleq W+\\\\frac{\\\\log n}{\\\\rho\\\\gamma} , this simplifies to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Crucially, the confidence-based rule eliminates the requirement for a pre-specified exploration budget without incurring an asymptotic penalty. Under the mixing condition of Definition 1 , the same visitation guarantees that justify the LEX budget ensure that confidence intervals become disjoint within the same order of time. The fundamental distinction lies in the termination mechanism: LEX utilizes an external budget, whereas CB-LEX certifies stopping through observed data. The primary technical novelty in the analysis is the coupling to a perpetual exploration walk, which enables the evaluation of a random stopping rule using the deterministic-time bounds previously established for the canonical walk.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C RALEX: reward-aware exploration with a canonical floor\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While CB-LEX uses data only for its stopping rule, its movement remains reward-indifferent. RALEX introduces reward-awareness by biasing transitions toward promising arms. The difficulty is that a purely reward-driven kernel often fails to preserve the stationary distribution π \\\\pi required by Definition 1. RALEX instead employs a mixture that retains a fixed floor of canonical movement.At round t t , the learner uses the empirical means μ ^ t − 1 \\\\hat{\\\\mu}_{t-1} and widths w t − 1 w_{t-1} to form the local optimistic score ξ t − 1 ( a ) ≜ μ ^ t − 1 ( a ) + w t − 1 ( a ) \\\\xi_{t-1}(a)\\\\triangleq\\\\hat{\\\\mu}_{t-1}(a)+w_{t-1}(a) . We define a softmax proposal on the neighborhood L t ( a t − 1 ) L_{t}(a_{t-1}) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where λ ≥ 0 \\\\lambda\\\\geq 0 controls the bias. To ensure mixing, RALEX defines the weight ϵ t ≜ max { ϵ 0 , 2 w ¯ t − 1 / ( 1 + 2 w ¯ t − 1 ) } \\\\epsilon_{t}\\\\triangleq\\\\max\\\\{\\\\epsilon_{0},2\\\\bar{w}_{t-1}/(1+2\\\\bar{w}_{t-1})\\\\} , where w ¯ t − 1 ≜ max a ∈ A w t − 1 ( a ) \\\\bar{w}_{t-1}\\\\triangleq\\\\max_{a\\\\in A}w_{t-1}(a) and ϵ 0 ∈ ( 0 , 1 ] \\\\epsilon_{0}\\\\in(0,1] is a fixed floor. The exploration kernel is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RALEX adopts the same stopping rule as CB-LEX, terminating exploration when the leader’s lower bound separates from all competing upper bounds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm 3 details the RALEX exploration phase. Each round begins by calculating the optimistic proposal ξ t − 1 \\\\xi_{t-1} from the preceding round’s estimates and determining the canonical mixing weight ϵ t \\\\epsilon_{t} . Following the state transition and subsequent reward update, the algorithm refreshes the confidence widths and evaluates the stopping certificate established for CB-LEX. Although the exploration kernel is the primary algorithmic departure, the theoretical analysis must rigorously address the reward-dependence of the resulting visitation counts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The deterministic floor ϵ 0 \\\\epsilon_{0} in ( 10 ) provides the key structural guarantee for RALEX. It implies the pointwise lower bound:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The worst-case analysis relies exclusively on this inequality, ensuring the subsequent theorem holds for any bias parameter λ ≥ 0 \\\\lambda\\\\geq 0 . While this generality leads to a conservative bound; treating the biased component as an arbitrary perturbation; it preserves the rigorous structural properties established for the canonical walk.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume the graph sequence is ( W , ρ , γ , π ) (W,\\\\rho,\\\\gamma,\\\\pi) -mixing. For bias λ ≥ 0 \\\\lambda\\\\geq 0 and floor ε 0 ∈ ( 0 , 1 ] \\\\varepsilon_{0}\\\\in(0,1] , let τ 0 \\\\tau_{0} be the mixing time from Theorem 1 and ϑ 0 ≜ 1 2 ε 0 τ 0 \\\\vartheta_{0}\\\\triangleq\\\\frac{1}{2}\\\\varepsilon_{0}^{\\\\tau_{0}} . Define the complexity:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a constant C > 0 C>0 , if T ≥ T nom ≜ τ 0 ⌈ C Ψ RA ( δ ) ⌉ T\\\\geq T_{\\\\mathrm{nom}}\\\\triangleq\\\\tau_{0}\\\\lceil C\\\\Psi_{\\\\mathrm{RA}}(\\\\delta)\\\\rceil , then with probability at least 1 − δ 1-\\\\delta : a ^ ⋆ = a ⋆ \\\\hat{a}^{\\\\star}=a^{\\\\star} , the algorithm stops at σ ≤ T nom \\\\sigma\\\\leq T_{\\\\mathrm{nom}} , and R learn ( T ) ≤ T nom R_{\\\\mathrm{learn}}(T)\\\\leq T_{\\\\mathrm{nom}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The analysis of RALEX introduces two primary technical refinements beyond the CB-LEX framework. The first is statistical: because the exploration kernel is reward-dependent, sample counts are no longer independent of the reward history. We address this in Appendix LABEL:app:ralex-proof using a uniform confidence lemma for adaptive sampling derived from a reward-table construction. The second is structural: since RALEX deviates from the canonical walk, the standard visitation lemma for U t U_{t} does not apply directly. Instead, we show that each block of τ 0 \\\\tau_{0} rounds behaves as a canonical walk with probability at least ε 0 τ 0 \\\\varepsilon_{0}^{\\\\tau_{0}} , yielding a thinned visitation guarantee with success rate ϑ 0 π ( a ) \\\\vartheta_{0}\\\\pi(a) . With these ingredients, the stopping-time analysis follows the CB-LEX pattern.The resulting bound is conservative compared to CB-LEX, incurring a factor ϑ 0 − 1 = 2 ε 0 − τ 0 \\\\vartheta_{0}^{-1}=2\\\\varepsilon_{0}^{-\\\\tau_{0}} by relying strictly on the canonical component from ( 11 ). This constitutes the theoretical price for permitting the biased kernel component to depend arbitrarily on observations. At ε 0 = 1 \\\\varepsilon_{0}=1 , RALEX simplifies to CB-LEX and the theorem recovers the CB-LEX rate. Notably, this result generalizes to any local proposal kernel mixed with the same canonical floor.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the conditions of Theorem 3 , let δ = T − 2 \\\\delta=T^{-2} and T nom ≜ τ 0 ⌈ C Ψ RA ( T − 2 ) ⌉ T_{\\\\mathrm{nom}}\\\\triangleq\\\\tau_{0}\\\\lceil C\\\\Psi_{\\\\mathrm{RA}}(T^{-2})\\\\rceil . The expected regret of RALEX satisfies:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Asymptotically, RALEX achieves sublinear regret 𝔼 [ R ( T ) ] = o ( T ) \\\\mathbb{E}[R(T)]=o(T) bounded by:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RALEX represents a distinct trade-off within the algorithmic design space. While LEX and CB-LEX achieve the sharpest worst-case guarantees by adhering strictly to the canonical walk, RALEX sacrifices a portion of this theoretical sharpness to prioritize transitions toward locally promising arms. Theorem 3 serves as a worst-case baseline; it establishes that reward-aware bias is compatible with sublinear regret, though it does not formally certify a performance gain over the baseline. A genuine improvement requires an additional condition: the optimistic proposal must successfully allocate probability mass toward the specific arms that determine the stopping time. The following result isolates this mechanism.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-D When reward awareness yields a provable gain\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Theorem 3 establishes that optimistic bias is compatible with canonical exploration, but it does not specify a regime where this bias provably enhances performance. To characterize these gains, we analyze the perpetual exploration process { X ~ t } t ≥ 0 \\\\{\\\\widetilde{X}_{t}\\\\}_{t\\\\geq 0} , which follows the RALEX kernel indefinitely without transitioning to navigation.Fix a deterministic burn-in time t gain ∈ { 0 , … , T } t_{\\\\mathrm{gain}}\\\\in\\\\{0,\\\\dots,T\\\\} and define the mixing parameters:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where π ∗ ≜ min a ∈ A π ( a ) \\\\pi_{*}\\\\triangleq\\\\min_{a\\\\in A}\\\\pi(a) . For each block index j ≥ 1 j\\\\geq 1 , let s j ≜ t gain + ( j − 1 ) τ 1 s_{j}\\\\triangleq t_{\\\\mathrm{gain}}+(j-1)\\\\tau_{1} and define the filtration 𝒢 ~ j − 1 ≜ σ ( ℱ ~ s j ) \\\\widetilde{\\\\mathcal{G}}_{j-1}\\\\triangleq\\\\sigma(\\\\widetilde{\\\\mathcal{F}}_{s_{j}}) . The quantity Pr ( X ~ s j + τ 1 = a ∣ 𝒢 ~ j − 1 ) \\\\Pr(\\\\widetilde{X}_{s_{j}+\\\\tau_{1}}=a\\\\mid\\\\widetilde{\\\\mathcal{G}}_{j-1}) represents the probability that the j j -th post-burn-in block terminates at arm a a . Because optimistic scores are recomputed after every reward observation within a block, a one-step bound on the proposal kernel is insufficient for the online algorithm. Consequently, our analysis utilizes these block-end probabilities directly to quantify the visitation benefits of the reward-aware bias.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume the conditions of Theorem 3 and let { X ~ t } \\\\{\\\\widetilde{X}_{t}\\\\} be the perpetual exploration process. Fix a subset B ⊆ A B\\\\subseteq A containing a ⋆ a^{\\\\star} and constants κ gain ( a ) ≥ ϑ 1 π ( a ) \\\\kappa_{\\\\mathrm{gain}}(a)\\\\geq\\\\vartheta_{1}\\\\pi(a) for a ∈ B a\\\\in B . Suppose that for every block index j ≥ 1 j\\\\geq 1 with s j + τ 1 ≤ T s_{j}+\\\\tau_{1}\\\\leq T :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Define κ ( a ) ≜ κ gain ( a ) \\\\kappa(a)\\\\triangleq\\\\kappa_{\\\\mathrm{gain}}(a) for a ∈ B a\\\\in B and κ ( a ) ≜ ϑ 1 π ( a ) \\\\kappa(a)\\\\triangleq\\\\vartheta_{1}\\\\pi(a) otherwise. Let the complexity be:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a constant C > 0 C>0 , if T ≥ T nom ≜ t gain + τ 1 ⌈ C Ψ RA gain ( δ ) ⌉ T\\\\geq T_{\\\\mathrm{nom}}\\\\triangleq t_{\\\\mathrm{gain}}+\\\\tau_{1}\\\\lceil C\\\\Psi_{\\\\mathrm{RA}}^{\\\\mathrm{gain}}(\\\\delta)\\\\rceil , then with probability at least 1 − δ 1-\\\\delta : a ^ ⋆ = a ⋆ \\\\hat{a}^{\\\\star}=a^{\\\\star} , σ ≤ T nom \\\\sigma\\\\leq T_{\\\\mathrm{nom}} , and R learn ( T ) ≤ T nom R_{\\\\mathrm{learn}}(T)\\\\leq T_{\\\\mathrm{nom}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The theorem is formulated using block-end probabilities, as these are the fundamental objects in our analysis. Appendix LABEL:app:ralex-gain-proof provides a sufficient condition for verifying this gain. Let 𝒞 j \\\\mathcal{C}_{j} be the event that the first τ 1 − 1 \\\\tau_{1}-1 rounds of block j j use the canonical component of RALEX. If the final RALEX step in that block satisfies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"then the block-end coefficient can be taken to be κ gain ( a ) = ( 1 + 2 β ( a ) / ε 0 ) ϑ 1 π ( a ) . \\\\kappa_{\\\\mathrm{gain}}(a)=\\\\left(1+2\\\\beta(a)/\\\\varepsilon_{0}\\\\right)\\\\vartheta_{1}\\\\pi(a). In other words, the improvement factor is determined by the amount of extra endpoint mass that the last reward-aware step contributes above the canonical floor.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To interpret when this yields a genuine reduction of the global learning term, define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let Λ max := max a ∈ A Λ ( a ) \\\\Lambda_{\\\\max}:=\\\\max_{a\\\\in A}\\\\Lambda(a) . Thus the leading learning term in the safe RALEX bound is proportional to ϑ 1 − 1 log ( 4 n T / δ ) Λ max . \\\\vartheta_{1}^{-1}\\\\log(4nT/\\\\delta)\\\\,\\\\Lambda_{\\\\max}. For a factor ζ > 0 \\\\zeta>0 , define the ζ \\\\zeta -near-bottleneck set\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These are the arms whose safe contribution lies within a factor 1 + ζ 1+\\\\zeta of the worst one. We want to control this set as improving only the exact maximizers need not reduce the global maximum, because a near-bottleneck arm left unimproved can become the new dominant term after the gain is applied.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assume the conditions of Theorem 4 , fix ζ > 0 \\\\zeta>0 , and suppose that ℋ ζ ⊆ B \\\\mathcal{H}_{\\\\zeta}\\\\subseteq B and κ gain ( a ) ≥ ( 1 + ζ ) ϑ 1 π ( a ) for every a ∈ ℋ ζ . \\\\kappa_{\\\\mathrm{gain}}(a)\\\\geq(1+\\\\zeta)\\\\vartheta_{1}\\\\pi(a)\\\\text{ for every }a\\\\in\\\\mathcal{H}_{\\\\zeta}. Then\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consequently,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Relative to the safety theorem for RALEX, the leading learning term decreases by the factor ( 1 + ζ ) − 1 (1+\\\\zeta)^{-1} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is the strongest multiplicative improvement that Theorem 4 certifies without additional structure on the reward-aware proposal: the gain must cover every arm that could otherwise remain, or become, the dominant term.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV Numerical Simulations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We empirically ground our theoretical guarantees with a series of simulations designed to evaluate the practical performance of the proposed algorithm family. Our experiments, summarized in Figure 2 , focus on isolating the spatial behavior of different exploration strategies and quantifying their efficiency in identifying optimal regions. Specifically, we evaluate (a) mean cumulative regret on the Hard instance and (b) the distribution of adaptive stopping times across both Easy and Hard instances.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Environment and Reward Model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The environment is modeled as a dynamic graph G t = ( A , E t ) G_{t}=(A,E_{t}) defined over a discrete set of n = | A | = 205 n=|A|=205 nodes, simulated over a total horizon of T = 70 , 000 T=70,000 rounds. The network topology evolves via degree-preserving edge switches, maintaining a fixed degree sequence across all snapshots and thus a common stationary distribution π \\\\pi as required by Definition 1 . Upon visiting a node a ∈ A a\\\\in A , the agent observes a stochastic reward r t ( a ) ∈ [ 0 , 1 ] r_{t}(a)\\\\in[0,1] characterized by an underlying mean μ ( a ) \\\\mu(a) perturbed by bounded, zero-mean noise.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Experimental Design.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate performance on two distinct reward instances. The hard instance features a small suboptimality gap, with a single optimal node μ ( a ⋆ ) = 0.95 \\\\mu(a^{\\\\star})=0.95 and several competing arms in the range [ 0.45 , 0.65 ] [0.45,0.65] , requiring significant evidence to distinguish the true optimum. Conversely, the easy instance features a large gap where suboptimal arms have low rewards in the range [ 0.1 , 0.2 ] [0.1,0.2] , allowing for rapid identification. For the dynamic graph, we set the edge appearance probability to 0.1 and the disappearance probability to 0.01. For the learning algorithms, we set the reward-bias to λ = 10 \\\\lambda=10 for RA-LEX . Results are averaged over five random seeds, with shaded regions in regret curves representing one standard deviation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Results and Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Visualizing Exploration Strategies: Figure 3 provides the visual intuition behind the performance differences by comparing exploration-phase visitation patterns. The heatmap for CB-LEX is relatively uniform, showing a “blind” search of the entire node set. In stark contrast, the RA-LEX heatmap is highly concentrated, resembling a “searchlight” focused on the true high-reward zones. This visually confirms that its biased random walk successfully prioritizes promising regions, explaining its superior sample efficiency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Efficiency of Reward-Aware Exploration: As shown in Figure 2 a, both algorithms achieve sublinear regret, indicated by the eventual flattening of their regret curves. However, the reward-aware RA-LEX achieves significantly lower final regret compared to the uniform exploration of CB-LEX , corroborating the benefit of biasing the exploration walk toward high-reward regions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Validation of the Adaptive Mechanism: Figure 2 b confirms that the adaptive stopping rule correctly responds to statistical difficulty, with both algorithms stopping significantly earlier on the Easy instance. On the Hard instance, efficiency gains are pronounced: RA-LEX stops at a median of 1,850 rounds with a remarkably tight interquartile range (IQR) of 1,600 to 1,900. In contrast, CB-LEX requires a median of 6,000 rounds and exhibits a much wider variance (IQR of 4,500 to 7,500). This confirms that reward-aware biasing accelerates best-arm identification by nearly 70% and makes the required exploration duration significantly more predictable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Robustness: Further sensitivity analysis identifies a “sweet spot” for the bias parameter λ \\\\lambda that successfully balances exploiting high-reward regions with maintaining sufficient stochasticity to escape local optima.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We investigated stochastic multi-armed bandits under local movement constraints on dynamic graphs, a setting where the structural necessity of post-exploration navigation decouples identification from exploitation. Because exploration concludes at an uncertain stopping time, whole-horizon connectivity is an insufficient metric for learnability; instead, structural properties must be controlled over the subsequent random suffix. This requirement led to our process-agnostic approach of placing assumptions directly on the graph’s intrinsic walk.By imposing a mixing condition on the canonical walk, we established sublinear expected regret for a family of explore-then-commit algorithms. Our framework includes a fixed-budget baseline (LEX), an adaptive variant (CB-LEX) that eliminates oracle dependence, and a reward-aware strategy (RALEX) that maintains worst-case safety while providing provable performance gains.Future research will focus on weakening these structural requirements and extending the analysis to broader local kernels, such as Metropolis-type walks, to further expand the positive regime for bandit learning on dynamic networks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"new_app\"\n }\n]","isConversational":false}}},{"rowIdx":390,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10569"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"When Does Restricting a Coding Agent to execute_code Help? A Regime $\\times$ Agent-Design Ablation"},"authors":{"kind":"list like","value":["Hong Yang","Qi Yu","Travis Desell"],"string":"[\n \"Hong Yang\",\n \"Qi Yu\",\n \"Travis Desell\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10569v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1. Introduction"},{"type":"para","text":"Coding agents now ship with overlapping tool surfaces—Read, Grep, Glob, Edit, Write, and Bash on Claude Code; ACI primitives on SWE-agent; an MCP execute_code tool on emerging stacks—and the field has settled on three mutually incompatible claims about which surface a coding agent should expose. Specialised IDE primitives are required: SWE-agent argues that the Agent–Computer Interface (ACI) is load-bearing, with an ACI-vs-shell ablation that produced a gain of + 10.7 +10.7 percentage points on SWE-bench at the time of publication (Yang et al . , 2024 ) . Bash alone is sufficient: mini-SWE-agent is a ∼ \\sim 100-line bash-only scaffold that posts > 74 % {>}74\\% on SWE-bench Verified with no IDE primitives at all (SWE-agent contributors, 2025 ) . Replace the tool surface with code execution: industry write-ups report ∼ 98 {\\sim}98 – 99 % 99\\% token reductions by routing through an MCP execute_code tool rather than calling the native tools directly (Anthropic, 2025 ; Varda and Pai, 2025 ; Cloudflare, 2026 ) . The three prescriptions have never been crossed on a single harness, on the same model, with regime stratification, so the field cannot tell which condition each applies under, or whether they contradict each other at all."},{"type":"para","text":"We run the missing crossed comparison: the same harness, the same models, and the same prompts, evaluated under three tool surfaces ( baseline , bash_only , code_only ) crossed with two task regimes (computation, modification) and two agents (Claude Code, Codex CLI), all under an integrity-clean SWE-bench protocol with post-agent test_patch application (§ 3.3 ) (Jimenez et al . , 2024 ) . The baseline arm exposes each agent’s default tool surface; bash_only restricts to native bash plus read/glob/grep, with Edit and Write removed; code_only replaces the entire tool surface with a single MCP tool— mcp__codebox__execute_code on a persistent Python+Bash REPL—and disallows every native built-in (Anthropic, 2024 ) . The modification cell is SWE-bench Mini ( n = 100 n=100 , canonical post-agent test_patch protocol); the computation cell is a set of self-contained tasks each defined by a (workspace, hidden grader, reference output) triple with deterministic offline scoring (§ 3.6 ). The crossed 3 × 2 × 2 3\\times 2\\times 2 design is the minimum that disambiguates the three claims of § 1 , because each claim’s scope (which surface, which regime, which agent) is exactly what is in dispute. The computation cell is a methodological vehicle, not a benchmark contribution—the artifact-benchmark space is already crowded (§ 2.4 )."},{"type":"para","text":"None of the prior data points on this axis is the crossed comparison we run. SWE-agent’s two-arm ACI-vs-shell ablation (Yang et al . , 2024 ) and mini-SWE-agent’s single bash-only arm on SWE-bench Verified (SWE-agent contributors, 2025 ) both fix a single tool-surface axis on the modification regime, with no code-execution arm and no regime stratification. The industry write-ups on code-execution surfaces (Anthropic, 2025 ; Varda and Pai, 2025 ; Cloudflare, 2026 ) target external-MCP tool surfaces (Drive, Salesforce, Stripe, Cloudflare’s API) rather than internal IDE primitives and report no benchmark and no regime split. Liu et al. offer an architectural taxonomy of Claude Code’s primitives explicitly without ablation (Liu et al . , 2026b ) ; the Verdent technical report describes a single-vendor informal ablation without public methodology or regime stratification (Verdent, 2025 ) . No prior work has crossed three internal IDE surfaces under regime stratification with both a Claude- and a Codex-family agent, so the three claims of § 1 have remained unfalsifiable at the population level."},{"type":"para","text":"We make three contributions."},{"type":"heading","level":4,"text":"(1) A four-cell agent-conditional cost structure."},{"type":"para","text":"The four cells are the cross of the two task regimes with the two agents—{computation (Artifact), modification (SWE-bench)} × \\times {Claude, Codex}. Across these four (regime × \\times agent) cells, code_only is cheaper in three cells (significantly in two, directionally in one) and directionally more expensive in the fourth—Artifact/Claude ( Δ cost adj. = − 24.595 857 141 034 71 % \\Delta_{\\text{cost adj.}}=$-24.595\\,857\\,141\\,034\\,71$\\% , p = 7.365 478 099 521 983 × 10 − 14 p=$7.365\\,478\\,099\\,521\\,983\\text{\\times}{10}^{-14}$ ), Artifact/Codex ( − 6.697 940 133 090 269 % $-6.697\\,940\\,133\\,090\\,269$\\% , directional), and SWE-bench/Codex ( − 19.908 625 337 034 56 % $-19.908\\,625\\,337\\,034\\,56$\\% , p = 2.016 919 093 053 276 2 × 10 − 09 p=$2.016\\,919\\,093\\,053\\,276\\,2\\text{\\times}{10}^{-09}$ ) all favour code_only , while SWE-bench/Claude reverses ( 14.437 541 589 685 27 % $14.437\\,541\\,589\\,685\\,27$\\% , NS; full table at § 5 , Table 2 ). The same restriction is cheaper under three (regime, agent) combinations (significantly under two) and costlier under one, so surface choice is jointly determined by regime AND agent design, not by either alone; the three prior claims of § 1 coexist because each is true under its specific (regime, agent) combination and false under the others."},{"type":"heading","level":4,"text":"(2) A three-mechanism causal decomposition."},{"type":"para","text":"We decompose the four-cell structure into three causally distinct mechanisms whose joint footprint matches it; no single mechanism predicts the full table. Path-cost (edit friction): code_only must express every file edit as a Python script, and on Claude SWE-bench this scales output tokens with edit volume (per-instance Spearman ρ = 0.488 054 269 591 964 4 \\rho=$0.488\\,054\\,269\\,591\\,964\\,4$ , p = 2.594 788 687 130 005 3 × 10 − 07 p=$2.594\\,788\\,687\\,130\\,005\\,3\\text{\\times}{10}^{-07}$ ), driving the lone Claude × \\times SWE-bench output-token blow-up (§ 6.1 ). Failure-cost: the Claude × \\times SWE-bench cost overrun localises to unanimous-fail and split instances; on the unanimous-pass subset the gap collapses sharply (§ 6.3 ). Pricing asymmetry: on Codex, execute_code batches operations per LLM call ( ∼ 2.5 \\sim\\!2.5 tool calls per LLM step vs. ∼ 1.0 \\sim\\!1.0 for baseline ) and lets the agent constrain per-call output volume programmatically, which produces the SWE-bench/Codex win without changing LLM-call count—an effect absent on Claude (§ 6.2 )."},{"type":"heading","level":4,"text":"(3) Capability invariance, with empirical receipt."},{"type":"para","text":"Pass rates differ by less than three percentage points across all four cells × \\times three arms, with all four headline pass-rate contrasts NS (Table 2 ); and on a strict 9 / 9 9/9 -trial unanimity criterion the majority of instances in every cell are unanimously decided, so task outcome does not vary with tool surface for the bulk of the corpus. Tool surface changes the path and the cost, not the answer. This dissociation is what licenses contributions (1) and (2) as cost-engineering claims rather than capability claims, and it is consistent with the Capability Overlap Principle (Zhang et al . , 2026 ) ; the load-bearing claim is the empirical agreement-matrix result, not the theoretical frame."},{"type":"para","text":"Table 1 condenses these four cells into an at-a-glance map: the code_only cost outcome against its cheapest rival, whether that outcome is statistically significant or merely directional, and the operational lesson each cell licenses."},{"type":"heading","level":1,"text":"2. Related Work"},{"type":"heading","level":2,"text":"2.1. IDE-tool-surface ablation on coding agents"},{"type":"para","text":"The closest precedent is SWE-agent, whose two-arm Agent–Computer Interface (ACI) vs. raw shell ablation established that structured IDE primitives beat free-form shell at the model size of late 2024 2024 (Yang et al . , 2024 ) . Two subsequent points have left the conclusion partly stale: mini-SWE-agent, a ∼ 100 {\\sim}\\!100 -line bash-only scaffold, posts competitive numbers on SWE-bench Verified with no IDE primitives at all (SWE-agent contributors, 2025 ) , and Live-SWE-agent shows a self-evolving scaffold seeded from it can climb higher (Xia et al . , 2025 ) . Engineered surfaces continue to be explored on the other end, layering AST-aware retrieval, semantic search, or fixed localize–repair–validate pipelines on the SWE-agent line (Zhang et al . , 2024 ; Örwall, 2025 ; Xia et al . , 2024 ) . None of these crosses three internal IDE surfaces under a single integrity-clean harness with regime stratification; each fixes a single axis. Descriptive surveys of Claude Code’s primitive surface (Liu et al . , 2026b ; Rombaut, 2026 ; Bui, 2026 ) and component-evolution work on coding-agent harnesses (Wong et al . , 2025 ; Lin et al . , 2026 ) do not swap the entire surface either. The vendor signal closest in spirit to ours is the Verdent technical report (Verdent, 2025 ) , which informally notes that removing advanced tools while keeping bash + + read + + write + + edit produced little SWE-bench Verified change; single-vendor and without public methodology, we treat it as motivation rather than a scoop."},{"type":"heading","level":2,"text":"2.2. Code execution as the action interface"},{"type":"para","text":"The code_only arm sits in a line of work that replaces structured tool calls with executable code. PAL and Program-of-Thoughts established Python as the medium of model reasoning for arithmetic and symbolic computation (Gao et al . , 2023 ; Chen et al . , 2023 ) ; CodeAct extended the move to the action layer of agentic systems, contrasting executable Python with JSON tool calls on general-purpose agent benchmarks (Wang et al . , 2024 ) , and the OpenHands platform (Wang et al . , 2025 ) ships the CodeAct interface at scale. The most directly relevant industry signals are Anthropic’s “code execution with MCP” write-up and Cloudflare’s two “Code Mode” posts, which report large token reductions when an LLM writes code that calls external MCP tools rather than calling them directly (Anthropic, 2025 ; Varda and Pai, 2025 ; Cloudflare, 2026 ) . All four target external tool surfaces — Drive, Salesforce, Stripe, Cloudflare’s REST API — not the agent’s internal IDE primitives, and none reports a coding-benchmark evaluation. Bechard et al. (Bechard et al . , 2026 ) argue the analogous “less is more” point for enterprise API automation, again outside the code-repair domain. Our code_only arm ports code-as-action into the agent’s IDE-primitive layer and evaluates it on the benchmarks where Claude Code, Codex CLI, and their peers actually run."},{"type":"heading","level":2,"text":"2.3. Tool-use tax and capability overlap"},{"type":"para","text":"We frame the surface choice through Zhang et al.’s “tool-use tax” inequality: a tool delivers net benefit only when its task-specific capability gain exceeds the per-call cost of carrying its definition in the prompt (Zhang et al . , 2026 ) . Their evidence is on general math and question-answering; ours supplies the coding-agent receipt. The orthogonal tool-pruning and tool-budgeting literature — Budget-Aware Tool Use (Liu et al . , 2025 ) , ToolTree (Yang et al . , 2026 ) , Trajectory Reduction (Xiao et al . , 2025 ) , ToolScope (Liu et al . , 2026a ) , and the MCP description audit of Hasan et al. (Hasan et al . , 2026 ) — all ask which k k tools to select from a large pool, holding the surface design itself fixed. We instead compare three fixed surfaces on the same harness. Fan et al.’s SWE-Effi (Fan et al . , 2025 ) introduces resource-aware effectiveness metrics for coding agents and reports that failed runs consume an order of magnitude more compute than successes; this “expensive failures” framing motivates the cache-adjusted cost methodology in § 3.5 and is consistent with the failure-cost mechanism we identify on SWE-bench/Claude (§ 6.3 )."},{"type":"heading","level":2,"text":"2.4. Coding-agent benchmarks"},{"type":"para","text":"Our paired evaluation spans both halves of an established benchmark axis. On the modification side, SWE-bench (Jimenez et al . , 2024 ) and its human-validated Verified subset (OpenAI Preparedness Team, 2024 ) are the foundation; later benchmarks that build on them — SWE-bench Pro (Deng et al . , 2025 ) , Multi-SWE-bench (Zan et al . , 2025 ) , SWE-PolyBench (Rashid et al . , 2025 ) , and SWE Atlas (Raghavendra et al . , 2026 ) — extend SWE-bench along difficulty, language, and task-shape axes, but never along the agent’s tool surface. Ganhotra’s audit (Ganhotra, 2025 ) that the bulk of SWE-bench Verified is single-file motivates our targeted multi-file sampling. On the computation side, MLE-Bench (Chan et al . , 2024 ) and MLAgentBench (Huang et al . , 2024 ) grade Kaggle-style submissions by the numerical output of agent-written code, RE-Bench (Wijk et al . , 2024 ) pushes the same shape to longer-horizon ML R&D, CORE-Bench (Siegel et al . , 2024 ) grades reproduction of published numerical results, and SWE-Lancer (Miserendino et al . , 2025 ) mixes modification and computation in a single corpus. We do not propose a new computation benchmark — the space is crowded. The artifact suite we run is the methodological vehicle that makes the four-cell agent-conditional cost structure of § 5 observable at all: without a computation cell on the same harness, every (regime, agent) contrast collapses to a single column of SWE-bench numbers, and the same-regime divergence between Claude and Codex on SWE-bench cannot be read against any computation baseline."},{"type":"para","text":"This paper is the empirical complement to Liu et al.’s architectural taxonomy of Claude Code’s primitive surface (Liu et al . , 2026b ) : they describe the design space; we ablate three points in it under regime and agent-design stratification, on the benchmark pair that spans the field’s two graded regimes. Observational mining of in-the-wild agent pull requests, exemplified by the SE 3.0 anchor dataset AIDev (Li et al . , 2025 ) , holds neither the model nor the tool surface fixed and so does not substitute for the controlled ablation we report."},{"type":"heading","level":1,"text":"3. Method"},{"type":"heading","level":2,"text":"3.1. Three-arm tool surface"},{"type":"para","text":"Each arm is defined by what tools are exposed to the agent at process launch; the same Claude Code or Codex CLI binary runs all three. The baseline arm passes no restriction flag and exposes the agent’s default surface — Read, Grep, Glob, Edit, Write, Bash, and the agent-built-in subagents on Claude Code; the analogous file primitives, shell tool, and structured apply_patch on Codex. The bash_only arm restricts to bash plus read-only browsing primitives, disallowing every file-editing built-in ( Edit / Write / MultiEdit / NotebookEdit on Claude; shell_tool with structured patching disabled on Codex). The code_only arm replaces the entire surface with a single MCP tool, mcp__codebox__execute_code , served by our exec-server stack as a persistent Python+Bash REPL keyed by working directory across calls; all native built-ins are explicitly disallowed."},{"type":"para","text":"Two implementation asymmetries are worth surfacing. First, Codex exposes no feature flag for disabling its structured apply_patch tool, so on Codex the bash_only and code_only arms enforce restriction with a prompt-prefix directive rather than a hard flag; this is soft enforcement and model compliance is not guaranteed. We measure and report explicit apply_patch leakage in § 4.2 and find it empirically negligible ( < 1 % {<}1\\% on both restricted arms)—a bound on registered tool calls, not behavioral compliance (see Limitations). Second, code_only depends on REPL state persisting between execute_code invocations: without a persistent kernel, the round-trip savings of “one script per task” collapse, so the exec-server pins the persistent-kernel environment flag for the runs that feed § 5 ."},{"type":"para","text":"The bash_only arm matches the tool surface of mini-SWE-agent (SWE-agent contributors, 2025 ) but does not reuse its loop, prompt template, or scaffold; we apply the bash-only restriction to the same Claude Code and Codex binaries the other two arms use, so the contrast isolates the tool surface variable rather than comparing scaffold implementations."},{"type":"heading","level":2,"text":"3.2. Harness design"},{"type":"heading","level":4,"text":"Per-instance setup (SWE-bench)."},{"type":"para","text":"Each SWE-bench instance is cloned at its base_commit and a per-instance Python virtual environment is built against it. The repository and the virtual environment are each given their own read-only fuse-overlayfs lower directory, so a per-arm upper directory captures every write and is discarded between arms — no re-clone, no pip install -e . between arms. The virtual-environment overlay is mounted back at its original creation path because console-script shebangs hard-code that absolute path at install time. Artifact tasks are self-contained and skip this step."},{"type":"heading","level":4,"text":"Git-history strip."},{"type":"para","text":"Before the agent runs, the worktree is collapsed to a single orphan commit at its current tree; all refs, packed refs, reflogs, and alternates are deleted, and git gc --prune=now repacks the resulting object set. The agent cannot recover the upstream reference fix via git log , git show , or the reflog. Fixed author and committer dates make the orphan SHA deterministic for a given tree, so repeated runs strip to the same state."},{"type":"heading","level":4,"text":"Overlay refresh between arms."},{"type":"para","text":"git reset cannot un-create files an earlier arm added (fuse-overlayfs copy-up plus EEXIST semantics), so the harness unmounts, deletes the upper and work directories, recreates them, remounts, and re-strips history between arms. The agent’s view of the merged mount path is stable; the underlying state is fresh."},{"type":"heading","level":4,"text":"Per-arm subprocess isolation."},{"type":"para","text":"Each agent invocation creates a fresh temporary configuration directory containing only the credentials and minimal config needed for the run, and points the binary at it via CLAUDE_CONFIG_DIR or CODEX_HOME . No session-persistence files, no cross-run state. Both binaries run in headless mode with permission prompts and sandbox / approval gates bypassed."},{"type":"heading","level":4,"text":"Wall-time cap."},{"type":"para","text":"Each invocation runs in a fresh process group; on timeout the harness sends SIGKILL to the entire group (so subagents and spawned shells die with the parent) and the run is recorded FAIL with a synthetic wall_timeout JSONL record. The numerical wall budget is specified in § 4 ."},{"type":"heading","level":2,"text":"3.3. Evaluation integrity"},{"type":"para","text":"The canonical SWE-bench protocol applies the gold test_patch only after the agent terminates: at run time the agent observes the repository at base_commit , and the tests used for grading are not present in the worktree (Jimenez et al . , 2024 ) . Our harness implements deferred application explicitly. The test_patch is applied only after the agent’s process exits, and a pytest --collect-only gate then runs against the patched tree to confirm that the held-out tests are collectible. A zero-item collection scores FAIL rather than letting the test runner silently report success against a non-existent suite."},{"type":"para","text":"Because our agents — unlike the bash-only patch-submission scaffold used in the original SWE-bench evaluation — can write arbitrary files anywhere in the worktree, the canonical “apply the patch” step is preceded by two cleanup operations. Every file that the test_patch targets is restored from HEAD to discard any agent edits to it, and every path the patch lists as a new file is removed from the worktree so an agent-created file cannot collide with a held-out test and block the apply. git apply is then invoked against the resulting tree. If the apply still fails — almost always because an agent-created file could not safely be removed in the second step — the instance is recorded FAIL with reason post-agent git apply failed: agent edits conflicted with the held-out test patch , rather than scored against a tree we cannot trust. This is strictly more conservative than the canonical protocol, which assumes a clean apply."},{"type":"heading","level":2,"text":"3.4. The Capability Overlap framing"},{"type":"para","text":"We adopt the tool-use tax framing of Zhang et al. (Zhang et al . , 2026 ) : a tool delivers net benefit iff its task-specific capability gain exceeds the per-call cost of carrying its definition in the prompt. Most Claude Code IDE primitives are bash-subsets in capability (Read, Grep, Glob, and Write reduce to cat , grep , find , and shell-heredoc); Edit alone has non-overlapping capability — atomic byte-precise replacement with linting — awkward to express in bash. The per-call tax is task-invariant while the gain term is regime- and call-pattern-dependent, so § 5 reads the four-cell cost structure as the empirical outcome of this inequality and § 6 decomposes it into three mechanisms the framework alone does not determine."},{"type":"heading","level":2,"text":"3.5. Cache isolation and cost reporting"},{"type":"para","text":"Across the four surfaces this paper touches — the Anthropic Messages API, the Claude Code CLI, the OpenAI Responses API, and the OpenAI Codex CLI — no documented parameter, flag, environment variable, or namespace setting forces a server-side cache miss or scopes cached tokens to a single caller. The only documented mechanism is implicit: change the cached content so the cache key changes. The vendor cache primitives we surveyed — Anthropic’s cache_control and context-editing, OpenAI’s prompt_cache_key and Responses stored-state, and the two CLIs’ session flags — set cache breakpoints, routing hints, or local session state, none of which bypass the cache or scope it per caller (Anthropic, 2026b ; OpenAI, 2026b ) . We therefore cannot guarantee per-task cache isolation in the reported numbers and instead address the resulting non-stationarity in the cost-reporting methodology."},{"type":"para","text":"Cost in § 5 is reported as a single cache-adjusted column, derived from the underlying token-billed formula cost = input_tokens ⋅ r in + output_tokens ⋅ r out \\text{cost}=\\text{input\\_tokens}\\cdot r_{\\text{in}}+\\text{output\\_tokens}\\cdot r_{\\text{out}} (every token billed at non-cached rates; the per-model rates r in , r out r_{\\text{in}},r_{\\text{out}} are pinned in § 4 ) plus a per-arm median-floor adjustment. For each arm, let M arm M_{\\text{arm}} be the median across that arm’s instances of first-turn cache_read_input_tokens — empirically the floor representing “system prompt plus tool definitions are warm in cache.” Any instance whose first-turn cache_read falls below M arm M_{\\text{arm}} is bumped up to M arm M_{\\text{arm}} for accounting; instances already at or above the floor and all multi-turn cache reads are unchanged. The adjustment normalises the system-prompt warm/cold lottery without claiming we achieved isolation."},{"type":"para","text":"The adjustment stops at the shared system-prompt+tool-definition prefix; we do not claim control over post-prefix cache sharing. Such sharing is empirically negligible: prompt caches key on a prefix hash invalidated by any byte difference, and past the system prompt every turn diverges per task (fresh task statement, task-conditional response, task-specific tool-call payloads). Per-instance first-turn cache_read clusters tightly around each arm’s system-prompt token count, with no bimodal tail consistent with cross-task content reuse. We do not report raw API-billed cost: it is sensitive to run order and ambient cache state we cannot control."},{"type":"heading","level":2,"text":"3.6. The artifact-suite contract"},{"type":"para","text":"The artifact suite is a new benchmark; this subsection is the methodology layer that defends it the way § 3.3 defends our SWE-bench protocol. Per-category counts and what each category probes are reported in § 4 ; this section specifies the contract, not the inventory."},{"type":"heading","level":4,"text":"Task contract."},{"type":"para","text":"Every artifact instance is a four-part bundle: a task.yaml (problem statement, budget fields, and pointers), a workspace/ of public files copied into the agent’s scratch dir, a grader/hidden.py exposing a single grade(scratch_dir) entry point, and a grader/reference_output.* artifact used to verify the grader itself. The declared execution-budget fields ( max_code_runs , max_wall_seconds ) are not enforced in the corpus that feeds § 5 ; the only wall-clock cap is the harness budget from § 3.2 ."},{"type":"heading","level":4,"text":"Grader contract."},{"type":"para","text":"Each grade(scratch_dir) returns a score in [ 0 , 1 ] [0,1] with a pass flag and must be deterministic , offline (no network, keys, or clock-dependent oracles), and seeded-random only (fixed declared seed). It may not write to the scratch directory, which would contaminate re-grading. A reference-output round-trip — grading workspace/ merged with reference_output/ must score one — is the grader’s self-test, run by CI on every push."},{"type":"heading","level":4,"text":"Subprocess isolation."},{"type":"para","text":"Graders run in a fresh per-task subprocess, so a grader-side exception cannot kill the harness — the artifact-side analogue of the pytest --collect-only gate in § 3.3 ."},{"type":"heading","level":4,"text":"No-leak invariant."},{"type":"para","text":"Before the agent runs, the materializer scans the scratch tree for any grader/hidden.py or reference_output.* ; a match aborts the run before the agent starts, so no run can complete with the answer key in its working directory. Catching a leak post-hoc would already have biased the grade."},{"type":"heading","level":1,"text":"4. Experimental Setup"},{"type":"heading","level":2,"text":"4.1. Benchmark inventory"},{"type":"para","text":"We evaluate on two benchmarks chosen to cover disjoint task regimes; the methodology for each lives in § 3.3 (SWE-bench protocol) and § 3.6 (artifact-suite contract), and is not repeated here."},{"type":"heading","level":4,"text":"Artifact suite (computation regime)."},{"type":"para","text":"A self-contained suite of n = 93 n{=}$93$ tasks across nine categories, three at n = 15 n{=}$15$ (data engineering, data science, ML engineering) and six at n = 8 n{=}$8$ (algorithmic, data processing, enumeration, iterative numerical, stateful reasoning, verification-heavy). Each task is a (workspace, hidden grader, reference output) triple under the contract of § 3.6 ; the reference-output round-trip self-test passes on all 93 93 tasks in CI. The suite is a new benchmark rather than an externally validated one; we scope its generalization in Limitations, and note here that two design properties keep it from favoring the code_only arm by construction. First, grading is outcome-based: the hidden grader of § 3.6 scores the produced artifact in the scratch directory, identically across arms and blind to which tool surface produced it, so it cannot reward code execution as a method. Second, all three arms can already execute code ( baseline via Bash , bash_only via bash, code_only via execute_code ); they differ only in the file-manipulation and browsing primitives layered around that shared capability, so the regime contrast against the external SWE-bench benchmark isolates the tool surface rather than the ability to compute."},{"type":"heading","level":4,"text":"SWE-bench Mini (modification regime)."},{"type":"para","text":"A curated n = 100 n{=}$100$ subset of SWE-bench (Jimenez et al . , 2024 ) in two sub-corpora. verified-mini ( n = 50 n{=}$50$ ): Django ( 25 25 ), Sphinx ( 25 25 ), drawn from SWE-bench Verified (OpenAI Preparedness Team, 2024 ) . datasci-mini ( n = 50 n{=}$50$ ): sklearn ( 15 15 ), matplotlib ( 12 12 ), xarray ( 8 8 ), sympy ( 7 7 ), seaborn ( 5 5 ), astropy ( 3 3 ). The split spans the canonical modification regime and the data-science / scientific-Python tail underrepresented in SWE-bench Verified."},{"type":"heading","level":2,"text":"4.2. Agents"},{"type":"para","text":"We instantiate the three-arm ablation of § 3.1 ( baseline , bash_only , code_only ) on two production coding agents: Claude Code (Anthropic, 2026a ) (model claude-sonnet-4-6 , Claude Code 2.1.139) and Codex CLI (OpenAI, 2026a ) (model gpt-5.5 ). Both are invoked under the per-arm subprocess-isolation protocol of § 3.2 (per-run isolated config dir, no session-persistence, sandbox/permission bypass). Codex is reported as a co-headline finding, not a footnote-level generalisation probe: the agent-design contrast on SWE-bench (Codex p = p= 2.016 919 × 10 − 09 2.016\\,919\\text{\\times}{10}^{-09} vs. Claude p = p= 0.120 155 1 0.120\\,155\\,1 on the code_only cost contrast) is one of the four cells that Table 2 in § 5 reports."},{"type":"heading","level":4,"text":"Codex apply_patch leakage audit."},{"type":"para","text":"The § 3.1 soft-disable was audited on all 110 110 Codex SWE-bench JSONLs (seed 1): file_change events fire 7 7 times on bash_only and 4 4 times on code_only (both < 1 % {<}1\\% , the 4 4 concentrated in scikit-learn__scikit-learn-11596 ) vs. 344 344 on baseline . This audit counts explicit apply_patch / file_change tool-call events, so it bounds registered tool leakage rather than behavioral compliance: a model could in principle reach equivalent file-editing behavior through bash without emitting a file_change event, which this metric would not capture. We read it as a tight bound on explicit tool use—the isolation the crossed design assumes on Codex holds at the level of registered tool calls—and, because Codex restriction is prompt-enforced rather than hard-flagged (§ 3.1 ), treat the two significant Codex-side results as conditional on soft compliance (see Limitations)."},{"type":"heading","level":2,"text":"4.3. Seeds and the unit of inference"},{"type":"heading","level":4,"text":"Seeds."},{"type":"para","text":"Each (instance, arm) cell is run with 3 seeds, drawn as independent harness invocations on disjoint container snapshots. Per-seed raw records and the analysis scripts that produced every \\result{} macro in this paper are released with the harness; we do not reproduce them in this PDF."},{"type":"heading","level":4,"text":"The unit of inference is the task, not the seed."},{"type":"para","text":"A seed-marginal standard error (SE) would pool over instances of dramatically heterogeneous difficulty and confuse between-task variance with within-task replication noise. Empirically, on Claude SWE-bench the cross-instance SD of cache-adjusted cost is roughly an order of magnitude larger than the paired per-task contrast SE, and pairing flips several headline cells from NS to significant."},{"type":"heading","level":4,"text":"Procedure."},{"type":"para","text":"For every comparison reported in § 5 : (i) collapse the 3 seeds within each (task, arm) cell to a per-task mean; (ii) for each ordered arm pair ( A , B ) (A,B) within a benchmark, compute the per-task vector of differences Δ t = mean A ( t ) − mean B ( t ) \\Delta_{t}=\\mathrm{mean}_{A}(t)-\\mathrm{mean}_{B}(t) and report mean Δ \\Delta , SE Δ = SD ( Δ ) / n tasks \\mathrm{SE}_{\\Delta}=\\mathrm{SD}(\\Delta)/\\sqrt{n_{\\text{tasks}}} , and a 95% normal-approximation CI ( n tasks ∈ { 93 , 100 } n_{\\text{tasks}}\\in\\{93,100\\} supports the approximation); (iii) pair the CI with a paired Wilcoxon signed-rank p p (Wilcoxon, 1945 ) on the Δ \\Delta vector for continuous metrics (cost, input tokens, output tokens). Wilcoxon is preferred over the paired t t -test because the per-task Δ \\Delta distributions are heavy-tailed—a handful of pathological SWE-bench instances dominate parametric variance while the median direction is clear in every cell. Pass rate uses the same machinery on per-task pass rates ∈ { 0 , 1 / 3 , 2 / 3 , 1 } \\in\\{0,1/3,2/3,1\\} , a Wilcoxon-on-rates analogue of McNemar that respects the within-task seed structure. The marginal per-arm summary (the “row” in Table 2 ) is the mean over tasks of the per-task mean, ± SD task / n tasks \\pm\\mathrm{SD}_{\\text{task}}/\\sqrt{n_{\\text{tasks}}} —the SE that respects task as the unit of replication. Per-seed values are derivable from the released data but are not the inferential unit. Implementation: paper/data/scripts/paired_contrasts.py , fully reproducible from the released JSONL records."},{"type":"heading","level":2,"text":"4.4. Metrics, cost rates, and exclusions"},{"type":"heading","level":4,"text":"Metric surface."},{"type":"para","text":"Table 2 in § 5 reports four columns per (benchmark, agent) cell. Pass rate , defined as PASS / ( PASS + FAIL ) \\mathrm{PASS}/(\\mathrm{PASS}+\\mathrm{FAIL}) with env_fail instances excluded from the denominator (see below), reported as an absolute Δ \\Delta in percentage points. Cache-adjusted cost : the median-floor-adjusted variant defined in § 3.5 , which charges the shared per-arm prefix at the cache-read rate using the per-arm median first-turn cache_read_input_tokens as the floor; reported as a relative Δ \\Delta %. Input tokens and output tokens : per-run sums of the corresponding usage fields across user-role API turns (cached and uncached pooled for input); reported as relative Δ \\Delta %. Turns, dollars per turn, and median per-instance cost are diagnostic rather than headline; the raw token-based cost is the underlying input to the cache-adjusted column, not a parallel reporting axis (§ 3.5 )."},{"type":"heading","level":4,"text":"Cost rates."},{"type":"para","text":"The cache-adjusted cost formula plugs in the vendor-published per-million-token rates for input, cached-input, and output, separately for claude-sonnet-4-6 and gpt-5.5 , as of the access date in § 3.5 . Pinning the rate constants in the setup section is what makes every cost column in § 5 reproducible from the released JSONL records."},{"type":"heading","level":4,"text":"Wall budget."},{"type":"para","text":"Each ( instance , arm , run ) (\\text{instance},\\text{arm},\\text{run}) is capped at 60 minutes of wall time (§ 3.2 ); per-task execution-budget fields reserved in the artifact task schema are not enforced."},{"type":"heading","level":1,"text":"5. Main Results"},{"type":"heading","level":2,"text":"5.1. Code-only is cheaper in three of four cells (significant in two)"},{"type":"para","text":"Reading down the cost-adjusted column of Table 2 , code_only is cheaper than its cheapest rival in three of the four (regime, agent) cells. The two significant advantages are Artifact/Claude ( − 24.595 857 141 034 71 -24.595\\,857\\,141\\,034\\,71 %, p = p= 7.365 478 099 521 983 × 10 − 14 7.365\\,478\\,099\\,521\\,983\\text{\\times}{10}^{-14} ) and SWE-bench/Codex ( − 19.908 625 337 034 56 -19.908\\,625\\,337\\,034\\,56 %, p = p= 2.016 919 093 053 276 2 × 10 − 09 2.016\\,919\\,093\\,053\\,276\\,2\\text{\\times}{10}^{-09} ); the third is a directional Artifact/Codex advantage ( − 6.697 940 133 090 269 -6.697\\,940\\,133\\,090\\,269 %, NS). The lone exception is SWE-bench/Claude at 14.437 541 589 685 27 14.437\\,541\\,589\\,685\\,27 %(NS, p = p= 0.120 155 120 631 986 1 0.120\\,155\\,120\\,631\\,986\\,1 ). The two significant advantages span both benchmarks and both agents; the single cell where the advantage disappears motivates the edit-friction analysis in § 6.1 . Same model, same harness, same prompts — the per-cell magnitude and sign of the code_only advantage depend jointly on regime and agent design. These are cost-engineering results under the cache-adjusted accounting model of § 3.5 , not universal cache-isolated savings. Under a raw, cache-blind accounting—every token at the non-cached rate, no median-floor—the two significant advantages stay significant and same-signed (Artifact/Claude − 24.442 041 468 004 63 -24.442\\,041\\,468\\,004\\,63 %, SWE-bench/Codex − 17.849 487 778 369 -17.849\\,487\\,778\\,369 %) and SWE-bench/Claude stays directionally costlier; only the directional Artifact/Codex cell is accounting-sensitive, which is why we report it as directional rather than robust."},{"type":"para","text":"The pass column is statistically tied in every cell: the cost asymmetry lives in how the agent arrives at the answer, not whether it arrives. The token columns localize the SWE-bench surfaces to opposite axes. SWE-bench/Claude’s directionally costlier outcome is concentrated in output tokens ( 39.860 282 899 670 75 39.860\\,282\\,899\\,670\\,75 %, p = p= 4.760 390 853 563 277 × 10 − 10 4.760\\,390\\,853\\,563\\,277\\text{\\times}{10}^{-10} ), anticipating the edit-friction mechanism (§ 6.1 ). SWE-bench/Codex’s significant cost advantage is concentrated in input tokens ( − 24.803 972 165 942 06 -24.803\\,972\\,165\\,942\\,06 %, p = p= 5.968 225 746 376 303 × 10 − 09 5.968\\,225\\,746\\,376\\,303\\text{\\times}{10}^{-09} ) with output statistically flat, anticipating the per-call input-budget mechanism (§ 6.2 )."},{"type":"heading","level":4,"text":"Where bash_only sits."},{"type":"para","text":"The intermediate bash_only surface — native shell without the file-editing primitives — is a monotone waypoint between baseline and code_only only in the computation regime, where it undercuts baseline and code_only undercuts it in turn (Table 2 ). The modification regime breaks the ordering: on SWE-bench/Codex bash_only is the costliest arm, significantly above baseline ( 4.832 386 750 896 251 4.832\\,386\\,750\\,896\\,251 %, p = p= 0.004 658 564 912 524 6 0.004\\,658\\,564\\,912\\,524\\,6 ), even as code_only is the cheapest. Because bash_only can be the single most expensive arm, the code_only advantage reflects consolidation into one execute_code round-trip rather than tool removal per se — which is why the headline contrast is baseline vs. code_only ."},{"type":"heading","level":2,"text":"5.2. Per-cell cost-difference distributions"},{"type":"para","text":"Figure 2 sorts per-instance Δ cost adj. \\Delta_{\\text{cost adj.}} (code-only minus the cheapest rival, averaged across three seeds) within each cell; negative values favor code-only. Artifact/Claude is a uniform shift: 85.0 85.0 of 93.0 93.0 instances win, median − $ 0.023 -\\mathdollar$0.023$ per instance. SWE-bench/Codex shows the same shape: 76.0 76.0 of 100.0 100.0 win, median − $ 0.092 -\\mathdollar$0.092$ . The two NS cells differ in structure. Artifact/Codex is symmetric near parity — 48.0 48.0 of 93.0 93.0 win, IQR − $ 0.021 -\\mathdollar$0.021$ to $ 0.013 \\mathdollar$0.013$ — consistent with the Table 2 NS row. SWE-bench/Claude inverts: only 44.0 44.0 of 100.0 100.0 favor code-only, and the loss concentrates in a right tail ( p 75 = $ 0.097 p_{75}=\\mathdollar$0.097$ , max = $ 2.33 \\max=\\mathdollar$2.33$ ). The directionally costlier mean for that cell is task-structural, not outlier-driven; the shape of panels (c) and (d) motivates the two mechanism questions in § 6 ."},{"type":"heading","level":2,"text":"5.3. Cost surface depends jointly on regime and agent design"},{"type":"para","text":"Figure 1 reduces the four cost-adjusted contrasts to one panel. Left to right, the per-cell ratios ( code-only / cheapest rival ) (\\text{code-only}\\,/\\,\\text{cheapest rival}) are 0.754 041 0.754\\,041 (Artifact/Claude, p = p= 7.365 478 × 10 − 14 7.365\\,478\\text{\\times}{10}^{-14} ), 0.933 021 0.933\\,021 (Artifact/Codex, p = p= 0.254 328 4 0.254\\,328\\,4 ), 1.144 375 1.144\\,375 (SWE-bench/Claude, p = p= 0.120 155 1 0.120\\,155\\,1 ), and 0.800 914 0.800\\,914 (SWE-bench/Codex, p = p= 2.016 919 × 10 − 09 2.016\\,919\\text{\\times}{10}^{-09} ). The unity-parity line is crossed by the SWE-bench/Claude bar alone."},{"type":"para","text":"The SWE-bench/Claude cell is the only one where restricting the agent to a single execute_code tool makes it directionally costlier than its cheapest rival, while SWE-bench/Codex is the cell where the same restriction yields the matrix’s largest significant cost win. Same regime, same restriction, opposite outcomes: surface choice is jointly determined by regime and agent design. Codex’s significant cost win is the confirmed effect; Claude’s directional loss is the asymmetric companion that the discussion unpacks separately. The Claude direction (edit friction on Edit / Write -heavy tasks) is analyzed in § 6.1 ; the Codex direction (tool-call batching and upper-tail output suppression) in § 6.2 ."},{"type":"heading","level":2,"text":"5.4. Agreement matrix and conditional cost"},{"type":"para","text":"Three seeds across three arms give nine outcomes per instance. The agreement matrix (Table 3 ) reports how often arms agree per instance, under two definitions. Strict (every arm passes on every seed, or every arm fails on every seed) ranges from 74.0 74.0 % on SWE-bench/Claude to 94.62 94.62 % on Artifact/Codex. Majority (per-arm pass-rate ≥ 2 / 3 {\\geq}\\,2/3 , unanimously across arms) ranges from 91.0 91.0 % to 98.92 98.92 %. The strict-to-majority gap on SWE-bench cells ( ≈ 13 {\\approx}13 pp) matches the seed leave-one-out reclassification rate, so majority counts the structurally split instances while strict mixes structural and seed-noise splits. Split structure is mostly graded difficulty rather than arm-specific easy subsets: only 1.0 1.0 of 9.0 9.0 Claude SWE-bench splits has a single arm carrying all passes."},{"type":"para","text":"Restricting the cost contrast to the unanimous-pass subset (Table 4 ) localizes where the full-set gap comes from. In three of four cells the gap is preserved: Artifact/Claude holds at − 24.7747 -24.7747 % (vs. − 24.5959 -24.5959 % on the full set), Artifact/Codex at − 6.7541 -6.7541 %, and SWE-bench/Codex at − 15.4971 -15.4971 % (still significant). SWE-bench/Claude collapses: the full-set 14.4375 14.4375 % gap (NS) drops to 4.0601 4.0601 % (NS) on the subset where every arm passes. The full-set gap is therefore a failure-cost effect concentrated on doomed-run trajectories, not a per-edit tax on successful runs; the mechanism is analyzed in § 6.3 . The cache-floor median is recomputed on the subset per § 3.5 ; it matches the full-set floor in 34.0 34.0 of 36.0 36.0 (benchmark, seed, agent, arm) groups, with both exceptions on rival arms (not the contrast arm) for a single SWE-bench/Claude seed."},{"type":"heading","level":1,"text":"6. Discussion"},{"type":"heading","level":2,"text":"6.1. Path-cost: edit friction on Claude SWE-bench"},{"type":"para","text":"The lone cell of Table 2 in which code_only is directionally costlier than its rival is Claude × \\times SWE-bench: cost runs + 14 % +14\\% (NS) but output tokens are + 40 % +40\\% at p < 10 − 9 p{<}10^{-9} —the extra cost scales with output volume rather than reflecting a fixed per-run overhead. The working hypothesis is edit friction: under code_only , every file modification must be expressed as a Python script rather than as a single native Edit or Write call. The per-instance Spearman correlation between Δ edit chars \\Delta_{\\text{edit chars}} ( code_only − - baseline , computed from JSONL tool-use blocks) and Δ output tokens \\Delta_{\\text{output tokens}} is ρ = 0.488 054 269 591 964 4 \\rho=$0.488\\,054\\,269\\,591\\,964\\,4$ ( p = 2.594 788 687 130 005 3 × 10 − 07 p=$2.594\\,788\\,687\\,130\\,005\\,3\\text{\\times}{10}^{-07}$ , n = 100.0 n=$100.0$ ); a median split confirms that the per-instance penalty scales with edit volume, with Δ output tokens = 2650.333 333 333 333 \\Delta_{\\text{output tokens}}=$2650.333\\,333\\,333\\,333$ on the low-patch half vs. 6378.434 027 777 778 6378.434\\,027\\,777\\,778 on the high-patch half (one-sided Mann–Whitney p = 0.024 831 066 506 004 6 p=$0.024\\,831\\,066\\,506\\,004\\,6$ ). About 3992.612 162 581 266 6 3992.612\\,162\\,581\\,266\\,6 tokens of the aggregate gap is fixed-cost regime verbosity (OLS intercept at zero patch size); edit friction is the additional per-line tax, not the whole gap."},{"type":"heading","level":2,"text":"6.2. Pricing asymmetry: batching and programmatic output control on Codex"},{"type":"para","text":"Why does Codex’s code_only save ∼ 25 % {\\sim}25\\% on input tokens on SWE-bench while LLM-call counts stay essentially flat ( 18.2 18.2 for baseline vs. 18.0 18.0 for code_only )? Two mechanisms, neither reproduced by the bash_only control, jointly account for the win (conditional on the soft-enforced Codex arm; see Limitations). Tool-call batching: execute_code issues 17.1 17.1 tool calls per run vs. 22.9 22.9 for baseline , so a single Python script subsumes what exec_command requires several calls to do (Figure 3 shows one such step on a named instance). Upper-tail suppression: median per-call output is essentially unchanged ( 1747 1747 vs. 1648 1648 chars), but the p 99 p99 collapses from the exec_command ceiling of 40 154 40\\,154 chars to 16 736 16\\,736 —the agent constrains per-call output volume programmatically, by tightening query scope, slicing returned text, or staging output across multiple prints, rather than receiving whatever the native tool emits (Figure 4 catches the cap-pinning event on a named instance). This refines the industry token-reduction claim (Anthropic, 2025 ; Varda and Pai, 2025 ) : the gain materialises when the agent’s API expresses per-call batching and its native rival exposes a hard output ceiling, not from the execute_code surface alone. Claude’s API empirically does not batch on SWE-bench ( code_only issues 30.1 30.1 calls per run vs. 27.9 27.9 for baseline ), and the round-count inflation is the input-token mirror of the edit-friction effect in § 6.1 —one mechanism viewed at two granularities."},{"type":"heading","level":2,"text":"6.3. Failure-cost and capability invariance"},{"type":"para","text":"Pass rates differ by less than three percentage points across every cell while costs swing 20 20 – 40 % 40\\% (Table 2 ). The straightforward interpretation is that tool restriction is a harness-efficiency axis orthogonal to model capability: the model solves the task with any reasonable interface; the surface changes the path, not the answer. This matches the Capability Overlap Principle (Zhang et al . , 2026 ) and the pass-rate invariance reported under tool ablation in (Verdent, 2025 ) . The agreement matrix (Table 3 ) confirms it: unanimous-outcome instances dominate every cell at ≥ 91.0 % {\\geq}$91.0$\\% under majority vote and ≥ 74.0 % {\\geq}$74.0$\\% on the strictest 9 / 9 9/9 -trials criterion. The Claude × \\times SWE-bench cost gap decomposes cleanly against this anchor (Table 4 ): on the unanimous-pass subset Δ cost adj. \\Delta_{\\text{cost adj.}} collapses from + 14.4375 % +$14.4375$\\% to + 4.0601 % +$4.0601$\\% (NS), Δ input tokens \\Delta_{\\text{input tokens}} from + 25.9719 % +$25.9719$\\% to + 16.5215 % +$16.5215$\\% (NS), but Δ output tokens \\Delta_{\\text{output tokens}} stays at + 44.2414 % +$44.2414$\\% (highly significant). Two mechanisms cohabit the cell: the § 6.1 edit-friction path-cost is the output-side tax that persists on successful runs; the remaining input-side blow-up is a failure-cost—extended doomed-run trajectories on instances no surface can solve. Conditioning on success separates them. The cache-floor median is byte-identical on subset and full set in 34.0 34.0 of 36.0 36.0 (benchmark, seed, agent, arm) groups (§ 3.5 ), so the collapse is not a cost-adjustment artefact."},{"type":"para","text":"Agent-specific vs. interface-general. These mechanisms do not generalize equally, which bounds how far the four-cell result should be read as a rule for coding agents at large. Two are agent-specific , tied to one vendor’s tool implementation or pricing rather than to the execute_code surface itself: the tool-call batching and upper-tail output suppression behind the Codex × \\times SWE-bench win (§ 6.2 ) depend on how Codex’s native exec_command batches and caps output, and the edit-friction path-cost behind the Claude × \\times SWE-bench output-token blow-up (§ 6.1 ) depends on Claude’s per- Edit pricing and an API that does not batch. A different agent could move either sign, so neither transfers unchanged and both should be re-measured per agent. Two are more plausibly interface-general : the pass-rate invariance across surfaces (§ 6.3 )—the surface changes the path and cost, not the answer—which holds in every cell for both agents, and its corollary that restricting to a code-execution surface tends to reduce or at worst hold cost when edit volume is low, since the edit-friction and failure-cost taxes both scale with modification volume. The latter two are properties of the regime-by-surface interaction rather than of a single agent, and are the parts most likely to generalize."},{"type":"heading","level":2,"text":"6.4. Where the method works, and where it does not"},{"type":"para","text":"The harness quantifies tool-surface tax under cache noise (§ 3.5 ), isolates regime effects via the artifact-vs-SWE-bench split, and surfaces agent-design coupling: the same restriction ( code_only ) produces a significant cost win for Codex on SWE-bench ( p < 10 − 8 p{<}10^{-8} ) and a non-significant directionally-costlier result for Claude in the same cell — the sharpest agent-design contrast in the four-cell structure. It does not support model-capability claims, because pass rates are statistically tied; the path-vs-answer dissociation in § 6.3 licenses the cost contrasts as cost-engineering rather than capability claims. We make no per-repo SWE-bench claims at n ≈ 12 n{\\approx}12 – 15 15 ; the remaining headline caveat is the NS Claude × \\times SWE-bench cell at p ≈ 0.12 p{\\approx}0.12 ."},{"type":"heading","level":2,"text":"6.5. Implications for agent design"},{"type":"para","text":"The four-cell structure adjudicates between the prior prescriptions of § 1 as follows. Computation-dominated workloads: drop the IDE surface; code_only wins on cost at parity capability (Artifact × \\times Claude, Table 2 ). Modification-dominated workloads with a high solve-rate: the surface choice is roughly cost-neutral on Claude—the headline + 14.4375 % +$14.4375$\\% penalty is a failure-cost effect on doomed runs, not a per-edit tax on successful ones (Table 4 ). Modification-dominated workloads with a low solve-rate: keep Edit and Write at minimum—the edit-friction path-cost in § 6.1 compounds with the failure-cost in § 6.3 on instances no surface can solve. Tool surfaces that admit batching and programmatic output control: expect input-token savings of the kind Codex realises in § 6.2 , provided the workload does not push LLM-call count up in the process."},{"type":"heading","level":1,"text":"Limitations"},{"type":"para","text":"Two agent surfaces. The 3 × 2 × 2 3\\,{\\times}\\,2\\,{\\times}\\,2 design exercises Claude Code ( claude-sonnet-4-6 ) and Codex CLI ( gpt-5.5 ); GPT-5, Gemini 2.5, and open-weight scaffolds are out of scope."},{"type":"para","text":"Soft tool enforcement on Codex. Claude Code hard-disables tools at launch via a feature flag; the Codex CLI exposes no such flag, so its bash_only and code_only restrictions are imposed by a prompt-prefix directive (§ 3.1 ). The § 4.2 audit finds explicit apply_patch leakage < 1 % {<}1\\% , but this counts registered tool-call events, not behavioral compliance—a model could reach equivalent file editing through bash without registering as leakage. The two significant Codex-side findings—the SWE-bench/Codex cost advantage (Table 2 ) and the batching/output-control mechanism (§ 6.2 )—are therefore conditional on soft compliance rather than guaranteed isolation. A hard-enforced replication on an open harness with explicit programmatic tool registration—e.g. OpenHands (Wang et al . , 2025 ) or a LangGraph-style declarative allowlist that holds regardless of model intent—is the primary follow-up."},{"type":"para","text":"Sample size. Per-cell n n is 93 93 (Artifact) and 100 100 (SWE-bench Mini), three seeds each; the directional Claude/SWE-bench gap motivates Verified-scale replication as follow-up."},{"type":"para","text":"Artifact-suite scope and external validity. Tasks are single-language (Python) on Linux, skewed toward analytical and ML-flavoured computation; the grader contract in § 3.6 has not been exercised on Rust, systems-engineering tasks, or long-horizon scenarios. The suite is a new benchmark rather than an externally validated one, and several conclusions rest on contrasting it with SWE-bench; we therefore read the artifact-side results as early evidence, not a general rule, and claim no transfer to systems programming, multi-language repositories, or long-horizon agent workflows. § 4.1 argues the suite does not favor code_only by construction; external validation on independently authored computation benchmarks is future work."},{"type":"para","text":"Cost is cache-adjusted, not cache-isolated. Neither the Anthropic Messages nor OpenAI Responses API exposes a documented way to disable the server-side prompt cache (§ 3.5 ); within-task multi-turn cache reads are not adjusted. The two significant cost advantages survive a raw, cache-blind accounting (§ 5.1 )."},{"type":"para","text":"Single-shot, internal-surface scope. Multi-shot prompting, scaffold evolution, and external MCP tool surfaces (§ 2 ) are out of scope."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Coding agents now ship with overlapping tool surfaces—Read, Grep, Glob, Edit, Write, and Bash on Claude Code; ACI primitives on SWE-agent; an MCP execute_code tool on emerging stacks—and the field has settled on three mutually incompatible claims about which surface a coding agent should expose. Specialised IDE primitives are required: SWE-agent argues that the Agent–Computer Interface (ACI) is load-bearing, with an ACI-vs-shell ablation that produced a gain of + 10.7 +10.7 percentage points on SWE-bench at the time of publication (Yang et al . , 2024 ) . Bash alone is sufficient: mini-SWE-agent is a ∼ \\\\sim 100-line bash-only scaffold that posts > 74 % {>}74\\\\% on SWE-bench Verified with no IDE primitives at all (SWE-agent contributors, 2025 ) . Replace the tool surface with code execution: industry write-ups report ∼ 98 {\\\\sim}98 – 99 % 99\\\\% token reductions by routing through an MCP execute_code tool rather than calling the native tools directly (Anthropic, 2025 ; Varda and Pai, 2025 ; Cloudflare, 2026 ) . The three prescriptions have never been crossed on a single harness, on the same model, with regime stratification, so the field cannot tell which condition each applies under, or whether they contradict each other at all.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We run the missing crossed comparison: the same harness, the same models, and the same prompts, evaluated under three tool surfaces ( baseline , bash_only , code_only ) crossed with two task regimes (computation, modification) and two agents (Claude Code, Codex CLI), all under an integrity-clean SWE-bench protocol with post-agent test_patch application (§ 3.3 ) (Jimenez et al . , 2024 ) . The baseline arm exposes each agent’s default tool surface; bash_only restricts to native bash plus read/glob/grep, with Edit and Write removed; code_only replaces the entire tool surface with a single MCP tool— mcp__codebox__execute_code on a persistent Python+Bash REPL—and disallows every native built-in (Anthropic, 2024 ) . The modification cell is SWE-bench Mini ( n = 100 n=100 , canonical post-agent test_patch protocol); the computation cell is a set of self-contained tasks each defined by a (workspace, hidden grader, reference output) triple with deterministic offline scoring (§ 3.6 ). The crossed 3 × 2 × 2 3\\\\times 2\\\\times 2 design is the minimum that disambiguates the three claims of § 1 , because each claim’s scope (which surface, which regime, which agent) is exactly what is in dispute. The computation cell is a methodological vehicle, not a benchmark contribution—the artifact-benchmark space is already crowded (§ 2.4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"None of the prior data points on this axis is the crossed comparison we run. SWE-agent’s two-arm ACI-vs-shell ablation (Yang et al . , 2024 ) and mini-SWE-agent’s single bash-only arm on SWE-bench Verified (SWE-agent contributors, 2025 ) both fix a single tool-surface axis on the modification regime, with no code-execution arm and no regime stratification. The industry write-ups on code-execution surfaces (Anthropic, 2025 ; Varda and Pai, 2025 ; Cloudflare, 2026 ) target external-MCP tool surfaces (Drive, Salesforce, Stripe, Cloudflare’s API) rather than internal IDE primitives and report no benchmark and no regime split. Liu et al. offer an architectural taxonomy of Claude Code’s primitives explicitly without ablation (Liu et al . , 2026b ) ; the Verdent technical report describes a single-vendor informal ablation without public methodology or regime stratification (Verdent, 2025 ) . No prior work has crossed three internal IDE surfaces under regime stratification with both a Claude- and a Codex-family agent, so the three claims of § 1 have remained unfalsifiable at the population level.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We make three contributions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(1) A four-cell agent-conditional cost structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The four cells are the cross of the two task regimes with the two agents—{computation (Artifact), modification (SWE-bench)} × \\\\times {Claude, Codex}. Across these four (regime × \\\\times agent) cells, code_only is cheaper in three cells (significantly in two, directionally in one) and directionally more expensive in the fourth—Artifact/Claude ( Δ cost adj. = − 24.595 857 141 034 71 % \\\\Delta_{\\\\text{cost adj.}}=$-24.595\\\\,857\\\\,141\\\\,034\\\\,71$\\\\% , p = 7.365 478 099 521 983 × 10 − 14 p=$7.365\\\\,478\\\\,099\\\\,521\\\\,983\\\\text{\\\\times}{10}^{-14}$ ), Artifact/Codex ( − 6.697 940 133 090 269 % $-6.697\\\\,940\\\\,133\\\\,090\\\\,269$\\\\% , directional), and SWE-bench/Codex ( − 19.908 625 337 034 56 % $-19.908\\\\,625\\\\,337\\\\,034\\\\,56$\\\\% , p = 2.016 919 093 053 276 2 × 10 − 09 p=$2.016\\\\,919\\\\,093\\\\,053\\\\,276\\\\,2\\\\text{\\\\times}{10}^{-09}$ ) all favour code_only , while SWE-bench/Claude reverses ( 14.437 541 589 685 27 % $14.437\\\\,541\\\\,589\\\\,685\\\\,27$\\\\% , NS; full table at § 5 , Table 2 ). The same restriction is cheaper under three (regime, agent) combinations (significantly under two) and costlier under one, so surface choice is jointly determined by regime AND agent design, not by either alone; the three prior claims of § 1 coexist because each is true under its specific (regime, agent) combination and false under the others.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(2) A three-mechanism causal decomposition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We decompose the four-cell structure into three causally distinct mechanisms whose joint footprint matches it; no single mechanism predicts the full table. Path-cost (edit friction): code_only must express every file edit as a Python script, and on Claude SWE-bench this scales output tokens with edit volume (per-instance Spearman ρ = 0.488 054 269 591 964 4 \\\\rho=$0.488\\\\,054\\\\,269\\\\,591\\\\,964\\\\,4$ , p = 2.594 788 687 130 005 3 × 10 − 07 p=$2.594\\\\,788\\\\,687\\\\,130\\\\,005\\\\,3\\\\text{\\\\times}{10}^{-07}$ ), driving the lone Claude × \\\\times SWE-bench output-token blow-up (§ 6.1 ). Failure-cost: the Claude × \\\\times SWE-bench cost overrun localises to unanimous-fail and split instances; on the unanimous-pass subset the gap collapses sharply (§ 6.3 ). Pricing asymmetry: on Codex, execute_code batches operations per LLM call ( ∼ 2.5 \\\\sim\\\\!2.5 tool calls per LLM step vs. ∼ 1.0 \\\\sim\\\\!1.0 for baseline ) and lets the agent constrain per-call output volume programmatically, which produces the SWE-bench/Codex win without changing LLM-call count—an effect absent on Claude (§ 6.2 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"(3) Capability invariance, with empirical receipt.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pass rates differ by less than three percentage points across all four cells × \\\\times three arms, with all four headline pass-rate contrasts NS (Table 2 ); and on a strict 9 / 9 9/9 -trial unanimity criterion the majority of instances in every cell are unanimously decided, so task outcome does not vary with tool surface for the bulk of the corpus. Tool surface changes the path and the cost, not the answer. This dissociation is what licenses contributions (1) and (2) as cost-engineering claims rather than capability claims, and it is consistent with the Capability Overlap Principle (Zhang et al . , 2026 ) ; the load-bearing claim is the empirical agreement-matrix result, not the theoretical frame.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 condenses these four cells into an at-a-glance map: the code_only cost outcome against its cheapest rival, whether that outcome is statistically significant or merely directional, and the operational lesson each cell licenses.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2. Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1. IDE-tool-surface ablation on coding agents\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The closest precedent is SWE-agent, whose two-arm Agent–Computer Interface (ACI) vs. raw shell ablation established that structured IDE primitives beat free-form shell at the model size of late 2024 2024 (Yang et al . , 2024 ) . Two subsequent points have left the conclusion partly stale: mini-SWE-agent, a ∼ 100 {\\\\sim}\\\\!100 -line bash-only scaffold, posts competitive numbers on SWE-bench Verified with no IDE primitives at all (SWE-agent contributors, 2025 ) , and Live-SWE-agent shows a self-evolving scaffold seeded from it can climb higher (Xia et al . , 2025 ) . Engineered surfaces continue to be explored on the other end, layering AST-aware retrieval, semantic search, or fixed localize–repair–validate pipelines on the SWE-agent line (Zhang et al . , 2024 ; Örwall, 2025 ; Xia et al . , 2024 ) . None of these crosses three internal IDE surfaces under a single integrity-clean harness with regime stratification; each fixes a single axis. Descriptive surveys of Claude Code’s primitive surface (Liu et al . , 2026b ; Rombaut, 2026 ; Bui, 2026 ) and component-evolution work on coding-agent harnesses (Wong et al . , 2025 ; Lin et al . , 2026 ) do not swap the entire surface either. The vendor signal closest in spirit to ours is the Verdent technical report (Verdent, 2025 ) , which informally notes that removing advanced tools while keeping bash + + read + + write + + edit produced little SWE-bench Verified change; single-vendor and without public methodology, we treat it as motivation rather than a scoop.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2. Code execution as the action interface\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The code_only arm sits in a line of work that replaces structured tool calls with executable code. PAL and Program-of-Thoughts established Python as the medium of model reasoning for arithmetic and symbolic computation (Gao et al . , 2023 ; Chen et al . , 2023 ) ; CodeAct extended the move to the action layer of agentic systems, contrasting executable Python with JSON tool calls on general-purpose agent benchmarks (Wang et al . , 2024 ) , and the OpenHands platform (Wang et al . , 2025 ) ships the CodeAct interface at scale. The most directly relevant industry signals are Anthropic’s “code execution with MCP” write-up and Cloudflare’s two “Code Mode” posts, which report large token reductions when an LLM writes code that calls external MCP tools rather than calling them directly (Anthropic, 2025 ; Varda and Pai, 2025 ; Cloudflare, 2026 ) . All four target external tool surfaces — Drive, Salesforce, Stripe, Cloudflare’s REST API — not the agent’s internal IDE primitives, and none reports a coding-benchmark evaluation. Bechard et al. (Bechard et al . , 2026 ) argue the analogous “less is more” point for enterprise API automation, again outside the code-repair domain. Our code_only arm ports code-as-action into the agent’s IDE-primitive layer and evaluates it on the benchmarks where Claude Code, Codex CLI, and their peers actually run.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3. Tool-use tax and capability overlap\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We frame the surface choice through Zhang et al.’s “tool-use tax” inequality: a tool delivers net benefit only when its task-specific capability gain exceeds the per-call cost of carrying its definition in the prompt (Zhang et al . , 2026 ) . Their evidence is on general math and question-answering; ours supplies the coding-agent receipt. The orthogonal tool-pruning and tool-budgeting literature — Budget-Aware Tool Use (Liu et al . , 2025 ) , ToolTree (Yang et al . , 2026 ) , Trajectory Reduction (Xiao et al . , 2025 ) , ToolScope (Liu et al . , 2026a ) , and the MCP description audit of Hasan et al. (Hasan et al . , 2026 ) — all ask which k k tools to select from a large pool, holding the surface design itself fixed. We instead compare three fixed surfaces on the same harness. Fan et al.’s SWE-Effi (Fan et al . , 2025 ) introduces resource-aware effectiveness metrics for coding agents and reports that failed runs consume an order of magnitude more compute than successes; this “expensive failures” framing motivates the cache-adjusted cost methodology in § 3.5 and is consistent with the failure-cost mechanism we identify on SWE-bench/Claude (§ 6.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.4. Coding-agent benchmarks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our paired evaluation spans both halves of an established benchmark axis. On the modification side, SWE-bench (Jimenez et al . , 2024 ) and its human-validated Verified subset (OpenAI Preparedness Team, 2024 ) are the foundation; later benchmarks that build on them — SWE-bench Pro (Deng et al . , 2025 ) , Multi-SWE-bench (Zan et al . , 2025 ) , SWE-PolyBench (Rashid et al . , 2025 ) , and SWE Atlas (Raghavendra et al . , 2026 ) — extend SWE-bench along difficulty, language, and task-shape axes, but never along the agent’s tool surface. Ganhotra’s audit (Ganhotra, 2025 ) that the bulk of SWE-bench Verified is single-file motivates our targeted multi-file sampling. On the computation side, MLE-Bench (Chan et al . , 2024 ) and MLAgentBench (Huang et al . , 2024 ) grade Kaggle-style submissions by the numerical output of agent-written code, RE-Bench (Wijk et al . , 2024 ) pushes the same shape to longer-horizon ML R&D, CORE-Bench (Siegel et al . , 2024 ) grades reproduction of published numerical results, and SWE-Lancer (Miserendino et al . , 2025 ) mixes modification and computation in a single corpus. We do not propose a new computation benchmark — the space is crowded. The artifact suite we run is the methodological vehicle that makes the four-cell agent-conditional cost structure of § 5 observable at all: without a computation cell on the same harness, every (regime, agent) contrast collapses to a single column of SWE-bench numbers, and the same-regime divergence between Claude and Codex on SWE-bench cannot be read against any computation baseline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper is the empirical complement to Liu et al.’s architectural taxonomy of Claude Code’s primitive surface (Liu et al . , 2026b ) : they describe the design space; we ablate three points in it under regime and agent-design stratification, on the benchmark pair that spans the field’s two graded regimes. Observational mining of in-the-wild agent pull requests, exemplified by the SE 3.0 anchor dataset AIDev (Li et al . , 2025 ) , holds neither the model nor the tool surface fixed and so does not substitute for the controlled ablation we report.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3. Method\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1. Three-arm tool surface\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each arm is defined by what tools are exposed to the agent at process launch; the same Claude Code or Codex CLI binary runs all three. The baseline arm passes no restriction flag and exposes the agent’s default surface — Read, Grep, Glob, Edit, Write, Bash, and the agent-built-in subagents on Claude Code; the analogous file primitives, shell tool, and structured apply_patch on Codex. The bash_only arm restricts to bash plus read-only browsing primitives, disallowing every file-editing built-in ( Edit / Write / MultiEdit / NotebookEdit on Claude; shell_tool with structured patching disabled on Codex). The code_only arm replaces the entire surface with a single MCP tool, mcp__codebox__execute_code , served by our exec-server stack as a persistent Python+Bash REPL keyed by working directory across calls; all native built-ins are explicitly disallowed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two implementation asymmetries are worth surfacing. First, Codex exposes no feature flag for disabling its structured apply_patch tool, so on Codex the bash_only and code_only arms enforce restriction with a prompt-prefix directive rather than a hard flag; this is soft enforcement and model compliance is not guaranteed. We measure and report explicit apply_patch leakage in § 4.2 and find it empirically negligible ( < 1 % {<}1\\\\% on both restricted arms)—a bound on registered tool calls, not behavioral compliance (see Limitations). Second, code_only depends on REPL state persisting between execute_code invocations: without a persistent kernel, the round-trip savings of “one script per task” collapse, so the exec-server pins the persistent-kernel environment flag for the runs that feed § 5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The bash_only arm matches the tool surface of mini-SWE-agent (SWE-agent contributors, 2025 ) but does not reuse its loop, prompt template, or scaffold; we apply the bash-only restriction to the same Claude Code and Codex binaries the other two arms use, so the contrast isolates the tool surface variable rather than comparing scaffold implementations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2. Harness design\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Per-instance setup (SWE-bench).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each SWE-bench instance is cloned at its base_commit and a per-instance Python virtual environment is built against it. The repository and the virtual environment are each given their own read-only fuse-overlayfs lower directory, so a per-arm upper directory captures every write and is discarded between arms — no re-clone, no pip install -e . between arms. The virtual-environment overlay is mounted back at its original creation path because console-script shebangs hard-code that absolute path at install time. Artifact tasks are self-contained and skip this step.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Git-history strip.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before the agent runs, the worktree is collapsed to a single orphan commit at its current tree; all refs, packed refs, reflogs, and alternates are deleted, and git gc --prune=now repacks the resulting object set. The agent cannot recover the upstream reference fix via git log , git show , or the reflog. Fixed author and committer dates make the orphan SHA deterministic for a given tree, so repeated runs strip to the same state.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Overlay refresh between arms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"git reset cannot un-create files an earlier arm added (fuse-overlayfs copy-up plus EEXIST semantics), so the harness unmounts, deletes the upper and work directories, recreates them, remounts, and re-strips history between arms. The agent’s view of the merged mount path is stable; the underlying state is fresh.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Per-arm subprocess isolation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each agent invocation creates a fresh temporary configuration directory containing only the credentials and minimal config needed for the run, and points the binary at it via CLAUDE_CONFIG_DIR or CODEX_HOME . No session-persistence files, no cross-run state. Both binaries run in headless mode with permission prompts and sandbox / approval gates bypassed.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Wall-time cap.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each invocation runs in a fresh process group; on timeout the harness sends SIGKILL to the entire group (so subagents and spawned shells die with the parent) and the run is recorded FAIL with a synthetic wall_timeout JSONL record. The numerical wall budget is specified in § 4 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3. Evaluation integrity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The canonical SWE-bench protocol applies the gold test_patch only after the agent terminates: at run time the agent observes the repository at base_commit , and the tests used for grading are not present in the worktree (Jimenez et al . , 2024 ) . Our harness implements deferred application explicitly. The test_patch is applied only after the agent’s process exits, and a pytest --collect-only gate then runs against the patched tree to confirm that the held-out tests are collectible. A zero-item collection scores FAIL rather than letting the test runner silently report success against a non-existent suite.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because our agents — unlike the bash-only patch-submission scaffold used in the original SWE-bench evaluation — can write arbitrary files anywhere in the worktree, the canonical “apply the patch” step is preceded by two cleanup operations. Every file that the test_patch targets is restored from HEAD to discard any agent edits to it, and every path the patch lists as a new file is removed from the worktree so an agent-created file cannot collide with a held-out test and block the apply. git apply is then invoked against the resulting tree. If the apply still fails — almost always because an agent-created file could not safely be removed in the second step — the instance is recorded FAIL with reason post-agent git apply failed: agent edits conflicted with the held-out test patch , rather than scored against a tree we cannot trust. This is strictly more conservative than the canonical protocol, which assumes a clean apply.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4. The Capability Overlap framing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We adopt the tool-use tax framing of Zhang et al. (Zhang et al . , 2026 ) : a tool delivers net benefit iff its task-specific capability gain exceeds the per-call cost of carrying its definition in the prompt. Most Claude Code IDE primitives are bash-subsets in capability (Read, Grep, Glob, and Write reduce to cat , grep , find , and shell-heredoc); Edit alone has non-overlapping capability — atomic byte-precise replacement with linting — awkward to express in bash. The per-call tax is task-invariant while the gain term is regime- and call-pattern-dependent, so § 5 reads the four-cell cost structure as the empirical outcome of this inequality and § 6 decomposes it into three mechanisms the framework alone does not determine.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5. Cache isolation and cost reporting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across the four surfaces this paper touches — the Anthropic Messages API, the Claude Code CLI, the OpenAI Responses API, and the OpenAI Codex CLI — no documented parameter, flag, environment variable, or namespace setting forces a server-side cache miss or scopes cached tokens to a single caller. The only documented mechanism is implicit: change the cached content so the cache key changes. The vendor cache primitives we surveyed — Anthropic’s cache_control and context-editing, OpenAI’s prompt_cache_key and Responses stored-state, and the two CLIs’ session flags — set cache breakpoints, routing hints, or local session state, none of which bypass the cache or scope it per caller (Anthropic, 2026b ; OpenAI, 2026b ) . We therefore cannot guarantee per-task cache isolation in the reported numbers and instead address the resulting non-stationarity in the cost-reporting methodology.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cost in § 5 is reported as a single cache-adjusted column, derived from the underlying token-billed formula cost = input_tokens ⋅ r in + output_tokens ⋅ r out \\\\text{cost}=\\\\text{input\\\\_tokens}\\\\cdot r_{\\\\text{in}}+\\\\text{output\\\\_tokens}\\\\cdot r_{\\\\text{out}} (every token billed at non-cached rates; the per-model rates r in , r out r_{\\\\text{in}},r_{\\\\text{out}} are pinned in § 4 ) plus a per-arm median-floor adjustment. For each arm, let M arm M_{\\\\text{arm}} be the median across that arm’s instances of first-turn cache_read_input_tokens — empirically the floor representing “system prompt plus tool definitions are warm in cache.” Any instance whose first-turn cache_read falls below M arm M_{\\\\text{arm}} is bumped up to M arm M_{\\\\text{arm}} for accounting; instances already at or above the floor and all multi-turn cache reads are unchanged. The adjustment normalises the system-prompt warm/cold lottery without claiming we achieved isolation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The adjustment stops at the shared system-prompt+tool-definition prefix; we do not claim control over post-prefix cache sharing. Such sharing is empirically negligible: prompt caches key on a prefix hash invalidated by any byte difference, and past the system prompt every turn diverges per task (fresh task statement, task-conditional response, task-specific tool-call payloads). Per-instance first-turn cache_read clusters tightly around each arm’s system-prompt token count, with no bimodal tail consistent with cross-task content reuse. We do not report raw API-billed cost: it is sensitive to run order and ambient cache state we cannot control.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.6. The artifact-suite contract\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The artifact suite is a new benchmark; this subsection is the methodology layer that defends it the way § 3.3 defends our SWE-bench protocol. Per-category counts and what each category probes are reported in § 4 ; this section specifies the contract, not the inventory.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Task contract.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every artifact instance is a four-part bundle: a task.yaml (problem statement, budget fields, and pointers), a workspace/ of public files copied into the agent’s scratch dir, a grader/hidden.py exposing a single grade(scratch_dir) entry point, and a grader/reference_output.* artifact used to verify the grader itself. The declared execution-budget fields ( max_code_runs , max_wall_seconds ) are not enforced in the corpus that feeds § 5 ; the only wall-clock cap is the harness budget from § 3.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Grader contract.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each grade(scratch_dir) returns a score in [ 0 , 1 ] [0,1] with a pass flag and must be deterministic , offline (no network, keys, or clock-dependent oracles), and seeded-random only (fixed declared seed). It may not write to the scratch directory, which would contaminate re-grading. A reference-output round-trip — grading workspace/ merged with reference_output/ must score one — is the grader’s self-test, run by CI on every push.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Subprocess isolation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Graders run in a fresh per-task subprocess, so a grader-side exception cannot kill the harness — the artifact-side analogue of the pytest --collect-only gate in § 3.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"No-leak invariant.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before the agent runs, the materializer scans the scratch tree for any grader/hidden.py or reference_output.* ; a match aborts the run before the agent starts, so no run can complete with the answer key in its working directory. Catching a leak post-hoc would already have biased the grade.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4. Experimental Setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1. Benchmark inventory\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate on two benchmarks chosen to cover disjoint task regimes; the methodology for each lives in § 3.3 (SWE-bench protocol) and § 3.6 (artifact-suite contract), and is not repeated here.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Artifact suite (computation regime).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A self-contained suite of n = 93 n{=}$93$ tasks across nine categories, three at n = 15 n{=}$15$ (data engineering, data science, ML engineering) and six at n = 8 n{=}$8$ (algorithmic, data processing, enumeration, iterative numerical, stateful reasoning, verification-heavy). Each task is a (workspace, hidden grader, reference output) triple under the contract of § 3.6 ; the reference-output round-trip self-test passes on all 93 93 tasks in CI. The suite is a new benchmark rather than an externally validated one; we scope its generalization in Limitations, and note here that two design properties keep it from favoring the code_only arm by construction. First, grading is outcome-based: the hidden grader of § 3.6 scores the produced artifact in the scratch directory, identically across arms and blind to which tool surface produced it, so it cannot reward code execution as a method. Second, all three arms can already execute code ( baseline via Bash , bash_only via bash, code_only via execute_code ); they differ only in the file-manipulation and browsing primitives layered around that shared capability, so the regime contrast against the external SWE-bench benchmark isolates the tool surface rather than the ability to compute.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"SWE-bench Mini (modification regime).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A curated n = 100 n{=}$100$ subset of SWE-bench (Jimenez et al . , 2024 ) in two sub-corpora. verified-mini ( n = 50 n{=}$50$ ): Django ( 25 25 ), Sphinx ( 25 25 ), drawn from SWE-bench Verified (OpenAI Preparedness Team, 2024 ) . datasci-mini ( n = 50 n{=}$50$ ): sklearn ( 15 15 ), matplotlib ( 12 12 ), xarray ( 8 8 ), sympy ( 7 7 ), seaborn ( 5 5 ), astropy ( 3 3 ). The split spans the canonical modification regime and the data-science / scientific-Python tail underrepresented in SWE-bench Verified.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2. Agents\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We instantiate the three-arm ablation of § 3.1 ( baseline , bash_only , code_only ) on two production coding agents: Claude Code (Anthropic, 2026a ) (model claude-sonnet-4-6 , Claude Code 2.1.139) and Codex CLI (OpenAI, 2026a ) (model gpt-5.5 ). Both are invoked under the per-arm subprocess-isolation protocol of § 3.2 (per-run isolated config dir, no session-persistence, sandbox/permission bypass). Codex is reported as a co-headline finding, not a footnote-level generalisation probe: the agent-design contrast on SWE-bench (Codex p = p= 2.016 919 × 10 − 09 2.016\\\\,919\\\\text{\\\\times}{10}^{-09} vs. Claude p = p= 0.120 155 1 0.120\\\\,155\\\\,1 on the code_only cost contrast) is one of the four cells that Table 2 in § 5 reports.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Codex apply_patch leakage audit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The § 3.1 soft-disable was audited on all 110 110 Codex SWE-bench JSONLs (seed 1): file_change events fire 7 7 times on bash_only and 4 4 times on code_only (both < 1 % {<}1\\\\% , the 4 4 concentrated in scikit-learn__scikit-learn-11596 ) vs. 344 344 on baseline . This audit counts explicit apply_patch / file_change tool-call events, so it bounds registered tool leakage rather than behavioral compliance: a model could in principle reach equivalent file-editing behavior through bash without emitting a file_change event, which this metric would not capture. We read it as a tight bound on explicit tool use—the isolation the crossed design assumes on Codex holds at the level of registered tool calls—and, because Codex restriction is prompt-enforced rather than hard-flagged (§ 3.1 ), treat the two significant Codex-side results as conditional on soft compliance (see Limitations).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3. Seeds and the unit of inference\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Seeds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each (instance, arm) cell is run with 3 seeds, drawn as independent harness invocations on disjoint container snapshots. Per-seed raw records and the analysis scripts that produced every \\\\result{} macro in this paper are released with the harness; we do not reproduce them in this PDF.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The unit of inference is the task, not the seed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A seed-marginal standard error (SE) would pool over instances of dramatically heterogeneous difficulty and confuse between-task variance with within-task replication noise. Empirically, on Claude SWE-bench the cross-instance SD of cache-adjusted cost is roughly an order of magnitude larger than the paired per-task contrast SE, and pairing flips several headline cells from NS to significant.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Procedure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For every comparison reported in § 5 : (i) collapse the 3 seeds within each (task, arm) cell to a per-task mean; (ii) for each ordered arm pair ( A , B ) (A,B) within a benchmark, compute the per-task vector of differences Δ t = mean A ( t ) − mean B ( t ) \\\\Delta_{t}=\\\\mathrm{mean}_{A}(t)-\\\\mathrm{mean}_{B}(t) and report mean Δ \\\\Delta , SE Δ = SD ( Δ ) / n tasks \\\\mathrm{SE}_{\\\\Delta}=\\\\mathrm{SD}(\\\\Delta)/\\\\sqrt{n_{\\\\text{tasks}}} , and a 95% normal-approximation CI ( n tasks ∈ { 93 , 100 } n_{\\\\text{tasks}}\\\\in\\\\{93,100\\\\} supports the approximation); (iii) pair the CI with a paired Wilcoxon signed-rank p p (Wilcoxon, 1945 ) on the Δ \\\\Delta vector for continuous metrics (cost, input tokens, output tokens). Wilcoxon is preferred over the paired t t -test because the per-task Δ \\\\Delta distributions are heavy-tailed—a handful of pathological SWE-bench instances dominate parametric variance while the median direction is clear in every cell. Pass rate uses the same machinery on per-task pass rates ∈ { 0 , 1 / 3 , 2 / 3 , 1 } \\\\in\\\\{0,1/3,2/3,1\\\\} , a Wilcoxon-on-rates analogue of McNemar that respects the within-task seed structure. The marginal per-arm summary (the “row” in Table 2 ) is the mean over tasks of the per-task mean, ± SD task / n tasks \\\\pm\\\\mathrm{SD}_{\\\\text{task}}/\\\\sqrt{n_{\\\\text{tasks}}} —the SE that respects task as the unit of replication. Per-seed values are derivable from the released data but are not the inferential unit. Implementation: paper/data/scripts/paired_contrasts.py , fully reproducible from the released JSONL records.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4. Metrics, cost rates, and exclusions\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Metric surface.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 in § 5 reports four columns per (benchmark, agent) cell. Pass rate , defined as PASS / ( PASS + FAIL ) \\\\mathrm{PASS}/(\\\\mathrm{PASS}+\\\\mathrm{FAIL}) with env_fail instances excluded from the denominator (see below), reported as an absolute Δ \\\\Delta in percentage points. Cache-adjusted cost : the median-floor-adjusted variant defined in § 3.5 , which charges the shared per-arm prefix at the cache-read rate using the per-arm median first-turn cache_read_input_tokens as the floor; reported as a relative Δ \\\\Delta %. Input tokens and output tokens : per-run sums of the corresponding usage fields across user-role API turns (cached and uncached pooled for input); reported as relative Δ \\\\Delta %. Turns, dollars per turn, and median per-instance cost are diagnostic rather than headline; the raw token-based cost is the underlying input to the cache-adjusted column, not a parallel reporting axis (§ 3.5 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Cost rates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cache-adjusted cost formula plugs in the vendor-published per-million-token rates for input, cached-input, and output, separately for claude-sonnet-4-6 and gpt-5.5 , as of the access date in § 3.5 . Pinning the rate constants in the setup section is what makes every cost column in § 5 reproducible from the released JSONL records.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Wall budget.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each ( instance , arm , run ) (\\\\text{instance},\\\\text{arm},\\\\text{run}) is capped at 60 minutes of wall time (§ 3.2 ); per-task execution-budget fields reserved in the artifact task schema are not enforced.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5. Main Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1. Code-only is cheaper in three of four cells (significant in two)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reading down the cost-adjusted column of Table 2 , code_only is cheaper than its cheapest rival in three of the four (regime, agent) cells. The two significant advantages are Artifact/Claude ( − 24.595 857 141 034 71 -24.595\\\\,857\\\\,141\\\\,034\\\\,71 %, p = p= 7.365 478 099 521 983 × 10 − 14 7.365\\\\,478\\\\,099\\\\,521\\\\,983\\\\text{\\\\times}{10}^{-14} ) and SWE-bench/Codex ( − 19.908 625 337 034 56 -19.908\\\\,625\\\\,337\\\\,034\\\\,56 %, p = p= 2.016 919 093 053 276 2 × 10 − 09 2.016\\\\,919\\\\,093\\\\,053\\\\,276\\\\,2\\\\text{\\\\times}{10}^{-09} ); the third is a directional Artifact/Codex advantage ( − 6.697 940 133 090 269 -6.697\\\\,940\\\\,133\\\\,090\\\\,269 %, NS). The lone exception is SWE-bench/Claude at 14.437 541 589 685 27 14.437\\\\,541\\\\,589\\\\,685\\\\,27 %(NS, p = p= 0.120 155 120 631 986 1 0.120\\\\,155\\\\,120\\\\,631\\\\,986\\\\,1 ). The two significant advantages span both benchmarks and both agents; the single cell where the advantage disappears motivates the edit-friction analysis in § 6.1 . Same model, same harness, same prompts — the per-cell magnitude and sign of the code_only advantage depend jointly on regime and agent design. These are cost-engineering results under the cache-adjusted accounting model of § 3.5 , not universal cache-isolated savings. Under a raw, cache-blind accounting—every token at the non-cached rate, no median-floor—the two significant advantages stay significant and same-signed (Artifact/Claude − 24.442 041 468 004 63 -24.442\\\\,041\\\\,468\\\\,004\\\\,63 %, SWE-bench/Codex − 17.849 487 778 369 -17.849\\\\,487\\\\,778\\\\,369 %) and SWE-bench/Claude stays directionally costlier; only the directional Artifact/Codex cell is accounting-sensitive, which is why we report it as directional rather than robust.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pass column is statistically tied in every cell: the cost asymmetry lives in how the agent arrives at the answer, not whether it arrives. The token columns localize the SWE-bench surfaces to opposite axes. SWE-bench/Claude’s directionally costlier outcome is concentrated in output tokens ( 39.860 282 899 670 75 39.860\\\\,282\\\\,899\\\\,670\\\\,75 %, p = p= 4.760 390 853 563 277 × 10 − 10 4.760\\\\,390\\\\,853\\\\,563\\\\,277\\\\text{\\\\times}{10}^{-10} ), anticipating the edit-friction mechanism (§ 6.1 ). SWE-bench/Codex’s significant cost advantage is concentrated in input tokens ( − 24.803 972 165 942 06 -24.803\\\\,972\\\\,165\\\\,942\\\\,06 %, p = p= 5.968 225 746 376 303 × 10 − 09 5.968\\\\,225\\\\,746\\\\,376\\\\,303\\\\text{\\\\times}{10}^{-09} ) with output statistically flat, anticipating the per-call input-budget mechanism (§ 6.2 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Where bash_only sits.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The intermediate bash_only surface — native shell without the file-editing primitives — is a monotone waypoint between baseline and code_only only in the computation regime, where it undercuts baseline and code_only undercuts it in turn (Table 2 ). The modification regime breaks the ordering: on SWE-bench/Codex bash_only is the costliest arm, significantly above baseline ( 4.832 386 750 896 251 4.832\\\\,386\\\\,750\\\\,896\\\\,251 %, p = p= 0.004 658 564 912 524 6 0.004\\\\,658\\\\,564\\\\,912\\\\,524\\\\,6 ), even as code_only is the cheapest. Because bash_only can be the single most expensive arm, the code_only advantage reflects consolidation into one execute_code round-trip rather than tool removal per se — which is why the headline contrast is baseline vs. code_only .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2. Per-cell cost-difference distributions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 sorts per-instance Δ cost adj. \\\\Delta_{\\\\text{cost adj.}} (code-only minus the cheapest rival, averaged across three seeds) within each cell; negative values favor code-only. Artifact/Claude is a uniform shift: 85.0 85.0 of 93.0 93.0 instances win, median − $ 0.023 -\\\\mathdollar$0.023$ per instance. SWE-bench/Codex shows the same shape: 76.0 76.0 of 100.0 100.0 win, median − $ 0.092 -\\\\mathdollar$0.092$ . The two NS cells differ in structure. Artifact/Codex is symmetric near parity — 48.0 48.0 of 93.0 93.0 win, IQR − $ 0.021 -\\\\mathdollar$0.021$ to $ 0.013 \\\\mathdollar$0.013$ — consistent with the Table 2 NS row. SWE-bench/Claude inverts: only 44.0 44.0 of 100.0 100.0 favor code-only, and the loss concentrates in a right tail ( p 75 = $ 0.097 p_{75}=\\\\mathdollar$0.097$ , max = $ 2.33 \\\\max=\\\\mathdollar$2.33$ ). The directionally costlier mean for that cell is task-structural, not outlier-driven; the shape of panels (c) and (d) motivates the two mechanism questions in § 6 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3. Cost surface depends jointly on regime and agent design\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1 reduces the four cost-adjusted contrasts to one panel. Left to right, the per-cell ratios ( code-only / cheapest rival ) (\\\\text{code-only}\\\\,/\\\\,\\\\text{cheapest rival}) are 0.754 041 0.754\\\\,041 (Artifact/Claude, p = p= 7.365 478 × 10 − 14 7.365\\\\,478\\\\text{\\\\times}{10}^{-14} ), 0.933 021 0.933\\\\,021 (Artifact/Codex, p = p= 0.254 328 4 0.254\\\\,328\\\\,4 ), 1.144 375 1.144\\\\,375 (SWE-bench/Claude, p = p= 0.120 155 1 0.120\\\\,155\\\\,1 ), and 0.800 914 0.800\\\\,914 (SWE-bench/Codex, p = p= 2.016 919 × 10 − 09 2.016\\\\,919\\\\text{\\\\times}{10}^{-09} ). The unity-parity line is crossed by the SWE-bench/Claude bar alone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The SWE-bench/Claude cell is the only one where restricting the agent to a single execute_code tool makes it directionally costlier than its cheapest rival, while SWE-bench/Codex is the cell where the same restriction yields the matrix’s largest significant cost win. Same regime, same restriction, opposite outcomes: surface choice is jointly determined by regime and agent design. Codex’s significant cost win is the confirmed effect; Claude’s directional loss is the asymmetric companion that the discussion unpacks separately. The Claude direction (edit friction on Edit / Write -heavy tasks) is analyzed in § 6.1 ; the Codex direction (tool-call batching and upper-tail output suppression) in § 6.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4. Agreement matrix and conditional cost\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Three seeds across three arms give nine outcomes per instance. The agreement matrix (Table 3 ) reports how often arms agree per instance, under two definitions. Strict (every arm passes on every seed, or every arm fails on every seed) ranges from 74.0 74.0 % on SWE-bench/Claude to 94.62 94.62 % on Artifact/Codex. Majority (per-arm pass-rate ≥ 2 / 3 {\\\\geq}\\\\,2/3 , unanimously across arms) ranges from 91.0 91.0 % to 98.92 98.92 %. The strict-to-majority gap on SWE-bench cells ( ≈ 13 {\\\\approx}13 pp) matches the seed leave-one-out reclassification rate, so majority counts the structurally split instances while strict mixes structural and seed-noise splits. Split structure is mostly graded difficulty rather than arm-specific easy subsets: only 1.0 1.0 of 9.0 9.0 Claude SWE-bench splits has a single arm carrying all passes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Restricting the cost contrast to the unanimous-pass subset (Table 4 ) localizes where the full-set gap comes from. In three of four cells the gap is preserved: Artifact/Claude holds at − 24.7747 -24.7747 % (vs. − 24.5959 -24.5959 % on the full set), Artifact/Codex at − 6.7541 -6.7541 %, and SWE-bench/Codex at − 15.4971 -15.4971 % (still significant). SWE-bench/Claude collapses: the full-set 14.4375 14.4375 % gap (NS) drops to 4.0601 4.0601 % (NS) on the subset where every arm passes. The full-set gap is therefore a failure-cost effect concentrated on doomed-run trajectories, not a per-edit tax on successful runs; the mechanism is analyzed in § 6.3 . The cache-floor median is recomputed on the subset per § 3.5 ; it matches the full-set floor in 34.0 34.0 of 36.0 36.0 (benchmark, seed, agent, arm) groups, with both exceptions on rival arms (not the contrast arm) for a single SWE-bench/Claude seed.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6. Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.1. Path-cost: edit friction on Claude SWE-bench\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The lone cell of Table 2 in which code_only is directionally costlier than its rival is Claude × \\\\times SWE-bench: cost runs + 14 % +14\\\\% (NS) but output tokens are + 40 % +40\\\\% at p < 10 − 9 p{<}10^{-9} —the extra cost scales with output volume rather than reflecting a fixed per-run overhead. The working hypothesis is edit friction: under code_only , every file modification must be expressed as a Python script rather than as a single native Edit or Write call. The per-instance Spearman correlation between Δ edit chars \\\\Delta_{\\\\text{edit chars}} ( code_only − - baseline , computed from JSONL tool-use blocks) and Δ output tokens \\\\Delta_{\\\\text{output tokens}} is ρ = 0.488 054 269 591 964 4 \\\\rho=$0.488\\\\,054\\\\,269\\\\,591\\\\,964\\\\,4$ ( p = 2.594 788 687 130 005 3 × 10 − 07 p=$2.594\\\\,788\\\\,687\\\\,130\\\\,005\\\\,3\\\\text{\\\\times}{10}^{-07}$ , n = 100.0 n=$100.0$ ); a median split confirms that the per-instance penalty scales with edit volume, with Δ output tokens = 2650.333 333 333 333 \\\\Delta_{\\\\text{output tokens}}=$2650.333\\\\,333\\\\,333\\\\,333$ on the low-patch half vs. 6378.434 027 777 778 6378.434\\\\,027\\\\,777\\\\,778 on the high-patch half (one-sided Mann–Whitney p = 0.024 831 066 506 004 6 p=$0.024\\\\,831\\\\,066\\\\,506\\\\,004\\\\,6$ ). About 3992.612 162 581 266 6 3992.612\\\\,162\\\\,581\\\\,266\\\\,6 tokens of the aggregate gap is fixed-cost regime verbosity (OLS intercept at zero patch size); edit friction is the additional per-line tax, not the whole gap.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.2. Pricing asymmetry: batching and programmatic output control on Codex\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why does Codex’s code_only save ∼ 25 % {\\\\sim}25\\\\% on input tokens on SWE-bench while LLM-call counts stay essentially flat ( 18.2 18.2 for baseline vs. 18.0 18.0 for code_only )? Two mechanisms, neither reproduced by the bash_only control, jointly account for the win (conditional on the soft-enforced Codex arm; see Limitations). Tool-call batching: execute_code issues 17.1 17.1 tool calls per run vs. 22.9 22.9 for baseline , so a single Python script subsumes what exec_command requires several calls to do (Figure 3 shows one such step on a named instance). Upper-tail suppression: median per-call output is essentially unchanged ( 1747 1747 vs. 1648 1648 chars), but the p 99 p99 collapses from the exec_command ceiling of 40 154 40\\\\,154 chars to 16 736 16\\\\,736 —the agent constrains per-call output volume programmatically, by tightening query scope, slicing returned text, or staging output across multiple prints, rather than receiving whatever the native tool emits (Figure 4 catches the cap-pinning event on a named instance). This refines the industry token-reduction claim (Anthropic, 2025 ; Varda and Pai, 2025 ) : the gain materialises when the agent’s API expresses per-call batching and its native rival exposes a hard output ceiling, not from the execute_code surface alone. Claude’s API empirically does not batch on SWE-bench ( code_only issues 30.1 30.1 calls per run vs. 27.9 27.9 for baseline ), and the round-count inflation is the input-token mirror of the edit-friction effect in § 6.1 —one mechanism viewed at two granularities.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.3. Failure-cost and capability invariance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pass rates differ by less than three percentage points across every cell while costs swing 20 20 – 40 % 40\\\\% (Table 2 ). The straightforward interpretation is that tool restriction is a harness-efficiency axis orthogonal to model capability: the model solves the task with any reasonable interface; the surface changes the path, not the answer. This matches the Capability Overlap Principle (Zhang et al . , 2026 ) and the pass-rate invariance reported under tool ablation in (Verdent, 2025 ) . The agreement matrix (Table 3 ) confirms it: unanimous-outcome instances dominate every cell at ≥ 91.0 % {\\\\geq}$91.0$\\\\% under majority vote and ≥ 74.0 % {\\\\geq}$74.0$\\\\% on the strictest 9 / 9 9/9 -trials criterion. The Claude × \\\\times SWE-bench cost gap decomposes cleanly against this anchor (Table 4 ): on the unanimous-pass subset Δ cost adj. \\\\Delta_{\\\\text{cost adj.}} collapses from + 14.4375 % +$14.4375$\\\\% to + 4.0601 % +$4.0601$\\\\% (NS), Δ input tokens \\\\Delta_{\\\\text{input tokens}} from + 25.9719 % +$25.9719$\\\\% to + 16.5215 % +$16.5215$\\\\% (NS), but Δ output tokens \\\\Delta_{\\\\text{output tokens}} stays at + 44.2414 % +$44.2414$\\\\% (highly significant). Two mechanisms cohabit the cell: the § 6.1 edit-friction path-cost is the output-side tax that persists on successful runs; the remaining input-side blow-up is a failure-cost—extended doomed-run trajectories on instances no surface can solve. Conditioning on success separates them. The cache-floor median is byte-identical on subset and full set in 34.0 34.0 of 36.0 36.0 (benchmark, seed, agent, arm) groups (§ 3.5 ), so the collapse is not a cost-adjustment artefact.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Agent-specific vs. interface-general. These mechanisms do not generalize equally, which bounds how far the four-cell result should be read as a rule for coding agents at large. Two are agent-specific , tied to one vendor’s tool implementation or pricing rather than to the execute_code surface itself: the tool-call batching and upper-tail output suppression behind the Codex × \\\\times SWE-bench win (§ 6.2 ) depend on how Codex’s native exec_command batches and caps output, and the edit-friction path-cost behind the Claude × \\\\times SWE-bench output-token blow-up (§ 6.1 ) depends on Claude’s per- Edit pricing and an API that does not batch. A different agent could move either sign, so neither transfers unchanged and both should be re-measured per agent. Two are more plausibly interface-general : the pass-rate invariance across surfaces (§ 6.3 )—the surface changes the path and cost, not the answer—which holds in every cell for both agents, and its corollary that restricting to a code-execution surface tends to reduce or at worst hold cost when edit volume is low, since the edit-friction and failure-cost taxes both scale with modification volume. The latter two are properties of the regime-by-surface interaction rather than of a single agent, and are the parts most likely to generalize.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.4. Where the method works, and where it does not\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The harness quantifies tool-surface tax under cache noise (§ 3.5 ), isolates regime effects via the artifact-vs-SWE-bench split, and surfaces agent-design coupling: the same restriction ( code_only ) produces a significant cost win for Codex on SWE-bench ( p < 10 − 8 p{<}10^{-8} ) and a non-significant directionally-costlier result for Claude in the same cell — the sharpest agent-design contrast in the four-cell structure. It does not support model-capability claims, because pass rates are statistically tied; the path-vs-answer dissociation in § 6.3 licenses the cost contrasts as cost-engineering rather than capability claims. We make no per-repo SWE-bench claims at n ≈ 12 n{\\\\approx}12 – 15 15 ; the remaining headline caveat is the NS Claude × \\\\times SWE-bench cell at p ≈ 0.12 p{\\\\approx}0.12 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.5. Implications for agent design\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The four-cell structure adjudicates between the prior prescriptions of § 1 as follows. Computation-dominated workloads: drop the IDE surface; code_only wins on cost at parity capability (Artifact × \\\\times Claude, Table 2 ). Modification-dominated workloads with a high solve-rate: the surface choice is roughly cost-neutral on Claude—the headline + 14.4375 % +$14.4375$\\\\% penalty is a failure-cost effect on doomed runs, not a per-edit tax on successful ones (Table 4 ). Modification-dominated workloads with a low solve-rate: keep Edit and Write at minimum—the edit-friction path-cost in § 6.1 compounds with the failure-cost in § 6.3 on instances no surface can solve. Tool surfaces that admit batching and programmatic output control: expect input-token savings of the kind Codex realises in § 6.2 , provided the workload does not push LLM-call count up in the process.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two agent surfaces. The 3 × 2 × 2 3\\\\,{\\\\times}\\\\,2\\\\,{\\\\times}\\\\,2 design exercises Claude Code ( claude-sonnet-4-6 ) and Codex CLI ( gpt-5.5 ); GPT-5, Gemini 2.5, and open-weight scaffolds are out of scope.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Soft tool enforcement on Codex. Claude Code hard-disables tools at launch via a feature flag; the Codex CLI exposes no such flag, so its bash_only and code_only restrictions are imposed by a prompt-prefix directive (§ 3.1 ). The § 4.2 audit finds explicit apply_patch leakage < 1 % {<}1\\\\% , but this counts registered tool-call events, not behavioral compliance—a model could reach equivalent file editing through bash without registering as leakage. The two significant Codex-side findings—the SWE-bench/Codex cost advantage (Table 2 ) and the batching/output-control mechanism (§ 6.2 )—are therefore conditional on soft compliance rather than guaranteed isolation. A hard-enforced replication on an open harness with explicit programmatic tool registration—e.g. OpenHands (Wang et al . , 2025 ) or a LangGraph-style declarative allowlist that holds regardless of model intent—is the primary follow-up.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sample size. Per-cell n n is 93 93 (Artifact) and 100 100 (SWE-bench Mini), three seeds each; the directional Claude/SWE-bench gap motivates Verified-scale replication as follow-up.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Artifact-suite scope and external validity. Tasks are single-language (Python) on Linux, skewed toward analytical and ML-flavoured computation; the grader contract in § 3.6 has not been exercised on Rust, systems-engineering tasks, or long-horizon scenarios. The suite is a new benchmark rather than an externally validated one, and several conclusions rest on contrasting it with SWE-bench; we therefore read the artifact-side results as early evidence, not a general rule, and claim no transfer to systems programming, multi-language repositories, or long-horizon agent workflows. § 4.1 argues the suite does not favor code_only by construction; external validation on independently authored computation benchmarks is future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cost is cache-adjusted, not cache-isolated. Neither the Anthropic Messages nor OpenAI Responses API exposes a documented way to disable the server-side prompt cache (§ 3.5 ); within-task multi-turn cache reads are not adjusted. The two significant cost advantages survive a raw, cache-blind accounting (§ 5.1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Single-shot, internal-surface scope. Multi-shot prompting, scaffold evolution, and external MCP tool surfaces (§ 2 ) are out of scope.\"\n }\n]","isConversational":false}}},{"rowIdx":391,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10548"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Beyond Looking Up, Try Looking Around: Harmonizing Global Structure and Local Consistency in Optimal Transport for Short Text Clustering"},"authors":{"kind":"list like","value":["Zhihao Yao","Yuxuan Gu","Jixuan Yin","Bo Li"],"string":"[\n \"Zhihao Yao\",\n \"Yuxuan Gu\",\n \"Jixuan Yin\",\n \"Bo Li\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10548v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"In the era of massive digital communication, short texts such as messages, queries, and comments are generated on an unprecedented scale. Effectively clustering and organizing these short texts is crucial for numerous applications (Ma and Zheng, 2025 ) . This urgency is further amplified by the prevalence of Large Language Models (LLMs), since LLM-driven applications, such as chatbots (Kang and Ki, 2025 ) and virtual assistants (Kweon et al. , 2025 ) , have further fueled the proliferation of short texts. Existing methods typically adopt an EM-like optimization framework to achieve clustering (Zheng et al. , 2023 ; Wu et al. , 2025 ) , which alternates between estimating pseudo-labels (E-step) and updating model parameters based on the generated pseudo-labels (M-step). Under this framework, early methods often adopt a greedy strategy to estimate pseudo-labels directly from model predictions (Sohn et al. , 2020 ) . However, they ignore both global and local information of the data distribution (Qin et al. , 2026 ) , resulting in unreliable pseudo-labels that may misguide model optimization in the subsequent M-step (Yin et al. , 2025 ; Cui et al. , 2025 ) ."},{"type":"para","text":"Instead of assigning pseudo-labels greedily, recent studies introduce Optimal Transport (OT) to estimate pseudo-labels by minimizing the overall cost of transporting the sample distribution to the cluster distribution (Zheng et al. , 2023 ; Cui et al. , 2025 ) , integrating sample-cluster relationships from a global perspective (Zhang et al. , 2024 ; Zheng et al. , 2023 ) , i.e., the sample-to-cluster global structure . However, there is still a critical limitation: they ignore predictive consistency between semantically similar samples, i.e., sample-to-sample semantic consistency . Relying only on minimizing overall transport cost, these methods may assign different pseudo-labels to semantically similar samples when the transport costs to candidate pseudo-labels are not sufficiently distinct or accurate. As illustrated in Figure 1 (a), the blue triangles inside the red box exhibit nearly identical transport costs to both hexagons. Due to this ambiguity, existing OT methods struggle to make correct estimation, potentially assigning these samples to erroneous green pseudo-labels, even though they are surrounded by a dense neighborhood of correct blue peers. This issue is common in practice because feature spaces derived from pretrained encoders often lack clear category boundaries (Zhang et al. , 2021 ) , causing samples near boundaries to have similar transport costs to different candidate pseudo-labels. Once these noisy pseudo-labels are introduced into the M-step, the initial embedding biases are amplified, leading to widespread mislabeling and significantly degrading clustering quality."},{"type":"para","text":"To address this limitation, we propose a novel OT method to enhance pseudo-label estimation in the E-step, termed consistency-aware adaptive optimal transport ( CAOT ). In contrast to existing OT methods, CAOT incorporates the semantic relationships among surrounding samples into the transport process, correcting erroneous estimations caused by ambiguous transport costs, as shown in Figure 1 (b). To achieve this, we first propose a novel instance-level attention network to capture semantic relationships between samples. These learned relationships are integrated into CAOT as a semantic consistency constraint, encouraging semantically similar samples to be assigned identical pseudo-labels. During EM iterations, the instance-level attention network and the clustering assignments are jointly optimized in the M-step, using pseudo-labels inferred by CAOT (E-step) as supervisory signals. Through successive EM iterations, the attention network becomes increasingly effective in modeling semantic relationships, which in turn provides accurate guidance for CAOT to promote semantic consistency. As a result, the pseudo-labels generated by CAOT capture both the sample-to-cluster global structure and the sample-to-sample semantic consistency. These reliable pseudo-labels can effectively optimize model parameters to achieve satisfactory clustering performance."},{"type":"para","text":"We conduct comprehensive experiments on eight benchmark datasets. Experimental results demonstrate that the proposed approach outperforms state-of-the-art methods. Complementing these quantitative results, visualization comparisons of transport results further demonstrate that pseudo-labels produced by the proposed CAOT are more accurate and confident than those produced by existing OT methods. Beyond the field of short-text, the proposed approach can be generalized to long-text and image domains, demonstrating its potential as a general solution for various clustering tasks. Further analysis demonstrates that the proposed approach also has computational advantages over existing methods, which makes it suitable for real-world clustering tasks."},{"type":"heading","level":1,"text":"2 Preliminary"},{"type":"para","text":"This section reviews the commonly adopted OT formulation and its application in pseudo-labeling. Given a source distribution 𝒂 \\bm{a} and a target distribution 𝒃 \\bm{b} , the objective of OT is to find a matrix 𝑸 \\bm{Q} to transform the distribution 𝒂 \\bm{a} to 𝒃 \\bm{b} with minimal total cost. The optimization problem in OT is:"},{"type":"para","text":"where ⟨ ⋅ , ⋅ ⟩ \\langle\\cdot,\\cdot\\rangle represents the Frobenius inner product, ε 1 \\varepsilon_{1} is a weighting coefficient, 𝑪 ∈ ℝ | 𝒂 | × | 𝒃 | \\bm{C}\\in\\mathbb{R}^{|\\bm{a}|\\times|\\bm{b}|} is a cost matrix, and | 𝒂 | |\\bm{a}| and | 𝒃 | |\\bm{b}| are the dimensions of 𝒂 \\bm{a} and 𝒃 \\bm{b} , respectively."},{"type":"para","text":"In OT-based pseudo-labeling methods, the matrix 𝑪 \\bm{C} is defined as − log ( 𝑷 ) \\!-\\text{log}(\\bm{P}) , where 𝑷 ∈ ℝ n × k \\bm{P}\\!\\!\\in\\!\\!\\mathbb{R}^{n\\!\\times\\!k} is a matrix containing the predicted probabilities of samples, with n n the batch size and k k the number of categories. The optimization problem is:"},{"type":"para","text":"where 𝑸 𝟏 k × 1 = 1 n 𝟏 n × 1 \\bm{Q}\\bm{1}_{\\scriptscriptstyle k\\times 1}=\\frac{1}{n}\\bm{1}{\\scriptscriptstyle n\\times 1} indicates that all samples are equally important, and 𝑸 T 𝟏 n × 1 = 1 k 𝟏 k × 1 \\bm{Q}^{T}\\bm{1}_{\\scriptscriptstyle n\\times 1}=\\frac{1}{k}\\bm{1}_{\\scriptscriptstyle k\\times 1} ensures that each category in the target distribution is assigned the same probability. After obtaining the transport matrix 𝑸 \\bm{Q} , the pseudo-label for the i -th sample is calculated as y ^ i = arg max ( 𝑸 i : ) \\hat{y}_{i}=\\arg\\max(\\bm{Q}_{i:}) ."},{"type":"heading","level":1,"text":"3 The Proposed Method"},{"type":"para","text":"This section provides a detailed description of the proposed approach. § 3.1 gives a high-level overview of the entire framework. § 3.2 , 3.3 , and 3.4 offer an in-depth description of three sub-components, where the proposed CAOT is introduced in § 3.2 to address the limitation of existing OT methods. § 3.5 introduces the training pipeline."},{"type":"heading","level":2,"text":"3.1 The Overall Structure"},{"type":"para","text":"The structure of the proposed model is shown in Figure 2 . It consists of three components: the Pseudo-label Generation Module (PGM), the Semantic Similarity Construction Module (SSCM), and the Supervised Guidance Module (SGM)."},{"type":"para","text":"During model training, the PGM functions in the E-step, where it performs CAOT using the similarity matrix 𝑺 a t t \\bm{S}^{att} to generate pseudo-labels 𝒚 ^ \\hat{\\bm{y}} . The SSCM and SGM operate in the M-step to update model parameters. The role of SSCM is to capture the semantic similarities between samples and construct the similarity matrix 𝑺 a t t \\bm{S}^{att} . To this end, contrastive learning is first applied to enhance the discriminability of sample representations ( L I L_{I} loss), after which pseudo-labels 𝒚 ^ \\hat{\\bm{y}} are leveraged to train the instance-level attention network G h G_{h} to yield the matrix 𝑺 a t t \\bm{S}^{att} ( L A L_{A} loss). The role of SGM is to leverage pseudo-labels 𝒚 ^ \\hat{\\bm{y}} to optimize cluster assignments."},{"type":"heading","level":2,"text":"3.2 The Pseudo-label Generation Module"},{"type":"para","text":"The Pseudo-label Generation Module is expected to produce reliable pseudo-labels. To this end, we propose a novel OT formulation, referred to as CAOT. Solving the optimization problem in CAOT yields pseudo-labels that simultaneously integrate sample-to-cluster global structural information and sample-to-sample semantic consistency."},{"type":"para","text":"The structure of the Pseudo-label Generation Module is shown in Figure 2 (a). Given a batch of n n text samples X ( 0 ) = [ x 1 ( 0 ) ; … ; x n ( 0 ) ] X^{(0)}=[x_{1}^{(0)};\\dots;x_{n}^{(0)}] , the text representations are obtained by the Encoder Φ \\mathnormal{\\Phi} as 𝑽 ( 0 ) ∈ ℝ n × d 1 \\bm{V}^{(0)}\\in\\mathbb{R}^{n\\times d_{1}} , where d 1 d_{1} is the dimension of the representations. Then, the Clustering Network G p G_{p} is used to predict probability assignments 𝑷 ( 0 ) ∈ ℝ n × k \\bm{P}^{(0)}\\in\\mathbb{R}^{n\\times k} ( G p G_{p} is an MLP network with dimensions 768-768-768- k k ). After that, pseudo-labels can be generated by solving the optimization problem in CAOT, which is defined as follows:"},{"type":"para","text":"where ε 1 \\varepsilon_{1} , ε 2 \\varepsilon_{2} and ε 3 \\varepsilon_{3} are weighting coefficients, 𝒂 = 1 n 𝟏 n × 1 \\bm{a}=\\frac{1}{n}\\bm{1}_{\\scriptscriptstyle n\\times 1} , and 𝒃 \\bm{b} remains to be determined. 𝑺 = 𝑺 c o s + 𝑺 a t t \\bm{S}=\\bm{S}^{cos}+\\bm{S}^{att} is the semantic similarity matrix, where 𝑺 c o s \\bm{S}^{cos} is the cosine similarity calculated from 𝑷 ( 0 ) \\bm{P}^{(0)} , and 𝑺 a t t \\bm{S}^{att} is the attention similarity generated from the instance-level attention network ( G h G_{h} ) via Eq. ( 8 ). Each term in Eq. ( 3 ) is explained as follows:"},{"type":"para","text":"H ( 𝑸 ) = ⟨ 𝑸 , log ( 𝑸 ) − 𝟏 n × k ⟩ H(\\bm{Q})=\\langle\\bm{Q},\\text{log}(\\bm{Q})-\\bm{1}_{\\scriptscriptstyle n\\times k}\\rangle is the same as Eq. ( LABEL:eq2 ), which prevents 𝑸 \\bm{Q} from becoming too sparse while ensuring that all elements remain strictly non-negative."},{"type":"para","text":"Ψ ( 𝒃 ) = − log ( 𝒃 ) − log ( 𝟏 k × 1 − 𝒃 ) \\Psi(\\bm{b})=-\\text{log}(\\bm{b})-\\text{log}(\\bm{1}_{\\scriptscriptstyle k\\times 1}-\\bm{b}) . Unlike Eq. ( LABEL:eq2 ), which strictly enforces the hard constraint 𝑸 T 𝟏 n × 1 = 1 k 𝟏 k × 1 \\bm{Q}^{T}\\bm{1}{\\scriptscriptstyle n\\times 1}=\\frac{1}{k}\\bm{1}_{\\scriptscriptstyle k\\times 1} , CAOT adopts Ψ ( 𝒃 ) T 𝟏 k × 1 \\Psi(\\bm{b})^{T}\\bm{1}{\\scriptscriptstyle k\\times 1} as an adjustable penalty item to encourage 𝑸 T 𝟏 n × 1 \\bm{Q}^{T}\\bm{1}{\\scriptscriptstyle n\\times 1} to approach a uniform distribution. By adjusting the strength of this item, CAOT can flexibly adapt to datasets with varying degrees of category imbalance."},{"type":"para","text":"⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\langle\\bm{S},\\bm{Q}\\bm{Q}^{T}\\rangle incorporates semantic consistency into CAOT, encouraging the transport matrix 𝑸 \\bm{Q} to align with the intrinsic semantic relationships between samples. Specifically, this design encourages the transport vector 𝑸 i : \\bm{Q}_{i:} to be similar to 𝑸 j : \\bm{Q}_{j:} when S i j S_{ij} is large. In other words, CAOT drives semantically similar samples towards similar transport vectors, thereby producing the same pseudo-labels."},{"type":"para","text":"In the proposed CAOT, ⟨ 𝑸 , − log ( 𝑷 ( 0 ) ) ⟩ \\langle\\bm{Q},-\\text{log}(\\bm{P}^{\\scriptstyle(0)})\\rangle incorporates the sample-to-cluster global structure into 𝑸 \\bm{Q} through the global minimum-cost matching constraint, enabling CAOT to exploit the joint confidence over the entire batch to correct erroneous predictions for some samples in 𝑷 ( 0 ) \\bm{P}^{\\scriptstyle(0)} . Simultaneously, ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\langle\\bm{S},\\bm{Q}\\bm{Q}^{T}\\rangle integrates sample-to-sample semantic consistency in 𝑸 \\bm{Q} , ensuring that semantically similar samples are assigned the same pseudo-labels. These two properties in 𝑸 \\bm{Q} empower the generation of reliable pseudo-labels."},{"type":"para","text":"To solve the proposed optimization problem in CAOT, we design an iterative method that integrates Taylor expansion and Lagrange multiplier algorithm. The detailed solution is provided in Appendix A.1 , and the convergence analysis of the proposed solution is given in Appendix A.2 . Once the transport matrix 𝑸 \\bm{Q} is available, the pseudo-label for the i -th sample can be generated as y ^ i = arg max j Q i j . \\hat{y}_{i}=\\arg\\max_{j}\\,Q_{ij}. The resulting pseudo-labels are then utilized in SSCM and SGM to update the model parameters during the M-step."},{"type":"heading","level":2,"text":"3.3 Semantic Similarity Construction Module"},{"type":"para","text":"The Semantic Similarity Construction Module is designed to identify and quantify semantic similarities between samples. The resulting similarity information is integrated into Eq. ( 3 ), guiding CAOT to incorporate sample-to-sample semantic consistency into the E-step."},{"type":"para","text":"The structure of the Semantic Similarity Construction Module is shown in Figure 2 (b). Inspired by Socrates’ saying “To know others, first know yourself.”, we first enhance the discriminability of each sample representation using contrastive learning, providing a foundation for similarity modeling. Given a batch of texts X ( 0 ) X^{(0)} , contextual augmenter technique (Kobayashi, 2018 ) is used to generate two augmented texts, denoted as X ( 1 ) X^{(1)} and X ( 2 ) X^{(2)} . The representations of the augmented texts are then obtained by the Encoder Φ \\mathnormal{\\Phi} as 𝑽 ( 1 ) ∈ ℝ n × d 1 \\bm{V}^{(1)}\\in\\mathbb{R}^{n\\times d_{1}} and 𝑽 ( 2 ) ∈ ℝ n × d 1 \\bm{V}^{(2)}\\in\\mathbb{R}^{n\\times d_{1}} , respectively. Subsequently, the Projecting Network G z G_{z} is employed to reduce the dimensionality of the representations ( G z G_{z} is an MLP network with dimensions 768-768-128). The outputs are 𝒁 ( 1 ) ∈ ℝ n × d 2 \\bm{Z}^{(1)}\\in\\mathbb{R}^{n\\times d_{2}} and 𝒁 ( 2 ) ∈ ℝ n × d 2 \\bm{Z}^{(2)}\\in\\mathbb{R}^{n\\times d_{2}} , respectively."},{"type":"para","text":"Then, contrastive learning improves the discriminability of each sample representation by pulling positive pairs together and pushing negative pairs apart. Let 𝒛 i ( 1 ) \\bm{z}_{i}^{(1)} and 𝒛 i ( 2 ) \\bm{z}_{i}^{(2)} be the i i -th rows in 𝒁 ( 1 ) \\bm{Z}^{(1)} and 𝒁 ( 2 ) \\bm{Z}^{(2)} , respectively. For 𝒛 i ( 1 ) \\bm{z}_{i}^{(1)} , its corresponding positive pair is { 𝒛 i ( 1 ) , 𝒛 i ( 2 ) } \\{{\\bm{z}_{i}^{(1)}\\!,\\bm{z}_{i}^{(2)}}\\} , while the negative pairs are { 𝒛 i ( 1 ) , 𝒛 v ( 1 ) } \\{{\\bm{z}_{i}^{(1)}\\!,\\bm{z}_{v}^{(1)}}\\} and { 𝒛 i ( 1 ) , 𝒛 v ( 2 ) } \\{{\\bm{z}_{i}^{(1)}\\!,\\bm{z}_{v}^{(2)}}\\} for all v ≠ i v\\neq i . For 𝒛 i ( 2 ) \\bm{z}_{i}^{(2)} , the positive and negative pairs are constructed in the same manner. The loss function of contrastive learning for the i -th sample is defined as follows:"},{"type":"para","text":"where sim ( 𝒛 i ( 1 ) , 𝒛 v ( 2 ) ) \\textup{sim}(\\bm{z}_{i}^{(1)},\\bm{z}_{v}^{(2)}) is the cosine similarity between two representations, and τ I \\tau_{I} is the temperature parameter. The final loss function of contrastive learning is calculated over a batch and defined as L I = 1 2 n ∑ i = 1 n l I , i L_{I}=\\frac{1}{2n}\\sum_{i=1}^{n}l_{I,i} ."},{"type":"para","text":"By treating all samples with v ≠ i v\\neq i as negative pairs, contrastive learning encourages sample representations to be uniformly distributed in the feature space. Thus, a highly discriminative representation can be learned for each sample. This design facilitates the modeling of semantic similarity between samples, as discriminative representations possess the necessary clarity to distinguish each other (Zhang et al. , 2021 ) . Based on this, we propose an instance-level attention network to explore similarity relationships between samples. The structure of this attention network G h G_{h} is shown in Figure 3 . It takes 𝒁 ( 1 ) \\bm{Z}^{(1)} and 𝒁 ( 2 ) \\bm{Z}^{(2)} as inputs. For 𝒁 ( 1 ) \\bm{Z}^{(1)} , the network performs three linear transformations:"},{"type":"para","text":"after which the similarity matrix 𝑺 ( 1 ) \\bm{S}^{(1)} is computed:"},{"type":"para","text":"Finally, the rows of 𝑻 ( 1 ) \\bm{T}^{(1)} are linearly combined according to the similarity matrix 𝑺 ( 1 ) \\bm{S}^{(1)} as follows:"},{"type":"para","text":"where 𝒕 j ( 1 ) \\bm{t}_{j}^{(1)} is the j j -th row of 𝑻 ( 1 ) \\bm{T}^{(1)} , and S i j ( 1 ) S_{ij}^{(1)} is the ( i , j ) (i,j) -th element of 𝑺 ( 1 ) \\bm{S}^{(1)} . Similarly, 𝑺 ( 2 ) \\bm{S}^{(2)} and 𝑯 ( 2 ) \\bm{H}^{(2)} are obtained by passing 𝒁 ( 2 ) \\bm{Z}^{(2)} through the Attention Network. The attention similarity matrix is calculated as follows:"},{"type":"para","text":"To ensure that the similarity matrix 𝑺 a t t \\bm{S}^{att} can accurately reflects the underlying semantic relationships between samples, we apply targeted training to G h G_{h} . Let 𝑯 = [ 𝑯 ( 1 ) , 𝑯 ( 2 ) ] ∈ ℝ 2 n × d 2 \\bm{H}=[\\bm{H}^{(1)},\\bm{H}^{(2)}]\\in\\mathbb{R}^{2n\\times d_{2}} . As indicated in Eq. ( 7 ), 𝑺 a t t \\bm{S}^{att} is intrinsically linked to 𝑯 \\bm{H} . Based on this, we optimize 𝑺 a t t \\bm{S}^{att} by applying targeted training to 𝑯 \\bm{H} as a proxy. We achieve this by enhancing the intra-cluster compactness and inter-cluster separation of 𝑯 \\bm{H} , which implicitly guides 𝑺 a t t \\bm{S}^{att} to assign high attention weights to samples belonging to the same cluster. In practice, we first use pseudo-labels to identify samples belonging to the same cluster. For the i -th sample, the set of indices belonging to the same cluster is defined as follows:"},{"type":"para","text":"where y ^ i \\hat{y}_{i} is the pseudo-label of the i -th sample."},{"type":"para","text":"After that, the loss for the i i -th sample is defined as follows:"},{"type":"para","text":"where S i j a t t S_{ij}^{att} is the ( i , j ) (i,j) -th element of 𝑺 a t t \\bm{S}^{att} , 𝒉 i \\bm{h}_{i} is the i -th row of 𝑯 \\bm{H} , and τ A \\tau_{A} is the temperature parameter. The final loss function of attention optimization is calculated over a batch and defined as L A = 1 2 n ∑ i = 1 n l A , i L_{A}\\!=\\!\\frac{1}{2n}\\!\\sum_{i=1}^{n}\\!l_{A,i} ."},{"type":"para","text":"By minimizing L A L_{A} , samples within the same cluster are assigned high weights in 𝑺 a t t \\bm{S}^{att} , which in turn facilitates CAOT in the subsequent E-step. In the training process, accurate pseudo-labels help establish an accurate 𝑺 a t t \\bm{S}^{att} (M-step), which in turn facilitates the generation of more reliable pseudo-labels than those in the previous iteration (E-step)."},{"type":"heading","level":2,"text":"3.4 Supervised Guidance Module"},{"type":"para","text":"The Supervised Guidance Module leverages pseudo-labels generated by CAOT as supervision targets to optimize the model in clustering. The structure of the Supervised Guidance Module is shown in Figure 2 (c). Given a batch of representations 𝑽 ( 1 ) \\bm{V}^{(1)} and 𝑽 ( 2 ) \\bm{V}^{(2)} , the corresponding probability assignments 𝑷 ( 1 ) \\bm{P}^{(1)} and 𝑷 ( 2 ) \\bm{P}^{(2)} are obtained. Then, following the paradigm of supervised classification, the model is optimized by aligning its predictive probabilities with the supervision targets. The loss function is defined as follows:"},{"type":"para","text":"where OneHot( ⋅ \\cdot ) is a one-hot encoding operator, y ^ i \\hat{y}_{i} is the pseudo-label for the i i -th sample, 𝒑 i ( 1 ) \\bm{p}_{i}^{(1)} and 𝒑 i ( 2 ) \\bm{p}_{i}^{(2)} are the i -th rows of the probability matrix 𝑷 ( 1 ) \\bm{P}^{(1)} and 𝑷 ( 2 ) \\bm{P}^{(2)} , respectively."},{"type":"heading","level":2,"text":"3.5 Training Procedure"},{"type":"para","text":"To prevent error accumulation in EM iterative optimization, we introduce a warm-up stage to provide a stable starting point for the training process. In this stage, pseudo-labels are generated via K-means on the representations 𝑽 ( 0 ) \\bm{V}^{(0)} , rather than the proposed CAOT. The warm-up loss combines L I L_{I} and L A L_{A} , defined as follows:"},{"type":"para","text":"where λ \\lambda is a balancing hyperparameter. The stop gradient technique is applied to the backpropagation of L A L_{A} , ensuring that L A L_{A} only optimizes the Attention Network G h G_{h} . After the warm-up stage, CAOT is utilized to generate pseudo-labels. The overall loss at this stage is formulated as follows:"},{"type":"para","text":"Input: dataset 𝒳 ( 0 ) \\mathcal{X}^{(0)} ; warm-up epochs E w a r m E_{warm} ; total epochs E t o t a l E_{total} ; Encoder Φ \\mathit{\\Phi} ; Network G z G_{z} , G h G_{h} , and G p G_{p} . Output: model parameters. Procedure:"},{"type":"para","text":"After training, for a given text x ( 0 ) x^{(0)} , its clustering result is given by arg max ( 𝒑 ( 0 ) ) \\arg\\max(\\bm{p}^{(0)}) , where 𝒑 ( 0 ) \\bm{p}^{(0)} is the output of the Clustering Network G p G_{p} . The pseudo-code of the overall training process is provided in Algorithm 1 ."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"para","text":"In this section, we present systematic evaluation experiments. § 4.1 introduces the configurations. § 4.2 analyzes the performance of clustering. § 4.3 compares the quality of feature spaces learned by the proposed model and the baseline methods. § 4.4 presents ablation studies to evaluate the contribution of each component of the model. § 4.5 conducts sensitivity analysis to examine the stability of the model under different hyperparameter settings."},{"type":"heading","level":2,"text":"4.1 Experiment Setups"},{"type":"para","text":"Datasets. We conduct experiments on eight benchmark datasets: AgNews, StackOverflow, Biomedical, SearchSnippets, GoogleNews-TS, GoogleNews-T, GoogleNews-S, and Tweet. Table 1 summarizes the characteristics of each dataset. Depending on the degree of category imbalance (R), AgNews, StackOverflow, and Biomedical are regarded as balanced datasets; SearchSnippets as a slightly imbalanced dataset; GoogleNews-TS, GoogleNews-T, and GoogleNews-S as imbalanced datasets; Tweet as a severely imbalanced dataset. Detailed information about these datasets is provided in Appendix C.1 ."},{"type":"para","text":"Experiment Settings. We use distilbert-base-nli-stsb-mean-tokens (SBERT) from the Sentence Transformers library as the Encoder (Reimers and Gurevych, 2019 ) . All parameters are optimized using the Adam optimizer. The learning rate of the Encoder is 5 × 10 − 6 5\\times 10^{-6} , and that of other networks is 5 × 10 − 4 5\\times 10^{-4} . The maximum input length for the Encoder is 32. The output dimensions of the Encoder and the Projection Network are set to d 1 = 768 d_{1}=768 and d 2 = 128 d_{2}=128 , respectively. We adopt Accuracy (ACC) and Normalized Mutual Information (NMI) to evaluate the model. The definitions of the evaluation metrics and other settings are provided in Appendix C.2 and Appendix C.3 , respectively."},{"type":"para","text":"Baselines. We compare the proposed method with the following baselines: TF-IDF (Chowdhury, 2010 ) extracts text representations using the TF-IDF technique. Back-Trans (Shen et al. , 2019 ) generates representations via contrastive learning with back-translation. STCC (Xu et al. , 2017 ) uses a CNN to optimize the output of word2vec as representations. Self-Train (Hadifar et al. , 2019 ) learns representations with an auto-encoder and updates them according to cluster assignments. SimCSE (Gao et al. , 2021 ) generates representations through contrastive learning with Dropout. For all the above methods, K-means is applied on the learned representations to yield clustering results. ESimCSE (Wu et al. , 2022 ) extends SimCSE by addressing the influence of sentence length on positional embeddings in transformers. SCCL (Zhang et al. , 2021 ) employs contrastive learning to refine the output of SBERT as representations and obtains clustering results using the DEC algorithm (Xie et al. , 2016 ) . RSTC (Zheng et al. , 2023 ) constructs pseudo-labels using optimal transport to assist clustering. SCL (Yong et al. , 2024 ) learns a projection matrix to map samples into a contrastive subspace, aligning original and projected embeddings to improve clustering. SCPCL (Liu et al. , 2025 ) employs optimized adaptive optimal transport to generate pseudo-labels, which guide a contrastive learning module to learn robust representations. FNSCC (Zheng et al. , 2025 ) enhances contrastive clustering by integrating fuzzy neighborhood information to refine both instance representations and soft cluster assignments."},{"type":"heading","level":2,"text":"4.2 Main Results"},{"type":"para","text":"The results are presented in Table 2 . The results for Back-Trans, SimCSE, ESimCSE, and SCL are from the SCL paper. RSTC, SCPCL, and FNSCC are reproduced using the configurations provided in corresponding original papers, and the remaining baseline results are from the RSTC paper. According to the results, one can see that: (1) Conventional methods, including TF-IDF and Back-Trans , perform poorly due to their inability to produce meaningful representations. (2) Deep neural network-based methods ( STCC , Self-Train , SimCSE , ESimCSE , and SCL ) outperform conventional methods in learning effective representations. However, the decoupled feature learning and clustering processes lead to suboptimal results. (3) Deep joint clustering methods ( SCCL , RSTC , SCPCL , and FNSCC ) improve the clustering performance significantly by adopting contrastive learning to fine-tune pre-trained models. However, they ignore the relationships between samples, resulting in suboptimal performance. (4) The results of the proposed approach ranks first in eleven indicators and second in the remaining five indicators. Notably, on the StackOverflow dataset, the proposed method achieves a substantial accuracy gain of 5.01% . In comparison, existing methods often perform well on some datasets but exhibit performance degradation on others, indicating limited generalization across diverse datasets. These results validate the effectiveness and generalizability of the proposed approach."},{"type":"heading","level":2,"text":"4.3 Comparison of representation quality"},{"type":"para","text":"This section investigates the efficacy of the proposed approach by examining the quality of the learned feature space. We evaluate the proposed approach along with SCCL and RSTC on the SearchSnippets and GoogleNews-TS datasets. The normalized average cosine similarity of samples within the same category is used as the evaluation metric, which is defined as follows:"},{"type":"para","text":"where 𝟙 ( ⋅ ) \\mathds{1}(\\cdot) denotes the indicator function."},{"type":"para","text":"As shown in Figure 4 , the proposed approach substantially outperforms both SCCL and RSTC. These results demonstrate that the representations learned by the proposed method form a feature space with high intra-cluster compactness and clear inter-cluster separability."},{"type":"heading","level":2,"text":"4.4 Ablation Study"},{"type":"para","text":"This section presents three groups of ablation experiments, covering all key components in the proposed model."},{"type":"para","text":"Effectiveness of semantic consistency in CAOT. As formulated in Eq. ( 3 ), the semantic consistency term ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\langle\\bm{S},\\bm{Q}\\bm{Q}^{T}\\rangle incorporates both the learnable attention similarity 𝑺 a t t \\bm{S}^{att} and the static cosine similarity 𝑺 c o s \\bm{S}^{cos} . To evaluate the contribution of each component, we conduct ablation studies by selectively removing these similarities, i.e. w/o Cos, w/o Att and w/o All. The results are shown in Table 3 . The comparison between OURS and w/o All shows that incorporating sample-to-sample semantic consistency into the OT formulation increases accuracy significantly. The comparison between OURS with w/o Att and w/o Cos shows that the proposed method outperforms those using a single similarity. This is because exploring similarities in different spaces can capture authentic relationships between samples. Finally, the comparison between w/o Att and w/o Cos indicates that the proposed attention mechanism is more effective than the current method in capturing semantic relationships."},{"type":"para","text":"Effectiveness of contrastive learning . As detailed in Section 3.3 , contrastive learning is first used to enhance the discriminability of sample representations, which serves as a prerequisite for similarity modeling. To evaluate its contribution, we perform ablation experiments by removing the contrastive learning component, i.e., w/o CL . As shown in Table 3 , removing contrastive learning leads to a substantial performance degradation. These results validate that discriminative representations are essential for capturing the semantic similarity between samples, as they provided the necessary clarity to distinguish between samples."},{"type":"para","text":"Effectiveness of Pseudo-Label Optimization . We investigate the effectiveness of pseudo-labels for optimizing cluster assignments by removing the L P L_{P} loss function, i.e., w/o PL . In this variant, the K-means algorithm is used for clustering instead. As shown in Table 3 , the performance drops significantly without the L P L_{P} loss. This demonstrates that the pseudo-labels generated by CAOT can effectively guide the clustering process and yield a more accurate partition than static clustering algorithms."},{"type":"heading","level":2,"text":"4.5 Hyperparameter Analysis"},{"type":"para","text":"This section presents experiments to validate the effects of hyperparameters ε 1 \\varepsilon_{1} , ε 2 \\varepsilon_{2} , ε 3 \\varepsilon_{3} and λ \\lambda . Candidate sets for these parameters are { 0.5 , 1 , 5 , 10 } \\left\\{0.5,1,5,10\\right\\} for ε 1 \\varepsilon_{1} , { 0.03 , 0.06 , 0.1 , \\{0.03,0.06,0.1, 1 , 3.5 , 7 , 10 , 100 } 1,3.5,7,10,100\\} for ε 2 \\varepsilon_{2} , { 10 , 15 , 20 , 25 , 30 } \\{10,15,20,25,30\\} for ε 3 \\varepsilon_{3} , and { 1 , 5 , 10 , 15 , 20 } \\left\\{1,5,10,15,20\\right\\} for λ \\lambda . Experiments are conducted on the StackOverflow, SearchSnippets, GoogleNews-T, and Tweet datasets, which correspond to a balanced dataset, a slightly imbalanced dataset, an imbalanced dataset, and a severely imbalanced dataset, respectively."},{"type":"para","text":"The results are presented in Figure 5 . As illustrated in Figures 5 (a), (c), and (d), the proposed model exhibits robust performance across a wide range of values for ε 1 \\varepsilon_{1} , ε 3 \\varepsilon_{3} , and λ \\lambda , demonstrating that the model is insensitive to these hyperparameters. The hyperparameter ε 2 \\varepsilon_{2} governs the strength of the uniformity constraint in Eq. ( 3 ), and it needs to be adjusted according to the specific imbalance level of dataset. Fortunately, Figure 5 (b) shows that ε 2 \\varepsilon_{2} allows a flexible adjustment range, i.e., the model remains stable when ε 2 \\varepsilon_{2} ranges from 3.5 to 100 on balanced dataset (StackOverflow), from 0.03 to 3.5 on slightly imbalanced dataset (SearchSnippets), and from 0.03 to 0.1 on imbalanced dataset (GoogleNews-TS) and severely imbalanced dataset (Tweet). Therefore, an approximate estimate of the degree of category imbalance is sufficient to determine a suitable value of ε 2 \\varepsilon_{2} ."},{"type":"para","text":"Building on this observation, we introduce an estimation method based on the Coefficient of Variation (CV). Specifically, we first use bge-large-en-v1.5 to obtain text embeddings, and then compute the CV value of the K-means clustering results based on these embeddings. The results are shown in Table 4 . It can be seen that CV values exhibit significant variation across datasets with different imbalance levels. Based on this, we establish a step-wise mapping function that consider datasets with CV values in the ranges of [ 0 , 0.2 ) , [ 0.2 , 0.4 ) , [ 0.4 , 0.6 ) [0,0.2),[0.2,0.4),[0.4,0.6) , and [ 0.6 , 1.0 ] [0.6,1.0] to be balanced, slightly imbalanced, imbalanced, and severely imbalanced, respectively. This protocol allows the proposed model to adaptively configure ε 2 \\varepsilon_{2} for unseen datasets. Overall, the experiments demonstrate that the proposed model achieves adaptive hyperparameter configuration without manual hyperparameter tuning."},{"type":"heading","level":1,"text":"5 Cross-Domain Generalization"},{"type":"para","text":"This section extends the proposed method to long-text datasets and image datasets to verify its cross-domain generalization capability."},{"type":"para","text":"Long-text clustering. We validate the generalizability of the proposed model in the long-text clustering using four datasets: the SyskillWebert (Pazzani, 1999 ) , 20Newsgroups (Mitchell, 1997 ) , MN-DS (Petukhova and Fachada, 2023 ) , and Reuters (Lewis, 1997 ) . We compare the model with TCLLM (Petukhova et al. , 2025 ) , a recent SOTA method for long-text clustering. The performance comparison is shown in Table 5 . With the same encoder, the proposed model achieves better clustering performance than TCLLM. This demonstrates that the proposed approach generalizes well to long-document and exhibits cross-domain applicability."},{"type":"para","text":"Image clustering. In image clustering, we compare the proposed model with three methods: CC (Li et al. , 2021 ) , ICC-SPC (Guo et al. , 2023 ) , and discDC (Cai et al. , 2025 ) . The experiments are conducted on the STL-10 (Coates et al. , 2011 ) , CIFAR-10 (Krizhevsky et al. , 2009 ) and CIFAR-100 (Krizhevsky et al. , 2009 ) datasets. To ensure a fair comparison, we follow the feature extraction pipeline used in discDC. The result is reported in Table 6 . It can be observed that the proposed method outperforms CC and ICC-SPC, and achieves performance comparable to discDC (a recent SOTA method specifically tailored for image clustering). These results show that our method generalizes well to image clustering tasks and exhibits cross-domain adaptability."},{"type":"heading","level":1,"text":"6 Related Work"},{"type":"heading","level":4,"text":"Pseudo-labeling based on optimal transport."},{"type":"para","text":"Optimal transport is a mathematical theory concerned with finding a transport matrix from one distribution to another while minimizing the total transport cost (Peyré et al. , 2019 ; Laclau et al. , 2017 ; Liu et al. , 2023 ) . Compared with greedy pseudo-labeling methods, OT-based methods consider the overall structure between sample distributions and cluster distributions to generate pseudo-labels (Qin et al. , 2026 ; Caron et al. , 2020 ; Cui et al. , 2025 ) , rather than assigning pseudo-labels to each sample in isolation (Berthelot et al. , 2020 ; Del Barrio et al. , 2019 ) . SELA (Asano et al. , 2020 ) is a representative work that uses OT to generate pseudo-labels by enforcing a uniform distribution over clusters. However, this method is not suitable for imbalanced datasets. PPOT (Zhang et al. , 2024 ) and RSTC (Zheng et al. , 2023 ) introduce a regularizer based on the imbalance level of predicted cluster probabilities, enabling them to handle imbalanced datasets. However, all these methods only consider relationships between samples and clusters, but ignore the inherent consistency between samples, which may assign different pseudo-labels to semantically similar samples."},{"type":"para","text":"Short Text Clustering. Previous research can be divided into three categories: conventional methods, deep embedding methods, and deep joint clustering methods. Conventional methods rely on text statistical methods, such as BOW and TF-IDF, to extract features (Chowdhury, 2010 ) . However, the representations produced by these methods are typically sparse and lack semantic information (Hadifar et al. , 2019 ; Gao et al. , 2021 ) . Deep embedding methods use neural networks to extract features, after which clustering methods, such as K-means, are applied (Xu et al. , 2017 ; Yong et al. , 2024 ) . However, feature extraction and clustering are often decoupled in these methods, which makes extracted representations unsuitable for clustering (Yao et al. , 2025 ) . Deep joint clustering methods integrate representation learning and clustering into a unified framework, allowing representation learning to be guided by the clustering objective (Xie et al. , 2016 ; Zhang et al. , 2021 ; Zheng et al. , 2023 ; Yao et al. , 2025 ) ."},{"type":"heading","level":1,"text":"7 Conclusion"},{"type":"para","text":"In this paper, we propose an end-to-end short text clustering framework. Its effectiveness mainly stems from the proposed CAOT, which addresses the limitation of existing OT methods that ignore sample-to-sample semantic consistency. Extensive experiments on eight benchmark datasets demonstrate that the proposed framework outperforms state-of-the-art methods by a large margin. Beyond short texts clustering, the proposed framework can be generalized to long texts and image clustering tasks. Experiments also demonstrate that the proposed framework is robust to hyperparameter variations and has computational advantages over existing methods. The results suggest that the proposed framework can be applied to real-world clustering tasks."},{"type":"heading","level":1,"text":"Impact Statement"},{"type":"para","text":"This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here."},{"type":"heading","level":2,"text":"A.1 Formulation of the Solution"},{"type":"para","text":"As mentioned in Section 3.2 , the optimization problem in the proposed CAOT is:"},{"type":"para","text":"where ⟨ ⋅ , ⋅ ⟩ \\langle\\cdot,\\cdot\\rangle is the Frobenius inner product; ε 1 \\varepsilon_{1} , ε 2 \\varepsilon_{2} and ε 3 \\varepsilon_{3} are weighting coefficients; 𝒂 = 1 N 𝟏 n × 1 \\bm{a}=\\frac{1}{N}\\bm{1}_{\\scriptscriptstyle n\\times 1} ; H ( 𝑸 ) = ⟨ 𝑸 , log ( 𝑸 ) − 1 ⟩ H(\\bm{Q})=\\langle\\bm{Q},\\text{log}(\\bm{Q})-1\\rangle , and Ψ ( 𝒃 ) = − log ( 𝒃 ) − log ( 𝟏 k × 1 − 𝒃 ) \\Psi(\\bm{b})=-\\text{log}(\\bm{b})-\\text{log}(\\bm{1}_{\\scriptscriptstyle k\\times 1}-\\bm{b}) ."},{"type":"para","text":"To solve the problem, we propose an iterative method that integrates the Taylor expansion and the Lagrange multiplier. In the t t -th ( t ≥ 1 t\\geq 1 ) iteration, the Taylor expansion of ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\langle\\bm{S},\\bm{QQ}^{T}\\rangle truncated to the constant and linear terms is as follows:"},{"type":"para","text":"in which ∂ ⟨ 𝑺 , 𝑸 𝑸 T ⟩ ∂ 𝑸 = ( 𝑺 + 𝑺 T ) 𝑸 \\frac{\\partial\\langle\\bm{S},\\bm{QQ}^{T}\\rangle}{\\partial\\bm{Q}}=(\\bm{S}+\\bm{S}^{T})\\bm{Q} is used."},{"type":"para","text":"After substituting ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\langle\\bm{S},\\bm{QQ}^{T}\\rangle in the objective function of Eq. ( 16 ) with its first-order Taylor approximation, one can get the following optimization problem:"},{"type":"para","text":"In the t t -th iteration, with 𝑸 t − 1 \\bm{Q}_{t-1} available, the objective function in Eq. ( 17 ) can be rewritten as follows:"},{"type":"para","text":"in which C = ε 3 ⟨ ( 𝑺 + 𝑺 T ) 𝑸 t − 1 , 𝑸 t − 1 ⟩ − ε 3 ⟨ 𝑺 , 𝑸 t − 1 𝑸 t − 1 T ⟩ C=\\varepsilon_{3}\\langle(\\bm{S}+\\bm{S}^{T})\\bm{Q}_{t-1},\\bm{Q}_{t-1}\\rangle-\\varepsilon_{3}\\langle\\bm{S},\\bm{Q}_{t-1}\\bm{Q}_{t-1}^{T}\\rangle is a constant. Therefore, the optimization problem in Eq. ( 17 ) can be rewritten as follows:"},{"type":"para","text":"with 𝑴 = − log ( 𝑷 ( 0 ) ) − ε 3 ( 𝑺 + 𝑺 T ) 𝑸 t − 1 \\bm{M}=-\\text{log}(\\bm{P}^{(0)})-\\varepsilon_{3}(\\bm{S}+\\bm{S}^{T})\\bm{Q}_{t-1} ."},{"type":"para","text":"The Lagrange multiplier method is used to solve the optimization problem in Eq. ( 19 ). The Lagrangian function is:"},{"type":"para","text":"where 𝝀 \\bm{\\lambda} , 𝝉 \\bm{\\tau} and ν \\nu are Lagrange multipliers. The partial derivative of F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) F(\\bm{Q},\\bm{b},\\bm{\\lambda},\\bm{\\tau},\\nu) with respect to (w.r.t.) Q i j Q_{ij} is:"},{"type":"para","text":"where λ i \\lambda_{i} denotes the i i -th element of 𝝀 \\bm{\\lambda} and τ j \\tau_{j} denotes the j j -th element of 𝝉 \\bm{\\tau} . M i j M_{ij} and Q i j Q_{ij} are the ( i , j ) (i,j) -th elements of 𝑴 \\bm{M} and 𝑸 \\bm{Q} , respectively. By setting ∂ F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) ∂ Q i j = 0 \\frac{\\partial F(\\bm{Q},\\bm{b},\\bm{\\lambda},\\bm{\\tau},\\nu)}{\\partial Q_{ij}}=0 , one can get:"},{"type":"para","text":"Due to the equality 𝑸 𝟏 k × 1 = 𝒂 \\bm{Q}\\bm{1}_{\\scriptscriptstyle k\\times 1}=\\bm{a} , one can get:"},{"type":"para","text":"where k k is the number of categories in the dataset. It is non-trivial to obtain:"},{"type":"para","text":"Similarly, due to the equality constraint 𝑸 T 𝟏 k × 1 = 𝒃 \\bm{Q}^{T}\\bm{1}_{\\scriptscriptstyle k\\times 1}=\\bm{b} , one can obtain:"},{"type":"para","text":"From Eq. ( 23 ) and Eq. ( 24 ), one can see that λ i \\lambda_{i} and τ j \\tau_{j} are functions of each other. When 𝒃 \\bm{b} is available, one can update λ i \\lambda_{i} and τ j \\tau_{j} in an iterative way."},{"type":"para","text":"Once 𝝀 \\bm{\\lambda} and 𝝉 \\bm{\\tau} are available, the next step is to get the value of 𝒃 \\bm{b} . Specifically, the partial derivative of F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) F(\\bm{Q},\\bm{b},\\bm{\\lambda},\\bm{\\tau},\\nu) in Eq. ( A.1 ) w.r.t. 𝒃 \\bm{b} is:"},{"type":"para","text":"where b j b_{j} is the j j -th element of 𝒃 \\bm{b} . By setting ∂ F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) ∂ b j = 0 \\frac{\\partial F(\\bm{Q},\\bm{b},\\bm{\\lambda},\\bm{\\tau},\\nu)}{\\partial b_{j}}=0 , one can get:"},{"type":"para","text":"which is a second-order polynomial equation with the variable b j b_{j} . It is straightforward to verify that Δ j = ( τ j − ν ) 2 + 4 ε 2 2 > 0 \\Delta_{j}=(\\tau_{j}-\\nu)^{2}+4\\varepsilon_{2}^{2}>0 . Thus, there are two feasible solutions to Eq. ( 26 ), which are: ( τ j − ν + 2 ε 2 ) + Δ j 2 ( τ j − ν ) \\frac{(\\tau_{j}-\\nu+2\\varepsilon_{2})+\\sqrt{\\Delta_{j}}}{2(\\tau_{j}-\\nu)} and ( τ j − ν + 2 ε 2 ) − Δ j 2 ( τ j − ν ) \\frac{(\\tau_{j}-\\nu+2\\varepsilon_{2})-\\sqrt{\\Delta_{j}}}{2(\\tau_{j}-\\nu)} , respectively. Since ( τ j − ν + 2 ε 2 ) + Δ j 2 ( τ j − ν ) ≥ 1 \\frac{(\\tau_{j}-\\nu+2\\varepsilon_{2})+\\sqrt{\\Delta_{j}}}{2(\\tau_{j}-\\nu)}\\geq 1 , it violates the domain constraint of Ψ ( 𝒃 ) = − log ( 𝒃 ) − log ( 𝟏 k × 1 − 𝒃 ) \\Psi(\\bm{b})=-\\log(\\bm{b})-\\log(\\bm{1}_{\\scriptscriptstyle k\\times 1}-\\bm{b}) . Thus, ( τ j − ν + 2 ε 2 ) + Δ j 2 ( τ j − ν ) \\frac{(\\tau_{j}-\\nu+2\\varepsilon_{2})+\\sqrt{\\Delta_{j}}}{2(\\tau_{j}-\\nu)} is not a valid solution. In contrast, since 0 < ( τ j − ν + 2 ε 2 ) − Δ j 2 ( τ j − ν ) < 1 0<\\frac{(\\tau_{j}-\\nu+2\\varepsilon_{2})-\\sqrt{\\Delta_{j}}}{2(\\tau_{j}-\\nu)}<1 , it is a valid solution. Therefore,"},{"type":"para","text":"Based on the constraint 𝒃 T 𝟏 k × 1 = 1 \\bm{b}^{T}\\bm{1}_{\\scriptscriptstyle k\\times 1}=1 , there is:"},{"type":"para","text":"which is a function of ν \\nu . One can use the Newton method to solve the equation in Eq. ( 28 ) to get the value of ν \\nu . Once ν \\nu is available, 𝒃 \\bm{b} can be calculated by Eq. ( 27 )."},{"type":"para","text":"By iteratively updating { 𝝀 , 𝝉 } \\left\\{\\bm{\\lambda},\\bm{\\tau}\\right\\} and { ν , 𝒃 } \\left\\{\\nu,\\bm{b}\\right\\} , one can get the solution for the optimization problem in Eq. ( 19 ). The pseudo-code for solving the optimization problem in CAOT is presented in Algorithm 2 ."},{"type":"heading","level":2,"text":"A.2 Convergence Verification of the Solution"},{"type":"para","text":"Algorithm 2 presents an iterative algorithm for solving the optimization problem in the proposed CAOT. This section empirically verifies its convergence. Specifically, we evaluate the convergence behavior of CAOT by monitoring the pseudo-label change rate, defined as the fraction of CAOT-generated pseudo-labels that differ between consecutive iterations."},{"type":"para","text":"The results are presented in Figure 6 . As shown in Figure 6 (a), the change rate exhibits a sharp decline during the early training stages, indicating that the model rapidly learns the underlying cluster structure. As training progresses, the curve gradually flattens, reflecting the stabilization of the optimization process. Figure 6 (b) provides a detailed view of the final iterations (1600–2000). One can observe that the change rate for most datasets consistently stabilizes below 4%, demonstrating that the proposed CAOT effectively reaches a steady and convergent state."},{"type":"para","text":"This section presents extensive supplementary experiments to further validate the effectiveness of the proposed model. § B.1 evaluates the effectiveness of CAOT, which is the core component that enables the proposed model to achieve SOTA performance. § B.2 examines the compatibility of the proposed framework with advanced encoders. § B.3 reports an objective comparison with existing methods in terms of computational budget. § B.4 investigates the reason why the model performs well on imbalanced datasets. § B.5 conducts comparative experiments on the feature space learned by the model."},{"type":"heading","level":2,"text":"B.1 Comparison of Pseduo-labeling Effectiveness"},{"type":"para","text":"To demonstrate the effectiveness of the proposed pseudo-labeling method ( CAOT-based PL ), we conduct comparative experiments against the prediction-based pseudo-labeling method ( Prediction-based PL ) and the AOT-based pseudo-labeling method ( AOT-based PL ). The AOT-based PL method is proposed in RSTC (Zheng et al. , 2023 ) , and remains the state-of-the-art OT method for short text clustering."},{"type":"para","text":"Experiments were conducted on the SearchSnippets, StackOverflow, and GoogleNews-TS datasets. We use ACC and NMI to evaluate the quality of pseudo-labels. The results are presented in Table 7 . It can be observed that: (1) The pseudo-labels generated by Prediction-based PL are of low quality, as this method assigns pseudo-labels to samples independently. (2) AOT-based PL produces better pseudo-labels than the Prediction-based PL method by capturing the sample-to-cluster global structure. However, this method ignores sample-to-sample semantic consistency, resulting in suboptimal pseudo-labels. (3) CAOT-based PL generates reliable pseudo-labels by simultaneously considering both sample-to-sample semantic consistency and sample-to-cluster global structure."},{"type":"para","text":"Furthermore, to provide a detailed comparison among the various pseudo-labeling methods, we visualize their “coupling matrix” when generating pseudo-labels. In the Prediction-based PL experiment, the “coupling matrix” refers to the predicted probability assignment matrix. In the AOT-based PL and CAOT-based PL experiments, the “coupling matrix” refers to the transport matrix. We randomly selected five samples from each category in the StackOverflow datasets for visualization. Results are illustrated in Figure 7 . One can observed that: (1) Prediction-based PL method produces incorrect soft-labels for some samples. (2) AOT-based PL achieves slightly better performance than Prediction-based PL. However, it exhibits a pronounced tendency to assign different pseudo-labels to semantically consistent samples, as evidenced by cases where samples belonging to the same category are frequently assigned different pseudo-labels. The reason for this issue is that existing OT methods ignore sample-to-sample semantic consistency. (3) CAOT-based PL method alleviates ambiguous predictions, and confidently assigns pseudo-labels to samples."},{"type":"heading","level":2,"text":"B.2 Compatibility with Advanced Encoders"},{"type":"para","text":"Experiments in this paper adopt SBERT as the backbone encoder, under which the proposed method already achieves state-of-the-art performance on most benchmark datasets. To further evaluate the generalizability of the proposed framework across different encoders, we conduct additional experiments by replacing SBERT with advanced text encoders, including bge-small-en-V1.5, bge-base-en-v1.5 (Xiao et al. , 2023 ) , and Qwen-3-Embedding (Zhang et al. , 2025 ) . The corresponding results are reported in Table 8 ."},{"type":"para","text":"As shown in Table 8 , replacing SBERT with stronger encoders consistently improves the clustering accuracy of the proposed method across all datasets. In particular, compared with the SBERT-based variant, the Qwen3-Embedding-based model achieves substantial performance gains on several datasets, including StackOverflow, Biomedical, and Tweet. These improvements demonstrate that the proposed framework is not limited to a specific encoder, but can leverage advanced representation models."},{"type":"para","text":"It is worth noting that although the SBERT-based version already achieves highly competitive performance, the use of more powerful encoders further enhances the model performance substantially. This demonstrates the strong compatibility and scalability of the proposed method with advanced representation models, and further suggests its potential for future short text clustering scenarios where stronger pretrained encoders are available."},{"type":"heading","level":2,"text":"B.3 Computation Budget"},{"type":"para","text":"In this paper, all experiments are conducted on an NVIDIA GeForce RTX 3090 Ti GPU with the PyTorch framework. We compare the proposed model with SCCL and RSTC in terms of model size and training time. Detailed results are presented in Table 9 . Three models are nearly the same in terms of model size, indicating that the proposed approach does not introduce additional computational overhead. In terms of training time, SCCL consumes more time than both RSTC and the proposed method. Moreover, compared with RSTC, the proposed method requires lower training time on large-scale datasets, including StackOverflow and Biomedical. The reason is that RSTC constructs and solves the optimal transport problem over the entire dataset in each iteration, whereas the proposed method operates on mini-batches. As a result, the per-iteration computation time of RSTC is longer than that of the proposed method."},{"type":"para","text":"To further evaluate how runtime scales with data size, we compare the proposed method with RSTC on expanded versions of the StackOverflow dataset. Specifically, we increase the dataset size from 20,000 samples to 40,000 and 80,000. As shown in Table 10 , the runtime of RSTC rises significantly as the dataset size grows, revealing its limited scalability on large datasets. In contrast, the runtime of the proposed method remains stable regardless of dataset size. These results demonstrate that the proposed method achieves superior computational efficiency and scalability, making it well suited for large-scale data clustering."},{"type":"heading","level":2,"text":"B.4 Analysis on Imbalanced Datasets"},{"type":"para","text":"As shown in Table 2 , the proposed model can be effectively applied to category-imbalanced datasets, including SearchSnippets, GoogleNews-TS, GoogleNews-T, GoogleNews-S, and Tweet. Given that contrastive learning can alleviate class imbalance to a certain extent, we conduct experiments to explore whether the model’s adaptability to imbalanced datasets stems from CAOT or contrastive learning. In the experiment, CAOT is replaced by conventional OT while other components remain unchanged. The performance degradation on imbalanced datasets indicates that CAOT solves the class imbalance problem, while stable performance suggests that contrastive learning plays the decisive role. As shown in Table 11 , clustering performance declines significantly when CAOT is removed, which confirms that the proposed model’s adaptability to imbalanced datasets is primarily attributed to CAOT."},{"type":"heading","level":2,"text":"B.5 Comparison of Representation Quality"},{"type":"para","text":"In Section 4.3 , we evaluate the learned representations via cosine similarity. This section provides a complementary analysis from another perspective. Since K-means clustering results reflect the intrinsic quality and separability of the learned feature space, we apply K-means clustering to quantitatively assess the representation quality."},{"type":"para","text":"In this experiment, RSTC is used as a comparison. As shown in Table 12 , K-means clustering on the optimized representations outperforms that on untrained ones, demonstrating that both methods can learn high-quality representations. In particular, the results show that representations learned by the proposed method achieves better performance. It can also be observed that the K-means clustering is inferior to the original approach. This reveals that although applying K-means to high-quality representations achieves good clustering performance, developing advanced clustering algorithms still remains essential."},{"type":"para","text":"This section provides supplementary details of the experimental setup. § C.1 elaborates on the datasets used in this work. § C.2 provides detailed definitions of evaluation metrics, and § C.3 presents the main experimental configurations."},{"type":"heading","level":2,"text":"C.1 Datasets"},{"type":"para","text":"We conducted experiments on eight datasets: AgNews, StackOverflow, Biomedical, SearchSnippets, GoogleNews-TS, GoogleNews-T, GoogleNews-S and Tweet. Brief descriptions of these datasets are as follows:"},{"type":"para","text":"AgNews is extracted from the AG news corpus (Zhang et al. , 2015 ) , which contains 8,000 news headlines and is categorized into four topics (Rakib et al. , 2020 ) ."},{"type":"para","text":"SearchSnippets is extracted from web search transactions. It consists of 12,340 web search results and is divided into eight categories (Phan et al. , 2008 ) ."},{"type":"para","text":"StackOverflow consists of 20,000 question titles drawn from 20 technical categories (Xu et al. , 2017 ) . These samples are randomly selected from Kaggle competition data and include a mix of technical discussions and programming queries."},{"type":"para","text":"Biomedical contains 20,000 research paper titles from 20 scientific topics (Xu et al. , 2017 ) . The data is sourced from BioASQ and reflects the typical professional vocabulary and structure of scientific literature."},{"type":"para","text":"GoogleNews comprises 11,109 articles covering 152 events (Yin and Wang, 2016 ) . The dataset is provided in three variants: full articles ( GoogleNews-TS ), titles only ( GoogleNews-T ), and snippets only ( GoogleNews-S )."},{"type":"para","text":"Tweet contains 2,472 tweets linked to 89 distinct queries (Yin and Wang, 2016 ) , collected from the Text Retrieval Conference’s microblog tracks in 2011 and 2012, reflecting the informality of social media communication."},{"type":"heading","level":2,"text":"C.2 Evaluation Metrics"},{"type":"para","text":"Consistent with previous works (Rakib et al. , 2020 ; Zheng et al. , 2023 ) , we employ two standard metrics to evaluate clustering performance: Accuracy (ACC) and Normalized Mutual Information (NMI). Accuracy measures the proportion of correct clustered texts, which is defined as follows:"},{"type":"para","text":"where N N is the dataset size, y i y_{i} denotes the true label, and y i ~ \\tilde{y_{i}} denotes the predicted label. 𝟙 ( ⋅ ) \\mathds{1}(\\cdot) is the indicator function. map ( ⋅ ) \\textup{map}(\\cdot) is a function that employs the Hungarian algorithm to align predicted labels with true labels (Papadimitriou and Steiglitz, 1998 ) ."},{"type":"para","text":"Normalized Mutual Information measures the mutual dependence between ground-truth and predicted labels by normalizing mutual information with their entropy:"},{"type":"para","text":"where 𝒀 \\bm{Y} denotes the set of true labels, 𝒀 ~ \\tilde{\\bm{Y}} denotes the set of predicted labels, I ( ⋅ , ⋅ ) I(\\cdot,\\cdot) denotes mutual information, and H ( ⋅ ) H(\\cdot) represents entropy."},{"type":"heading","level":2,"text":"C.3 Experiment Settings"},{"type":"para","text":"The Clustering Network G p G_{p} is an MLP network with dimensions 768-768-768- k k , where the first two layers use ReLU activation function, and the last layer uses a softmax activation function. The Projecting Network G z G_{z} is an MLP network with dimensions 768-768-128, where the first layer uses the ReLU activation function and, there are no activation functions in the second layer. The temperature parameters for attention loss and contrastive learning loss are τ A = 1 \\tau_{A}=1 and τ I = 1 \\tau_{I}=1 , respectively. The number of iterations, including both T 1 T_{1} and T 2 T_{2} in the solution to CAOT, is set to 10. The batch size n n is 200. The total number of training iterations E t o t a l E_{total} is 2,000. The number of warm-up iterations E w a r m E_{warm} is 600 for all datasets except Stackoverflow and Biomedical, where the warm-up stage is omitted. The hyperparameters ε 1 = 1 \\varepsilon_{1}=1 , ε 3 = 25 \\varepsilon_{3}=25 , and λ = 5 \\lambda=5 are fixed across all datasets, while ε 2 \\varepsilon_{2} is set to 100, 3.5, 0.06, and 0.03 for balanced, slightly imbalanced, imbalanced, and severely imbalanced datasets, respectively. When employing data augmentation, the contextual augmenter (Kobayashi, 2018 ) is configured as in SCCL (Zhang et al. , 2021 ) and RSTC (Zheng et al. , 2023 ) to generate two augmented texts by substituting the top-n suitable words of the input text."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the era of massive digital communication, short texts such as messages, queries, and comments are generated on an unprecedented scale. Effectively clustering and organizing these short texts is crucial for numerous applications (Ma and Zheng, 2025 ) . This urgency is further amplified by the prevalence of Large Language Models (LLMs), since LLM-driven applications, such as chatbots (Kang and Ki, 2025 ) and virtual assistants (Kweon et al. , 2025 ) , have further fueled the proliferation of short texts. Existing methods typically adopt an EM-like optimization framework to achieve clustering (Zheng et al. , 2023 ; Wu et al. , 2025 ) , which alternates between estimating pseudo-labels (E-step) and updating model parameters based on the generated pseudo-labels (M-step). Under this framework, early methods often adopt a greedy strategy to estimate pseudo-labels directly from model predictions (Sohn et al. , 2020 ) . However, they ignore both global and local information of the data distribution (Qin et al. , 2026 ) , resulting in unreliable pseudo-labels that may misguide model optimization in the subsequent M-step (Yin et al. , 2025 ; Cui et al. , 2025 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instead of assigning pseudo-labels greedily, recent studies introduce Optimal Transport (OT) to estimate pseudo-labels by minimizing the overall cost of transporting the sample distribution to the cluster distribution (Zheng et al. , 2023 ; Cui et al. , 2025 ) , integrating sample-cluster relationships from a global perspective (Zhang et al. , 2024 ; Zheng et al. , 2023 ) , i.e., the sample-to-cluster global structure . However, there is still a critical limitation: they ignore predictive consistency between semantically similar samples, i.e., sample-to-sample semantic consistency . Relying only on minimizing overall transport cost, these methods may assign different pseudo-labels to semantically similar samples when the transport costs to candidate pseudo-labels are not sufficiently distinct or accurate. As illustrated in Figure 1 (a), the blue triangles inside the red box exhibit nearly identical transport costs to both hexagons. Due to this ambiguity, existing OT methods struggle to make correct estimation, potentially assigning these samples to erroneous green pseudo-labels, even though they are surrounded by a dense neighborhood of correct blue peers. This issue is common in practice because feature spaces derived from pretrained encoders often lack clear category boundaries (Zhang et al. , 2021 ) , causing samples near boundaries to have similar transport costs to different candidate pseudo-labels. Once these noisy pseudo-labels are introduced into the M-step, the initial embedding biases are amplified, leading to widespread mislabeling and significantly degrading clustering quality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address this limitation, we propose a novel OT method to enhance pseudo-label estimation in the E-step, termed consistency-aware adaptive optimal transport ( CAOT ). In contrast to existing OT methods, CAOT incorporates the semantic relationships among surrounding samples into the transport process, correcting erroneous estimations caused by ambiguous transport costs, as shown in Figure 1 (b). To achieve this, we first propose a novel instance-level attention network to capture semantic relationships between samples. These learned relationships are integrated into CAOT as a semantic consistency constraint, encouraging semantically similar samples to be assigned identical pseudo-labels. During EM iterations, the instance-level attention network and the clustering assignments are jointly optimized in the M-step, using pseudo-labels inferred by CAOT (E-step) as supervisory signals. Through successive EM iterations, the attention network becomes increasingly effective in modeling semantic relationships, which in turn provides accurate guidance for CAOT to promote semantic consistency. As a result, the pseudo-labels generated by CAOT capture both the sample-to-cluster global structure and the sample-to-sample semantic consistency. These reliable pseudo-labels can effectively optimize model parameters to achieve satisfactory clustering performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We conduct comprehensive experiments on eight benchmark datasets. Experimental results demonstrate that the proposed approach outperforms state-of-the-art methods. Complementing these quantitative results, visualization comparisons of transport results further demonstrate that pseudo-labels produced by the proposed CAOT are more accurate and confident than those produced by existing OT methods. Beyond the field of short-text, the proposed approach can be generalized to long-text and image domains, demonstrating its potential as a general solution for various clustering tasks. Further analysis demonstrates that the proposed approach also has computational advantages over existing methods, which makes it suitable for real-world clustering tasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Preliminary\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reviews the commonly adopted OT formulation and its application in pseudo-labeling. Given a source distribution 𝒂 \\\\bm{a} and a target distribution 𝒃 \\\\bm{b} , the objective of OT is to find a matrix 𝑸 \\\\bm{Q} to transform the distribution 𝒂 \\\\bm{a} to 𝒃 \\\\bm{b} with minimal total cost. The optimization problem in OT is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ⟨ ⋅ , ⋅ ⟩ \\\\langle\\\\cdot,\\\\cdot\\\\rangle represents the Frobenius inner product, ε 1 \\\\varepsilon_{1} is a weighting coefficient, 𝑪 ∈ ℝ | 𝒂 | × | 𝒃 | \\\\bm{C}\\\\in\\\\mathbb{R}^{|\\\\bm{a}|\\\\times|\\\\bm{b}|} is a cost matrix, and | 𝒂 | |\\\\bm{a}| and | 𝒃 | |\\\\bm{b}| are the dimensions of 𝒂 \\\\bm{a} and 𝒃 \\\\bm{b} , respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In OT-based pseudo-labeling methods, the matrix 𝑪 \\\\bm{C} is defined as − log ( 𝑷 ) \\\\!-\\\\text{log}(\\\\bm{P}) , where 𝑷 ∈ ℝ n × k \\\\bm{P}\\\\!\\\\!\\\\in\\\\!\\\\!\\\\mathbb{R}^{n\\\\!\\\\times\\\\!k} is a matrix containing the predicted probabilities of samples, with n n the batch size and k k the number of categories. The optimization problem is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝑸 𝟏 k × 1 = 1 n 𝟏 n × 1 \\\\bm{Q}\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1}=\\\\frac{1}{n}\\\\bm{1}{\\\\scriptscriptstyle n\\\\times 1} indicates that all samples are equally important, and 𝑸 T 𝟏 n × 1 = 1 k 𝟏 k × 1 \\\\bm{Q}^{T}\\\\bm{1}_{\\\\scriptscriptstyle n\\\\times 1}=\\\\frac{1}{k}\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1} ensures that each category in the target distribution is assigned the same probability. After obtaining the transport matrix 𝑸 \\\\bm{Q} , the pseudo-label for the i -th sample is calculated as y ^ i = arg max ( 𝑸 i : ) \\\\hat{y}_{i}=\\\\arg\\\\max(\\\\bm{Q}_{i:}) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The Proposed Method\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section provides a detailed description of the proposed approach. § 3.1 gives a high-level overview of the entire framework. § 3.2 , 3.3 , and 3.4 offer an in-depth description of three sub-components, where the proposed CAOT is introduced in § 3.2 to address the limitation of existing OT methods. § 3.5 introduces the training pipeline.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 The Overall Structure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The structure of the proposed model is shown in Figure 2 . It consists of three components: the Pseudo-label Generation Module (PGM), the Semantic Similarity Construction Module (SSCM), and the Supervised Guidance Module (SGM).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"During model training, the PGM functions in the E-step, where it performs CAOT using the similarity matrix 𝑺 a t t \\\\bm{S}^{att} to generate pseudo-labels 𝒚 ^ \\\\hat{\\\\bm{y}} . The SSCM and SGM operate in the M-step to update model parameters. The role of SSCM is to capture the semantic similarities between samples and construct the similarity matrix 𝑺 a t t \\\\bm{S}^{att} . To this end, contrastive learning is first applied to enhance the discriminability of sample representations ( L I L_{I} loss), after which pseudo-labels 𝒚 ^ \\\\hat{\\\\bm{y}} are leveraged to train the instance-level attention network G h G_{h} to yield the matrix 𝑺 a t t \\\\bm{S}^{att} ( L A L_{A} loss). The role of SGM is to leverage pseudo-labels 𝒚 ^ \\\\hat{\\\\bm{y}} to optimize cluster assignments.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 The Pseudo-label Generation Module\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Pseudo-label Generation Module is expected to produce reliable pseudo-labels. To this end, we propose a novel OT formulation, referred to as CAOT. Solving the optimization problem in CAOT yields pseudo-labels that simultaneously integrate sample-to-cluster global structural information and sample-to-sample semantic consistency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The structure of the Pseudo-label Generation Module is shown in Figure 2 (a). Given a batch of n n text samples X ( 0 ) = [ x 1 ( 0 ) ; … ; x n ( 0 ) ] X^{(0)}=[x_{1}^{(0)};\\\\dots;x_{n}^{(0)}] , the text representations are obtained by the Encoder Φ \\\\mathnormal{\\\\Phi} as 𝑽 ( 0 ) ∈ ℝ n × d 1 \\\\bm{V}^{(0)}\\\\in\\\\mathbb{R}^{n\\\\times d_{1}} , where d 1 d_{1} is the dimension of the representations. Then, the Clustering Network G p G_{p} is used to predict probability assignments 𝑷 ( 0 ) ∈ ℝ n × k \\\\bm{P}^{(0)}\\\\in\\\\mathbb{R}^{n\\\\times k} ( G p G_{p} is an MLP network with dimensions 768-768-768- k k ). After that, pseudo-labels can be generated by solving the optimization problem in CAOT, which is defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ε 1 \\\\varepsilon_{1} , ε 2 \\\\varepsilon_{2} and ε 3 \\\\varepsilon_{3} are weighting coefficients, 𝒂 = 1 n 𝟏 n × 1 \\\\bm{a}=\\\\frac{1}{n}\\\\bm{1}_{\\\\scriptscriptstyle n\\\\times 1} , and 𝒃 \\\\bm{b} remains to be determined. 𝑺 = 𝑺 c o s + 𝑺 a t t \\\\bm{S}=\\\\bm{S}^{cos}+\\\\bm{S}^{att} is the semantic similarity matrix, where 𝑺 c o s \\\\bm{S}^{cos} is the cosine similarity calculated from 𝑷 ( 0 ) \\\\bm{P}^{(0)} , and 𝑺 a t t \\\\bm{S}^{att} is the attention similarity generated from the instance-level attention network ( G h G_{h} ) via Eq. ( 8 ). Each term in Eq. ( 3 ) is explained as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"H ( 𝑸 ) = ⟨ 𝑸 , log ( 𝑸 ) − 𝟏 n × k ⟩ H(\\\\bm{Q})=\\\\langle\\\\bm{Q},\\\\text{log}(\\\\bm{Q})-\\\\bm{1}_{\\\\scriptscriptstyle n\\\\times k}\\\\rangle is the same as Eq. ( LABEL:eq2 ), which prevents 𝑸 \\\\bm{Q} from becoming too sparse while ensuring that all elements remain strictly non-negative.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ψ ( 𝒃 ) = − log ( 𝒃 ) − log ( 𝟏 k × 1 − 𝒃 ) \\\\Psi(\\\\bm{b})=-\\\\text{log}(\\\\bm{b})-\\\\text{log}(\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1}-\\\\bm{b}) . Unlike Eq. ( LABEL:eq2 ), which strictly enforces the hard constraint 𝑸 T 𝟏 n × 1 = 1 k 𝟏 k × 1 \\\\bm{Q}^{T}\\\\bm{1}{\\\\scriptscriptstyle n\\\\times 1}=\\\\frac{1}{k}\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1} , CAOT adopts Ψ ( 𝒃 ) T 𝟏 k × 1 \\\\Psi(\\\\bm{b})^{T}\\\\bm{1}{\\\\scriptscriptstyle k\\\\times 1} as an adjustable penalty item to encourage 𝑸 T 𝟏 n × 1 \\\\bm{Q}^{T}\\\\bm{1}{\\\\scriptscriptstyle n\\\\times 1} to approach a uniform distribution. By adjusting the strength of this item, CAOT can flexibly adapt to datasets with varying degrees of category imbalance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\\\langle\\\\bm{S},\\\\bm{Q}\\\\bm{Q}^{T}\\\\rangle incorporates semantic consistency into CAOT, encouraging the transport matrix 𝑸 \\\\bm{Q} to align with the intrinsic semantic relationships between samples. Specifically, this design encourages the transport vector 𝑸 i : \\\\bm{Q}_{i:} to be similar to 𝑸 j : \\\\bm{Q}_{j:} when S i j S_{ij} is large. In other words, CAOT drives semantically similar samples towards similar transport vectors, thereby producing the same pseudo-labels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the proposed CAOT, ⟨ 𝑸 , − log ( 𝑷 ( 0 ) ) ⟩ \\\\langle\\\\bm{Q},-\\\\text{log}(\\\\bm{P}^{\\\\scriptstyle(0)})\\\\rangle incorporates the sample-to-cluster global structure into 𝑸 \\\\bm{Q} through the global minimum-cost matching constraint, enabling CAOT to exploit the joint confidence over the entire batch to correct erroneous predictions for some samples in 𝑷 ( 0 ) \\\\bm{P}^{\\\\scriptstyle(0)} . Simultaneously, ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\\\langle\\\\bm{S},\\\\bm{Q}\\\\bm{Q}^{T}\\\\rangle integrates sample-to-sample semantic consistency in 𝑸 \\\\bm{Q} , ensuring that semantically similar samples are assigned the same pseudo-labels. These two properties in 𝑸 \\\\bm{Q} empower the generation of reliable pseudo-labels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To solve the proposed optimization problem in CAOT, we design an iterative method that integrates Taylor expansion and Lagrange multiplier algorithm. The detailed solution is provided in Appendix A.1 , and the convergence analysis of the proposed solution is given in Appendix A.2 . Once the transport matrix 𝑸 \\\\bm{Q} is available, the pseudo-label for the i -th sample can be generated as y ^ i = arg max j Q i j . \\\\hat{y}_{i}=\\\\arg\\\\max_{j}\\\\,Q_{ij}. The resulting pseudo-labels are then utilized in SSCM and SGM to update the model parameters during the M-step.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Semantic Similarity Construction Module\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Semantic Similarity Construction Module is designed to identify and quantify semantic similarities between samples. The resulting similarity information is integrated into Eq. ( 3 ), guiding CAOT to incorporate sample-to-sample semantic consistency into the E-step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The structure of the Semantic Similarity Construction Module is shown in Figure 2 (b). Inspired by Socrates’ saying “To know others, first know yourself.”, we first enhance the discriminability of each sample representation using contrastive learning, providing a foundation for similarity modeling. Given a batch of texts X ( 0 ) X^{(0)} , contextual augmenter technique (Kobayashi, 2018 ) is used to generate two augmented texts, denoted as X ( 1 ) X^{(1)} and X ( 2 ) X^{(2)} . The representations of the augmented texts are then obtained by the Encoder Φ \\\\mathnormal{\\\\Phi} as 𝑽 ( 1 ) ∈ ℝ n × d 1 \\\\bm{V}^{(1)}\\\\in\\\\mathbb{R}^{n\\\\times d_{1}} and 𝑽 ( 2 ) ∈ ℝ n × d 1 \\\\bm{V}^{(2)}\\\\in\\\\mathbb{R}^{n\\\\times d_{1}} , respectively. Subsequently, the Projecting Network G z G_{z} is employed to reduce the dimensionality of the representations ( G z G_{z} is an MLP network with dimensions 768-768-128). The outputs are 𝒁 ( 1 ) ∈ ℝ n × d 2 \\\\bm{Z}^{(1)}\\\\in\\\\mathbb{R}^{n\\\\times d_{2}} and 𝒁 ( 2 ) ∈ ℝ n × d 2 \\\\bm{Z}^{(2)}\\\\in\\\\mathbb{R}^{n\\\\times d_{2}} , respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then, contrastive learning improves the discriminability of each sample representation by pulling positive pairs together and pushing negative pairs apart. Let 𝒛 i ( 1 ) \\\\bm{z}_{i}^{(1)} and 𝒛 i ( 2 ) \\\\bm{z}_{i}^{(2)} be the i i -th rows in 𝒁 ( 1 ) \\\\bm{Z}^{(1)} and 𝒁 ( 2 ) \\\\bm{Z}^{(2)} , respectively. For 𝒛 i ( 1 ) \\\\bm{z}_{i}^{(1)} , its corresponding positive pair is { 𝒛 i ( 1 ) , 𝒛 i ( 2 ) } \\\\{{\\\\bm{z}_{i}^{(1)}\\\\!,\\\\bm{z}_{i}^{(2)}}\\\\} , while the negative pairs are { 𝒛 i ( 1 ) , 𝒛 v ( 1 ) } \\\\{{\\\\bm{z}_{i}^{(1)}\\\\!,\\\\bm{z}_{v}^{(1)}}\\\\} and { 𝒛 i ( 1 ) , 𝒛 v ( 2 ) } \\\\{{\\\\bm{z}_{i}^{(1)}\\\\!,\\\\bm{z}_{v}^{(2)}}\\\\} for all v ≠ i v\\\\neq i . For 𝒛 i ( 2 ) \\\\bm{z}_{i}^{(2)} , the positive and negative pairs are constructed in the same manner. The loss function of contrastive learning for the i -th sample is defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where sim ( 𝒛 i ( 1 ) , 𝒛 v ( 2 ) ) \\\\textup{sim}(\\\\bm{z}_{i}^{(1)},\\\\bm{z}_{v}^{(2)}) is the cosine similarity between two representations, and τ I \\\\tau_{I} is the temperature parameter. The final loss function of contrastive learning is calculated over a batch and defined as L I = 1 2 n ∑ i = 1 n l I , i L_{I}=\\\\frac{1}{2n}\\\\sum_{i=1}^{n}l_{I,i} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By treating all samples with v ≠ i v\\\\neq i as negative pairs, contrastive learning encourages sample representations to be uniformly distributed in the feature space. Thus, a highly discriminative representation can be learned for each sample. This design facilitates the modeling of semantic similarity between samples, as discriminative representations possess the necessary clarity to distinguish each other (Zhang et al. , 2021 ) . Based on this, we propose an instance-level attention network to explore similarity relationships between samples. The structure of this attention network G h G_{h} is shown in Figure 3 . It takes 𝒁 ( 1 ) \\\\bm{Z}^{(1)} and 𝒁 ( 2 ) \\\\bm{Z}^{(2)} as inputs. For 𝒁 ( 1 ) \\\\bm{Z}^{(1)} , the network performs three linear transformations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"after which the similarity matrix 𝑺 ( 1 ) \\\\bm{S}^{(1)} is computed:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, the rows of 𝑻 ( 1 ) \\\\bm{T}^{(1)} are linearly combined according to the similarity matrix 𝑺 ( 1 ) \\\\bm{S}^{(1)} as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝒕 j ( 1 ) \\\\bm{t}_{j}^{(1)} is the j j -th row of 𝑻 ( 1 ) \\\\bm{T}^{(1)} , and S i j ( 1 ) S_{ij}^{(1)} is the ( i , j ) (i,j) -th element of 𝑺 ( 1 ) \\\\bm{S}^{(1)} . Similarly, 𝑺 ( 2 ) \\\\bm{S}^{(2)} and 𝑯 ( 2 ) \\\\bm{H}^{(2)} are obtained by passing 𝒁 ( 2 ) \\\\bm{Z}^{(2)} through the Attention Network. The attention similarity matrix is calculated as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To ensure that the similarity matrix 𝑺 a t t \\\\bm{S}^{att} can accurately reflects the underlying semantic relationships between samples, we apply targeted training to G h G_{h} . Let 𝑯 = [ 𝑯 ( 1 ) , 𝑯 ( 2 ) ] ∈ ℝ 2 n × d 2 \\\\bm{H}=[\\\\bm{H}^{(1)},\\\\bm{H}^{(2)}]\\\\in\\\\mathbb{R}^{2n\\\\times d_{2}} . As indicated in Eq. ( 7 ), 𝑺 a t t \\\\bm{S}^{att} is intrinsically linked to 𝑯 \\\\bm{H} . Based on this, we optimize 𝑺 a t t \\\\bm{S}^{att} by applying targeted training to 𝑯 \\\\bm{H} as a proxy. We achieve this by enhancing the intra-cluster compactness and inter-cluster separation of 𝑯 \\\\bm{H} , which implicitly guides 𝑺 a t t \\\\bm{S}^{att} to assign high attention weights to samples belonging to the same cluster. In practice, we first use pseudo-labels to identify samples belonging to the same cluster. For the i -th sample, the set of indices belonging to the same cluster is defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where y ^ i \\\\hat{y}_{i} is the pseudo-label of the i -th sample.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After that, the loss for the i i -th sample is defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where S i j a t t S_{ij}^{att} is the ( i , j ) (i,j) -th element of 𝑺 a t t \\\\bm{S}^{att} , 𝒉 i \\\\bm{h}_{i} is the i -th row of 𝑯 \\\\bm{H} , and τ A \\\\tau_{A} is the temperature parameter. The final loss function of attention optimization is calculated over a batch and defined as L A = 1 2 n ∑ i = 1 n l A , i L_{A}\\\\!=\\\\!\\\\frac{1}{2n}\\\\!\\\\sum_{i=1}^{n}\\\\!l_{A,i} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By minimizing L A L_{A} , samples within the same cluster are assigned high weights in 𝑺 a t t \\\\bm{S}^{att} , which in turn facilitates CAOT in the subsequent E-step. In the training process, accurate pseudo-labels help establish an accurate 𝑺 a t t \\\\bm{S}^{att} (M-step), which in turn facilitates the generation of more reliable pseudo-labels than those in the previous iteration (E-step).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Supervised Guidance Module\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Supervised Guidance Module leverages pseudo-labels generated by CAOT as supervision targets to optimize the model in clustering. The structure of the Supervised Guidance Module is shown in Figure 2 (c). Given a batch of representations 𝑽 ( 1 ) \\\\bm{V}^{(1)} and 𝑽 ( 2 ) \\\\bm{V}^{(2)} , the corresponding probability assignments 𝑷 ( 1 ) \\\\bm{P}^{(1)} and 𝑷 ( 2 ) \\\\bm{P}^{(2)} are obtained. Then, following the paradigm of supervised classification, the model is optimized by aligning its predictive probabilities with the supervision targets. The loss function is defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where OneHot( ⋅ \\\\cdot ) is a one-hot encoding operator, y ^ i \\\\hat{y}_{i} is the pseudo-label for the i i -th sample, 𝒑 i ( 1 ) \\\\bm{p}_{i}^{(1)} and 𝒑 i ( 2 ) \\\\bm{p}_{i}^{(2)} are the i -th rows of the probability matrix 𝑷 ( 1 ) \\\\bm{P}^{(1)} and 𝑷 ( 2 ) \\\\bm{P}^{(2)} , respectively.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5 Training Procedure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To prevent error accumulation in EM iterative optimization, we introduce a warm-up stage to provide a stable starting point for the training process. In this stage, pseudo-labels are generated via K-means on the representations 𝑽 ( 0 ) \\\\bm{V}^{(0)} , rather than the proposed CAOT. The warm-up loss combines L I L_{I} and L A L_{A} , defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where λ \\\\lambda is a balancing hyperparameter. The stop gradient technique is applied to the backpropagation of L A L_{A} , ensuring that L A L_{A} only optimizes the Attention Network G h G_{h} . After the warm-up stage, CAOT is utilized to generate pseudo-labels. The overall loss at this stage is formulated as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Input: dataset 𝒳 ( 0 ) \\\\mathcal{X}^{(0)} ; warm-up epochs E w a r m E_{warm} ; total epochs E t o t a l E_{total} ; Encoder Φ \\\\mathit{\\\\Phi} ; Network G z G_{z} , G h G_{h} , and G p G_{p} . Output: model parameters. Procedure:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After training, for a given text x ( 0 ) x^{(0)} , its clustering result is given by arg max ( 𝒑 ( 0 ) ) \\\\arg\\\\max(\\\\bm{p}^{(0)}) , where 𝒑 ( 0 ) \\\\bm{p}^{(0)} is the output of the Clustering Network G p G_{p} . The pseudo-code of the overall training process is provided in Algorithm 1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we present systematic evaluation experiments. § 4.1 introduces the configurations. § 4.2 analyzes the performance of clustering. § 4.3 compares the quality of feature spaces learned by the proposed model and the baseline methods. § 4.4 presents ablation studies to evaluate the contribution of each component of the model. § 4.5 conducts sensitivity analysis to examine the stability of the model under different hyperparameter settings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Experiment Setups\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Datasets. We conduct experiments on eight benchmark datasets: AgNews, StackOverflow, Biomedical, SearchSnippets, GoogleNews-TS, GoogleNews-T, GoogleNews-S, and Tweet. Table 1 summarizes the characteristics of each dataset. Depending on the degree of category imbalance (R), AgNews, StackOverflow, and Biomedical are regarded as balanced datasets; SearchSnippets as a slightly imbalanced dataset; GoogleNews-TS, GoogleNews-T, and GoogleNews-S as imbalanced datasets; Tweet as a severely imbalanced dataset. Detailed information about these datasets is provided in Appendix C.1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Experiment Settings. We use distilbert-base-nli-stsb-mean-tokens (SBERT) from the Sentence Transformers library as the Encoder (Reimers and Gurevych, 2019 ) . All parameters are optimized using the Adam optimizer. The learning rate of the Encoder is 5 × 10 − 6 5\\\\times 10^{-6} , and that of other networks is 5 × 10 − 4 5\\\\times 10^{-4} . The maximum input length for the Encoder is 32. The output dimensions of the Encoder and the Projection Network are set to d 1 = 768 d_{1}=768 and d 2 = 128 d_{2}=128 , respectively. We adopt Accuracy (ACC) and Normalized Mutual Information (NMI) to evaluate the model. The definitions of the evaluation metrics and other settings are provided in Appendix C.2 and Appendix C.3 , respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baselines. We compare the proposed method with the following baselines: TF-IDF (Chowdhury, 2010 ) extracts text representations using the TF-IDF technique. Back-Trans (Shen et al. , 2019 ) generates representations via contrastive learning with back-translation. STCC (Xu et al. , 2017 ) uses a CNN to optimize the output of word2vec as representations. Self-Train (Hadifar et al. , 2019 ) learns representations with an auto-encoder and updates them according to cluster assignments. SimCSE (Gao et al. , 2021 ) generates representations through contrastive learning with Dropout. For all the above methods, K-means is applied on the learned representations to yield clustering results. ESimCSE (Wu et al. , 2022 ) extends SimCSE by addressing the influence of sentence length on positional embeddings in transformers. SCCL (Zhang et al. , 2021 ) employs contrastive learning to refine the output of SBERT as representations and obtains clustering results using the DEC algorithm (Xie et al. , 2016 ) . RSTC (Zheng et al. , 2023 ) constructs pseudo-labels using optimal transport to assist clustering. SCL (Yong et al. , 2024 ) learns a projection matrix to map samples into a contrastive subspace, aligning original and projected embeddings to improve clustering. SCPCL (Liu et al. , 2025 ) employs optimized adaptive optimal transport to generate pseudo-labels, which guide a contrastive learning module to learn robust representations. FNSCC (Zheng et al. , 2025 ) enhances contrastive clustering by integrating fuzzy neighborhood information to refine both instance representations and soft cluster assignments.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Main Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results are presented in Table 2 . The results for Back-Trans, SimCSE, ESimCSE, and SCL are from the SCL paper. RSTC, SCPCL, and FNSCC are reproduced using the configurations provided in corresponding original papers, and the remaining baseline results are from the RSTC paper. According to the results, one can see that: (1) Conventional methods, including TF-IDF and Back-Trans , perform poorly due to their inability to produce meaningful representations. (2) Deep neural network-based methods ( STCC , Self-Train , SimCSE , ESimCSE , and SCL ) outperform conventional methods in learning effective representations. However, the decoupled feature learning and clustering processes lead to suboptimal results. (3) Deep joint clustering methods ( SCCL , RSTC , SCPCL , and FNSCC ) improve the clustering performance significantly by adopting contrastive learning to fine-tune pre-trained models. However, they ignore the relationships between samples, resulting in suboptimal performance. (4) The results of the proposed approach ranks first in eleven indicators and second in the remaining five indicators. Notably, on the StackOverflow dataset, the proposed method achieves a substantial accuracy gain of 5.01% . In comparison, existing methods often perform well on some datasets but exhibit performance degradation on others, indicating limited generalization across diverse datasets. These results validate the effectiveness and generalizability of the proposed approach.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Comparison of representation quality\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section investigates the efficacy of the proposed approach by examining the quality of the learned feature space. We evaluate the proposed approach along with SCCL and RSTC on the SearchSnippets and GoogleNews-TS datasets. The normalized average cosine similarity of samples within the same category is used as the evaluation metric, which is defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝟙 ( ⋅ ) \\\\mathds{1}(\\\\cdot) denotes the indicator function.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Figure 4 , the proposed approach substantially outperforms both SCCL and RSTC. These results demonstrate that the representations learned by the proposed method form a feature space with high intra-cluster compactness and clear inter-cluster separability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Ablation Study\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section presents three groups of ablation experiments, covering all key components in the proposed model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Effectiveness of semantic consistency in CAOT. As formulated in Eq. ( 3 ), the semantic consistency term ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\\\langle\\\\bm{S},\\\\bm{Q}\\\\bm{Q}^{T}\\\\rangle incorporates both the learnable attention similarity 𝑺 a t t \\\\bm{S}^{att} and the static cosine similarity 𝑺 c o s \\\\bm{S}^{cos} . To evaluate the contribution of each component, we conduct ablation studies by selectively removing these similarities, i.e. w/o Cos, w/o Att and w/o All. The results are shown in Table 3 . The comparison between OURS and w/o All shows that incorporating sample-to-sample semantic consistency into the OT formulation increases accuracy significantly. The comparison between OURS with w/o Att and w/o Cos shows that the proposed method outperforms those using a single similarity. This is because exploring similarities in different spaces can capture authentic relationships between samples. Finally, the comparison between w/o Att and w/o Cos indicates that the proposed attention mechanism is more effective than the current method in capturing semantic relationships.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Effectiveness of contrastive learning . As detailed in Section 3.3 , contrastive learning is first used to enhance the discriminability of sample representations, which serves as a prerequisite for similarity modeling. To evaluate its contribution, we perform ablation experiments by removing the contrastive learning component, i.e., w/o CL . As shown in Table 3 , removing contrastive learning leads to a substantial performance degradation. These results validate that discriminative representations are essential for capturing the semantic similarity between samples, as they provided the necessary clarity to distinguish between samples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Effectiveness of Pseudo-Label Optimization . We investigate the effectiveness of pseudo-labels for optimizing cluster assignments by removing the L P L_{P} loss function, i.e., w/o PL . In this variant, the K-means algorithm is used for clustering instead. As shown in Table 3 , the performance drops significantly without the L P L_{P} loss. This demonstrates that the pseudo-labels generated by CAOT can effectively guide the clustering process and yield a more accurate partition than static clustering algorithms.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Hyperparameter Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section presents experiments to validate the effects of hyperparameters ε 1 \\\\varepsilon_{1} , ε 2 \\\\varepsilon_{2} , ε 3 \\\\varepsilon_{3} and λ \\\\lambda . Candidate sets for these parameters are { 0.5 , 1 , 5 , 10 } \\\\left\\\\{0.5,1,5,10\\\\right\\\\} for ε 1 \\\\varepsilon_{1} , { 0.03 , 0.06 , 0.1 , \\\\{0.03,0.06,0.1, 1 , 3.5 , 7 , 10 , 100 } 1,3.5,7,10,100\\\\} for ε 2 \\\\varepsilon_{2} , { 10 , 15 , 20 , 25 , 30 } \\\\{10,15,20,25,30\\\\} for ε 3 \\\\varepsilon_{3} , and { 1 , 5 , 10 , 15 , 20 } \\\\left\\\\{1,5,10,15,20\\\\right\\\\} for λ \\\\lambda . Experiments are conducted on the StackOverflow, SearchSnippets, GoogleNews-T, and Tweet datasets, which correspond to a balanced dataset, a slightly imbalanced dataset, an imbalanced dataset, and a severely imbalanced dataset, respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results are presented in Figure 5 . As illustrated in Figures 5 (a), (c), and (d), the proposed model exhibits robust performance across a wide range of values for ε 1 \\\\varepsilon_{1} , ε 3 \\\\varepsilon_{3} , and λ \\\\lambda , demonstrating that the model is insensitive to these hyperparameters. The hyperparameter ε 2 \\\\varepsilon_{2} governs the strength of the uniformity constraint in Eq. ( 3 ), and it needs to be adjusted according to the specific imbalance level of dataset. Fortunately, Figure 5 (b) shows that ε 2 \\\\varepsilon_{2} allows a flexible adjustment range, i.e., the model remains stable when ε 2 \\\\varepsilon_{2} ranges from 3.5 to 100 on balanced dataset (StackOverflow), from 0.03 to 3.5 on slightly imbalanced dataset (SearchSnippets), and from 0.03 to 0.1 on imbalanced dataset (GoogleNews-TS) and severely imbalanced dataset (Tweet). Therefore, an approximate estimate of the degree of category imbalance is sufficient to determine a suitable value of ε 2 \\\\varepsilon_{2} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Building on this observation, we introduce an estimation method based on the Coefficient of Variation (CV). Specifically, we first use bge-large-en-v1.5 to obtain text embeddings, and then compute the CV value of the K-means clustering results based on these embeddings. The results are shown in Table 4 . It can be seen that CV values exhibit significant variation across datasets with different imbalance levels. Based on this, we establish a step-wise mapping function that consider datasets with CV values in the ranges of [ 0 , 0.2 ) , [ 0.2 , 0.4 ) , [ 0.4 , 0.6 ) [0,0.2),[0.2,0.4),[0.4,0.6) , and [ 0.6 , 1.0 ] [0.6,1.0] to be balanced, slightly imbalanced, imbalanced, and severely imbalanced, respectively. This protocol allows the proposed model to adaptively configure ε 2 \\\\varepsilon_{2} for unseen datasets. Overall, the experiments demonstrate that the proposed model achieves adaptive hyperparameter configuration without manual hyperparameter tuning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Cross-Domain Generalization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section extends the proposed method to long-text datasets and image datasets to verify its cross-domain generalization capability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Long-text clustering. We validate the generalizability of the proposed model in the long-text clustering using four datasets: the SyskillWebert (Pazzani, 1999 ) , 20Newsgroups (Mitchell, 1997 ) , MN-DS (Petukhova and Fachada, 2023 ) , and Reuters (Lewis, 1997 ) . We compare the model with TCLLM (Petukhova et al. , 2025 ) , a recent SOTA method for long-text clustering. The performance comparison is shown in Table 5 . With the same encoder, the proposed model achieves better clustering performance than TCLLM. This demonstrates that the proposed approach generalizes well to long-document and exhibits cross-domain applicability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Image clustering. In image clustering, we compare the proposed model with three methods: CC (Li et al. , 2021 ) , ICC-SPC (Guo et al. , 2023 ) , and discDC (Cai et al. , 2025 ) . The experiments are conducted on the STL-10 (Coates et al. , 2011 ) , CIFAR-10 (Krizhevsky et al. , 2009 ) and CIFAR-100 (Krizhevsky et al. , 2009 ) datasets. To ensure a fair comparison, we follow the feature extraction pipeline used in discDC. The result is reported in Table 6 . It can be observed that the proposed method outperforms CC and ICC-SPC, and achieves performance comparable to discDC (a recent SOTA method specifically tailored for image clustering). These results show that our method generalizes well to image clustering tasks and exhibits cross-domain adaptability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Pseudo-labeling based on optimal transport.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optimal transport is a mathematical theory concerned with finding a transport matrix from one distribution to another while minimizing the total transport cost (Peyré et al. , 2019 ; Laclau et al. , 2017 ; Liu et al. , 2023 ) . Compared with greedy pseudo-labeling methods, OT-based methods consider the overall structure between sample distributions and cluster distributions to generate pseudo-labels (Qin et al. , 2026 ; Caron et al. , 2020 ; Cui et al. , 2025 ) , rather than assigning pseudo-labels to each sample in isolation (Berthelot et al. , 2020 ; Del Barrio et al. , 2019 ) . SELA (Asano et al. , 2020 ) is a representative work that uses OT to generate pseudo-labels by enforcing a uniform distribution over clusters. However, this method is not suitable for imbalanced datasets. PPOT (Zhang et al. , 2024 ) and RSTC (Zheng et al. , 2023 ) introduce a regularizer based on the imbalance level of predicted cluster probabilities, enabling them to handle imbalanced datasets. However, all these methods only consider relationships between samples and clusters, but ignore the inherent consistency between samples, which may assign different pseudo-labels to semantically similar samples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Short Text Clustering. Previous research can be divided into three categories: conventional methods, deep embedding methods, and deep joint clustering methods. Conventional methods rely on text statistical methods, such as BOW and TF-IDF, to extract features (Chowdhury, 2010 ) . However, the representations produced by these methods are typically sparse and lack semantic information (Hadifar et al. , 2019 ; Gao et al. , 2021 ) . Deep embedding methods use neural networks to extract features, after which clustering methods, such as K-means, are applied (Xu et al. , 2017 ; Yong et al. , 2024 ) . However, feature extraction and clustering are often decoupled in these methods, which makes extracted representations unsuitable for clustering (Yao et al. , 2025 ) . Deep joint clustering methods integrate representation learning and clustering into a unified framework, allowing representation learning to be guided by the clustering objective (Xie et al. , 2016 ; Zhang et al. , 2021 ; Zheng et al. , 2023 ; Yao et al. , 2025 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, we propose an end-to-end short text clustering framework. Its effectiveness mainly stems from the proposed CAOT, which addresses the limitation of existing OT methods that ignore sample-to-sample semantic consistency. Extensive experiments on eight benchmark datasets demonstrate that the proposed framework outperforms state-of-the-art methods by a large margin. Beyond short texts clustering, the proposed framework can be generalized to long texts and image clustering tasks. Experiments also demonstrate that the proposed framework is robust to hyperparameter variations and has computational advantages over existing methods. The results suggest that the proposed framework can be applied to real-world clustering tasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Impact Statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.1 Formulation of the Solution\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As mentioned in Section 3.2 , the optimization problem in the proposed CAOT is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ⟨ ⋅ , ⋅ ⟩ \\\\langle\\\\cdot,\\\\cdot\\\\rangle is the Frobenius inner product; ε 1 \\\\varepsilon_{1} , ε 2 \\\\varepsilon_{2} and ε 3 \\\\varepsilon_{3} are weighting coefficients; 𝒂 = 1 N 𝟏 n × 1 \\\\bm{a}=\\\\frac{1}{N}\\\\bm{1}_{\\\\scriptscriptstyle n\\\\times 1} ; H ( 𝑸 ) = ⟨ 𝑸 , log ( 𝑸 ) − 1 ⟩ H(\\\\bm{Q})=\\\\langle\\\\bm{Q},\\\\text{log}(\\\\bm{Q})-1\\\\rangle , and Ψ ( 𝒃 ) = − log ( 𝒃 ) − log ( 𝟏 k × 1 − 𝒃 ) \\\\Psi(\\\\bm{b})=-\\\\text{log}(\\\\bm{b})-\\\\text{log}(\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1}-\\\\bm{b}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To solve the problem, we propose an iterative method that integrates the Taylor expansion and the Lagrange multiplier. In the t t -th ( t ≥ 1 t\\\\geq 1 ) iteration, the Taylor expansion of ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\\\langle\\\\bm{S},\\\\bm{QQ}^{T}\\\\rangle truncated to the constant and linear terms is as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"in which ∂ ⟨ 𝑺 , 𝑸 𝑸 T ⟩ ∂ 𝑸 = ( 𝑺 + 𝑺 T ) 𝑸 \\\\frac{\\\\partial\\\\langle\\\\bm{S},\\\\bm{QQ}^{T}\\\\rangle}{\\\\partial\\\\bm{Q}}=(\\\\bm{S}+\\\\bm{S}^{T})\\\\bm{Q} is used.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After substituting ⟨ 𝑺 , 𝑸 𝑸 T ⟩ \\\\langle\\\\bm{S},\\\\bm{QQ}^{T}\\\\rangle in the objective function of Eq. ( 16 ) with its first-order Taylor approximation, one can get the following optimization problem:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the t t -th iteration, with 𝑸 t − 1 \\\\bm{Q}_{t-1} available, the objective function in Eq. ( 17 ) can be rewritten as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"in which C = ε 3 ⟨ ( 𝑺 + 𝑺 T ) 𝑸 t − 1 , 𝑸 t − 1 ⟩ − ε 3 ⟨ 𝑺 , 𝑸 t − 1 𝑸 t − 1 T ⟩ C=\\\\varepsilon_{3}\\\\langle(\\\\bm{S}+\\\\bm{S}^{T})\\\\bm{Q}_{t-1},\\\\bm{Q}_{t-1}\\\\rangle-\\\\varepsilon_{3}\\\\langle\\\\bm{S},\\\\bm{Q}_{t-1}\\\\bm{Q}_{t-1}^{T}\\\\rangle is a constant. Therefore, the optimization problem in Eq. ( 17 ) can be rewritten as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with 𝑴 = − log ( 𝑷 ( 0 ) ) − ε 3 ( 𝑺 + 𝑺 T ) 𝑸 t − 1 \\\\bm{M}=-\\\\text{log}(\\\\bm{P}^{(0)})-\\\\varepsilon_{3}(\\\\bm{S}+\\\\bm{S}^{T})\\\\bm{Q}_{t-1} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Lagrange multiplier method is used to solve the optimization problem in Eq. ( 19 ). The Lagrangian function is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝝀 \\\\bm{\\\\lambda} , 𝝉 \\\\bm{\\\\tau} and ν \\\\nu are Lagrange multipliers. The partial derivative of F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) F(\\\\bm{Q},\\\\bm{b},\\\\bm{\\\\lambda},\\\\bm{\\\\tau},\\\\nu) with respect to (w.r.t.) Q i j Q_{ij} is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where λ i \\\\lambda_{i} denotes the i i -th element of 𝝀 \\\\bm{\\\\lambda} and τ j \\\\tau_{j} denotes the j j -th element of 𝝉 \\\\bm{\\\\tau} . M i j M_{ij} and Q i j Q_{ij} are the ( i , j ) (i,j) -th elements of 𝑴 \\\\bm{M} and 𝑸 \\\\bm{Q} , respectively. By setting ∂ F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) ∂ Q i j = 0 \\\\frac{\\\\partial F(\\\\bm{Q},\\\\bm{b},\\\\bm{\\\\lambda},\\\\bm{\\\\tau},\\\\nu)}{\\\\partial Q_{ij}}=0 , one can get:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Due to the equality 𝑸 𝟏 k × 1 = 𝒂 \\\\bm{Q}\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1}=\\\\bm{a} , one can get:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where k k is the number of categories in the dataset. It is non-trivial to obtain:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Similarly, due to the equality constraint 𝑸 T 𝟏 k × 1 = 𝒃 \\\\bm{Q}^{T}\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1}=\\\\bm{b} , one can obtain:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From Eq. ( 23 ) and Eq. ( 24 ), one can see that λ i \\\\lambda_{i} and τ j \\\\tau_{j} are functions of each other. When 𝒃 \\\\bm{b} is available, one can update λ i \\\\lambda_{i} and τ j \\\\tau_{j} in an iterative way.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Once 𝝀 \\\\bm{\\\\lambda} and 𝝉 \\\\bm{\\\\tau} are available, the next step is to get the value of 𝒃 \\\\bm{b} . Specifically, the partial derivative of F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) F(\\\\bm{Q},\\\\bm{b},\\\\bm{\\\\lambda},\\\\bm{\\\\tau},\\\\nu) in Eq. ( A.1 ) w.r.t. 𝒃 \\\\bm{b} is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where b j b_{j} is the j j -th element of 𝒃 \\\\bm{b} . By setting ∂ F ( 𝑸 , 𝒃 , 𝝀 , 𝝉 , ν ) ∂ b j = 0 \\\\frac{\\\\partial F(\\\\bm{Q},\\\\bm{b},\\\\bm{\\\\lambda},\\\\bm{\\\\tau},\\\\nu)}{\\\\partial b_{j}}=0 , one can get:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which is a second-order polynomial equation with the variable b j b_{j} . It is straightforward to verify that Δ j = ( τ j − ν ) 2 + 4 ε 2 2 > 0 \\\\Delta_{j}=(\\\\tau_{j}-\\\\nu)^{2}+4\\\\varepsilon_{2}^{2}>0 . Thus, there are two feasible solutions to Eq. ( 26 ), which are: ( τ j − ν + 2 ε 2 ) + Δ j 2 ( τ j − ν ) \\\\frac{(\\\\tau_{j}-\\\\nu+2\\\\varepsilon_{2})+\\\\sqrt{\\\\Delta_{j}}}{2(\\\\tau_{j}-\\\\nu)} and ( τ j − ν + 2 ε 2 ) − Δ j 2 ( τ j − ν ) \\\\frac{(\\\\tau_{j}-\\\\nu+2\\\\varepsilon_{2})-\\\\sqrt{\\\\Delta_{j}}}{2(\\\\tau_{j}-\\\\nu)} , respectively. Since ( τ j − ν + 2 ε 2 ) + Δ j 2 ( τ j − ν ) ≥ 1 \\\\frac{(\\\\tau_{j}-\\\\nu+2\\\\varepsilon_{2})+\\\\sqrt{\\\\Delta_{j}}}{2(\\\\tau_{j}-\\\\nu)}\\\\geq 1 , it violates the domain constraint of Ψ ( 𝒃 ) = − log ( 𝒃 ) − log ( 𝟏 k × 1 − 𝒃 ) \\\\Psi(\\\\bm{b})=-\\\\log(\\\\bm{b})-\\\\log(\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1}-\\\\bm{b}) . Thus, ( τ j − ν + 2 ε 2 ) + Δ j 2 ( τ j − ν ) \\\\frac{(\\\\tau_{j}-\\\\nu+2\\\\varepsilon_{2})+\\\\sqrt{\\\\Delta_{j}}}{2(\\\\tau_{j}-\\\\nu)} is not a valid solution. In contrast, since 0 < ( τ j − ν + 2 ε 2 ) − Δ j 2 ( τ j − ν ) < 1 0<\\\\frac{(\\\\tau_{j}-\\\\nu+2\\\\varepsilon_{2})-\\\\sqrt{\\\\Delta_{j}}}{2(\\\\tau_{j}-\\\\nu)}<1 , it is a valid solution. Therefore,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Based on the constraint 𝒃 T 𝟏 k × 1 = 1 \\\\bm{b}^{T}\\\\bm{1}_{\\\\scriptscriptstyle k\\\\times 1}=1 , there is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which is a function of ν \\\\nu . One can use the Newton method to solve the equation in Eq. ( 28 ) to get the value of ν \\\\nu . Once ν \\\\nu is available, 𝒃 \\\\bm{b} can be calculated by Eq. ( 27 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By iteratively updating { 𝝀 , 𝝉 } \\\\left\\\\{\\\\bm{\\\\lambda},\\\\bm{\\\\tau}\\\\right\\\\} and { ν , 𝒃 } \\\\left\\\\{\\\\nu,\\\\bm{b}\\\\right\\\\} , one can get the solution for the optimization problem in Eq. ( 19 ). The pseudo-code for solving the optimization problem in CAOT is presented in Algorithm 2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.2 Convergence Verification of the Solution\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm 2 presents an iterative algorithm for solving the optimization problem in the proposed CAOT. This section empirically verifies its convergence. Specifically, we evaluate the convergence behavior of CAOT by monitoring the pseudo-label change rate, defined as the fraction of CAOT-generated pseudo-labels that differ between consecutive iterations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results are presented in Figure 6 . As shown in Figure 6 (a), the change rate exhibits a sharp decline during the early training stages, indicating that the model rapidly learns the underlying cluster structure. As training progresses, the curve gradually flattens, reflecting the stabilization of the optimization process. Figure 6 (b) provides a detailed view of the final iterations (1600–2000). One can observe that the change rate for most datasets consistently stabilizes below 4%, demonstrating that the proposed CAOT effectively reaches a steady and convergent state.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section presents extensive supplementary experiments to further validate the effectiveness of the proposed model. § B.1 evaluates the effectiveness of CAOT, which is the core component that enables the proposed model to achieve SOTA performance. § B.2 examines the compatibility of the proposed framework with advanced encoders. § B.3 reports an objective comparison with existing methods in terms of computational budget. § B.4 investigates the reason why the model performs well on imbalanced datasets. § B.5 conducts comparative experiments on the feature space learned by the model.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 Comparison of Pseduo-labeling Effectiveness\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To demonstrate the effectiveness of the proposed pseudo-labeling method ( CAOT-based PL ), we conduct comparative experiments against the prediction-based pseudo-labeling method ( Prediction-based PL ) and the AOT-based pseudo-labeling method ( AOT-based PL ). The AOT-based PL method is proposed in RSTC (Zheng et al. , 2023 ) , and remains the state-of-the-art OT method for short text clustering.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Experiments were conducted on the SearchSnippets, StackOverflow, and GoogleNews-TS datasets. We use ACC and NMI to evaluate the quality of pseudo-labels. The results are presented in Table 7 . It can be observed that: (1) The pseudo-labels generated by Prediction-based PL are of low quality, as this method assigns pseudo-labels to samples independently. (2) AOT-based PL produces better pseudo-labels than the Prediction-based PL method by capturing the sample-to-cluster global structure. However, this method ignores sample-to-sample semantic consistency, resulting in suboptimal pseudo-labels. (3) CAOT-based PL generates reliable pseudo-labels by simultaneously considering both sample-to-sample semantic consistency and sample-to-cluster global structure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Furthermore, to provide a detailed comparison among the various pseudo-labeling methods, we visualize their “coupling matrix” when generating pseudo-labels. In the Prediction-based PL experiment, the “coupling matrix” refers to the predicted probability assignment matrix. In the AOT-based PL and CAOT-based PL experiments, the “coupling matrix” refers to the transport matrix. We randomly selected five samples from each category in the StackOverflow datasets for visualization. Results are illustrated in Figure 7 . One can observed that: (1) Prediction-based PL method produces incorrect soft-labels for some samples. (2) AOT-based PL achieves slightly better performance than Prediction-based PL. However, it exhibits a pronounced tendency to assign different pseudo-labels to semantically consistent samples, as evidenced by cases where samples belonging to the same category are frequently assigned different pseudo-labels. The reason for this issue is that existing OT methods ignore sample-to-sample semantic consistency. (3) CAOT-based PL method alleviates ambiguous predictions, and confidently assigns pseudo-labels to samples.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 Compatibility with Advanced Encoders\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Experiments in this paper adopt SBERT as the backbone encoder, under which the proposed method already achieves state-of-the-art performance on most benchmark datasets. To further evaluate the generalizability of the proposed framework across different encoders, we conduct additional experiments by replacing SBERT with advanced text encoders, including bge-small-en-V1.5, bge-base-en-v1.5 (Xiao et al. , 2023 ) , and Qwen-3-Embedding (Zhang et al. , 2025 ) . The corresponding results are reported in Table 8 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Table 8 , replacing SBERT with stronger encoders consistently improves the clustering accuracy of the proposed method across all datasets. In particular, compared with the SBERT-based variant, the Qwen3-Embedding-based model achieves substantial performance gains on several datasets, including StackOverflow, Biomedical, and Tweet. These improvements demonstrate that the proposed framework is not limited to a specific encoder, but can leverage advanced representation models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is worth noting that although the SBERT-based version already achieves highly competitive performance, the use of more powerful encoders further enhances the model performance substantially. This demonstrates the strong compatibility and scalability of the proposed method with advanced representation models, and further suggests its potential for future short text clustering scenarios where stronger pretrained encoders are available.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.3 Computation Budget\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, all experiments are conducted on an NVIDIA GeForce RTX 3090 Ti GPU with the PyTorch framework. We compare the proposed model with SCCL and RSTC in terms of model size and training time. Detailed results are presented in Table 9 . Three models are nearly the same in terms of model size, indicating that the proposed approach does not introduce additional computational overhead. In terms of training time, SCCL consumes more time than both RSTC and the proposed method. Moreover, compared with RSTC, the proposed method requires lower training time on large-scale datasets, including StackOverflow and Biomedical. The reason is that RSTC constructs and solves the optimal transport problem over the entire dataset in each iteration, whereas the proposed method operates on mini-batches. As a result, the per-iteration computation time of RSTC is longer than that of the proposed method.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To further evaluate how runtime scales with data size, we compare the proposed method with RSTC on expanded versions of the StackOverflow dataset. Specifically, we increase the dataset size from 20,000 samples to 40,000 and 80,000. As shown in Table 10 , the runtime of RSTC rises significantly as the dataset size grows, revealing its limited scalability on large datasets. In contrast, the runtime of the proposed method remains stable regardless of dataset size. These results demonstrate that the proposed method achieves superior computational efficiency and scalability, making it well suited for large-scale data clustering.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.4 Analysis on Imbalanced Datasets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Table 2 , the proposed model can be effectively applied to category-imbalanced datasets, including SearchSnippets, GoogleNews-TS, GoogleNews-T, GoogleNews-S, and Tweet. Given that contrastive learning can alleviate class imbalance to a certain extent, we conduct experiments to explore whether the model’s adaptability to imbalanced datasets stems from CAOT or contrastive learning. In the experiment, CAOT is replaced by conventional OT while other components remain unchanged. The performance degradation on imbalanced datasets indicates that CAOT solves the class imbalance problem, while stable performance suggests that contrastive learning plays the decisive role. As shown in Table 11 , clustering performance declines significantly when CAOT is removed, which confirms that the proposed model’s adaptability to imbalanced datasets is primarily attributed to CAOT.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.5 Comparison of Representation Quality\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In Section 4.3 , we evaluate the learned representations via cosine similarity. This section provides a complementary analysis from another perspective. Since K-means clustering results reflect the intrinsic quality and separability of the learned feature space, we apply K-means clustering to quantitatively assess the representation quality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this experiment, RSTC is used as a comparison. As shown in Table 12 , K-means clustering on the optimized representations outperforms that on untrained ones, demonstrating that both methods can learn high-quality representations. In particular, the results show that representations learned by the proposed method achieves better performance. It can also be observed that the K-means clustering is inferior to the original approach. This reveals that although applying K-means to high-quality representations achieves good clustering performance, developing advanced clustering algorithms still remains essential.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section provides supplementary details of the experimental setup. § C.1 elaborates on the datasets used in this work. § C.2 provides detailed definitions of evaluation metrics, and § C.3 presents the main experimental configurations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.1 Datasets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We conducted experiments on eight datasets: AgNews, StackOverflow, Biomedical, SearchSnippets, GoogleNews-TS, GoogleNews-T, GoogleNews-S and Tweet. Brief descriptions of these datasets are as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AgNews is extracted from the AG news corpus (Zhang et al. , 2015 ) , which contains 8,000 news headlines and is categorized into four topics (Rakib et al. , 2020 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SearchSnippets is extracted from web search transactions. It consists of 12,340 web search results and is divided into eight categories (Phan et al. , 2008 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"StackOverflow consists of 20,000 question titles drawn from 20 technical categories (Xu et al. , 2017 ) . These samples are randomly selected from Kaggle competition data and include a mix of technical discussions and programming queries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Biomedical contains 20,000 research paper titles from 20 scientific topics (Xu et al. , 2017 ) . The data is sourced from BioASQ and reflects the typical professional vocabulary and structure of scientific literature.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GoogleNews comprises 11,109 articles covering 152 events (Yin and Wang, 2016 ) . The dataset is provided in three variants: full articles ( GoogleNews-TS ), titles only ( GoogleNews-T ), and snippets only ( GoogleNews-S ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Tweet contains 2,472 tweets linked to 89 distinct queries (Yin and Wang, 2016 ) , collected from the Text Retrieval Conference’s microblog tracks in 2011 and 2012, reflecting the informality of social media communication.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.2 Evaluation Metrics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consistent with previous works (Rakib et al. , 2020 ; Zheng et al. , 2023 ) , we employ two standard metrics to evaluate clustering performance: Accuracy (ACC) and Normalized Mutual Information (NMI). Accuracy measures the proportion of correct clustered texts, which is defined as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where N N is the dataset size, y i y_{i} denotes the true label, and y i ~ \\\\tilde{y_{i}} denotes the predicted label. 𝟙 ( ⋅ ) \\\\mathds{1}(\\\\cdot) is the indicator function. map ( ⋅ ) \\\\textup{map}(\\\\cdot) is a function that employs the Hungarian algorithm to align predicted labels with true labels (Papadimitriou and Steiglitz, 1998 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Normalized Mutual Information measures the mutual dependence between ground-truth and predicted labels by normalizing mutual information with their entropy:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝒀 \\\\bm{Y} denotes the set of true labels, 𝒀 ~ \\\\tilde{\\\\bm{Y}} denotes the set of predicted labels, I ( ⋅ , ⋅ ) I(\\\\cdot,\\\\cdot) denotes mutual information, and H ( ⋅ ) H(\\\\cdot) represents entropy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.3 Experiment Settings\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Clustering Network G p G_{p} is an MLP network with dimensions 768-768-768- k k , where the first two layers use ReLU activation function, and the last layer uses a softmax activation function. The Projecting Network G z G_{z} is an MLP network with dimensions 768-768-128, where the first layer uses the ReLU activation function and, there are no activation functions in the second layer. The temperature parameters for attention loss and contrastive learning loss are τ A = 1 \\\\tau_{A}=1 and τ I = 1 \\\\tau_{I}=1 , respectively. The number of iterations, including both T 1 T_{1} and T 2 T_{2} in the solution to CAOT, is set to 10. The batch size n n is 200. The total number of training iterations E t o t a l E_{total} is 2,000. The number of warm-up iterations E w a r m E_{warm} is 600 for all datasets except Stackoverflow and Biomedical, where the warm-up stage is omitted. The hyperparameters ε 1 = 1 \\\\varepsilon_{1}=1 , ε 3 = 25 \\\\varepsilon_{3}=25 , and λ = 5 \\\\lambda=5 are fixed across all datasets, while ε 2 \\\\varepsilon_{2} is set to 100, 3.5, 0.06, and 0.03 for balanced, slightly imbalanced, imbalanced, and severely imbalanced datasets, respectively. When employing data augmentation, the contextual augmenter (Kobayashi, 2018 ) is configured as in SCCL (Zhang et al. , 2021 ) and RSTC (Zheng et al. , 2023 ) to generate two augmented texts by substituting the top-n suitable words of the input text.\"\n }\n]","isConversational":false}}},{"rowIdx":392,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10546"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation"},"authors":{"kind":"list like","value":["Jinyang Du","Hao Ma","Xiaohu Shi","Bo Yang","Yanchun Liang","Heow Pueh Lee","Chunguo Wu"],"string":"[\n \"Jinyang Du\",\n \"Hao Ma\",\n \"Xiaohu Shi\",\n \"Bo Yang\",\n \"Yanchun Liang\",\n \"Heow Pueh Lee\",\n \"Chunguo Wu\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10546v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"A central pursuit in modern computational science is bridging the gap between abundant empirical measurements and the underlying mathematical laws that dictate system dynamics [ 1 , 2 , 3 , 4 , 5 ] . While the formalization of this problem through symbolic regression (SR) has spurred significant advances [ 6 , 7 , 8 ] , traditional data-driven methodologies ranging from sparse regression [ 9 , 10 ] to genetic programming [ 11 , 12 , 13 ] exhibit critical limitations. Constrained by predefined function libraries, these approaches struggle to scale across vast combinatorial hypothesis spaces and lack the architectural flexibility to incorporate domain-specific physical priors [ 14 ] ."},{"type":"para","text":"Recently, Large Language Models (LLMs) have emerged as highly promising engines for this task [ 15 , 16 , 17 , 18 , 19 ] . By treating equations as executable structures, LLMs leverage their pre-trained scientific knowledge to propose physically plausible structural hypotheses [ 20 ] . However, extending these LLM-driven paradigms to the discovery of complex, time-varying PDEs is bottlenecked by severe numerical instabilities. To compute the necessary spatial derivatives for evaluating PDE hypotheses, traditional optimization frameworks predominantly rely on standard finite difference (FD) methods and pointwise residual evaluations [ 21 , 22 , 23 ] . These discrete operations inherently amplify high-frequency measurement noise, distorting the optimization landscape [ 24 , 25 ] . When coupled with an LLM, this numerical failure is fatal: it feeds deceptive fitness scores back to the language model, derailing the evolutionary search and preventing it from distinguishing true physical mechanisms from mathematical artifacts."},{"type":"para","text":"To resolve this critical bottleneck, we introduce LLM-PDESR (Large Language Model-guided Partial Differential Equation Symbolic Regression), a novel framework that integrates the structural hypothesis generation of LLMs with a mathematically rigorous evaluation environment. First, we replace noise-sensitive FD methods with quintic spline smoothing. By guaranteeing 𝒞 4 \\mathcal{C}^{4} continuity, quintic splines provide noise-robust spatial derivatives up to the fourth order [ 26 , 27 ] . Second, we transition from pointwise residuals to a subdomain weighted residual(SWR) evaluation. By integrating the PDE residual against smooth test functions over localized sub-domains, the SWR formulation acts as a natural low-pass filter [ 28 ] . This mathematically principled smoothing mitigates fitness landscape distortion, ensuring the internal continuous optimizer converges reliably and provides the LLM with accurate fitness signals. Consequently, a Pareto-driven feedback loop efficiently balances equation complexity and predictive accuracy, naturally pruning unphysical bloat [ 29 , 30 ] ."},{"type":"para","text":"We evaluate LLM-PDESR on a comprehensive benchmark of 23 canonical PDEs, alongside five structurally novel equations explicitly formulated to preclude dataset memorization and validate genuine symbolic discovery capabilities. Demonstrating real-world applicability, our framework successfully extracts a novel coupled atmospheric circulation surrogate (1D-CACE) directly from noisy ERA5 climate data [ 31 ] . Since capturing such intricate multivariate interactions severely degrades or exceeds existing baseline capabilities, we validate the discovered equations through rigorous out-of-distribution (OOD) testing across distinct spatiotemporal domains. This confirms they capture invariant dynamical mechanisms rather than localized statistical artifacts. Extensive empirical comparisons demonstrate that LLM-PDESR significantly outperforms state-of-the-art methods (including SGA-PDE [ 32 ] , DISCOVER [ 33 ] , WSINDy_PDE [ 34 ] , and EqGPT [ 35 ] ) in structural recovery, noise resilience, and the avoidance of spurious equations."},{"type":"para","text":"The main contributions of this work are summarized as follows:"},{"type":"para","text":"We propose LLM-PDESR, an end-to-end framework that integrates the structural hypothesis generation of LLMs with continuous numerical optimization and evolutionary search to automate PDE discovery."},{"type":"para","text":"We resolve the derivative noise bottleneck by leveraging 𝒞 4 \\mathcal{C}^{4} continuous quintic splines and SWR evaluations. This approach mitigates fitness landscape distortion, while a Pareto-driven feedback loop optimally balances predictive accuracy with structural parsimony."},{"type":"para","text":"We introduce a rigorous benchmark comprising 23 canonical PDEs and five structurally novel equations. This benchmark is explicitly designed to preclude dataset memorization and validate genuine symbolic discovery capabilities."},{"type":"para","text":"We demonstrate real-world applicability by extracting an interpretable 1D dynamical surrogate (1D-CACE) directly from noisy ERA5 climate data. Rigorous spatiotemporal OOD evaluations confirm that the framework captures invariant dynamical mechanisms rather than localized statistical artifacts."},{"type":"heading","level":1,"text":"2 Related work"},{"type":"heading","level":2,"text":"2.1 Dictionary-based methods and deep learning surrogates"},{"type":"para","text":"Data-driven PDE discovery initially leveraged sparse regression, exemplified by SINDy [ 9 ] , PDE-FIND, and STRidge [ 1 ] , which identify governing laws by selecting terms from predefined mathematical libraries. To mitigate the acute sensitivity of numerical differentiation to observation noise, integral-based approaches like WSINDy and WSINDy_PDE [ 34 ] were developed, successfully utilizing weak forms as natural low-pass filters. Concurrently, deep learning architectures such as Physics-Informed Neural Networks (PINNs) [ 3 ] , Fourier Neural Operators (FNO) [ 36 ] , and DeepONet [ 37 ] have demonstrated remarkable predictive capabilities. However, these neural surrogates typically act as black boxes, precluding explicit equation extraction. Moreover, while sparse regression offers interpretability, its reliance on handcrafted dictionaries imposes a rigid structural bottleneck. This rigidity not only triggers an intractable combinatorial explosion when scaling to highly nonlinear and high-dimensional physical systems, but also lacks the compositional flexibility to incorporate complex, domain-specific priors required for discovering novel, free-form dynamics."},{"type":"heading","level":2,"text":"2.2 Evolutionary and neural symbolic regression"},{"type":"para","text":"To overcome the structural rigidity of fixed dictionaries, Genetic Programming (GP) tools like Eureqa [ 38 ] and PySR [ 39 ] , alongside hybrid frameworks such as SGA-PDE [ 32 ] and EPDE [ 40 ] , explore free-form equations via stochastic mutation. However, these methods are often hindered by expression bloat and inefficient convergence within vast combinatorial search spaces. Neural Symbolic Regression (NSR) paradigms address these search inefficiencies by leveraging reinforcement learning and transformer architectures, as demonstrated by DSR [ 41 ] , MORL4PDEs [ 42 ] , and DISCOVER [ 33 ] . More recently, EqGPT [ 35 ] trained a specialized transformer on mathematical corpora to construct an automated generation-evaluation loop. Yet, these autoregressive models operate within purely syntactic spaces lacking explicit physical priors, and predominantly rely on discrete FDs for evaluation. This reliance drastically amplifies high-frequency noise—particularly in high-order or cross-derivatives—distorting the fitness landscape and degrading robustness. In contrast, LLM-PDESR resolves both limitations: it repurposes the LLM as an active reasoning agent bridging syntax and physics via Pareto-driven feedback, while employing a continuous, noise-robust evaluation environment."},{"type":"heading","level":1,"text":"3 The LLM-PDESR framework"},{"type":"para","text":"We consider a physical system evolving over a spatial domain Ω x \\Omega_{x} and temporal domain Ω t \\Omega_{t} , characterized by a state field 𝐮 ( x , t ) ∈ ℝ D \\mathbf{u}(x,t)\\in\\mathbb{R}^{D} (e.g., D = 1 D=1 for canonical PDEs like Burgers equation; D > 1 D>1 for coupled fluid dynamics). Given discrete, noisy spatiotemporal observations, our objective is to discover the underlying partial differential operator governing its temporal evolution:"},{"type":"para","text":"where 𝒩 \\mathcal{N} represents the unknown structural skeleton to be generated, and θ \\theta denotes the optimizable physical parameters."},{"type":"para","text":"To prevent noise amplification, the target temporal derivative ( ∂ 𝐮 / ∂ t \\partial\\mathbf{u}/\\partial t ) is pre-computed via a continuous cubic spline approximation along the time axis."},{"type":"para","text":"Traditionally, parameter optimization involves minimizing a pointwise mean squared error (MSE) loss between the observed and predicted temporal derivatives [ 43 , 44 ] . Evaluating this objective function, however, fundamentally requires approximating the requisite spatial derivatives using discrete FD methods. This dependency introduces a critical evaluation bottleneck: discrete FD inherently amplifies high-frequency observation noise ( 𝒪 ( σ / ( Δ x ) k ) \\mathcal{O}(\\sigma/(\\Delta x)^{k}) for k k -th order derivatives [ 45 ] ). This numerical instability distorts the fitness landscape, yielding deceptive evaluation scores that derail search algorithms and prevent convergence to the underlying physical laws."},{"type":"para","text":"To resolve this bottleneck, we propose LLM-PDESR. As illustrated in Figure 1 and formalized later in Algorithm 1 , this automated, closed-loop pipeline addresses PDE discovery through three core modules: LLM-Driven Hypothesis Generation, Robust Parameter Identification, and Pareto-Driven Evaluation."},{"type":"heading","level":2,"text":"3.1 LLM-driven hypothesis generation"},{"type":"para","text":"Framing the LLM as an automated hypothesis generator, our prompt emulates the empirical discovery pipeline by mapping physical observations (e.g., 1D heat conduction data) to candidate governing equations. To preclude the generation of syntactically or physically invalid structures, this search space is strictly bounded by predefined constraint conditions (e.g., enforcing the spatial derivative operator d_dx(u, order) ). The complete prompts and constraint specifications are detailed in Appendix A ."},{"type":"para","text":"Conditioned on this prompt, the LLM acts as a structural architect, proposing a population of n n candidate executable equations (represented as Python Abstract Syntax Trees). Crucially, the LLM generates only the symbolic skeletons, utilizing parameter placeholders (e.g., params[0] ) for all numeric coefficients. These n n candidate equations are subsequently forwarded to the numerical environment to evaluate their empirical fitness."},{"type":"heading","level":2,"text":"3.2 Robust parameter identification"},{"type":"para","text":"The parameter identification module bridges the gap between discrete symbolic generation and continuous observational data through a rigorous three-step numerical pipeline:"},{"type":"para","text":"1. Derivative Computation (Quintic Spline): To eliminate the aforementioned FD noise bottleneck, we reconstruct the noisy spatial field at each time step t t using quintic B-spline smoothing. This enables analytical and noise-robust spatial differentiation:"},{"type":"para","text":"where c j ( t ) c_{j}(t) are the spline coefficients and B j 5 ( x ) B_{j}^{5}(x) denotes the quintic basis functions."},{"type":"para","text":"2. Subdomain Weighted Residual Construction: To further fortify the evaluation against localized anomalies, we compute the integral residual ℛ i ( t ; θ ) \\mathcal{R}_{i}(t;\\theta) over K K overlapping local sub-domains Ω i \\Omega_{i} . By integrating against a compactly supported smooth test function ϕ i ( x ) \\phi_{i}(x) , this formulation inherently acts as a low-pass filter, significantly attenuating high-frequency noise prior to error calculation:"},{"type":"para","text":"Here, c i c_{i} and ω \\omega define the center and width of the local window Ω i \\Omega_{i} , respectively, and u t _ pred u_{t\\_\\text{pred}} is the time derivative predicted by the candidate equation parameterized by θ \\theta ."},{"type":"para","text":"3. Parameter Optimization via BFGS: The objective function ℒ ( θ ) \\mathcal{L}(\\theta) is explicitly formulated as the MSE of the integrated residuals. To minimize this non-convex loss, we deploy the quasi-Newton BFGS algorithm [ 46 , 47 ] , which iteratively updates the parameters using an approximate inverse Hessian matrix H k H_{k} :"},{"type":"para","text":"where M M and K K denote the number of temporal snapshots and spatial sub-domains, respectively; α k \\alpha_{k} is the step size, and H k H_{k} is the approximate inverse Hessian updated via BFGS."},{"type":"para","text":"Rigorous theoretical proofs establishing the error bounds of quintic splines over FD, the noise-attenuation properties of the SWR versus point-wise MSE, and the convergence advantages of BFGS in this context are detailed in Appendix B ."},{"type":"heading","level":2,"text":"3.3 Pareto-driven evaluation and automated selection"},{"type":"para","text":"Following parameter identification, the optimized candidates are assessed in the Evaluation module. In accordance with Occam’s razor, equation discovery is modeled as a multi-objective optimization problem. We map each candidate into a 2D objective space using two strictly defined metrics:"},{"type":"para","text":"Score: Defined as the negative MSE ( − MSE -\\text{MSE} ) of the optimized SWR."},{"type":"para","text":"Complexity: Quantified as the total number of nodes in the candidate’s Abstract Syntax Tree ( C ( ℱ ) = ∑ v ∈ AST c v C(\\mathcal{F})=\\sum_{v\\in\\text{AST}}c_{v} )."},{"type":"para","text":"To guide the LLM’s evolutionary search while maintaining context efficiency, we extract three elite solutions from the Pareto frontier 𝒫 \\mathcal{P} (highest score, lowest complexity, and balanced) and dynamically inject their serializations into the subsequent prompt. For the final parsimony-guided extraction, we employ a programmatic selection heuristic based on a logarithmic error tolerance. Specifically, we define a subset of highly accurate candidates 𝒫 ∗ = { ℱ ∈ 𝒫 ∣ MSE ( ℱ ) ≤ 10 × MSE min } \\mathcal{P}^{*}=\\{\\mathcal{F}\\in\\mathcal{P}\\mid\\text{MSE}(\\mathcal{F})\\leq 10\\times\\text{MSE}_{\\min}\\} , representing equations within one order of magnitude of the global minimum error. The algorithm ultimately outputs ℱ ∗ = arg min ℱ ∈ 𝒫 ∗ C ( ℱ ) \\mathcal{F}^{*}=\\arg\\min_{\\mathcal{F}\\in\\mathcal{P}^{*}}C(\\mathcal{F}) . This mechanism systematically prunes over-parameterized models while guaranteeing near-optimal predictive accuracy."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"heading","level":2,"text":"4.1 Experimental setup and baselines"},{"type":"para","text":"Datasets and Scenarios. Our evaluation encompasses three distinct regimes:"},{"type":"para","text":"(1) Benchmark PDEs: A library of 23 canonical 1D PDEs (e.g., linear, semilinear, Burgers, and high-order nonlinear systems; detailed in Appendix C ). To simulate realistic measurement imperfections, the clean spatial fields u u are corrupted with additive Gaussian noise: u ~ = u + γ σ u ϵ \\tilde{u}=u+\\gamma\\sigma_{u}\\epsilon , where σ u \\sigma_{u} is the standard deviation of u u , ϵ ∼ 𝒩 ( 0 , 1 ) \\epsilon\\sim\\mathcal{N}(0,1) , and γ ∈ { 0.05 , 0.10 , 0.20 } \\gamma\\in\\{0.05,0.10,0.20\\} denotes the noise level."},{"type":"para","text":"(2) Structurally Novel Equations: Five synthetically designed physical systems featuring highly nonlinear spatial terms and unconventional interactions. These are explicitly formulated to preclude dataset memorization and validate genuine symbolic discovery capabilities."},{"type":"para","text":"(3) Real-World Observations: Hourly macroscopic wind velocity fields from the ERA5 atmospheric reanalysis dataset [ 31 ] , utilized to extract the interpretable 1D dynamical surrogate (1D-CACE)."},{"type":"para","text":"Baselines and Metrics. We benchmark our framework against state-of-the-art methodologies spanning evolutionary algorithms, sparse regression, and neural symbolic generation: SGA-PDE [ 32 ] , DISCOVER [ 33 ] , WSINDy_PDE [ 34 ] , and EqGPT [ 35 ] . Performance is quantitatively assessed using two rigorous metrics: Structural Success Rate , strictly defined as the exact symbolic recovery of the true PDE skeleton; and Relative Coefficient Error , computed via the L 2 L_{2} norm ‖ θ ^ − θ ‖ 2 / ‖ θ ‖ 2 \\|\\hat{\\theta}-\\theta\\|_{2}/\\|\\theta\\|_{2} exclusively for successfully identified structures. Detailed experimental configurations and parameter settings for all evaluated methods are provided in Appendix D ."},{"type":"heading","level":2,"text":"4.2 Robustness on benchmark PDEs"},{"type":"para","text":"We first evaluate the Structural Success Rate (Figure 2 a). LLM-PDESR achieves a 100% success rate at 0 % 0\\% noise and maintains consistently high robustness as noise increases ( 5 % 5\\% – 20 % 20\\% ). Most baselines (e.g., SGA-PDE, DISCOVER, WSINDy_PDE) suffer severe degradation, while EqGPT retains only partial robustness via explicit data denoising. LLM-PDESR consistently outperforms all competitors, demonstrating the superior noise resilience of our evaluation environment."},{"type":"para","text":"Regarding parameter accuracy, Figure 2 b reports the Relative Coefficient Error exclusively for successful identifications. At 0 % 0\\% noise, our framework exhibits a low coefficient error. The nominally lower baseline errors in this regime stem from a statistical survivorship bias : these methods only converge on the simplest equations. Under noisy conditions, EqGPT exhibits the lowest error due to its external data preprocessing. In contrast, LLM-PDESR ingests raw, noisy observations directly to preserve high-fidelity physical dynamics. Despite omitting heuristic denoising, our framework maintains highly competitive error bounds, confirming that the SWR’s intrinsic low-pass filtering effectively isolates true parameters from noise."},{"type":"heading","level":2,"text":"4.3 Structurally novel equations"},{"type":"para","text":"Evaluating LLMs exclusively on canonical PDEs inherently risks testing training data memorization rather than genuine symbolic reasoning. To rigorously probe the generalization capabilities of our framework and explicitly mitigate the risk of training data contamination, we designed five structurally novel equations:"},{"type":"para","text":"Topography-Constrained Chemotaxis ( u t = 0.1 u x x − 0.5 ( u ( 1 − u ) cos ( x ) ) x + u ( 1 − u ) u_{t}=0.1u_{xx}-0.5(u(1-u)\\cos(x))_{x}+u(1-u) ): Models biological foraging on undulating terrain with logistic crowding. The nested derivative structure strictly prevents simple additive token generation."},{"type":"para","text":"Morphogenesis with Michaelis-Menten Kinetics ( u t = 0.1 u x x − 1.0 u / ( 0.5 + u ) + exp ( − x 2 ) u_{t}=0.1u_{xx}-1.0u/(0.5+u)+\\exp(-x^{2}) ): Features a non-polynomial rational fraction and a spatial Gaussian source, a structure typically intractable for traditional polynomial-biased sparse regression."},{"type":"para","text":"Forced Quintic Swift-Hohenberg ( u t = 1.5 exp ( − 0.1 x 2 ) u − 2 u x x − u x x x x − u 3 + u 5 u_{t}=1.5\\exp(-0.1x^{2})u-2u_{xx}-u_{xxxx}-u^{3}+u^{5} ): A synthetic, out-of-distribution model inspired by optical microcavity pattern formation. It features a 5th-order nonlinearity and 4th-order spatial derivatives, explicitly designed to break standard physical symmetries."},{"type":"para","text":"Traffic Flow with Spatial Bottleneck ( u t = − ( u ( 1 − u ) ( 1 − 0.5 tanh ( x ) ) ) x + 0.05 ( u x / u ) x u_{t}=-(u(1-u)(1-0.5\\tanh(x)))_{x}+0.05(u_{x}/u)_{x} ): A macroscopic traffic model incorporating a physical road narrowing ( tanh \\tanh ) and driver anticipation gradients."},{"type":"para","text":"Cross-Chemotactic Predator-Prey System ( u t = 0.05 u x x + u ( 1 − u − 0.5 v ) u_{t}=0.05u_{xx}+u(1-u-0.5v) and v t = 0.01 v x x − 0.2 ( v u x ) x + 0.8 v ( u − 1 ) v_{t}=0.01v_{xx}-0.2(vu_{x})_{x}+0.8v(u-1) ): A coupled multivariate system where predators ( v v ) actively hunt by moving along the prey’s ( u u ) density gradient."},{"type":"para","text":"As Table 1 reports, the evaluated baselines universally fail to recover these structurally novel equations. Specifically, WSINDy_PDE succumbs to combinatorial explosion and extreme overfitting (e.g., yielding an AST complexity of 203 with high MSE for Predator-Prey). Meanwhile, SGA-PDE, DISCOVER, and EqGPT suffer from structural collapse and fundamentally fail to discover the correct forms. Furthermore, as most baselines lack the architectural support for multivariate systems, they are inherently inapplicable to the coupled Predator-Prey dynamics (denoted as N/A)."},{"type":"para","text":"Conversely, LLM-PDESR successfully navigates this vast hypothesis space. By accommodating inherent structural complexities, our framework extracts the exact mathematical skeletons ( ★ \\color[rgb]{0,0.88,0}\\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\\bigstar} ) for both Morphogenesis and the coupled Predator-Prey dynamics without generating spurious terms. Notably, even when exact symbolic recovery is unachieved for highly nonlinear systems like the Traffic Flow Bottleneck ( × \\color[rgb]{1,0,0}\\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\\boldsymbol{\\times}} ), LLM-PDESR consistently secures the lowest MSE across all five systems, demonstrating superior capability in approximating underlying physical dynamics."},{"type":"para","text":"Remark on Partial Recovery: For Topography Chemotaxis and Forced Swift-Hohenberg, LLM-PDESR achieves partial recovery ( ≈ \\color[rgb]{1,.5,0}\\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{\\mathbf{\\approx}} ), strictly defined as correctly identifying ≥ 50 % \\geq 50\\% of the exact governing terms. Rather than producing unphysical terms when faced with complex interactions, our framework exhibits exceptional structural robustness via Pareto-driven optimization. It reliably isolates core physical mechanisms and prevents mathematically invalid bloat, even when the complete skeleton proves exceptionally intricate. A comprehensive, term-by-term analysis and further detailed discussions regarding the recovered structures for all five novel systems are provided in Appendix E ."},{"type":"heading","level":2,"text":"4.4 Real-world discovery: coupled atmospheric dynamics"},{"type":"para","text":"Evaluating symbolic regression on authentic reanalysis data like ERA5 is notoriously challenging, as the algorithm must contend with severe dimensionality truncation (unresolved 3D thermodynamic variables) and high-frequency data assimilation noise. To demonstrate our framework’s capability in this extreme regime, we deployed LLM-PDESR on real-world ERA5 atmospheric data (North Pacific domain; Jan 2023) [ 48 , 49 , 50 ] , autonomously extracting an interpretable 1D coupled atmospheric circulation surrogate (1D-CACE):"},{"type":"para","text":"where x x is the spatial coordinate and θ 1 … θ 14 \\theta_{1}\\dots\\theta_{14} are optimizable physical parameters. Although applicable to multivariate systems, WSINDy_PDE yields a severely overfitted, unphysical expression here, omitted for brevity."},{"type":"para","text":"Unlike standard 1D models (e.g., Burgers equation) limited to self-advection, 1D-CACE explicitly captures cross-wind momentum transfer ( v u x , u v x vu_{x},uv_{x} ) and linear coupling ( v x , u x v_{x},u_{x} ). From a Reduced Order Modeling (ROM) perspective, projecting complex 3D atmospheric flows onto a 1D manifold inevitably loses external physical forcings. Remarkably, LLM-PDESR autonomously compensates for this by discovering spatially dependent terms ( x v xv and x u xu ). Rather than just overfitting localized noise, these act as vital mathematical closure terms, implicitly proxying unresolved macroscopic background gradients (analogous to the well-known β \\beta -plane approximation of the Coriolis force). Detailed physical derivations of these compensation terms are provided in Appendix F ."},{"type":"para","text":"This dynamical consistency ensures strong out-of-distribution (OOD) generalization. By freezing the structural skeleton of Eq. ( 5 ) and re-optimizing only the numeric coefficients, the equation transfers effectively to distinct spatiotemporal domains (Table 2 )."},{"type":"para","text":"Notably, while temporal drift to February causes a performance drop ( R 2 = 58.64 % R^{2}=58.64\\% ) due to transitional late-winter dynamics, the geographically extreme Hemispheric OOD setting (South Indian Ocean, July) rebounds significantly to 72.54 % 72.54\\% . Because July represents the austral winter, this recovery empirically confirms that 1D-CACE successfully captures the macroscopic dynamics of winter atmospheric circulation , rather than memorizing geographical artifacts. Figure 3 visualizes this high-fidelity hemispheric reconstruction, verifying that LLM-PDESR effectively distills interpretable governing dynamics from chaotic real-world data."},{"type":"heading","level":2,"text":"4.5 Ablation studies"},{"type":"para","text":"To systematically evaluate the individual contribution of each core component, we conduct an ablation study using noise-free data. We compare the full LLM-PDESR framework against three ablated variants: (1) w/o Spline (replacing quintic splines with standard FDs), (2) w/o SWR (replacing local integral residuals with pointwise MSE), and (3) w/o Pareto (removing the multi-objective selection, retaining only MSE as the fitness metric)."},{"type":"para","text":"Figure 4 (a) illustrates the significant impact of removing any single module on the Structural Success Rate . While the full framework achieves a 100 % 100\\% recovery on the benchmark PDEs, reverting to discrete FDs (w/o Spline) or pointwise evaluation (w/o SWR) significantly degrades success rates to 69.6 % 69.6\\% and 60.9 % 60.9\\% , respectively."},{"type":"para","text":"Figure 4 (b) elucidates the underlying optimization dynamics through the MSE distributions. Without splines, numerical differentiation errors rapidly accumulate. Similarly, without the SWR’s inherent smoothing, the fitness landscape is severely distorted by high-frequency numerical artifacts and truncation errors (even in the absence of observation noise), resulting in high MSE variances and optimization failures. Most crucially, the w/o Pareto variant exhibits a deceptive phenomenon: while its median MSE appears competitively low, its structural recovery remains poor. This corroborates our theoretical analysis: without a strict complexity penalty, the LLM inevitably falls into an overfitting trap. It produces bloated, mathematically spurious equations that artificially minimize the residual error, completely failing to execute genuine symbolic discovery."},{"type":"heading","level":1,"text":"5 Conclusion and future work"},{"type":"para","text":"In this work, we introduce LLM-PDESR, a novel paradigm bridging the structural hypothesis generation of LLMs with the numerical rigor required for robust PDE discovery. By integrating 𝒞 4 \\mathcal{C}^{4} continuous quintic splines, SWR evaluations, and a Pareto-driven automated selection loop, our framework effectively mitigates the fitness landscape distortion and the generation of spurious equations that plague traditional methods. Empirically, it achieves state-of-the-art performance across canonical benchmarks and structurally novel systems (explicitly designed to mitigate the risk of training data contamination), while demonstrating practical utility by extracting an interpretable 1D coupled atmospheric circulation surrogate (1D-CACE) directly from noisy ERA5 observations. While our current formulation successfully distills macroscopic dynamics from noisy real-world environments, future work will focus on extending this framework to high-dimensional (2D/3D) physical systems on unstructured grids and optimizing the computational overhead of iterative LLM inference, thereby further advancing the frontier of automated scientific discovery."},{"type":"heading","level":1,"text":"6 Acknowledgments"},{"type":"para","text":"This work was supported by the National Science and Technology Major Project under Grant 2021ZD0112500. We thank the Computing Center of Jilin Province for technical support."},{"type":"para","text":"This section details the operating mechanism of LLM-PDESR. The process is decoupled into two core components: the structured prompt template used to guide the LLM in generating candidate equations, and the rigorous numerical pipeline used to evaluate and optimize the parameters of these candidates."},{"type":"heading","level":2,"text":"A.1 Structured LLM prompt for equation generation"},{"type":"para","text":"To concretize our methodology, we illustrate the prompt structure using a representative and highly challenging governing equation from our experiments: Topography-Constrained Chemotaxis . The target PDE is given by:"},{"type":"para","text":"This equation models biological foraging on an undulating terrain coupled with logistic crowding. Crucially, its nested derivative structure—specifically the advection term ( u ( 1 − u ) cos ( x ) ) x (u(1-u)\\cos(x))_{x} —inherently prevents simple additive token generation, thereby severely testing the LLM’s capacity for deep physical reasoning and symbolic manipulation."},{"type":"para","text":"To prevent the generation of physically meaningless or computationally intractable expressions, we constrain the LLM’s action space through a highly structured prompt. The prompt enforces strict mathematical vocabularies and introduces the Pareto evolution mechanism, which organically penalizes bloated equations. The exact prompt template tailored for this specific biological task is provided in Prompt A.1 ."},{"type":"para","text":"To rigorously evaluate the generalizability of our framework and ensure reproducibility, we apply this standardized prompt template across all structurally novel systems. Crucially, the Strict Constraints and Pareto Evolution Mechanism remain completely identical across all trials to prevent task-specific prompt engineering bias. The LLM’s initial context is altered solely by modifying the physical background provided in the Task description."},{"type":"para","text":"The specific task contexts supplied for the remaining five novel systems are detailed below:"},{"type":"para","text":"Morphogenesis with Michaelis-Menten Kinetics: \"Find the mathematical function skeleton that represents the time derivative of the field variable ( u t u_{t} ), given data on spatial coordinates ( x x ) and the field variable ( u u ) in a morphogen diffusion from a localized Gaussian source with Michaelis-Menten absorption.\""},{"type":"para","text":"Forced Quintic Swift-Hohenberg: \"Find the mathematical function skeleton that represents the time derivative of the field variable ( u t u_{t} ), given data on spatial coordinates ( x x ) and the field variable ( u u ) in a subcritical optical medium driven by a Gaussian laser and stabilized by hyper-viscosity.\""},{"type":"para","text":"Traffic Flow with Spatial Bottleneck: \"Find the mathematical function skeleton that represents the time derivative of the field variable ( u t u_{t} ), given data on spatial coordinates ( x x ) and the field variable ( u u ) in a traffic flow with a spatial bottleneck and anticipation-driven relative gradient diffusion.\""},{"type":"para","text":"Cross-Chemotactic Predator-Prey System (Multivariate): \"Find the mathematical function skeletons that represent the time derivatives of the field variables ( u t u_{t} and v t v_{t} ), given data on spatial coordinates ( x x ) and the field variables ( u u and v v ) in a coupled system: For u t u_{t} : diffusing prey population suppressed by local predator encounters. For v t v_{t} : diffusing predator population migrating toward prey density gradients.\""},{"type":"para","text":"Real-World Application: 1D Coupled Atmospheric Circulation (1D-CACE): \"Find the mathematical function skeletons that represent the time derivatives of the field variables ( u t u_{t} and v t v_{t} ), given data on spatial coordinates ( x x ) and the field variables ( u u and v v ) from real-world Earth atmospheric reanalysis: For u t u_{t} (zonal wind): longitudinal velocity of a viscous advective atmospheric flow under planetary rotation. For v t v_{t} (meridional wind): latitudinal velocity of a viscous advective atmospheric flow under planetary rotation.\""},{"type":"heading","level":2,"text":"A.2 Numerical evaluation and parameter optimization algorithm"},{"type":"para","text":"Once a candidate equation skeleton 𝒩 ( u ; θ ) \\mathcal{N}(u;\\theta) is proposed by the LLM, its fitness must be evaluated. To bridge the gap between discrete, noisy observational data and continuous spatial derivatives, we employ a pipeline combining quintic B-spline smoothing, SWR integration, and BFGS optimization. This robust evaluation process is summarized in Algorithm 2 ."},{"type":"para","text":"In this appendix, we establish the theoretical foundations of the proposed parameter identification module. We provide rigorous error bounds for the quintic spline formulation compared to classical FDs, prove the spectral noise-attenuation properties of the SWR, and demonstrate the convergence advantages of the BFGS algorithm within our specifically smoothed loss landscape."},{"type":"heading","level":2,"text":"B.1 Error bounds and noise robustness of regularized quintic B-spline smoothing"},{"type":"para","text":"Let the underlying true spatial field be u ∈ 𝒞 6 ( Ω ) u\\in\\mathcal{C}^{6}(\\Omega) . The observed data is assumed to be corrupted by additive noise, denoted as u ~ j = u ( x j ) + ϵ j \\tilde{u}_{j}=u(x_{j})+\\epsilon_{j} , where ϵ j ∼ 𝒩 ( 0 , σ 2 ) \\epsilon_{j}\\sim\\mathcal{N}(0,\\sigma^{2}) are independent and identically distributed (i.i.d.) random variables, and h h is the uniform spatial discretization step."},{"type":"para","text":"Let S 5 ( x ) S_{5}(x) be the quintic B-spline constructed from the noisy observations u ~ \\tilde{u} via regularized smoothing. The total expected error of the n n -th derivative approximation ( n ≤ 4 ) (n\\leq 4) in the Chebyshev norm is bounded by:"},{"type":"para","text":"where C 1 C_{1} and C 2 C_{2} are constants independent of h h and σ \\sigma . Furthermore, the expected squared L 2 L^{2} norm of the noise-induced derivative variance scales as 𝒪 ( σ 2 h − 2 n ) \\mathcal{O}(\\sigma^{2}h^{-2n}) ."},{"type":"para","text":"The total error is bounded by the triangle inequality, separating the approximation error (bias) and the noise propagation error (variance):"},{"type":"para","text":"For the bias term, classical Schoenberg approximation theory for a spline of degree d = 5 d=5 guarantees the Peano kernel bound:"},{"type":"para","text":"For the variance term, S 5 , noise ( x ) = ∑ j ϵ j B j 5 ( x ) S_{5,\\text{noise}}(x)=\\sum_{j}\\epsilon_{j}B_{j}^{5}(x) . Because the quintic B-spline basis functions B j 5 ( x ) B_{j}^{5}(x) form a partition of unity, are non-negative, and strictly possess compact support over exactly 6 consecutive intervals, the n n -th derivative obeys the Markov-type inequality ‖ d n d x n B j 5 ‖ ∞ ≤ C h − n ||\\frac{d^{n}}{dx^{n}}B_{j}^{5}||_{\\infty}\\leq Ch^{-n} ."},{"type":"para","text":"Crucially, the uniform norm evaluates the supremum over the entire spatial domain, which effectively consists of N ∝ L / h N\\propto L/h independent local regions. By extreme value theory, the expected maximum of these Gaussian random variables introduces a logarithmic growth factor. Therefore, taking the expectation of the maximum deviation yields:"},{"type":"para","text":"To analyze the global variance accumulation in the L 2 L^{2} sense, we integrate the squared noise function over the entire domain of length L L . Because 𝔼 [ ϵ j ϵ k ] = σ 2 δ j k \\mathbb{E}[\\epsilon_{j}\\epsilon_{k}]=\\sigma^{2}\\delta_{jk} , the cross-terms vanish in expectation:"},{"type":"para","text":"For a single localized basis function, the integral of its squared n n -th derivative scales as 𝒪 ( h − 2 n + 1 ) \\mathcal{O}(h^{-2n+1}) . Since the domain is spanned by N ∝ L / h N\\propto L/h basis functions, the total expected squared L 2 L^{2} error evaluates to:"},{"type":"para","text":"This mathematically confirms that the root-mean-square asymptotic scaling with respect to h h is 𝒪 ( h − n ) \\mathcal{O}(h^{-n}) , which identically matches that of discrete finite differences."},{"type":"para","text":"However, despite sharing the identical asymptotic rate, the regularized spline formulation provides two indispensable practical advantages over standard FDs. First, the smoothing penalty strictly bounds the error coefficient (the constant factor), substantially reducing the absolute variance compared to point-wise differences. Second, and most crucially, whereas FDs generate discontinuous grid-scale oscillations that fatally distort subsequent gradient-based optimization, S 5 ( x ) ∈ 𝒞 4 ( Ω ) S_{5}(x)\\in\\mathcal{C}^{4}(\\Omega) produces an intrinsically smooth, globally differentiable vector field, which is a fundamental prerequisite for evaluating the Subdomain Weighted Residuals (SWR) accurately. ∎"},{"type":"heading","level":2,"text":"B.2 Spectral noise attenuation via the SWR"},{"type":"para","text":"We now prove that the SWR method inherently acts as an optimal low-pass filter, dramatically reducing the variance of the loss gradient compared to point-wise MSE."},{"type":"para","text":"Let the observed temporal derivative be u ~ t ( x ) = u t , true ( x ) + η ( x ) \\tilde{u}_{t}(x)=u_{t,\\text{true}}(x)+\\eta(x) . To maintain dimensional consistency between discrete grid observations (with variance σ 2 \\sigma^{2} and grid spacing h h ) and continuous integration, we define η ( x ) \\eta(x) as a zero-mean spatial white noise with an equivalent continuous autocovariance 𝔼 [ η ( x ) η ( y ) ] = σ 2 h δ ( x − y ) \\mathbb{E}[\\eta(x)\\eta(y)]=\\sigma^{2}h\\delta(x-y) ."},{"type":"para","text":"The variance of the normalized localized integral residual ℛ ¯ i ( θ ) \\bar{\\mathcal{R}}_{i}(\\theta) evaluated with the test function ϕ i ( x ) = sin 4 ( π x − c i + ω / 2 ω ) \\phi_{i}(x)=\\sin^{4}\\left(\\pi\\frac{x-c_{i}+\\omega/2}{\\omega}\\right) is attenuated by a factor inversely proportional to the number of data points within the local window, compared to the pointwise residual."},{"type":"para","text":"In a purely point-wise collocation framework, the expected MSE loss incorporates the full noise variance directly: 𝔼 [ | u t , true − u t , pred + η | 2 ] = MSE true + σ 2 \\mathbb{E}[|u_{t,\\text{true}}-u_{t,\\text{pred}}+\\eta|^{2}]=\\text{MSE}_{\\text{true}}+\\sigma^{2} ."},{"type":"para","text":"In our locally integrated strong form, we rigorously define the normalized residual to evaluate the true localized tendency. The noise contribution to the normalized residual at domain Ω i \\Omega_{i} is explicitly:"},{"type":"para","text":"Given 𝔼 [ η ( x ) ] = 0 \\mathbb{E}[\\eta(x)]=0 , the expected value 𝔼 [ ϵ ℛ ¯ ] = 0 \\mathbb{E}[\\epsilon_{\\bar{\\mathcal{R}}}]=0 . The variance of this noise contribution is computed as:"},{"type":"para","text":"Applying the equivalent continuous white noise property 𝔼 [ η ( x ) η ( y ) ] = σ 2 h δ ( x − y ) \\mathbb{E}[\\eta(x)\\eta(y)]=\\sigma^{2}h\\delta(x-y) :"},{"type":"para","text":"Substituting ϕ i ( x ) = sin 4 ( π x − c i + ω / 2 ω ) \\phi_{i}(x)=\\sin^{4}\\left(\\pi\\frac{x-c_{i}+\\omega/2}{\\omega}\\right) and applying the change of variables z = x − c i + ω / 2 ω z=\\frac{x-c_{i}+\\omega/2}{\\omega} , we obtain:"},{"type":"para","text":"Using the standard reduction formula for even powers of sine (Wallis’ integrals), we evaluate the definite integral exactly:"},{"type":"para","text":"Thus, Var ( ϵ ℛ ¯ ) = 35 128 ( h ω ) σ 2 \\text{Var}(\\epsilon_{\\bar{\\mathcal{R}}})=\\frac{35}{128}\\left(\\frac{h}{\\omega}\\right)\\sigma^{2} . Let N ω = ω / h N_{\\omega}=\\omega/h represent the number of discrete grid points inside the window. The effective variance is strictly reduced to:"},{"type":"para","text":"This establishes a rigorous theoretical guarantee: the variance is substantially suppressed proportional to the window density, effectively resolving the severe noise amplification endemic to standard collocation methods. ∎"},{"type":"heading","level":4,"text":"Boundary Handling and Support Preservation."},{"type":"para","text":"As correctly noted in the framework’s implementation, a naive truncation of the test functions at x min x_{\\min} or x max x_{\\max} would induce a step discontinuity, degrading the spectral noise roll-off. To preserve the high-order smoothness required for optimal noise attenuation, LLM-PDESR ensures that the support of every test function is strictly contained within the observational domain, i.e., supp ( ϕ i ) ⊆ [ x min , x max ] \\text{supp}(\\phi_{i})\\subseteq[x_{\\min},x_{\\max}] . Specifically, the boundary-adjacent window centers c 1 c_{1} and c K c_{K} are shifted inward such that the function reaches zero exactly at x min x_{\\min} and x max x_{\\max} . This configuration guarantees that ϕ i ∈ 𝒞 3 ( [ x min , x max ] ) \\phi_{i}\\in\\mathcal{C}^{3}([x_{\\min},x_{\\max}]) for all i i , ensuring the 𝒪 ( | ξ | − 4 ) \\mathcal{O}(|\\xi|^{-4}) spectral decay holds uniformly."},{"type":"heading","level":2,"text":"B.3 Superlinear Convergence of the BFGS Optimization in Smoothed Landscapes"},{"type":"para","text":"Finally, we establish that the combination of our noise-attenuating SWR and the BFGS quasi-Newton method leads to guaranteed superlinear convergence, circumventing the ill-conditioning typical of inverse PDE problems."},{"type":"para","text":"Let the SWR objective ℒ ( θ ) = 1 M K ∑ j = 1 M ∑ i = 1 K | ℛ ¯ i , j ( θ ) | 2 \\mathcal{L}(\\theta)=\\frac{1}{MK}\\sum_{j=1}^{M}\\sum_{i=1}^{K}|\\bar{\\mathcal{R}}_{i,j}(\\theta)|^{2} be twice continuously differentiable, where M M and K K denote the number of temporal snapshots and spatial sub-domains, respectively. Under the assumptions of PDE identifiability and sufficiently small integrated observation noise (the small residual condition), ℒ ( θ ) \\mathcal{L}(\\theta) satisfies local strong convexity in a neighborhood 𝒩 \\mathcal{N} of the optimal parameters θ ∗ \\theta^{*} . The BFGS update, generating a sequence of approximate inverse Hessians { H k } \\{H_{k}\\} , guarantees superlinear convergence to θ ∗ \\theta^{*} ."},{"type":"para","text":"As established in Theorem 2 , the SWR formulation acts as a low-pass filter by integrating residuals against 𝒞 3 \\mathcal{C}^{3} test functions. By employing regularized quintic spline smoothing instead of exact interpolation, the Jacobian of the residuals J ( θ ) J(\\theta) remains numerically stable. The Hessian of the scalar objective ℒ ( θ ) \\mathcal{L}(\\theta) is approximated via the Gauss-Newton formulation:"},{"type":"para","text":"In the presence of observation data, the true parameters θ ∗ \\theta^{*} do not yield a strictly zero residual; rather, ℛ ¯ i , j ( θ ∗ ) \\bar{\\mathcal{R}}_{i,j}(\\theta^{*}) equals the integrated observation noise. Consequently, the second-order summation acts as an indefinite random perturbation matrix proportional to the noise level. To ensure local strong convexity, the optimization must satisfy a small residual assumption , where the noise is sufficiently small such that the positive-definite first-order term ( J ( θ ) T J ( θ ) J(\\theta)^{T}J(\\theta) ) strictly dominates the second-order term."},{"type":"para","text":"Crucially, as proven in Theorem 2 , our SWR formulation inherently shrinks the variance of this residual noise by a factor of 𝒪 ( 1 / N ω ) \\mathcal{O}(1/N_{\\omega}) . By explicitly attenuating the residual magnitude at θ ∗ \\theta^{*} , the SWR framework mathematically ensures that the small residual condition holds over a much broader range of physical noise levels compared to pointwise formulations. Thus, the positive-definite Gram matrix robustly dominates:"},{"type":"para","text":"Since the test functions and spline smoothing mollify the ruggedness caused by data noise, ∇ 2 ℒ ( θ ) \\nabla^{2}\\mathcal{L}(\\theta) is locally strictly positive definite, fulfilling the strong convexity requirement:"},{"type":"para","text":"where 0 < m ≤ M 0<m\\leq M ."},{"type":"para","text":"The BFGS algorithm iteratively updates the approximate inverse Hessian H k H_{k} via the Sherman-Morrison-Woodbury formula:"},{"type":"para","text":"where s k = θ k + 1 − θ k s_{k}=\\theta_{k+1}-\\theta_{k} and y k = ∇ ℒ ( θ k + 1 ) − ∇ ℒ ( θ k ) y_{k}=\\nabla\\mathcal{L}(\\theta_{k+1})-\\nabla\\mathcal{L}(\\theta_{k}) . Let B k = H k − 1 B_{k}=H_{k}^{-1} denote the corresponding approximate Hessian. Under the established conditions of strong convexity and smoothness, the sequence { B k } \\{B_{k}\\} satisfies the Dennis-Moré condition:"},{"type":"para","text":"This condition is necessary and sufficient for the sequence of parameters { θ k } \\{\\theta_{k}\\} to converge to θ ∗ \\theta^{*} at a superlinear rate :"},{"type":"para","text":"By dynamically approximating the local curvature, BFGS implicitly resolves the severe ill-conditioning between parameters representing different spatial derivative orders (e.g., advection vs. hyper-diffusion terms), circumventing the 𝒪 ( d 3 ) \\mathcal{O}(d^{3}) computational bottleneck per iteration required by exact Newton methods. ∎"},{"type":"para","text":"Our evaluation relies on a benchmark suite of 23 diverse 1D PDEs, which are grouped into five mathematical categories: Linear Convection-Diffusion, Semilinear Reaction-Diffusion, Quasilinear Diffusion, Viscous Burgers, and Higher-Order Nonlinear Evolution Equations (see Table 3 for exact formulations and domains). To guarantee the fidelity of the reference data, we simulated these systems using high-order spectral methods paired with precise time integration, thereby effectively suppressing numerical artifacts. For each specific PDE, the spatial resolution ( N x N_{x} ) and temporal step size ( N t N_{t} ) were individually optimized to capture its unique dynamic features while strictly maintaining numerical stability. Furthermore, the selected domains, bounded by [ x min , x max ] [x_{\\min},x_{\\max}] and [ t min , t max ] [t_{\\min},t_{\\max}] , encompass a wide variety of boundary conditions and evolutionary timescales, offering a rigorous testbed for model generalizability."},{"type":"para","text":"To ensure strict reproducibility, we detail the hardware environment and hyperparameter configurations for all evaluated methodologies. All experiments were conducted on a Linux server equipped with a 48-core CPU, 128 GB RAM, and 4 × \\times NVIDIA GeForce RTX 3090 GPUs."},{"type":"heading","level":4,"text":"SGA-PDE."},{"type":"para","text":"The symbolic genetic algorithm was configured with a population size of 100 and executed for 500 generations. Tree structure constraints were set to a maximum depth of 4 and a maximum width of 5. The evolutionary probabilities were set to P v a r = 0.5 P_{var}=0.5 , P m u t e = 0.3 P_{mute}=0.3 , and P c r o = 0.5 P_{cro}=0.5 ."},{"type":"heading","level":4,"text":"DISCOVER."},{"type":"para","text":"The RNN controller was trained on 200,000 samples with a batch size of 1,000, a learning rate of 0.0025, and an entropy weight of 0.03. The function library included standard arithmetic operators and up to fourth-order derivatives. To constrain the search space, we applied structural priors (e.g., inverse, trigonometric, and differentiation descendant rules), alongside a sequence length constraint bounded between 3 and 256."},{"type":"heading","level":4,"text":"WSINDy_PDE."},{"type":"para","text":"We utilized piecewise polynomial test functions with query point spacings s x = 4 s_{x}=4 and s t = 4 s_{t}=4 . The candidate library comprised polynomials up to degree 3, spatial derivatives up to 4th order, and trigonometric terms. Sparse regression was executed using Sequential Threshold Ridge regression (STRidge), searching over 100 logarithmically spaced λ \\lambda values in [ 10 − 4 , 1 ] [10^{-4},1] ."},{"type":"heading","level":4,"text":"EqGPT."},{"type":"para","text":"Following its original proof-of-concept setup, target structures were masked from the dataset during training. The comprehensive candidate library contained 43 terms (up to 3rd-order polynomials and 4th-order derivatives). The Generational Genetic Algorithm (GGA) module utilized a population of 400, a maximum of 200 generations, a mutation rate of 0.3, and an L 0 L_{0} penalty strength of 3 × 10 − 4 3\\times 10^{-4} ."},{"type":"heading","level":4,"text":"LLM-PDESR (Ours)."},{"type":"para","text":"Our framework utilized DeepSeek-V3 ( deepseek-chat ) as the generative backbone, accessed via the official DeepSeek API. The optimization spanned 1,000 iterations. In each iteration, the LLM generated 4 candidate hypotheses. For in-context evolution, the top 3 equation templates from the Pareto frontier were fed back into the prompt. The sampling temperature was initialized at 0.1 and periodically adjusted every 30,000 tokens to balance exploration and exploitation. To evaluate hypotheses via the SWR formulation, we employed quintic splines for up to 4th-order spatial derivatives and evaluated the integral loss across K = 10 K=10 local domains using a sin 4 \\sin^{4} window function. Equation coefficients were optimized via BFGS with a strict 30-second timeout per evaluation to prevent infinite execution loops in generated code."},{"type":"para","text":"Evaluating LLMs exclusively on canonical PDEs (e.g., Burgers, KdV, Kuramoto-Sivashinsky) inherently risks testing “memorization” rather than genuine symbolic “discovery.” Because canonical equations are heavily represented in pre-training corpora, high performance on these tasks often reflects a model’s ability to interpolate within its training distribution rather than extrapolate to novel physical laws."},{"type":"para","text":"To rigorously probe the generalization capabilities of our framework and explicitly mitigate the risk of data contamination, we designed a benchmark suite of five structurally novel, out-of-distribution physical equations. These equations do not exist in standard textbooks in these exact forms, yet they are synthesized from fundamental physical and biological phenomena. Below, we provide an in-depth physical motivation for each system, compare the equations discovered by our LLM-PDESR framework against state-of-the-art baselines (SGA-PDE, DISCOVER, WSINDy_PDE, and EqGPT), and theoretically analyze the resulting spatiotemporal dynamics."},{"type":"para","text":"Note: The equations discovered by WSINDy_PDE are omitted from the detailed text below due to extreme algorithmic bloating (generating excessively long, uninterpretable polynomial chains comprising dozens of terms), though the method consistently failed to recover the true underlying physical structures."},{"type":"para","text":"Before detailing the mathematical formulations of each novel system, we first address a fundamental methodological question regarding baseline evaluation."},{"type":"heading","level":2,"text":"E.1 Information asymmetry and baseline evaluation constraints"},{"type":"para","text":"A natural question arises regarding the evaluation of these structurally novel equations: could deterministic baselines like WSINDy [ 34 ] achieve comparable performance if their basis libraries were manually augmented to reflect the semantic physical hints provided to LLM-PDESR (e.g., explicitly adding Gaussians or rational fractions)?"},{"type":"para","text":"We elected not to augment the deterministic baselines with these specific terms, as doing so mathematically violates the foundational assumptions of sparse dictionary regression and inevitably leads to algorithmic failure:"},{"type":"para","text":"Incompatibility with Non-linear Parameterization: Incorporating a semantic feature like a “localized Gaussian source” necessitates a non-linear parameterized structure, such as θ 1 exp ( − ( x − θ 2 ) 2 / θ 3 ) \\theta_{1}\\exp(-(x-\\theta_{2})^{2}/\\theta_{3}) . Dictionary-based methods construct a static, linear-in-the-parameters feature matrix. They cannot dynamically optimize internal parameters (such as the mean θ 2 \\theta_{2} or variance θ 3 \\theta_{3} ) during the sparse regression phase without resorting to an intractable pre-computed grid of candidate columns."},{"type":"para","text":"Multicollinearity and Ill-conditioning: Naively expanding the dictionary with complex rational terms (e.g., u / ( 1 + u ) u/(1+u) to represent logistic crowding) and their cross-products triggers severe multicollinearity. For instance, candidate columns like u u and u / ( 1 + u ) u/(1+u) become numerically indistinguishable under moderate observational noise. This causes thresholding algorithms (e.g., STRidge or LASSO) to suffer from severe ill-conditioning, resulting in the selection of spurious terms and catastrophic failure."},{"type":"para","text":"Therefore, this “information asymmetry” actually highlights a fundamental architectural advantage of our framework. LLM-PDESR explicitly decouples structural hypothesis generation from non-linear parameter identification, possessing the unique capacity to ingest abstract semantic priors and instantiate them as optimizable non-linear structures."},{"type":"heading","level":2,"text":"E.2 Topography-constrained chemotaxis"},{"type":"para","text":"Ground Truth Equation:"},{"type":"para","text":"Physical Motivation: This equation models the population dynamics of a biological species foraging on an undulating terrain. The reaction term u ( 1 − u ) u(1-u) represents logistic crowding, enforcing a strict environmental carrying capacity. The advection term, featuring a nested derivative structure ( u ( 1 − u ) cos ( x ) ) x (u(1-u)\\cos(x))_{x} , models topography-driven drift where the periodic terrain cos ( x ) \\cos(x) interacts nonlinearly with the population density. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ 0 , 2 π ] × [ 0 , 10 ] (x,t)\\in[0,2\\pi]\\times[0,10] with a resolution of 100 × 1000 100\\times 1000 . This nested structure strictly prevents simple additive token generation, challenging the LLM’s capacity for deep spatial reasoning and chain-rule expansion."},{"type":"para","text":"Baseline Discoveries:"},{"type":"para","text":"SGA-PDE: u t = 0.1584 u x + 0.9636 ( u − u 2 ) u_{t}=0.1584u_{x}+0.9636(u-u^{2})"},{"type":"para","text":"DISCOVER: u t = 0.1584 u x − 0.9710 sin ( u 2 − u ) u_{t}=0.1584u_{x}-0.9710\\sin(u^{2}-u)"},{"type":"para","text":"EqGPT: u t = 0.0082 u 2 − 0.9691 u x x + 0.9977 u u_{t}=0.0082u^{2}-0.9691u_{xx}+0.9977u"},{"type":"para","text":"All baselines completely failed to identify the explicit spatial dependence cos ( x ) \\cos(x) . They either collapsed the complex topography-constrained advection into simple linear convection (SGA-PDE) or hallucinated unphysical trigonometric reaction terms (DISCOVER)."},{"type":"para","text":"LLM-PDESR Discovery (Effective Macroscopic Approximation):"},{"type":"para","text":"Analysis: While our algorithm did not recover the exact explicit spatial forcing cos ( x ) \\cos(x) , it successfully distilled an effective macroscopic approximation of the underlying physics. It perfectly identified the logistic growth skeleton u ( 1 − u / K ) u(1-u/K) with a carrying capacity K ≈ 1.0032 K\\approx 1.0032 , closely matching the theoretical K = 1 K=1 . Furthermore, the framework approximated the complex topography-driven flux with an effective linear advection term ( 0.1580 u x 0.1580u_{x} ). In homogenization theory, this represents the “effective drift velocity” of the population averaged over the periodic terrain. As seen in Figure 5 , despite missing the micro-scale spatial forcing, the discovered dynamics (b) and cross-sections (c) capture the macroscopic wave propagation and logistic saturation remarkably well, providing a highly robust phenomenological surrogate."},{"type":"heading","level":2,"text":"E.3 Morphogenesis with Michaelis-Menten kinetics"},{"type":"para","text":"Ground Truth Equation:"},{"type":"para","text":"Physical Motivation: This system captures the biochemical concentration of a morphogen during embryonic development. It features a non-polynomial rational fraction u 0.5 + u \\frac{u}{0.5+u} representing saturated enzyme decay (Michaelis-Menten kinetics), alongside a highly localized spatial Gaussian source exp ( − x 2 ) \\exp(-x^{2}) . To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ − 5 , 5 ] × [ 0 , 10 ] (x,t)\\in[-5,5]\\times[0,10] with a resolution of 128 × 1000 128\\times 1000 . It is specifically designed to completely defy traditional sparse regression methods, which rely heavily on predefined polynomial libraries."},{"type":"para","text":"Baseline Discoveries:"},{"type":"para","text":"SGA-PDE: u t = 0.0238 ( exp ( cos ( x ) ) + u ) u_{t}=0.0238(\\exp(\\cos(x))+u)"},{"type":"para","text":"DISCOVER: u t = 0.2080 ( ( u − 2 cos ( x ) ) cos ( x ) ) 2 u_{t}=0.2080((u-2\\cos(x))\\cos(x))^{2}"},{"type":"para","text":"EqGPT: u t = 0.0443 u + 0.4735 ( u x ) 2 / u − 0.4745 u x x u_{t}=0.0443u+0.4735(u_{x})^{2}/u-0.4745u_{xx}"},{"type":"para","text":"Because a rational fraction requires an infinite Taylor series to be represented by polynomials, polynomial-biased baselines fail catastrophically, hallucinating absurd trigonometric compositions or numerically unstable divisions by u u ."},{"type":"para","text":"LLM-PDESR Discovery (Algebraically Equivalent Full Recovery):"},{"type":"para","text":"Analysis: Our algorithm achieved a complete structural discovery . Notably, this result showcases the framework’s capacity for algebraic invariant learning. The discovered rational fraction 1.9887 u 1 + 1.9847 u \\frac{1.9887u}{1+1.9847u} is mathematically equivalent to the ground truth. By dividing the numerator and denominator by 1.9847 1.9847 , we obtain ≈ 1.002 u 0.503 + u \\approx\\frac{1.002u}{0.503+u} , which is virtually identical to the true term 1.0 u 0.5 + u \\frac{1.0u}{0.5+u} . Figure 6 demonstrates a perfect visual and quantitative overlap between the true and discovered spatial cross-sections, proving the framework’s unique ability to handle non-standard rational kinetics without requiring manually crafted basis libraries."},{"type":"heading","level":2,"text":"E.4 Forced quintic Swift-Hohenberg"},{"type":"para","text":"Ground Truth Equation:"},{"type":"para","text":"Physical Motivation & Dynamical Justification: This synthetic system is inspired by pattern formation in an optical microcavity pumped by a spatially inhomogeneous Gaussian laser. It features a 4 th 4^{\\text{th}} -order spatial derivative ( u x x x x u_{xxxx} ) acting as a high-frequency spectral cut-off, which, balanced against the 2 nd 2^{\\text{nd}} -order diffusion, dictates the specific wavelength of emergent Turing-like patterns. Crucially, we introduce an extreme, unconventional − u 3 + u 5 -u^{3}+u^{5} nonlinearity. This specific algebraic structure deliberately breaks standard canonical symmetries, ensuring the equation does not exist in standard physical corpora. This serves as an ultimate stress test for both the framework’s numerical differentiation stability and its genuine out-of-distribution (OOD) symbolic reasoning. To safely navigate the inherent instability of the positive quintic term, the true system is strictly evaluated in a bounded amplitude regime ( | u | < 1 |u|<1 ) to prevent finite-time blow-up. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ − 10 , 10 ] × [ 0 , 1.2 ] (x,t)\\in[-10,10]\\times[0,1.2] with a resolution of 256 × 500 256\\times 500 ."},{"type":"para","text":"Baseline Discoveries:"},{"type":"para","text":"SGA-PDE: u t = 1.8135 u u_{t}=1.8135u"},{"type":"para","text":"DISCOVER: u t = − 1.1938 u x x + 0.9625 sin ( u ( u − u 2 ) + u ) − 0.6296 u x x x x + 0.4197 u u_{t}=-1.1938u_{xx}+0.9625\\sin(u(u-u^{2})+u)-0.6296u_{xxxx}+0.4197u"},{"type":"para","text":"EqGPT: u t = − 0.1346 u + 0.4798 u x x u_{t}=-0.1346u+0.4798u_{xx}"},{"type":"para","text":"LLM-PDESR Discovery (High-Order Structural Recovery):"},{"type":"para","text":"Analysis: The model correctly identified the core pattern-forming skeleton. Remarkably, it successfully extracted the notoriously difficult 4 th 4^{\\text{th}} -order spatial operator u x x x x u_{xxxx} from noisy data, demonstrating the extreme robustness of our quintic B-spline evaluation module. While it missed the 5 th 5^{\\text{th}} -order term and decoupled the Gaussian pump from u u , the discovered positive cubic nonlinearity ( + 0.2120 u 3 +0.2120u^{3} ) acts as an effective phenomenological compensation within the coupled dynamics. As evidenced by Figure 7 , this discovered skeleton is highly capable of inducing the correct spontaneous symmetry breaking, preserving the spatial frequency selection and qualitative stability of the true optical pattern within the evaluation window."},{"type":"heading","level":2,"text":"E.5 Traffic flow with spatial bottleneck"},{"type":"para","text":"Ground Truth Equation:"},{"type":"para","text":"Physical Motivation: A macroscopic traffic model incorporating a physical road narrowing represented by tanh ( x ) \\tanh(x) , and highly unconventional fractional driver anticipation gradient ( u x / u ) x (u_{x}/u)_{x} . These terms create kinematic shockwaves (traffic jams) that propagate backward."},{"type":"para","text":"Baseline Discoveries:"},{"type":"para","text":"SGA-PDE: u t = − 0.1385 u x / u u_{t}=-0.1385u_{x}/u"},{"type":"para","text":"DISCOVER: u t = 0.2932 tanh ( ( ( tanh ( x ) tanh ( tanh … u_{t}=0.2932\\tanh(((\\tanh(x)\\tanh(\\tanh\\dots (Severe algorithmic loop hallucination)"},{"type":"para","text":"EqGPT: u t = 0.6818 u u x + 0.1589 u x u_{t}=0.6818\\sqrt{u}u_{x}+0.1589u_{x}"},{"type":"para","text":"LLM-PDESR Discovery (Algorithmic Failure):"},{"type":"para","text":"Analysis: We transparently report that our framework failed to reconstruct this equation. The combination of a fractional flux inside a nested derivative and a highly localized tanh ( x ) \\tanh(x) spatial bottleneck creates near-discontinuous shock fronts. Such steep gradients induce severe numerical stiffness during the SWR integration step, corrupting the gradient-based parameter optimization. Consequently, the Pareto front discarded the true functional forms due to poor numerical convergence. As clearly demonstrated in Figure 8 (c), the discovered equation (dotted black line) deviates completely from the true kinematic shockwave profile. This failure highlights the current boundary of SR when confronting strongly heterogeneous, shock-forming conservation laws. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ − 15 , 15 ] × [ 0 , 5 ] (x,t)\\in[-15,15]\\times[0,5] with a resolution of 256 × 400 256\\times 400 ."},{"type":"heading","level":2,"text":"E.6 Cross-chemotactic predator-prey system"},{"type":"para","text":"Ground Truth Equations:"},{"type":"para","text":"Physical Motivation: A complex multivariable system modeling active biological hunting. The predators ( v v ) do not merely diffuse randomly; they exhibit targeted chemotaxis by moving against the concentration gradient of the prey ( u u ). The nonlinear cross-diffusion term ( v u x ) x (vu_{x})_{x} forces the algorithm to recognize that the temporal evolution of one variable depends directly on the spatial gradient of another, entirely breaking the variable independence assumption. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ 0 , 10 ] × [ 0 , 10 ] (x,t)\\in[0,10]\\times[0,10] with a resolution of 128 × 1000 128\\times 1000 ."},{"type":"para","text":"Baseline Discoveries: None. SGA-PDE, DISCOVER, and EqGPT fundamentally lack the architectural support to process, couple, or evaluate multivariable PDE systems simultaneously, rendering them blind to cross-variable derivatives. Conversely, while WSINDy_PDE theoretically supports multivariate extraction, applying it to this coupled system yields an intractable, severely overfitted expression characterized by extreme structural bloat. As this generated equation entirely lacks physical interpretability, we omit it here for brevity."},{"type":"para","text":"LLM-PDESR Discovery (Flawless Structural Recovery):"},{"type":"para","text":"Analysis: LLM-PDESR achieved a flawless structural recovery of this highly complex coupled system. The framework successfully deduced the intricate cross-chemotaxis dependency out of an exponentially vast multivariable search space (where combinations of u , v , u x , v x , u x x , v x x u,v,u_{x},v_{x},u_{xx},v_{xx} combinatorially explode). Figure 9 (c) presents the ultimate validation of this capability: at t = 10.0 t=10.0 , the discovered profiles for both the prey (dashed blue line) and the tracking predator (dashed red line) perfectly overlay the exact ground truth across the entire spatial domain. The explicit localization of the predator’s density peaks precisely where the prey gradients are steepest proves that the LLM is genuinely executing multivariable symbolic reasoning, rather than merely fitting isolated geometric shapes."},{"type":"para","text":"In the main text, we introduced the 1D Coupled Atmospheric Circulation Equation (1D-CACE) extracted by LLM-PDESR directly from chaotic ERA5 observational data:"},{"type":"para","text":"Evaluating symbolic models on genuine atmospheric data presents a fundamental closure problem. To extract the 1D spatiotemporal fields u ( x , t ) u(x,t) and v ( x , t ) v(x,t) from the raw 2D geographic bounding boxes, we apply a meridional average —specifically, computing the spatial mean across the specified latitude band for each longitude coordinate x x . However, atmospheric turbulence remains inherently chaotic and is governed by latent 3D thermodynamic variables (e.g., vertical convection, baroclinic instability, latent heat release) that are inevitably omitted from such a 1D kinematic extraction. Faced with this severe dimensionality truncation, traditional methods often fail catastrophically. Although applicable to multivariate systems, WSINDy_PDE yields a severely overfitted, unphysical expression here, which we omit for brevity. Rather than claiming the discovery of a complete fundamental law, we rigorously demonstrate below how the discovered structural skeleton serves as a highly effective, physically consistent dynamical surrogate . We detail how this framework compresses macroscopic 3D dynamics into an interpretable 1D manifold, and subsequently validate how this physical robustness ensures structural generalization across distinct spatiotemporal regimes."},{"type":"heading","level":2,"text":"F.1 Effective dynamical mechanics and spatial closure"},{"type":"para","text":"The discovered structural skeleton aligns remarkably well with the principles of geophysical fluid dynamics (GFD), breaking down into several effective governing mechanisms:"},{"type":"heading","level":4,"text":"Effective eddy viscosity and kinematic dissipation ( θ 1 u x x , θ 8 v x x \\theta_{1}u_{xx},\\theta_{8}v_{xx} ):"},{"type":"para","text":"Analogous to the viscous terms in the Navier–Stokes equations, these second-order spatial derivatives function as effective sub-grid scale turbulent diffusion operators. For 10-meter wind fields, they mathematically parameterize the irreversible dissipation of kinetic energy and the smoothing of sharp velocity gradients induced by planetary boundary layer turbulence and surface friction."},{"type":"heading","level":4,"text":"Nonlinear advection and zonal-meridional momentum exchange ( θ 2 u u x , θ 9 v v x \\theta_{2}uu_{x},\\theta_{9}vv_{x} and θ 3 v u x , θ 10 u v x \\theta_{3}vu_{x},\\theta_{10}uv_{x} ):"},{"type":"para","text":"While traditional 1D surrogate models (e.g., the 1D Burgers equation) are strictly limited to self-advection, LLM-PDESR identifies a deeply coupled advective system. Retaining the cross-advection terms ( θ 3 , θ 10 \\theta_{3},\\theta_{10} ) is critical; it captures the physical reality that the meridional wind ( v v ) actively transports zonal momentum ( u u ) and vice versa, maintaining physically plausible momentum balances within the planar formulation."},{"type":"heading","level":4,"text":"Linear coupling and implicit geopotential gradients ( θ 4 v x , θ 11 u x \\theta_{4}v_{x},\\theta_{11}u_{x} ):"},{"type":"para","text":"In simplified shallow water frameworks, spatial gradients of one velocity component often act as kinematic proxies for geopotential height perturbations that drive the orthogonal flow. Here, these terms serve as essential linear coupling mechanisms, providing the mathematical degrees of freedom necessary to approximate the propagation of inertial-gravity waves."},{"type":"heading","level":4,"text":"Rayleigh drag and background state relaxation ( θ 5 u , θ 6 v \\theta_{5}u,\\theta_{6}v and θ 12 v , θ 13 u \\theta_{12}v,\\theta_{13}u ):"},{"type":"para","text":"These zeroth-order terms function as linear damping mechanisms, acting analogously to Rayleigh friction (surface drag), which continuously dissipates kinetic energy and relaxes the velocity anomalies toward a local equilibrium. The cross-linear terms provide necessary adjustments to the steady-state balance between the orthogonal components."},{"type":"heading","level":4,"text":"First-order spatial closure for macroscopic forcings ( θ 7 x v , θ 14 x u \\theta_{7}xv,\\theta_{14}xu ):"},{"type":"para","text":"From a strict fluid dynamics perspective, fundamental physical laws demand spatial translational symmetry. However, our model operates on a reduced 1D domain ( x x ), rendering it inherently unable to explicitly resolve meridional variations (e.g., the latitude-dependent Coriolis parameter, f ≈ f 0 + β y f\\approx f_{0}+\\beta y ) or persistent 2D geographical forcings. Mathematically, when a spatially heterogeneous 2D/3D background state is projected onto a 1D manifold, the unresolved spatial variations can be optimally parameterized via a first-order Taylor expansion with respect to the spatial coordinate. Thus, the emergence of the explicit spatially dependent terms ( θ 7 x v , θ 14 x u \\theta_{7}xv,\\theta_{14}xu ) is not an overfitting artifact, but a rigorous first-order spatial closure . These terms effectively break local translational symmetry to mathematically compensate for missing macroscopic background gradients (such as persistent zonal pressure asymmetries or the projected imprint of the β \\beta -effect)."},{"type":"heading","level":2,"text":"F.2 Empirical validation via structural invariance"},{"type":"para","text":"The theoretical soundness of this spatial parameterization is precisely what enables the framework to transcend localized statistical memorization. In GFD, the nonlinear differential operators dictate the invariant physical rules (fluid motion), while the algebraic coefficients dictate the local environmental state (boundary conditions and forcings). To empirically validate this, we demonstrate robust generalization by strictly freezing the differential skeleton and only re-optimizing the scalar coefficients θ \\theta for new domains. Note that fitting the temporal tendency ( ∂ u / ∂ t , ∂ v / ∂ t \\partial u/\\partial t,\\partial v/\\partial t ) is orders of magnitude more challenging than predicting the state variables, as the derivative exponentially amplifies high-frequency chaotic noise."},{"type":"heading","level":3,"text":"F.2.1 In-distribution baseline: North Pacific (Jan 1–10)"},{"type":"para","text":"The base regime spans the North Pacific ( 25 ∘ 25^{\\circ} – 45 ∘ N 45^{\\circ}\\text{N} , 160 ∘ 160^{\\circ} – 130 ∘ W 130^{\\circ}\\text{W} ), a dynamically active region during boreal winter characterized by the intense westerly jet stream."},{"type":"para","text":"Analysis: As depicted in Figure 10 , the ground truth tendency exhibits severe high-frequency spatiotemporal turbulence. When parameterized for this region, the frozen 1D-CACE skeleton achieves an R 2 R^{2} of 76.28% . The prediction closely tracks the dominant macroscopic Rossby wave manifolds while preserving meso-scale fluctuations, successfully avoiding the over-smoothing typical of deep neural networks. The unstructured residual error rigorously represents the latent influence of unresolvable 3D thermodynamic forcings."},{"type":"heading","level":3,"text":"F.2.2 Spatial transferability: West Pacific (Jan 1–10)"},{"type":"para","text":"To rigorously test spatial robustness, we transfer the frozen skeleton to the West Pacific ( 140 ∘ 140^{\\circ} – 170 ∘ E 170^{\\circ}\\text{E} ). Dominated by the Kuroshio current’s intense sea surface temperature gradients, this cyclogenesis zone presents boundary conditions fundamentally distinct from the Central Pacific."},{"type":"para","text":"Analysis: Transferring a PDE to an unseen geographical domain typically risks numerical instability. However, simply by re-optimizing the scalar coefficients to absorb the new background state, our skeleton successfully reconstructs the dominant tendency manifolds ( R 2 = R^{2}= 60.97% , Figure 11 ). This stability confirms that the framework has successfully isolated the globally consistent advective-diffusive balance from localized geographical noise."},{"type":"heading","level":3,"text":"F.2.3 Temporal phase shift: North Pacific (Feb 1–10)"},{"type":"para","text":"Finally, we evaluate the model under Temporal Drift by shifting the window forward by one month. Due to rapid internal atmospheric variability, the flow states and storm tracks in February represent a phase shift from early January."},{"type":"para","text":"Analysis: Despite this heavily altered topology (Figure 12 ), the 1D-CACE skeleton demonstrates robust temporal resilience ( R 2 = R^{2}= 58.64% ). While purely statistical models trained on narrow temporal windows often suffer severe degradation under such phase shifts, LLM-PDESR responds to the invariant physical state gradients rather than sequentially memorized patterns. This confirms that the framework extracts genuine dynamical abstractions governing mid-latitude transport, effectively unifying mathematical interpretability with real-world predictive robustness."},{"type":"para","text":"While LLM-PDESR demonstrates robust symbolic discovery capabilities across diverse physical systems, several limitations warrant future investigation. First, as analyzed in the traffic flow failure case, the framework struggles with PDEs exhibiting extreme numerical stiffness or near-discontinuous shock fronts. Such steep spatial gradients challenge the SWR integration step and disrupt the gradient-based BFGS parameter optimization. Second, the reliance on iterative LLM inference within the Pareto evolutionary loop introduces a significant computational overhead compared to traditional sparse regression algorithms, making it highly dependent on GPU resources and inference latencies. Finally, the current numerical evaluation pipeline is built upon 1D quintic B-splines. Extending this high-order continuous formulation to 2D or 3D dynamical systems on unstructured grids introduces non-trivial topological complexities. Addressing these constraints—potentially through adaptive mesh refinement and deploying smaller, domain-distilled language models—remains an active area for our future work."},{"type":"para","text":"While our work focuses on foundational algorithmic advancements in scientific machine learning, we acknowledge its broader impacts here. On a positive note, automating PDE discovery can significantly accelerate research in critical societal domains like climate modeling (as demonstrated with our ERA5 application). Conversely, deploying LLM-driven scientific tools in high-stakes engineering systems without proper human oversight poses risks, as mathematically deceptive but physically flawed hypotheses could mislead downstream applications."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A central pursuit in modern computational science is bridging the gap between abundant empirical measurements and the underlying mathematical laws that dictate system dynamics [ 1 , 2 , 3 , 4 , 5 ] . While the formalization of this problem through symbolic regression (SR) has spurred significant advances [ 6 , 7 , 8 ] , traditional data-driven methodologies ranging from sparse regression [ 9 , 10 ] to genetic programming [ 11 , 12 , 13 ] exhibit critical limitations. Constrained by predefined function libraries, these approaches struggle to scale across vast combinatorial hypothesis spaces and lack the architectural flexibility to incorporate domain-specific physical priors [ 14 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recently, Large Language Models (LLMs) have emerged as highly promising engines for this task [ 15 , 16 , 17 , 18 , 19 ] . By treating equations as executable structures, LLMs leverage their pre-trained scientific knowledge to propose physically plausible structural hypotheses [ 20 ] . However, extending these LLM-driven paradigms to the discovery of complex, time-varying PDEs is bottlenecked by severe numerical instabilities. To compute the necessary spatial derivatives for evaluating PDE hypotheses, traditional optimization frameworks predominantly rely on standard finite difference (FD) methods and pointwise residual evaluations [ 21 , 22 , 23 ] . These discrete operations inherently amplify high-frequency measurement noise, distorting the optimization landscape [ 24 , 25 ] . When coupled with an LLM, this numerical failure is fatal: it feeds deceptive fitness scores back to the language model, derailing the evolutionary search and preventing it from distinguishing true physical mechanisms from mathematical artifacts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To resolve this critical bottleneck, we introduce LLM-PDESR (Large Language Model-guided Partial Differential Equation Symbolic Regression), a novel framework that integrates the structural hypothesis generation of LLMs with a mathematically rigorous evaluation environment. First, we replace noise-sensitive FD methods with quintic spline smoothing. By guaranteeing 𝒞 4 \\\\mathcal{C}^{4} continuity, quintic splines provide noise-robust spatial derivatives up to the fourth order [ 26 , 27 ] . Second, we transition from pointwise residuals to a subdomain weighted residual(SWR) evaluation. By integrating the PDE residual against smooth test functions over localized sub-domains, the SWR formulation acts as a natural low-pass filter [ 28 ] . This mathematically principled smoothing mitigates fitness landscape distortion, ensuring the internal continuous optimizer converges reliably and provides the LLM with accurate fitness signals. Consequently, a Pareto-driven feedback loop efficiently balances equation complexity and predictive accuracy, naturally pruning unphysical bloat [ 29 , 30 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate LLM-PDESR on a comprehensive benchmark of 23 canonical PDEs, alongside five structurally novel equations explicitly formulated to preclude dataset memorization and validate genuine symbolic discovery capabilities. Demonstrating real-world applicability, our framework successfully extracts a novel coupled atmospheric circulation surrogate (1D-CACE) directly from noisy ERA5 climate data [ 31 ] . Since capturing such intricate multivariate interactions severely degrades or exceeds existing baseline capabilities, we validate the discovered equations through rigorous out-of-distribution (OOD) testing across distinct spatiotemporal domains. This confirms they capture invariant dynamical mechanisms rather than localized statistical artifacts. Extensive empirical comparisons demonstrate that LLM-PDESR significantly outperforms state-of-the-art methods (including SGA-PDE [ 32 ] , DISCOVER [ 33 ] , WSINDy_PDE [ 34 ] , and EqGPT [ 35 ] ) in structural recovery, noise resilience, and the avoidance of spurious equations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The main contributions of this work are summarized as follows:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose LLM-PDESR, an end-to-end framework that integrates the structural hypothesis generation of LLMs with continuous numerical optimization and evolutionary search to automate PDE discovery.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We resolve the derivative noise bottleneck by leveraging 𝒞 4 \\\\mathcal{C}^{4} continuous quintic splines and SWR evaluations. This approach mitigates fitness landscape distortion, while a Pareto-driven feedback loop optimally balances predictive accuracy with structural parsimony.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce a rigorous benchmark comprising 23 canonical PDEs and five structurally novel equations. This benchmark is explicitly designed to preclude dataset memorization and validate genuine symbolic discovery capabilities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate real-world applicability by extracting an interpretable 1D dynamical surrogate (1D-CACE) directly from noisy ERA5 climate data. Rigorous spatiotemporal OOD evaluations confirm that the framework captures invariant dynamical mechanisms rather than localized statistical artifacts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Dictionary-based methods and deep learning surrogates\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data-driven PDE discovery initially leveraged sparse regression, exemplified by SINDy [ 9 ] , PDE-FIND, and STRidge [ 1 ] , which identify governing laws by selecting terms from predefined mathematical libraries. To mitigate the acute sensitivity of numerical differentiation to observation noise, integral-based approaches like WSINDy and WSINDy_PDE [ 34 ] were developed, successfully utilizing weak forms as natural low-pass filters. Concurrently, deep learning architectures such as Physics-Informed Neural Networks (PINNs) [ 3 ] , Fourier Neural Operators (FNO) [ 36 ] , and DeepONet [ 37 ] have demonstrated remarkable predictive capabilities. However, these neural surrogates typically act as black boxes, precluding explicit equation extraction. Moreover, while sparse regression offers interpretability, its reliance on handcrafted dictionaries imposes a rigid structural bottleneck. This rigidity not only triggers an intractable combinatorial explosion when scaling to highly nonlinear and high-dimensional physical systems, but also lacks the compositional flexibility to incorporate complex, domain-specific priors required for discovering novel, free-form dynamics.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Evolutionary and neural symbolic regression\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To overcome the structural rigidity of fixed dictionaries, Genetic Programming (GP) tools like Eureqa [ 38 ] and PySR [ 39 ] , alongside hybrid frameworks such as SGA-PDE [ 32 ] and EPDE [ 40 ] , explore free-form equations via stochastic mutation. However, these methods are often hindered by expression bloat and inefficient convergence within vast combinatorial search spaces. Neural Symbolic Regression (NSR) paradigms address these search inefficiencies by leveraging reinforcement learning and transformer architectures, as demonstrated by DSR [ 41 ] , MORL4PDEs [ 42 ] , and DISCOVER [ 33 ] . More recently, EqGPT [ 35 ] trained a specialized transformer on mathematical corpora to construct an automated generation-evaluation loop. Yet, these autoregressive models operate within purely syntactic spaces lacking explicit physical priors, and predominantly rely on discrete FDs for evaluation. This reliance drastically amplifies high-frequency noise—particularly in high-order or cross-derivatives—distorting the fitness landscape and degrading robustness. In contrast, LLM-PDESR resolves both limitations: it repurposes the LLM as an active reasoning agent bridging syntax and physics via Pareto-driven feedback, while employing a continuous, noise-robust evaluation environment.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The LLM-PDESR framework\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We consider a physical system evolving over a spatial domain Ω x \\\\Omega_{x} and temporal domain Ω t \\\\Omega_{t} , characterized by a state field 𝐮 ( x , t ) ∈ ℝ D \\\\mathbf{u}(x,t)\\\\in\\\\mathbb{R}^{D} (e.g., D = 1 D=1 for canonical PDEs like Burgers equation; D > 1 D>1 for coupled fluid dynamics). Given discrete, noisy spatiotemporal observations, our objective is to discover the underlying partial differential operator governing its temporal evolution:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝒩 \\\\mathcal{N} represents the unknown structural skeleton to be generated, and θ \\\\theta denotes the optimizable physical parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To prevent noise amplification, the target temporal derivative ( ∂ 𝐮 / ∂ t \\\\partial\\\\mathbf{u}/\\\\partial t ) is pre-computed via a continuous cubic spline approximation along the time axis.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traditionally, parameter optimization involves minimizing a pointwise mean squared error (MSE) loss between the observed and predicted temporal derivatives [ 43 , 44 ] . Evaluating this objective function, however, fundamentally requires approximating the requisite spatial derivatives using discrete FD methods. This dependency introduces a critical evaluation bottleneck: discrete FD inherently amplifies high-frequency observation noise ( 𝒪 ( σ / ( Δ x ) k ) \\\\mathcal{O}(\\\\sigma/(\\\\Delta x)^{k}) for k k -th order derivatives [ 45 ] ). This numerical instability distorts the fitness landscape, yielding deceptive evaluation scores that derail search algorithms and prevent convergence to the underlying physical laws.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To resolve this bottleneck, we propose LLM-PDESR. As illustrated in Figure 1 and formalized later in Algorithm 1 , this automated, closed-loop pipeline addresses PDE discovery through three core modules: LLM-Driven Hypothesis Generation, Robust Parameter Identification, and Pareto-Driven Evaluation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 LLM-driven hypothesis generation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Framing the LLM as an automated hypothesis generator, our prompt emulates the empirical discovery pipeline by mapping physical observations (e.g., 1D heat conduction data) to candidate governing equations. To preclude the generation of syntactically or physically invalid structures, this search space is strictly bounded by predefined constraint conditions (e.g., enforcing the spatial derivative operator d_dx(u, order) ). The complete prompts and constraint specifications are detailed in Appendix A .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conditioned on this prompt, the LLM acts as a structural architect, proposing a population of n n candidate executable equations (represented as Python Abstract Syntax Trees). Crucially, the LLM generates only the symbolic skeletons, utilizing parameter placeholders (e.g., params[0] ) for all numeric coefficients. These n n candidate equations are subsequently forwarded to the numerical environment to evaluate their empirical fitness.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Robust parameter identification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The parameter identification module bridges the gap between discrete symbolic generation and continuous observational data through a rigorous three-step numerical pipeline:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"1. Derivative Computation (Quintic Spline): To eliminate the aforementioned FD noise bottleneck, we reconstruct the noisy spatial field at each time step t t using quintic B-spline smoothing. This enables analytical and noise-robust spatial differentiation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where c j ( t ) c_{j}(t) are the spline coefficients and B j 5 ( x ) B_{j}^{5}(x) denotes the quintic basis functions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"2. Subdomain Weighted Residual Construction: To further fortify the evaluation against localized anomalies, we compute the integral residual ℛ i ( t ; θ ) \\\\mathcal{R}_{i}(t;\\\\theta) over K K overlapping local sub-domains Ω i \\\\Omega_{i} . By integrating against a compactly supported smooth test function ϕ i ( x ) \\\\phi_{i}(x) , this formulation inherently acts as a low-pass filter, significantly attenuating high-frequency noise prior to error calculation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Here, c i c_{i} and ω \\\\omega define the center and width of the local window Ω i \\\\Omega_{i} , respectively, and u t _ pred u_{t\\\\_\\\\text{pred}} is the time derivative predicted by the candidate equation parameterized by θ \\\\theta .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"3. Parameter Optimization via BFGS: The objective function ℒ ( θ ) \\\\mathcal{L}(\\\\theta) is explicitly formulated as the MSE of the integrated residuals. To minimize this non-convex loss, we deploy the quasi-Newton BFGS algorithm [ 46 , 47 ] , which iteratively updates the parameters using an approximate inverse Hessian matrix H k H_{k} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where M M and K K denote the number of temporal snapshots and spatial sub-domains, respectively; α k \\\\alpha_{k} is the step size, and H k H_{k} is the approximate inverse Hessian updated via BFGS.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Rigorous theoretical proofs establishing the error bounds of quintic splines over FD, the noise-attenuation properties of the SWR versus point-wise MSE, and the convergence advantages of BFGS in this context are detailed in Appendix B .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Pareto-driven evaluation and automated selection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following parameter identification, the optimized candidates are assessed in the Evaluation module. In accordance with Occam’s razor, equation discovery is modeled as a multi-objective optimization problem. We map each candidate into a 2D objective space using two strictly defined metrics:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Score: Defined as the negative MSE ( − MSE -\\\\text{MSE} ) of the optimized SWR.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Complexity: Quantified as the total number of nodes in the candidate’s Abstract Syntax Tree ( C ( ℱ ) = ∑ v ∈ AST c v C(\\\\mathcal{F})=\\\\sum_{v\\\\in\\\\text{AST}}c_{v} ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To guide the LLM’s evolutionary search while maintaining context efficiency, we extract three elite solutions from the Pareto frontier 𝒫 \\\\mathcal{P} (highest score, lowest complexity, and balanced) and dynamically inject their serializations into the subsequent prompt. For the final parsimony-guided extraction, we employ a programmatic selection heuristic based on a logarithmic error tolerance. Specifically, we define a subset of highly accurate candidates 𝒫 ∗ = { ℱ ∈ 𝒫 ∣ MSE ( ℱ ) ≤ 10 × MSE min } \\\\mathcal{P}^{*}=\\\\{\\\\mathcal{F}\\\\in\\\\mathcal{P}\\\\mid\\\\text{MSE}(\\\\mathcal{F})\\\\leq 10\\\\times\\\\text{MSE}_{\\\\min}\\\\} , representing equations within one order of magnitude of the global minimum error. The algorithm ultimately outputs ℱ ∗ = arg min ℱ ∈ 𝒫 ∗ C ( ℱ ) \\\\mathcal{F}^{*}=\\\\arg\\\\min_{\\\\mathcal{F}\\\\in\\\\mathcal{P}^{*}}C(\\\\mathcal{F}) . This mechanism systematically prunes over-parameterized models while guaranteeing near-optimal predictive accuracy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Experimental setup and baselines\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Datasets and Scenarios. Our evaluation encompasses three distinct regimes:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(1) Benchmark PDEs: A library of 23 canonical 1D PDEs (e.g., linear, semilinear, Burgers, and high-order nonlinear systems; detailed in Appendix C ). To simulate realistic measurement imperfections, the clean spatial fields u u are corrupted with additive Gaussian noise: u ~ = u + γ σ u ϵ \\\\tilde{u}=u+\\\\gamma\\\\sigma_{u}\\\\epsilon , where σ u \\\\sigma_{u} is the standard deviation of u u , ϵ ∼ 𝒩 ( 0 , 1 ) \\\\epsilon\\\\sim\\\\mathcal{N}(0,1) , and γ ∈ { 0.05 , 0.10 , 0.20 } \\\\gamma\\\\in\\\\{0.05,0.10,0.20\\\\} denotes the noise level.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(2) Structurally Novel Equations: Five synthetically designed physical systems featuring highly nonlinear spatial terms and unconventional interactions. These are explicitly formulated to preclude dataset memorization and validate genuine symbolic discovery capabilities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(3) Real-World Observations: Hourly macroscopic wind velocity fields from the ERA5 atmospheric reanalysis dataset [ 31 ] , utilized to extract the interpretable 1D dynamical surrogate (1D-CACE).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baselines and Metrics. We benchmark our framework against state-of-the-art methodologies spanning evolutionary algorithms, sparse regression, and neural symbolic generation: SGA-PDE [ 32 ] , DISCOVER [ 33 ] , WSINDy_PDE [ 34 ] , and EqGPT [ 35 ] . Performance is quantitatively assessed using two rigorous metrics: Structural Success Rate , strictly defined as the exact symbolic recovery of the true PDE skeleton; and Relative Coefficient Error , computed via the L 2 L_{2} norm ‖ θ ^ − θ ‖ 2 / ‖ θ ‖ 2 \\\\|\\\\hat{\\\\theta}-\\\\theta\\\\|_{2}/\\\\|\\\\theta\\\\|_{2} exclusively for successfully identified structures. Detailed experimental configurations and parameter settings for all evaluated methods are provided in Appendix D .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Robustness on benchmark PDEs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first evaluate the Structural Success Rate (Figure 2 a). LLM-PDESR achieves a 100% success rate at 0 % 0\\\\% noise and maintains consistently high robustness as noise increases ( 5 % 5\\\\% – 20 % 20\\\\% ). Most baselines (e.g., SGA-PDE, DISCOVER, WSINDy_PDE) suffer severe degradation, while EqGPT retains only partial robustness via explicit data denoising. LLM-PDESR consistently outperforms all competitors, demonstrating the superior noise resilience of our evaluation environment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Regarding parameter accuracy, Figure 2 b reports the Relative Coefficient Error exclusively for successful identifications. At 0 % 0\\\\% noise, our framework exhibits a low coefficient error. The nominally lower baseline errors in this regime stem from a statistical survivorship bias : these methods only converge on the simplest equations. Under noisy conditions, EqGPT exhibits the lowest error due to its external data preprocessing. In contrast, LLM-PDESR ingests raw, noisy observations directly to preserve high-fidelity physical dynamics. Despite omitting heuristic denoising, our framework maintains highly competitive error bounds, confirming that the SWR’s intrinsic low-pass filtering effectively isolates true parameters from noise.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Structurally novel equations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluating LLMs exclusively on canonical PDEs inherently risks testing training data memorization rather than genuine symbolic reasoning. To rigorously probe the generalization capabilities of our framework and explicitly mitigate the risk of training data contamination, we designed five structurally novel equations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Topography-Constrained Chemotaxis ( u t = 0.1 u x x − 0.5 ( u ( 1 − u ) cos ( x ) ) x + u ( 1 − u ) u_{t}=0.1u_{xx}-0.5(u(1-u)\\\\cos(x))_{x}+u(1-u) ): Models biological foraging on undulating terrain with logistic crowding. The nested derivative structure strictly prevents simple additive token generation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Morphogenesis with Michaelis-Menten Kinetics ( u t = 0.1 u x x − 1.0 u / ( 0.5 + u ) + exp ( − x 2 ) u_{t}=0.1u_{xx}-1.0u/(0.5+u)+\\\\exp(-x^{2}) ): Features a non-polynomial rational fraction and a spatial Gaussian source, a structure typically intractable for traditional polynomial-biased sparse regression.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Forced Quintic Swift-Hohenberg ( u t = 1.5 exp ( − 0.1 x 2 ) u − 2 u x x − u x x x x − u 3 + u 5 u_{t}=1.5\\\\exp(-0.1x^{2})u-2u_{xx}-u_{xxxx}-u^{3}+u^{5} ): A synthetic, out-of-distribution model inspired by optical microcavity pattern formation. It features a 5th-order nonlinearity and 4th-order spatial derivatives, explicitly designed to break standard physical symmetries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traffic Flow with Spatial Bottleneck ( u t = − ( u ( 1 − u ) ( 1 − 0.5 tanh ( x ) ) ) x + 0.05 ( u x / u ) x u_{t}=-(u(1-u)(1-0.5\\\\tanh(x)))_{x}+0.05(u_{x}/u)_{x} ): A macroscopic traffic model incorporating a physical road narrowing ( tanh \\\\tanh ) and driver anticipation gradients.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cross-Chemotactic Predator-Prey System ( u t = 0.05 u x x + u ( 1 − u − 0.5 v ) u_{t}=0.05u_{xx}+u(1-u-0.5v) and v t = 0.01 v x x − 0.2 ( v u x ) x + 0.8 v ( u − 1 ) v_{t}=0.01v_{xx}-0.2(vu_{x})_{x}+0.8v(u-1) ): A coupled multivariate system where predators ( v v ) actively hunt by moving along the prey’s ( u u ) density gradient.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As Table 1 reports, the evaluated baselines universally fail to recover these structurally novel equations. Specifically, WSINDy_PDE succumbs to combinatorial explosion and extreme overfitting (e.g., yielding an AST complexity of 203 with high MSE for Predator-Prey). Meanwhile, SGA-PDE, DISCOVER, and EqGPT suffer from structural collapse and fundamentally fail to discover the correct forms. Furthermore, as most baselines lack the architectural support for multivariate systems, they are inherently inapplicable to the coupled Predator-Prey dynamics (denoted as N/A).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conversely, LLM-PDESR successfully navigates this vast hypothesis space. By accommodating inherent structural complexities, our framework extracts the exact mathematical skeletons ( ★ \\\\color[rgb]{0,0.88,0}\\\\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\\\\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\\\\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}{\\\\bigstar} ) for both Morphogenesis and the coupled Predator-Prey dynamics without generating spurious terms. Notably, even when exact symbolic recovery is unachieved for highly nonlinear systems like the Traffic Flow Bottleneck ( × \\\\color[rgb]{1,0,0}\\\\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\\\\boldsymbol{\\\\times}} ), LLM-PDESR consistently secures the lowest MSE across all five systems, demonstrating superior capability in approximating underlying physical dynamics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Remark on Partial Recovery: For Topography Chemotaxis and Forced Swift-Hohenberg, LLM-PDESR achieves partial recovery ( ≈ \\\\color[rgb]{1,.5,0}\\\\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{\\\\mathbf{\\\\approx}} ), strictly defined as correctly identifying ≥ 50 % \\\\geq 50\\\\% of the exact governing terms. Rather than producing unphysical terms when faced with complex interactions, our framework exhibits exceptional structural robustness via Pareto-driven optimization. It reliably isolates core physical mechanisms and prevents mathematically invalid bloat, even when the complete skeleton proves exceptionally intricate. A comprehensive, term-by-term analysis and further detailed discussions regarding the recovered structures for all five novel systems are provided in Appendix E .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Real-world discovery: coupled atmospheric dynamics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluating symbolic regression on authentic reanalysis data like ERA5 is notoriously challenging, as the algorithm must contend with severe dimensionality truncation (unresolved 3D thermodynamic variables) and high-frequency data assimilation noise. To demonstrate our framework’s capability in this extreme regime, we deployed LLM-PDESR on real-world ERA5 atmospheric data (North Pacific domain; Jan 2023) [ 48 , 49 , 50 ] , autonomously extracting an interpretable 1D coupled atmospheric circulation surrogate (1D-CACE):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where x x is the spatial coordinate and θ 1 … θ 14 \\\\theta_{1}\\\\dots\\\\theta_{14} are optimizable physical parameters. Although applicable to multivariate systems, WSINDy_PDE yields a severely overfitted, unphysical expression here, omitted for brevity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unlike standard 1D models (e.g., Burgers equation) limited to self-advection, 1D-CACE explicitly captures cross-wind momentum transfer ( v u x , u v x vu_{x},uv_{x} ) and linear coupling ( v x , u x v_{x},u_{x} ). From a Reduced Order Modeling (ROM) perspective, projecting complex 3D atmospheric flows onto a 1D manifold inevitably loses external physical forcings. Remarkably, LLM-PDESR autonomously compensates for this by discovering spatially dependent terms ( x v xv and x u xu ). Rather than just overfitting localized noise, these act as vital mathematical closure terms, implicitly proxying unresolved macroscopic background gradients (analogous to the well-known β \\\\beta -plane approximation of the Coriolis force). Detailed physical derivations of these compensation terms are provided in Appendix F .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This dynamical consistency ensures strong out-of-distribution (OOD) generalization. By freezing the structural skeleton of Eq. ( 5 ) and re-optimizing only the numeric coefficients, the equation transfers effectively to distinct spatiotemporal domains (Table 2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Notably, while temporal drift to February causes a performance drop ( R 2 = 58.64 % R^{2}=58.64\\\\% ) due to transitional late-winter dynamics, the geographically extreme Hemispheric OOD setting (South Indian Ocean, July) rebounds significantly to 72.54 % 72.54\\\\% . Because July represents the austral winter, this recovery empirically confirms that 1D-CACE successfully captures the macroscopic dynamics of winter atmospheric circulation , rather than memorizing geographical artifacts. Figure 3 visualizes this high-fidelity hemispheric reconstruction, verifying that LLM-PDESR effectively distills interpretable governing dynamics from chaotic real-world data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Ablation studies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To systematically evaluate the individual contribution of each core component, we conduct an ablation study using noise-free data. We compare the full LLM-PDESR framework against three ablated variants: (1) w/o Spline (replacing quintic splines with standard FDs), (2) w/o SWR (replacing local integral residuals with pointwise MSE), and (3) w/o Pareto (removing the multi-objective selection, retaining only MSE as the fitness metric).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 (a) illustrates the significant impact of removing any single module on the Structural Success Rate . While the full framework achieves a 100 % 100\\\\% recovery on the benchmark PDEs, reverting to discrete FDs (w/o Spline) or pointwise evaluation (w/o SWR) significantly degrades success rates to 69.6 % 69.6\\\\% and 60.9 % 60.9\\\\% , respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 (b) elucidates the underlying optimization dynamics through the MSE distributions. Without splines, numerical differentiation errors rapidly accumulate. Similarly, without the SWR’s inherent smoothing, the fitness landscape is severely distorted by high-frequency numerical artifacts and truncation errors (even in the absence of observation noise), resulting in high MSE variances and optimization failures. Most crucially, the w/o Pareto variant exhibits a deceptive phenomenon: while its median MSE appears competitively low, its structural recovery remains poor. This corroborates our theoretical analysis: without a strict complexity penalty, the LLM inevitably falls into an overfitting trap. It produces bloated, mathematically spurious equations that artificially minimize the residual error, completely failing to execute genuine symbolic discovery.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Conclusion and future work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this work, we introduce LLM-PDESR, a novel paradigm bridging the structural hypothesis generation of LLMs with the numerical rigor required for robust PDE discovery. By integrating 𝒞 4 \\\\mathcal{C}^{4} continuous quintic splines, SWR evaluations, and a Pareto-driven automated selection loop, our framework effectively mitigates the fitness landscape distortion and the generation of spurious equations that plague traditional methods. Empirically, it achieves state-of-the-art performance across canonical benchmarks and structurally novel systems (explicitly designed to mitigate the risk of training data contamination), while demonstrating practical utility by extracting an interpretable 1D coupled atmospheric circulation surrogate (1D-CACE) directly from noisy ERA5 observations. While our current formulation successfully distills macroscopic dynamics from noisy real-world environments, future work will focus on extending this framework to high-dimensional (2D/3D) physical systems on unstructured grids and optimizing the computational overhead of iterative LLM inference, thereby further advancing the frontier of automated scientific discovery.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Acknowledgments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This work was supported by the National Science and Technology Major Project under Grant 2021ZD0112500. We thank the Computing Center of Jilin Province for technical support.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section details the operating mechanism of LLM-PDESR. The process is decoupled into two core components: the structured prompt template used to guide the LLM in generating candidate equations, and the rigorous numerical pipeline used to evaluate and optimize the parameters of these candidates.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.1 Structured LLM prompt for equation generation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To concretize our methodology, we illustrate the prompt structure using a representative and highly challenging governing equation from our experiments: Topography-Constrained Chemotaxis . The target PDE is given by:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This equation models biological foraging on an undulating terrain coupled with logistic crowding. Crucially, its nested derivative structure—specifically the advection term ( u ( 1 − u ) cos ( x ) ) x (u(1-u)\\\\cos(x))_{x} —inherently prevents simple additive token generation, thereby severely testing the LLM’s capacity for deep physical reasoning and symbolic manipulation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To prevent the generation of physically meaningless or computationally intractable expressions, we constrain the LLM’s action space through a highly structured prompt. The prompt enforces strict mathematical vocabularies and introduces the Pareto evolution mechanism, which organically penalizes bloated equations. The exact prompt template tailored for this specific biological task is provided in Prompt A.1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To rigorously evaluate the generalizability of our framework and ensure reproducibility, we apply this standardized prompt template across all structurally novel systems. Crucially, the Strict Constraints and Pareto Evolution Mechanism remain completely identical across all trials to prevent task-specific prompt engineering bias. The LLM’s initial context is altered solely by modifying the physical background provided in the Task description.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The specific task contexts supplied for the remaining five novel systems are detailed below:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Morphogenesis with Michaelis-Menten Kinetics: \\\"Find the mathematical function skeleton that represents the time derivative of the field variable ( u t u_{t} ), given data on spatial coordinates ( x x ) and the field variable ( u u ) in a morphogen diffusion from a localized Gaussian source with Michaelis-Menten absorption.\\\"\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Forced Quintic Swift-Hohenberg: \\\"Find the mathematical function skeleton that represents the time derivative of the field variable ( u t u_{t} ), given data on spatial coordinates ( x x ) and the field variable ( u u ) in a subcritical optical medium driven by a Gaussian laser and stabilized by hyper-viscosity.\\\"\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traffic Flow with Spatial Bottleneck: \\\"Find the mathematical function skeleton that represents the time derivative of the field variable ( u t u_{t} ), given data on spatial coordinates ( x x ) and the field variable ( u u ) in a traffic flow with a spatial bottleneck and anticipation-driven relative gradient diffusion.\\\"\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cross-Chemotactic Predator-Prey System (Multivariate): \\\"Find the mathematical function skeletons that represent the time derivatives of the field variables ( u t u_{t} and v t v_{t} ), given data on spatial coordinates ( x x ) and the field variables ( u u and v v ) in a coupled system: For u t u_{t} : diffusing prey population suppressed by local predator encounters. For v t v_{t} : diffusing predator population migrating toward prey density gradients.\\\"\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Real-World Application: 1D Coupled Atmospheric Circulation (1D-CACE): \\\"Find the mathematical function skeletons that represent the time derivatives of the field variables ( u t u_{t} and v t v_{t} ), given data on spatial coordinates ( x x ) and the field variables ( u u and v v ) from real-world Earth atmospheric reanalysis: For u t u_{t} (zonal wind): longitudinal velocity of a viscous advective atmospheric flow under planetary rotation. For v t v_{t} (meridional wind): latitudinal velocity of a viscous advective atmospheric flow under planetary rotation.\\\"\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.2 Numerical evaluation and parameter optimization algorithm\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Once a candidate equation skeleton 𝒩 ( u ; θ ) \\\\mathcal{N}(u;\\\\theta) is proposed by the LLM, its fitness must be evaluated. To bridge the gap between discrete, noisy observational data and continuous spatial derivatives, we employ a pipeline combining quintic B-spline smoothing, SWR integration, and BFGS optimization. This robust evaluation process is summarized in Algorithm 2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this appendix, we establish the theoretical foundations of the proposed parameter identification module. We provide rigorous error bounds for the quintic spline formulation compared to classical FDs, prove the spectral noise-attenuation properties of the SWR, and demonstrate the convergence advantages of the BFGS algorithm within our specifically smoothed loss landscape.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 Error bounds and noise robustness of regularized quintic B-spline smoothing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let the underlying true spatial field be u ∈ 𝒞 6 ( Ω ) u\\\\in\\\\mathcal{C}^{6}(\\\\Omega) . The observed data is assumed to be corrupted by additive noise, denoted as u ~ j = u ( x j ) + ϵ j \\\\tilde{u}_{j}=u(x_{j})+\\\\epsilon_{j} , where ϵ j ∼ 𝒩 ( 0 , σ 2 ) \\\\epsilon_{j}\\\\sim\\\\mathcal{N}(0,\\\\sigma^{2}) are independent and identically distributed (i.i.d.) random variables, and h h is the uniform spatial discretization step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let S 5 ( x ) S_{5}(x) be the quintic B-spline constructed from the noisy observations u ~ \\\\tilde{u} via regularized smoothing. The total expected error of the n n -th derivative approximation ( n ≤ 4 ) (n\\\\leq 4) in the Chebyshev norm is bounded by:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where C 1 C_{1} and C 2 C_{2} are constants independent of h h and σ \\\\sigma . Furthermore, the expected squared L 2 L^{2} norm of the noise-induced derivative variance scales as 𝒪 ( σ 2 h − 2 n ) \\\\mathcal{O}(\\\\sigma^{2}h^{-2n}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The total error is bounded by the triangle inequality, separating the approximation error (bias) and the noise propagation error (variance):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the bias term, classical Schoenberg approximation theory for a spline of degree d = 5 d=5 guarantees the Peano kernel bound:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the variance term, S 5 , noise ( x ) = ∑ j ϵ j B j 5 ( x ) S_{5,\\\\text{noise}}(x)=\\\\sum_{j}\\\\epsilon_{j}B_{j}^{5}(x) . Because the quintic B-spline basis functions B j 5 ( x ) B_{j}^{5}(x) form a partition of unity, are non-negative, and strictly possess compact support over exactly 6 consecutive intervals, the n n -th derivative obeys the Markov-type inequality ‖ d n d x n B j 5 ‖ ∞ ≤ C h − n ||\\\\frac{d^{n}}{dx^{n}}B_{j}^{5}||_{\\\\infty}\\\\leq Ch^{-n} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Crucially, the uniform norm evaluates the supremum over the entire spatial domain, which effectively consists of N ∝ L / h N\\\\propto L/h independent local regions. By extreme value theory, the expected maximum of these Gaussian random variables introduces a logarithmic growth factor. Therefore, taking the expectation of the maximum deviation yields:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To analyze the global variance accumulation in the L 2 L^{2} sense, we integrate the squared noise function over the entire domain of length L L . Because 𝔼 [ ϵ j ϵ k ] = σ 2 δ j k \\\\mathbb{E}[\\\\epsilon_{j}\\\\epsilon_{k}]=\\\\sigma^{2}\\\\delta_{jk} , the cross-terms vanish in expectation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a single localized basis function, the integral of its squared n n -th derivative scales as 𝒪 ( h − 2 n + 1 ) \\\\mathcal{O}(h^{-2n+1}) . Since the domain is spanned by N ∝ L / h N\\\\propto L/h basis functions, the total expected squared L 2 L^{2} error evaluates to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This mathematically confirms that the root-mean-square asymptotic scaling with respect to h h is 𝒪 ( h − n ) \\\\mathcal{O}(h^{-n}) , which identically matches that of discrete finite differences.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, despite sharing the identical asymptotic rate, the regularized spline formulation provides two indispensable practical advantages over standard FDs. First, the smoothing penalty strictly bounds the error coefficient (the constant factor), substantially reducing the absolute variance compared to point-wise differences. Second, and most crucially, whereas FDs generate discontinuous grid-scale oscillations that fatally distort subsequent gradient-based optimization, S 5 ( x ) ∈ 𝒞 4 ( Ω ) S_{5}(x)\\\\in\\\\mathcal{C}^{4}(\\\\Omega) produces an intrinsically smooth, globally differentiable vector field, which is a fundamental prerequisite for evaluating the Subdomain Weighted Residuals (SWR) accurately. ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 Spectral noise attenuation via the SWR\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now prove that the SWR method inherently acts as an optimal low-pass filter, dramatically reducing the variance of the loss gradient compared to point-wise MSE.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let the observed temporal derivative be u ~ t ( x ) = u t , true ( x ) + η ( x ) \\\\tilde{u}_{t}(x)=u_{t,\\\\text{true}}(x)+\\\\eta(x) . To maintain dimensional consistency between discrete grid observations (with variance σ 2 \\\\sigma^{2} and grid spacing h h ) and continuous integration, we define η ( x ) \\\\eta(x) as a zero-mean spatial white noise with an equivalent continuous autocovariance 𝔼 [ η ( x ) η ( y ) ] = σ 2 h δ ( x − y ) \\\\mathbb{E}[\\\\eta(x)\\\\eta(y)]=\\\\sigma^{2}h\\\\delta(x-y) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The variance of the normalized localized integral residual ℛ ¯ i ( θ ) \\\\bar{\\\\mathcal{R}}_{i}(\\\\theta) evaluated with the test function ϕ i ( x ) = sin 4 ( π x − c i + ω / 2 ω ) \\\\phi_{i}(x)=\\\\sin^{4}\\\\left(\\\\pi\\\\frac{x-c_{i}+\\\\omega/2}{\\\\omega}\\\\right) is attenuated by a factor inversely proportional to the number of data points within the local window, compared to the pointwise residual.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In a purely point-wise collocation framework, the expected MSE loss incorporates the full noise variance directly: 𝔼 [ | u t , true − u t , pred + η | 2 ] = MSE true + σ 2 \\\\mathbb{E}[|u_{t,\\\\text{true}}-u_{t,\\\\text{pred}}+\\\\eta|^{2}]=\\\\text{MSE}_{\\\\text{true}}+\\\\sigma^{2} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In our locally integrated strong form, we rigorously define the normalized residual to evaluate the true localized tendency. The noise contribution to the normalized residual at domain Ω i \\\\Omega_{i} is explicitly:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given 𝔼 [ η ( x ) ] = 0 \\\\mathbb{E}[\\\\eta(x)]=0 , the expected value 𝔼 [ ϵ ℛ ¯ ] = 0 \\\\mathbb{E}[\\\\epsilon_{\\\\bar{\\\\mathcal{R}}}]=0 . The variance of this noise contribution is computed as:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Applying the equivalent continuous white noise property 𝔼 [ η ( x ) η ( y ) ] = σ 2 h δ ( x − y ) \\\\mathbb{E}[\\\\eta(x)\\\\eta(y)]=\\\\sigma^{2}h\\\\delta(x-y) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Substituting ϕ i ( x ) = sin 4 ( π x − c i + ω / 2 ω ) \\\\phi_{i}(x)=\\\\sin^{4}\\\\left(\\\\pi\\\\frac{x-c_{i}+\\\\omega/2}{\\\\omega}\\\\right) and applying the change of variables z = x − c i + ω / 2 ω z=\\\\frac{x-c_{i}+\\\\omega/2}{\\\\omega} , we obtain:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using the standard reduction formula for even powers of sine (Wallis’ integrals), we evaluate the definite integral exactly:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus, Var ( ϵ ℛ ¯ ) = 35 128 ( h ω ) σ 2 \\\\text{Var}(\\\\epsilon_{\\\\bar{\\\\mathcal{R}}})=\\\\frac{35}{128}\\\\left(\\\\frac{h}{\\\\omega}\\\\right)\\\\sigma^{2} . Let N ω = ω / h N_{\\\\omega}=\\\\omega/h represent the number of discrete grid points inside the window. The effective variance is strictly reduced to:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This establishes a rigorous theoretical guarantee: the variance is substantially suppressed proportional to the window density, effectively resolving the severe noise amplification endemic to standard collocation methods. ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Boundary Handling and Support Preservation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As correctly noted in the framework’s implementation, a naive truncation of the test functions at x min x_{\\\\min} or x max x_{\\\\max} would induce a step discontinuity, degrading the spectral noise roll-off. To preserve the high-order smoothness required for optimal noise attenuation, LLM-PDESR ensures that the support of every test function is strictly contained within the observational domain, i.e., supp ( ϕ i ) ⊆ [ x min , x max ] \\\\text{supp}(\\\\phi_{i})\\\\subseteq[x_{\\\\min},x_{\\\\max}] . Specifically, the boundary-adjacent window centers c 1 c_{1} and c K c_{K} are shifted inward such that the function reaches zero exactly at x min x_{\\\\min} and x max x_{\\\\max} . This configuration guarantees that ϕ i ∈ 𝒞 3 ( [ x min , x max ] ) \\\\phi_{i}\\\\in\\\\mathcal{C}^{3}([x_{\\\\min},x_{\\\\max}]) for all i i , ensuring the 𝒪 ( | ξ | − 4 ) \\\\mathcal{O}(|\\\\xi|^{-4}) spectral decay holds uniformly.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.3 Superlinear Convergence of the BFGS Optimization in Smoothed Landscapes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, we establish that the combination of our noise-attenuating SWR and the BFGS quasi-Newton method leads to guaranteed superlinear convergence, circumventing the ill-conditioning typical of inverse PDE problems.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let the SWR objective ℒ ( θ ) = 1 M K ∑ j = 1 M ∑ i = 1 K | ℛ ¯ i , j ( θ ) | 2 \\\\mathcal{L}(\\\\theta)=\\\\frac{1}{MK}\\\\sum_{j=1}^{M}\\\\sum_{i=1}^{K}|\\\\bar{\\\\mathcal{R}}_{i,j}(\\\\theta)|^{2} be twice continuously differentiable, where M M and K K denote the number of temporal snapshots and spatial sub-domains, respectively. Under the assumptions of PDE identifiability and sufficiently small integrated observation noise (the small residual condition), ℒ ( θ ) \\\\mathcal{L}(\\\\theta) satisfies local strong convexity in a neighborhood 𝒩 \\\\mathcal{N} of the optimal parameters θ ∗ \\\\theta^{*} . The BFGS update, generating a sequence of approximate inverse Hessians { H k } \\\\{H_{k}\\\\} , guarantees superlinear convergence to θ ∗ \\\\theta^{*} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As established in Theorem 2 , the SWR formulation acts as a low-pass filter by integrating residuals against 𝒞 3 \\\\mathcal{C}^{3} test functions. By employing regularized quintic spline smoothing instead of exact interpolation, the Jacobian of the residuals J ( θ ) J(\\\\theta) remains numerically stable. The Hessian of the scalar objective ℒ ( θ ) \\\\mathcal{L}(\\\\theta) is approximated via the Gauss-Newton formulation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the presence of observation data, the true parameters θ ∗ \\\\theta^{*} do not yield a strictly zero residual; rather, ℛ ¯ i , j ( θ ∗ ) \\\\bar{\\\\mathcal{R}}_{i,j}(\\\\theta^{*}) equals the integrated observation noise. Consequently, the second-order summation acts as an indefinite random perturbation matrix proportional to the noise level. To ensure local strong convexity, the optimization must satisfy a small residual assumption , where the noise is sufficiently small such that the positive-definite first-order term ( J ( θ ) T J ( θ ) J(\\\\theta)^{T}J(\\\\theta) ) strictly dominates the second-order term.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Crucially, as proven in Theorem 2 , our SWR formulation inherently shrinks the variance of this residual noise by a factor of 𝒪 ( 1 / N ω ) \\\\mathcal{O}(1/N_{\\\\omega}) . By explicitly attenuating the residual magnitude at θ ∗ \\\\theta^{*} , the SWR framework mathematically ensures that the small residual condition holds over a much broader range of physical noise levels compared to pointwise formulations. Thus, the positive-definite Gram matrix robustly dominates:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since the test functions and spline smoothing mollify the ruggedness caused by data noise, ∇ 2 ℒ ( θ ) \\\\nabla^{2}\\\\mathcal{L}(\\\\theta) is locally strictly positive definite, fulfilling the strong convexity requirement:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 0 < m ≤ M 0<m\\\\leq M .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The BFGS algorithm iteratively updates the approximate inverse Hessian H k H_{k} via the Sherman-Morrison-Woodbury formula:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where s k = θ k + 1 − θ k s_{k}=\\\\theta_{k+1}-\\\\theta_{k} and y k = ∇ ℒ ( θ k + 1 ) − ∇ ℒ ( θ k ) y_{k}=\\\\nabla\\\\mathcal{L}(\\\\theta_{k+1})-\\\\nabla\\\\mathcal{L}(\\\\theta_{k}) . Let B k = H k − 1 B_{k}=H_{k}^{-1} denote the corresponding approximate Hessian. Under the established conditions of strong convexity and smoothness, the sequence { B k } \\\\{B_{k}\\\\} satisfies the Dennis-Moré condition:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This condition is necessary and sufficient for the sequence of parameters { θ k } \\\\{\\\\theta_{k}\\\\} to converge to θ ∗ \\\\theta^{*} at a superlinear rate :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By dynamically approximating the local curvature, BFGS implicitly resolves the severe ill-conditioning between parameters representing different spatial derivative orders (e.g., advection vs. hyper-diffusion terms), circumventing the 𝒪 ( d 3 ) \\\\mathcal{O}(d^{3}) computational bottleneck per iteration required by exact Newton methods. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our evaluation relies on a benchmark suite of 23 diverse 1D PDEs, which are grouped into five mathematical categories: Linear Convection-Diffusion, Semilinear Reaction-Diffusion, Quasilinear Diffusion, Viscous Burgers, and Higher-Order Nonlinear Evolution Equations (see Table 3 for exact formulations and domains). To guarantee the fidelity of the reference data, we simulated these systems using high-order spectral methods paired with precise time integration, thereby effectively suppressing numerical artifacts. For each specific PDE, the spatial resolution ( N x N_{x} ) and temporal step size ( N t N_{t} ) were individually optimized to capture its unique dynamic features while strictly maintaining numerical stability. Furthermore, the selected domains, bounded by [ x min , x max ] [x_{\\\\min},x_{\\\\max}] and [ t min , t max ] [t_{\\\\min},t_{\\\\max}] , encompass a wide variety of boundary conditions and evolutionary timescales, offering a rigorous testbed for model generalizability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To ensure strict reproducibility, we detail the hardware environment and hyperparameter configurations for all evaluated methodologies. All experiments were conducted on a Linux server equipped with a 48-core CPU, 128 GB RAM, and 4 × \\\\times NVIDIA GeForce RTX 3090 GPUs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"SGA-PDE.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The symbolic genetic algorithm was configured with a population size of 100 and executed for 500 generations. Tree structure constraints were set to a maximum depth of 4 and a maximum width of 5. The evolutionary probabilities were set to P v a r = 0.5 P_{var}=0.5 , P m u t e = 0.3 P_{mute}=0.3 , and P c r o = 0.5 P_{cro}=0.5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"DISCOVER.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The RNN controller was trained on 200,000 samples with a batch size of 1,000, a learning rate of 0.0025, and an entropy weight of 0.03. The function library included standard arithmetic operators and up to fourth-order derivatives. To constrain the search space, we applied structural priors (e.g., inverse, trigonometric, and differentiation descendant rules), alongside a sequence length constraint bounded between 3 and 256.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"WSINDy_PDE.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We utilized piecewise polynomial test functions with query point spacings s x = 4 s_{x}=4 and s t = 4 s_{t}=4 . The candidate library comprised polynomials up to degree 3, spatial derivatives up to 4th order, and trigonometric terms. Sparse regression was executed using Sequential Threshold Ridge regression (STRidge), searching over 100 logarithmically spaced λ \\\\lambda values in [ 10 − 4 , 1 ] [10^{-4},1] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"EqGPT.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following its original proof-of-concept setup, target structures were masked from the dataset during training. The comprehensive candidate library contained 43 terms (up to 3rd-order polynomials and 4th-order derivatives). The Generational Genetic Algorithm (GGA) module utilized a population of 400, a maximum of 200 generations, a mutation rate of 0.3, and an L 0 L_{0} penalty strength of 3 × 10 − 4 3\\\\times 10^{-4} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"LLM-PDESR (Ours).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our framework utilized DeepSeek-V3 ( deepseek-chat ) as the generative backbone, accessed via the official DeepSeek API. The optimization spanned 1,000 iterations. In each iteration, the LLM generated 4 candidate hypotheses. For in-context evolution, the top 3 equation templates from the Pareto frontier were fed back into the prompt. The sampling temperature was initialized at 0.1 and periodically adjusted every 30,000 tokens to balance exploration and exploitation. To evaluate hypotheses via the SWR formulation, we employed quintic splines for up to 4th-order spatial derivatives and evaluated the integral loss across K = 10 K=10 local domains using a sin 4 \\\\sin^{4} window function. Equation coefficients were optimized via BFGS with a strict 30-second timeout per evaluation to prevent infinite execution loops in generated code.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluating LLMs exclusively on canonical PDEs (e.g., Burgers, KdV, Kuramoto-Sivashinsky) inherently risks testing “memorization” rather than genuine symbolic “discovery.” Because canonical equations are heavily represented in pre-training corpora, high performance on these tasks often reflects a model’s ability to interpolate within its training distribution rather than extrapolate to novel physical laws.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To rigorously probe the generalization capabilities of our framework and explicitly mitigate the risk of data contamination, we designed a benchmark suite of five structurally novel, out-of-distribution physical equations. These equations do not exist in standard textbooks in these exact forms, yet they are synthesized from fundamental physical and biological phenomena. Below, we provide an in-depth physical motivation for each system, compare the equations discovered by our LLM-PDESR framework against state-of-the-art baselines (SGA-PDE, DISCOVER, WSINDy_PDE, and EqGPT), and theoretically analyze the resulting spatiotemporal dynamics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note: The equations discovered by WSINDy_PDE are omitted from the detailed text below due to extreme algorithmic bloating (generating excessively long, uninterpretable polynomial chains comprising dozens of terms), though the method consistently failed to recover the true underlying physical structures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before detailing the mathematical formulations of each novel system, we first address a fundamental methodological question regarding baseline evaluation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.1 Information asymmetry and baseline evaluation constraints\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A natural question arises regarding the evaluation of these structurally novel equations: could deterministic baselines like WSINDy [ 34 ] achieve comparable performance if their basis libraries were manually augmented to reflect the semantic physical hints provided to LLM-PDESR (e.g., explicitly adding Gaussians or rational fractions)?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We elected not to augment the deterministic baselines with these specific terms, as doing so mathematically violates the foundational assumptions of sparse dictionary regression and inevitably leads to algorithmic failure:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Incompatibility with Non-linear Parameterization: Incorporating a semantic feature like a “localized Gaussian source” necessitates a non-linear parameterized structure, such as θ 1 exp ( − ( x − θ 2 ) 2 / θ 3 ) \\\\theta_{1}\\\\exp(-(x-\\\\theta_{2})^{2}/\\\\theta_{3}) . Dictionary-based methods construct a static, linear-in-the-parameters feature matrix. They cannot dynamically optimize internal parameters (such as the mean θ 2 \\\\theta_{2} or variance θ 3 \\\\theta_{3} ) during the sparse regression phase without resorting to an intractable pre-computed grid of candidate columns.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Multicollinearity and Ill-conditioning: Naively expanding the dictionary with complex rational terms (e.g., u / ( 1 + u ) u/(1+u) to represent logistic crowding) and their cross-products triggers severe multicollinearity. For instance, candidate columns like u u and u / ( 1 + u ) u/(1+u) become numerically indistinguishable under moderate observational noise. This causes thresholding algorithms (e.g., STRidge or LASSO) to suffer from severe ill-conditioning, resulting in the selection of spurious terms and catastrophic failure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Therefore, this “information asymmetry” actually highlights a fundamental architectural advantage of our framework. LLM-PDESR explicitly decouples structural hypothesis generation from non-linear parameter identification, possessing the unique capacity to ingest abstract semantic priors and instantiate them as optimizable non-linear structures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.2 Topography-constrained chemotaxis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ground Truth Equation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Physical Motivation: This equation models the population dynamics of a biological species foraging on an undulating terrain. The reaction term u ( 1 − u ) u(1-u) represents logistic crowding, enforcing a strict environmental carrying capacity. The advection term, featuring a nested derivative structure ( u ( 1 − u ) cos ( x ) ) x (u(1-u)\\\\cos(x))_{x} , models topography-driven drift where the periodic terrain cos ( x ) \\\\cos(x) interacts nonlinearly with the population density. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ 0 , 2 π ] × [ 0 , 10 ] (x,t)\\\\in[0,2\\\\pi]\\\\times[0,10] with a resolution of 100 × 1000 100\\\\times 1000 . This nested structure strictly prevents simple additive token generation, challenging the LLM’s capacity for deep spatial reasoning and chain-rule expansion.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline Discoveries:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SGA-PDE: u t = 0.1584 u x + 0.9636 ( u − u 2 ) u_{t}=0.1584u_{x}+0.9636(u-u^{2})\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DISCOVER: u t = 0.1584 u x − 0.9710 sin ( u 2 − u ) u_{t}=0.1584u_{x}-0.9710\\\\sin(u^{2}-u)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"EqGPT: u t = 0.0082 u 2 − 0.9691 u x x + 0.9977 u u_{t}=0.0082u^{2}-0.9691u_{xx}+0.9977u\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All baselines completely failed to identify the explicit spatial dependence cos ( x ) \\\\cos(x) . They either collapsed the complex topography-constrained advection into simple linear convection (SGA-PDE) or hallucinated unphysical trigonometric reaction terms (DISCOVER).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LLM-PDESR Discovery (Effective Macroscopic Approximation):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: While our algorithm did not recover the exact explicit spatial forcing cos ( x ) \\\\cos(x) , it successfully distilled an effective macroscopic approximation of the underlying physics. It perfectly identified the logistic growth skeleton u ( 1 − u / K ) u(1-u/K) with a carrying capacity K ≈ 1.0032 K\\\\approx 1.0032 , closely matching the theoretical K = 1 K=1 . Furthermore, the framework approximated the complex topography-driven flux with an effective linear advection term ( 0.1580 u x 0.1580u_{x} ). In homogenization theory, this represents the “effective drift velocity” of the population averaged over the periodic terrain. As seen in Figure 5 , despite missing the micro-scale spatial forcing, the discovered dynamics (b) and cross-sections (c) capture the macroscopic wave propagation and logistic saturation remarkably well, providing a highly robust phenomenological surrogate.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.3 Morphogenesis with Michaelis-Menten kinetics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ground Truth Equation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Physical Motivation: This system captures the biochemical concentration of a morphogen during embryonic development. It features a non-polynomial rational fraction u 0.5 + u \\\\frac{u}{0.5+u} representing saturated enzyme decay (Michaelis-Menten kinetics), alongside a highly localized spatial Gaussian source exp ( − x 2 ) \\\\exp(-x^{2}) . To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ − 5 , 5 ] × [ 0 , 10 ] (x,t)\\\\in[-5,5]\\\\times[0,10] with a resolution of 128 × 1000 128\\\\times 1000 . It is specifically designed to completely defy traditional sparse regression methods, which rely heavily on predefined polynomial libraries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline Discoveries:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SGA-PDE: u t = 0.0238 ( exp ( cos ( x ) ) + u ) u_{t}=0.0238(\\\\exp(\\\\cos(x))+u)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DISCOVER: u t = 0.2080 ( ( u − 2 cos ( x ) ) cos ( x ) ) 2 u_{t}=0.2080((u-2\\\\cos(x))\\\\cos(x))^{2}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"EqGPT: u t = 0.0443 u + 0.4735 ( u x ) 2 / u − 0.4745 u x x u_{t}=0.0443u+0.4735(u_{x})^{2}/u-0.4745u_{xx}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because a rational fraction requires an infinite Taylor series to be represented by polynomials, polynomial-biased baselines fail catastrophically, hallucinating absurd trigonometric compositions or numerically unstable divisions by u u .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LLM-PDESR Discovery (Algebraically Equivalent Full Recovery):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: Our algorithm achieved a complete structural discovery . Notably, this result showcases the framework’s capacity for algebraic invariant learning. The discovered rational fraction 1.9887 u 1 + 1.9847 u \\\\frac{1.9887u}{1+1.9847u} is mathematically equivalent to the ground truth. By dividing the numerator and denominator by 1.9847 1.9847 , we obtain ≈ 1.002 u 0.503 + u \\\\approx\\\\frac{1.002u}{0.503+u} , which is virtually identical to the true term 1.0 u 0.5 + u \\\\frac{1.0u}{0.5+u} . Figure 6 demonstrates a perfect visual and quantitative overlap between the true and discovered spatial cross-sections, proving the framework’s unique ability to handle non-standard rational kinetics without requiring manually crafted basis libraries.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.4 Forced quintic Swift-Hohenberg\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ground Truth Equation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Physical Motivation & Dynamical Justification: This synthetic system is inspired by pattern formation in an optical microcavity pumped by a spatially inhomogeneous Gaussian laser. It features a 4 th 4^{\\\\text{th}} -order spatial derivative ( u x x x x u_{xxxx} ) acting as a high-frequency spectral cut-off, which, balanced against the 2 nd 2^{\\\\text{nd}} -order diffusion, dictates the specific wavelength of emergent Turing-like patterns. Crucially, we introduce an extreme, unconventional − u 3 + u 5 -u^{3}+u^{5} nonlinearity. This specific algebraic structure deliberately breaks standard canonical symmetries, ensuring the equation does not exist in standard physical corpora. This serves as an ultimate stress test for both the framework’s numerical differentiation stability and its genuine out-of-distribution (OOD) symbolic reasoning. To safely navigate the inherent instability of the positive quintic term, the true system is strictly evaluated in a bounded amplitude regime ( | u | < 1 |u|<1 ) to prevent finite-time blow-up. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ − 10 , 10 ] × [ 0 , 1.2 ] (x,t)\\\\in[-10,10]\\\\times[0,1.2] with a resolution of 256 × 500 256\\\\times 500 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline Discoveries:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SGA-PDE: u t = 1.8135 u u_{t}=1.8135u\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DISCOVER: u t = − 1.1938 u x x + 0.9625 sin ( u ( u − u 2 ) + u ) − 0.6296 u x x x x + 0.4197 u u_{t}=-1.1938u_{xx}+0.9625\\\\sin(u(u-u^{2})+u)-0.6296u_{xxxx}+0.4197u\"\n },\n {\n \"type\": \"para\",\n \"text\": \"EqGPT: u t = − 0.1346 u + 0.4798 u x x u_{t}=-0.1346u+0.4798u_{xx}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LLM-PDESR Discovery (High-Order Structural Recovery):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: The model correctly identified the core pattern-forming skeleton. Remarkably, it successfully extracted the notoriously difficult 4 th 4^{\\\\text{th}} -order spatial operator u x x x x u_{xxxx} from noisy data, demonstrating the extreme robustness of our quintic B-spline evaluation module. While it missed the 5 th 5^{\\\\text{th}} -order term and decoupled the Gaussian pump from u u , the discovered positive cubic nonlinearity ( + 0.2120 u 3 +0.2120u^{3} ) acts as an effective phenomenological compensation within the coupled dynamics. As evidenced by Figure 7 , this discovered skeleton is highly capable of inducing the correct spontaneous symmetry breaking, preserving the spatial frequency selection and qualitative stability of the true optical pattern within the evaluation window.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.5 Traffic flow with spatial bottleneck\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ground Truth Equation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Physical Motivation: A macroscopic traffic model incorporating a physical road narrowing represented by tanh ( x ) \\\\tanh(x) , and highly unconventional fractional driver anticipation gradient ( u x / u ) x (u_{x}/u)_{x} . These terms create kinematic shockwaves (traffic jams) that propagate backward.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline Discoveries:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SGA-PDE: u t = − 0.1385 u x / u u_{t}=-0.1385u_{x}/u\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DISCOVER: u t = 0.2932 tanh ( ( ( tanh ( x ) tanh ( tanh … u_{t}=0.2932\\\\tanh(((\\\\tanh(x)\\\\tanh(\\\\tanh\\\\dots (Severe algorithmic loop hallucination)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"EqGPT: u t = 0.6818 u u x + 0.1589 u x u_{t}=0.6818\\\\sqrt{u}u_{x}+0.1589u_{x}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LLM-PDESR Discovery (Algorithmic Failure):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: We transparently report that our framework failed to reconstruct this equation. The combination of a fractional flux inside a nested derivative and a highly localized tanh ( x ) \\\\tanh(x) spatial bottleneck creates near-discontinuous shock fronts. Such steep gradients induce severe numerical stiffness during the SWR integration step, corrupting the gradient-based parameter optimization. Consequently, the Pareto front discarded the true functional forms due to poor numerical convergence. As clearly demonstrated in Figure 8 (c), the discovered equation (dotted black line) deviates completely from the true kinematic shockwave profile. This failure highlights the current boundary of SR when confronting strongly heterogeneous, shock-forming conservation laws. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ − 15 , 15 ] × [ 0 , 5 ] (x,t)\\\\in[-15,15]\\\\times[0,5] with a resolution of 256 × 400 256\\\\times 400 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"E.6 Cross-chemotactic predator-prey system\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ground Truth Equations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Physical Motivation: A complex multivariable system modeling active biological hunting. The predators ( v v ) do not merely diffuse randomly; they exhibit targeted chemotaxis by moving against the concentration gradient of the prey ( u u ). The nonlinear cross-diffusion term ( v u x ) x (vu_{x})_{x} forces the algorithm to recognize that the temporal evolution of one variable depends directly on the spatial gradient of another, entirely breaking the variable independence assumption. To generate the observational data, the continuous system is sampled on a uniform spatiotemporal grid ( x , t ) ∈ [ 0 , 10 ] × [ 0 , 10 ] (x,t)\\\\in[0,10]\\\\times[0,10] with a resolution of 128 × 1000 128\\\\times 1000 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baseline Discoveries: None. SGA-PDE, DISCOVER, and EqGPT fundamentally lack the architectural support to process, couple, or evaluate multivariable PDE systems simultaneously, rendering them blind to cross-variable derivatives. Conversely, while WSINDy_PDE theoretically supports multivariate extraction, applying it to this coupled system yields an intractable, severely overfitted expression characterized by extreme structural bloat. As this generated equation entirely lacks physical interpretability, we omit it here for brevity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"LLM-PDESR Discovery (Flawless Structural Recovery):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: LLM-PDESR achieved a flawless structural recovery of this highly complex coupled system. The framework successfully deduced the intricate cross-chemotaxis dependency out of an exponentially vast multivariable search space (where combinations of u , v , u x , v x , u x x , v x x u,v,u_{x},v_{x},u_{xx},v_{xx} combinatorially explode). Figure 9 (c) presents the ultimate validation of this capability: at t = 10.0 t=10.0 , the discovered profiles for both the prey (dashed blue line) and the tracking predator (dashed red line) perfectly overlay the exact ground truth across the entire spatial domain. The explicit localization of the predator’s density peaks precisely where the prey gradients are steepest proves that the LLM is genuinely executing multivariable symbolic reasoning, rather than merely fitting isolated geometric shapes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the main text, we introduced the 1D Coupled Atmospheric Circulation Equation (1D-CACE) extracted by LLM-PDESR directly from chaotic ERA5 observational data:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Evaluating symbolic models on genuine atmospheric data presents a fundamental closure problem. To extract the 1D spatiotemporal fields u ( x , t ) u(x,t) and v ( x , t ) v(x,t) from the raw 2D geographic bounding boxes, we apply a meridional average —specifically, computing the spatial mean across the specified latitude band for each longitude coordinate x x . However, atmospheric turbulence remains inherently chaotic and is governed by latent 3D thermodynamic variables (e.g., vertical convection, baroclinic instability, latent heat release) that are inevitably omitted from such a 1D kinematic extraction. Faced with this severe dimensionality truncation, traditional methods often fail catastrophically. Although applicable to multivariate systems, WSINDy_PDE yields a severely overfitted, unphysical expression here, which we omit for brevity. Rather than claiming the discovery of a complete fundamental law, we rigorously demonstrate below how the discovered structural skeleton serves as a highly effective, physically consistent dynamical surrogate . We detail how this framework compresses macroscopic 3D dynamics into an interpretable 1D manifold, and subsequently validate how this physical robustness ensures structural generalization across distinct spatiotemporal regimes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"F.1 Effective dynamical mechanics and spatial closure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The discovered structural skeleton aligns remarkably well with the principles of geophysical fluid dynamics (GFD), breaking down into several effective governing mechanisms:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Effective eddy viscosity and kinematic dissipation ( θ 1 u x x , θ 8 v x x \\\\theta_{1}u_{xx},\\\\theta_{8}v_{xx} ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analogous to the viscous terms in the Navier–Stokes equations, these second-order spatial derivatives function as effective sub-grid scale turbulent diffusion operators. For 10-meter wind fields, they mathematically parameterize the irreversible dissipation of kinetic energy and the smoothing of sharp velocity gradients induced by planetary boundary layer turbulence and surface friction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Nonlinear advection and zonal-meridional momentum exchange ( θ 2 u u x , θ 9 v v x \\\\theta_{2}uu_{x},\\\\theta_{9}vv_{x} and θ 3 v u x , θ 10 u v x \\\\theta_{3}vu_{x},\\\\theta_{10}uv_{x} ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While traditional 1D surrogate models (e.g., the 1D Burgers equation) are strictly limited to self-advection, LLM-PDESR identifies a deeply coupled advective system. Retaining the cross-advection terms ( θ 3 , θ 10 \\\\theta_{3},\\\\theta_{10} ) is critical; it captures the physical reality that the meridional wind ( v v ) actively transports zonal momentum ( u u ) and vice versa, maintaining physically plausible momentum balances within the planar formulation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Linear coupling and implicit geopotential gradients ( θ 4 v x , θ 11 u x \\\\theta_{4}v_{x},\\\\theta_{11}u_{x} ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In simplified shallow water frameworks, spatial gradients of one velocity component often act as kinematic proxies for geopotential height perturbations that drive the orthogonal flow. Here, these terms serve as essential linear coupling mechanisms, providing the mathematical degrees of freedom necessary to approximate the propagation of inertial-gravity waves.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Rayleigh drag and background state relaxation ( θ 5 u , θ 6 v \\\\theta_{5}u,\\\\theta_{6}v and θ 12 v , θ 13 u \\\\theta_{12}v,\\\\theta_{13}u ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These zeroth-order terms function as linear damping mechanisms, acting analogously to Rayleigh friction (surface drag), which continuously dissipates kinetic energy and relaxes the velocity anomalies toward a local equilibrium. The cross-linear terms provide necessary adjustments to the steady-state balance between the orthogonal components.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"First-order spatial closure for macroscopic forcings ( θ 7 x v , θ 14 x u \\\\theta_{7}xv,\\\\theta_{14}xu ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From a strict fluid dynamics perspective, fundamental physical laws demand spatial translational symmetry. However, our model operates on a reduced 1D domain ( x x ), rendering it inherently unable to explicitly resolve meridional variations (e.g., the latitude-dependent Coriolis parameter, f ≈ f 0 + β y f\\\\approx f_{0}+\\\\beta y ) or persistent 2D geographical forcings. Mathematically, when a spatially heterogeneous 2D/3D background state is projected onto a 1D manifold, the unresolved spatial variations can be optimally parameterized via a first-order Taylor expansion with respect to the spatial coordinate. Thus, the emergence of the explicit spatially dependent terms ( θ 7 x v , θ 14 x u \\\\theta_{7}xv,\\\\theta_{14}xu ) is not an overfitting artifact, but a rigorous first-order spatial closure . These terms effectively break local translational symmetry to mathematically compensate for missing macroscopic background gradients (such as persistent zonal pressure asymmetries or the projected imprint of the β \\\\beta -effect).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"F.2 Empirical validation via structural invariance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The theoretical soundness of this spatial parameterization is precisely what enables the framework to transcend localized statistical memorization. In GFD, the nonlinear differential operators dictate the invariant physical rules (fluid motion), while the algebraic coefficients dictate the local environmental state (boundary conditions and forcings). To empirically validate this, we demonstrate robust generalization by strictly freezing the differential skeleton and only re-optimizing the scalar coefficients θ \\\\theta for new domains. Note that fitting the temporal tendency ( ∂ u / ∂ t , ∂ v / ∂ t \\\\partial u/\\\\partial t,\\\\partial v/\\\\partial t ) is orders of magnitude more challenging than predicting the state variables, as the derivative exponentially amplifies high-frequency chaotic noise.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"F.2.1 In-distribution baseline: North Pacific (Jan 1–10)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The base regime spans the North Pacific ( 25 ∘ 25^{\\\\circ} – 45 ∘ N 45^{\\\\circ}\\\\text{N} , 160 ∘ 160^{\\\\circ} – 130 ∘ W 130^{\\\\circ}\\\\text{W} ), a dynamically active region during boreal winter characterized by the intense westerly jet stream.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: As depicted in Figure 10 , the ground truth tendency exhibits severe high-frequency spatiotemporal turbulence. When parameterized for this region, the frozen 1D-CACE skeleton achieves an R 2 R^{2} of 76.28% . The prediction closely tracks the dominant macroscopic Rossby wave manifolds while preserving meso-scale fluctuations, successfully avoiding the over-smoothing typical of deep neural networks. The unstructured residual error rigorously represents the latent influence of unresolvable 3D thermodynamic forcings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"F.2.2 Spatial transferability: West Pacific (Jan 1–10)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To rigorously test spatial robustness, we transfer the frozen skeleton to the West Pacific ( 140 ∘ 140^{\\\\circ} – 170 ∘ E 170^{\\\\circ}\\\\text{E} ). Dominated by the Kuroshio current’s intense sea surface temperature gradients, this cyclogenesis zone presents boundary conditions fundamentally distinct from the Central Pacific.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: Transferring a PDE to an unseen geographical domain typically risks numerical instability. However, simply by re-optimizing the scalar coefficients to absorb the new background state, our skeleton successfully reconstructs the dominant tendency manifolds ( R 2 = R^{2}= 60.97% , Figure 11 ). This stability confirms that the framework has successfully isolated the globally consistent advective-diffusive balance from localized geographical noise.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"F.2.3 Temporal phase shift: North Pacific (Feb 1–10)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, we evaluate the model under Temporal Drift by shifting the window forward by one month. Due to rapid internal atmospheric variability, the flow states and storm tracks in February represent a phase shift from early January.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Analysis: Despite this heavily altered topology (Figure 12 ), the 1D-CACE skeleton demonstrates robust temporal resilience ( R 2 = R^{2}= 58.64% ). While purely statistical models trained on narrow temporal windows often suffer severe degradation under such phase shifts, LLM-PDESR responds to the invariant physical state gradients rather than sequentially memorized patterns. This confirms that the framework extracts genuine dynamical abstractions governing mid-latitude transport, effectively unifying mathematical interpretability with real-world predictive robustness.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While LLM-PDESR demonstrates robust symbolic discovery capabilities across diverse physical systems, several limitations warrant future investigation. First, as analyzed in the traffic flow failure case, the framework struggles with PDEs exhibiting extreme numerical stiffness or near-discontinuous shock fronts. Such steep spatial gradients challenge the SWR integration step and disrupt the gradient-based BFGS parameter optimization. Second, the reliance on iterative LLM inference within the Pareto evolutionary loop introduces a significant computational overhead compared to traditional sparse regression algorithms, making it highly dependent on GPU resources and inference latencies. Finally, the current numerical evaluation pipeline is built upon 1D quintic B-splines. Extending this high-order continuous formulation to 2D or 3D dynamical systems on unstructured grids introduces non-trivial topological complexities. Addressing these constraints—potentially through adaptive mesh refinement and deploying smaller, domain-distilled language models—remains an active area for our future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While our work focuses on foundational algorithmic advancements in scientific machine learning, we acknowledge its broader impacts here. On a positive note, automating PDE discovery can significantly accelerate research in critical societal domains like climate modeling (as demonstrated with our ERA5 application). Conversely, deploying LLM-driven scientific tools in high-stakes engineering systems without proper human oversight poses risks, as mathematically deceptive but physically flawed hypotheses could mislead downstream applications.\"\n }\n]","isConversational":false}}},{"rowIdx":393,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10540"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Representation Learning for Semiparametric Causal Mediation Analysis under No Essential Heterogeneity"},"authors":{"kind":"list like","value":["Roberto Faleh","Sofia Morelli","Holger Brandt"],"string":"[\n \"Roberto Faleh\",\n \"Sofia Morelli\",\n \"Holger Brandt\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10540v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Keywords: causal mediation, G-estimation, no essential heterogeneity, representation learning, structural mean models, TARNet."},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"The primary goal of many randomized experiments is to estimate the effect of a treatment on an outcome (Hernán and Robins, 2020 ) . In psychology, economics, and biology, researchers are often interested not only in estimating overall treatment effects, but also in studying the underlying causal mechanisms and understanding how an intervention exerts its effects (VanderWeele, 2015 ) . Such an investigation is especially relevant when the objective is, for instance, to evaluate a scientific theory or to guide and inform the design of future interventions. A frequently studied setting (Rijnhart et al. , 2021 ; Holland, 1988 ) involves a mediator M M , an intermediate variable that is influenced by the treatment and, in turn, affects the outcome, potentially together with a set of baseline covariates. Randomization of the treatment T T removes confounding between the treatment and the outcome. However, for mediator variables, the situation becomes more complex. Randomization of the treatment does not, in general, imply randomization of the mediator, since the mediator may be influenced by processes that occur after the initial treatment assignment (Bullock et al. , 2010 ) . The consequence is that the causal effect of the mediator on the outcome can be confounded. In this context, several technical approaches are available and, in general, this does not represent a major methodological concern (Wodtke and Zhou, 2026 ) . A more serious problem arises when some of these confounding variables are not measured (Ten Have and Joffe, 2012 ; Imai et al. , 2010 ) . This is a plausible scenario when, for example, the trial was originally designed to estimate only the overall treatment effect and was not designed to support a causal mediation analysis. In other circumstances, the confounders can be potentially known but unethical, prohibitively expensive, or difficult to measure (Brandt, 2020 ) . As a result, researchers could have extensive knowledge about the causal mechanism and a rich set of covariates but still lack confounders needed for the identification of the indirect effects. For example, an online retail platform may be interested in examining the impact of a banner on customers’ spending, with this relationship potentially mediated by user engagement. The platform could collect customer-level data, including individual preferences and demographic information, yet still lacks covariates necessary for decomposition of the total causal effect. These unobserved variables may include sensitive attributes or information that is difficult to collect, such as the presence of other individuals in the room while the customer is interacting with the platform."},{"type":"para","text":"In such cases, the classical assumption of Sequential Ignorability (SI) (Imai et al. , 2010 ) , which requires the absence of unmeasured mediator–outcome confounding conditional on observed covariates, is not satisfied. Methods that rely on the sequential ignorability assumption, then, would deliver inconsistent estimates (Imai et al. , 2010 ; Ten Have et al. , 2007 ) ."},{"type":"para","text":"An alternative framework that avoids relying on the SI assumption is offered by Structural Nested Models (SNMs) together with the corresponding G-estimation approach introduced by Robins ( 1994 ) . This approach offers a framework for estimating causal effects in settings where confounders may themselves be influenced by the treatment, as in the case of confounders measured post-treatment. SNMs have considerable advantages over other methods for handling time-varying and unmeasured confounding (Loh and Ren, 2025 ) : they can accommodate flexible structural specification, have doubly robust properties, and can exploit the instrumental variable approach for identification (Vansteelandt and Joffe, 2014 ) . Despite these strengths, the application of G-estimation in empirical research has been relatively infrequent, historically generally due to the complexity of implementation and the limited availability of \"off-the-shelf\" software (Vansteelandt and Joffe, 2014 ) . In the causal mediation context, Ten Have et al. ( 2007 ) applied G-estimation under a rank-preserving assumption. This assumption requires that the idiosyncratic error does not depend on the counterfactual regime at the individual level, meaning that each person’s deviation from the structural model is identical regardless of what treatment and mediator values are hypothetically assigned."},{"type":"para","text":"While this allows the existence of unmeasured confounders in the mediator–outcome mechanism, it rules out individual-level heterogeneity in treatment or mediator effects through unobserved confounders. In many settings this assumption could be difficult to satisfy and often it is an excessively strong constraint:it implies, for instance, that a person whose outcome deviates unusually from the structural prediction under one treatment–mediator regime must deviate by exactly the same amount under every other regime. Indeed, not even non-confounding, idiosyncratic effect heterogeneity is admitted under rank preservation. In many behavioural and biomedical settings, such rank-preservation is implausible and has been criticised accordingly (Hernán and Robins, 2020 ) ."},{"type":"para","text":"To overcome this limitation, Zheng and Zhou ( 2015 ) propose a more general structural mean model in which the Ten Have et al. ( 2007 ) formulation appears as a special case. They alleviate rank preservation by replacing it with a weaker mean-level condition closely related to the econometric “No Essential Heterogeneity” (NEH) assumption, building on the marginal treatment effect framework of Heckman and Vytlacil ( 2005 ) and the definition of essential heterogeneity introduced in Heckman et al. ( 2006 ) . This permits arbitrarily rich effect modification through observed covariates, and even idiosyncratic, non-confounding individual heterogeneity, while still accommodating unmeasured confounders that shift both mediator and outcome."},{"type":"para","text":"Additionally, the semiparametric structure of the Zheng and Zhou ( 2015 ) model makes it especially suitable for applied settings in which researchers are not merely interested in the magnitude of a treatment effect but also seek to parametrically characterize the causal mechanism through which it operates. A practical challenge in implementing the Zheng and Zhou ( 2015 ) framework is the estimation of the optimal weight function that enters the G-estimating equation. Zheng and Zhou ( 2015 ) employed parametric formulations, Brandt ( 2020 ) adopted a Thin Plate Spline (Wood, 2003 ) to introduce flexibility, and Morelli et al. ( 2025 ) broadened the original framework to incorporate latent variables."},{"type":"para","text":"However, when covariate-mediator relationships are nonlinear or the covariate space is high-dimensional, the available tools for weight estimation are limited, and poor weight estimation translates directly into efficiency loss Zheng and Zhou ( 2015 ) . We will show that that the weights depend on the conditional average treatment effect (CATE) of the treatment on the mediator, which must be estimated from the observed data. We will show, both theoretically and with a simulation study, that when the covariate-mediator relationship is nonlinear and high-dimensional, linear approaches may incur in an estimation error for the mediator CATE (Curth and Van der Schaar, 2021 ) , which in turn degrades the quality of the plug-in weight necessary for the G-estimation step, reducing the precision of the downstream structural parameter estimates."},{"type":"para","text":"We address this challenge by proposing a representation-learning approach (Bengio et al. , 2013 ) for the estimation of the weights, thereby broadening the empirical applicability of a framework that already enjoys strong semiparametric identification guarantees. We use a Treatment-Agnostic Representation Network (TARNet; Shalit et al. , 2017 ) , a neural network architecture with shared representation layers and treatment-specific output heads, to estimate the CATE on the mediator. The shared representation Φ ( X ) \\Phi(X) enforces a common feature space across treatment arms. Because the CATE is computed as a difference of two functions of the same representation, the smoothness and dimensionality of Φ \\Phi provide an inductive bias that may help regularise the estimated treatment effect surface, particularly when μ 0 \\mu_{0} and μ 1 \\mu_{1} , the arm-specific conditional expectations of the mediator, share common features (Curth and Van der Schaar, 2021 ) . The TARNet’s CATE output is then embedded as the weight function in the G-estimating equation based on the model originally developed by Zheng and Zhou ( 2015 ) . A detailed description is provided in 3.1"},{"type":"heading","level":4,"text":"Contributions."},{"type":"para","text":"Our contributions are threefold. First, we illustrate a clear link between the quality of the learned representation Φ \\Phi and the efficiency of the structural parameter estimator: better CATE estimation results in a more informative weight matrix, which reduces the asymptotic variance of θ ^ \\hat{\\theta} . Second, we propose UNIT (Unmeasured-confounding-robust NEH-based Identification with TARNet), a two-stage algorithm combining cross-fitted TARNet estimation and G-estimation. This approach requires the No Essential Heterogeneity assumption rather than the classical, and stronger, Sequential Ignorability as the key identifying condition. Third, the practical advantages of this approach are demonstrated through a simulation study with non-Gaussian covariates, complex nonlinear treatment effects, and unmeasured confounding. We show that TARNet-based weights can improve efficiency."},{"type":"heading","level":4,"text":"Outline."},{"type":"para","text":"Section 2 introduces the causal framework, the model and the NEH assumption. Section 3 develops the representation-learning approach to weight estimation and the two-stage algorithm. Section 4 presents a simulation study that is used to investigate the properties of the proposed two-stage algorithm under empirically relevant conditions. Section 4.5 presents the results. Section 5 discusses the findings and limitations."},{"type":"heading","level":1,"text":"2 Causal Framework"},{"type":"heading","level":2,"text":"2.1 Model and potential outcomes"},{"type":"para","text":"Following the potential outcomes framework introduced by (Rubin, 1974 ) and the model specifications introduced Zheng and Zhou ( 2015 ) , let Y ( t , m ) Y(t,m) denote the potential outcome under treatment level t t and mediator level m m , and let M ( t ) M(t) denote the potential mediator under treatment t t (Robins and Greenland, 1992 ) . We observe n n independent units ( X i , T i , M i , Y i ) i = 1 n (X_{i},T_{i},M_{i},Y_{i})_{i=1}^{n} , where X ∈ ℝ p X\\in\\mathbb{R}^{p} is a vector of pre-treatment covariates, T ∈ { 0 , 1 } T\\in\\{0,1\\} is the binary treatment, M ∈ ℝ M\\in\\mathbb{R} is the mediator, and Y ∈ ℝ Y\\in\\mathbb{R} is the outcome."},{"type":"para","text":"For observed covariates X X and unobserved confounders U U , write"},{"type":"para","text":"where μ ( ⋅ ) \\mu(\\cdot) captures the systematic (identified) component and ε ( t , m ; U , X ) \\varepsilon(t,m;\\,U,X) is an idiosyncratic error satisfying 𝔼 [ ε ( t , m ; U , X ) ∣ X ] = 0 \\mathbb{E}[\\varepsilon(t,m;\\,U,X)\\mid X]=0 for all ( t , m ) (t,m) . The semicolon separates the counterfactual regime ( t , m ) (t,m) from the random components ( U , X ) (U,X) . To streamline the notation, the ( U , X ) (U,X) component will be omitted whenever it is not required for the derivations."},{"type":"para","text":"We specify a structural mean model of the form"},{"type":"para","text":"where g ( X ) g(X) is a baseline outcome model and the h k ( ⋅ ) , k = 1 , 2 , … , K , h_{k}(\\cdot),\\;k=1,2,\\ldots,K, are known functions, which satisfy h k ( 0 , 0 , X ) = 0 h_{k}(0,0,X)=0 and describe the causal effects of the treatment and the mediator."},{"type":"para","text":"By consistency assumption (Assumption 1 ) the observed outcome satisfies"},{"type":"para","text":"where ε i := ε ( T i , M i ; U i , X i ) \\varepsilon_{i}:=\\varepsilon(T_{i},M_{i};\\,U_{i},X_{i}) is the structural error evaluated at the realised treatment and mediator. Defining the transformed outcome"},{"type":"para","text":"the true parameter θ 0 \\theta_{0} satisfies 𝔼 [ Y ~ ( θ 0 ) ∣ T , X ] = E [ Y ~ ( θ 0 ) ∣ X ] = g ( X ) \\mathbb{E}[\\tilde{Y}(\\theta_{0})\\mid T,X]=E[\\tilde{Y}(\\theta_{0})\\mid X]=g(X) (under assumption 3 ). For suitable weight functions A ( T , X ) A(T,X) satisfying 𝔼 [ A ( T , X ) ∣ X ] = 0 \\mathbb{E}[A(T,X)\\mid X]=0 , the estimating equation"},{"type":"para","text":"yields a consistent estimator θ ^ \\hat{\\theta} (Zheng and Zhou, 2015 ) ."},{"type":"para","text":"Throughout the remainder of this paper, we refer to the model with K = 2 K=2 , h 1 ( T , M , X ) = T h_{1}(T,M,X)=T , and h 2 ( T , M , X ) = M h_{2}(T,M,X)=M , as an illustrative example, so that"},{"type":"para","text":"This implies no treatment-by-mediator interaction in the outcome and constant controlled effects θ 1 \\theta_{1} , θ 2 \\theta_{2} . This keeps the structural parameters interpretable and preserves comparability with the results reported in Ten Have et al. ( 2007 ) , Zheng and Zhou ( 2015 ) , Brandt ( 2020 ) , and Morelli et al. ( 2025 ) . Richer specifications in the structural model can be easily accommodated (Zheng and Zhou, 2015 ) and additional details will be developed in Section 2.5 ."},{"type":"heading","level":2,"text":"2.2 Overview on 2-stage approach"},{"type":"para","text":"Parsimony in the structural outcome model does not require parsimony in the rest of the data-generating process. Our methodological focus is on settings where the covariate vector X ∈ ℝ p X\\in\\mathbb{R}^{p} is high-dimensional and the mediator CATE"},{"type":"para","text":"is a non-trivial function to learn, involving nonlinearities or interactions that cannot be captured by simple linear models. The setting we are describing is the combination of a complex and heterogeneous treatment effect on the mediator, paired with a relatively simple, structural outcome model, and could generally arise in many applied settings. For example, baseline individual characteristics (e.g., genetic profile, comorbidities) may interact in a complex manner in determining the level of a biomarker (e.g., blood pressure), whereas the effect of the biomarker on the outcome (e.g., cardiovascular risk) per unit change in the biomarker may be relatively simple, invariant among the population, and approximately constant."},{"type":"para","text":"In the specific formulation made, the weights are"},{"type":"para","text":"details about the weights and the CATE will be discussed in Section 2.5 ."},{"type":"para","text":"We refer to this step, the estimation of the weights necessary for the estimating equations, as Stage 1 . Solving the estimating equations is Stage 2. We summarize the algorithm below; the full development is deferred to Section 3 ."},{"type":"heading","level":2,"text":"2.3 General Algorithm Overview"},{"type":"para","text":"Collect the basis functions of Eq. ( 2 ) into the vector"},{"type":"para","text":"so that the structural component of the model is ∑ k = 1 K θ k h k ( T i , M i , X i ) = H i ⊤ θ \\sum_{k=1}^{K}\\theta_{k}\\,h_{k}(T_{i},M_{i},X_{i})=H_{i}^{\\top}\\theta , with θ = ( θ 1 , … , θ K ) ⊤ \\theta=(\\theta_{1},\\dots,\\theta_{K})^{\\top} . In the structural mean (mediation) model the basis becomes H i = ( T i , M i ) ⊤ H_{i}=(T_{i},M_{i})^{\\top} and θ = ( θ 1 , θ 2 ) ⊤ \\theta=(\\theta_{1},\\theta_{2})^{\\top} are the controlled direct and mediator effects."},{"type":"heading","level":4,"text":"Stage 1: weight estimation."},{"type":"para","text":"We estimate the optimal weight W ∗ ( X ) = ( 1 , τ M ( X ) ) ⊤ W^{*}(X)=(1,\\tau_{M}(X))^{\\top} by replacing the unknown mediator CATE with a cross-fitted estimate τ ^ M ( − ) \\hat{\\tau}_{M}^{(-)} of ( 6 ). Any CATE learner may be used; we use a cross-fitted TARNet (Section 3.2 ). This yields the plug-in centred instrument"},{"type":"para","text":"which satisfies the centering condition 𝔼 [ A ^ i ∣ X i ] = 0 \\mathbb{E}[\\hat{A}_{i}\\mid X_{i}]=0 under randomisation. The weight A ^ i \\hat{A}_{i} is held fixed for Stage 2."},{"type":"heading","level":4,"text":"Stage 2: structural estimation."},{"type":"para","text":"Given the Stage 1 weight A ^ i \\hat{A}_{i} and a cross-fitted estimate g ^ ( − ) \\hat{g}^{(-)} of the baseline, the structural estimator θ ^ \\hat{\\theta} is defined as the solution of the empirical G-estimating equation in Eq. ( 4 ):"},{"type":"para","text":"It is important to highlight two aspects about Stage 2;"},{"type":"para","text":"First, because dim A ^ i = dim H i = K \\dim\\hat{A}_{i}=\\dim H_{i}=K , Eq. ( 8 ) is a just-identified linear system in θ \\theta , with closed-form solution"},{"type":"para","text":"This is an instrumental-variable estimator: the centred weight A ^ i \\hat{A}_{i} acts as an instrument for the structural regressors H i H_{i} , and invertibility of ∑ i A ^ i H i ⊤ \\sum_{i}\\hat{A}_{i}H_{i}^{\\top} is exactly the relevance (non-degeneracy) condition of Assumption 5 ."},{"type":"para","text":"Second, because the instrument is centred, the equation is orthogonal to the baseline: 𝔼 [ A ^ i f ( X i ) ] = 0 \\mathbb{E}[\\hat{A}_{i}\\,f(X_{i})]=0 for any function f f of the covariates. Consistency of θ ^ \\hat{\\theta} therefore does not require g ^ ( − ) \\hat{g}^{(-)} to be correctly specified; the baseline enters only as a variance-reducing nuisance (cf. the working-model property of Eq. ( 4 ))."},{"type":"para","text":"The baseline is itself fit by cross-fitting on the transformed outcome Y ~ i ( θ ) = Y i − H i ⊤ θ \\tilde{Y}_{i}(\\theta)=Y_{i}-H_{i}^{\\top}\\theta , and so depends on the current value of θ \\theta . In the general case g ^ ( − ) \\hat{g}^{(-)} and θ ^ \\hat{\\theta} are thus determined jointly, by the estimation scheme of Zheng and Zhou ( 2015 ) . Under the linear working model for the baseline adopted here the joint system is linear and Eq. ( 9 ) is obtained in a single step, consistent with the one-step recommendation of Zheng and Zhou ( 2015 ) ; the concrete cross-fitted solver and this reduction are given in Algorithm 1 ."},{"type":"heading","level":2,"text":"2.4 Causal Assumptions"},{"type":"para","text":"The model rests on five assumptions, introduced below. The first two: consistency/SUTVA (Assumption 1 ) and treatment ignorability with positivity (Assumption 2 ) are standard in the causal inference literature and hold by design in a randomised trial. The third, no essential heterogeneity (Assumption 3 ), is the key identifying condition that distinguishes our approach from methods based on sequential ignorability: it permits unmeasured mediator–outcome confounding at the cost of restricting how unobserved confounders may modify treatment and mediator effects in the mean. The final two assumptions, which are of a statistical nature, are necessary for estimation. In particular, the fourth assumption, concerning the specification of the structural model (Assumption 4 ), is necessary to guarantee consistency. The fifth assumption, the non-degeneracy of the weight matrix (Assumption 5 ), ensures that the structural parameters are identifiable from the estimating equation."},{"type":"para","text":"(No interference) The potential outcomes and potential mediators of unit i i do not depend on the treatment or mediator values assigned to other units."},{"type":"para","text":"(No hidden treatment versions) There is a single version of each treatment level, so that the mapping from treatment assignment to potential outcome is well-defined."},{"type":"para","text":"(Consistency) Observed outcomes equal the relevant potential outcomes under realised treatment and mediator values: M i = M i ( T i ) M_{i}=M_{i}(T_{i}) and Y i = Y i ( T i , M i ) Y_{i}=Y_{i}(T_{i},M_{i}) ."},{"type":"para","text":"The three conditions constituting Assumption 1 arise directly from the experimental design (Morgan and Winship, 2014 ) . Their validity must be evaluated prior to the initiation of the experiment, as any violation would render both estimation and identification infeasible."},{"type":"para","text":"(Ignorability) Treatment assignment is independent of all potential outcomes and potential mediators given pre-treatment covariates:"},{"type":"para","text":"(Positivity) Every unit has a positive probability of receiving each treatment level: 0 < Pr ( T = t ∣ X ) < 1 0<\\Pr(T=t\\mid X)<1 for each t ∈ { 0 , 1 } t\\in\\{0,1\\} ."},{"type":"para","text":"In observational settings, Assumption 2 would require that all confounders of the T − M T-M and T − Y T-Y relationships are measured and included in X X . The extension of UNIT to this case is discussed in Section 5.3 . It should be noted that Assumption 2 concerns only the treatment assignment mechanism. It does not require the absence of unmeasured confounding between M M and Y Y ."},{"type":"para","text":"For all treatment and mediator levels t , m t,m ,"},{"type":"para","text":"where F F is the same function of ( M , X , T ) (M,X,T) across all counterfactual regimes ( t , m ) (t,m) ."},{"type":"para","text":"Equivalently, for any ( t 1 , m 1 ) (t_{1},m_{1}) and ( t 0 , m 0 ) (t_{0},m_{0}) ,"},{"type":"para","text":"so that differences in conditional means across regimes are driven entirely by the systematic component μ \\mu . Assumption 3 allows for rich heterogeneity through X X (including X X -specific effect modification via μ \\mu ) and arbitrary residual noise. However, it excludes unmeasured effect modification in the mean: once we condition on M , T , X M,T,X , unobserved confounders do not influence the average gain from switching t , m t,m . From an econometric perspective, this is the equivalent of the negation of “Essential Heterogeneity”, as formally defined by Heckman et al. ( 2006 ) ."},{"type":"para","text":"We emphasize that, although the NEH assumption is weaker than SI (Small, 2012 ; Zheng and Zhou, 2015 ) , it is suboptimal in settings where SI actually holds. This loss of optimality manifests as an increased variance in the resulting parameter estimates relative to estimators that target the SI semiparametric efficiency bound (e.g., Tchetgen Tchetgen and Shpitser, 2012 ). Moreover, if SI holds, nonparametric identification is possible, adding flexibility to the estimation procedure (Imai et al. , 2010 ; Pearl, 2001 ) ."},{"type":"para","text":"The basis functions { h k ( t , m , X ) } k = 1 K \\{h_{k}(t,m,X)\\}_{k=1}^{K} in the structural mean model in Eq. ( 2 ) are assumed to be correctly specified up to unknown parameters, the nuisance function g ( ⋅ ) g(\\cdot) can be misspecified."},{"type":"para","text":"This assumption is only partially testable: since potential outcomes are not jointly observed, and only the corresponding model for the observed outcome can be examined empirically."},{"type":"para","text":"Assumption 4 is necessary for consistency. If structural terms are omitted, the baseline model g ( X ) g(X) can not absorb them, introducing bias in the estimated causal parameters θ ^ \\hat{\\theta} . On the other hand, if g g is incorrectly specified, only efficiency is affected, but not consistency. This is the working-model property of G-estimation (see the centering argument in Appendix A , Eq. ( 38 ))."},{"type":"para","text":"To estimate the model, an additional assumption regarding the regularity properties of the weight matrix needs to be introduced."},{"type":"para","text":"The weight vector W ( X ) W(X) is non-degenerate such that its second moment matrix 𝔼 [ W ( X ) W ( X ) ⊤ ] \\mathbb{E}[W(X)W(X)^{\\top}] is positive definite."},{"type":"para","text":"For example the weight vector introduced for the illustrative model earlier is W ( X ) = ( 1 , τ M ( X ) ) ⊤ W(X)=(1,\\tau_{M}(X))^{\\top} . In particular, we require that Var ( τ M ( X ) ) > 0 \\operatorname{Var}(\\tau_{M}(X))>0 , implying that the treatment has a heterogeneous effect on the mediator across the covariate space. This suggests that if the treatment has no heterogeneous effects on the mediator (i.e., τ M ( X ) \\tau_{M}(X) is constant), the second row of the weight matrix is collinear with the first, then the estimating equation has no unique solution for θ 2 \\theta_{2} , and the mediation parameter is not identified. Assumption 5 is the mediation analogue of the instrumental-variable relevance condition: this condition can be assessed empirically by testing whether treatment-by-covariate interactions in the mediator model are significant (Morelli et al. , 2025 ) ."},{"type":"para","text":"In practice, τ M ( X ) \\tau_{M}(X) is unknown and must be estimated from the data. Even if the model is formally identified with Var ( τ M ( X ) ) > 0 \\operatorname{Var}(\\tau_{M}(X))>0 , the actual size of Var ( τ M ( X ) ) \\operatorname{Var}(\\tau_{M}(X)) affects the efficiency of the estimator in Stage-2; this is the task we address in Section 3 ."},{"type":"heading","level":2,"text":"2.5 The mediator CATE and the optimal weight function"},{"type":"para","text":"The G-estimating Eq. ( 4 ) holds for any weight function A ( T , X ) A(T,X) satisfying the centering condition 𝔼 [ A ( T , X ) ∣ X ] = 0 \\mathbb{E}[A(T,X)\\mid X]=0 . Different choices of A A yield consistent estimators with different asymptotic variances. Theorem 3 of Zheng and Zhou ( 2015 ) characterises the semiparametrically efficient choice: the optimal weight is"},{"type":"para","text":"where Ω ( X ) = Var ( ε ( T , M ; U , X ) ∣ X ) \\Omega(X)=\\operatorname{Var}(\\varepsilon(T,M;\\,U,X)\\mid X) . UNIT adopts the Ω − 1 ≡ 𝐈 p \\Omega^{-1}\\equiv\\mathbf{I}_{p} simplification (valid under the linear- g g working model), so the deployed weight is ( T − p ) W ∗ ( X ) (T-p)\\,W^{*}(X) ; see Appendix B.3 . For our illustrative model specification the partial derivative ∂ Y ~ / ∂ θ \\partial\\tilde{Y}/\\partial\\theta equals − H i -H_{i} with H i = ( T i , M i ) ⊤ H_{i}=(T_{i},M_{i})^{\\top} , and the difference of conditional expectations of H i H_{i} simplifies to"},{"type":"para","text":"where the first entry is trivial ( 𝔼 [ T ∣ X , T ] − 𝔼 [ T ∣ X ] = T − p \\mathbb{E}[T\\mid X,T]-\\mathbb{E}[T\\mid X]=T-p ) and the second follows from 𝔼 [ M ∣ X , T = 1 ] − 𝔼 [ M ∣ X ] = ( 1 − p ) τ M ( X ) \\mathbb{E}[M\\mid X,T=1]-\\mathbb{E}[M\\mid X]=(1-p)\\,\\tau_{M}(X) and analogously for T = 0 T=0 . The optimal weight therefore takes the multiplicative form"},{"type":"para","text":"Under Assumptions 1 and 2 , the mediator CATE is identified from observed data:"},{"type":"para","text":"Two features of Eq. ( 12 ) need additional details. First, the weight vector W ∗ ( X ) W^{*}(X) depends on X X only through the mediator CATE τ M ( X ) \\tau_{M}(X) . This means that the entire problem of constructing efficient instruments for θ \\theta reduces to the problem of estimating a single conditional average treatment effect, which is a well-studied task in the causal inference literature (Künzel et al. , 2019 ) . Second, the centering factor ( T − p ) (T-p) ensures that 𝔼 [ A ∗ ( T , X ) ∣ X ] = 0 \\mathbb{E}[A^{*}(T,X)\\mid X]=0 automatically under randomisation, regardless of the quality of τ ^ M \\hat{\\tau}_{M} . Consequently, any plug-in weight A ^ i = ( T i − p ) W ^ ( X i ) \\hat{A}_{i}=(T_{i}-p)\\hat{W}(X_{i}) preserves the consistency of θ ^ \\hat{\\theta} ; the quality of τ ^ M \\hat{\\tau}_{M} affects only efficiency, not identification."},{"type":"para","text":"Because τ M ( X ) \\tau_{M}(X) is a CATE, with the mediator M M playing the role of outcome and the randomised treatment T T as the intervention, the rich literature on CATE estimation applies directly to Stage 1 of our algorithm. Existing theoretical results on learner-specific error bounds and the function of shared representations (Curth and Van der Schaar, 2021 ) , systematic comparisons of metalearners (Künzel et al. , 2019 ) , the use of cross-fitting for nuisance-function estimation (Chernozhukov et al. , 2018 ) , as well as analyses of the dependence of learner performance on the underlying data-generating process (Curth and Van der Schaar, 2021 ) , are all directly pertinent to our setting. These considerations motivate the adoption of representation-learning architectures (Johansson et al. , 2016 ) , such as TARnet (Shalit et al. , 2017 ) ."},{"type":"para","text":"For richer structural models, additional structural components can be added. If the basis includes a treatment-by-mediator interaction h 3 = T M h_{3}=TM , the corresponding row of the optimal weight involves the arm-specific conditional mean μ 1 M ( X ) = 𝔼 [ M ∣ T = 1 , X ] \\mu_{1}^{M}(X)=\\mathbb{E}[M\\mid T{=}1,X] , not just the difference. More generally, for any basis function that is linear in M M without T × M T\\times M interaction, including covariate modulated effects such as h k = M ⋅ f ( X ) h_{k}=M\\cdot f(X) , the weight continues to depend on X X through τ M ( X ) \\tau_{M}(X) alone. In all cases where arm-specific means are required, TARNet provides them directly through its two output heads h 0 ( Φ ( X ) ) h_{0}(\\Phi(X)) and h 1 ( Φ ( X ) ) h_{1}(\\Phi(X)) , so the extension of UNIT to richer structural specifications does not require a change of Stage 1 architecture, only the construction of the weight vector W ( X ) W(X) and the basis vector H i H_{i} must be adapted accordingly."},{"type":"heading","level":1,"text":"3 Methods: Representation Learning for Weight Estimation"},{"type":"heading","level":2,"text":"3.1 From separate regressions to shared representations"},{"type":"para","text":"The most direct approach to estimating the mediator CATE is the T-learner (Künzel et al. , 2019 ) : fit two separate response surfaces μ ^ 0 ( x ) \\widehat{\\mu}_{0}(x) and μ ^ 1 ( x ) \\widehat{\\mu}_{1}(x) on the control and treated groups respectively, and define τ ^ T ( x ) = μ ^ 1 ( x ) − μ ^ 0 ( x ) \\widehat{\\tau}_{T}(x)=\\widehat{\\mu}_{1}(x)-\\widehat{\\mu}_{0}(x) . While consistent, the T-learner can exhibit larger finite-sample estimation error than methods that share information across treatment arms. Because μ ^ 0 \\widehat{\\mu}_{0} and μ ^ 1 \\widehat{\\mu}_{1} are estimated on disjoint subsets, each model is regularised independently, and when the two groups differ in size, the regularisation schedules diverge. This regularisation mismatch produces variation in τ ^ T ( x ) \\widehat{\\tau}_{T}(x) that does not correspond to treatment-effect heterogeneity (Künzel et al. , 2019 ) . More formally, the error bound for the T-learner scales as 2 { ε ( μ ^ 1 ) + ε ( μ ^ 0 ) } 2\\{\\varepsilon(\\hat{\\mu}_{1})+\\varepsilon(\\hat{\\mu}_{0})\\} , where ε ( μ ^ w ) \\varepsilon(\\hat{\\mu}_{w}) denotes the estimation error of each of the w = 0 , 1 w=0,1 arm-specific model (Curth and Van der Schaar, 2021 ) . Learners that share structure between arms can achieve tighter bounds by exploiting a common signal Curth and Van der Schaar ( 2021 ) ."},{"type":"para","text":"A more fundamental limitation is that the T-learner cannot exploit structural similarities between the two response surfaces. In many applications, the conditional expectations μ 1 ( x ) \\mu_{1}(x) and μ 0 ( x ) \\mu_{0}(x) share substantial low-dimensional structure; the treatment shifts the outcome without radically altering which features are relevant (Kennedy, 2023 ) . Separate models must therefore recover the common signal independently in each arm, even though that signal is largely shared, and under balanced 1:1 treatment allocation, each model uses at most half of the available data. These considerations motivate the use of representation learning as an inductive bias for CATE estimation. Instead of separate models, we learn a shared feature map Φ : ℝ p → ℝ r \\Phi\\colon\\mathbb{R}^{p}\\to\\mathbb{R}^{r} that compresses the covariate space into a common representation for both treatment arms. Treatment-specific heads h 0 h_{0} and h 1 h_{1} then map from this representation to the arm-specific conditional expectations. The key structural assumption, formalised by Shalit et al. ( 2017 ) , is that there exists a common feature space underlying both { μ w ( x ) } w ∈ { 0 , 1 } \\{\\mu_{w}(x)\\}_{w\\in\\{0,1\\}} , so that Φ ( X ) \\Phi(X) preserves all identifying conditional independence relationships (Curth and Van der Schaar, 2021 ; Shalit et al. , 2017 ) ."},{"type":"para","text":"By pooling all observations through the shared layers, the representation Φ \\Phi is estimated with the full sample, regardless of treatment assignment. This yields two advantages directly relevant for the G-estimation framework. First, when treatment groups are of unequal size, the shared layers stabilise estimation by pooling information across arms (variance reduction under imbalance; Curth and Van der Schaar, 2021 ) . Second, because the CATE is computed as a difference of two functions of the same representation, the smoothness and dimensionality of Φ \\Phi provide an inductive bias that may help regularise the treatment effect surface, particularly when μ 0 \\mu_{0} and μ 1 \\mu_{1} share substantial low-dimensional structure (Kennedy, 2023 ; Shalit et al. , 2017 ) ."},{"type":"heading","level":2,"text":"3.2 Direct TARNet estimation of the mediator CATE"},{"type":"para","text":"We adopt a TARNet (Shalit et al. , 2017 ) as implemented in the SNet-1 architecture of Curth and Van der Schaar ( 2021 ) . The network consists of three components:"},{"type":"para","text":"A shared representation Φ : ℝ p → ℝ r \\Phi\\colon\\mathbb{R}^{p}\\to\\mathbb{R}^{r} , parameterised as a multi-layer feed-forward network with ELU activations, mapping the raw covariate vector X X into a latent feature space."},{"type":"para","text":"Two treatment-specific heads h 0 , h 1 : ℝ r → ℝ h_{0},h_{1}\\colon\\mathbb{R}^{r}\\to\\mathbb{R} , each a smaller dense network, predicting the arm-specific conditional expectations:"},{"type":"para","text":"The mediator CATE is obtained directly as the contrast:"},{"type":"para","text":"The entire network is trained end-to-end by minimising the factual loss"},{"type":"para","text":"where ℛ ( ⋅ ) \\mathcal{R}(\\cdot) is an L 2 L_{2} -regularisation penalty on all network weights. Each observation contributes through exactly one head (corresponding to its observed treatment), while both treated and control units contribute to the shared representation Φ \\Phi ."},{"type":"heading","level":2,"text":"3.3 Cross-fitting both nuisance components"},{"type":"para","text":"To separate nuisance estimation from score evaluation, we cross-fit both nuisance components entering Stage 2: the mediator CATE τ M ( X ) \\tau_{M}(X) and the baseline outcome function g ( X ) g(X) . We model the nuisance baseline as a linear working model, g ( X ) = β ⊤ b ( X ) g(X)=\\beta^{\\top}b(X) , where b ( X ) b(X) is a fixed basis expansion that may include polynomial terms of the raw covariates. Thus g g is linear in its coefficients even when it is polynomial in the original variables, avoiding introducing an additional nonlinear nuisance layer in Stage 2."},{"type":"para","text":"In principle, g ( x ) g(x) can be nonlinear and specified more flexibly, for example with a second neural network. The main consequences would be two. First, the baseline nuisance is fit to the transformed outcome"},{"type":"para","text":"so the target of the baseline regression changes whenever θ \\theta is updated. This implies that it becomes necessary to adopt a general estimation strategy: alternating update of g g and θ \\theta (Zheng and Zhou, 2015 ) . The second consequence is that it becomes necessary to estimate the Ω − 1 ( X ) \\Omega^{-1}(X) component. Under our linear in-parameters specification both complications collapse, the update of θ \\theta reduces to a one-step G-estimation equation, and Ω − 1 ( X ) \\Omega^{-1}(X) need not be estimated explicitly. However, we present the alternating algorithm, as it is the more general method and remains applicable when g ( x ) g(x) is extended beyond the linear working model b ( x ) b(x) . A more detailed discussion of these aspects is provided in B.3 . Let { I k } k = 1 K \\{I_{k}\\}_{k=1}^{K} be a partition of the sample into K K folds, and let I k c I_{k}^{c} denote the complementary training set."},{"type":"heading","level":4,"text":"Stage 1 nuisance: mediator CATE."},{"type":"para","text":"For each fold k = 1 , … , K k=1,\\dots,K , we train a TARNet on I k c I_{k}^{c} by minimising the factual loss in Eq. ( 15 ), obtaining fold-specific estimators Φ ( − k ) , h 0 ( − k ) , h 1 ( − k ) \\Phi^{(-k)},h_{0}^{(-k)},h_{1}^{(-k)} . For every unit i ∈ I k i\\in I_{k} , we compute the out-of-fold mediator CATE"},{"type":"para","text":"Stacking across folds yields cross-fitted estimates τ ^ M ( − ) ( X i ) \\hat{\\tau}_{M}^{(-)}(X_{i}) for all i i ."},{"type":"heading","level":4,"text":"Stage 2 nuisance: baseline outcome model."},{"type":"para","text":"The baseline function is estimated foldwise inside Stage 2 as well. Given a current iterate θ ( t ) \\theta^{(t)} , define the transformed outcome Y ~ i ( θ ( t ) ) = Y i − H i ⊤ θ ( t ) \\tilde{Y}_{i}(\\theta^{(t)})=Y_{i}-H_{i}^{\\top}\\theta^{(t)} with H i = ( T i , M i ) ⊤ H_{i}=(T_{i},M_{i})^{\\top} . For each fold k k , fit the baseline learner on I k c I_{k}^{c} using { Y ~ i ( θ ( t ) ) , X i : i ∈ I k c } \\{\\tilde{Y}_{i}(\\theta^{(t)}),X_{i}:i\\in I_{k}^{c}\\} , and denote the resulting estimator by g ^ θ ( t ) ( − k ) \\hat{g}_{\\theta^{(t)}}^{(-k)} . For each held-out observation i ∈ I k i\\in I_{k} , define the out-of-fold prediction g ^ θ ( t ) ( − ) ( X i ) = g ^ θ ( t ) ( − k ) ( X i ) \\hat{g}_{\\theta^{(t)}}^{(-)}(X_{i})=\\hat{g}_{\\theta^{(t)}}^{(-k)}(X_{i}) ."},{"type":"para","text":"Cross-fitting is relevant for the asymptotic argument in Appendix B . In particular, when the same sample is used both to estimate g g and to evaluate the score, the empirical process remainder from plugging in g ^ \\hat{g} requires additional regularity conditions (e.g., Donsker-class membership; Chernozhukov et al. , 2018 ) that may not hold for flexible learners."},{"type":"para","text":"Cross-fitting the baseline removes this dependence: conditional on the training folds, the held-out score contributions are mean-zero, and the baseline remainder becomes of order ‖ g ^ θ ( − ) − g 0 , θ ‖ L 2 ( P ) \\|\\hat{g}_{\\theta}^{(-)}-g_{0,\\theta}\\|_{L_{2}(P)} , which is negligible under consistency (Chernozhukov et al. , 2018 ) ."},{"type":"heading","level":2,"text":"3.4 G-estimation weights"},{"type":"para","text":"We now explicitly specify how the CATE on the mediator output of the TARNet model is incorporated into the G-estimation equation. Substituting the cross-fitted estimates into Eq. ( 12 ), the plug-in weight vector for unit i i is"},{"type":"para","text":"and the empirical optimal weight becomes"},{"type":"para","text":"where p p is the known allocation probability (or T ¯ \\bar{T} if estimated)."},{"type":"para","text":"The representation Φ \\Phi determines the quality of τ ^ M ( X ) \\hat{\\tau}_{M}(X) and thereby the informativeness of the weight matrix A ^ \\hat{A} . If Φ \\Phi captures all relevant heterogeneity in the T → M T\\to M effect, the resulting weights provide a good plug-in approximation to the heterogeneity component of the semiparametrically efficient instrument (cf. Theorem 3 in Zheng and Zhou, 2015 ) , thereby improving the efficiency of θ ^ \\hat{\\theta} . Conversely, if Φ \\Phi is chosen too coarsely and discards features that generate true effect heterogeneity, the weights lose informative signal and θ ^ \\hat{\\theta} becomes less efficient, although it still remains consistent under the NEH assumption."},{"type":"heading","level":4,"text":"Inference."},{"type":"para","text":"Under the regularity conditions stated in Appendix B , including an n − 1 / 4 n^{-1/4} rate for the cross-fitted mediator CATE and consistency of the cross-fitted baseline, the G-estimator θ ^ \\hat{\\theta} is asymptotically normal with a sandwich-type variance (see Proposition 2 and Appendix A for additional details and derivations). A consistent estimator of the asymptotic variance is"},{"type":"para","text":"where G ^ θ = n − 1 ∑ i A ^ i H i ⊤ \\hat{G}_{\\theta}=n^{-1}\\sum_{i}\\hat{A}_{i}H_{i}^{\\top} and S ^ = n − 1 ∑ i ψ ^ i 2 A ^ i A ^ i ⊤ \\hat{S}=n^{-1}\\sum_{i}\\hat{\\psi}_{i}^{2}\\,\\hat{A}_{i}\\hat{A}_{i}^{\\top} with out-of-fold residuals ψ ^ i = Y i − g ^ θ ^ ( − ) ( X i ) − H i ⊤ θ ^ \\hat{\\psi}_{i}=Y_{i}-\\hat{g}_{\\hat{\\theta}}^{(-)}(X_{i})-H_{i}^{\\top}\\hat{\\theta} ."},{"type":"heading","level":4,"text":"Quality of the estimator."},{"type":"para","text":"It is useful to separate three distinct roles that govern the quality of the final estimator:"},{"type":"para","text":"Identification of θ \\theta requires treatment ignorability (Assumption 2 ), the NEH assumption (Assumption 3 ), correct structural specification (Assumption 4 ), and non-degeneracy of the weight matrix (Assumption 5 ). The centering condition 𝔼 [ ( T − p ) ∣ X ] = 0 \\mathbb{E}[(T-p)\\mid X]=0 follows from Assumption 2 under randomisation."},{"type":"para","text":"Estimation quality of τ M ( X ) \\tau_{M}(X) determines how closely the plug-in weight W ^ ( X ) \\hat{W}(X) approximates the oracle optimal weight W ∗ ( X ) W^{*}(X) . This is governed by the CATE estimation error, which depends on the learner architecture, the sample size, and the complexity of the true τ M \\tau_{M} (Curth and Van der Schaar, 2021 ) ."},{"type":"para","text":"Instrument strength determines how much precision the weight matrix delivers, even at the oracle. The centred weights ( T − p ) W ( X ) (T-p)W(X) function as instruments for the structural parameters. If the underlying DGP has weak treatment-by-covariate interaction on the mediator, i.e., Var ( τ M ( X ) ) \\operatorname{Var}(\\tau_{M}(X)) is small, the instruments are weak, and θ ^ 2 \\hat{\\theta}_{2} will be imprecise regardless of how well τ ^ M \\hat{\\tau}_{M} approximates the truth."},{"type":"heading","level":1,"text":"4 Simulation Study"},{"type":"para","text":"We construct a DGP that mimics typical features of modern and challenging psychological and biomedical data (Micceri, 1989 ; Blanca et al. , 2013 ; Bono et al. , 2017 ) : non-Gaussian covariates, complex nonlinear effects, and unmeasured mediator–outcome confounding (Rudolph et al. , 2019 ) . We observe i.i.d. tuples ( X i , T i , M i , Y i ) (X_{i},T_{i},M_{i},Y_{i}) , i = 1 , … , n i=1,\\dots,n : covariates X ∈ ℝ 25 X\\in\\mathbb{R}^{25} , a randomized binary treatment T ∈ { 0 , 1 } T\\in\\{0,1\\} , a continuous mediator M M , and a continuous outcome Y Y . Each of the 7 scenarios that we study is derived by modifying one characteristic of the following base model:"},{"type":"para","text":"Where τ ( X ) \\tau(X) is the heterogeneous effect of T T on M M (the mediator CATE, target of Stage 1); μ 0 ( X ) \\mu_{0}(X) a nonlinear baseline mediator level; K K an unmeasured confounder entering both structural equations (loading λ K \\lambda_{K} ); g ( X ) = ∑ j = 0 6 [ 0.4 X j + 0.2 X j 2 𝟏 { j even } ] g(X)=\\sum_{j=0}^{6}[0.4X_{j}+0.2X_{j}^{2}\\mathbf{1}\\{j\\text{ even}\\}] a nonlinear nuisance. Parameters θ 1 = 1 \\theta_{1}=1 (direct effect), θ 2 = 0.5 \\theta_{2}=0.5 (the mediation coefficient of interest), λ K = 1 \\lambda_{K}=1 . Treatment is randomized, T ⟂ ⟂ X T\\perp\\!\\!\\!\\perp X , with known e ( X ) ≡ Pr ( T = 1 ∣ X ) = 0.5 e(X)\\equiv\\Pr(T{=}1\\mid X)=0.5 . The total effect of T T on Y Y decomposes as TE ( x ) = θ 1 + θ 2 τ ( x ) \\mathrm{TE}(x)=\\theta_{1}+\\theta_{2}\\,\\tau(x) . Below, we introduce the base generating process for each component, the specific modifications will be made explicit in each of the respective scenarios."},{"type":"heading","level":4,"text":"Covariates."},{"type":"para","text":"Each X ∈ ℝ 25 X\\in\\mathbb{R}^{25} is generated via a Gaussian copula with Gaussian mixture model (GMM) marginals. In particular we draw Z i ∼ 𝒩 25 ( 0 , Σ ) Z_{i}\\sim\\mathcal{N}_{25}(0,\\Sigma) with Σ j l = ρ | j − l | \\Sigma_{jl}=\\rho^{|j-l|} ( ρ = 0.5 \\rho=0.5 ), transform to uniform margins U i j = Φ ( Z i j ) U_{ij}=\\Phi(Z_{ij}) , and apply the inverse GMM CDF componentwise to obtain X i j = F j − 1 ( U i j ) X_{ij}=F_{j}^{-1}(U_{ij}) . Each marginal F j F_{j} is a mixture of n j ∈ { 5 , … , 10 } n_{j}\\in\\{5,\\ldots,10\\} Gaussian components (independently configured per feature, and number of components sampled uniformly), with component means drawn from Uniform ( − 2 , 2 ) \\mathrm{Uniform}(-2,2) , standard deviations from Uniform ( 0.5 , 1.5 ) \\mathrm{Uniform}(0.5,1.5) , and weights proportional to Uniform ( 0.5 , 2 ) \\mathrm{Uniform}(0.5,2) draws; the resulting distribution is min-max scaled to the interval [ 0 , 5 ] [0,5] . This yields non-Gaussian, skewed, and multimodal marginals linked by moderate AR(1) copula dependence."},{"type":"para","text":"The 25 features are partitioned into four blocks following the benchmark design of Curth and Van der Schaar ( 2021 ) as depicted in Table 1 :"},{"type":"heading","level":2,"text":"4.1 Latent-score construction"},{"type":"para","text":"The treatment effect and baseline outcome are built from three fixed linear projections of the standardized covariates: z = ( X − X ¯ ) / s z=(X-\\bar{X}_{\\mathrm{}})/s_{\\mathrm{}} :"},{"type":"para","text":"The scores s 1 s_{1} and s 2 s_{2} are shared entering both τ ( X ) \\tau(X) and μ 0 ( X ) \\mu_{0}(X) and provide the joint latent structure. The treatment-effect function and baseline outcome are:"},{"type":"para","text":"where σ ( ⋅ ) \\sigma(\\cdot) is the logistic function and softplus ( x ) = log ( 1 + e x ) \\mathrm{softplus}(x)=\\log(1+e^{x}) ."},{"type":"heading","level":4,"text":"Unmeasured confounder."},{"type":"para","text":"The unmeasured confounder shares the same distributional family as the observed confounders. In particular we construct the unmeasured confounder K K as a GMM draw mixed with Gaussian noise:"},{"type":"para","text":"where K gmm K_{\\mathrm{gmm}} is drawn from a three-component Gaussian mixture with means ( − 1.2 , 0.3 , 1.5 ) (-1.2,\\,0.3,\\,1.5) , standard deviations ( 0.8 , 0.6 , 0.9 ) (0.8,\\,0.6,\\,0.9) , and mixture weights ( 0.40 , 0.35 , 0.25 ) (0.40,\\,0.35,\\,0.25) ."},{"type":"heading","level":4,"text":"Variance calibration and scenarios."},{"type":"para","text":"The components ( τ ⋅ T , μ 0 , K , ε M ) (\\tau\\!\\cdot\\!T,\\,\\mu_{0},\\,K,\\,\\varepsilon_{M}) are independently rescaled to achieve target variance shares."},{"type":"heading","level":2,"text":"4.2 The seven scenarios"},{"type":"para","text":"We design the simulation study to investigate two orthogonal dimensions: identification and efficiency. Specifically, we examine the efficiency of the methods by varying the degree of nonlinearity and the magnitude of τ \\tau , while holding the identification structure fixed; this is implemented in scenarios A, B, E, and Wtau. Conversely, we assess the impact of identification by varying the identification conditions while keeping the baseline surface characteristics fixed in scenarios C and D."},{"type":"para","text":"Throughout, ε M , i ∼ 𝒩 ( 0 , 1 ) \\varepsilon_{M,i}\\sim\\mathcal{N}(0,1) , ε Y , i ∼ 𝒩 ( 0 , σ ε Y 2 ) \\varepsilon_{Y,i}\\sim\\mathcal{N}(0,\\sigma_{\\varepsilon_{Y}}^{2}) , K ¯ i = a K K i \\bar{K}_{i}=a_{K}K_{i} , and ( a τ , a D , a K , a ε ) (a_{\\tau},a_{D},a_{K},a_{\\varepsilon}) are the calibration scales for the cell’s shares."},{"type":"heading","level":4,"text":"A: realistic heterogeneity ( τ \\tau -share 25 % 25\\% )."},{"type":"para","text":"This is the reference scenario NEH holds; predicted bias 0 ."},{"type":"heading","level":4,"text":"B: strong heterogeneity ( τ \\tau -share 40 % 40\\% )."},{"type":"para","text":"This case has exactly the same structure as in Eq. ( 27 ), but assigns a much larger weight to the τ \\tau component, namely ( 0.40 , 0.28 , 0.18 , 0.14 ) (0.40,0.28,0.18,0.14) , thus corresponding to an “ideal instrument” setting. Under this specification, NEH is satisfied and the predicted bias equals 0 ."},{"type":"heading","level":4,"text":"C: rank preservation fails, NEH holds."},{"type":"para","text":"The mediator is specified as in Eq. ( 27 ). The outcome slope with respect to M M becomes idiosyncratic through the inclusion of the term ξ i \\xi_{i} :"},{"type":"para","text":"The individual rank preservation assumption fails, but NEH holds. Predicted bias 0 ."},{"type":"heading","level":4,"text":"D: essential heterogeneity, NEH violated ( γ = κ = 0.4 \\gamma=\\kappa=0.4 )."},{"type":"para","text":"A latent L i ∼ 𝒩 ( 0 , 1 ) L_{i}\\sim\\mathcal{N}(0,1) , independent of ( X , T , K , ε M , ε Y ) (X,T,K,\\varepsilon_{M},\\varepsilon_{Y}) , enters both equations:"},{"type":"para","text":"The same L L modulates the mediator’s treatment response ( κ L \\kappa L ) and the outcome’s mediator slope ( γ L \\gamma L ). NEH violated; predicted bias γ κ = 0.16 \\gamma\\kappa=0.16 ."},{"type":"heading","level":4,"text":"Dsmall: half amplitude ( γ = κ = 0.2 \\gamma=\\kappa=0.2 )."},{"type":"para","text":"Eq. ( 29 ) with γ = κ = 0.2 \\gamma=\\kappa=0.2 , predicted bias 0.04 0.04 ."},{"type":"heading","level":4,"text":"E: CATE (mostly) linear, NEH holds."},{"type":"para","text":"The response surface is replaced, by an orthogonalized combination of linear and nonlinear components. τ E \\tau_{E} ; μ 0 , K \\mu_{0},K , shares, noise are those of scenario A:"},{"type":"para","text":"z nl z_{\\mathrm{nl}} is built as in Eq. ( 24 ), β = Cov ( z nl , z lin ) / Var ( z lin ) \\beta=\\operatorname{Cov}(z_{\\mathrm{nl}},z_{\\mathrm{lin}})/\\operatorname{Var}(z_{\\mathrm{lin}}) and the reference SDs computed once on a large population draw (DGP identical across n n ), ( w nl , w lin ) = ( 0.30 , 0.90 ) (w_{\\mathrm{nl}},w_{\\mathrm{lin}})=(0.30,0.90) . Realized full-linear R 2 R^{2} of τ E \\tau_{E} averages 0.89 0.89 (vs 0.19 0.19 in scenario A)."},{"type":"heading","level":4,"text":"Wtau: weak instrument, NEH holds."},{"type":"para","text":"Eq. ( 27 ), shares ( 0.08 , 0.50 , 0.20 , 0.22 ) (0.08,0.50,0.20,0.22) : the τ \\tau -share is 8 % 8\\% , It corresponds to a weak-instrument regime, translating in a weak signal for Stage-1. NEH holds; predicted bias 0 ."},{"type":"heading","level":2,"text":"4.3 Stage 1 methods"},{"type":"para","text":"All Stage 1 methods use 5-fold cross-fitting; implementation details and hyperparameters are given in the Estimator Hyperparameters appendix C . We compare 1 1 1 Thin Plate Splines were considered as a candidate middle-tier learner, motivated by their use in semiparametric mediator modeling (Brandt, 2020 ) . However, in preliminary runs at d = 25 d=25 TPS was numerically unstable despite regularisation, so we excluded it and do not report TPS results Kalogridis ( 2026 ) :"},{"type":"para","text":"TARNet : SNet-1 architecture (Curth and Van der Schaar, 2021 ) with shared representation layers and treatment-specific heads. The CATE is computed directly as τ ^ ( X ) = h 1 ( Φ ( X ) ) − h 0 ( Φ ( X ) ) \\hat{\\tau}(X)=h_{1}(\\Phi(X))-h_{0}(\\Phi(X)) ."},{"type":"para","text":"TNet (no shared representation) : the same SNet-1 architecture as TARNet (Curth and Van der Schaar, 2021 ) but without the shared representation layers, so each treatment arm is fitted separately. The CATE is computed as τ ^ ( X ) = h 1 ( X ) − h 0 ( X ) \\hat{\\tau}(X)=h_{1}(X)-h_{0}(X) . This learner serves as the ablation that isolates the value of the shared representation."},{"type":"para","text":"Ridge (T-learner) : Separate Ridge regressions for T = 0 T=0 and T = 1 T=1 , with τ ^ ( X ) = μ ^ 1 ( X ) − μ ^ 0 ( X ) \\hat{\\tau}(X)=\\hat{\\mu}_{1}(X)-\\hat{\\mu}_{0}(X) . Near-optimal when τ \\tau is linear but unable to capture interactions or nonlinearities."},{"type":"para","text":"Random Forest (T-learner) Breiman ( 2001 ) : Separate random forests for T = 0 T=0 and T = 1 T=1 . Hyperparameters (number of estimators, maximum depth, maximum features fraction) are selected by 3-fold cross-validation with 6 random configurations per fold. Captures nonlinearity and feature interactions via tree splits but lacks TARNet’s shared-representation inductive bias."},{"type":"para","text":"Baron–Kenny estimator. The naive baseline is the Baron and Kenny ( 1986 ) two-regression estimator. Because it omits the unmeasured confounder K K from the outcome regression, it is biased for θ 2 \\theta_{2} by a positive amount that does not vanish with n n . Empirically (see Table 5 ), this bias is + 0.27 +0.27 to + 0.39 +0.39 on the true θ 2 = 0.5 \\theta_{2}=0.5 , with 0 % 0\\% coverage in every cell, irrespective of sample size or signal strength."},{"type":"heading","level":2,"text":"4.4 Monte-Carlo design and evaluation metrics"},{"type":"para","text":"Each scenario is replicated over R = 200 R=200 random seeds, for sample sizes n ∈ { 500 , 1000 , 2000 , 5000 , 10000 } n\\in\\{500,\\,1000,\\,2000,\\,5000,\\,10000\\} . The low-sample-size settings n ∈ { 500 , 1000 } n\\in\\{500,\\,1000\\} are included to assess the behaviour of the methods in small-sample regimes. 2 2 2 However, for applied analyses, we recommend avoiding sample sizes below 1000. The structural parameters θ 1 = 1.0 \\theta_{1}=1.0 and θ 2 = 0.5 \\theta_{2}=0.5 are fixed across replications. Each replication produces a single point estimate θ ^ 2 ( r ) \\hat{\\theta}_{2}^{(r)} , the sandwich estimator SE r \\mathrm{SE}_{r} from Eq. ( 18 ), and a confidence interval CI r = θ ^ 2 ( r ) ± 1.96 SE r \\mathrm{CI}_{r}=\\hat{\\theta}_{2}^{(r)}\\pm 1.96\\,\\mathrm{SE}_{r} . The reported per-cell, per-method metrics are depicted in Table 3 :"},{"type":"para","text":"A replication’s Stage 2 is skipped if the Stage-1 NRMSE > 1.3 >1.3 , or sd ( τ ^ ) < 10 − 4 \\mathrm{sd}(\\hat{\\tau})<10^{-4} . A replication is likewise excluded if Stage 2 does not converge, any θ ^ \\hat{\\theta} or SE is non-finite. Table 4 reports skip counts at n = 500 n=500 ; at n ≥ 1000 n\\geq 1000 skips fall below 2 % 2\\% except for Wtau, and are zero at n ≥ 5000 n\\geq 5000 ."},{"type":"heading","level":2,"text":"4.5 Results"},{"type":"heading","level":4,"text":"Stage-1 CATE quality."},{"type":"para","text":"Stage-1 quality is measured as the learner’s normalized RMSE for τ ^ \\hat{\\tau} (see Figure 2 ). In the nonlinear surface scenarios (A, B, C, D, Dsmall), TARNet’s NRMSE decreases as n n grows, whereas Ridge levels off above 0.77 0.77 (reflecting the irreducible error from linear misspecification). In scenario E (linear surface), Ridge’s NRMSE falls to 0.36 0.36 , while RF, which lacks a linear inductive bias, flattens at 0.61 0.61 . In the Wtau (weak instrument) scenario, the NRMSE increases for all learners, with TARNet exceeding the 1.3 1.3 threshold at n = 500 n{=}500 ."},{"type":"para","text":"The correlation Corr ( τ ^ , τ ) \\operatorname{Corr}(\\hat{\\tau},\\tau) (Figure 3 ) between true and estimated tau provides additional information about the performance of the learners. Across all scenarios from A through Dsmall, a very similar pattern can be observed. TARNet yields the highest correlations across all sample sizes n n , with correlations close to or above 0.8 0.8 for n ≥ 1000 n\\geq 1000 . For n = 500 n=500 , all learners perform similarly in these scenarios, with relatively low correlations ranging between 0.4 0.4 and 0.6 0.6 ."},{"type":"para","text":"In scenario E, corresponding to the linear setting, both TARNet and Ridge achieve similarly high correlations above 0.9 0.9 for n ≥ 1000 n\\geq 1000 . In scenario Wtau, corresponding to the weak-instrument setting, a similar ranking of the learners is observed, although correlations are generally lower. In this case, TARNet reaches correlations close to 0.8 0.8 only for n ≥ 5000 n\\geq 5000 ."},{"type":"heading","level":4,"text":"Stage-2 estimation."},{"type":"para","text":"Table 5 summarizes the main patterns. In the settings where NEH holds (A, B, C, E, Wtau), all learners are unbiased (albeit the presence of a minor empirical bias is to be expected) and achieve nominal coverage for n ≥ 1000 n\\geq 1000 , whereas BK exhibits a positive bias of + 0.27 +0.27 to + 0.33 +0.33 and 0 % 0\\% coverage at all sample sizes. When NEH is violated (D, Dsmall), each learner converges to the expected γ κ \\gamma\\kappa bias, and coverage collapses as n n increases. This breakdown is most rapid for TARNet, whose narrower intervals concentrate around the wrong parameter. The efficiency ratios indicate that TARNet achieves the smallest SE in every nonlinear design (R/T, RF/T ≈ 1.4 \\approx 1.4 – 1.6 1.6 for n ≥ 1000 n\\geq 1000 ) and remains at roughly ≈ 1.0 \\approx 1.0 on the linear setting E. The TT/T column isolates the penalty of removing the shared representation: TNet lags TARNet by 30 30 – 40 % 40\\% in SE at moderate n n ( 1.3 1.3 – 1.4 1.4 at n = 1000 n{=}1000 – 2000 2000 in A, B, C, Dsmall), but this difference shrinks to about ≈ 1.0 \\approx 1.0 by n = 10 , 000 n{=}10{,}000 , indicating a finite-sample benefit of the shared representation rather than an asymptotic one."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"heading","level":2,"text":"5.1 Summary of findings"},{"type":"para","text":"Our results trace a two-stage efficiency chain,"},{"type":"para","text":"Across the nonlinear scenarios the TARNet weight delivers Stage 2 standard errors about 1.5 × 1.5\\times smaller than a linear T-learner (median Ridge/TARNet ratio 1.45 1.45 – 1.51 1.51 at n ≥ 2 , 000 n\\geq 2{,}000 )."},{"type":"para","text":"Identification behaves exactly as the NEH theory predicts, and independently of the Stage-1 learner. Where NEH holds, every G-estimator is approximately unbiased with near-nominal coverage; where it fails (Scenario D) all learners converge to the same predicted γ κ \\gamma\\kappa offset, and coverage degrades predictably as the sample grows and the interval tightens around the biased target. Scenario C confirms that the method requires only NEH and not rank preservation: its results are not significantly different from the baseline scenario. The naive Baron–Kenny estimator, carries a persistent bias of + 0.27 +0.27 to + 0.39 +0.39 on θ 2 = 0.5 \\theta_{2}=0.5 with 0 % 0\\% coverage in every cell, regardless of sample size or signal strength."},{"type":"heading","level":2,"text":"5.2 Shared representation versus separate arms"},{"type":"para","text":"Given the design of the simulation we can separate the two distinct mechanisms that drive the efficiency gains."},{"type":"para","text":"First, flexibility by itself is insufficient. Although the Random Forest is fully nonparametric, it only slightly outperforms Ridge, with Corr ( τ ^ M , τ M ) ≈ 0.65 \\operatorname{Corr}(\\hat{\\tau}_{M},\\tau_{M})\\approx 0.65 – 0.70 0.70 even at n = 10 , 000 n=10{,}000 , whereas the neural learners reach 0.84 0.84 – 0.96 0.96 . The core issue is structural: the mediator CATE is defined via smooth functions of dense linear projections ( s 1 , s 2 s_{1},s_{2} ) of the covariates, and an axis-aligned splitting scheme cannot capture these projections. This also provides a mechanistic explanation for how a highly flexible tree ensemble can fail to beat a linear model when the underlying structure is governed by latent linear relationships."},{"type":"para","text":"Second, the shared representation contributes beyond the neural function class. TNet uses the same neural architecture as TARNet but fits each treatment arm separately. Although it learns the latent projections, because each arm only sees the units in its own treatment group it approaches the shared TARNet only asymptotically. At n = 1 , 000 n=1{,}000 it lags substantially ( Corr ≈ 0.46 \\operatorname{Corr}\\approx 0.46 vs 0.76 0.76 in Scenario A), but the gap closes by n = 10 , 000 n=10{,}000 ( 0.88 0.88 vs 0.94 0.94 )."},{"type":"heading","level":2,"text":"5.3 Limitations and Future Research"},{"type":"para","text":"While the UNIT framework provides a robust integration of deep representation learning with G-estimation, it is important to acknowledge some limitations and to highlight promising directions for future research."},{"type":"heading","level":4,"text":"Estimation of the Nuisance Baseline"},{"type":"para","text":"In our current implementation, the baseline outcome model g ( X ) g(X) is estimated via 5-fold cross-fitted Ridge regression within Stage 2, as prescribed in Section 2.3 . For the linear specifications of g g employed in our simulations, the numerical discrepancy between cross-fitted and in-sample estimates was negligible. Consequently, the DML correction Chernozhukov et al. ( 2018 ) provided primarily theoretical asymptotic safeguards rather than substantial finite-sample gains in these specific settings. Future research should evaluate the performance of UNIT in contexts where g ( X ) g(X) is highly non-linear, a detailed discussion of the complications that arise under nonlinear g g is given in Section B.3 ."},{"type":"heading","level":4,"text":"Extension to Observational Data"},{"type":"para","text":"The methodology developed in this work is specifically designed for contexts in which the treatment assignment mechanism is randomized (for example, randomized clinical trials or A/B testing environments). Extending this framework to observational study designs would necessitate the simultaneous estimation of the propensity score e ( X ) e(X) and its explicit incorporation into the construction of the weight function A ^ i \\hat{A}_{i} ."},{"type":"heading","level":4,"text":"Empirical Benchmarking"},{"type":"para","text":"In contrast to standard CATE estimation, where performance can be assessed using widely adopted benchmarks such as IHDP (Hill, 2011 ) or ACIC (Hahn et al. , 2019 ) , to the best of our knowledge, no public benchmark datasets are currently available for the mediation setting considered here. Developing and analyzing experimental datasets to further evaluate the empirical performance of UNIT and competing methods, therefore, constitutes an important direction for future research."},{"type":"heading","level":2,"text":"5.4 Conclusion"},{"type":"para","text":"We introduced UNIT, a two-stage estimator that pairs a TARNet-learned mediator CATE with the Zheng and Zhou ( 2015 ) G-estimation of a structural mean model under no essential heterogeneity. Identification of the structural parameters is inherited from the NEH framework and holds regardless of the Stage-1 learner; our contribution is to show, both theoretically and in simulation, that a more accurate first-stage representation yields a more informative plug-in weight and thereby a more efficient structural estimator, in high-dimensional and non linear settings. Across nonlinear mediator-effect surfaces, the shared-representation learner reduces the Stage-2 standard error of the mediation coefficient by roughly one third relative to a linear T-learner, while maintaining low bias and close to nomimal coverage rates as long as NEH assumption holds. For applied mediation analyses with randomised treatment and complex covariate–mediator relationships, UNIT offers a substantive efficiency gain compared to other learners (eg. Ridge, RF or TNet) within a semiparametrically principled framework."},{"type":"heading","level":1,"text":"Data and Code Availability"},{"type":"para","text":"All code and frozen simulation data needed to reproduce every table and figure in this paper are available at https://github.com/PsychometricsMZ/UNIT . Each cell aggregates R = 200 R=200 Monte-Carlo seeds; the seed scheme and all hyperparameters are documented in the Appendix. The software stack is Python (NumPy, scikit-learn, and JAX/CATENets for the TARNet learner)."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Keywords: causal mediation, G-estimation, no essential heterogeneity, representation learning, structural mean models, TARNet.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The primary goal of many randomized experiments is to estimate the effect of a treatment on an outcome (Hernán and Robins, 2020 ) . In psychology, economics, and biology, researchers are often interested not only in estimating overall treatment effects, but also in studying the underlying causal mechanisms and understanding how an intervention exerts its effects (VanderWeele, 2015 ) . Such an investigation is especially relevant when the objective is, for instance, to evaluate a scientific theory or to guide and inform the design of future interventions. A frequently studied setting (Rijnhart et al. , 2021 ; Holland, 1988 ) involves a mediator M M , an intermediate variable that is influenced by the treatment and, in turn, affects the outcome, potentially together with a set of baseline covariates. Randomization of the treatment T T removes confounding between the treatment and the outcome. However, for mediator variables, the situation becomes more complex. Randomization of the treatment does not, in general, imply randomization of the mediator, since the mediator may be influenced by processes that occur after the initial treatment assignment (Bullock et al. , 2010 ) . The consequence is that the causal effect of the mediator on the outcome can be confounded. In this context, several technical approaches are available and, in general, this does not represent a major methodological concern (Wodtke and Zhou, 2026 ) . A more serious problem arises when some of these confounding variables are not measured (Ten Have and Joffe, 2012 ; Imai et al. , 2010 ) . This is a plausible scenario when, for example, the trial was originally designed to estimate only the overall treatment effect and was not designed to support a causal mediation analysis. In other circumstances, the confounders can be potentially known but unethical, prohibitively expensive, or difficult to measure (Brandt, 2020 ) . As a result, researchers could have extensive knowledge about the causal mechanism and a rich set of covariates but still lack confounders needed for the identification of the indirect effects. For example, an online retail platform may be interested in examining the impact of a banner on customers’ spending, with this relationship potentially mediated by user engagement. The platform could collect customer-level data, including individual preferences and demographic information, yet still lacks covariates necessary for decomposition of the total causal effect. These unobserved variables may include sensitive attributes or information that is difficult to collect, such as the presence of other individuals in the room while the customer is interacting with the platform.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In such cases, the classical assumption of Sequential Ignorability (SI) (Imai et al. , 2010 ) , which requires the absence of unmeasured mediator–outcome confounding conditional on observed covariates, is not satisfied. Methods that rely on the sequential ignorability assumption, then, would deliver inconsistent estimates (Imai et al. , 2010 ; Ten Have et al. , 2007 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An alternative framework that avoids relying on the SI assumption is offered by Structural Nested Models (SNMs) together with the corresponding G-estimation approach introduced by Robins ( 1994 ) . This approach offers a framework for estimating causal effects in settings where confounders may themselves be influenced by the treatment, as in the case of confounders measured post-treatment. SNMs have considerable advantages over other methods for handling time-varying and unmeasured confounding (Loh and Ren, 2025 ) : they can accommodate flexible structural specification, have doubly robust properties, and can exploit the instrumental variable approach for identification (Vansteelandt and Joffe, 2014 ) . Despite these strengths, the application of G-estimation in empirical research has been relatively infrequent, historically generally due to the complexity of implementation and the limited availability of \\\"off-the-shelf\\\" software (Vansteelandt and Joffe, 2014 ) . In the causal mediation context, Ten Have et al. ( 2007 ) applied G-estimation under a rank-preserving assumption. This assumption requires that the idiosyncratic error does not depend on the counterfactual regime at the individual level, meaning that each person’s deviation from the structural model is identical regardless of what treatment and mediator values are hypothetically assigned.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While this allows the existence of unmeasured confounders in the mediator–outcome mechanism, it rules out individual-level heterogeneity in treatment or mediator effects through unobserved confounders. In many settings this assumption could be difficult to satisfy and often it is an excessively strong constraint:it implies, for instance, that a person whose outcome deviates unusually from the structural prediction under one treatment–mediator regime must deviate by exactly the same amount under every other regime. Indeed, not even non-confounding, idiosyncratic effect heterogeneity is admitted under rank preservation. In many behavioural and biomedical settings, such rank-preservation is implausible and has been criticised accordingly (Hernán and Robins, 2020 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To overcome this limitation, Zheng and Zhou ( 2015 ) propose a more general structural mean model in which the Ten Have et al. ( 2007 ) formulation appears as a special case. They alleviate rank preservation by replacing it with a weaker mean-level condition closely related to the econometric “No Essential Heterogeneity” (NEH) assumption, building on the marginal treatment effect framework of Heckman and Vytlacil ( 2005 ) and the definition of essential heterogeneity introduced in Heckman et al. ( 2006 ) . This permits arbitrarily rich effect modification through observed covariates, and even idiosyncratic, non-confounding individual heterogeneity, while still accommodating unmeasured confounders that shift both mediator and outcome.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Additionally, the semiparametric structure of the Zheng and Zhou ( 2015 ) model makes it especially suitable for applied settings in which researchers are not merely interested in the magnitude of a treatment effect but also seek to parametrically characterize the causal mechanism through which it operates. A practical challenge in implementing the Zheng and Zhou ( 2015 ) framework is the estimation of the optimal weight function that enters the G-estimating equation. Zheng and Zhou ( 2015 ) employed parametric formulations, Brandt ( 2020 ) adopted a Thin Plate Spline (Wood, 2003 ) to introduce flexibility, and Morelli et al. ( 2025 ) broadened the original framework to incorporate latent variables.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, when covariate-mediator relationships are nonlinear or the covariate space is high-dimensional, the available tools for weight estimation are limited, and poor weight estimation translates directly into efficiency loss Zheng and Zhou ( 2015 ) . We will show that that the weights depend on the conditional average treatment effect (CATE) of the treatment on the mediator, which must be estimated from the observed data. We will show, both theoretically and with a simulation study, that when the covariate-mediator relationship is nonlinear and high-dimensional, linear approaches may incur in an estimation error for the mediator CATE (Curth and Van der Schaar, 2021 ) , which in turn degrades the quality of the plug-in weight necessary for the G-estimation step, reducing the precision of the downstream structural parameter estimates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We address this challenge by proposing a representation-learning approach (Bengio et al. , 2013 ) for the estimation of the weights, thereby broadening the empirical applicability of a framework that already enjoys strong semiparametric identification guarantees. We use a Treatment-Agnostic Representation Network (TARNet; Shalit et al. , 2017 ) , a neural network architecture with shared representation layers and treatment-specific output heads, to estimate the CATE on the mediator. The shared representation Φ ( X ) \\\\Phi(X) enforces a common feature space across treatment arms. Because the CATE is computed as a difference of two functions of the same representation, the smoothness and dimensionality of Φ \\\\Phi provide an inductive bias that may help regularise the estimated treatment effect surface, particularly when μ 0 \\\\mu_{0} and μ 1 \\\\mu_{1} , the arm-specific conditional expectations of the mediator, share common features (Curth and Van der Schaar, 2021 ) . The TARNet’s CATE output is then embedded as the weight function in the G-estimating equation based on the model originally developed by Zheng and Zhou ( 2015 ) . A detailed description is provided in 3.1\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our contributions are threefold. First, we illustrate a clear link between the quality of the learned representation Φ \\\\Phi and the efficiency of the structural parameter estimator: better CATE estimation results in a more informative weight matrix, which reduces the asymptotic variance of θ ^ \\\\hat{\\\\theta} . Second, we propose UNIT (Unmeasured-confounding-robust NEH-based Identification with TARNet), a two-stage algorithm combining cross-fitted TARNet estimation and G-estimation. This approach requires the No Essential Heterogeneity assumption rather than the classical, and stronger, Sequential Ignorability as the key identifying condition. Third, the practical advantages of this approach are demonstrated through a simulation study with non-Gaussian covariates, complex nonlinear treatment effects, and unmeasured confounding. We show that TARNet-based weights can improve efficiency.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Outline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 2 introduces the causal framework, the model and the NEH assumption. Section 3 develops the representation-learning approach to weight estimation and the two-stage algorithm. Section 4 presents a simulation study that is used to investigate the properties of the proposed two-stage algorithm under empirically relevant conditions. Section 4.5 presents the results. Section 5 discusses the findings and limitations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Causal Framework\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Model and potential outcomes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following the potential outcomes framework introduced by (Rubin, 1974 ) and the model specifications introduced Zheng and Zhou ( 2015 ) , let Y ( t , m ) Y(t,m) denote the potential outcome under treatment level t t and mediator level m m , and let M ( t ) M(t) denote the potential mediator under treatment t t (Robins and Greenland, 1992 ) . We observe n n independent units ( X i , T i , M i , Y i ) i = 1 n (X_{i},T_{i},M_{i},Y_{i})_{i=1}^{n} , where X ∈ ℝ p X\\\\in\\\\mathbb{R}^{p} is a vector of pre-treatment covariates, T ∈ { 0 , 1 } T\\\\in\\\\{0,1\\\\} is the binary treatment, M ∈ ℝ M\\\\in\\\\mathbb{R} is the mediator, and Y ∈ ℝ Y\\\\in\\\\mathbb{R} is the outcome.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For observed covariates X X and unobserved confounders U U , write\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where μ ( ⋅ ) \\\\mu(\\\\cdot) captures the systematic (identified) component and ε ( t , m ; U , X ) \\\\varepsilon(t,m;\\\\,U,X) is an idiosyncratic error satisfying 𝔼 [ ε ( t , m ; U , X ) ∣ X ] = 0 \\\\mathbb{E}[\\\\varepsilon(t,m;\\\\,U,X)\\\\mid X]=0 for all ( t , m ) (t,m) . The semicolon separates the counterfactual regime ( t , m ) (t,m) from the random components ( U , X ) (U,X) . To streamline the notation, the ( U , X ) (U,X) component will be omitted whenever it is not required for the derivations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We specify a structural mean model of the form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where g ( X ) g(X) is a baseline outcome model and the h k ( ⋅ ) , k = 1 , 2 , … , K , h_{k}(\\\\cdot),\\\\;k=1,2,\\\\ldots,K, are known functions, which satisfy h k ( 0 , 0 , X ) = 0 h_{k}(0,0,X)=0 and describe the causal effects of the treatment and the mediator.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By consistency assumption (Assumption 1 ) the observed outcome satisfies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ε i := ε ( T i , M i ; U i , X i ) \\\\varepsilon_{i}:=\\\\varepsilon(T_{i},M_{i};\\\\,U_{i},X_{i}) is the structural error evaluated at the realised treatment and mediator. Defining the transformed outcome\"\n },\n {\n \"type\": \"para\",\n \"text\": \"the true parameter θ 0 \\\\theta_{0} satisfies 𝔼 [ Y ~ ( θ 0 ) ∣ T , X ] = E [ Y ~ ( θ 0 ) ∣ X ] = g ( X ) \\\\mathbb{E}[\\\\tilde{Y}(\\\\theta_{0})\\\\mid T,X]=E[\\\\tilde{Y}(\\\\theta_{0})\\\\mid X]=g(X) (under assumption 3 ). For suitable weight functions A ( T , X ) A(T,X) satisfying 𝔼 [ A ( T , X ) ∣ X ] = 0 \\\\mathbb{E}[A(T,X)\\\\mid X]=0 , the estimating equation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"yields a consistent estimator θ ^ \\\\hat{\\\\theta} (Zheng and Zhou, 2015 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Throughout the remainder of this paper, we refer to the model with K = 2 K=2 , h 1 ( T , M , X ) = T h_{1}(T,M,X)=T , and h 2 ( T , M , X ) = M h_{2}(T,M,X)=M , as an illustrative example, so that\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This implies no treatment-by-mediator interaction in the outcome and constant controlled effects θ 1 \\\\theta_{1} , θ 2 \\\\theta_{2} . This keeps the structural parameters interpretable and preserves comparability with the results reported in Ten Have et al. ( 2007 ) , Zheng and Zhou ( 2015 ) , Brandt ( 2020 ) , and Morelli et al. ( 2025 ) . Richer specifications in the structural model can be easily accommodated (Zheng and Zhou, 2015 ) and additional details will be developed in Section 2.5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Overview on 2-stage approach\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Parsimony in the structural outcome model does not require parsimony in the rest of the data-generating process. Our methodological focus is on settings where the covariate vector X ∈ ℝ p X\\\\in\\\\mathbb{R}^{p} is high-dimensional and the mediator CATE\"\n },\n {\n \"type\": \"para\",\n \"text\": \"is a non-trivial function to learn, involving nonlinearities or interactions that cannot be captured by simple linear models. The setting we are describing is the combination of a complex and heterogeneous treatment effect on the mediator, paired with a relatively simple, structural outcome model, and could generally arise in many applied settings. For example, baseline individual characteristics (e.g., genetic profile, comorbidities) may interact in a complex manner in determining the level of a biomarker (e.g., blood pressure), whereas the effect of the biomarker on the outcome (e.g., cardiovascular risk) per unit change in the biomarker may be relatively simple, invariant among the population, and approximately constant.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the specific formulation made, the weights are\"\n },\n {\n \"type\": \"para\",\n \"text\": \"details about the weights and the CATE will be discussed in Section 2.5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We refer to this step, the estimation of the weights necessary for the estimating equations, as Stage 1 . Solving the estimating equations is Stage 2. We summarize the algorithm below; the full development is deferred to Section 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 General Algorithm Overview\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Collect the basis functions of Eq. ( 2 ) into the vector\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so that the structural component of the model is ∑ k = 1 K θ k h k ( T i , M i , X i ) = H i ⊤ θ \\\\sum_{k=1}^{K}\\\\theta_{k}\\\\,h_{k}(T_{i},M_{i},X_{i})=H_{i}^{\\\\top}\\\\theta , with θ = ( θ 1 , … , θ K ) ⊤ \\\\theta=(\\\\theta_{1},\\\\dots,\\\\theta_{K})^{\\\\top} . In the structural mean (mediation) model the basis becomes H i = ( T i , M i ) ⊤ H_{i}=(T_{i},M_{i})^{\\\\top} and θ = ( θ 1 , θ 2 ) ⊤ \\\\theta=(\\\\theta_{1},\\\\theta_{2})^{\\\\top} are the controlled direct and mediator effects.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Stage 1: weight estimation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We estimate the optimal weight W ∗ ( X ) = ( 1 , τ M ( X ) ) ⊤ W^{*}(X)=(1,\\\\tau_{M}(X))^{\\\\top} by replacing the unknown mediator CATE with a cross-fitted estimate τ ^ M ( − ) \\\\hat{\\\\tau}_{M}^{(-)} of ( 6 ). Any CATE learner may be used; we use a cross-fitted TARNet (Section 3.2 ). This yields the plug-in centred instrument\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which satisfies the centering condition 𝔼 [ A ^ i ∣ X i ] = 0 \\\\mathbb{E}[\\\\hat{A}_{i}\\\\mid X_{i}]=0 under randomisation. The weight A ^ i \\\\hat{A}_{i} is held fixed for Stage 2.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Stage 2: structural estimation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given the Stage 1 weight A ^ i \\\\hat{A}_{i} and a cross-fitted estimate g ^ ( − ) \\\\hat{g}^{(-)} of the baseline, the structural estimator θ ^ \\\\hat{\\\\theta} is defined as the solution of the empirical G-estimating equation in Eq. ( 4 ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is important to highlight two aspects about Stage 2;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, because dim A ^ i = dim H i = K \\\\dim\\\\hat{A}_{i}=\\\\dim H_{i}=K , Eq. ( 8 ) is a just-identified linear system in θ \\\\theta , with closed-form solution\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is an instrumental-variable estimator: the centred weight A ^ i \\\\hat{A}_{i} acts as an instrument for the structural regressors H i H_{i} , and invertibility of ∑ i A ^ i H i ⊤ \\\\sum_{i}\\\\hat{A}_{i}H_{i}^{\\\\top} is exactly the relevance (non-degeneracy) condition of Assumption 5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, because the instrument is centred, the equation is orthogonal to the baseline: 𝔼 [ A ^ i f ( X i ) ] = 0 \\\\mathbb{E}[\\\\hat{A}_{i}\\\\,f(X_{i})]=0 for any function f f of the covariates. Consistency of θ ^ \\\\hat{\\\\theta} therefore does not require g ^ ( − ) \\\\hat{g}^{(-)} to be correctly specified; the baseline enters only as a variance-reducing nuisance (cf. the working-model property of Eq. ( 4 )).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The baseline is itself fit by cross-fitting on the transformed outcome Y ~ i ( θ ) = Y i − H i ⊤ θ \\\\tilde{Y}_{i}(\\\\theta)=Y_{i}-H_{i}^{\\\\top}\\\\theta , and so depends on the current value of θ \\\\theta . In the general case g ^ ( − ) \\\\hat{g}^{(-)} and θ ^ \\\\hat{\\\\theta} are thus determined jointly, by the estimation scheme of Zheng and Zhou ( 2015 ) . Under the linear working model for the baseline adopted here the joint system is linear and Eq. ( 9 ) is obtained in a single step, consistent with the one-step recommendation of Zheng and Zhou ( 2015 ) ; the concrete cross-fitted solver and this reduction are given in Algorithm 1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.4 Causal Assumptions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model rests on five assumptions, introduced below. The first two: consistency/SUTVA (Assumption 1 ) and treatment ignorability with positivity (Assumption 2 ) are standard in the causal inference literature and hold by design in a randomised trial. The third, no essential heterogeneity (Assumption 3 ), is the key identifying condition that distinguishes our approach from methods based on sequential ignorability: it permits unmeasured mediator–outcome confounding at the cost of restricting how unobserved confounders may modify treatment and mediator effects in the mean. The final two assumptions, which are of a statistical nature, are necessary for estimation. In particular, the fourth assumption, concerning the specification of the structural model (Assumption 4 ), is necessary to guarantee consistency. The fifth assumption, the non-degeneracy of the weight matrix (Assumption 5 ), ensures that the structural parameters are identifiable from the estimating equation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(No interference) The potential outcomes and potential mediators of unit i i do not depend on the treatment or mediator values assigned to other units.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(No hidden treatment versions) There is a single version of each treatment level, so that the mapping from treatment assignment to potential outcome is well-defined.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(Consistency) Observed outcomes equal the relevant potential outcomes under realised treatment and mediator values: M i = M i ( T i ) M_{i}=M_{i}(T_{i}) and Y i = Y i ( T i , M i ) Y_{i}=Y_{i}(T_{i},M_{i}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The three conditions constituting Assumption 1 arise directly from the experimental design (Morgan and Winship, 2014 ) . Their validity must be evaluated prior to the initiation of the experiment, as any violation would render both estimation and identification infeasible.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(Ignorability) Treatment assignment is independent of all potential outcomes and potential mediators given pre-treatment covariates:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(Positivity) Every unit has a positive probability of receiving each treatment level: 0 < Pr ( T = t ∣ X ) < 1 0<\\\\Pr(T=t\\\\mid X)<1 for each t ∈ { 0 , 1 } t\\\\in\\\\{0,1\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In observational settings, Assumption 2 would require that all confounders of the T − M T-M and T − Y T-Y relationships are measured and included in X X . The extension of UNIT to this case is discussed in Section 5.3 . It should be noted that Assumption 2 concerns only the treatment assignment mechanism. It does not require the absence of unmeasured confounding between M M and Y Y .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For all treatment and mediator levels t , m t,m ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where F F is the same function of ( M , X , T ) (M,X,T) across all counterfactual regimes ( t , m ) (t,m) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Equivalently, for any ( t 1 , m 1 ) (t_{1},m_{1}) and ( t 0 , m 0 ) (t_{0},m_{0}) ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so that differences in conditional means across regimes are driven entirely by the systematic component μ \\\\mu . Assumption 3 allows for rich heterogeneity through X X (including X X -specific effect modification via μ \\\\mu ) and arbitrary residual noise. However, it excludes unmeasured effect modification in the mean: once we condition on M , T , X M,T,X , unobserved confounders do not influence the average gain from switching t , m t,m . From an econometric perspective, this is the equivalent of the negation of “Essential Heterogeneity”, as formally defined by Heckman et al. ( 2006 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We emphasize that, although the NEH assumption is weaker than SI (Small, 2012 ; Zheng and Zhou, 2015 ) , it is suboptimal in settings where SI actually holds. This loss of optimality manifests as an increased variance in the resulting parameter estimates relative to estimators that target the SI semiparametric efficiency bound (e.g., Tchetgen Tchetgen and Shpitser, 2012 ). Moreover, if SI holds, nonparametric identification is possible, adding flexibility to the estimation procedure (Imai et al. , 2010 ; Pearl, 2001 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The basis functions { h k ( t , m , X ) } k = 1 K \\\\{h_{k}(t,m,X)\\\\}_{k=1}^{K} in the structural mean model in Eq. ( 2 ) are assumed to be correctly specified up to unknown parameters, the nuisance function g ( ⋅ ) g(\\\\cdot) can be misspecified.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This assumption is only partially testable: since potential outcomes are not jointly observed, and only the corresponding model for the observed outcome can be examined empirically.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Assumption 4 is necessary for consistency. If structural terms are omitted, the baseline model g ( X ) g(X) can not absorb them, introducing bias in the estimated causal parameters θ ^ \\\\hat{\\\\theta} . On the other hand, if g g is incorrectly specified, only efficiency is affected, but not consistency. This is the working-model property of G-estimation (see the centering argument in Appendix A , Eq. ( 38 )).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To estimate the model, an additional assumption regarding the regularity properties of the weight matrix needs to be introduced.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The weight vector W ( X ) W(X) is non-degenerate such that its second moment matrix 𝔼 [ W ( X ) W ( X ) ⊤ ] \\\\mathbb{E}[W(X)W(X)^{\\\\top}] is positive definite.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For example the weight vector introduced for the illustrative model earlier is W ( X ) = ( 1 , τ M ( X ) ) ⊤ W(X)=(1,\\\\tau_{M}(X))^{\\\\top} . In particular, we require that Var ( τ M ( X ) ) > 0 \\\\operatorname{Var}(\\\\tau_{M}(X))>0 , implying that the treatment has a heterogeneous effect on the mediator across the covariate space. This suggests that if the treatment has no heterogeneous effects on the mediator (i.e., τ M ( X ) \\\\tau_{M}(X) is constant), the second row of the weight matrix is collinear with the first, then the estimating equation has no unique solution for θ 2 \\\\theta_{2} , and the mediation parameter is not identified. Assumption 5 is the mediation analogue of the instrumental-variable relevance condition: this condition can be assessed empirically by testing whether treatment-by-covariate interactions in the mediator model are significant (Morelli et al. , 2025 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In practice, τ M ( X ) \\\\tau_{M}(X) is unknown and must be estimated from the data. Even if the model is formally identified with Var ( τ M ( X ) ) > 0 \\\\operatorname{Var}(\\\\tau_{M}(X))>0 , the actual size of Var ( τ M ( X ) ) \\\\operatorname{Var}(\\\\tau_{M}(X)) affects the efficiency of the estimator in Stage-2; this is the task we address in Section 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.5 The mediator CATE and the optimal weight function\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The G-estimating Eq. ( 4 ) holds for any weight function A ( T , X ) A(T,X) satisfying the centering condition 𝔼 [ A ( T , X ) ∣ X ] = 0 \\\\mathbb{E}[A(T,X)\\\\mid X]=0 . Different choices of A A yield consistent estimators with different asymptotic variances. Theorem 3 of Zheng and Zhou ( 2015 ) characterises the semiparametrically efficient choice: the optimal weight is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where Ω ( X ) = Var ( ε ( T , M ; U , X ) ∣ X ) \\\\Omega(X)=\\\\operatorname{Var}(\\\\varepsilon(T,M;\\\\,U,X)\\\\mid X) . UNIT adopts the Ω − 1 ≡ 𝐈 p \\\\Omega^{-1}\\\\equiv\\\\mathbf{I}_{p} simplification (valid under the linear- g g working model), so the deployed weight is ( T − p ) W ∗ ( X ) (T-p)\\\\,W^{*}(X) ; see Appendix B.3 . For our illustrative model specification the partial derivative ∂ Y ~ / ∂ θ \\\\partial\\\\tilde{Y}/\\\\partial\\\\theta equals − H i -H_{i} with H i = ( T i , M i ) ⊤ H_{i}=(T_{i},M_{i})^{\\\\top} , and the difference of conditional expectations of H i H_{i} simplifies to\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the first entry is trivial ( 𝔼 [ T ∣ X , T ] − 𝔼 [ T ∣ X ] = T − p \\\\mathbb{E}[T\\\\mid X,T]-\\\\mathbb{E}[T\\\\mid X]=T-p ) and the second follows from 𝔼 [ M ∣ X , T = 1 ] − 𝔼 [ M ∣ X ] = ( 1 − p ) τ M ( X ) \\\\mathbb{E}[M\\\\mid X,T=1]-\\\\mathbb{E}[M\\\\mid X]=(1-p)\\\\,\\\\tau_{M}(X) and analogously for T = 0 T=0 . The optimal weight therefore takes the multiplicative form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under Assumptions 1 and 2 , the mediator CATE is identified from observed data:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two features of Eq. ( 12 ) need additional details. First, the weight vector W ∗ ( X ) W^{*}(X) depends on X X only through the mediator CATE τ M ( X ) \\\\tau_{M}(X) . This means that the entire problem of constructing efficient instruments for θ \\\\theta reduces to the problem of estimating a single conditional average treatment effect, which is a well-studied task in the causal inference literature (Künzel et al. , 2019 ) . Second, the centering factor ( T − p ) (T-p) ensures that 𝔼 [ A ∗ ( T , X ) ∣ X ] = 0 \\\\mathbb{E}[A^{*}(T,X)\\\\mid X]=0 automatically under randomisation, regardless of the quality of τ ^ M \\\\hat{\\\\tau}_{M} . Consequently, any plug-in weight A ^ i = ( T i − p ) W ^ ( X i ) \\\\hat{A}_{i}=(T_{i}-p)\\\\hat{W}(X_{i}) preserves the consistency of θ ^ \\\\hat{\\\\theta} ; the quality of τ ^ M \\\\hat{\\\\tau}_{M} affects only efficiency, not identification.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because τ M ( X ) \\\\tau_{M}(X) is a CATE, with the mediator M M playing the role of outcome and the randomised treatment T T as the intervention, the rich literature on CATE estimation applies directly to Stage 1 of our algorithm. Existing theoretical results on learner-specific error bounds and the function of shared representations (Curth and Van der Schaar, 2021 ) , systematic comparisons of metalearners (Künzel et al. , 2019 ) , the use of cross-fitting for nuisance-function estimation (Chernozhukov et al. , 2018 ) , as well as analyses of the dependence of learner performance on the underlying data-generating process (Curth and Van der Schaar, 2021 ) , are all directly pertinent to our setting. These considerations motivate the adoption of representation-learning architectures (Johansson et al. , 2016 ) , such as TARnet (Shalit et al. , 2017 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For richer structural models, additional structural components can be added. If the basis includes a treatment-by-mediator interaction h 3 = T M h_{3}=TM , the corresponding row of the optimal weight involves the arm-specific conditional mean μ 1 M ( X ) = 𝔼 [ M ∣ T = 1 , X ] \\\\mu_{1}^{M}(X)=\\\\mathbb{E}[M\\\\mid T{=}1,X] , not just the difference. More generally, for any basis function that is linear in M M without T × M T\\\\times M interaction, including covariate modulated effects such as h k = M ⋅ f ( X ) h_{k}=M\\\\cdot f(X) , the weight continues to depend on X X through τ M ( X ) \\\\tau_{M}(X) alone. In all cases where arm-specific means are required, TARNet provides them directly through its two output heads h 0 ( Φ ( X ) ) h_{0}(\\\\Phi(X)) and h 1 ( Φ ( X ) ) h_{1}(\\\\Phi(X)) , so the extension of UNIT to richer structural specifications does not require a change of Stage 1 architecture, only the construction of the weight vector W ( X ) W(X) and the basis vector H i H_{i} must be adapted accordingly.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Methods: Representation Learning for Weight Estimation\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 From separate regressions to shared representations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The most direct approach to estimating the mediator CATE is the T-learner (Künzel et al. , 2019 ) : fit two separate response surfaces μ ^ 0 ( x ) \\\\widehat{\\\\mu}_{0}(x) and μ ^ 1 ( x ) \\\\widehat{\\\\mu}_{1}(x) on the control and treated groups respectively, and define τ ^ T ( x ) = μ ^ 1 ( x ) − μ ^ 0 ( x ) \\\\widehat{\\\\tau}_{T}(x)=\\\\widehat{\\\\mu}_{1}(x)-\\\\widehat{\\\\mu}_{0}(x) . While consistent, the T-learner can exhibit larger finite-sample estimation error than methods that share information across treatment arms. Because μ ^ 0 \\\\widehat{\\\\mu}_{0} and μ ^ 1 \\\\widehat{\\\\mu}_{1} are estimated on disjoint subsets, each model is regularised independently, and when the two groups differ in size, the regularisation schedules diverge. This regularisation mismatch produces variation in τ ^ T ( x ) \\\\widehat{\\\\tau}_{T}(x) that does not correspond to treatment-effect heterogeneity (Künzel et al. , 2019 ) . More formally, the error bound for the T-learner scales as 2 { ε ( μ ^ 1 ) + ε ( μ ^ 0 ) } 2\\\\{\\\\varepsilon(\\\\hat{\\\\mu}_{1})+\\\\varepsilon(\\\\hat{\\\\mu}_{0})\\\\} , where ε ( μ ^ w ) \\\\varepsilon(\\\\hat{\\\\mu}_{w}) denotes the estimation error of each of the w = 0 , 1 w=0,1 arm-specific model (Curth and Van der Schaar, 2021 ) . Learners that share structure between arms can achieve tighter bounds by exploiting a common signal Curth and Van der Schaar ( 2021 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A more fundamental limitation is that the T-learner cannot exploit structural similarities between the two response surfaces. In many applications, the conditional expectations μ 1 ( x ) \\\\mu_{1}(x) and μ 0 ( x ) \\\\mu_{0}(x) share substantial low-dimensional structure; the treatment shifts the outcome without radically altering which features are relevant (Kennedy, 2023 ) . Separate models must therefore recover the common signal independently in each arm, even though that signal is largely shared, and under balanced 1:1 treatment allocation, each model uses at most half of the available data. These considerations motivate the use of representation learning as an inductive bias for CATE estimation. Instead of separate models, we learn a shared feature map Φ : ℝ p → ℝ r \\\\Phi\\\\colon\\\\mathbb{R}^{p}\\\\to\\\\mathbb{R}^{r} that compresses the covariate space into a common representation for both treatment arms. Treatment-specific heads h 0 h_{0} and h 1 h_{1} then map from this representation to the arm-specific conditional expectations. The key structural assumption, formalised by Shalit et al. ( 2017 ) , is that there exists a common feature space underlying both { μ w ( x ) } w ∈ { 0 , 1 } \\\\{\\\\mu_{w}(x)\\\\}_{w\\\\in\\\\{0,1\\\\}} , so that Φ ( X ) \\\\Phi(X) preserves all identifying conditional independence relationships (Curth and Van der Schaar, 2021 ; Shalit et al. , 2017 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By pooling all observations through the shared layers, the representation Φ \\\\Phi is estimated with the full sample, regardless of treatment assignment. This yields two advantages directly relevant for the G-estimation framework. First, when treatment groups are of unequal size, the shared layers stabilise estimation by pooling information across arms (variance reduction under imbalance; Curth and Van der Schaar, 2021 ) . Second, because the CATE is computed as a difference of two functions of the same representation, the smoothness and dimensionality of Φ \\\\Phi provide an inductive bias that may help regularise the treatment effect surface, particularly when μ 0 \\\\mu_{0} and μ 1 \\\\mu_{1} share substantial low-dimensional structure (Kennedy, 2023 ; Shalit et al. , 2017 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Direct TARNet estimation of the mediator CATE\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We adopt a TARNet (Shalit et al. , 2017 ) as implemented in the SNet-1 architecture of Curth and Van der Schaar ( 2021 ) . The network consists of three components:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A shared representation Φ : ℝ p → ℝ r \\\\Phi\\\\colon\\\\mathbb{R}^{p}\\\\to\\\\mathbb{R}^{r} , parameterised as a multi-layer feed-forward network with ELU activations, mapping the raw covariate vector X X into a latent feature space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two treatment-specific heads h 0 , h 1 : ℝ r → ℝ h_{0},h_{1}\\\\colon\\\\mathbb{R}^{r}\\\\to\\\\mathbb{R} , each a smaller dense network, predicting the arm-specific conditional expectations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mediator CATE is obtained directly as the contrast:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The entire network is trained end-to-end by minimising the factual loss\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ℛ ( ⋅ ) \\\\mathcal{R}(\\\\cdot) is an L 2 L_{2} -regularisation penalty on all network weights. Each observation contributes through exactly one head (corresponding to its observed treatment), while both treated and control units contribute to the shared representation Φ \\\\Phi .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Cross-fitting both nuisance components\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To separate nuisance estimation from score evaluation, we cross-fit both nuisance components entering Stage 2: the mediator CATE τ M ( X ) \\\\tau_{M}(X) and the baseline outcome function g ( X ) g(X) . We model the nuisance baseline as a linear working model, g ( X ) = β ⊤ b ( X ) g(X)=\\\\beta^{\\\\top}b(X) , where b ( X ) b(X) is a fixed basis expansion that may include polynomial terms of the raw covariates. Thus g g is linear in its coefficients even when it is polynomial in the original variables, avoiding introducing an additional nonlinear nuisance layer in Stage 2.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In principle, g ( x ) g(x) can be nonlinear and specified more flexibly, for example with a second neural network. The main consequences would be two. First, the baseline nuisance is fit to the transformed outcome\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so the target of the baseline regression changes whenever θ \\\\theta is updated. This implies that it becomes necessary to adopt a general estimation strategy: alternating update of g g and θ \\\\theta (Zheng and Zhou, 2015 ) . The second consequence is that it becomes necessary to estimate the Ω − 1 ( X ) \\\\Omega^{-1}(X) component. Under our linear in-parameters specification both complications collapse, the update of θ \\\\theta reduces to a one-step G-estimation equation, and Ω − 1 ( X ) \\\\Omega^{-1}(X) need not be estimated explicitly. However, we present the alternating algorithm, as it is the more general method and remains applicable when g ( x ) g(x) is extended beyond the linear working model b ( x ) b(x) . A more detailed discussion of these aspects is provided in B.3 . Let { I k } k = 1 K \\\\{I_{k}\\\\}_{k=1}^{K} be a partition of the sample into K K folds, and let I k c I_{k}^{c} denote the complementary training set.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Stage 1 nuisance: mediator CATE.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each fold k = 1 , … , K k=1,\\\\dots,K , we train a TARNet on I k c I_{k}^{c} by minimising the factual loss in Eq. ( 15 ), obtaining fold-specific estimators Φ ( − k ) , h 0 ( − k ) , h 1 ( − k ) \\\\Phi^{(-k)},h_{0}^{(-k)},h_{1}^{(-k)} . For every unit i ∈ I k i\\\\in I_{k} , we compute the out-of-fold mediator CATE\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stacking across folds yields cross-fitted estimates τ ^ M ( − ) ( X i ) \\\\hat{\\\\tau}_{M}^{(-)}(X_{i}) for all i i .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Stage 2 nuisance: baseline outcome model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The baseline function is estimated foldwise inside Stage 2 as well. Given a current iterate θ ( t ) \\\\theta^{(t)} , define the transformed outcome Y ~ i ( θ ( t ) ) = Y i − H i ⊤ θ ( t ) \\\\tilde{Y}_{i}(\\\\theta^{(t)})=Y_{i}-H_{i}^{\\\\top}\\\\theta^{(t)} with H i = ( T i , M i ) ⊤ H_{i}=(T_{i},M_{i})^{\\\\top} . For each fold k k , fit the baseline learner on I k c I_{k}^{c} using { Y ~ i ( θ ( t ) ) , X i : i ∈ I k c } \\\\{\\\\tilde{Y}_{i}(\\\\theta^{(t)}),X_{i}:i\\\\in I_{k}^{c}\\\\} , and denote the resulting estimator by g ^ θ ( t ) ( − k ) \\\\hat{g}_{\\\\theta^{(t)}}^{(-k)} . For each held-out observation i ∈ I k i\\\\in I_{k} , define the out-of-fold prediction g ^ θ ( t ) ( − ) ( X i ) = g ^ θ ( t ) ( − k ) ( X i ) \\\\hat{g}_{\\\\theta^{(t)}}^{(-)}(X_{i})=\\\\hat{g}_{\\\\theta^{(t)}}^{(-k)}(X_{i}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cross-fitting is relevant for the asymptotic argument in Appendix B . In particular, when the same sample is used both to estimate g g and to evaluate the score, the empirical process remainder from plugging in g ^ \\\\hat{g} requires additional regularity conditions (e.g., Donsker-class membership; Chernozhukov et al. , 2018 ) that may not hold for flexible learners.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cross-fitting the baseline removes this dependence: conditional on the training folds, the held-out score contributions are mean-zero, and the baseline remainder becomes of order ‖ g ^ θ ( − ) − g 0 , θ ‖ L 2 ( P ) \\\\|\\\\hat{g}_{\\\\theta}^{(-)}-g_{0,\\\\theta}\\\\|_{L_{2}(P)} , which is negligible under consistency (Chernozhukov et al. , 2018 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 G-estimation weights\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We now explicitly specify how the CATE on the mediator output of the TARNet model is incorporated into the G-estimation equation. Substituting the cross-fitted estimates into Eq. ( 12 ), the plug-in weight vector for unit i i is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and the empirical optimal weight becomes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where p p is the known allocation probability (or T ¯ \\\\bar{T} if estimated).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The representation Φ \\\\Phi determines the quality of τ ^ M ( X ) \\\\hat{\\\\tau}_{M}(X) and thereby the informativeness of the weight matrix A ^ \\\\hat{A} . If Φ \\\\Phi captures all relevant heterogeneity in the T → M T\\\\to M effect, the resulting weights provide a good plug-in approximation to the heterogeneity component of the semiparametrically efficient instrument (cf. Theorem 3 in Zheng and Zhou, 2015 ) , thereby improving the efficiency of θ ^ \\\\hat{\\\\theta} . Conversely, if Φ \\\\Phi is chosen too coarsely and discards features that generate true effect heterogeneity, the weights lose informative signal and θ ^ \\\\hat{\\\\theta} becomes less efficient, although it still remains consistent under the NEH assumption.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Inference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the regularity conditions stated in Appendix B , including an n − 1 / 4 n^{-1/4} rate for the cross-fitted mediator CATE and consistency of the cross-fitted baseline, the G-estimator θ ^ \\\\hat{\\\\theta} is asymptotically normal with a sandwich-type variance (see Proposition 2 and Appendix A for additional details and derivations). A consistent estimator of the asymptotic variance is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where G ^ θ = n − 1 ∑ i A ^ i H i ⊤ \\\\hat{G}_{\\\\theta}=n^{-1}\\\\sum_{i}\\\\hat{A}_{i}H_{i}^{\\\\top} and S ^ = n − 1 ∑ i ψ ^ i 2 A ^ i A ^ i ⊤ \\\\hat{S}=n^{-1}\\\\sum_{i}\\\\hat{\\\\psi}_{i}^{2}\\\\,\\\\hat{A}_{i}\\\\hat{A}_{i}^{\\\\top} with out-of-fold residuals ψ ^ i = Y i − g ^ θ ^ ( − ) ( X i ) − H i ⊤ θ ^ \\\\hat{\\\\psi}_{i}=Y_{i}-\\\\hat{g}_{\\\\hat{\\\\theta}}^{(-)}(X_{i})-H_{i}^{\\\\top}\\\\hat{\\\\theta} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Quality of the estimator.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"It is useful to separate three distinct roles that govern the quality of the final estimator:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Identification of θ \\\\theta requires treatment ignorability (Assumption 2 ), the NEH assumption (Assumption 3 ), correct structural specification (Assumption 4 ), and non-degeneracy of the weight matrix (Assumption 5 ). The centering condition 𝔼 [ ( T − p ) ∣ X ] = 0 \\\\mathbb{E}[(T-p)\\\\mid X]=0 follows from Assumption 2 under randomisation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Estimation quality of τ M ( X ) \\\\tau_{M}(X) determines how closely the plug-in weight W ^ ( X ) \\\\hat{W}(X) approximates the oracle optimal weight W ∗ ( X ) W^{*}(X) . This is governed by the CATE estimation error, which depends on the learner architecture, the sample size, and the complexity of the true τ M \\\\tau_{M} (Curth and Van der Schaar, 2021 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Instrument strength determines how much precision the weight matrix delivers, even at the oracle. The centred weights ( T − p ) W ( X ) (T-p)W(X) function as instruments for the structural parameters. If the underlying DGP has weak treatment-by-covariate interaction on the mediator, i.e., Var ( τ M ( X ) ) \\\\operatorname{Var}(\\\\tau_{M}(X)) is small, the instruments are weak, and θ ^ 2 \\\\hat{\\\\theta}_{2} will be imprecise regardless of how well τ ^ M \\\\hat{\\\\tau}_{M} approximates the truth.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Simulation Study\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We construct a DGP that mimics typical features of modern and challenging psychological and biomedical data (Micceri, 1989 ; Blanca et al. , 2013 ; Bono et al. , 2017 ) : non-Gaussian covariates, complex nonlinear effects, and unmeasured mediator–outcome confounding (Rudolph et al. , 2019 ) . We observe i.i.d. tuples ( X i , T i , M i , Y i ) (X_{i},T_{i},M_{i},Y_{i}) , i = 1 , … , n i=1,\\\\dots,n : covariates X ∈ ℝ 25 X\\\\in\\\\mathbb{R}^{25} , a randomized binary treatment T ∈ { 0 , 1 } T\\\\in\\\\{0,1\\\\} , a continuous mediator M M , and a continuous outcome Y Y . Each of the 7 scenarios that we study is derived by modifying one characteristic of the following base model:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Where τ ( X ) \\\\tau(X) is the heterogeneous effect of T T on M M (the mediator CATE, target of Stage 1); μ 0 ( X ) \\\\mu_{0}(X) a nonlinear baseline mediator level; K K an unmeasured confounder entering both structural equations (loading λ K \\\\lambda_{K} ); g ( X ) = ∑ j = 0 6 [ 0.4 X j + 0.2 X j 2 𝟏 { j even } ] g(X)=\\\\sum_{j=0}^{6}[0.4X_{j}+0.2X_{j}^{2}\\\\mathbf{1}\\\\{j\\\\text{ even}\\\\}] a nonlinear nuisance. Parameters θ 1 = 1 \\\\theta_{1}=1 (direct effect), θ 2 = 0.5 \\\\theta_{2}=0.5 (the mediation coefficient of interest), λ K = 1 \\\\lambda_{K}=1 . Treatment is randomized, T ⟂ ⟂ X T\\\\perp\\\\!\\\\!\\\\!\\\\perp X , with known e ( X ) ≡ Pr ( T = 1 ∣ X ) = 0.5 e(X)\\\\equiv\\\\Pr(T{=}1\\\\mid X)=0.5 . The total effect of T T on Y Y decomposes as TE ( x ) = θ 1 + θ 2 τ ( x ) \\\\mathrm{TE}(x)=\\\\theta_{1}+\\\\theta_{2}\\\\,\\\\tau(x) . Below, we introduce the base generating process for each component, the specific modifications will be made explicit in each of the respective scenarios.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Covariates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each X ∈ ℝ 25 X\\\\in\\\\mathbb{R}^{25} is generated via a Gaussian copula with Gaussian mixture model (GMM) marginals. In particular we draw Z i ∼ 𝒩 25 ( 0 , Σ ) Z_{i}\\\\sim\\\\mathcal{N}_{25}(0,\\\\Sigma) with Σ j l = ρ | j − l | \\\\Sigma_{jl}=\\\\rho^{|j-l|} ( ρ = 0.5 \\\\rho=0.5 ), transform to uniform margins U i j = Φ ( Z i j ) U_{ij}=\\\\Phi(Z_{ij}) , and apply the inverse GMM CDF componentwise to obtain X i j = F j − 1 ( U i j ) X_{ij}=F_{j}^{-1}(U_{ij}) . Each marginal F j F_{j} is a mixture of n j ∈ { 5 , … , 10 } n_{j}\\\\in\\\\{5,\\\\ldots,10\\\\} Gaussian components (independently configured per feature, and number of components sampled uniformly), with component means drawn from Uniform ( − 2 , 2 ) \\\\mathrm{Uniform}(-2,2) , standard deviations from Uniform ( 0.5 , 1.5 ) \\\\mathrm{Uniform}(0.5,1.5) , and weights proportional to Uniform ( 0.5 , 2 ) \\\\mathrm{Uniform}(0.5,2) draws; the resulting distribution is min-max scaled to the interval [ 0 , 5 ] [0,5] . This yields non-Gaussian, skewed, and multimodal marginals linked by moderate AR(1) copula dependence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 25 features are partitioned into four blocks following the benchmark design of Curth and Van der Schaar ( 2021 ) as depicted in Table 1 :\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Latent-score construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The treatment effect and baseline outcome are built from three fixed linear projections of the standardized covariates: z = ( X − X ¯ ) / s z=(X-\\\\bar{X}_{\\\\mathrm{}})/s_{\\\\mathrm{}} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The scores s 1 s_{1} and s 2 s_{2} are shared entering both τ ( X ) \\\\tau(X) and μ 0 ( X ) \\\\mu_{0}(X) and provide the joint latent structure. The treatment-effect function and baseline outcome are:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where σ ( ⋅ ) \\\\sigma(\\\\cdot) is the logistic function and softplus ( x ) = log ( 1 + e x ) \\\\mathrm{softplus}(x)=\\\\log(1+e^{x}) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Unmeasured confounder.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The unmeasured confounder shares the same distributional family as the observed confounders. In particular we construct the unmeasured confounder K K as a GMM draw mixed with Gaussian noise:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where K gmm K_{\\\\mathrm{gmm}} is drawn from a three-component Gaussian mixture with means ( − 1.2 , 0.3 , 1.5 ) (-1.2,\\\\,0.3,\\\\,1.5) , standard deviations ( 0.8 , 0.6 , 0.9 ) (0.8,\\\\,0.6,\\\\,0.9) , and mixture weights ( 0.40 , 0.35 , 0.25 ) (0.40,\\\\,0.35,\\\\,0.25) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Variance calibration and scenarios.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The components ( τ ⋅ T , μ 0 , K , ε M ) (\\\\tau\\\\!\\\\cdot\\\\!T,\\\\,\\\\mu_{0},\\\\,K,\\\\,\\\\varepsilon_{M}) are independently rescaled to achieve target variance shares.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 The seven scenarios\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We design the simulation study to investigate two orthogonal dimensions: identification and efficiency. Specifically, we examine the efficiency of the methods by varying the degree of nonlinearity and the magnitude of τ \\\\tau , while holding the identification structure fixed; this is implemented in scenarios A, B, E, and Wtau. Conversely, we assess the impact of identification by varying the identification conditions while keeping the baseline surface characteristics fixed in scenarios C and D.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Throughout, ε M , i ∼ 𝒩 ( 0 , 1 ) \\\\varepsilon_{M,i}\\\\sim\\\\mathcal{N}(0,1) , ε Y , i ∼ 𝒩 ( 0 , σ ε Y 2 ) \\\\varepsilon_{Y,i}\\\\sim\\\\mathcal{N}(0,\\\\sigma_{\\\\varepsilon_{Y}}^{2}) , K ¯ i = a K K i \\\\bar{K}_{i}=a_{K}K_{i} , and ( a τ , a D , a K , a ε ) (a_{\\\\tau},a_{D},a_{K},a_{\\\\varepsilon}) are the calibration scales for the cell’s shares.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"A: realistic heterogeneity ( τ \\\\tau -share 25 % 25\\\\% ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is the reference scenario NEH holds; predicted bias 0 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"B: strong heterogeneity ( τ \\\\tau -share 40 % 40\\\\% ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This case has exactly the same structure as in Eq. ( 27 ), but assigns a much larger weight to the τ \\\\tau component, namely ( 0.40 , 0.28 , 0.18 , 0.14 ) (0.40,0.28,0.18,0.14) , thus corresponding to an “ideal instrument” setting. Under this specification, NEH is satisfied and the predicted bias equals 0 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"C: rank preservation fails, NEH holds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mediator is specified as in Eq. ( 27 ). The outcome slope with respect to M M becomes idiosyncratic through the inclusion of the term ξ i \\\\xi_{i} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The individual rank preservation assumption fails, but NEH holds. Predicted bias 0 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"D: essential heterogeneity, NEH violated ( γ = κ = 0.4 \\\\gamma=\\\\kappa=0.4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A latent L i ∼ 𝒩 ( 0 , 1 ) L_{i}\\\\sim\\\\mathcal{N}(0,1) , independent of ( X , T , K , ε M , ε Y ) (X,T,K,\\\\varepsilon_{M},\\\\varepsilon_{Y}) , enters both equations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The same L L modulates the mediator’s treatment response ( κ L \\\\kappa L ) and the outcome’s mediator slope ( γ L \\\\gamma L ). NEH violated; predicted bias γ κ = 0.16 \\\\gamma\\\\kappa=0.16 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Dsmall: half amplitude ( γ = κ = 0.2 \\\\gamma=\\\\kappa=0.2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Eq. ( 29 ) with γ = κ = 0.2 \\\\gamma=\\\\kappa=0.2 , predicted bias 0.04 0.04 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"E: CATE (mostly) linear, NEH holds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The response surface is replaced, by an orthogonalized combination of linear and nonlinear components. τ E \\\\tau_{E} ; μ 0 , K \\\\mu_{0},K , shares, noise are those of scenario A:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"z nl z_{\\\\mathrm{nl}} is built as in Eq. ( 24 ), β = Cov ( z nl , z lin ) / Var ( z lin ) \\\\beta=\\\\operatorname{Cov}(z_{\\\\mathrm{nl}},z_{\\\\mathrm{lin}})/\\\\operatorname{Var}(z_{\\\\mathrm{lin}}) and the reference SDs computed once on a large population draw (DGP identical across n n ), ( w nl , w lin ) = ( 0.30 , 0.90 ) (w_{\\\\mathrm{nl}},w_{\\\\mathrm{lin}})=(0.30,0.90) . Realized full-linear R 2 R^{2} of τ E \\\\tau_{E} averages 0.89 0.89 (vs 0.19 0.19 in scenario A).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Wtau: weak instrument, NEH holds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Eq. ( 27 ), shares ( 0.08 , 0.50 , 0.20 , 0.22 ) (0.08,0.50,0.20,0.22) : the τ \\\\tau -share is 8 % 8\\\\% , It corresponds to a weak-instrument regime, translating in a weak signal for Stage-1. NEH holds; predicted bias 0 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Stage 1 methods\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All Stage 1 methods use 5-fold cross-fitting; implementation details and hyperparameters are given in the Estimator Hyperparameters appendix C . We compare 1 1 1 Thin Plate Splines were considered as a candidate middle-tier learner, motivated by their use in semiparametric mediator modeling (Brandt, 2020 ) . However, in preliminary runs at d = 25 d=25 TPS was numerically unstable despite regularisation, so we excluded it and do not report TPS results Kalogridis ( 2026 ) :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TARNet : SNet-1 architecture (Curth and Van der Schaar, 2021 ) with shared representation layers and treatment-specific heads. The CATE is computed directly as τ ^ ( X ) = h 1 ( Φ ( X ) ) − h 0 ( Φ ( X ) ) \\\\hat{\\\\tau}(X)=h_{1}(\\\\Phi(X))-h_{0}(\\\\Phi(X)) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TNet (no shared representation) : the same SNet-1 architecture as TARNet (Curth and Van der Schaar, 2021 ) but without the shared representation layers, so each treatment arm is fitted separately. The CATE is computed as τ ^ ( X ) = h 1 ( X ) − h 0 ( X ) \\\\hat{\\\\tau}(X)=h_{1}(X)-h_{0}(X) . This learner serves as the ablation that isolates the value of the shared representation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ridge (T-learner) : Separate Ridge regressions for T = 0 T=0 and T = 1 T=1 , with τ ^ ( X ) = μ ^ 1 ( X ) − μ ^ 0 ( X ) \\\\hat{\\\\tau}(X)=\\\\hat{\\\\mu}_{1}(X)-\\\\hat{\\\\mu}_{0}(X) . Near-optimal when τ \\\\tau is linear but unable to capture interactions or nonlinearities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Random Forest (T-learner) Breiman ( 2001 ) : Separate random forests for T = 0 T=0 and T = 1 T=1 . Hyperparameters (number of estimators, maximum depth, maximum features fraction) are selected by 3-fold cross-validation with 6 random configurations per fold. Captures nonlinearity and feature interactions via tree splits but lacks TARNet’s shared-representation inductive bias.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Baron–Kenny estimator. The naive baseline is the Baron and Kenny ( 1986 ) two-regression estimator. Because it omits the unmeasured confounder K K from the outcome regression, it is biased for θ 2 \\\\theta_{2} by a positive amount that does not vanish with n n . Empirically (see Table 5 ), this bias is + 0.27 +0.27 to + 0.39 +0.39 on the true θ 2 = 0.5 \\\\theta_{2}=0.5 , with 0 % 0\\\\% coverage in every cell, irrespective of sample size or signal strength.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Monte-Carlo design and evaluation metrics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each scenario is replicated over R = 200 R=200 random seeds, for sample sizes n ∈ { 500 , 1000 , 2000 , 5000 , 10000 } n\\\\in\\\\{500,\\\\,1000,\\\\,2000,\\\\,5000,\\\\,10000\\\\} . The low-sample-size settings n ∈ { 500 , 1000 } n\\\\in\\\\{500,\\\\,1000\\\\} are included to assess the behaviour of the methods in small-sample regimes. 2 2 2 However, for applied analyses, we recommend avoiding sample sizes below 1000. The structural parameters θ 1 = 1.0 \\\\theta_{1}=1.0 and θ 2 = 0.5 \\\\theta_{2}=0.5 are fixed across replications. Each replication produces a single point estimate θ ^ 2 ( r ) \\\\hat{\\\\theta}_{2}^{(r)} , the sandwich estimator SE r \\\\mathrm{SE}_{r} from Eq. ( 18 ), and a confidence interval CI r = θ ^ 2 ( r ) ± 1.96 SE r \\\\mathrm{CI}_{r}=\\\\hat{\\\\theta}_{2}^{(r)}\\\\pm 1.96\\\\,\\\\mathrm{SE}_{r} . The reported per-cell, per-method metrics are depicted in Table 3 :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A replication’s Stage 2 is skipped if the Stage-1 NRMSE > 1.3 >1.3 , or sd ( τ ^ ) < 10 − 4 \\\\mathrm{sd}(\\\\hat{\\\\tau})<10^{-4} . A replication is likewise excluded if Stage 2 does not converge, any θ ^ \\\\hat{\\\\theta} or SE is non-finite. Table 4 reports skip counts at n = 500 n=500 ; at n ≥ 1000 n\\\\geq 1000 skips fall below 2 % 2\\\\% except for Wtau, and are zero at n ≥ 5000 n\\\\geq 5000 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Stage-1 CATE quality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stage-1 quality is measured as the learner’s normalized RMSE for τ ^ \\\\hat{\\\\tau} (see Figure 2 ). In the nonlinear surface scenarios (A, B, C, D, Dsmall), TARNet’s NRMSE decreases as n n grows, whereas Ridge levels off above 0.77 0.77 (reflecting the irreducible error from linear misspecification). In scenario E (linear surface), Ridge’s NRMSE falls to 0.36 0.36 , while RF, which lacks a linear inductive bias, flattens at 0.61 0.61 . In the Wtau (weak instrument) scenario, the NRMSE increases for all learners, with TARNet exceeding the 1.3 1.3 threshold at n = 500 n{=}500 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The correlation Corr ( τ ^ , τ ) \\\\operatorname{Corr}(\\\\hat{\\\\tau},\\\\tau) (Figure 3 ) between true and estimated tau provides additional information about the performance of the learners. Across all scenarios from A through Dsmall, a very similar pattern can be observed. TARNet yields the highest correlations across all sample sizes n n , with correlations close to or above 0.8 0.8 for n ≥ 1000 n\\\\geq 1000 . For n = 500 n=500 , all learners perform similarly in these scenarios, with relatively low correlations ranging between 0.4 0.4 and 0.6 0.6 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In scenario E, corresponding to the linear setting, both TARNet and Ridge achieve similarly high correlations above 0.9 0.9 for n ≥ 1000 n\\\\geq 1000 . In scenario Wtau, corresponding to the weak-instrument setting, a similar ranking of the learners is observed, although correlations are generally lower. In this case, TARNet reaches correlations close to 0.8 0.8 only for n ≥ 5000 n\\\\geq 5000 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Stage-2 estimation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 summarizes the main patterns. In the settings where NEH holds (A, B, C, E, Wtau), all learners are unbiased (albeit the presence of a minor empirical bias is to be expected) and achieve nominal coverage for n ≥ 1000 n\\\\geq 1000 , whereas BK exhibits a positive bias of + 0.27 +0.27 to + 0.33 +0.33 and 0 % 0\\\\% coverage at all sample sizes. When NEH is violated (D, Dsmall), each learner converges to the expected γ κ \\\\gamma\\\\kappa bias, and coverage collapses as n n increases. This breakdown is most rapid for TARNet, whose narrower intervals concentrate around the wrong parameter. The efficiency ratios indicate that TARNet achieves the smallest SE in every nonlinear design (R/T, RF/T ≈ 1.4 \\\\approx 1.4 – 1.6 1.6 for n ≥ 1000 n\\\\geq 1000 ) and remains at roughly ≈ 1.0 \\\\approx 1.0 on the linear setting E. The TT/T column isolates the penalty of removing the shared representation: TNet lags TARNet by 30 30 – 40 % 40\\\\% in SE at moderate n n ( 1.3 1.3 – 1.4 1.4 at n = 1000 n{=}1000 – 2000 2000 in A, B, C, Dsmall), but this difference shrinks to about ≈ 1.0 \\\\approx 1.0 by n = 10 , 000 n{=}10{,}000 , indicating a finite-sample benefit of the shared representation rather than an asymptotic one.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Summary of findings\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our results trace a two-stage efficiency chain,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across the nonlinear scenarios the TARNet weight delivers Stage 2 standard errors about 1.5 × 1.5\\\\times smaller than a linear T-learner (median Ridge/TARNet ratio 1.45 1.45 – 1.51 1.51 at n ≥ 2 , 000 n\\\\geq 2{,}000 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Identification behaves exactly as the NEH theory predicts, and independently of the Stage-1 learner. Where NEH holds, every G-estimator is approximately unbiased with near-nominal coverage; where it fails (Scenario D) all learners converge to the same predicted γ κ \\\\gamma\\\\kappa offset, and coverage degrades predictably as the sample grows and the interval tightens around the biased target. Scenario C confirms that the method requires only NEH and not rank preservation: its results are not significantly different from the baseline scenario. The naive Baron–Kenny estimator, carries a persistent bias of + 0.27 +0.27 to + 0.39 +0.39 on θ 2 = 0.5 \\\\theta_{2}=0.5 with 0 % 0\\\\% coverage in every cell, regardless of sample size or signal strength.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Shared representation versus separate arms\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given the design of the simulation we can separate the two distinct mechanisms that drive the efficiency gains.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, flexibility by itself is insufficient. Although the Random Forest is fully nonparametric, it only slightly outperforms Ridge, with Corr ( τ ^ M , τ M ) ≈ 0.65 \\\\operatorname{Corr}(\\\\hat{\\\\tau}_{M},\\\\tau_{M})\\\\approx 0.65 – 0.70 0.70 even at n = 10 , 000 n=10{,}000 , whereas the neural learners reach 0.84 0.84 – 0.96 0.96 . The core issue is structural: the mediator CATE is defined via smooth functions of dense linear projections ( s 1 , s 2 s_{1},s_{2} ) of the covariates, and an axis-aligned splitting scheme cannot capture these projections. This also provides a mechanistic explanation for how a highly flexible tree ensemble can fail to beat a linear model when the underlying structure is governed by latent linear relationships.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, the shared representation contributes beyond the neural function class. TNet uses the same neural architecture as TARNet but fits each treatment arm separately. Although it learns the latent projections, because each arm only sees the units in its own treatment group it approaches the shared TARNet only asymptotically. At n = 1 , 000 n=1{,}000 it lags substantially ( Corr ≈ 0.46 \\\\operatorname{Corr}\\\\approx 0.46 vs 0.76 0.76 in Scenario A), but the gap closes by n = 10 , 000 n=10{,}000 ( 0.88 0.88 vs 0.94 0.94 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Limitations and Future Research\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While the UNIT framework provides a robust integration of deep representation learning with G-estimation, it is important to acknowledge some limitations and to highlight promising directions for future research.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Estimation of the Nuisance Baseline\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In our current implementation, the baseline outcome model g ( X ) g(X) is estimated via 5-fold cross-fitted Ridge regression within Stage 2, as prescribed in Section 2.3 . For the linear specifications of g g employed in our simulations, the numerical discrepancy between cross-fitted and in-sample estimates was negligible. Consequently, the DML correction Chernozhukov et al. ( 2018 ) provided primarily theoretical asymptotic safeguards rather than substantial finite-sample gains in these specific settings. Future research should evaluate the performance of UNIT in contexts where g ( X ) g(X) is highly non-linear, a detailed discussion of the complications that arise under nonlinear g g is given in Section B.3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Extension to Observational Data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The methodology developed in this work is specifically designed for contexts in which the treatment assignment mechanism is randomized (for example, randomized clinical trials or A/B testing environments). Extending this framework to observational study designs would necessitate the simultaneous estimation of the propensity score e ( X ) e(X) and its explicit incorporation into the construction of the weight function A ^ i \\\\hat{A}_{i} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Empirical Benchmarking\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In contrast to standard CATE estimation, where performance can be assessed using widely adopted benchmarks such as IHDP (Hill, 2011 ) or ACIC (Hahn et al. , 2019 ) , to the best of our knowledge, no public benchmark datasets are currently available for the mediation setting considered here. Developing and analyzing experimental datasets to further evaluate the empirical performance of UNIT and competing methods, therefore, constitutes an important direction for future research.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduced UNIT, a two-stage estimator that pairs a TARNet-learned mediator CATE with the Zheng and Zhou ( 2015 ) G-estimation of a structural mean model under no essential heterogeneity. Identification of the structural parameters is inherited from the NEH framework and holds regardless of the Stage-1 learner; our contribution is to show, both theoretically and in simulation, that a more accurate first-stage representation yields a more informative plug-in weight and thereby a more efficient structural estimator, in high-dimensional and non linear settings. Across nonlinear mediator-effect surfaces, the shared-representation learner reduces the Stage-2 standard error of the mediation coefficient by roughly one third relative to a linear T-learner, while maintaining low bias and close to nomimal coverage rates as long as NEH assumption holds. For applied mediation analyses with randomised treatment and complex covariate–mediator relationships, UNIT offers a substantive efficiency gain compared to other learners (eg. Ridge, RF or TNet) within a semiparametrically principled framework.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Data and Code Availability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All code and frozen simulation data needed to reproduce every table and figure in this paper are available at https://github.com/PsychometricsMZ/UNIT . Each cell aggregates R = 200 R=200 Monte-Carlo seeds; the seed scheme and all hyperparameters are documented in the Appendix. The software stack is Python (NumPy, scikit-learn, and JAX/CATENets for the TARNet learner).\"\n }\n]","isConversational":false}}},{"rowIdx":394,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10534"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Cross-Layer Misalignment Detection in Agent Skills: A Progressive Loading-Aware Contrastive Learning Approach"},"authors":{"kind":"list like","value":["Chengjun Zhang","Yang Gao","Jianna Hur","Jingjing Zhang","Sagar Samtani"],"string":"[\n \"Chengjun Zhang\",\n \"Yang Gao\",\n \"Jianna Hur\",\n \"Jingjing Zhang\",\n \"Sagar Samtani\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10534v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1. Introduction"},{"type":"para","text":"Large Language Models (LLMs) are evolving from single-turn text generators into agentic Artificial Intelligence (AI) systems capable of reasoning, tool use, and interaction with external environments (Yao et al . , 2023 ; Schick et al . , 2023 ) . Agent Skills have become essential supply-side components of agentic AI systems to extend an agent’s capabilities (Anthropic, 2025 ; Li et al . , 2026 ) . Agent Skills encapsulate reusable procedural knowledge into loadable modules, which typically include natural language instructions, scripts, and resources to support specific tasks or workflows (Anthropic, 2025 ) ."},{"type":"para","text":"In contrast to standard prompts or standalone tools, Agent Skills are hierarchical artifacts in which a surface metadata layer sits above layers of instructions, code, files, and executables. Users and agents access metadata before progressively accessing deeper components. This progressive loading mechanism introduces the risk that the metadata used for skill selection may not correspond to the behaviors actually supported or enabled by the underlying components (Greshake et al . , 2023 ; Zhan et al . , 2024 ; Debenedetti et al . , 2024 ) . For example, a skill may claim in its metadata to be a harmless productivity assistant while containing prompt injection, credential leakage, data exfiltration, or insecure command execution within its instruction or script layers (Liu et al . , 2026b , a ) . Conversely, a skill may claim to support a specific capability in its description, yet fail to implement it within its instructions or resources. This reflects a broader documentation-implementation consistency problem, where natural-language descriptions can diverge from the executable or procedural artifacts they are intended to summarize (Tan et al . , 2012 ; Zhou et al . , 2017 ; Liu et al . , 2018 ) . In Agent Skill packages, such inconsistency can be described as cross-layer misalignment. In this case, user-facing claims may diverge from the behaviors supported, requested, or enabled by the underlying instructions, resources, and executable components."},{"type":"para","text":"This mismatch warrants attention because trust decisions regarding Agent Skills are frequently made before the package is fully inspected or executed. Marketplace descriptions, repository metadata, and skill names influence the selection process. However, the underlying instruction and resource layers that dictate behavior may not be accessible until later loading or execution stages. Consequently, the trustworthiness of Agent Skills cannot be assessed solely by examining the security of a single layer or by evaluating task success rates after execution. Past research has shown that agent and skill evaluation often focuses either on security vulnerabilities in specific artifacts or on downstream task performance, while cross-layer consistency between surface claims and deeper package evidence remains a distinct evaluation target (Debenedetti et al . , 2024 ; Liu et al . , 2026a ; Li et al . , 2026 ; Mialon et al . , 2023 ; Han et al . , 2026 ) . Instead, it is necessary to determine whether a consistent behavioral contract exists across the different layers prior to execution."},{"type":"para","text":"Current evaluations of agentic AI focus primarily on runtime performance, including task success rates, reasoning-action trajectories, web browsing, multimodal reasoning, or tool proficiency (Mialon et al . , 2023 ; Yao et al . , 2023 ) . Although these paradigms are essential for understanding the execution process, they generally assume that the relevant tools or skills have already been selected and loaded (Mialon et al . , 2023 ; Li et al . , 2026 ; Han et al . , 2026 ; Debenedetti et al . , 2024 ) . The key question is whether the components within a skill package are internally consistent prior to execution. This “pre-execution” setting requires the model to compare heterogeneous forms of evidence, specifically metadata, instructions, and resources. Although general LLMs or cybersecurity LLMs may detect suspicious text or code patterns, they have not been explicitly trained to verify whether high-level claims are genuinely supported by the evidence contained in the underlying package components. In light of these issues, we propose a Progressive Loading-Aware Hierarchical Contrastive Learning (PL-HCL) framework that aims to evaluate cross-layer misalignment in Agent Skills before execution. PL-HCL treats each skill package as a layered artifact consisting of metadata, instructions, and resources, and trains the model to learn whether user-facing claims are supported by deeper package evidence. The framework integrates two key components:"},{"type":"para","text":"A two-stage continued pretraining process that adapts base LLMs to the vocabulary, formatting, and progressive structure of Agent Skill packages. The first stage trains on short metadata-instruction views, and the second stage trains on full package views that include resources."},{"type":"para","text":"A progressive loading-aware hierarchical contrastive learning process learns cross-layer consistency by contrasting aligned same-skill layers with metadata-swapped and corrupted variants. This design directly targets the mismatch between surface claims and deeper behavioral evidence."},{"type":"para","text":"This paper makes several key contributions. First, we construct a large-scale open-source Agent Skill corpus and a human-verified challenge set for aligned/misaligned skill evaluation, enabling systematic study of skill-level trustworthiness before execution. Second, we introduce PL-HCL, a contrastive pretraining framework that learns cross-layer consistency from aligned packages and synthetic pairs. Finally, we evaluate base LLMs, two-stage continued pretraining, and PL-HCL across general and cybersecurity backbones to demonstrate that skill-format adaptation alone is insufficient while explicit cross-layer contrastive learning substantially improves misaligned-class detection."},{"type":"para","text":"The remainder of this paper is organized as follows. We first review prior work on Agent Skill security, continued pretraining and curriculum adaptation, and contrastive learning for hierarchical and multi-view representations. Second, we identify research gaps and present the research questions for this study. Third, we define cross-layer misalignment and describe the dataset. We then present the PL-HCL framework and report the experimental results. Finally, we discuss implications for trustworthy Agent Skill evaluation and conclude with limitations and future directions."},{"type":"heading","level":1,"text":"2. Related Work"},{"type":"para","text":"This section reviews three streams of related research that motivate and ground our problem formulation and proposed method: 1) work on tool-augmented agents and Agent Skill security explains why reusable skill components create a new trustworthiness surface for agentic AI; 2) continued pretraining and curriculum adaptation motivate our use of staged skill-format adaptation before contrastive learning, and 3) contrastive and hierarchical representation learning provide the methodological foundation for learning cross-layer consistency within agent skills."},{"type":"heading","level":2,"text":"2.1. Tool-Augmented Agents and Agent Skill Security"},{"type":"heading","level":3,"text":"2.1.1. Tool-Augmented Agent Evaluation"},{"type":"para","text":"Research on tool-augmented agents explores how LLMs extend their reasoning and task-completion capabilities through external tools, APIs, and environment actions. For example, ReAct demonstrated an agent pattern that interleaves reasoning traces with actions, enabling LLMs to invoke external information or tools during task execution (Yao et al . , 2023 ) . Toolformer further investigated how LLMs can learn to call APIs at appropriate junctures, thereby integrating tool use into model behavior (Schick et al . , 2023 ) . As agent systems shift from isolated text generation to external interaction, evaluation has increasingly focused on agent performance in complex task environments. GAIA evaluates the task-completion capabilities of general AI assistants in dimensions such as reasoning, web browsing, multimodal understanding, and tool-use proficiency (Mialon et al . , 2023 ) . Recent red-teaming work further shows that autonomous agents equipped with persistent memory, communication channels, file-system access, or shell execution can produce failures that involve security, privacy, and governance risks. These risks include information disclosure, destructive actions, identity spoofing, and partial system takeover (Shapira et al . , 2026 ) . Such research demonstrates that evaluating agentic AI requires attention not only to the quality of the final answer but also to the broader trustworthiness risks arising from the interaction between agents and external capabilities. Therefore, we review the characteristics of Agent Skill."},{"type":"heading","level":3,"text":"2.1.2. Agent Skills as Reusable Evaluation Artifacts"},{"type":"para","text":"As indicated in the introduction, Agent Skills encapsulate procedural knowledge into reusable packages, enabling agents to load task-specific capabilities at inference time (Anthropic, 2025 ; Li et al . , 2026 ) . Unlike a single API endpoint or function call, Agent Skill packages typically comprise marketplace metadata, natural-language instructions, scripts, configuration files, dependencies, and resources (Anthropic, 2025 ) . Thus, Agent Skills are not merely external interfaces invoked by agents. Instead, skills are structured artifacts containing claims, instructions, and executable or resource-based evidence. Recent work has begun to investigate Agent Skills as independent objects of evaluation. For example, SkillsBench examines whether curated or self-generated skills enhance agent performance across tasks, finding that curated skills increase average pass rates, but their effectiveness depends on the domain and task (Li et al . , 2026 ) . Such benchmarks isolate agent skills from general tool-use settings and emphasize the impact of the skill package itself on agent performance. For skill-based agent ecosystems, this demonstrates that reusable skill packages are both auxiliary materials and evaluation artifacts capable of substantially altering agent behavior and task outcomes."},{"type":"heading","level":3,"text":"2.1.3. Security and Trustworthiness Risks in Agent Skills"},{"type":"para","text":"As Agent Skills are increasingly used to extend agent capabilities, security studies have begun to systematically analyze their risks. Liu et al. (Liu et al . , 2026a ) conducted a security analysis of a large-scale set of skills from community registries, finding that malicious skills can facilitate credential theft, remote code execution, agent manipulation, and other attack behaviors through natural-language instructions and helper scripts. Other studies have also highlighted issues such as prompt injection, credential leakage, data exfiltration, privilege escalation, unsafe code execution, and hidden malicious behavior in open-source skills (Liu et al . , 2026b ; Schmotz et al . , 2026 ; Jin et al . , 2026 ) . Taken together, these studies show that Agent Skills play a vital role in establishing trustworthiness in agentic AI from the supply side. Evaluating the structure, content, and potential security issues of reusable skill packages is a crucial prerequisite for understanding and improving the trustworthiness of agentic AI within skill-based agent ecosystems. However, the evolving and hierarchical nature of Agent Skills necessitates appropriate training and representation learning strategies."},{"type":"heading","level":2,"text":"2.2. Continued Pretraining and Curriculum Adaptation"},{"type":"para","text":"Agent Skill packages are layered artifacts comprising metadata, instructions, and resources. Metadata and skill descriptions are often surface-level capability claims, while the underlying SKILL.md file, resources, and executables are longer, heterogeneous, and contain clearer behavioral evidence. Consequently, models need to adapt to a skill’s high-level descriptions before engaging with the more complex, resource-level information. Curriculum learning has emerged as a training strategy in which a model learns from easy to hard samples, and holds significant promise for modelling Agent Skill structures. Early work indicated that models could first learn fundamental structures from simpler or clearer examples before gradually transitioning to more complex ones, thereby improving optimization and generalization (Bengio et al . , 2009 ) . Subsequent research extended into self-paced learning and competence-based curriculum learning. These approaches allow models to encounter data with varying difficulty levels, determined by sample difficulty or model competence, throughout training stages (Kumar et al . , 2010 ; Platanios et al . , 2019 ) . This body of work demonstrates that the training process is not merely a matter of repeated sampling from a fixed data distribution; the training order itself serves as a crucial learning signal, helping the model progressively adapt to complex structures."},{"type":"para","text":"While continued pretraining addresses different issues than curriculum learning, the two are complementary in the context of model adaptation. Continued pretraining typically involves further training a pre-existing language model on domain-specific or task-relevant corpora, enabling it to adapt to the target domain’s vocabulary, style, and discourse patterns prior to downstream supervised learning. Gururangan et al. (Gururangan et al . , 2020 ) conducted a systematic study of domain-adaptive pretraining (DAPT) and task-adaptive pretraining (TAPT), demonstrating that adaptive pretraining can enhance downstream performance across various domains, such as biomedicine, computer science, news, and reviews. Related works have also applied domain-adaptive methods to specialized text-analysis settings, including legal, scientific, biomedical, and news-analysis tasks (Lee et al . , 2020 ; Beltagy et al . , 2019 ; Chalkidis et al . , 2020 ; Zhang et al . , 2026 ) . These studies collectively indicate that model adaptation often requires considering both what data the model sees and in what order it sees them. In the context of Agent Skill analysis, this requires a strategy to carefully represent the hierarchical and multi-faceted structure of skills."},{"type":"heading","level":2,"text":"2.3. Contrastive Learning for Hierarchical and Multi-View Representations"},{"type":"para","text":"For Agent Skill packages, metadata, instructions, and resources can be seen as distinct layers or views of the same skill artifact. Metadata provides user-facing capability claims, instructions offer agent-facing procedural guidance, and resources, along with executable artifacts, serve as evidence that is closer to actual behavior. Therefore, hierarchical and multi-view contrastive learning provide a methodological foundation for cross-layer representation learning. Contrastive learning is a class of self-supervised representation learning that seeks to pull related views or semantically similar samples closer together while pushing unrelated samples further apart. A prevailing approach is SimCLR, which constructs positive pairs from different augmentations of the same image and employs a contrastive objective to learn transferable visual representations (Chen et al . , 2020 ) . Supervised contrastive learning further incorporates label information into the contrastive objective, bringing samples of the same class closer together in the representation space while increasing the separation between samples of different classes (Khosla et al . , 2020 ) . In the realm of language-image representation learning, CLIP demonstrates the effectiveness of contrastive objectives for cross-modal alignment and transferable representation learning by aligning image-text pairs (Radford et al . , 2021 ) . These works show that contrastive learning, when using carefully defined positive and negative samples, can shift representation learning away from a singular focus on reconstruction or classification and toward modeling relationships."},{"type":"para","text":"Beyond general contrastive learning, multi-view contrastive learning focuses on how different views of the same underlying object share semantic factors. Contrastive Multiview Coding (CMC) treats different sensory channels as complementary sources of information, learning view-invariant representations by maximizing cross-view agreement for the same instance (Tian et al . , 2020 ) . Similarly, CLIP views images and text as distinct modalities of the same semantic content, learning aligned multimodal embeddings through large-scale contrastive pretraining (Radford et al . , 2021 ) . These studies show that contrastive learning facilitates learning both individual representations and shared semantics when objects have multiple views."},{"type":"para","text":"Hierarchical contrastive learning extends the contrastive objective to data characterized by hierarchies or multi-level structures. In hierarchical text classification, label taxonomies provide a class-level hierarchy; models can leverage parent-child label relationships, label paths, or hierarchy-aware positive samples to learn text representations that better align with the underlying label structure. HGCLR integrates the label hierarchy into the text encoder, learning hierarchy-aware representations through hierarchy-guided positives (Wang et al . , 2022 ) . Hierarchical multi-label contrastive learning further leverages multi-level labels and label relations to preserve the class hierarchy, ensuring the representation space reflects the structural relationships between different labels (Zhang et al . , 2022 ) . In relation extraction and recommendation settings, hierarchical contrastive objectives are also employed to combine global structure with local interactions, thereby learning multi-granularity representations (Li et al . , 2022 ; Wu et al . , 2023 ) . These studies show that contrastive learning supports structure-aware representations, with positive and negative pairs defined by labels, modalities, views, or hierarchical relations, not just data augmentation. Within the context of Agent Skill analysis, these approaches can help models learn shared skill semantics by aligning layers belonging to the same skill, while also learning to detect signals of inconsistency between layers through mismatched or corrupted layer combinations."},{"type":"heading","level":1,"text":"3. Research Gaps and Questions"},{"type":"para","text":"We identified several research gaps from our literature review. First, although research on tool-augmented agents and Agent Skill security is expanding, current evaluation frameworks offer limited mechanisms for assessing the trustworthiness of a skill package prior to execution. Most agentic AI benchmarks focus on evaluating task completion, reasoning processes, or tool-use proficiency only after tools or skills have been selected and loaded. Consequently, there remains an unresolved challenge in evaluating the skill artifact itself at the stage when users or agents must make trust decisions based solely on partial or surface-level information. Second, existing Agent Skill security research has identified specific risks, including prompt injection, credential leakage, data exfiltration, unsafe code execution, and concealed malicious behavior. However, these studies predominantly address vulnerabilities or malicious intent, whereas Agent Skill trustworthiness also relies on the alignment between user-facing descriptions and the underlying instruction and resource layers. As a result, a skill may be misaligned without being overtly malicious, such as when its metadata exaggerates or inaccurately represents capabilities not supported by the underlying package. Third, although continued pretraining helps LLMs learn skill package formats, it does not train models to compare claims across different layers. Finally, contrastive and hierarchical methods leverage structure from labels or graphs, but Agent Skills require a unique structural signal. Their hierarchy arises from progressive loading, with metadata first and instructions or resources providing deeper behavioral evidence. Based on these gaps, we pose the following research questions:"},{"type":"para","text":"RQ1: How can cross-layer misalignment in Agent Skills be formalized as a pre-execution, artifact-level evaluation problem?"},{"type":"para","text":"RQ2: To what extent can base general and cybersecurity LLMs detect cross-layer misalignment without skill-specific adaptation?"},{"type":"para","text":"RQ3: How does two-stage continued pretraining improve misalignment detection?"},{"type":"para","text":"RQ4: How does progressive loading-aware hierarchical contrastive learning improve misaligned-skill detection beyond base LLMs and CPT-only checkpoints?"},{"type":"heading","level":1,"text":"4. Progressive Loading-Aware Hierarchical Contrastive Learning (PL-HCL) Framework for Detecting Misaligned Agent Skills"},{"type":"para","text":"We propose a PL-HCL framework for Agent Skill misalignment detection, illustrated in Figure 1 . The proposed framework consists of four major components: (1) public Agent Skill data is collected and processed; (2) two-stage continued pretraining adapts the base LLM to these packages; (3) progressive loading-aware hierarchical contrastive learning then distinguishes between aligned and misaligned skill layers; and (4) the model determines whether user-facing metadata is supported by underlying instructions and resources. Each component is described in the following subsections."},{"type":"heading","level":2,"text":"4.1. Skill Corpus Construction"},{"type":"para","text":"We build an open-source Agent Skill corpus by linking public records from SkillsMP (SkillsMP, 2026 ) , the largest open-source Agent Skills platform, to their GitHub repositories and normalizing each retrieved package into a layered representation S = ( M , I , R ) S=(M,I,R) : the metadata layer M M contains user-facing fields such as the skill name, description, categories, tags, repository identifiers, and declared resource references; the instruction layer I I contains agent-facing procedural guidance; and the resource layer R R contains scripts, configuration files, dependencies, examples, assets, and other supporting files. From 273,657 SkillsMP catalog entries, 264,937 packages were successfully downloaded and normalized; after applying the 4,096-token metadata–instruction budget and the 10,240-token full-package budget, 248,473 packages fit at least one training stage and form the unlabeled training corpus, while the held-out Challenge Set is excluded from both CPT and PL-HCL training. 1 1 1 Detailed corpus construction, token-budget coverage, and filtering statistics are provided in the supplementary materials."},{"type":"para","text":"This representation preserves the distinction between user-facing claims and deeper package evidence, supporting our pre-execution, artifact-level task: the detector observes package artifacts available at inspection time and assesses whether user-facing metadata is supported by the instruction and resource layers. We do not verify runtime-only behaviors unless the package provides observable evidence."},{"type":"heading","level":2,"text":"4.2. Progressive CPT"},{"type":"para","text":"The prediction target of our proposed approach is whether the layers of a skill package are mutually consistent: 0 if aligned, 1 if misaligned. A skill is considered misaligned when its metadata describes a capability or intent not supported by the instruction or resource layers. We consider two forms of misalignment: (1) malicious misalignment , which masks harmful behavior in deeper layers, and (2) benign misalignment , when the metadata overstates or misdescribes capabilities that the deeper layers do not substantiate. The two-stage continued pretraining component adapts the backbone of a decoder-only LLM to Agent Skill packages prior to training for misalignment detection. This stage enables the model to acquire familiarity with the vocabulary, formatting conventions, and hierarchical structure of Agent Skills, instead of immediately requiring the base LLM to assess the correspondence between metadata and underlying artifacts. We employ a curriculum-inspired adaptation strategy in which the model initially learns from concise, structured representations of a skill and subsequently progresses to more comprehensive and heterogeneous package contexts."},{"type":"para","text":"For each normalized skill S = ( M , I , R ) S=(M,I,R) , we construct two layer-delimited views: 1) The short view comprises metadata M M and instructions I I , representing surface-level claims and agent-facing guidance available during initial skill inspection; and 2) The full view includes metadata M M , instructions I I , and resources R R , thereby exposing the model to scripts, configuration files, dependencies, assets, and other resource-level evidence. Sequential training on the short view followed by the full view provides the model with a progressive adaptation trajectory from surface skill descriptions to complete multi-file packages. Both stages optimize the standard causal language modeling objective. The first stage adapts the model to metadata and instruction structure. The second stage adapts it to the full package context. The resulting checkpoint, θ CPT \\theta_{\\mathrm{CPT}} , serves as the skill-aware backbone for PL-HCL. 2 2 2 Full training details and pseudocode are provided in the supplementary materials."},{"type":"heading","level":2,"text":"4.3. Progressive Loading-Aware Hierarchical Contrastive Learning (PL-HCL)"},{"type":"para","text":"PL-HCL serves as the consistency-learning component of the workflow. After two-stage CPT, the model has adapted to the vocabulary, formatting, and overall structure of Agent Skills. However, causal language modeling does not explicitly require the model to assess whether user-facing metadata is substantiated by underlying instruction and resource evidence. To this end, PL-HCL transforms cross-layer consistency into a contrastive training signal. For each normalized skill S i = ( M i , I i , R i ) S_{i}=(M_{i},I_{i},R_{i}) , we first construct an aligned view in which metadata, instructions, and resources come from the same package:"},{"type":"para","text":"This positive view represents a skill package in which the surface claims and deeper artifacts are expected to be mutually consistent. We then construct two types of negative views. The first is swap negative , which replaces the metadata layer with metadata from a different skill:"},{"type":"para","text":"This negative preserves the original instruction and resource evidence but alters the surface claim, resulting in a mismatch between the advertised capability and the underlying package behavior."},{"type":"para","text":"The second type is corruption negative , which perturbs the original metadata while maintaining the deeper layers unchanged:"},{"type":"para","text":"This view simulates misleading, exaggerated, or unsupported capability claims."},{"type":"para","text":"PL-HCL encodes the metadata, instruction, and resource layers into a shared contrastive space. The objective is to score the aligned view P i P_{i} higher than its swap and corruption negatives N i A N_{i}^{A} and N i B N_{i}^{B} , thus bringing the same-skill layers closer together and separating mismatched layer combinations. The training signal evaluates whether the surface description is consistent with the evidence provided by the deeper layers of the package, rather than simply identifying suspicious files. This approach mirrors the progressive loading structure of Agent Skills, in which metadata is typically encountered before the complete instruction and resource context is available."},{"type":"para","text":"PL-HCL is trained in two sub-stages that correspond to the short-view and full-view settings used in CPT. The first sub-stage utilizes metadata–instruction views and includes 96,762 positive training pairs, 386,128 swap negatives, and 334,867 corruption negatives. The second sub-stage employs full metadata–instruction–resource views and comprises 182,135 positive training pairs, 646,945 swap negatives, and 258,368 corruption negatives. The Challenge Set is a held-out, human-verified gold-standard evaluation dataset with 1,150 aligned and 294 misaligned skills, designed to test whether models trained on synthetic contrastive pairs can generalize to realistic in-the-wild misalignment cases. 3 3 3 Additional details on dataset construction and verification are provided in the supplementary material This design follows common benchmark practice in which a curated challenge subset is reserved for evaluating model robustness under more realistic or difficult conditions, such as evasive-malware evaluation in EMBER2024 and in-the-wild distribution-shift evaluation in WILDS (Joyce et al . , 2025 ; Koh et al . , 2021 ) ."},{"type":"heading","level":2,"text":"4.4. Evaluation"},{"type":"para","text":"We evaluate the proposed PL-HCL framework through three experiments designed to assess cross-layer misalignment detection in Agent Skills, each aligned with one of our research questions (E0 addresses RQ2, E1 addresses RQ3, and E2 addresses RQ4). In Experiment 0 (E0), we evaluate whether base LLMs can identify misaligned skills without skill-specific adaptation. This experiment compares general and cybersecurity backbones under zero-shot, two-shot, and five-shot prompting settings, allowing us to examine whether general instruction-following ability or security-oriented pretraining is sufficient for this task. In Experiment 1 (E1), we evaluate the effect of two-stage continued pretraining by comparing base checkpoints with CPT-only checkpoints. This experiment assesses whether adapting the model to the vocabulary, formatting, and layered structure of Agent Skill packages improves misalignment detection. In Experiment 2 (E2), we evaluate the full PL-HCL framework by comparing CPT-only checkpoints with CPT+PL-HCL checkpoints. This experiment tests whether explicitly learning cross-layer consistency from aligned, metadata-swapped, and corrupted-claim views provides measurable gains beyond format adaptation."},{"type":"para","text":"Across these experiments, we use both intrinsic and extrinsic evaluation tasks. The intrinsic tasks assess whether training improves the model’s adaptation to Agent Skill artifacts before downstream inference. Specifically, next-token prediction measures whether CPT improves the model’s ability to model layer-delimited skill text, while contrastive pair detection evaluates whether PL-HCL learns to distinguish aligned cross-layer views from metadata-swapped or corrupted-claim variants. The extrinsic task assesses the pre-execution misalignment detection . In this task, the model receives a normalized skill package and predicts whether the metadata is supported by the instruction and resource layers under zero-shot, two-shot, and five-shot prompting conditions."},{"type":"para","text":"The evaluation utilizes a human-verified Challenge Set comprising 1,444 Agent Skills, with 1,150 aligned and 294 misaligned packages (196 are SAFE, 38 are SUSPICIOUS, and 60 are MALICIOUS.) All Challenge Set packages are actual skills collected from SkillsMP (rather than synthetically perturbed variants); misaligned examples are naturally occurring, in-the-wild packages whose metadata is not supported by their instruction or resource layers. Candidate skills were screened automatically and subsequently verified by human annotators (see Figure 1 ). This Challenge Set is excluded from pretraining and contrastive learning and is reserved solely for final evaluation. Because all PL-HCL training negatives are synthetically constructed, evaluation on this set tests whether the learned consistency signal generalizes from synthetic contrast pairs to naturally occurring misalignment. Given the Challenge Set’s imbalance toward aligned skills, both overall and misalignment-sensitive metrics are reported. Accuracy (Acc) measures the overall proportion of correct predictions. Since accuracy may be inflated by the aligned-class majority, misaligned-class F1 (F1 m ) treats misaligned as the positive class and measures the model’s ability to correctly identify the safety-critical class. Macro-F1 computes the unweighted average of aligned-class and misaligned-class F1, providing a balanced perspective across both labels. Area under the ROC curve (AUC) and average precision for the misaligned class (AP m ) are also reported to evaluate ranking quality under varying decision thresholds."},{"type":"heading","level":1,"text":"5. Results"},{"type":"para","text":"This section presents the evaluation results on the human-verified Challenge Set following the E0–E2 evaluation design described in Section 4.4 . E0 evaluates whether unadapted base LLMs can detect cross-layer misalignment without skill-specific adaptation. E1 isolates the effect of two-stage CPT by comparing base checkpoints with CPT-only checkpoints. E2 evaluates the incremental contribution of PL-HCL by comparing CPT-only checkpoints with CPT+PL-HCL checkpoints. 4 4 4 k denotes the number of labeled in-context examples. F1 m and AP m are computed for the misaligned class. Macro-F1 is the unweighted average of aligned-class and misaligned-class F1. Base, CPT, and CPT+PL-HCL denote the unadapted backbone, the two-stage continued-pretraining checkpoint, and the full contrastive checkpoint, respectively. We first summarize the main findings across E0–E2, and then report the results for E1 and E2 in detail. 5 5 5 Additional intrinsic evaluations, full prompting-setting results, and extended metric tables are provided in the supplementary materials."},{"type":"heading","level":2,"text":"5.1. Main Findings across E0-E2"},{"type":"para","text":"Table 1 provides a compact summary of the strongest prompting setting for each backbone and adapter on the Challenge Set. To avoid relying on a single test-selected configuration, Tables 2 and 3 report the full results for all k ∈ 0 , 2 , 5 k\\in{0,2,5} prompting settings. We use Table 1 only as a high-level summary, while the main comparisons in 5.2 and 5.3 are based on the full prompting-setting results. Unless otherwise stated, the decision rule is fixed across models, and no model-specific threshold is tuned on the Challenge Set. In deployment, both the prompting configuration and decision threshold should be calibrated using a held-out validation set from the target skill ecosystem."},{"type":"para","text":"The E0 results (base rows of Table 1 ) show that base LLMs demonstrate limited performance on the misaligned class, despite achieving accuracy levels close to the aligned-class majority baseline. For instance, both base Llama-3.1-8B and base Foundation-Sec-8B attain approximately 0.81 accuracy in their optimal prompting configurations; however, their F1 m remains below 0.15. These results suggest that the models frequently classify plausible skill packages as aligned without consistently verifying whether the surface claim corresponds to deeper package evidence. The CPT+PL-HCL approach helps improve performance. On Llama-3.1-8B, CPT+PL-HCL achieves 0.929 accuracy, 0.788 F1 m , and 0.872 Macro-F1. On Foundation-Sec-8B, CPT+PL-HCL attains its highest performance, with 0.937 accuracy, 0.817 F1 m , and 0.889 Macro-F1 in the two-shot setting. The improvements are concentrated in F1 m , which represents the safety-critical class for this task. Base-cybersecurity LLMs do not independently bridge the performance gap. WhiteRabbitNeo-2-8B, RedSage-Qwen3-8B-DPO, and Foundation-Sec-8B, when used without skill-specific adaptation, exhibit Macro-F1 scores close to the majority-class baseline. Although certain security-tuned models achieve higher area under the curve (AUC) or AP m , indicating some ability to rank risky cases. However, this signal does not translate into reliable classification performance without PL-HCL."},{"type":"heading","level":2,"text":"5.2. E1: Evaluation of CPT"},{"type":"para","text":"Table 2 reports E1 by comparing unadapted base LLMs with CPT-only checkpoints across k = 0 k=0 , k = 2 k=2 , and k = 5 k=5 prompting settings. This comparison directly evaluates whether continued pretraining on Agent Skill packages improves misalignment detection, or whether it primarily helps the model adapt to the vocabulary, formatting, and layered structure of skill artifacts."},{"type":"para","text":"CPT improves over base checkpoints in the zero-shot setting, suggesting that continued pretraining helps the model adapt to the vocabulary, formatting, and layered structure of Agent Skill packages. For Llama-3.1-8B, zero-shot F1 m increases from 0.032 to 0.130 and Macro-F1 increases from 0.457 to 0.511 after CPT. Foundation-Sec-8B shows a similar pattern, with zero-shot F1 m increasing from 0.026 to 0.134 and Macro-F1 from 0.454 to 0.513. The improvement, however, remains limited. Across prompting settings, CPT-only Macro-F1 stays near 0.48–0.52, and F1 m remains below 0.15. Thus, skill-format adaptation alone does not provide a reliable misalignment detector. CPT makes the model more familiar with skill packages, but it does not explicitly train the model to compare advertised claims with deeper behavioral evidence."},{"type":"heading","level":2,"text":"5.3. E2: Evaluation of PL-HCL"},{"type":"para","text":"Table 3 reports E2 by comparing CPT-only checkpoints with CPT+PL-HCL checkpoints across k = 0 k=0 , k = 2 k=2 , and k = 5 k=5 prompting settings. This comparison tests whether explicitly learning cross-layer consistency from aligned, metadata-swapped, and corrupted-claim variants improves misaligned-skill detection beyond the skill-format adaptation provided by CPT."},{"type":"para","text":"Adding PL-HCL produces the largest gains across all prompting settings. On Llama-3.1-8B, Macro-F1 increases from 0.511, 0.520, and 0.494 to 0.717, 0.872, and 0.867 for k = 0 , 2 , 5 k=0,2,5 , respectively. On Foundation-Sec-8B, Macro-F1 increases from 0.513, 0.522, and 0.483 to 0.779, 0.889, and 0.887. The largest gains appear in F1 m . CPT-only never exceeds 0.150 F1 m , while CPT+PL-HCL reaches 0.788 on Llama-3.1-8B and 0.817 on Foundation-Sec-8B in the two-shot setting. These results show that PL-HCL supplies the discriminative signal missing from ordinary continued pretraining. By contrasting aligned packages against swapped and corrupted layer combinations, PL-HCL turns cross-layer consistency into a usable detection cue. Notably, PL-HCL is trained exclusively on synthetic negatives, yet these gains are measured on naturally occurring, in-the-wild misaligned skills, indicating that the learned claim-versus-evidence signal transfers beyond the perturbation distribution used for training."},{"type":"heading","level":2,"text":"5.4. Misaligned Examples In-the-wild"},{"type":"para","text":"While aggregate metrics indicate that PL-HCL significantly improves detection for the misaligned category, looking only at average results leaves unclear what signals the model has learned. To investigate, we examined two representative skills from the Challenge Set."},{"type":"para","text":"For both, the zero-shot base model Foundation-Sec-8B-Reasoning classified them as aligned with high confidence. After PL-HCL training, the same model reclassified them as misaligned . Although both are correctly labeled misaligned , they represent different safety issues. The first example is an obviously unsafe Skill involving a Chinese-language prompt injection attack. Its metadata and instruction body contain commands designed to override the system prompt, and it has broad Bash() and WebFetch() permissions. The second example demonstrates misalignment but is not inherently malicious (benign but misaligned). Its metadata claims to provide an inline TypeScript assertion function, but the instruction body actually invokes a third-party recipe URL using curl . As a result, it does not support the claimed tiny-invariant dependency or TypeScript-specific behaviors."},{"type":"para","text":"The base model yielded positive “aligned” logit differences for these examples (+4.33 and +5.81), both exceeding the model’s average for the aligned class on the Challenge Set (+3.29). This matches the “majority class bias” observed in Section 5 : base LLMs tend to classify plausible skill packages as aligned . After PL-HCL, the logit differences shifted to negative values (-0.13 and -1.50), crossing the decision boundary and resulting in misaligned classification. Interestingly, the benign yet mismatched example produced a stronger mismatch signal despite lacking obvious jailbreak indicators like Bash(*) , “ignore previous instructions,” or shell execution hooks. This shows PL-HCL does not just react to malicious tokens or prompt injection keywords; it learns a “claim-versus-evidence” signal, evaluating if the surface description is genuinely supported by the instruction and resource layers."},{"type":"para","text":"These examples clarify why mismatch detection for Agent Skills cannot rely solely on general instruction-following or specialized cybersecurity expertise. Whether using a general or cybersecurity LLM, a plausible Skill package might still get a high “aligned” confidence score even if its components contradict its claimed capabilities or hide unsafe behaviors. While base cybersecurity LLMs sometimes achieve higher AUC or AP scores and can rank risky examples to some extent, this does not consistently lead to accurate “aligned” versus “misaligned” classification."},{"type":"heading","level":1,"text":"6. Discussion"},{"type":"para","text":"This article aims to establish a pre-execution screening capability for skill-augmented agentic AI systems. Rather than viewing Agent Skills solely as reusable modules for task performance, this work conceptualizes Agent Skills as layered supply-side artifacts, where metadata, instructions, and resource or executable artifacts collectively determine trustworthiness. The primary practical contribution of PL-HCL is its ability to transform cross-layer consistency into a learnable and deployable detection signal, enabling screening for downstream decision making prior to skill loading or execution. This perspective holds direct relevance for three key stakeholder groups: skill marketplaces, skill users, and security analysts. We describe each in turn below."},{"type":"para","text":"For skill marketplaces, PL-HCL can function as a pre-publication or pre-listing governance mechanism. In open skill ecosystems, skills are frequently submitted by third-party contributors, and marketplace metadata serves as the primary basis for users and agents to assess trustworthiness. Absent installed-base screening, misalignment risks may be transferred to downstream users. PL-HCL evaluates whether metadata is substantiated by instructions and resources before a skill is released or recommended, directing high-risk skills to manual review, suspending release, or marking approved skills with a verified-alignment badge. PL-HCL would not replace manual auditing or runtime sandboxing, but would offer a scalable initial filter for marketplace-level governance."},{"type":"para","text":"For skill users, PL-HCL enables lightweight client-side screening. Most users do not examine files such as SKILL.md , scripts, dependencies, or external references individually when installing Agent Skills. Instead, they often rely on the skill name, description, and repository metadata. PL-HCL can be deployed as a local plugin or command-line scanner that analyzes the local skill directory post-installation and issues warnings when surface claims are inconsistent with deeper package evidence. This approach provides users with direct alignment signals, offering additional trust cues prior to skill execution and reducing reliance on marketplace curation or post-execution observation."},{"type":"para","text":"For security analysts, PL-HCL offers an operational tool for retrieval and triage. Analysts frequently need to identify high-risk subsets within extensive collections of open-source packages and determine which warrant resource-intensive sandboxed analysis or manual inspection. Because PL-HCL learns cross-layer representations rather than producing only binary classification scores, it facilitates grouping of similar suspicious skills, prioritization of high-risk clusters, and identification of skill families exhibiting similar metadata and behavioral inconsistencies. Thus, PL-HCL can serve as a foundational representation for large-scale skill ecosystem analysis, supporting incident response and adversarial analytics workflows."},{"type":"para","text":"Methodologically, PL-HCL’s broader implication is its treatment of progressively loaded, hierarchical artifacts as novel evaluation objects. Similar structural patterns are present in MCP servers, browser extensions, VS Code extensions, and software package registries. These artifacts typically include user-facing descriptions, configuration files, permission declarations, scripts, or executable resources, with users generally encountering surface claims before deeper behavioral elements. Consequently, this study’s guiding design principle is that, when trust decisions must be made prior to full execution, models should explicitly learn the consistency between surface claims and deeper evidence. This departs from approaches that rely solely on generic language modeling, keyword-based security scanning, or post-execution behavioral observation."},{"type":"para","text":"As with any research, this work has its limitations. First, the focus is on pre-execution, artifact-level assessment. Our proposed approach relies exclusively on information available to the skill package prior to execution, and therefore cannot address failure modes that manifest only during live interaction, dynamic remote payload execution, post-deployment mutations, or environment-specific side effects. Runtime sandboxing and trace-level agent evaluation are thus complementary to this approach. Second, the dataset is primarily derived from SkillsMP and open-source skills available on GitHub. Closed skill ecosystems, internal enterprise skills, and dynamically generated skills may exhibit different package structures, documentation practices, and risk profiles. Third, our experiments are conducted primarily on 8B-scale backbones. This design allows us to compare general and cybersecurity LLMs under a controlled and computationally feasible setting, but it does not fully characterize how model scale affects Agent Skill misalignment detection. Larger models may exhibit stronger cross-layer reasoning or better few-shot sensitivity, while smaller models may require different adaptation strategies. Hence, we cannot ensure that the observed performance trends fully generalize across model sizes."},{"type":"heading","level":1,"text":"7. Conclusion and Future Directions"},{"type":"para","text":"Agent Skills have become an integral component of modern agentic AI systems. Despite their growing prevalence, Agent Skills often suffer from misalignment between user-facing claims and deeper instructional or executable evidence. By conceptualizing each skill as a layered artifact comprising metadata, instructions, and resources, this work proposes a pre-execution evaluation target for trustworthy agentic AI. Specifically, before a skill is loaded or executed, evaluators should assess whether its advertised capabilities are substantiated by package evidence available at inspection time. A large-scale open-source Agent Skill corpus and a human-verified Challenge Set were constructed, and PL-HCL, a progressive loading-aware contrastive framework for learning cross-layer consistency among metadata, instructions, and resources, was proposed. Experimental results indicate that both general and cybersecurity LLMs struggle to reliably identify misaligned skills. The two-stage CPT approach primarily enhances skill-format adaptation but is insufficient as a standalone detector. In contrast, PL-HCL significantly improves misaligned-class detection by explicitly modeling the relationship between surface claims and deeper package evidence."},{"type":"para","text":"Future research can expand along both empirical and methodological dimensions. First, cross-layer misalignment detection may be integrated with runtime sandboxing and trajectory-level evaluation to capture behaviors not directly observable from package artifacts, such as dynamic remote payloads, environment-specific side effects, or interaction-dependent failures. Second, PL-HCL could be extended into directed hierarchical contrastive learning, enabling the model to assess not only layer consistency but also the direction of support among layers. In this framework, metadata provides surface claims, instructions offer procedural support, and resources supply behavioral evidence. A directed objective would distinguish whether misalignment results from metadata overclaiming unsupported capabilities or from deeper artifacts introducing undisclosed behaviors. Third, future models could incorporate richer artifact representations, including dependency graphs, permission structures, external endpoint analysis, and execution traces, to better capture behavioral evidence that is only partially expressed in text. Additionally, future evaluations should examine whether these approaches generalize across closed-source or enterprise skill ecosystems and across different model scales. In summary, Agent Skill evaluation should address not only improvements in task performance but also the consistency between surface claims and the deeper artifacts that influence agent behavior."},{"type":"para","text":"This supplement provides additional reproducibility details, including dataset construction, training settings, pseudocode, and extended evaluation results. The main paper is self-contained and does not rely on the supplement for its core argument or primary findings."},{"type":"heading","level":2,"text":"A.1. Package Retrieval and Normalization"},{"type":"para","text":"Our corpus is built by linking public skill records from SkillsMP to the GitHub repositories that host their implementations. From each SkillsMP record, we extract the skill name, marketplace description, category or tag information when available, repository owner, repository name, branch, and relative package path. We then download the corresponding repository snapshot from GitHub and extract the skill package from the declared path."},{"type":"para","text":"Each downloaded package is normalized into the tuple S = ( M , I , R ) S=(M,I,R) . The metadata layer M M contains marketplace and repository-level fields, including the skill name, description, tags or categories when available, repository identifiers, and source path. The instruction layer I I is primarily derived from SKILL.md and related instruction files. The resource layer R R contains scripts, configuration files, examples, assets, dependency files, API wrappers, and external references."},{"type":"para","text":"Packages are retained only when the GitHub repository is accessible, the declared package path exists, and a valid instruction artifact is present. If a package contains only SKILL.md , it is still retained; its resource layer is represented as empty or minimal. This preserves instruction-only skills while making the absence of supporting resources explicit."},{"type":"heading","level":2,"text":"A.2. Corpus Construction Details"},{"type":"para","text":"We constructed the Agent Skill corpus from the largest open-source Agent Skill marketplace, SkillsMP . A REST API crawl produced 273,657 unique metadata listings after rolling deduplication. We then used repository links in these listings to download the corresponding public GitHub packages and retained packages with a parseable SKILL.md . Each retained package was normalized into metadata, instruction, and resource layers based on SKILL.md frontmatter, the procedural body of SKILL.md , and accessible package files under directories such as scripts/ , assets/ , and reference/ ."},{"type":"para","text":"Before training, we applied two corpus-level filters. First, packages whose normalized context exceeded the maximum input budget of the corresponding training stage were excluded from that stage. Stage 1 uses a 4,096-token metadata-plus-instruction rendering, while Stage 2 uses a 10,240-token full-package rendering that also includes resource files. Second, exact overlaps with benchmark or supervision sources were excluded using repository owner, repository name, and skill name to reduce the risk of label leakage. After removing unavailable repositories, incomplete packages, duplicate skills, and unparseable files, 264,937 skill packages were successfully downloaded and normalized. Among them, 248,473 packages fit at least one training stage and were eligible for CPT or PL-HCL training; packages that fit neither token budget were excluded from training."},{"type":"para","text":"Stage 2 coverage is lower because full-package renderings include heterogeneous resource files when available. In particular, 149,150 normalized skills contain no package resource files; among the remaining 115,787 skills with resource files, 91,053 fit the Stage-2 full-package budget. The resulting token-budgeted corpus provides the unlabeled skill views used for progressive CPT and PL-HCL, while the Challenge Set is held out from both training stages and used only for extrinsic evaluation."},{"type":"heading","level":2,"text":"A.3. Reproducibility Manifest"},{"type":"para","text":"For reproducibility, each retained package is accompanied by a manifest entry. The manifest records source identifiers, repository owner, repository name, branch, package path, artifact types, file counts, token counts, filtering decisions, and normalization status. These records allow us to trace each training instance back to its original marketplace and repository source while preserving the normalized M M , I I , and R R representation used in training and evaluation."},{"type":"para","text":"Gold labels for the Challenge Set were produced through a three-stage pipeline. First, we used the MASB scanner to generate severity-ranked candidate flags from the collected corpus; the scanner served only as a candidate-generation tool, not as the source of final labels. Second, flagged skills were inspected in a Jetstream Docker sandbox with behavioral instrumentation for network connections, subprocess calls, filesystem writes, dependency installation, and potential exfiltration. Third, human coders compared each skill’s metadata claim with its instruction/resource behavior and assigned two independent labels: final-align , indicating whether the skill is ALIGNED or MISALIGNED, and final-malicious , indicating whether the skill is SAFE, SUSPICIOUS, or MALICIOUS."},{"type":"para","text":"Two human annotators conducted the verification and quality-assurance process. To assess label reliability, 144 randomly sampled Challenge Set examples were independently re-coded. The two annotators disagreed on 4 cases, yielding 97.2% agreement. Disagreements and uncertain cases were resolved through discussion and artifact re-inspection before finalizing the gold labels."},{"type":"para","text":"The corpus version used in this study was crawled from 2026-04-15 to 2026-04-17 using the public skillsmp listing endpoint and linked GitHub repositories. We retained packages with a parseable SKILL.md and non-empty name / description fields, and excluded deleted repositories, duplicates, forks of retained corpus members, and packages that redirected to non-skill repositories. The released corpus contains parsed structural fields and content hashes rather than raw bundled scripts; per-skill content remains governed by the upstream repository license. The malicious-misaligned portion of the Challenge Set is released only under a controlled-access data-use agreement. The corpus contains public marketplace metadata and committed repository content only; it includes no end-user telemetry, install logs, or private personal information. We’ve released the dataset on Huggingface : https://huggingface.co/datasets/anon-skillsalign-26/skill_align . Code files are hosted on: https://anonymous.4open.science/r/skill_align_anonymous_v2-C628/ ."},{"type":"heading","level":2,"text":"D.1. Two-Stage CPT Objective"},{"type":"para","text":"For a rendered token sequence x 1 , … , x T x_{1},\\ldots,x_{T} , let o t , k o_{t,k} denote the model logit for vocabulary token k k at position t t . The next-token probability is:"},{"type":"para","text":"The CPT loss is:"},{"type":"heading","level":2,"text":"D.2. PL-HCL Objective"},{"type":"para","text":"Let h θ ( X ) h_{\\theta}(X) be the hidden representation produced by the CPT-adapted model for layer X ∈ { M , I , R } X\\in\\{M,I,R\\} , and let g ϕ g_{\\phi} be a projection head. The projected layer embedding is:"},{"type":"para","text":"For a package S = ( M , I , R ) S=(M,I,R) , the cross-layer score is:"},{"type":"para","text":"where sim ( ⋅ , ⋅ ) \\operatorname{sim}(\\cdot,\\cdot) is cosine similarity and α M I + α M R + α I R = 1 \\alpha_{MI}+\\alpha_{MR}+\\alpha_{IR}=1 ."},{"type":"para","text":"The PL-HCL loss is:"},{"type":"para","text":"where τ \\tau is the temperature and 𝕀 i B \\mathbb{I}_{i}^{B} indicates whether a corruption negative is sampled for S i S_{i} ."},{"type":"heading","level":2,"text":"D.3. Training Data Splits"},{"type":"para","text":"Table 5 reports the train, validation, and test examples actually consumed during CPT, PL-HCL, and intrinsic evaluation after applying the 25% per-split subsampling used for compute feasibility. Table 6 reports the corresponding whole-corpus and anchor-package accounting. CPT rows report sampled skill examples, while PL-HCL rows report sampled contrastive examples generated from the corresponding T1/T2/T3 pools. The Challenge Set is test-only, excluded from all CPT and PL-HCL training stages, and used only for the main extrinsic evaluation."},{"type":"para","text":"Table 7 lists the hyperparameters used for the two pretraining phases. Values were held fixed across the two reference backbones unless otherwise noted."},{"type":"heading","level":4,"text":"Compute."},{"type":"para","text":"Each reference backbone was pretrained on a 2 × 2\\times NVIDIA H200 node using bf16 mixed precision and data parallelism. The full two-stage CPT and PL-HCL pipeline required approximately 48 H200-hours per backbone, and the Challenge Set evaluation was completed in under 2 H200-hours."},{"type":"para","text":"We run two intrinsic diagnostics to verify that the pretrained backbones learn the intermediate objectives used in PL-HCL. First, next-token prediction (NTP) on held-out skill text assesses whether CPT improves the modeling of Agent Skill packages. Second, the held-out PL-HCL contrastive evaluation assesses whether the model separates aligned pairs from constructed swap and corruption negatives. These diagnostics support the main results in § 5 : CPT improves skill-text modeling, and PL-HCL learns the contrastive objective, while extrinsic misalignment detection remains the primary evaluation."},{"type":"heading","level":4,"text":"1. Next-Token Prediction:"},{"type":"para","text":"Table 8 reports NTP metrics before and after CPT. For both full-pipeline backbones, CPT Sub-stage 2 enhances final held-out skill-text metrics, reducing perplexity and boosting top-1 accuracy."},{"type":"heading","level":4,"text":"2. PL-HCL Contrastive Objective:"},{"type":"para","text":"Table 9 reports held-out PL-HCL performance by sub-stage and pair kind. The pair-kind decomposition shows that corrupted type-B pairs are more difficult than swapped type-A pairs, particularly for Foundation-Sec-8B-R, while Llama-3.1-8B performs strongly across all pair types."},{"type":"para","text":"Table 10 reports supplementary per-class metrics and confusion counts for the Challenge Set evaluation. These metrics complement the macro-F1 results reported in § 5 . Because the Challenge Set is class-imbalanced, aligned-class performance alone can obscure failures on the misaligned class. We therefore report precision and recall for both classes, together with confusion counts using aligned as the positive class. Under this convention, false positives aligned correspond to missed misaligned skills, while true negatives aligned correspond to correctly detected misaligned skills."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Large Language Models (LLMs) are evolving from single-turn text generators into agentic Artificial Intelligence (AI) systems capable of reasoning, tool use, and interaction with external environments (Yao et al . , 2023 ; Schick et al . , 2023 ) . Agent Skills have become essential supply-side components of agentic AI systems to extend an agent’s capabilities (Anthropic, 2025 ; Li et al . , 2026 ) . Agent Skills encapsulate reusable procedural knowledge into loadable modules, which typically include natural language instructions, scripts, and resources to support specific tasks or workflows (Anthropic, 2025 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In contrast to standard prompts or standalone tools, Agent Skills are hierarchical artifacts in which a surface metadata layer sits above layers of instructions, code, files, and executables. Users and agents access metadata before progressively accessing deeper components. This progressive loading mechanism introduces the risk that the metadata used for skill selection may not correspond to the behaviors actually supported or enabled by the underlying components (Greshake et al . , 2023 ; Zhan et al . , 2024 ; Debenedetti et al . , 2024 ) . For example, a skill may claim in its metadata to be a harmless productivity assistant while containing prompt injection, credential leakage, data exfiltration, or insecure command execution within its instruction or script layers (Liu et al . , 2026b , a ) . Conversely, a skill may claim to support a specific capability in its description, yet fail to implement it within its instructions or resources. This reflects a broader documentation-implementation consistency problem, where natural-language descriptions can diverge from the executable or procedural artifacts they are intended to summarize (Tan et al . , 2012 ; Zhou et al . , 2017 ; Liu et al . , 2018 ) . In Agent Skill packages, such inconsistency can be described as cross-layer misalignment. In this case, user-facing claims may diverge from the behaviors supported, requested, or enabled by the underlying instructions, resources, and executable components.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This mismatch warrants attention because trust decisions regarding Agent Skills are frequently made before the package is fully inspected or executed. Marketplace descriptions, repository metadata, and skill names influence the selection process. However, the underlying instruction and resource layers that dictate behavior may not be accessible until later loading or execution stages. Consequently, the trustworthiness of Agent Skills cannot be assessed solely by examining the security of a single layer or by evaluating task success rates after execution. Past research has shown that agent and skill evaluation often focuses either on security vulnerabilities in specific artifacts or on downstream task performance, while cross-layer consistency between surface claims and deeper package evidence remains a distinct evaluation target (Debenedetti et al . , 2024 ; Liu et al . , 2026a ; Li et al . , 2026 ; Mialon et al . , 2023 ; Han et al . , 2026 ) . Instead, it is necessary to determine whether a consistent behavioral contract exists across the different layers prior to execution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Current evaluations of agentic AI focus primarily on runtime performance, including task success rates, reasoning-action trajectories, web browsing, multimodal reasoning, or tool proficiency (Mialon et al . , 2023 ; Yao et al . , 2023 ) . Although these paradigms are essential for understanding the execution process, they generally assume that the relevant tools or skills have already been selected and loaded (Mialon et al . , 2023 ; Li et al . , 2026 ; Han et al . , 2026 ; Debenedetti et al . , 2024 ) . The key question is whether the components within a skill package are internally consistent prior to execution. This “pre-execution” setting requires the model to compare heterogeneous forms of evidence, specifically metadata, instructions, and resources. Although general LLMs or cybersecurity LLMs may detect suspicious text or code patterns, they have not been explicitly trained to verify whether high-level claims are genuinely supported by the evidence contained in the underlying package components. In light of these issues, we propose a Progressive Loading-Aware Hierarchical Contrastive Learning (PL-HCL) framework that aims to evaluate cross-layer misalignment in Agent Skills before execution. PL-HCL treats each skill package as a layered artifact consisting of metadata, instructions, and resources, and trains the model to learn whether user-facing claims are supported by deeper package evidence. The framework integrates two key components:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A two-stage continued pretraining process that adapts base LLMs to the vocabulary, formatting, and progressive structure of Agent Skill packages. The first stage trains on short metadata-instruction views, and the second stage trains on full package views that include resources.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A progressive loading-aware hierarchical contrastive learning process learns cross-layer consistency by contrasting aligned same-skill layers with metadata-swapped and corrupted variants. This design directly targets the mismatch between surface claims and deeper behavioral evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper makes several key contributions. First, we construct a large-scale open-source Agent Skill corpus and a human-verified challenge set for aligned/misaligned skill evaluation, enabling systematic study of skill-level trustworthiness before execution. Second, we introduce PL-HCL, a contrastive pretraining framework that learns cross-layer consistency from aligned packages and synthetic pairs. Finally, we evaluate base LLMs, two-stage continued pretraining, and PL-HCL across general and cybersecurity backbones to demonstrate that skill-format adaptation alone is insufficient while explicit cross-layer contrastive learning substantially improves misaligned-class detection.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The remainder of this paper is organized as follows. We first review prior work on Agent Skill security, continued pretraining and curriculum adaptation, and contrastive learning for hierarchical and multi-view representations. Second, we identify research gaps and present the research questions for this study. Third, we define cross-layer misalignment and describe the dataset. We then present the PL-HCL framework and report the experimental results. Finally, we discuss implications for trustworthy Agent Skill evaluation and conclude with limitations and future directions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2. Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section reviews three streams of related research that motivate and ground our problem formulation and proposed method: 1) work on tool-augmented agents and Agent Skill security explains why reusable skill components create a new trustworthiness surface for agentic AI; 2) continued pretraining and curriculum adaptation motivate our use of staged skill-format adaptation before contrastive learning, and 3) contrastive and hierarchical representation learning provide the methodological foundation for learning cross-layer consistency within agent skills.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1. Tool-Augmented Agents and Agent Skill Security\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.1. Tool-Augmented Agent Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Research on tool-augmented agents explores how LLMs extend their reasoning and task-completion capabilities through external tools, APIs, and environment actions. For example, ReAct demonstrated an agent pattern that interleaves reasoning traces with actions, enabling LLMs to invoke external information or tools during task execution (Yao et al . , 2023 ) . Toolformer further investigated how LLMs can learn to call APIs at appropriate junctures, thereby integrating tool use into model behavior (Schick et al . , 2023 ) . As agent systems shift from isolated text generation to external interaction, evaluation has increasingly focused on agent performance in complex task environments. GAIA evaluates the task-completion capabilities of general AI assistants in dimensions such as reasoning, web browsing, multimodal understanding, and tool-use proficiency (Mialon et al . , 2023 ) . Recent red-teaming work further shows that autonomous agents equipped with persistent memory, communication channels, file-system access, or shell execution can produce failures that involve security, privacy, and governance risks. These risks include information disclosure, destructive actions, identity spoofing, and partial system takeover (Shapira et al . , 2026 ) . Such research demonstrates that evaluating agentic AI requires attention not only to the quality of the final answer but also to the broader trustworthiness risks arising from the interaction between agents and external capabilities. Therefore, we review the characteristics of Agent Skill.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.2. Agent Skills as Reusable Evaluation Artifacts\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As indicated in the introduction, Agent Skills encapsulate procedural knowledge into reusable packages, enabling agents to load task-specific capabilities at inference time (Anthropic, 2025 ; Li et al . , 2026 ) . Unlike a single API endpoint or function call, Agent Skill packages typically comprise marketplace metadata, natural-language instructions, scripts, configuration files, dependencies, and resources (Anthropic, 2025 ) . Thus, Agent Skills are not merely external interfaces invoked by agents. Instead, skills are structured artifacts containing claims, instructions, and executable or resource-based evidence. Recent work has begun to investigate Agent Skills as independent objects of evaluation. For example, SkillsBench examines whether curated or self-generated skills enhance agent performance across tasks, finding that curated skills increase average pass rates, but their effectiveness depends on the domain and task (Li et al . , 2026 ) . Such benchmarks isolate agent skills from general tool-use settings and emphasize the impact of the skill package itself on agent performance. For skill-based agent ecosystems, this demonstrates that reusable skill packages are both auxiliary materials and evaluation artifacts capable of substantially altering agent behavior and task outcomes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"2.1.3. Security and Trustworthiness Risks in Agent Skills\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As Agent Skills are increasingly used to extend agent capabilities, security studies have begun to systematically analyze their risks. Liu et al. (Liu et al . , 2026a ) conducted a security analysis of a large-scale set of skills from community registries, finding that malicious skills can facilitate credential theft, remote code execution, agent manipulation, and other attack behaviors through natural-language instructions and helper scripts. Other studies have also highlighted issues such as prompt injection, credential leakage, data exfiltration, privilege escalation, unsafe code execution, and hidden malicious behavior in open-source skills (Liu et al . , 2026b ; Schmotz et al . , 2026 ; Jin et al . , 2026 ) . Taken together, these studies show that Agent Skills play a vital role in establishing trustworthiness in agentic AI from the supply side. Evaluating the structure, content, and potential security issues of reusable skill packages is a crucial prerequisite for understanding and improving the trustworthiness of agentic AI within skill-based agent ecosystems. However, the evolving and hierarchical nature of Agent Skills necessitates appropriate training and representation learning strategies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2. Continued Pretraining and Curriculum Adaptation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Agent Skill packages are layered artifacts comprising metadata, instructions, and resources. Metadata and skill descriptions are often surface-level capability claims, while the underlying SKILL.md file, resources, and executables are longer, heterogeneous, and contain clearer behavioral evidence. Consequently, models need to adapt to a skill’s high-level descriptions before engaging with the more complex, resource-level information. Curriculum learning has emerged as a training strategy in which a model learns from easy to hard samples, and holds significant promise for modelling Agent Skill structures. Early work indicated that models could first learn fundamental structures from simpler or clearer examples before gradually transitioning to more complex ones, thereby improving optimization and generalization (Bengio et al . , 2009 ) . Subsequent research extended into self-paced learning and competence-based curriculum learning. These approaches allow models to encounter data with varying difficulty levels, determined by sample difficulty or model competence, throughout training stages (Kumar et al . , 2010 ; Platanios et al . , 2019 ) . This body of work demonstrates that the training process is not merely a matter of repeated sampling from a fixed data distribution; the training order itself serves as a crucial learning signal, helping the model progressively adapt to complex structures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While continued pretraining addresses different issues than curriculum learning, the two are complementary in the context of model adaptation. Continued pretraining typically involves further training a pre-existing language model on domain-specific or task-relevant corpora, enabling it to adapt to the target domain’s vocabulary, style, and discourse patterns prior to downstream supervised learning. Gururangan et al. (Gururangan et al . , 2020 ) conducted a systematic study of domain-adaptive pretraining (DAPT) and task-adaptive pretraining (TAPT), demonstrating that adaptive pretraining can enhance downstream performance across various domains, such as biomedicine, computer science, news, and reviews. Related works have also applied domain-adaptive methods to specialized text-analysis settings, including legal, scientific, biomedical, and news-analysis tasks (Lee et al . , 2020 ; Beltagy et al . , 2019 ; Chalkidis et al . , 2020 ; Zhang et al . , 2026 ) . These studies collectively indicate that model adaptation often requires considering both what data the model sees and in what order it sees them. In the context of Agent Skill analysis, this requires a strategy to carefully represent the hierarchical and multi-faceted structure of skills.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3. Contrastive Learning for Hierarchical and Multi-View Representations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For Agent Skill packages, metadata, instructions, and resources can be seen as distinct layers or views of the same skill artifact. Metadata provides user-facing capability claims, instructions offer agent-facing procedural guidance, and resources, along with executable artifacts, serve as evidence that is closer to actual behavior. Therefore, hierarchical and multi-view contrastive learning provide a methodological foundation for cross-layer representation learning. Contrastive learning is a class of self-supervised representation learning that seeks to pull related views or semantically similar samples closer together while pushing unrelated samples further apart. A prevailing approach is SimCLR, which constructs positive pairs from different augmentations of the same image and employs a contrastive objective to learn transferable visual representations (Chen et al . , 2020 ) . Supervised contrastive learning further incorporates label information into the contrastive objective, bringing samples of the same class closer together in the representation space while increasing the separation between samples of different classes (Khosla et al . , 2020 ) . In the realm of language-image representation learning, CLIP demonstrates the effectiveness of contrastive objectives for cross-modal alignment and transferable representation learning by aligning image-text pairs (Radford et al . , 2021 ) . These works show that contrastive learning, when using carefully defined positive and negative samples, can shift representation learning away from a singular focus on reconstruction or classification and toward modeling relationships.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond general contrastive learning, multi-view contrastive learning focuses on how different views of the same underlying object share semantic factors. Contrastive Multiview Coding (CMC) treats different sensory channels as complementary sources of information, learning view-invariant representations by maximizing cross-view agreement for the same instance (Tian et al . , 2020 ) . Similarly, CLIP views images and text as distinct modalities of the same semantic content, learning aligned multimodal embeddings through large-scale contrastive pretraining (Radford et al . , 2021 ) . These studies show that contrastive learning facilitates learning both individual representations and shared semantics when objects have multiple views.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hierarchical contrastive learning extends the contrastive objective to data characterized by hierarchies or multi-level structures. In hierarchical text classification, label taxonomies provide a class-level hierarchy; models can leverage parent-child label relationships, label paths, or hierarchy-aware positive samples to learn text representations that better align with the underlying label structure. HGCLR integrates the label hierarchy into the text encoder, learning hierarchy-aware representations through hierarchy-guided positives (Wang et al . , 2022 ) . Hierarchical multi-label contrastive learning further leverages multi-level labels and label relations to preserve the class hierarchy, ensuring the representation space reflects the structural relationships between different labels (Zhang et al . , 2022 ) . In relation extraction and recommendation settings, hierarchical contrastive objectives are also employed to combine global structure with local interactions, thereby learning multi-granularity representations (Li et al . , 2022 ; Wu et al . , 2023 ) . These studies show that contrastive learning supports structure-aware representations, with positive and negative pairs defined by labels, modalities, views, or hierarchical relations, not just data augmentation. Within the context of Agent Skill analysis, these approaches can help models learn shared skill semantics by aligning layers belonging to the same skill, while also learning to detect signals of inconsistency between layers through mismatched or corrupted layer combinations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3. Research Gaps and Questions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We identified several research gaps from our literature review. First, although research on tool-augmented agents and Agent Skill security is expanding, current evaluation frameworks offer limited mechanisms for assessing the trustworthiness of a skill package prior to execution. Most agentic AI benchmarks focus on evaluating task completion, reasoning processes, or tool-use proficiency only after tools or skills have been selected and loaded. Consequently, there remains an unresolved challenge in evaluating the skill artifact itself at the stage when users or agents must make trust decisions based solely on partial or surface-level information. Second, existing Agent Skill security research has identified specific risks, including prompt injection, credential leakage, data exfiltration, unsafe code execution, and concealed malicious behavior. However, these studies predominantly address vulnerabilities or malicious intent, whereas Agent Skill trustworthiness also relies on the alignment between user-facing descriptions and the underlying instruction and resource layers. As a result, a skill may be misaligned without being overtly malicious, such as when its metadata exaggerates or inaccurately represents capabilities not supported by the underlying package. Third, although continued pretraining helps LLMs learn skill package formats, it does not train models to compare claims across different layers. Finally, contrastive and hierarchical methods leverage structure from labels or graphs, but Agent Skills require a unique structural signal. Their hierarchy arises from progressive loading, with metadata first and instructions or resources providing deeper behavioral evidence. Based on these gaps, we pose the following research questions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RQ1: How can cross-layer misalignment in Agent Skills be formalized as a pre-execution, artifact-level evaluation problem?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RQ2: To what extent can base general and cybersecurity LLMs detect cross-layer misalignment without skill-specific adaptation?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RQ3: How does two-stage continued pretraining improve misalignment detection?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RQ4: How does progressive loading-aware hierarchical contrastive learning improve misaligned-skill detection beyond base LLMs and CPT-only checkpoints?\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4. Progressive Loading-Aware Hierarchical Contrastive Learning (PL-HCL) Framework for Detecting Misaligned Agent Skills\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose a PL-HCL framework for Agent Skill misalignment detection, illustrated in Figure 1 . The proposed framework consists of four major components: (1) public Agent Skill data is collected and processed; (2) two-stage continued pretraining adapts the base LLM to these packages; (3) progressive loading-aware hierarchical contrastive learning then distinguishes between aligned and misaligned skill layers; and (4) the model determines whether user-facing metadata is supported by underlying instructions and resources. Each component is described in the following subsections.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1. Skill Corpus Construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We build an open-source Agent Skill corpus by linking public records from SkillsMP (SkillsMP, 2026 ) , the largest open-source Agent Skills platform, to their GitHub repositories and normalizing each retrieved package into a layered representation S = ( M , I , R ) S=(M,I,R) : the metadata layer M M contains user-facing fields such as the skill name, description, categories, tags, repository identifiers, and declared resource references; the instruction layer I I contains agent-facing procedural guidance; and the resource layer R R contains scripts, configuration files, dependencies, examples, assets, and other supporting files. From 273,657 SkillsMP catalog entries, 264,937 packages were successfully downloaded and normalized; after applying the 4,096-token metadata–instruction budget and the 10,240-token full-package budget, 248,473 packages fit at least one training stage and form the unlabeled training corpus, while the held-out Challenge Set is excluded from both CPT and PL-HCL training. 1 1 1 Detailed corpus construction, token-budget coverage, and filtering statistics are provided in the supplementary materials.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This representation preserves the distinction between user-facing claims and deeper package evidence, supporting our pre-execution, artifact-level task: the detector observes package artifacts available at inspection time and assesses whether user-facing metadata is supported by the instruction and resource layers. We do not verify runtime-only behaviors unless the package provides observable evidence.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2. Progressive CPT\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The prediction target of our proposed approach is whether the layers of a skill package are mutually consistent: 0 if aligned, 1 if misaligned. A skill is considered misaligned when its metadata describes a capability or intent not supported by the instruction or resource layers. We consider two forms of misalignment: (1) malicious misalignment , which masks harmful behavior in deeper layers, and (2) benign misalignment , when the metadata overstates or misdescribes capabilities that the deeper layers do not substantiate. The two-stage continued pretraining component adapts the backbone of a decoder-only LLM to Agent Skill packages prior to training for misalignment detection. This stage enables the model to acquire familiarity with the vocabulary, formatting conventions, and hierarchical structure of Agent Skills, instead of immediately requiring the base LLM to assess the correspondence between metadata and underlying artifacts. We employ a curriculum-inspired adaptation strategy in which the model initially learns from concise, structured representations of a skill and subsequently progresses to more comprehensive and heterogeneous package contexts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each normalized skill S = ( M , I , R ) S=(M,I,R) , we construct two layer-delimited views: 1) The short view comprises metadata M M and instructions I I , representing surface-level claims and agent-facing guidance available during initial skill inspection; and 2) The full view includes metadata M M , instructions I I , and resources R R , thereby exposing the model to scripts, configuration files, dependencies, assets, and other resource-level evidence. Sequential training on the short view followed by the full view provides the model with a progressive adaptation trajectory from surface skill descriptions to complete multi-file packages. Both stages optimize the standard causal language modeling objective. The first stage adapts the model to metadata and instruction structure. The second stage adapts it to the full package context. The resulting checkpoint, θ CPT \\\\theta_{\\\\mathrm{CPT}} , serves as the skill-aware backbone for PL-HCL. 2 2 2 Full training details and pseudocode are provided in the supplementary materials.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3. Progressive Loading-Aware Hierarchical Contrastive Learning (PL-HCL)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PL-HCL serves as the consistency-learning component of the workflow. After two-stage CPT, the model has adapted to the vocabulary, formatting, and overall structure of Agent Skills. However, causal language modeling does not explicitly require the model to assess whether user-facing metadata is substantiated by underlying instruction and resource evidence. To this end, PL-HCL transforms cross-layer consistency into a contrastive training signal. For each normalized skill S i = ( M i , I i , R i ) S_{i}=(M_{i},I_{i},R_{i}) , we first construct an aligned view in which metadata, instructions, and resources come from the same package:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This positive view represents a skill package in which the surface claims and deeper artifacts are expected to be mutually consistent. We then construct two types of negative views. The first is swap negative , which replaces the metadata layer with metadata from a different skill:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This negative preserves the original instruction and resource evidence but alters the surface claim, resulting in a mismatch between the advertised capability and the underlying package behavior.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The second type is corruption negative , which perturbs the original metadata while maintaining the deeper layers unchanged:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This view simulates misleading, exaggerated, or unsupported capability claims.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PL-HCL encodes the metadata, instruction, and resource layers into a shared contrastive space. The objective is to score the aligned view P i P_{i} higher than its swap and corruption negatives N i A N_{i}^{A} and N i B N_{i}^{B} , thus bringing the same-skill layers closer together and separating mismatched layer combinations. The training signal evaluates whether the surface description is consistent with the evidence provided by the deeper layers of the package, rather than simply identifying suspicious files. This approach mirrors the progressive loading structure of Agent Skills, in which metadata is typically encountered before the complete instruction and resource context is available.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PL-HCL is trained in two sub-stages that correspond to the short-view and full-view settings used in CPT. The first sub-stage utilizes metadata–instruction views and includes 96,762 positive training pairs, 386,128 swap negatives, and 334,867 corruption negatives. The second sub-stage employs full metadata–instruction–resource views and comprises 182,135 positive training pairs, 646,945 swap negatives, and 258,368 corruption negatives. The Challenge Set is a held-out, human-verified gold-standard evaluation dataset with 1,150 aligned and 294 misaligned skills, designed to test whether models trained on synthetic contrastive pairs can generalize to realistic in-the-wild misalignment cases. 3 3 3 Additional details on dataset construction and verification are provided in the supplementary material This design follows common benchmark practice in which a curated challenge subset is reserved for evaluating model robustness under more realistic or difficult conditions, such as evasive-malware evaluation in EMBER2024 and in-the-wild distribution-shift evaluation in WILDS (Joyce et al . , 2025 ; Koh et al . , 2021 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4. Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate the proposed PL-HCL framework through three experiments designed to assess cross-layer misalignment detection in Agent Skills, each aligned with one of our research questions (E0 addresses RQ2, E1 addresses RQ3, and E2 addresses RQ4). In Experiment 0 (E0), we evaluate whether base LLMs can identify misaligned skills without skill-specific adaptation. This experiment compares general and cybersecurity backbones under zero-shot, two-shot, and five-shot prompting settings, allowing us to examine whether general instruction-following ability or security-oriented pretraining is sufficient for this task. In Experiment 1 (E1), we evaluate the effect of two-stage continued pretraining by comparing base checkpoints with CPT-only checkpoints. This experiment assesses whether adapting the model to the vocabulary, formatting, and layered structure of Agent Skill packages improves misalignment detection. In Experiment 2 (E2), we evaluate the full PL-HCL framework by comparing CPT-only checkpoints with CPT+PL-HCL checkpoints. This experiment tests whether explicitly learning cross-layer consistency from aligned, metadata-swapped, and corrupted-claim views provides measurable gains beyond format adaptation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across these experiments, we use both intrinsic and extrinsic evaluation tasks. The intrinsic tasks assess whether training improves the model’s adaptation to Agent Skill artifacts before downstream inference. Specifically, next-token prediction measures whether CPT improves the model’s ability to model layer-delimited skill text, while contrastive pair detection evaluates whether PL-HCL learns to distinguish aligned cross-layer views from metadata-swapped or corrupted-claim variants. The extrinsic task assesses the pre-execution misalignment detection . In this task, the model receives a normalized skill package and predicts whether the metadata is supported by the instruction and resource layers under zero-shot, two-shot, and five-shot prompting conditions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The evaluation utilizes a human-verified Challenge Set comprising 1,444 Agent Skills, with 1,150 aligned and 294 misaligned packages (196 are SAFE, 38 are SUSPICIOUS, and 60 are MALICIOUS.) All Challenge Set packages are actual skills collected from SkillsMP (rather than synthetically perturbed variants); misaligned examples are naturally occurring, in-the-wild packages whose metadata is not supported by their instruction or resource layers. Candidate skills were screened automatically and subsequently verified by human annotators (see Figure 1 ). This Challenge Set is excluded from pretraining and contrastive learning and is reserved solely for final evaluation. Because all PL-HCL training negatives are synthetically constructed, evaluation on this set tests whether the learned consistency signal generalizes from synthetic contrast pairs to naturally occurring misalignment. Given the Challenge Set’s imbalance toward aligned skills, both overall and misalignment-sensitive metrics are reported. Accuracy (Acc) measures the overall proportion of correct predictions. Since accuracy may be inflated by the aligned-class majority, misaligned-class F1 (F1 m ) treats misaligned as the positive class and measures the model’s ability to correctly identify the safety-critical class. Macro-F1 computes the unweighted average of aligned-class and misaligned-class F1, providing a balanced perspective across both labels. Area under the ROC curve (AUC) and average precision for the misaligned class (AP m ) are also reported to evaluate ranking quality under varying decision thresholds.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5. Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section presents the evaluation results on the human-verified Challenge Set following the E0–E2 evaluation design described in Section 4.4 . E0 evaluates whether unadapted base LLMs can detect cross-layer misalignment without skill-specific adaptation. E1 isolates the effect of two-stage CPT by comparing base checkpoints with CPT-only checkpoints. E2 evaluates the incremental contribution of PL-HCL by comparing CPT-only checkpoints with CPT+PL-HCL checkpoints. 4 4 4 k denotes the number of labeled in-context examples. F1 m and AP m are computed for the misaligned class. Macro-F1 is the unweighted average of aligned-class and misaligned-class F1. Base, CPT, and CPT+PL-HCL denote the unadapted backbone, the two-stage continued-pretraining checkpoint, and the full contrastive checkpoint, respectively. We first summarize the main findings across E0–E2, and then report the results for E1 and E2 in detail. 5 5 5 Additional intrinsic evaluations, full prompting-setting results, and extended metric tables are provided in the supplementary materials.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1. Main Findings across E0-E2\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 provides a compact summary of the strongest prompting setting for each backbone and adapter on the Challenge Set. To avoid relying on a single test-selected configuration, Tables 2 and 3 report the full results for all k ∈ 0 , 2 , 5 k\\\\in{0,2,5} prompting settings. We use Table 1 only as a high-level summary, while the main comparisons in 5.2 and 5.3 are based on the full prompting-setting results. Unless otherwise stated, the decision rule is fixed across models, and no model-specific threshold is tuned on the Challenge Set. In deployment, both the prompting configuration and decision threshold should be calibrated using a held-out validation set from the target skill ecosystem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The E0 results (base rows of Table 1 ) show that base LLMs demonstrate limited performance on the misaligned class, despite achieving accuracy levels close to the aligned-class majority baseline. For instance, both base Llama-3.1-8B and base Foundation-Sec-8B attain approximately 0.81 accuracy in their optimal prompting configurations; however, their F1 m remains below 0.15. These results suggest that the models frequently classify plausible skill packages as aligned without consistently verifying whether the surface claim corresponds to deeper package evidence. The CPT+PL-HCL approach helps improve performance. On Llama-3.1-8B, CPT+PL-HCL achieves 0.929 accuracy, 0.788 F1 m , and 0.872 Macro-F1. On Foundation-Sec-8B, CPT+PL-HCL attains its highest performance, with 0.937 accuracy, 0.817 F1 m , and 0.889 Macro-F1 in the two-shot setting. The improvements are concentrated in F1 m , which represents the safety-critical class for this task. Base-cybersecurity LLMs do not independently bridge the performance gap. WhiteRabbitNeo-2-8B, RedSage-Qwen3-8B-DPO, and Foundation-Sec-8B, when used without skill-specific adaptation, exhibit Macro-F1 scores close to the majority-class baseline. Although certain security-tuned models achieve higher area under the curve (AUC) or AP m , indicating some ability to rank risky cases. However, this signal does not translate into reliable classification performance without PL-HCL.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2. E1: Evaluation of CPT\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 reports E1 by comparing unadapted base LLMs with CPT-only checkpoints across k = 0 k=0 , k = 2 k=2 , and k = 5 k=5 prompting settings. This comparison directly evaluates whether continued pretraining on Agent Skill packages improves misalignment detection, or whether it primarily helps the model adapt to the vocabulary, formatting, and layered structure of skill artifacts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"CPT improves over base checkpoints in the zero-shot setting, suggesting that continued pretraining helps the model adapt to the vocabulary, formatting, and layered structure of Agent Skill packages. For Llama-3.1-8B, zero-shot F1 m increases from 0.032 to 0.130 and Macro-F1 increases from 0.457 to 0.511 after CPT. Foundation-Sec-8B shows a similar pattern, with zero-shot F1 m increasing from 0.026 to 0.134 and Macro-F1 from 0.454 to 0.513. The improvement, however, remains limited. Across prompting settings, CPT-only Macro-F1 stays near 0.48–0.52, and F1 m remains below 0.15. Thus, skill-format adaptation alone does not provide a reliable misalignment detector. CPT makes the model more familiar with skill packages, but it does not explicitly train the model to compare advertised claims with deeper behavioral evidence.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3. E2: Evaluation of PL-HCL\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 3 reports E2 by comparing CPT-only checkpoints with CPT+PL-HCL checkpoints across k = 0 k=0 , k = 2 k=2 , and k = 5 k=5 prompting settings. This comparison tests whether explicitly learning cross-layer consistency from aligned, metadata-swapped, and corrupted-claim variants improves misaligned-skill detection beyond the skill-format adaptation provided by CPT.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Adding PL-HCL produces the largest gains across all prompting settings. On Llama-3.1-8B, Macro-F1 increases from 0.511, 0.520, and 0.494 to 0.717, 0.872, and 0.867 for k = 0 , 2 , 5 k=0,2,5 , respectively. On Foundation-Sec-8B, Macro-F1 increases from 0.513, 0.522, and 0.483 to 0.779, 0.889, and 0.887. The largest gains appear in F1 m . CPT-only never exceeds 0.150 F1 m , while CPT+PL-HCL reaches 0.788 on Llama-3.1-8B and 0.817 on Foundation-Sec-8B in the two-shot setting. These results show that PL-HCL supplies the discriminative signal missing from ordinary continued pretraining. By contrasting aligned packages against swapped and corrupted layer combinations, PL-HCL turns cross-layer consistency into a usable detection cue. Notably, PL-HCL is trained exclusively on synthetic negatives, yet these gains are measured on naturally occurring, in-the-wild misaligned skills, indicating that the learned claim-versus-evidence signal transfers beyond the perturbation distribution used for training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4. Misaligned Examples In-the-wild\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While aggregate metrics indicate that PL-HCL significantly improves detection for the misaligned category, looking only at average results leaves unclear what signals the model has learned. To investigate, we examined two representative skills from the Challenge Set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For both, the zero-shot base model Foundation-Sec-8B-Reasoning classified them as aligned with high confidence. After PL-HCL training, the same model reclassified them as misaligned . Although both are correctly labeled misaligned , they represent different safety issues. The first example is an obviously unsafe Skill involving a Chinese-language prompt injection attack. Its metadata and instruction body contain commands designed to override the system prompt, and it has broad Bash() and WebFetch() permissions. The second example demonstrates misalignment but is not inherently malicious (benign but misaligned). Its metadata claims to provide an inline TypeScript assertion function, but the instruction body actually invokes a third-party recipe URL using curl . As a result, it does not support the claimed tiny-invariant dependency or TypeScript-specific behaviors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The base model yielded positive “aligned” logit differences for these examples (+4.33 and +5.81), both exceeding the model’s average for the aligned class on the Challenge Set (+3.29). This matches the “majority class bias” observed in Section 5 : base LLMs tend to classify plausible skill packages as aligned . After PL-HCL, the logit differences shifted to negative values (-0.13 and -1.50), crossing the decision boundary and resulting in misaligned classification. Interestingly, the benign yet mismatched example produced a stronger mismatch signal despite lacking obvious jailbreak indicators like Bash(*) , “ignore previous instructions,” or shell execution hooks. This shows PL-HCL does not just react to malicious tokens or prompt injection keywords; it learns a “claim-versus-evidence” signal, evaluating if the surface description is genuinely supported by the instruction and resource layers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These examples clarify why mismatch detection for Agent Skills cannot rely solely on general instruction-following or specialized cybersecurity expertise. Whether using a general or cybersecurity LLM, a plausible Skill package might still get a high “aligned” confidence score even if its components contradict its claimed capabilities or hide unsafe behaviors. While base cybersecurity LLMs sometimes achieve higher AUC or AP scores and can rank risky examples to some extent, this does not consistently lead to accurate “aligned” versus “misaligned” classification.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6. Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This article aims to establish a pre-execution screening capability for skill-augmented agentic AI systems. Rather than viewing Agent Skills solely as reusable modules for task performance, this work conceptualizes Agent Skills as layered supply-side artifacts, where metadata, instructions, and resource or executable artifacts collectively determine trustworthiness. The primary practical contribution of PL-HCL is its ability to transform cross-layer consistency into a learnable and deployable detection signal, enabling screening for downstream decision making prior to skill loading or execution. This perspective holds direct relevance for three key stakeholder groups: skill marketplaces, skill users, and security analysts. We describe each in turn below.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For skill marketplaces, PL-HCL can function as a pre-publication or pre-listing governance mechanism. In open skill ecosystems, skills are frequently submitted by third-party contributors, and marketplace metadata serves as the primary basis for users and agents to assess trustworthiness. Absent installed-base screening, misalignment risks may be transferred to downstream users. PL-HCL evaluates whether metadata is substantiated by instructions and resources before a skill is released or recommended, directing high-risk skills to manual review, suspending release, or marking approved skills with a verified-alignment badge. PL-HCL would not replace manual auditing or runtime sandboxing, but would offer a scalable initial filter for marketplace-level governance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For skill users, PL-HCL enables lightweight client-side screening. Most users do not examine files such as SKILL.md , scripts, dependencies, or external references individually when installing Agent Skills. Instead, they often rely on the skill name, description, and repository metadata. PL-HCL can be deployed as a local plugin or command-line scanner that analyzes the local skill directory post-installation and issues warnings when surface claims are inconsistent with deeper package evidence. This approach provides users with direct alignment signals, offering additional trust cues prior to skill execution and reducing reliance on marketplace curation or post-execution observation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For security analysts, PL-HCL offers an operational tool for retrieval and triage. Analysts frequently need to identify high-risk subsets within extensive collections of open-source packages and determine which warrant resource-intensive sandboxed analysis or manual inspection. Because PL-HCL learns cross-layer representations rather than producing only binary classification scores, it facilitates grouping of similar suspicious skills, prioritization of high-risk clusters, and identification of skill families exhibiting similar metadata and behavioral inconsistencies. Thus, PL-HCL can serve as a foundational representation for large-scale skill ecosystem analysis, supporting incident response and adversarial analytics workflows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Methodologically, PL-HCL’s broader implication is its treatment of progressively loaded, hierarchical artifacts as novel evaluation objects. Similar structural patterns are present in MCP servers, browser extensions, VS Code extensions, and software package registries. These artifacts typically include user-facing descriptions, configuration files, permission declarations, scripts, or executable resources, with users generally encountering surface claims before deeper behavioral elements. Consequently, this study’s guiding design principle is that, when trust decisions must be made prior to full execution, models should explicitly learn the consistency between surface claims and deeper evidence. This departs from approaches that rely solely on generic language modeling, keyword-based security scanning, or post-execution behavioral observation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As with any research, this work has its limitations. First, the focus is on pre-execution, artifact-level assessment. Our proposed approach relies exclusively on information available to the skill package prior to execution, and therefore cannot address failure modes that manifest only during live interaction, dynamic remote payload execution, post-deployment mutations, or environment-specific side effects. Runtime sandboxing and trace-level agent evaluation are thus complementary to this approach. Second, the dataset is primarily derived from SkillsMP and open-source skills available on GitHub. Closed skill ecosystems, internal enterprise skills, and dynamically generated skills may exhibit different package structures, documentation practices, and risk profiles. Third, our experiments are conducted primarily on 8B-scale backbones. This design allows us to compare general and cybersecurity LLMs under a controlled and computationally feasible setting, but it does not fully characterize how model scale affects Agent Skill misalignment detection. Larger models may exhibit stronger cross-layer reasoning or better few-shot sensitivity, while smaller models may require different adaptation strategies. Hence, we cannot ensure that the observed performance trends fully generalize across model sizes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7. Conclusion and Future Directions\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Agent Skills have become an integral component of modern agentic AI systems. Despite their growing prevalence, Agent Skills often suffer from misalignment between user-facing claims and deeper instructional or executable evidence. By conceptualizing each skill as a layered artifact comprising metadata, instructions, and resources, this work proposes a pre-execution evaluation target for trustworthy agentic AI. Specifically, before a skill is loaded or executed, evaluators should assess whether its advertised capabilities are substantiated by package evidence available at inspection time. A large-scale open-source Agent Skill corpus and a human-verified Challenge Set were constructed, and PL-HCL, a progressive loading-aware contrastive framework for learning cross-layer consistency among metadata, instructions, and resources, was proposed. Experimental results indicate that both general and cybersecurity LLMs struggle to reliably identify misaligned skills. The two-stage CPT approach primarily enhances skill-format adaptation but is insufficient as a standalone detector. In contrast, PL-HCL significantly improves misaligned-class detection by explicitly modeling the relationship between surface claims and deeper package evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Future research can expand along both empirical and methodological dimensions. First, cross-layer misalignment detection may be integrated with runtime sandboxing and trajectory-level evaluation to capture behaviors not directly observable from package artifacts, such as dynamic remote payloads, environment-specific side effects, or interaction-dependent failures. Second, PL-HCL could be extended into directed hierarchical contrastive learning, enabling the model to assess not only layer consistency but also the direction of support among layers. In this framework, metadata provides surface claims, instructions offer procedural support, and resources supply behavioral evidence. A directed objective would distinguish whether misalignment results from metadata overclaiming unsupported capabilities or from deeper artifacts introducing undisclosed behaviors. Third, future models could incorporate richer artifact representations, including dependency graphs, permission structures, external endpoint analysis, and execution traces, to better capture behavioral evidence that is only partially expressed in text. Additionally, future evaluations should examine whether these approaches generalize across closed-source or enterprise skill ecosystems and across different model scales. In summary, Agent Skill evaluation should address not only improvements in task performance but also the consistency between surface claims and the deeper artifacts that influence agent behavior.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This supplement provides additional reproducibility details, including dataset construction, training settings, pseudocode, and extended evaluation results. The main paper is self-contained and does not rely on the supplement for its core argument or primary findings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.1. Package Retrieval and Normalization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our corpus is built by linking public skill records from SkillsMP to the GitHub repositories that host their implementations. From each SkillsMP record, we extract the skill name, marketplace description, category or tag information when available, repository owner, repository name, branch, and relative package path. We then download the corresponding repository snapshot from GitHub and extract the skill package from the declared path.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each downloaded package is normalized into the tuple S = ( M , I , R ) S=(M,I,R) . The metadata layer M M contains marketplace and repository-level fields, including the skill name, description, tags or categories when available, repository identifiers, and source path. The instruction layer I I is primarily derived from SKILL.md and related instruction files. The resource layer R R contains scripts, configuration files, examples, assets, dependency files, API wrappers, and external references.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Packages are retained only when the GitHub repository is accessible, the declared package path exists, and a valid instruction artifact is present. If a package contains only SKILL.md , it is still retained; its resource layer is represented as empty or minimal. This preserves instruction-only skills while making the absence of supporting resources explicit.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.2. Corpus Construction Details\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We constructed the Agent Skill corpus from the largest open-source Agent Skill marketplace, SkillsMP . A REST API crawl produced 273,657 unique metadata listings after rolling deduplication. We then used repository links in these listings to download the corresponding public GitHub packages and retained packages with a parseable SKILL.md . Each retained package was normalized into metadata, instruction, and resource layers based on SKILL.md frontmatter, the procedural body of SKILL.md , and accessible package files under directories such as scripts/ , assets/ , and reference/ .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before training, we applied two corpus-level filters. First, packages whose normalized context exceeded the maximum input budget of the corresponding training stage were excluded from that stage. Stage 1 uses a 4,096-token metadata-plus-instruction rendering, while Stage 2 uses a 10,240-token full-package rendering that also includes resource files. Second, exact overlaps with benchmark or supervision sources were excluded using repository owner, repository name, and skill name to reduce the risk of label leakage. After removing unavailable repositories, incomplete packages, duplicate skills, and unparseable files, 264,937 skill packages were successfully downloaded and normalized. Among them, 248,473 packages fit at least one training stage and were eligible for CPT or PL-HCL training; packages that fit neither token budget were excluded from training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stage 2 coverage is lower because full-package renderings include heterogeneous resource files when available. In particular, 149,150 normalized skills contain no package resource files; among the remaining 115,787 skills with resource files, 91,053 fit the Stage-2 full-package budget. The resulting token-budgeted corpus provides the unlabeled skill views used for progressive CPT and PL-HCL, while the Challenge Set is held out from both training stages and used only for extrinsic evaluation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.3. Reproducibility Manifest\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For reproducibility, each retained package is accompanied by a manifest entry. The manifest records source identifiers, repository owner, repository name, branch, package path, artifact types, file counts, token counts, filtering decisions, and normalization status. These records allow us to trace each training instance back to its original marketplace and repository source while preserving the normalized M M , I I , and R R representation used in training and evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Gold labels for the Challenge Set were produced through a three-stage pipeline. First, we used the MASB scanner to generate severity-ranked candidate flags from the collected corpus; the scanner served only as a candidate-generation tool, not as the source of final labels. Second, flagged skills were inspected in a Jetstream Docker sandbox with behavioral instrumentation for network connections, subprocess calls, filesystem writes, dependency installation, and potential exfiltration. Third, human coders compared each skill’s metadata claim with its instruction/resource behavior and assigned two independent labels: final-align , indicating whether the skill is ALIGNED or MISALIGNED, and final-malicious , indicating whether the skill is SAFE, SUSPICIOUS, or MALICIOUS.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two human annotators conducted the verification and quality-assurance process. To assess label reliability, 144 randomly sampled Challenge Set examples were independently re-coded. The two annotators disagreed on 4 cases, yielding 97.2% agreement. Disagreements and uncertain cases were resolved through discussion and artifact re-inspection before finalizing the gold labels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The corpus version used in this study was crawled from 2026-04-15 to 2026-04-17 using the public skillsmp listing endpoint and linked GitHub repositories. We retained packages with a parseable SKILL.md and non-empty name / description fields, and excluded deleted repositories, duplicates, forks of retained corpus members, and packages that redirected to non-skill repositories. The released corpus contains parsed structural fields and content hashes rather than raw bundled scripts; per-skill content remains governed by the upstream repository license. The malicious-misaligned portion of the Challenge Set is released only under a controlled-access data-use agreement. The corpus contains public marketplace metadata and committed repository content only; it includes no end-user telemetry, install logs, or private personal information. We’ve released the dataset on Huggingface : https://huggingface.co/datasets/anon-skillsalign-26/skill_align . Code files are hosted on: https://anonymous.4open.science/r/skill_align_anonymous_v2-C628/ .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.1. Two-Stage CPT Objective\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a rendered token sequence x 1 , … , x T x_{1},\\\\ldots,x_{T} , let o t , k o_{t,k} denote the model logit for vocabulary token k k at position t t . The next-token probability is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The CPT loss is:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.2. PL-HCL Objective\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let h θ ( X ) h_{\\\\theta}(X) be the hidden representation produced by the CPT-adapted model for layer X ∈ { M , I , R } X\\\\in\\\\{M,I,R\\\\} , and let g ϕ g_{\\\\phi} be a projection head. The projected layer embedding is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a package S = ( M , I , R ) S=(M,I,R) , the cross-layer score is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where sim ( ⋅ , ⋅ ) \\\\operatorname{sim}(\\\\cdot,\\\\cdot) is cosine similarity and α M I + α M R + α I R = 1 \\\\alpha_{MI}+\\\\alpha_{MR}+\\\\alpha_{IR}=1 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The PL-HCL loss is:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where τ \\\\tau is the temperature and 𝕀 i B \\\\mathbb{I}_{i}^{B} indicates whether a corruption negative is sampled for S i S_{i} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.3. Training Data Splits\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 reports the train, validation, and test examples actually consumed during CPT, PL-HCL, and intrinsic evaluation after applying the 25% per-split subsampling used for compute feasibility. Table 6 reports the corresponding whole-corpus and anchor-package accounting. CPT rows report sampled skill examples, while PL-HCL rows report sampled contrastive examples generated from the corresponding T1/T2/T3 pools. The Challenge Set is test-only, excluded from all CPT and PL-HCL training stages, and used only for the main extrinsic evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 7 lists the hyperparameters used for the two pretraining phases. Values were held fixed across the two reference backbones unless otherwise noted.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Compute.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each reference backbone was pretrained on a 2 × 2\\\\times NVIDIA H200 node using bf16 mixed precision and data parallelism. The full two-stage CPT and PL-HCL pipeline required approximately 48 H200-hours per backbone, and the Challenge Set evaluation was completed in under 2 H200-hours.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We run two intrinsic diagnostics to verify that the pretrained backbones learn the intermediate objectives used in PL-HCL. First, next-token prediction (NTP) on held-out skill text assesses whether CPT improves the modeling of Agent Skill packages. Second, the held-out PL-HCL contrastive evaluation assesses whether the model separates aligned pairs from constructed swap and corruption negatives. These diagnostics support the main results in § 5 : CPT improves skill-text modeling, and PL-HCL learns the contrastive objective, while extrinsic misalignment detection remains the primary evaluation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"1. Next-Token Prediction:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 8 reports NTP metrics before and after CPT. For both full-pipeline backbones, CPT Sub-stage 2 enhances final held-out skill-text metrics, reducing perplexity and boosting top-1 accuracy.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"2. PL-HCL Contrastive Objective:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 9 reports held-out PL-HCL performance by sub-stage and pair kind. The pair-kind decomposition shows that corrupted type-B pairs are more difficult than swapped type-A pairs, particularly for Foundation-Sec-8B-R, while Llama-3.1-8B performs strongly across all pair types.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 10 reports supplementary per-class metrics and confusion counts for the Challenge Set evaluation. These metrics complement the macro-F1 results reported in § 5 . Because the Challenge Set is class-imbalanced, aligned-class performance alone can obscure failures on the misaligned class. We therefore report precision and recall for both classes, together with confusion counts using aligned as the positive class. Under this convention, false positives aligned correspond to missed misaligned skills, while true negatives aligned correspond to correctly detected misaligned skills.\"\n }\n]","isConversational":false}}},{"rowIdx":395,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10517"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Conditional Optimal Bridge for Riemannian Activation Steering"},"authors":{"kind":"list like","value":["Seyed Arshan Dalili","Ajay Narayanan Sridhar","Vijaykrishnan Narayanan","Mehrdad Mahdavi"],"string":"[\n \"Seyed Arshan Dalili\",\n \"Ajay Narayanan Sridhar\",\n \"Vijaykrishnan Narayanan\",\n \"Mehrdad Mahdavi\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10517v1"},"published":{"kind":"timestamp","value":"2026-07-12T00:00:00","string":"2026-07-12T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Activation steering, also termed representation engineering, has emerged as a compelling alternative to fine-tuning for shaping the behavior of large language models (LLMs) at inference time (Rimsky et al. , 2024 ; Li et al. , 2023 ; Wang et al. , 2025 ) . Its empirical success is often motivated by the linear representation hypothesis (Park et al. , 2023 ; Alain and Bengio, 2016 ) where many high-level properties encoded by an LLM, such as sentiment, truthfulness, or harmfulness, appear to correspond to approximately linear directions in activation space. Under this view, changing a model’s behavior need not require modifying its weights; it can instead be achieved by moving its internal activations along directions associated with the target property at inference time. A standard approach is activation addition, which modifies the original activations by adding either a fixed steering vector or one that depends on the current activation."},{"type":"para","text":"This perspective has inspired a growing body of work on inference-time control through activation interventions. Early methods typically construct a single steering direction from contrastive examples and then apply that direction uniformly across queries (Rimsky et al. , 2024 ) , while more recent approaches refine the dynamics of the intervention by integrating a continuous steering field over multiple steps (Zhao et al. , 2026 ; Wang et al. , 2025 ; Rodriguez et al. , 2025 ) or working on a more faithful manifold by rotating activations on the hypersphere You et al. ( 2026 ); Pham and Nguyen ( 2024 ) . These advances show that steering can be both effective and computationally lightweight. Despite this progress, existing methods share two limitations. First, most of the methods and their implicit objective are heuristic rather than derived from a well-posed optimization problem without formal justification for the objective itself. Second, these steering signals are query-independent . A steering vector that helps in-distribution prompts can make out-of-distribution (OOD) inputs wrong or gibberish, and our experiments confirm that several strong baselines improve TruthfulQA Lin et al. ( 2021 ) at the cost of severe MMLU Hendrycks et al. ( 2020 ) and GSM8K Cobbe et al. ( 2021 ) degradation. We address both gaps with Cobras , which casts steering as a Schrödinger Bridge, yielding both a principled derivation of the steering objective and an inherently query-adaptive steering direction."},{"type":"para","text":"Overall, our contributions are: (i) we show that the Schrödinger Bridge on the activation hypersphere provides, to our knowledge, the first principled derivation of the implicitly widely used log-density-ratio steering objective from an optimization problem; (ii) we propose Cobras (Conditional Optimal Bridge for Riemannian Activation Steering), a practical algorithm that combines entropic optimal-transport potentials, spherical kernel density estimation, and a gating mechanism to produce query-adaptive steering directions; and (iii) we demonstrate that Cobras consistently outperforms prior baselines on helpfulness, truthfulness, and detoxification while avoiding the out-of-distribution degradation observed in existing methods."},{"type":"heading","level":4,"text":"Organization."},{"type":"para","text":"Section 2 discusses the related work. Section 3 elaborates on the background, notation, setup, and the assumption of the manifold where LLM activations live. Section 4 then formulates the steering problem and its objective as the Schrödinger Bridge instance, and then solves this problem and derives the objective for steering, and proposes Cobras . Section 5 then does extensive experiments for both in-distribution and OOD and shows that Cobras outperforms baselines in both settings. Finally, Section 6 concludes the paper, discussing limitations and future work. The proof of technical results and additional experiments are reported in the appendix."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":4,"text":"Theory of activation steering."},{"type":"para","text":"Several recent works have sought to develop a unified theory of activation steering by interpreting interventions as structured transformations in representation space. Linear Activation Transportation (Rodriguez et al. , 2025 ) formulates steering as a transport map between source and target activation distributions, LiSeCo (Cheng and Alonso, 2024 ) analyzes steering through linear separability and constrained optimization, and ODESteer (Zhao et al. , 2026 ) views steering as a continuous-time dynamical process rather than a one-shot activation edit. These perspectives provide useful formal tools for reasoning about activation interventions. However, their guarantees are typically derived for surrogate objectives and under specific geometric constraints rather than from a principled perspective. Also, the Schrödinger Bridge (SB) problem has been widely adopted for generative modeling via forward–backward SDE solvers (De Bortoli et al. , 2021 ; Chen et al. , 2021 ; Shi et al. , 2023 ) . Cobras is the first to apply this framework for LLM activation steering."},{"type":"heading","level":4,"text":"Activation steering approaches."},{"type":"para","text":"Early and widely used steering approaches Zou et al. ( 2023 ); Li et al. ( 2023 ); Rimsky et al. ( 2024 ) identify behavior-relevant directions in activation space and apply additive edits. More recent methods refine this basic recipe by accounting for activation geometry or distributional structure, for example through minimally invasive affine maps (Singh et al. , 2024 ) , norm-preserving rotations (Pham and Nguyen, 2024 ) , transport-based maps (Rodriguez et al. , 2025 ) , hyperspherical interventions (You et al. , 2026 ) , or continuous dynamical updates (Zhao et al. , 2026 ) . Despite these advances, most existing methods are query-independent: once a direction or transformation is learned, essentially the same intervention is applied across prompts. Prior work shows that such fixed interventions can have limited OOD generalization, induce off-target side effects, and degrade performance when steering is unnecessary (Tan et al. , 2024 ; Raedler et al. , 2025 ; Da Silva et al. , 2025 ) . Our work instead proposes a query-adaptive steering method that modulates the intervention based on the input, preserving OOD performance while improving in-distribution performance."},{"type":"heading","level":1,"text":"3 Background"},{"type":"para","text":"This section establishes the background for the steering problem by defining key notation and activation populations. We provide empirical evidence that these activations concentrate on a hypersphere, justifying the Riemannian geometry utilized throughout this work."},{"type":"heading","level":4,"text":"Notation and setup."},{"type":"para","text":"Fix a decoder-only transformer with residual width d d , a layer index ℓ \\ell , and a token position index t t . For an input sequence s s , let 𝒉 s ∈ ℝ d \\bm{h}_{s}\\in\\mathbb{R}^{d} denote the residual-stream activation at layer-position pair ( ℓ , t ) (\\ell,t) when the model is run without steering. We consider two populations of input sequences, one associated with the desired behavior and one associated with the undesired behavior, and write p + p_{+} and p − p_{-} for the induced distributions of their activations. Given i.i.d. samples from these activation distributions 𝒟 + = { 𝒉 i + } i = 1 N + \\mathcal{D}^{+}=\\{\\bm{h}^{+}_{i}\\}_{i=1}^{N_{+}} with 𝒉 i + ∼ i . i . d . p + \\bm{h}^{+}_{i}\\overset{\\mathrm{i.i.d.}}{\\sim}p_{+} and 𝒟 − = { 𝒉 j − } j = 1 N − \\mathcal{D}^{-}=\\{\\bm{h}^{-}_{j}\\}_{j=1}^{N_{-}} with 𝒉 i − ∼ i . i . d . p − \\bm{h}^{-}_{i}\\overset{\\mathrm{i.i.d.}}{\\sim}p_{-} , we denote the corresponding empirical measures by p ^ + \\hat{p}_{+} and p ^ − \\hat{p}_{-} . At inference time, for a new input sequence q q , let 𝒉 q ∈ ℝ d \\bm{h}_{q}\\in\\mathbb{R}^{d} be its unsteered activation at ( ℓ , t ) (\\ell,t) . The steering map should return a modified activation that shifts 𝒉 q \\bm{h}_{q} toward regions of activation space with high density under p + p_{+} and low density under p − p_{-} , while remaining sufficiently close to 𝒉 q \\bm{h}_{q} to preserve the semantics of q q ."},{"type":"heading","level":4,"text":"Activation Manifold."},{"type":"para","text":"To leverage the inherent geometric structure of neural representations, we motivate our manifold-based approach by observing the distribution of activation magnitudes within transformer architectures. A line of recent work has shown that, once the layer index is fixed, residual-stream activations concentrate strongly in norm across inputs, both empirically (Pham and Nguyen, 2024 ; You et al. , 2026 ) and structurally (Yang et al. , 2023 ; Xie et al. , 2026 ) . Thus, for the present layer ℓ \\ell , most of the variability relevant for steering lies in the direction of the activation rather than in its Euclidean magnitude. This suggests that the natural state space is not all of ℝ d \\mathbb{R}^{d} but instead a sphere of approximately constant radius. We therefore model activations as living on the hypersphere 𝒮 = { 𝒙 ∈ ℝ d : ‖ 𝒙 ‖ = R } \\mathcal{S}=\\{\\bm{x}\\in\\mathbb{R}^{d}:\\|\\bm{x}\\|=R\\} . This norm concentration allows Cobras to cast the steering problem into hyperspherical Riemannian geometry, letting us leverage established tools and results from Riemannian optimal transport."},{"type":"para","text":"Figure 1 provides empirical support for this approximation by showing that activation norms cluster tightly around the layer mean radius. Accordingly, we project the query activation and the samples in 𝒟 ± \\mathcal{D}^{\\pm} onto 𝒮 \\mathcal{S} and formulate the steering problem on this manifold."},{"type":"para","text":"We equip 𝒮 \\mathcal{S} with the tangent projector Π 𝒙 = 𝑰 − R − 2 𝒙 𝒙 ⊤ \\Pi_{\\bm{x}}=\\bm{I}-R^{-2}\\bm{x}\\bm{x}^{\\top} , Riemannian gradient ∇ 𝒮 f = Π 𝒙 ∇ f \\nabla_{\\mathcal{S}}f=\\Pi_{\\bm{x}}\\nabla f , and geodesic distance d 𝒮 ( 𝒙 , 𝒚 ) = R arccos ( R − 2 𝒙 ⊤ 𝒚 ) d_{\\mathcal{S}}(\\bm{x},\\bm{y})=R\\arccos\\left(R^{-2}\\bm{x}^{\\top}\\bm{y}\\right) . The spherical exponential map exp 𝒙 \\exp_{\\bm{x}} walks from 𝒙 \\bm{x} along a tangent vector 𝒗 \\bm{v} for a geodesic arc of length ‖ 𝒗 ‖ \\|\\bm{v}\\| , while the logarithm log 𝒙 ( 𝒚 ) \\log_{\\bm{x}}(\\bm{y}) inverts this operation, returning the tangent vector at 𝒙 \\bm{x} that points toward 𝒚 \\bm{y} along the shortest great-circle arc, with magnitude equal to the geodesic distance. Explicitly,"},{"type":"para","text":"With the contrastive samples 𝒟 − ∼ p − \\mathcal{D}^{-}\\sim p_{-} and 𝒟 + ∼ p + \\mathcal{D}^{+}\\sim p_{+} projected onto 𝒮 \\mathcal{S} and the Riemannian toolkit in hand, we develop Cobras that operates entirely within this spherical geometry."},{"type":"heading","level":1,"text":"4 LLM Steering as Schrödinger Bridge"},{"type":"para","text":"A growing body of steering methods Rimsky et al. ( 2024 ); Zhao et al. ( 2026 ); Li et al. ( 2023 ) implicitly construct steering directions that approximate the gradient of the log-density ratio between desired and undesired activations:"},{"type":"para","text":"The term ∇ log p + \\nabla\\log p_{+} points toward regions favored by the desired behavior, while − ∇ log p − -\\nabla\\log p_{-} points away from regions associated with the undesired behavior. Together, they define the steepest-ascent direction of the density ratio. As observed by Zhao et al. ( 2026 ) , many existing steering methods can be reinterpreted as approximations to Eq. 2 . For example, Contrastive Activation Addition (CAA) (Rimsky et al. , 2024 ) gets its steering vector with p + = 𝒩 ( 𝝁 + , 𝑰 d ) p_{+}=\\mathcal{N}(\\bm{\\mu}_{+},\\bm{I}_{d}) and p − = 𝒩 ( 𝝁 − , 𝑰 d ) p_{-}=\\mathcal{N}(\\bm{\\mu}_{-},\\bm{I}_{d}) , leading to v ( 𝒙 ) = ∇ log p + p − = 𝝁 + − 𝝁 − v(\\bm{x})=\\nabla\\log\\frac{p_{+}}{p_{-}}=\\bm{\\mu}_{+}-\\bm{\\mu}_{-} . In Appendix A we extend this connection and show that several other prominent steering paradigms such as Spherical Steering (SS) (You et al. , 2026 ) , Inference-Time Intervention (ITI) (Li et al. , 2023 ) , Householder Reflection Probing (HPR) (Pham and Nguyen, 2024 ) , and ODESteer (Zhao et al. , 2026 ) can also be unified under a singular framework governed by log-density ratio optimization over a latent manifold."},{"type":"para","text":"Nevertheless, Eq. 2 is typically adopted as a plausible objective rather than derived from an optimization problem. To our knowledge, no existing work formulates an optimization problem on the activation manifold whose solution provably yields the log-density-ratio gradient of Eq. 2 . This gap raises a natural question: can such a optimization-based formulation be rigorously established?"},{"type":"para","text":"We show that the Schrödinger Bridge (SB) provides a generalized, principled framework for steering. Specifically, we cast steering as an SB problem between the two marginals p − p_{-} and p + p_{+} and seek the optimal stochastic process that steers the distribution of hidden state from p − p_{-} to p + p_{+} with minimal “effort,” measured as the Kullback-Leibler (KL) divergence from a reference stochastic process. The SB solution induces a stochastic differential equation (SDE). To find the drift of the corresponding SDE, we reduce SB to an entropic optimal transport and solve it using the iterative Sinkhorn algorithm by alternating between updating the dual potentials (Sec. 4.1 ). We then apply the probability flow ODE to the induced SDE drift to obtain a deterministic process with the same time marginals. The resulting ODE has drift proportional to ∇ 𝒮 log ( ψ ^ / ϕ ^ ) \\nabla_{\\mathcal{S}}\\log(\\hat{\\psi}/\\hat{\\phi}) , the Riemannian gradient of the log-ratio of two Schrödinger potentials (Sec. 4.2 ). Finally, we evaluate this drift at arbitrary query points using nonparametric kernel density estimation (Sec. 4.3 ), yielding the Cobras method (Sec. 4.4 )."},{"type":"para","text":"This derivation has two key consequences. First, the heuristic Eq. 2 emerges as a well-defined limiting case of the SB log-ratio log ( ψ ^ / ϕ ^ ) \\log(\\hat{\\psi}/\\hat{\\phi}) : when the Sinkhorn potentials are approximately uniform, ψ ^ \\hat{\\psi} and ϕ ^ \\hat{\\phi} reduce to kernel density estimates of p + p_{+} and p − p_{-} , and the drift becomes ∇ 𝒮 log ( p + / p − ) \\nabla_{\\mathcal{S}}\\log(p_{+}/p_{-}) (Prop. 1 ). To the best of our knowledge, this is the first principled derivation of the log-density-ratio steering objective from a rigorous optimization problem. Second, because the Schrödinger potentials are evaluated at the current activation, the resulting steering is inherently context dependent. Different queries receive different steering directions, with the weights adapting to the query’s location and naturally attenuating intervention for inputs far from the training, thus supporting OOD queries."},{"type":"heading","level":2,"text":"4.1 Formulating the objective of steering"},{"type":"para","text":"We cast the steering problem as finding the path measure P ⋆ P^{\\star} that transports the empirical distribution of undesired activations, p ^ − \\hat{p}_{-} , to the desired distribution, p ^ + \\hat{p}_{+} , while minimizing the KL divergence with respect to a reference path measure Q Q :"},{"type":"para","text":"where P 0 P_{0} and P T P_{T} denote the time- 0 and time- T T marginals of P P . The KL penalty ensures that the resulting transport plan is the minimal entropically regularized perturbation of the reference Q Q . We take Q Q as Brownian motion on 𝒮 \\mathcal{S} with diffusivity σ 2 > 0 \\sigma^{2}>0 . It is defined by the SDE"},{"type":"para","text":"where W t W_{t} is Brownian motion (random Gaussian noise) at time t t . We need the reference’s one-step transition kernel for solving Eq. 3 . In the short-time regime, the reference’s transition kernel is approximated as De Bortoli et al. ( 2022 ); Chen et al. ( 2023b )"},{"type":"para","text":"Intuitively, this kernel acts as a spatial transition probability. It penalizes large geodesic distances d 𝒮 ( 𝒙 , 𝒚 ) d_{\\mathcal{S}}(\\bm{x},\\bm{y}) , ensuring that a particle at 𝒙 \\bm{x} is most likely to diffuse to a nearby point 𝒚 \\bm{y} with spread σ 2 \\sigma^{2} ."},{"type":"para","text":"Having established the reference frame, we shift our focus to the practical resolution of solving Eq. 3 . In practice, the true distributions p − / p + p_{-}/p_{+} are typically inaccessible; consequently, we must operate within a data-driven framework, substituting these theoretical distributions with their empirical counterparts p ^ − / p ^ + \\hat{p}_{-}/\\hat{p}_{+} . With these discrete marginals, the static formulation of Eq. 3 reduces to an entropic optimal transport task with squared geodesic cost c j i = d 𝒮 ( 𝒉 j − , 𝒉 i + ) 2 / ( 2 σ 2 ) c_{ji}=d_{\\mathcal{S}}(\\bm{h}^{-}_{j},\\,\\bm{h}^{+}_{i})^{2}/{(2\\sigma^{2})} Nutz ( 2021 ) , that governs the geometry of the underlying manifold. Then the objective becomes to find a coupling that minimizes the transport cost while accounting for diffusive entropy"},{"type":"para","text":"and by Sinkhorn’s theorem (Cuturi, 2013 ) , the optimal coupling π ⋆ \\pi^{\\star} factorizes as"},{"type":"para","text":"where the dual potentials { ϕ j } j = 1 N − \\{\\phi_{j}\\}_{j=1}^{N_{-}} and { ψ i } i = 1 N + \\{\\psi_{i}\\}_{i=1}^{N_{+}} satisfy"},{"type":"para","text":"These dual potentials are efficiently computed using Sinkhorn iterations, alternating updates between ϕ \\phi and ψ \\psi until convergence and later will be used to obtain the drift of the SDE induced by Eq. 3 ."},{"type":"heading","level":2,"text":"4.2 Optimal Bridge SDE and its Probability Flow ODE"},{"type":"para","text":"With the dual potentials computed, we now derive the forward SDE of the optimal solution to Eq. 3 . The SB optimal bridge forward SDE is Chen et al. ( 2021 )"},{"type":"para","text":"where ψ ^ ( ⋅ , t ) \\hat{\\psi}(\\cdot,t) is the backward Schrödinger potential obtained by propagating the terminal potential ψ \\psi backward via a kernel-weighted sum over the positive samples, with bandwidth ( T − t ) σ 2 (T-t)\\sigma^{2} that shrinks to zero as t → T t\\to T , concentrating mass on p ^ + \\hat{p}_{+} :"},{"type":"para","text":"Similarly, the forward Schrödinger potential propagates ϕ \\phi from the starting margin as a kernel-weighted sum over the negative samples, with bandwidth t σ 2 t\\sigma^{2} that shrinks to zero as t → 0 t\\to 0 , concentrating mass on p ^ − \\hat{p}_{-} :"},{"type":"para","text":"A fundamental property of the SB when the reference process is Brownian motion is that its time-marginal density factorizes as p t ( ⋅ ) = ϕ ^ ( ⋅ , t ) ψ ^ ( ⋅ , t ) p_{t}(\\cdot)=\\hat{\\phi}(\\cdot,t)\\,\\hat{\\psi}(\\cdot,t) (Chen et al. , 2023a ) . Now with the SDE at Eq. 9 in hand, we seek a deterministic process sharing the same marginal evolution with it. The probability flow ODE provides such a process. For any diffusion d 𝒙 t = b d t + σ d W t d\\bm{x}_{t}=b\\,dt+\\sigma\\,dW_{t} with marginal density p t p_{t} , the corresponding probability flow ODE is d 𝒙 t = ( b − σ 2 2 ∇ 𝒮 log p t ) d t d\\bm{x}_{t}=\\left(b-\\tfrac{\\sigma^{2}}{2}\\nabla_{\\mathcal{S}}\\log p_{t}\\right)\\,dt (Song et al. , 2020 ) . Substituting b = σ 2 ∇ 𝒮 log ψ ^ b=\\sigma^{2}\\nabla_{\\mathcal{S}}\\log\\hat{\\psi} and log p t = log ϕ ^ + log ψ ^ \\log p_{t}=\\log\\hat{\\phi}+\\log\\hat{\\psi} gives"},{"type":"para","text":"or equivalently,"},{"type":"para","text":"The two terms in the drift have clear roles: ∇ 𝒮 log ψ ^ \\nabla_{\\mathcal{S}}\\log\\hat{\\psi} attracts toward p ^ + \\hat{p}_{+} , while − ∇ 𝒮 log ϕ ^ -\\nabla_{\\mathcal{S}}\\log\\hat{\\phi} repels from p ^ − \\hat{p}_{-} , and both arise from Eq. 3 . The following proposition shows that the heuristic objective of Eq. 2 is a limiting case of this drift."},{"type":"para","text":"Let c max = max j , i d 𝒮 ( 𝐡 j − , 𝐡 i + ) 2 / ( 2 σ 2 ) c_{\\max}=\\max_{j,i}\\,d_{\\mathcal{S}}(\\bm{h}^{-}_{j},\\bm{h}^{+}_{i})^{2}/(2\\sigma^{2}) . In the regime c max → 0 c_{\\max}\\to 0 , the Sinkhorn potentials converge to the uniform values ψ i → 1 N + \\psi_{i}\\to\\frac{1}{N_{+}} and ϕ j → 1 N − \\phi_{j}\\to\\frac{1}{N_{-}} , and the flow drift reduces to"},{"type":"para","text":"where the numerator and denominator are spherical kernel density estimates of p + p_{+} and p − p_{-} that converge to the true densities as N ± → ∞ N_{\\pm}\\to\\infty , recovering ∇ 𝒮 log ( p + / p − ) \\nabla_{\\mathcal{S}}\\log(p_{+}/p_{-}) ."},{"type":"para","text":"The proof is provided in Appendix B.1 . Prop. 1 shows that in the limit of no transportation cost, e.g., due to small σ \\sigma , the drift of the probability flow ODE converges to ∇ 𝒮 log ( p + / p − ) \\nabla_{\\mathcal{S}}\\log(p_{+}/p_{-}) , and optimizing ∇ 𝒮 log ( p + / p − ) \\nabla_{\\mathcal{S}}\\log(p_{+}/p_{-}) recovers the steering dictated by SB. To our knowledge, this is the first principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. Moreover, because ψ ^ \\hat{\\psi} and ϕ ^ \\hat{\\phi} are evaluated at the current activation, the drift is inherently query-adaptive, thus avoiding the performance degradation on OOD inputs that is commonly seen in fixed-direction steering methods."},{"type":"heading","level":2,"text":"4.3 Non-parametric Estimation of the Probability Flow ODE Drift"},{"type":"para","text":"To steer a hidden state 𝒉 q \\bm{h}_{q} , we need to solve the ODE in Eq. 12 with initial 𝒙 0 = 𝒉 q \\bm{x}_{0}=\\bm{h}_{q} which requires computing the Riemannian gradients ∇ 𝒮 log ψ ^ \\nabla_{\\mathcal{S}}\\log\\hat{\\psi} and ∇ 𝒮 log ϕ ^ \\nabla_{\\mathcal{S}}\\log\\hat{\\phi} as continuous functions on 𝒮 \\mathcal{S} . Here we derive these objects using Kernel Density Estimation (KDE). This requires us to extend the potential values ( ϕ j , ψ i ) (\\phi_{j},\\psi_{i}) , but they are only defined at the training points in 𝒟 − / 𝒟 + \\mathcal{D}^{-}/\\mathcal{D}^{+} . To evaluate them at a query 𝒉 q \\bm{h}_{q} , we marginalize through the opposite training points via the kernel and cost matrix:"},{"type":"para","text":"For a query 𝒉 q \\bm{h}_{q} , Eq. 15 weights each negative sample 𝒉 j − \\bm{h}^{-}_{j} by its proximity (via ϕ j \\phi_{j} and the kernel), then propagate through the cost matrix to assign an effective potential to each positive sample 𝒉 i + \\bm{h}^{+}_{i} ."},{"type":"heading","level":4,"text":"Riemannian gradient via kernel density estimation."},{"type":"para","text":"The gradient of log ψ ^ \\log\\hat{\\psi} at 𝒉 q \\bm{h}_{q} is a weighted Riemannian mean direction in the tangent space T 𝒉 q 𝒮 T_{\\bm{h}_{q}}\\mathcal{S} , yielding (the derivation is provided in Appendix B.2 ):"},{"type":"para","text":"with weights"},{"type":"para","text":"and symmetrically for ∇ 𝒮 log ϕ ^ ( 𝒉 q ) = ∑ j w ¯ j − ( 𝒉 q ) log 𝒉 q ( 𝒉 j − ) \\nabla_{\\mathcal{S}}\\log\\hat{\\phi}(\\bm{h}_{q})=\\sum_{j}\\bar{w}^{-}_{j}(\\bm{h}_{q})\\,\\log_{\\bm{h}_{q}}(\\bm{h}^{-}_{j}) with weights w ¯ j − ( 𝒉 q ) \\bar{w}^{-}_{j}(\\bm{h}_{q}) defined over 𝒟 − \\mathcal{D}^{-} using ϕ ~ j \\tilde{\\phi}_{j} . Substituting these expressions into the probability flow ODE in Eq. 12 and absorbing the constant σ 2 2 \\frac{\\sigma^{2}}{2} into the step size yields the Cobras vector field:"},{"type":"heading","level":2,"text":"4.4 The Overall Algorithm"},{"type":"para","text":"Given the vector field in Eq. 19 , we steer a query activation 𝒉 q \\bm{h}_{q} by integrating the probability flow ODE at Eq. 12 on 𝒮 \\mathcal{S} using K K geodesic Euler steps. At each step, we evaluate the field at the current iterate and update via the exponential map in Eq. 1 , ensuring that every iterate remains on 𝒮 \\mathcal{S} . After integration, we restore the original norm ‖ 𝒉 q ‖ \\|\\bm{h}_{q}\\| . Algorithm 1 summarizes the full procedure. The algorithm operates in two distinct phases: an offline optimization stage (Lines 1-3) to solve the static SB problem and an online inference stage (Lines 4-10) that evolves a query activation along a probability flow. In the offline phase, it projects sample data onto a hypersphere and solves an entropic optimal transport problem via the Sinkhorn algorithm to derive Schrödinger potentials. During the online phase, the algorithm iteratively steers a query activation 𝒉 q \\bm{h}_{q} by integrating a probability flow ODE. It calculates a velocity vector 𝒗 k \\bm{v}_{k} in the tangent space based on weighted geodesic distances between positive and negative samples, updating the intermediate state 𝒒 k \\bm{q}_{k} via the exponential map. The process concludes by rescaling the final activation 𝒒 K \\bm{q}_{K} to the activation’s original norm."},{"type":"para","text":"For OOD, an abstain gate attenuates steering by comparing the k abs k_{\\mathrm{abs}} -nearest-neighbor geodesic distance from 𝒒 0 \\bm{q}_{0} to 𝒟 − \\mathcal{D}^{-} against a reference radius ρ ref \\rho_{\\mathrm{ref}} (the p p -th percentile of intra-negative k abs k_{\\mathrm{abs}} -NN distances), ensuring that queries far from the training support are left largely unmodified. For per-query step size, a vMF strength gate s vMF ( 𝒒 0 ) s_{\\mathrm{vMF}}(\\bm{q}_{0}) measures the alignment of the query with the mean contrastive direction 𝒉 ¯ + − 𝒉 ¯ − \\bar{\\bm{h}}^{+}-\\bar{\\bm{h}}^{-} , so that queries already aligned receive less steering."},{"type":"heading","level":1,"text":"5 Experiments"},{"type":"para","text":"We conduct experiments to answer the following research questions. (RQ1): How does Cobras compare to other steering methods on In-distribution tasks? (RQ2): How does Cobras perform on OOD tasks? (RQ3): How does each parameter of Cobras impact its performance?"},{"type":"heading","level":4,"text":"Datasets."},{"type":"para","text":"We conduct our evaluations on three prevalent steering datasets: TruthfulQA Lin et al. ( 2021 ) (TQA) for Truthfulness, Ultrafeedback Cui et al. ( 2023 ) (UF) for Helpfulness and RealToxicityPrompts Gehman et al. ( 2020 ) for Detoxification. Further, to compare the performance of steering methods on Out-of-Distribution (OOD) tasks, we use MMLU Hendrycks et al. ( 2020 ) and GSM8K Cobbe et al. ( 2021 ) ."},{"type":"heading","level":4,"text":"Baselines."},{"type":"para","text":"Following ODESteer Zhao et al. ( 2026 ) , we compare Cobras against the following baselines: Representation Engineering (Zou et al. , 2023 ) (RepE), Inference-Time Intervention (Li et al. , 2023 ) (ITI), Contrastive Activation Addition (Rimsky et al. , 2024 ) (CAA), Minimally Modified Counterfactuals (Singh et al. , 2024 ) (MiMiC), Linear Activation Transport (Rodriguez et al. , 2025 ) (Linear-AcT), SphericalSteering You et al. ( 2026 ) and ODESteer Zhao et al. ( 2026 ) . We test the steering methods on four open source models: Falcon-7B (Almazrouei et al. , 2023 ) , Mistral-7B (Jiang et al. , 2023 ) , LLaMA3.1-8B (Meta AI, 2024 ) and Qwen2.5-7B Team ( 2024 ) ."},{"type":"heading","level":4,"text":"Metrics."},{"type":"para","text":"Following ODESteer Zhao et al. ( 2026 ) , we use Truthfulness x Informativeness % as the primary metric for TQA while reporting Truthfulness% and Informativeness%. For UF/Helpfulness task, we use win rate over no steer (primary metric), Mean Reward and 90th Percentile Reward as secondary metrics. For Detoxification task, we use Toxicity Score from Perspective API Jigsaw and Google ( 2026 ) as primary metric while reporting Perplexity and Dist-2 as secondary metrics. For GSM8K and MMLU, we use accuracy as the metric. See Appendix C.3 for details."},{"type":"heading","level":4,"text":"RQ1: How does Cobras compare to other steering methods on in-distribution tasks?"},{"type":"para","text":"We first evaluate Cobras in the standard in-distribution setting, where each steering method is trained and tested on the same task axis. We consider three representative alignment objectives: helpfulness on UltraFeedback, truthfulness on TruthfulQA, and detoxification on RealToxicityPrompts. Table 1 reports results across four base models."},{"type":"para","text":"Cobras consistently improves the primary metric for all three tasks across models. On helpfulness, Cobras achieves the highest UltraFeedback win rate for every model, improving over the original model by + 2.8 +2.8 , + 7.1 +7.1 , + 9.7 +9.7 , and + 1.9 +1.9 absolute points on Falcon-7B, Mistral-7B, LLaMA3.1-8B, and Qwen2.5-7B, respectively. On truthfulness, Cobras obtains the best TruthfulQA True × \\times Info score across all four models, with especially large gains over the original model on Mistral-7B ( + 29.5 ) (+29.5) and LLaMA3.1-8B ( + 25.1 ) (+25.1) . On detoxification, Cobras also achieves the lowest toxicity score for every model, reducing toxicity by 46 % 46\\% – 53 % 53\\% relative to the original models."},{"type":"para","text":"Compared to prior steering baselines, Cobras delivers the most consistent gains across objectives. While other methods can perform well on individual tasks, they often trade off informativeness, perplexity, or diversity. In contrast, Cobras improves target alignment metrics while largely preserving generation quality, making it a robust in-distribution steering method across helpfulness, truthfulness, and detoxification."},{"type":"heading","level":4,"text":"RQ2: How does Cobras perform on OOD tasks?"},{"type":"para","text":"Activation steering can be brittle under distribution shift, with prior work showing that steering vectors may be unreliable in-distribution and can misgeneralize out-of-distribution (OOD) (Tan et al. , 2024 ; Braun et al. , 2025 ; Korznikov et al. , 2025 ) . We therefore evaluate whether Cobras improves TruthfulQA performance without degrading unrelated capabilities."},{"type":"para","text":"We train all steering methods on TruthfulQA and evaluate OOD performance on MMLU Hendrycks et al. ( 2020 ) and GSM8K Cobbe et al. ( 2021 ) . Figure 2 plots OOD accuracy against TruthfulQA True × \\times Info; the ideal method appears in the upper-right, improving truthfulness while preserving OOD accuracy. Baselines show inconsistent trade-offs: several methods improve TruthfulQA but substantially reduce OOD performance, especially on GSM8K for LLaMA3.1-8B and Mistral-7B, and on MMLU for Mistral-7B. This suggests that some steering methods sacrifice general reasoning ability for in-distribution gains. By contrast, Cobras consistently lie near the upper-right across all models and OOD benchmarks. It improves TruthfulQA over the original model while maintaining OOD accuracy close to, or above, the unsteered model, avoiding the severe GSM8K drops seen in several baselines. Overall, Cobras achieves the most favorable trade-off between in-distribution steering effectiveness and OOD capability preservation."},{"type":"heading","level":4,"text":"RQ3: Ablation of Cobras Hyperparameters"},{"type":"para","text":"We ablate the abstention neighborhood size K K and abstention percentile; steering-strength ablations are deferred to Appendix D.1 . Cobras uses an abstention rule to avoid steering inputs far from the training distribution. For each training example, we compute the distance to its K K -th nearest neighbor and set the abstention threshold to a chosen percentile of these distances. At test time, Cobras abstains from steering when an input’s neighborhood distance exceeds this threshold. In-distribution inputs are expected to have nearby neighbors and therefore be steered, whereas OOD inputs should have larger distances and trigger abstention. Table 2 evaluate this mechanism on LLaMA3.1-8B-Base and Mistral-7B-Base using TruthfulQA as the in-distribution task and GSM8K as the OOD task. Increasing the abstention percentile makes Cobras steer more examples. This improves TruthfulQA True × \\times Info, but can reduce OOD accuracy, most notably for LLaMA3.1-8B-Base at the 99.5 99.5 th percentile. Increasing K K shows a similar trade-off: larger neighborhoods improve TruthfulQA, but can lower GSM8K accuracy, especially for LLaMA3.1-8B-Base. These results show that abstention helps balance effective in-distribution steering with preserving OOD capabilities."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"By bridging the gap between heuristic methods and the rigorous optimization framework of the Schrödinger Bridge, this paper provides the first principled derivation of the log-density-ratio steering objective for activation steering. Experiments across four models and three alignment axes show that Cobras consistently outperforms baselines in-distribution while preserving OOD performance (commonly seen in fixed-direction steering methods), thanks to its query-adaptive nature. Limitations include the 𝒪 ( N + ⋅ N − ) \\mathcal{O}(N_{+}\\cdot N_{-}) cost of pairwise geodesic distances and Sinkhorn, which may be prohibitive for very large contrastive sets, and the reliance on a fixed spherical geometry that may not hold for all architectures or layers. Promising future directions include hierarchical transport solvers to reduce computational overhead and extending the framework to learned or more general Riemannian manifolds."},{"type":"para","text":"In this section, we show that diverse activation steering methodologies can be unified under a single theoretical framework governed by log-density ratio optimization. Specifically, we analyze how these seemingly disparate techniques implicitly optimize the ratio log ( p + / p − ) \\log(p_{+}/p_{-}) by reinterpreting them as specific geometric configurations of a continuous global flow. Formally, let p + p_{+} and p − p_{-} define a pair of continuous densities corresponding to the contrastive behavior distributions 𝒟 + \\mathcal{D}^{+} and 𝒟 − \\mathcal{D}^{-} , respectively. Given an initial query activation 𝒉 q \\bm{h}_{q} on an underlying representation manifold ℳ \\mathcal{M} , the steering trajectory is obtained by integrating the continuous-time ordinary differential equation:"},{"type":"para","text":"for K K steps over a specified time horizon [ 0 , T ] [0,T] . Here, 𝒙 t \\bm{x}_{t} denotes the steered activation along the flow, initialized as 𝒙 0 = 𝒉 q \\bm{x}_{0}=\\bm{h}_{q} . By systematically varying the manifold geometry ℳ \\mathcal{M} , the step horizon K K , and the parameterization of the density functions, we can recover prominent steering paradigms from this single formulation 1 1 1 Zhao et al. [ 2026 ] presents a related unification of several of these methods, but their analysis is confined to Euclidean space; by treating the manifold ℳ \\mathcal{M} as an explicit design choice, however, our framework additionally captures geometry-aware methods such as SS and HPR. ."},{"type":"para","text":"For example, the following methods can be recalled from this perspective as summarized in Table 3 ."},{"type":"heading","level":4,"text":"Contrastive Activation Addition (CAA) [Rimsky et al. , 2024 ]"},{"type":"para","text":"The CAA algorithm can be framed as a specific, single-step optimization instantiation of a continuous global flow over the hidden state manifold. Specifically, CAA simplifies this trajectory by optimizing over a flat Euclidean space ℳ = ℝ d \\mathcal{M}=\\mathbb{R}^{d} via a single discrete step ( K = 1 K=1 ). By modeling the factual (positive) and counterfactual (negative) activation distributions as isotropic Gaussians sharing an identity covariance p + = 𝒩 ( 𝝁 + , 𝑰 d ) p_{+}=\\mathcal{N}(\\bm{\\mu}_{+},\\bm{I}_{d}) and p − = 𝒩 ( 𝝁 − , 𝑰 d ) p_{-}=\\mathcal{N}(\\bm{\\mu}_{-},\\bm{I}_{d}) , the complex continuous gradient field collapses into a constant shift vector defined by the difference of their respective means ∇ ℳ log p + p − = 𝝁 + − 𝝁 − \\nabla_{\\mathcal{M}}\\log\\frac{p_{+}}{p_{-}}=\\bm{\\mu}_{+}-\\bm{\\mu}_{-} . Consequently, the empirical practice of injecting a static difference vector during the forward pass is mathematically grounded as an exact, first-order approximation toward maximizing the target likelihood ratio."},{"type":"heading","level":4,"text":"Spherical Steering (SS) [You et al. , 2026 ]"},{"type":"para","text":"To relax the assumption of a flat Euclidean space and account for the directional nature of high-dimensional neural representations, Spherical Steering constrains the optimization trajectory to a unit hypersphere manifold, setting ℳ = 𝒮 d − 1 \\mathcal{M}=\\mathcal{S}^{d-1} . Under a single-step discrete horizon ( K = 1 K=1 ), the method models the target positive activations using a von Mises-Fisher (vMF) distribution p + = vMF ( ⋅ ; 𝝁 , κ ) p_{+}=\\mathrm{vMF}(\\cdot;\\bm{\\mu},\\kappa) which acts as the spherical analogue to an isotropic Gaussian with a mean direction vector 𝝁 \\bm{\\mu} and a concentration parameter κ \\kappa . Conversely, the negative behavior is modeled as a uniform distribution over the sphere p − = Unif ( 𝒮 ) p_{-}=\\mathrm{Unif}(\\mathcal{S}) . The unit prototype towards the positive activations 𝝁 \\bm{\\mu} is empirically defined by normalizing the difference between the mean positive and negative activations:"},{"type":"para","text":"When evaluating the global flow framework under these spherical constraints, the Riemannian gradient of the likelihood ratio yields a non-linear direction proportional to the logarithmic map on the sphere ∇ 𝒮 log p + p − ∝ log 𝒙 ( 𝝁 ) \\nabla_{\\mathcal{S}}\\log\\tfrac{p_{+}}{p_{-}}\\propto\\log_{\\bm{x}}(\\bm{\\mu}) which corresponds to the Slerp direction toward 𝝁 \\bm{\\mu} in the direction of this Riemannian gradient. By explicitly respecting the spherical geometry of hidden states, SS ensures that the magnitude of the activation vector is preserved, avoiding out-of-distribution distortion while performing directional steering."},{"type":"heading","level":4,"text":"Inference-Time Intervention (ITI) [Li et al. , 2023 ]"},{"type":"para","text":"Rather than assuming explicit parametric distributions for the hidden states, ITI leverages a discriminative classifier to implicitly estimate the underlying density ratio. Operating within a single-step horizon ( K = 1 K=1 ) over a flat Euclidean vector space ℳ = ℝ d \\mathcal{M}=\\mathbb{R}^{d} , ITI trains a linear logistic probe, p 𝜽 ( 𝒙 ) = σ ( 𝜽 ⊤ 𝒙 ) p_{\\bm{\\theta}}(\\bm{x})=\\sigma(\\bm{\\theta}^{\\top}\\bm{x}) , to distinguish between positive and negative activation vectors. By invoking the classic equivalence between logistic regression and density ratio estimation, the log-density ratio can be rewritten linearly as log p + ( 𝒙 ) p − ( 𝒙 ) = 𝜽 ⊤ 𝒙 + log N − N + \\log\\frac{p_{+}(\\bm{x})}{p_{-}(\\bm{x})}=\\bm{\\theta}^{\\top}\\bm{x}+\\log\\frac{N_{-}}{N_{+}} , where N + N_{+} and N − N_{-} represent the respective sample counts. Taking the gradient with respect to the manifold yields a constant vector field defined entirely by the probe’s learned weights:"},{"type":"para","text":"This formulation reveals a powerful geometric justification for ITI: the orthogonal vector to the classifier’s decision boundary acts as the exact gradient direction needed to maximize the factual likelihood ratio. Consequently, shifting activations along the direction of 𝜽 \\bm{\\theta} during inference provides a theoretically grounded, first-order trajectory for target behavior optimization."},{"type":"heading","level":4,"text":"Householder Reflection Probe (HPR) [Pham and Nguyen, 2024 ]"},{"type":"para","text":"To reconcile discriminative, probe-based steering with non-Euclidean representation spaces, the HPR method maps the log-density ratio directly onto a hyperspherical manifold ( ℳ = 𝒮 d − 1 \\mathcal{M}=\\mathcal{S}^{d-1} ) under a single-step horizon ( K = 1 K=1 ). Similar to ITI, the method first projects the log-density ratio linearly using a trained linear probe, log p + ( 𝒙 ) p − ( 𝒙 ) = 𝜽 ⊤ 𝒙 + log N − N + \\log\\frac{p_{+}(\\bm{x})}{p_{-}(\\bm{x})}=\\bm{\\theta}^{\\top}\\bm{x}+\\log\\frac{N_{-}}{N_{+}} . To respect the spherical constraints, HPR constructs an orthogonal Householder reflection matrix, 𝑯 = 𝑰 − 2 𝒖 𝒖 ⊤ \\bm{H}=\\bm{I}-2\\bm{u}\\bm{u}^{\\top} , where 𝒖 = 𝜽 / ‖ 𝜽 ‖ \\bm{u}=\\bm{\\theta}/\\|\\bm{\\theta}\\| represents the unit normal vector to the probe’s decision boundary. By reflecting the normalized activation vector 𝒙 ^ = 𝒙 / ‖ 𝒙 ‖ \\hat{\\bm{x}}=\\bm{x}/\\|\\bm{x}\\| across this boundary, the directional gradient on the sphere is formalized as ∇ 𝒮 log p + ( 𝒙 ) p − ( 𝒙 ) ∝ Π 𝒙 ^ ( 𝑯 𝒙 ^ − 𝒙 ^ ) \\nabla_{\\mathcal{S}}\\log\\frac{p_{+}(\\bm{x})}{p_{-}(\\bm{x})}\\propto\\Pi_{\\hat{\\bm{x}}}(\\bm{H}\\hat{\\bm{x}}-\\hat{\\bm{x}}) , where Π 𝒙 ^ \\Pi_{\\hat{\\bm{x}}} denotes the projection onto the tangent space at 𝒙 ^ \\hat{\\bm{x}} . Geometrically, this confirms that the optimal trajectory corresponds to a Spherical Linear Interpolation (Slerp) toward the reflected state 𝑯 𝒙 ^ \\bm{H}\\hat{\\bm{x}} . By leveraging a reflection operator, HPR effectively utilizes the probe’s decision boundaries to execute directional steering while intrinsically preserving the latent norm of the activation path."},{"type":"heading","level":4,"text":"ODESteer [Zhao et al. , 2026 ]"},{"type":"para","text":"Departing from single-step first-order approximations ( K = 1 K=1 ), ODESteer fully embraces the continuous-time formulation of the global flow framework by integrating across multiple steps ( K > 1 K>1 ) over a flat Euclidean vector space ℳ = ℝ d \\mathcal{M}=\\mathbb{R}^{d} . Rather than adopting simple parametric distributions or linear probes, this approach models complex, non-linear activation dynamics by parameterizing the log-density ratio with a polynomial count-sketch feature map ϕ \\phi and setting log p + ( 𝒙 ) p − ( 𝒙 ) = 𝒘 ⊤ ϕ ( 𝒙 ) + b \\log\\frac{p_{+}(\\bm{x})}{p_{-}(\\bm{x})}=\\bm{w}^{\\top}\\phi(\\bm{x})+b , where the parameter pair ( 𝒘 , b ) (\\bm{w},b) is fit empirically over the contrastive dataset. Under this non-linear projection, the steering direction is dictated by a dynamic, normalized vector field:"},{"type":"para","text":"where 𝑱 ϕ ( 𝒙 ) \\bm{J}_{\\phi}(\\bm{x}) denotes the Jacobian matrix of the feature map ϕ \\phi evaluated at the current state 𝒙 \\bm{x} . By evaluating and integrating this state-dependent vector field using discrete Euler steps over the time horizon [ 0 , T ] [0,T] , ODESteer traces a curved, non-linear path through the representation space. This multi-step integration allows the model to continuously adapt to the changing manifold geometry during the steering process, providing precise control while tracking complex latent boundaries."},{"type":"para","text":"In this section, we provide the theoretical analysis for Cobras ."},{"type":"heading","level":2,"text":"B.1 Proof of Proposition 1"},{"type":"para","text":"When c max → 0 c_{\\max}\\to 0 , every cost entry c j i → 0 c_{ji}\\to 0 . The Sinkhorn update for ϕ j \\phi_{j} (Eq. 8 ) becomes"},{"type":"para","text":"which is independent of j j . Hence, every ϕ j \\phi_{j} takes the same value. By the symmetric argument applied to the ψ i \\psi_{i} update, every ψ i \\psi_{i} is also equal. The marginal constraints ∑ i π j i ⋆ = 1 N − \\sum_{i}\\pi^{\\star}_{ji}=\\frac{1}{N_{-}} and ∑ j π j i ⋆ = 1 N + \\sum_{j}\\pi^{\\star}_{ji}=\\frac{1}{N_{+}} then fix ψ i = 1 N + \\psi_{i}=\\frac{1}{N_{+}} and ϕ j = 1 N − \\phi_{j}=\\frac{1}{N_{-}} ."},{"type":"para","text":"Substituting into Eqs. 10 - 11 :"},{"type":"para","text":"The log-ratio is"},{"type":"para","text":"The constant vanishes and yields Eq. 14 . The numerator and denominator are spherical kernel density estimates of p + p_{+} and p − p_{-} , which converge to the true densities as N ± → ∞ N_{\\pm}\\to\\infty [Pelletier, 2005 ] . ∎"},{"type":"heading","level":2,"text":"B.2 Derivation of the Riemannian Gradient of log ψ ^ \\log\\hat{\\psi} and log ϕ ^ \\log\\hat{\\phi}"},{"type":"para","text":"We derive Eq. ( 17 ) which is the closed-form Riemannian gradient of the log backward Schrödinger factor on 𝒮 \\mathcal{S} . The derivation for ∇ 𝒮 log φ ^ \\nabla_{\\mathcal{S}}\\log\\hat{\\varphi} is identical with ( 𝒟 − , φ ~ j ) (\\mathcal{D}^{-},\\tilde{\\varphi}_{j}) replacing ( 𝒟 + , ψ ~ i ) (\\mathcal{D}^{+},\\tilde{\\psi}_{i}) ."},{"type":"heading","level":2,"text":"B.3 Gradient of the Geodesic Gaussian Kernel"},{"type":"para","text":"Let K σ ( 𝒙 , 𝒉 ) = exp ( − d 𝒮 ( 𝒙 , 𝒉 ) 2 / ( 2 σ ) ) K_{\\sigma}(\\bm{x},\\bm{h})=\\exp\\!\\bigl(-d_{\\mathcal{S}}(\\bm{x},\\bm{h})^{2}/(2\\sigma)\\bigr) for a bandwidth σ > 0 \\sigma>0 . By the chain rule and the fact that ∇ 𝒮 d 𝒮 ( 𝒙 , 𝒉 ) 2 = − 2 log 𝒙 ( 𝒉 ) \\nabla_{\\mathcal{S}}\\,d_{\\mathcal{S}}(\\bm{x},\\bm{h})^{2}=-2\\,\\log_{\\bm{x}}(\\bm{h}) :"},{"type":"heading","level":2,"text":"B.4 Gradient of the Log Backward Potential"},{"type":"para","text":"Define the backward potential at 𝒙 ∈ 𝒮 \\bm{x}\\in\\mathcal{S} :"},{"type":"para","text":"where the extended potentials ψ ~ i ( 𝒙 ) \\tilde{\\psi}_{i}(\\bm{x}) are also treated as constants with respect to differentiation at 𝒙 \\bm{x} ."},{"type":"para","text":"Taking the Riemannian gradient of log ψ ^ \\log\\hat{\\psi} :"},{"type":"para","text":"Substituting the kernel gradient ( 24 ) with σ = σ 2 \\sigma=\\sigma^{2} :"},{"type":"para","text":"Define the normalized weights"},{"type":"para","text":"In log-space, the unnormalized log-weight of point i i is"},{"type":"para","text":"so normalization is a softmax:"},{"type":"para","text":"This recovers Eq. ( 18 ). Substituting into ( 27 ):"},{"type":"para","text":"This is a weighted Riemannian mean direction in the tangent space T 𝒙 𝒮 T_{\\bm{x}}\\mathcal{S} . Each log-map vector log 𝒙 ( 𝒉 i + ) \\log_{\\bm{x}}(\\bm{h}^{+}_{i}) points from the current query 𝒙 \\bm{x} toward the positive point 𝒉 i + \\bm{h}^{+}_{i} with magnitude equal to the geodesic distance, and the weight w ¯ i + ( 𝒙 ) \\bar{w}^{+}_{i}(\\bm{x}) combines geometric proximity (the kernel) with the Schrödinger potential (the transport plan)."},{"type":"heading","level":2,"text":"C.1 Base LLM Models"},{"type":"para","text":"Our base LLMs follow the experimental setup of Zhao et al. Zhao et al. [ 2026 ] . We describe them here for completeness. We evaluate all methods on four open-source models: (i) Falcon-7B, tiiuae/falcon-7b 2 2 2 https://huggingface.co/tiiuae/falcon-7b ; (ii) Mistral-7B-v0.3, mistralai/Mistral-7B-v0.3 3 3 3 https://huggingface.co/mistralai/Mistral-7B-v0.3 ; (iii) LLaMA3.1-8B, meta-llama/Llama-3.1-8B 4 4 4 https://huggingface.co/meta-llama/Llama-3.1-8B ; and (iv) Qwen2.5-7B, Qwen/Qwen2.5-7B 5 5 5 https://huggingface.co/Qwen/Qwen2.5-7B . We use the same generation configuration for all models and experiments: temperature = 0.7 =0.7 , top- p = 0.9 p=0.9 , and repetition penalty = 1.1 =1.1 ."},{"type":"heading","level":2,"text":"C.2 Datasets"},{"type":"para","text":"TruthfulQA [Lin et al. , 2021 ] is a benchmark consisting of 817 questions paired with true and false reference answers, along with source information. The questions cover 38 diverse categories and are commonly evaluated in a zero-shot setting. Following Zhao et al. Zhao et al. [ 2026 ] , we construct 5,918 question–answer pairs and use a 40/10/50 train/validation/test split with two-fold cross validation."},{"type":"para","text":"UltraFeedback [Cui et al. , 2023 ] is a large-scale alignment dataset covering instruction-following qualities such as truthfulness, honesty, and helpfulness. In our experiments, we use the UltraFeedback-Binarized dataset 6 6 6 https://huggingface.co/datasets/HuggingFaceH4/ultrafeedback_binarized . Specifically, we use the preference-modeling split, where each prompt is paired with an accepted response and a rejected response. Following Zhao et al. Zhao et al. [ 2026 ] , we randomly sample 10K training pairs, 500 validation pairs, and 500 test pairs."},{"type":"para","text":"RealToxicityPrompts [Gehman et al. , 2020 ] is a benchmark for evaluating toxic degeneration in language models. It consists of naturally occurring web prompts paired with toxicity annotations obtained from the Perspective API. The dataset contains approximately 21K toxic prompts with toxicity scores greater than 0.5 and 77K non-toxic prompts. Following Zhao et al. [Zhao et al. , 2026 ] , we train toxicity steering methods using the Jigsaw Unintended Bias in Toxicity Classification dataset 7 7 7 https://www.kaggle.com/c/jigsaw-unintended-bias-in-toxicity-classification and evaluate them on RealToxicityPrompts. Specifically, we construct a balanced Jigsaw training set of 10K sentences by sampling according to toxicity scores, with 5K toxic and 5K benign examples. For evaluation, we select 500 toxic prompts from RealToxicityPrompts and use them as inputs to the LLMs. This setup allows us to assess whether steering methods reduce toxic generations while preserving response quality."},{"type":"para","text":"GSM8K [Cobbe et al. , 2021 ] contains approximately 8.5K grade-school math word problems paired with natural-language solutions. The dataset contains roughly 7.5K training examples and 1K test examples. We use the original test split for out-of-distribution evaluation. All methods are trained on TruthfulQA as described above and then evaluated on the 1K GSM8K test examples in a 5-shot setting."},{"type":"para","text":"MMLU [Hendrycks et al. , 2020 ] is a large-scale multiple-choice benchmark covering a broad range of academic and professional subjects. It contains approximately 100K training examples and 14K test examples. We use the default test split for out-of-distribution evaluation and perform 5-shot generation."},{"type":"heading","level":2,"text":"C.3 Metrics"},{"type":"para","text":"Truthfulness (%). Truthfulness is evaluated on TruthfulQA Lin et al. [ 2021 ] using a publicly available fine-tuned LLaMA2-7B judge 8 8 8 https://huggingface.co/allenai/truthfulqa-truth-judge-llama2-7B . The judge predicts a binary truthfulness label for each generated answer using prompts of the form:"},{"type":"para","text":"We report the percentage of generations classified as truthful."},{"type":"para","text":"Informativeness (%). Informativeness is evaluated on TruthfulQA Lin et al. [ 2021 ] using a publicly available fine-tuned LLaMA2-7B judge 9 9 9 https://huggingface.co/allenai/truthfulqa-info-judge-llama2-7B . The judge predicts whether a response is informative or helpful using prompts of the form:"},{"type":"para","text":"We report the percentage of generations classified as informative."},{"type":"para","text":"Truthfulness × \\times Informativeness (%). Following prior work, we report the percentage of responses that are both truthful and informative. For each prompt, this metric is computed as the product of the binary truthfulness and informativeness labels, and the final score is averaged over all prompts."},{"type":"para","text":"Win Rate (%). We adopt the reward-model win-rate metric used in RewardBench [Lambert et al. , 2025 ] and ODESteer [Zhao et al. , 2026 ] . Given a set of prompts { x i } i = 1 N \\{x_{i}\\}_{i=1}^{N} and two candidate systems A A and B B , let s i A s_{i}^{A} and s i B s_{i}^{B} denote their reward-model scores under the same reward model. The win rate of A A over B B is defined as"},{"type":"para","text":"where 𝟙 ( ⋅ ) \\mathbbm{1}(\\cdot) is the indicator function. A value of 0.5 0.5 indicates parity with system B B , values greater than 0.5 0.5 indicate that A A outperforms B B , and ties contribute 0.5 0.5 by convention."},{"type":"para","text":"RM mean {}_{\\text{mean}} and RM 90 . For UltraFeedback, we use an LLM-as-a-judge reward model to score helpfulness. Specifically, we use Skywork/Skywork-Reward-V2-Llama-3.1-8B 10 10 10 https://huggingface.co/Skywork/Skywork-Reward-V2-Llama-3.1-8B . We report RM mean {}_{\\text{mean}} , the average reward score across examples and three random seeds, and RM 90 , the 90th percentile reward score."},{"type":"para","text":"Toxicity Score. Toxicity is evaluated using the Perspective API 11 11 11 https://perspectiveapi.com/ . The toxicity score is a probability-like score indicating the likelihood that a reader would perceive the generated text as toxic. Lower values therefore indicate less toxic generations."},{"type":"para","text":"Perplexity. We evaluate generation fluency using perplexity under GPT-2-XL Radford et al. [ 2019 ] . Given a generated response y = ( y 1 , … , y T ) y=(y_{1},\\ldots,y_{T}) , GPT-2-XL assigns a next-token negative log-likelihood"},{"type":"para","text":"The perplexity of y y is then, PPL ( y ) = exp ( ℒ ( y ) ) . \\mathrm{PPL}(y)=\\exp\\left(\\mathcal{L}(y)\\right). Lower perplexity indicates that the generated text is more fluent under GPT-2-XL."},{"type":"para","text":"Dist- n n . We measure lexical diversity using Dist- n n for n ∈ { 1 , 2 , 3 } n\\in\\{1,2,3\\} . For a generated response y y , let 𝒢 n ( y ) \\mathcal{G}_{n}(y) denote the multiset of all word-level n n -grams in y y , obtained by whitespace tokenization. Dist- n n is defined as"},{"type":"para","text":"If the response contains fewer than n n words, we set Dist - n ( y ) = 0 \\mathrm{Dist}\\text{-}n(y)=0 . Higher Dist- n n indicates greater lexical diversity."},{"type":"heading","level":2,"text":"C.4 Implementation Details"},{"type":"para","text":"All our experiments are conducted on a machine equipped with 1x H100 80GB. We extend ODESteer codebase 12 12 12 https://github.com/ZhaoHongjue/odesteer to implement our steering methods and baselines. Cobras uses the following default values: steering strength of 0.65 0.65 , abstention K K of 32 32 , abstention percentile of 98 98 , geodesic Euler steps K K = 10."},{"type":"heading","level":2,"text":"D.1 Effect of steering strength T T on Cobras"},{"type":"para","text":"Figure 3 shows that steering strength has a non-monotonic effect. On TruthfulQA, moderate T T generally improves True × \\times Info, with peaks around T = 0.6 T=0.6 for LLaMA3.1-8B, T = 0.7 T=0.7 for Mistral-7B, and T = 0.4 T=0.4 for Qwen2.5-7B; larger values often degrade performance, suggesting over-steering. Falcon-7B is flatter and weaker overall, with a sharp drop at high T T . On UltraFeedback, the trend is more model-dependent: LLaMA3.1-8B benefits from moderate T T , Falcon-7B is relatively stable, while Mistral-7B and Qwen2.5-7B degrade as T T increases. Overall, these results support using moderate steering, typically T ≈ 0.5 T\\approx 0.5 – 0.7 0.7 for TruthfulQA and lower-to-moderate values for UltraFeedback."},{"type":"heading","level":2,"text":"D.2 Diversity on Detoxification"},{"type":"para","text":"Table 4 shows the generation diversity of steering methods on the detoxification task using Dist- n n metrics. Overall, Cobras preserves lexical diversity close to the original model across all four backbones. The gap is small for Mistral-7B and LLaMA3.1-8B, where Dist-2 and Dist-3 remain nearly unchanged compared to the original model."},{"type":"para","text":"We also observe that some baselines reduce diversity more noticeably. For example, CAA lowers diversity on Falcon-7B and Qwen2.5-7B, while SphericalSteer shows a larger drop on Qwen2.5-7B. In contrast, Cobras maintains high Dist-2 and Dist-3 scores, suggesting that its detoxification effect does not come from generating overly repetitive or generic responses."},{"type":"heading","level":2,"text":"D.3 Impact of geodesic Euler steps K K"},{"type":"para","text":"Table 5 shows the effect of increasing the number of geodesic Euler steps K K in our proposed Cobras steering method on TruthfulQA. Overall, increasing K K improves the True × \\times Info score across all models, although the trend differs slightly by model."},{"type":"para","text":"For Falcon-7B, larger K K increases informativeness, but also reduces the True score, leading to only a modest gain in True × \\times Info. For Mistral-7B, performance is relatively stable across different values of K K , with a small improvement at K = 10 K=10 . LLaMA3.1-8B benefits the most from increasing K K , where True × \\times Info improves consistently from 64.6 64.6 at K = 1 K=1 to 68.5 68.5 at K = 10 K=10 . These results suggest that using more geodesic Euler steps generally improves steering quality, with K = 10 K=10 giving the best overall performance in our experiments."},{"type":"heading","level":2,"text":"D.4 Performance vs Steering Strength for baselines"},{"type":"para","text":"To select the steering strength for each baseline, we sweep over model-specific ranges: T = 20 T=20 – 23 23 for Falcon-7B, T = 2 T=2 – 5 5 for Mistral-7B, T = 3 T=3 – 6 6 for LLaMA3.1-8B, and T = 13 T=13 – 16 16 for Qwen2.5-7B. Figure 4 shows the performance of RepE, ITI, CAA, and ODESteer on TruthfulQA and UltraFeedback. For baselines without a comparable steering-strength parameter, we use the best values reported in their papers."},{"type":"para","text":"The results show that the best steering strength depends on both the model and the method. In general, some methods improve TruthfulQA performance at moderate values of T T , but larger steering strengths can hurt UltraFeedback performance. This trend is especially visible for ODESteer on Mistral-7B and LLaMA3.1-8B, where increasing T T leads to a large drop in UltraFeedback win rate. CAA is more stable on UltraFeedback and often improves with larger T T , while ITI tends to change more gradually across the sweep."},{"type":"para","text":"To analyze the quality of the steering, we show random samples obtained from UltraFeedback and TruthfulQA on steering methods."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Activation steering, also termed representation engineering, has emerged as a compelling alternative to fine-tuning for shaping the behavior of large language models (LLMs) at inference time (Rimsky et al. , 2024 ; Li et al. , 2023 ; Wang et al. , 2025 ) . Its empirical success is often motivated by the linear representation hypothesis (Park et al. , 2023 ; Alain and Bengio, 2016 ) where many high-level properties encoded by an LLM, such as sentiment, truthfulness, or harmfulness, appear to correspond to approximately linear directions in activation space. Under this view, changing a model’s behavior need not require modifying its weights; it can instead be achieved by moving its internal activations along directions associated with the target property at inference time. A standard approach is activation addition, which modifies the original activations by adding either a fixed steering vector or one that depends on the current activation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This perspective has inspired a growing body of work on inference-time control through activation interventions. Early methods typically construct a single steering direction from contrastive examples and then apply that direction uniformly across queries (Rimsky et al. , 2024 ) , while more recent approaches refine the dynamics of the intervention by integrating a continuous steering field over multiple steps (Zhao et al. , 2026 ; Wang et al. , 2025 ; Rodriguez et al. , 2025 ) or working on a more faithful manifold by rotating activations on the hypersphere You et al. ( 2026 ); Pham and Nguyen ( 2024 ) . These advances show that steering can be both effective and computationally lightweight. Despite this progress, existing methods share two limitations. First, most of the methods and their implicit objective are heuristic rather than derived from a well-posed optimization problem without formal justification for the objective itself. Second, these steering signals are query-independent . A steering vector that helps in-distribution prompts can make out-of-distribution (OOD) inputs wrong or gibberish, and our experiments confirm that several strong baselines improve TruthfulQA Lin et al. ( 2021 ) at the cost of severe MMLU Hendrycks et al. ( 2020 ) and GSM8K Cobbe et al. ( 2021 ) degradation. We address both gaps with Cobras , which casts steering as a Schrödinger Bridge, yielding both a principled derivation of the steering objective and an inherently query-adaptive steering direction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Overall, our contributions are: (i) we show that the Schrödinger Bridge on the activation hypersphere provides, to our knowledge, the first principled derivation of the implicitly widely used log-density-ratio steering objective from an optimization problem; (ii) we propose Cobras (Conditional Optimal Bridge for Riemannian Activation Steering), a practical algorithm that combines entropic optimal-transport potentials, spherical kernel density estimation, and a gating mechanism to produce query-adaptive steering directions; and (iii) we demonstrate that Cobras consistently outperforms prior baselines on helpfulness, truthfulness, and detoxification while avoiding the out-of-distribution degradation observed in existing methods.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Organization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 2 discusses the related work. Section 3 elaborates on the background, notation, setup, and the assumption of the manifold where LLM activations live. Section 4 then formulates the steering problem and its objective as the Schrödinger Bridge instance, and then solves this problem and derives the objective for steering, and proposes Cobras . Section 5 then does extensive experiments for both in-distribution and OOD and shows that Cobras outperforms baselines in both settings. Finally, Section 6 concludes the paper, discussing limitations and future work. The proof of technical results and additional experiments are reported in the appendix.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Theory of activation steering.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several recent works have sought to develop a unified theory of activation steering by interpreting interventions as structured transformations in representation space. Linear Activation Transportation (Rodriguez et al. , 2025 ) formulates steering as a transport map between source and target activation distributions, LiSeCo (Cheng and Alonso, 2024 ) analyzes steering through linear separability and constrained optimization, and ODESteer (Zhao et al. , 2026 ) views steering as a continuous-time dynamical process rather than a one-shot activation edit. These perspectives provide useful formal tools for reasoning about activation interventions. However, their guarantees are typically derived for surrogate objectives and under specific geometric constraints rather than from a principled perspective. Also, the Schrödinger Bridge (SB) problem has been widely adopted for generative modeling via forward–backward SDE solvers (De Bortoli et al. , 2021 ; Chen et al. , 2021 ; Shi et al. , 2023 ) . Cobras is the first to apply this framework for LLM activation steering.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Activation steering approaches.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Early and widely used steering approaches Zou et al. ( 2023 ); Li et al. ( 2023 ); Rimsky et al. ( 2024 ) identify behavior-relevant directions in activation space and apply additive edits. More recent methods refine this basic recipe by accounting for activation geometry or distributional structure, for example through minimally invasive affine maps (Singh et al. , 2024 ) , norm-preserving rotations (Pham and Nguyen, 2024 ) , transport-based maps (Rodriguez et al. , 2025 ) , hyperspherical interventions (You et al. , 2026 ) , or continuous dynamical updates (Zhao et al. , 2026 ) . Despite these advances, most existing methods are query-independent: once a direction or transformation is learned, essentially the same intervention is applied across prompts. Prior work shows that such fixed interventions can have limited OOD generalization, induce off-target side effects, and degrade performance when steering is unnecessary (Tan et al. , 2024 ; Raedler et al. , 2025 ; Da Silva et al. , 2025 ) . Our work instead proposes a query-adaptive steering method that modulates the intervention based on the input, preserving OOD performance while improving in-distribution performance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Background\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section establishes the background for the steering problem by defining key notation and activation populations. We provide empirical evidence that these activations concentrate on a hypersphere, justifying the Riemannian geometry utilized throughout this work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Notation and setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fix a decoder-only transformer with residual width d d , a layer index ℓ \\\\ell , and a token position index t t . For an input sequence s s , let 𝒉 s ∈ ℝ d \\\\bm{h}_{s}\\\\in\\\\mathbb{R}^{d} denote the residual-stream activation at layer-position pair ( ℓ , t ) (\\\\ell,t) when the model is run without steering. We consider two populations of input sequences, one associated with the desired behavior and one associated with the undesired behavior, and write p + p_{+} and p − p_{-} for the induced distributions of their activations. Given i.i.d. samples from these activation distributions 𝒟 + = { 𝒉 i + } i = 1 N + \\\\mathcal{D}^{+}=\\\\{\\\\bm{h}^{+}_{i}\\\\}_{i=1}^{N_{+}} with 𝒉 i + ∼ i . i . d . p + \\\\bm{h}^{+}_{i}\\\\overset{\\\\mathrm{i.i.d.}}{\\\\sim}p_{+} and 𝒟 − = { 𝒉 j − } j = 1 N − \\\\mathcal{D}^{-}=\\\\{\\\\bm{h}^{-}_{j}\\\\}_{j=1}^{N_{-}} with 𝒉 i − ∼ i . i . d . p − \\\\bm{h}^{-}_{i}\\\\overset{\\\\mathrm{i.i.d.}}{\\\\sim}p_{-} , we denote the corresponding empirical measures by p ^ + \\\\hat{p}_{+} and p ^ − \\\\hat{p}_{-} . At inference time, for a new input sequence q q , let 𝒉 q ∈ ℝ d \\\\bm{h}_{q}\\\\in\\\\mathbb{R}^{d} be its unsteered activation at ( ℓ , t ) (\\\\ell,t) . The steering map should return a modified activation that shifts 𝒉 q \\\\bm{h}_{q} toward regions of activation space with high density under p + p_{+} and low density under p − p_{-} , while remaining sufficiently close to 𝒉 q \\\\bm{h}_{q} to preserve the semantics of q q .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Activation Manifold.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To leverage the inherent geometric structure of neural representations, we motivate our manifold-based approach by observing the distribution of activation magnitudes within transformer architectures. A line of recent work has shown that, once the layer index is fixed, residual-stream activations concentrate strongly in norm across inputs, both empirically (Pham and Nguyen, 2024 ; You et al. , 2026 ) and structurally (Yang et al. , 2023 ; Xie et al. , 2026 ) . Thus, for the present layer ℓ \\\\ell , most of the variability relevant for steering lies in the direction of the activation rather than in its Euclidean magnitude. This suggests that the natural state space is not all of ℝ d \\\\mathbb{R}^{d} but instead a sphere of approximately constant radius. We therefore model activations as living on the hypersphere 𝒮 = { 𝒙 ∈ ℝ d : ‖ 𝒙 ‖ = R } \\\\mathcal{S}=\\\\{\\\\bm{x}\\\\in\\\\mathbb{R}^{d}:\\\\|\\\\bm{x}\\\\|=R\\\\} . This norm concentration allows Cobras to cast the steering problem into hyperspherical Riemannian geometry, letting us leverage established tools and results from Riemannian optimal transport.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1 provides empirical support for this approximation by showing that activation norms cluster tightly around the layer mean radius. Accordingly, we project the query activation and the samples in 𝒟 ± \\\\mathcal{D}^{\\\\pm} onto 𝒮 \\\\mathcal{S} and formulate the steering problem on this manifold.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We equip 𝒮 \\\\mathcal{S} with the tangent projector Π 𝒙 = 𝑰 − R − 2 𝒙 𝒙 ⊤ \\\\Pi_{\\\\bm{x}}=\\\\bm{I}-R^{-2}\\\\bm{x}\\\\bm{x}^{\\\\top} , Riemannian gradient ∇ 𝒮 f = Π 𝒙 ∇ f \\\\nabla_{\\\\mathcal{S}}f=\\\\Pi_{\\\\bm{x}}\\\\nabla f , and geodesic distance d 𝒮 ( 𝒙 , 𝒚 ) = R arccos ( R − 2 𝒙 ⊤ 𝒚 ) d_{\\\\mathcal{S}}(\\\\bm{x},\\\\bm{y})=R\\\\arccos\\\\left(R^{-2}\\\\bm{x}^{\\\\top}\\\\bm{y}\\\\right) . The spherical exponential map exp 𝒙 \\\\exp_{\\\\bm{x}} walks from 𝒙 \\\\bm{x} along a tangent vector 𝒗 \\\\bm{v} for a geodesic arc of length ‖ 𝒗 ‖ \\\\|\\\\bm{v}\\\\| , while the logarithm log 𝒙 ( 𝒚 ) \\\\log_{\\\\bm{x}}(\\\\bm{y}) inverts this operation, returning the tangent vector at 𝒙 \\\\bm{x} that points toward 𝒚 \\\\bm{y} along the shortest great-circle arc, with magnitude equal to the geodesic distance. Explicitly,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With the contrastive samples 𝒟 − ∼ p − \\\\mathcal{D}^{-}\\\\sim p_{-} and 𝒟 + ∼ p + \\\\mathcal{D}^{+}\\\\sim p_{+} projected onto 𝒮 \\\\mathcal{S} and the Riemannian toolkit in hand, we develop Cobras that operates entirely within this spherical geometry.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 LLM Steering as Schrödinger Bridge\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A growing body of steering methods Rimsky et al. ( 2024 ); Zhao et al. ( 2026 ); Li et al. ( 2023 ) implicitly construct steering directions that approximate the gradient of the log-density ratio between desired and undesired activations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The term ∇ log p + \\\\nabla\\\\log p_{+} points toward regions favored by the desired behavior, while − ∇ log p − -\\\\nabla\\\\log p_{-} points away from regions associated with the undesired behavior. Together, they define the steepest-ascent direction of the density ratio. As observed by Zhao et al. ( 2026 ) , many existing steering methods can be reinterpreted as approximations to Eq. 2 . For example, Contrastive Activation Addition (CAA) (Rimsky et al. , 2024 ) gets its steering vector with p + = 𝒩 ( 𝝁 + , 𝑰 d ) p_{+}=\\\\mathcal{N}(\\\\bm{\\\\mu}_{+},\\\\bm{I}_{d}) and p − = 𝒩 ( 𝝁 − , 𝑰 d ) p_{-}=\\\\mathcal{N}(\\\\bm{\\\\mu}_{-},\\\\bm{I}_{d}) , leading to v ( 𝒙 ) = ∇ log p + p − = 𝝁 + − 𝝁 − v(\\\\bm{x})=\\\\nabla\\\\log\\\\frac{p_{+}}{p_{-}}=\\\\bm{\\\\mu}_{+}-\\\\bm{\\\\mu}_{-} . In Appendix A we extend this connection and show that several other prominent steering paradigms such as Spherical Steering (SS) (You et al. , 2026 ) , Inference-Time Intervention (ITI) (Li et al. , 2023 ) , Householder Reflection Probing (HPR) (Pham and Nguyen, 2024 ) , and ODESteer (Zhao et al. , 2026 ) can also be unified under a singular framework governed by log-density ratio optimization over a latent manifold.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Nevertheless, Eq. 2 is typically adopted as a plausible objective rather than derived from an optimization problem. To our knowledge, no existing work formulates an optimization problem on the activation manifold whose solution provably yields the log-density-ratio gradient of Eq. 2 . This gap raises a natural question: can such a optimization-based formulation be rigorously established?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We show that the Schrödinger Bridge (SB) provides a generalized, principled framework for steering. Specifically, we cast steering as an SB problem between the two marginals p − p_{-} and p + p_{+} and seek the optimal stochastic process that steers the distribution of hidden state from p − p_{-} to p + p_{+} with minimal “effort,” measured as the Kullback-Leibler (KL) divergence from a reference stochastic process. The SB solution induces a stochastic differential equation (SDE). To find the drift of the corresponding SDE, we reduce SB to an entropic optimal transport and solve it using the iterative Sinkhorn algorithm by alternating between updating the dual potentials (Sec. 4.1 ). We then apply the probability flow ODE to the induced SDE drift to obtain a deterministic process with the same time marginals. The resulting ODE has drift proportional to ∇ 𝒮 log ( ψ ^ / ϕ ^ ) \\\\nabla_{\\\\mathcal{S}}\\\\log(\\\\hat{\\\\psi}/\\\\hat{\\\\phi}) , the Riemannian gradient of the log-ratio of two Schrödinger potentials (Sec. 4.2 ). Finally, we evaluate this drift at arbitrary query points using nonparametric kernel density estimation (Sec. 4.3 ), yielding the Cobras method (Sec. 4.4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This derivation has two key consequences. First, the heuristic Eq. 2 emerges as a well-defined limiting case of the SB log-ratio log ( ψ ^ / ϕ ^ ) \\\\log(\\\\hat{\\\\psi}/\\\\hat{\\\\phi}) : when the Sinkhorn potentials are approximately uniform, ψ ^ \\\\hat{\\\\psi} and ϕ ^ \\\\hat{\\\\phi} reduce to kernel density estimates of p + p_{+} and p − p_{-} , and the drift becomes ∇ 𝒮 log ( p + / p − ) \\\\nabla_{\\\\mathcal{S}}\\\\log(p_{+}/p_{-}) (Prop. 1 ). To the best of our knowledge, this is the first principled derivation of the log-density-ratio steering objective from a rigorous optimization problem. Second, because the Schrödinger potentials are evaluated at the current activation, the resulting steering is inherently context dependent. Different queries receive different steering directions, with the weights adapting to the query’s location and naturally attenuating intervention for inputs far from the training, thus supporting OOD queries.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Formulating the objective of steering\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We cast the steering problem as finding the path measure P ⋆ P^{\\\\star} that transports the empirical distribution of undesired activations, p ^ − \\\\hat{p}_{-} , to the desired distribution, p ^ + \\\\hat{p}_{+} , while minimizing the KL divergence with respect to a reference path measure Q Q :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where P 0 P_{0} and P T P_{T} denote the time- 0 and time- T T marginals of P P . The KL penalty ensures that the resulting transport plan is the minimal entropically regularized perturbation of the reference Q Q . We take Q Q as Brownian motion on 𝒮 \\\\mathcal{S} with diffusivity σ 2 > 0 \\\\sigma^{2}>0 . It is defined by the SDE\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where W t W_{t} is Brownian motion (random Gaussian noise) at time t t . We need the reference’s one-step transition kernel for solving Eq. 3 . In the short-time regime, the reference’s transition kernel is approximated as De Bortoli et al. ( 2022 ); Chen et al. ( 2023b )\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Intuitively, this kernel acts as a spatial transition probability. It penalizes large geodesic distances d 𝒮 ( 𝒙 , 𝒚 ) d_{\\\\mathcal{S}}(\\\\bm{x},\\\\bm{y}) , ensuring that a particle at 𝒙 \\\\bm{x} is most likely to diffuse to a nearby point 𝒚 \\\\bm{y} with spread σ 2 \\\\sigma^{2} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Having established the reference frame, we shift our focus to the practical resolution of solving Eq. 3 . In practice, the true distributions p − / p + p_{-}/p_{+} are typically inaccessible; consequently, we must operate within a data-driven framework, substituting these theoretical distributions with their empirical counterparts p ^ − / p ^ + \\\\hat{p}_{-}/\\\\hat{p}_{+} . With these discrete marginals, the static formulation of Eq. 3 reduces to an entropic optimal transport task with squared geodesic cost c j i = d 𝒮 ( 𝒉 j − , 𝒉 i + ) 2 / ( 2 σ 2 ) c_{ji}=d_{\\\\mathcal{S}}(\\\\bm{h}^{-}_{j},\\\\,\\\\bm{h}^{+}_{i})^{2}/{(2\\\\sigma^{2})} Nutz ( 2021 ) , that governs the geometry of the underlying manifold. Then the objective becomes to find a coupling that minimizes the transport cost while accounting for diffusive entropy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and by Sinkhorn’s theorem (Cuturi, 2013 ) , the optimal coupling π ⋆ \\\\pi^{\\\\star} factorizes as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the dual potentials { ϕ j } j = 1 N − \\\\{\\\\phi_{j}\\\\}_{j=1}^{N_{-}} and { ψ i } i = 1 N + \\\\{\\\\psi_{i}\\\\}_{i=1}^{N_{+}} satisfy\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These dual potentials are efficiently computed using Sinkhorn iterations, alternating updates between ϕ \\\\phi and ψ \\\\psi until convergence and later will be used to obtain the drift of the SDE induced by Eq. 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Optimal Bridge SDE and its Probability Flow ODE\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With the dual potentials computed, we now derive the forward SDE of the optimal solution to Eq. 3 . The SB optimal bridge forward SDE is Chen et al. ( 2021 )\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ψ ^ ( ⋅ , t ) \\\\hat{\\\\psi}(\\\\cdot,t) is the backward Schrödinger potential obtained by propagating the terminal potential ψ \\\\psi backward via a kernel-weighted sum over the positive samples, with bandwidth ( T − t ) σ 2 (T-t)\\\\sigma^{2} that shrinks to zero as t → T t\\\\to T , concentrating mass on p ^ + \\\\hat{p}_{+} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Similarly, the forward Schrödinger potential propagates ϕ \\\\phi from the starting margin as a kernel-weighted sum over the negative samples, with bandwidth t σ 2 t\\\\sigma^{2} that shrinks to zero as t → 0 t\\\\to 0 , concentrating mass on p ^ − \\\\hat{p}_{-} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A fundamental property of the SB when the reference process is Brownian motion is that its time-marginal density factorizes as p t ( ⋅ ) = ϕ ^ ( ⋅ , t ) ψ ^ ( ⋅ , t ) p_{t}(\\\\cdot)=\\\\hat{\\\\phi}(\\\\cdot,t)\\\\,\\\\hat{\\\\psi}(\\\\cdot,t) (Chen et al. , 2023a ) . Now with the SDE at Eq. 9 in hand, we seek a deterministic process sharing the same marginal evolution with it. The probability flow ODE provides such a process. For any diffusion d 𝒙 t = b d t + σ d W t d\\\\bm{x}_{t}=b\\\\,dt+\\\\sigma\\\\,dW_{t} with marginal density p t p_{t} , the corresponding probability flow ODE is d 𝒙 t = ( b − σ 2 2 ∇ 𝒮 log p t ) d t d\\\\bm{x}_{t}=\\\\left(b-\\\\tfrac{\\\\sigma^{2}}{2}\\\\nabla_{\\\\mathcal{S}}\\\\log p_{t}\\\\right)\\\\,dt (Song et al. , 2020 ) . Substituting b = σ 2 ∇ 𝒮 log ψ ^ b=\\\\sigma^{2}\\\\nabla_{\\\\mathcal{S}}\\\\log\\\\hat{\\\\psi} and log p t = log ϕ ^ + log ψ ^ \\\\log p_{t}=\\\\log\\\\hat{\\\\phi}+\\\\log\\\\hat{\\\\psi} gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"or equivalently,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The two terms in the drift have clear roles: ∇ 𝒮 log ψ ^ \\\\nabla_{\\\\mathcal{S}}\\\\log\\\\hat{\\\\psi} attracts toward p ^ + \\\\hat{p}_{+} , while − ∇ 𝒮 log ϕ ^ -\\\\nabla_{\\\\mathcal{S}}\\\\log\\\\hat{\\\\phi} repels from p ^ − \\\\hat{p}_{-} , and both arise from Eq. 3 . The following proposition shows that the heuristic objective of Eq. 2 is a limiting case of this drift.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let c max = max j , i d 𝒮 ( 𝐡 j − , 𝐡 i + ) 2 / ( 2 σ 2 ) c_{\\\\max}=\\\\max_{j,i}\\\\,d_{\\\\mathcal{S}}(\\\\bm{h}^{-}_{j},\\\\bm{h}^{+}_{i})^{2}/(2\\\\sigma^{2}) . In the regime c max → 0 c_{\\\\max}\\\\to 0 , the Sinkhorn potentials converge to the uniform values ψ i → 1 N + \\\\psi_{i}\\\\to\\\\frac{1}{N_{+}} and ϕ j → 1 N − \\\\phi_{j}\\\\to\\\\frac{1}{N_{-}} , and the flow drift reduces to\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the numerator and denominator are spherical kernel density estimates of p + p_{+} and p − p_{-} that converge to the true densities as N ± → ∞ N_{\\\\pm}\\\\to\\\\infty , recovering ∇ 𝒮 log ( p + / p − ) \\\\nabla_{\\\\mathcal{S}}\\\\log(p_{+}/p_{-}) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The proof is provided in Appendix B.1 . Prop. 1 shows that in the limit of no transportation cost, e.g., due to small σ \\\\sigma , the drift of the probability flow ODE converges to ∇ 𝒮 log ( p + / p − ) \\\\nabla_{\\\\mathcal{S}}\\\\log(p_{+}/p_{-}) , and optimizing ∇ 𝒮 log ( p + / p − ) \\\\nabla_{\\\\mathcal{S}}\\\\log(p_{+}/p_{-}) recovers the steering dictated by SB. To our knowledge, this is the first principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. Moreover, because ψ ^ \\\\hat{\\\\psi} and ϕ ^ \\\\hat{\\\\phi} are evaluated at the current activation, the drift is inherently query-adaptive, thus avoiding the performance degradation on OOD inputs that is commonly seen in fixed-direction steering methods.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Non-parametric Estimation of the Probability Flow ODE Drift\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To steer a hidden state 𝒉 q \\\\bm{h}_{q} , we need to solve the ODE in Eq. 12 with initial 𝒙 0 = 𝒉 q \\\\bm{x}_{0}=\\\\bm{h}_{q} which requires computing the Riemannian gradients ∇ 𝒮 log ψ ^ \\\\nabla_{\\\\mathcal{S}}\\\\log\\\\hat{\\\\psi} and ∇ 𝒮 log ϕ ^ \\\\nabla_{\\\\mathcal{S}}\\\\log\\\\hat{\\\\phi} as continuous functions on 𝒮 \\\\mathcal{S} . Here we derive these objects using Kernel Density Estimation (KDE). This requires us to extend the potential values ( ϕ j , ψ i ) (\\\\phi_{j},\\\\psi_{i}) , but they are only defined at the training points in 𝒟 − / 𝒟 + \\\\mathcal{D}^{-}/\\\\mathcal{D}^{+} . To evaluate them at a query 𝒉 q \\\\bm{h}_{q} , we marginalize through the opposite training points via the kernel and cost matrix:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a query 𝒉 q \\\\bm{h}_{q} , Eq. 15 weights each negative sample 𝒉 j − \\\\bm{h}^{-}_{j} by its proximity (via ϕ j \\\\phi_{j} and the kernel), then propagate through the cost matrix to assign an effective potential to each positive sample 𝒉 i + \\\\bm{h}^{+}_{i} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Riemannian gradient via kernel density estimation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The gradient of log ψ ^ \\\\log\\\\hat{\\\\psi} at 𝒉 q \\\\bm{h}_{q} is a weighted Riemannian mean direction in the tangent space T 𝒉 q 𝒮 T_{\\\\bm{h}_{q}}\\\\mathcal{S} , yielding (the derivation is provided in Appendix B.2 ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with weights\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and symmetrically for ∇ 𝒮 log ϕ ^ ( 𝒉 q ) = ∑ j w ¯ j − ( 𝒉 q ) log 𝒉 q ( 𝒉 j − ) \\\\nabla_{\\\\mathcal{S}}\\\\log\\\\hat{\\\\phi}(\\\\bm{h}_{q})=\\\\sum_{j}\\\\bar{w}^{-}_{j}(\\\\bm{h}_{q})\\\\,\\\\log_{\\\\bm{h}_{q}}(\\\\bm{h}^{-}_{j}) with weights w ¯ j − ( 𝒉 q ) \\\\bar{w}^{-}_{j}(\\\\bm{h}_{q}) defined over 𝒟 − \\\\mathcal{D}^{-} using ϕ ~ j \\\\tilde{\\\\phi}_{j} . Substituting these expressions into the probability flow ODE in Eq. 12 and absorbing the constant σ 2 2 \\\\frac{\\\\sigma^{2}}{2} into the step size yields the Cobras vector field:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 The Overall Algorithm\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given the vector field in Eq. 19 , we steer a query activation 𝒉 q \\\\bm{h}_{q} by integrating the probability flow ODE at Eq. 12 on 𝒮 \\\\mathcal{S} using K K geodesic Euler steps. At each step, we evaluate the field at the current iterate and update via the exponential map in Eq. 1 , ensuring that every iterate remains on 𝒮 \\\\mathcal{S} . After integration, we restore the original norm ‖ 𝒉 q ‖ \\\\|\\\\bm{h}_{q}\\\\| . Algorithm 1 summarizes the full procedure. The algorithm operates in two distinct phases: an offline optimization stage (Lines 1-3) to solve the static SB problem and an online inference stage (Lines 4-10) that evolves a query activation along a probability flow. In the offline phase, it projects sample data onto a hypersphere and solves an entropic optimal transport problem via the Sinkhorn algorithm to derive Schrödinger potentials. During the online phase, the algorithm iteratively steers a query activation 𝒉 q \\\\bm{h}_{q} by integrating a probability flow ODE. It calculates a velocity vector 𝒗 k \\\\bm{v}_{k} in the tangent space based on weighted geodesic distances between positive and negative samples, updating the intermediate state 𝒒 k \\\\bm{q}_{k} via the exponential map. The process concludes by rescaling the final activation 𝒒 K \\\\bm{q}_{K} to the activation’s original norm.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For OOD, an abstain gate attenuates steering by comparing the k abs k_{\\\\mathrm{abs}} -nearest-neighbor geodesic distance from 𝒒 0 \\\\bm{q}_{0} to 𝒟 − \\\\mathcal{D}^{-} against a reference radius ρ ref \\\\rho_{\\\\mathrm{ref}} (the p p -th percentile of intra-negative k abs k_{\\\\mathrm{abs}} -NN distances), ensuring that queries far from the training support are left largely unmodified. For per-query step size, a vMF strength gate s vMF ( 𝒒 0 ) s_{\\\\mathrm{vMF}}(\\\\bm{q}_{0}) measures the alignment of the query with the mean contrastive direction 𝒉 ¯ + − 𝒉 ¯ − \\\\bar{\\\\bm{h}}^{+}-\\\\bar{\\\\bm{h}}^{-} , so that queries already aligned receive less steering.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We conduct experiments to answer the following research questions. (RQ1): How does Cobras compare to other steering methods on In-distribution tasks? (RQ2): How does Cobras perform on OOD tasks? (RQ3): How does each parameter of Cobras impact its performance?\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Datasets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We conduct our evaluations on three prevalent steering datasets: TruthfulQA Lin et al. ( 2021 ) (TQA) for Truthfulness, Ultrafeedback Cui et al. ( 2023 ) (UF) for Helpfulness and RealToxicityPrompts Gehman et al. ( 2020 ) for Detoxification. Further, to compare the performance of steering methods on Out-of-Distribution (OOD) tasks, we use MMLU Hendrycks et al. ( 2020 ) and GSM8K Cobbe et al. ( 2021 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following ODESteer Zhao et al. ( 2026 ) , we compare Cobras against the following baselines: Representation Engineering (Zou et al. , 2023 ) (RepE), Inference-Time Intervention (Li et al. , 2023 ) (ITI), Contrastive Activation Addition (Rimsky et al. , 2024 ) (CAA), Minimally Modified Counterfactuals (Singh et al. , 2024 ) (MiMiC), Linear Activation Transport (Rodriguez et al. , 2025 ) (Linear-AcT), SphericalSteering You et al. ( 2026 ) and ODESteer Zhao et al. ( 2026 ) . We test the steering methods on four open source models: Falcon-7B (Almazrouei et al. , 2023 ) , Mistral-7B (Jiang et al. , 2023 ) , LLaMA3.1-8B (Meta AI, 2024 ) and Qwen2.5-7B Team ( 2024 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Metrics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following ODESteer Zhao et al. ( 2026 ) , we use Truthfulness x Informativeness % as the primary metric for TQA while reporting Truthfulness% and Informativeness%. For UF/Helpfulness task, we use win rate over no steer (primary metric), Mean Reward and 90th Percentile Reward as secondary metrics. For Detoxification task, we use Toxicity Score from Perspective API Jigsaw and Google ( 2026 ) as primary metric while reporting Perplexity and Dist-2 as secondary metrics. For GSM8K and MMLU, we use accuracy as the metric. See Appendix C.3 for details.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"RQ1: How does Cobras compare to other steering methods on in-distribution tasks?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We first evaluate Cobras in the standard in-distribution setting, where each steering method is trained and tested on the same task axis. We consider three representative alignment objectives: helpfulness on UltraFeedback, truthfulness on TruthfulQA, and detoxification on RealToxicityPrompts. Table 1 reports results across four base models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cobras consistently improves the primary metric for all three tasks across models. On helpfulness, Cobras achieves the highest UltraFeedback win rate for every model, improving over the original model by + 2.8 +2.8 , + 7.1 +7.1 , + 9.7 +9.7 , and + 1.9 +1.9 absolute points on Falcon-7B, Mistral-7B, LLaMA3.1-8B, and Qwen2.5-7B, respectively. On truthfulness, Cobras obtains the best TruthfulQA True × \\\\times Info score across all four models, with especially large gains over the original model on Mistral-7B ( + 29.5 ) (+29.5) and LLaMA3.1-8B ( + 25.1 ) (+25.1) . On detoxification, Cobras also achieves the lowest toxicity score for every model, reducing toxicity by 46 % 46\\\\% – 53 % 53\\\\% relative to the original models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Compared to prior steering baselines, Cobras delivers the most consistent gains across objectives. While other methods can perform well on individual tasks, they often trade off informativeness, perplexity, or diversity. In contrast, Cobras improves target alignment metrics while largely preserving generation quality, making it a robust in-distribution steering method across helpfulness, truthfulness, and detoxification.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"RQ2: How does Cobras perform on OOD tasks?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Activation steering can be brittle under distribution shift, with prior work showing that steering vectors may be unreliable in-distribution and can misgeneralize out-of-distribution (OOD) (Tan et al. , 2024 ; Braun et al. , 2025 ; Korznikov et al. , 2025 ) . We therefore evaluate whether Cobras improves TruthfulQA performance without degrading unrelated capabilities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train all steering methods on TruthfulQA and evaluate OOD performance on MMLU Hendrycks et al. ( 2020 ) and GSM8K Cobbe et al. ( 2021 ) . Figure 2 plots OOD accuracy against TruthfulQA True × \\\\times Info; the ideal method appears in the upper-right, improving truthfulness while preserving OOD accuracy. Baselines show inconsistent trade-offs: several methods improve TruthfulQA but substantially reduce OOD performance, especially on GSM8K for LLaMA3.1-8B and Mistral-7B, and on MMLU for Mistral-7B. This suggests that some steering methods sacrifice general reasoning ability for in-distribution gains. By contrast, Cobras consistently lie near the upper-right across all models and OOD benchmarks. It improves TruthfulQA over the original model while maintaining OOD accuracy close to, or above, the unsteered model, avoiding the severe GSM8K drops seen in several baselines. Overall, Cobras achieves the most favorable trade-off between in-distribution steering effectiveness and OOD capability preservation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"RQ3: Ablation of Cobras Hyperparameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We ablate the abstention neighborhood size K K and abstention percentile; steering-strength ablations are deferred to Appendix D.1 . Cobras uses an abstention rule to avoid steering inputs far from the training distribution. For each training example, we compute the distance to its K K -th nearest neighbor and set the abstention threshold to a chosen percentile of these distances. At test time, Cobras abstains from steering when an input’s neighborhood distance exceeds this threshold. In-distribution inputs are expected to have nearby neighbors and therefore be steered, whereas OOD inputs should have larger distances and trigger abstention. Table 2 evaluate this mechanism on LLaMA3.1-8B-Base and Mistral-7B-Base using TruthfulQA as the in-distribution task and GSM8K as the OOD task. Increasing the abstention percentile makes Cobras steer more examples. This improves TruthfulQA True × \\\\times Info, but can reduce OOD accuracy, most notably for LLaMA3.1-8B-Base at the 99.5 99.5 th percentile. Increasing K K shows a similar trade-off: larger neighborhoods improve TruthfulQA, but can lower GSM8K accuracy, especially for LLaMA3.1-8B-Base. These results show that abstention helps balance effective in-distribution steering with preserving OOD capabilities.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By bridging the gap between heuristic methods and the rigorous optimization framework of the Schrödinger Bridge, this paper provides the first principled derivation of the log-density-ratio steering objective for activation steering. Experiments across four models and three alignment axes show that Cobras consistently outperforms baselines in-distribution while preserving OOD performance (commonly seen in fixed-direction steering methods), thanks to its query-adaptive nature. Limitations include the 𝒪 ( N + ⋅ N − ) \\\\mathcal{O}(N_{+}\\\\cdot N_{-}) cost of pairwise geodesic distances and Sinkhorn, which may be prohibitive for very large contrastive sets, and the reliance on a fixed spherical geometry that may not hold for all architectures or layers. Promising future directions include hierarchical transport solvers to reduce computational overhead and extending the framework to learned or more general Riemannian manifolds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we show that diverse activation steering methodologies can be unified under a single theoretical framework governed by log-density ratio optimization. Specifically, we analyze how these seemingly disparate techniques implicitly optimize the ratio log ( p + / p − ) \\\\log(p_{+}/p_{-}) by reinterpreting them as specific geometric configurations of a continuous global flow. Formally, let p + p_{+} and p − p_{-} define a pair of continuous densities corresponding to the contrastive behavior distributions 𝒟 + \\\\mathcal{D}^{+} and 𝒟 − \\\\mathcal{D}^{-} , respectively. Given an initial query activation 𝒉 q \\\\bm{h}_{q} on an underlying representation manifold ℳ \\\\mathcal{M} , the steering trajectory is obtained by integrating the continuous-time ordinary differential equation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"for K K steps over a specified time horizon [ 0 , T ] [0,T] . Here, 𝒙 t \\\\bm{x}_{t} denotes the steered activation along the flow, initialized as 𝒙 0 = 𝒉 q \\\\bm{x}_{0}=\\\\bm{h}_{q} . By systematically varying the manifold geometry ℳ \\\\mathcal{M} , the step horizon K K , and the parameterization of the density functions, we can recover prominent steering paradigms from this single formulation 1 1 1 Zhao et al. [ 2026 ] presents a related unification of several of these methods, but their analysis is confined to Euclidean space; by treating the manifold ℳ \\\\mathcal{M} as an explicit design choice, however, our framework additionally captures geometry-aware methods such as SS and HPR. .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For example, the following methods can be recalled from this perspective as summarized in Table 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contrastive Activation Addition (CAA) [Rimsky et al. , 2024 ]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The CAA algorithm can be framed as a specific, single-step optimization instantiation of a continuous global flow over the hidden state manifold. Specifically, CAA simplifies this trajectory by optimizing over a flat Euclidean space ℳ = ℝ d \\\\mathcal{M}=\\\\mathbb{R}^{d} via a single discrete step ( K = 1 K=1 ). By modeling the factual (positive) and counterfactual (negative) activation distributions as isotropic Gaussians sharing an identity covariance p + = 𝒩 ( 𝝁 + , 𝑰 d ) p_{+}=\\\\mathcal{N}(\\\\bm{\\\\mu}_{+},\\\\bm{I}_{d}) and p − = 𝒩 ( 𝝁 − , 𝑰 d ) p_{-}=\\\\mathcal{N}(\\\\bm{\\\\mu}_{-},\\\\bm{I}_{d}) , the complex continuous gradient field collapses into a constant shift vector defined by the difference of their respective means ∇ ℳ log p + p − = 𝝁 + − 𝝁 − \\\\nabla_{\\\\mathcal{M}}\\\\log\\\\frac{p_{+}}{p_{-}}=\\\\bm{\\\\mu}_{+}-\\\\bm{\\\\mu}_{-} . Consequently, the empirical practice of injecting a static difference vector during the forward pass is mathematically grounded as an exact, first-order approximation toward maximizing the target likelihood ratio.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Spherical Steering (SS) [You et al. , 2026 ]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To relax the assumption of a flat Euclidean space and account for the directional nature of high-dimensional neural representations, Spherical Steering constrains the optimization trajectory to a unit hypersphere manifold, setting ℳ = 𝒮 d − 1 \\\\mathcal{M}=\\\\mathcal{S}^{d-1} . Under a single-step discrete horizon ( K = 1 K=1 ), the method models the target positive activations using a von Mises-Fisher (vMF) distribution p + = vMF ( ⋅ ; 𝝁 , κ ) p_{+}=\\\\mathrm{vMF}(\\\\cdot;\\\\bm{\\\\mu},\\\\kappa) which acts as the spherical analogue to an isotropic Gaussian with a mean direction vector 𝝁 \\\\bm{\\\\mu} and a concentration parameter κ \\\\kappa . Conversely, the negative behavior is modeled as a uniform distribution over the sphere p − = Unif ( 𝒮 ) p_{-}=\\\\mathrm{Unif}(\\\\mathcal{S}) . The unit prototype towards the positive activations 𝝁 \\\\bm{\\\\mu} is empirically defined by normalizing the difference between the mean positive and negative activations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When evaluating the global flow framework under these spherical constraints, the Riemannian gradient of the likelihood ratio yields a non-linear direction proportional to the logarithmic map on the sphere ∇ 𝒮 log p + p − ∝ log 𝒙 ( 𝝁 ) \\\\nabla_{\\\\mathcal{S}}\\\\log\\\\tfrac{p_{+}}{p_{-}}\\\\propto\\\\log_{\\\\bm{x}}(\\\\bm{\\\\mu}) which corresponds to the Slerp direction toward 𝝁 \\\\bm{\\\\mu} in the direction of this Riemannian gradient. By explicitly respecting the spherical geometry of hidden states, SS ensures that the magnitude of the activation vector is preserved, avoiding out-of-distribution distortion while performing directional steering.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Inference-Time Intervention (ITI) [Li et al. , 2023 ]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Rather than assuming explicit parametric distributions for the hidden states, ITI leverages a discriminative classifier to implicitly estimate the underlying density ratio. Operating within a single-step horizon ( K = 1 K=1 ) over a flat Euclidean vector space ℳ = ℝ d \\\\mathcal{M}=\\\\mathbb{R}^{d} , ITI trains a linear logistic probe, p 𝜽 ( 𝒙 ) = σ ( 𝜽 ⊤ 𝒙 ) p_{\\\\bm{\\\\theta}}(\\\\bm{x})=\\\\sigma(\\\\bm{\\\\theta}^{\\\\top}\\\\bm{x}) , to distinguish between positive and negative activation vectors. By invoking the classic equivalence between logistic regression and density ratio estimation, the log-density ratio can be rewritten linearly as log p + ( 𝒙 ) p − ( 𝒙 ) = 𝜽 ⊤ 𝒙 + log N − N + \\\\log\\\\frac{p_{+}(\\\\bm{x})}{p_{-}(\\\\bm{x})}=\\\\bm{\\\\theta}^{\\\\top}\\\\bm{x}+\\\\log\\\\frac{N_{-}}{N_{+}} , where N + N_{+} and N − N_{-} represent the respective sample counts. Taking the gradient with respect to the manifold yields a constant vector field defined entirely by the probe’s learned weights:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This formulation reveals a powerful geometric justification for ITI: the orthogonal vector to the classifier’s decision boundary acts as the exact gradient direction needed to maximize the factual likelihood ratio. Consequently, shifting activations along the direction of 𝜽 \\\\bm{\\\\theta} during inference provides a theoretically grounded, first-order trajectory for target behavior optimization.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Householder Reflection Probe (HPR) [Pham and Nguyen, 2024 ]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To reconcile discriminative, probe-based steering with non-Euclidean representation spaces, the HPR method maps the log-density ratio directly onto a hyperspherical manifold ( ℳ = 𝒮 d − 1 \\\\mathcal{M}=\\\\mathcal{S}^{d-1} ) under a single-step horizon ( K = 1 K=1 ). Similar to ITI, the method first projects the log-density ratio linearly using a trained linear probe, log p + ( 𝒙 ) p − ( 𝒙 ) = 𝜽 ⊤ 𝒙 + log N − N + \\\\log\\\\frac{p_{+}(\\\\bm{x})}{p_{-}(\\\\bm{x})}=\\\\bm{\\\\theta}^{\\\\top}\\\\bm{x}+\\\\log\\\\frac{N_{-}}{N_{+}} . To respect the spherical constraints, HPR constructs an orthogonal Householder reflection matrix, 𝑯 = 𝑰 − 2 𝒖 𝒖 ⊤ \\\\bm{H}=\\\\bm{I}-2\\\\bm{u}\\\\bm{u}^{\\\\top} , where 𝒖 = 𝜽 / ‖ 𝜽 ‖ \\\\bm{u}=\\\\bm{\\\\theta}/\\\\|\\\\bm{\\\\theta}\\\\| represents the unit normal vector to the probe’s decision boundary. By reflecting the normalized activation vector 𝒙 ^ = 𝒙 / ‖ 𝒙 ‖ \\\\hat{\\\\bm{x}}=\\\\bm{x}/\\\\|\\\\bm{x}\\\\| across this boundary, the directional gradient on the sphere is formalized as ∇ 𝒮 log p + ( 𝒙 ) p − ( 𝒙 ) ∝ Π 𝒙 ^ ( 𝑯 𝒙 ^ − 𝒙 ^ ) \\\\nabla_{\\\\mathcal{S}}\\\\log\\\\frac{p_{+}(\\\\bm{x})}{p_{-}(\\\\bm{x})}\\\\propto\\\\Pi_{\\\\hat{\\\\bm{x}}}(\\\\bm{H}\\\\hat{\\\\bm{x}}-\\\\hat{\\\\bm{x}}) , where Π 𝒙 ^ \\\\Pi_{\\\\hat{\\\\bm{x}}} denotes the projection onto the tangent space at 𝒙 ^ \\\\hat{\\\\bm{x}} . Geometrically, this confirms that the optimal trajectory corresponds to a Spherical Linear Interpolation (Slerp) toward the reflected state 𝑯 𝒙 ^ \\\\bm{H}\\\\hat{\\\\bm{x}} . By leveraging a reflection operator, HPR effectively utilizes the probe’s decision boundaries to execute directional steering while intrinsically preserving the latent norm of the activation path.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"ODESteer [Zhao et al. , 2026 ]\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Departing from single-step first-order approximations ( K = 1 K=1 ), ODESteer fully embraces the continuous-time formulation of the global flow framework by integrating across multiple steps ( K > 1 K>1 ) over a flat Euclidean vector space ℳ = ℝ d \\\\mathcal{M}=\\\\mathbb{R}^{d} . Rather than adopting simple parametric distributions or linear probes, this approach models complex, non-linear activation dynamics by parameterizing the log-density ratio with a polynomial count-sketch feature map ϕ \\\\phi and setting log p + ( 𝒙 ) p − ( 𝒙 ) = 𝒘 ⊤ ϕ ( 𝒙 ) + b \\\\log\\\\frac{p_{+}(\\\\bm{x})}{p_{-}(\\\\bm{x})}=\\\\bm{w}^{\\\\top}\\\\phi(\\\\bm{x})+b , where the parameter pair ( 𝒘 , b ) (\\\\bm{w},b) is fit empirically over the contrastive dataset. Under this non-linear projection, the steering direction is dictated by a dynamic, normalized vector field:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝑱 ϕ ( 𝒙 ) \\\\bm{J}_{\\\\phi}(\\\\bm{x}) denotes the Jacobian matrix of the feature map ϕ \\\\phi evaluated at the current state 𝒙 \\\\bm{x} . By evaluating and integrating this state-dependent vector field using discrete Euler steps over the time horizon [ 0 , T ] [0,T] , ODESteer traces a curved, non-linear path through the representation space. This multi-step integration allows the model to continuously adapt to the changing manifold geometry during the steering process, providing precise control while tracking complex latent boundaries.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we provide the theoretical analysis for Cobras .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.1 Proof of Proposition 1\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When c max → 0 c_{\\\\max}\\\\to 0 , every cost entry c j i → 0 c_{ji}\\\\to 0 . The Sinkhorn update for ϕ j \\\\phi_{j} (Eq. 8 ) becomes\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which is independent of j j . Hence, every ϕ j \\\\phi_{j} takes the same value. By the symmetric argument applied to the ψ i \\\\psi_{i} update, every ψ i \\\\psi_{i} is also equal. The marginal constraints ∑ i π j i ⋆ = 1 N − \\\\sum_{i}\\\\pi^{\\\\star}_{ji}=\\\\frac{1}{N_{-}} and ∑ j π j i ⋆ = 1 N + \\\\sum_{j}\\\\pi^{\\\\star}_{ji}=\\\\frac{1}{N_{+}} then fix ψ i = 1 N + \\\\psi_{i}=\\\\frac{1}{N_{+}} and ϕ j = 1 N − \\\\phi_{j}=\\\\frac{1}{N_{-}} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Substituting into Eqs. 10 - 11 :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The log-ratio is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The constant vanishes and yields Eq. 14 . The numerator and denominator are spherical kernel density estimates of p + p_{+} and p − p_{-} , which converge to the true densities as N ± → ∞ N_{\\\\pm}\\\\to\\\\infty [Pelletier, 2005 ] . ∎\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.2 Derivation of the Riemannian Gradient of log ψ ^ \\\\log\\\\hat{\\\\psi} and log ϕ ^ \\\\log\\\\hat{\\\\phi}\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We derive Eq. ( 17 ) which is the closed-form Riemannian gradient of the log backward Schrödinger factor on 𝒮 \\\\mathcal{S} . The derivation for ∇ 𝒮 log φ ^ \\\\nabla_{\\\\mathcal{S}}\\\\log\\\\hat{\\\\varphi} is identical with ( 𝒟 − , φ ~ j ) (\\\\mathcal{D}^{-},\\\\tilde{\\\\varphi}_{j}) replacing ( 𝒟 + , ψ ~ i ) (\\\\mathcal{D}^{+},\\\\tilde{\\\\psi}_{i}) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.3 Gradient of the Geodesic Gaussian Kernel\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let K σ ( 𝒙 , 𝒉 ) = exp ( − d 𝒮 ( 𝒙 , 𝒉 ) 2 / ( 2 σ ) ) K_{\\\\sigma}(\\\\bm{x},\\\\bm{h})=\\\\exp\\\\!\\\\bigl(-d_{\\\\mathcal{S}}(\\\\bm{x},\\\\bm{h})^{2}/(2\\\\sigma)\\\\bigr) for a bandwidth σ > 0 \\\\sigma>0 . By the chain rule and the fact that ∇ 𝒮 d 𝒮 ( 𝒙 , 𝒉 ) 2 = − 2 log 𝒙 ( 𝒉 ) \\\\nabla_{\\\\mathcal{S}}\\\\,d_{\\\\mathcal{S}}(\\\\bm{x},\\\\bm{h})^{2}=-2\\\\,\\\\log_{\\\\bm{x}}(\\\\bm{h}) :\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"B.4 Gradient of the Log Backward Potential\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Define the backward potential at 𝒙 ∈ 𝒮 \\\\bm{x}\\\\in\\\\mathcal{S} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the extended potentials ψ ~ i ( 𝒙 ) \\\\tilde{\\\\psi}_{i}(\\\\bm{x}) are also treated as constants with respect to differentiation at 𝒙 \\\\bm{x} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Taking the Riemannian gradient of log ψ ^ \\\\log\\\\hat{\\\\psi} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Substituting the kernel gradient ( 24 ) with σ = σ 2 \\\\sigma=\\\\sigma^{2} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Define the normalized weights\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In log-space, the unnormalized log-weight of point i i is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so normalization is a softmax:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This recovers Eq. ( 18 ). Substituting into ( 27 ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is a weighted Riemannian mean direction in the tangent space T 𝒙 𝒮 T_{\\\\bm{x}}\\\\mathcal{S} . Each log-map vector log 𝒙 ( 𝒉 i + ) \\\\log_{\\\\bm{x}}(\\\\bm{h}^{+}_{i}) points from the current query 𝒙 \\\\bm{x} toward the positive point 𝒉 i + \\\\bm{h}^{+}_{i} with magnitude equal to the geodesic distance, and the weight w ¯ i + ( 𝒙 ) \\\\bar{w}^{+}_{i}(\\\\bm{x}) combines geometric proximity (the kernel) with the Schrödinger potential (the transport plan).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.1 Base LLM Models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our base LLMs follow the experimental setup of Zhao et al. Zhao et al. [ 2026 ] . We describe them here for completeness. We evaluate all methods on four open-source models: (i) Falcon-7B, tiiuae/falcon-7b 2 2 2 https://huggingface.co/tiiuae/falcon-7b ; (ii) Mistral-7B-v0.3, mistralai/Mistral-7B-v0.3 3 3 3 https://huggingface.co/mistralai/Mistral-7B-v0.3 ; (iii) LLaMA3.1-8B, meta-llama/Llama-3.1-8B 4 4 4 https://huggingface.co/meta-llama/Llama-3.1-8B ; and (iv) Qwen2.5-7B, Qwen/Qwen2.5-7B 5 5 5 https://huggingface.co/Qwen/Qwen2.5-7B . We use the same generation configuration for all models and experiments: temperature = 0.7 =0.7 , top- p = 0.9 p=0.9 , and repetition penalty = 1.1 =1.1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.2 Datasets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"TruthfulQA [Lin et al. , 2021 ] is a benchmark consisting of 817 questions paired with true and false reference answers, along with source information. The questions cover 38 diverse categories and are commonly evaluated in a zero-shot setting. Following Zhao et al. Zhao et al. [ 2026 ] , we construct 5,918 question–answer pairs and use a 40/10/50 train/validation/test split with two-fold cross validation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"UltraFeedback [Cui et al. , 2023 ] is a large-scale alignment dataset covering instruction-following qualities such as truthfulness, honesty, and helpfulness. In our experiments, we use the UltraFeedback-Binarized dataset 6 6 6 https://huggingface.co/datasets/HuggingFaceH4/ultrafeedback_binarized . Specifically, we use the preference-modeling split, where each prompt is paired with an accepted response and a rejected response. Following Zhao et al. Zhao et al. [ 2026 ] , we randomly sample 10K training pairs, 500 validation pairs, and 500 test pairs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RealToxicityPrompts [Gehman et al. , 2020 ] is a benchmark for evaluating toxic degeneration in language models. It consists of naturally occurring web prompts paired with toxicity annotations obtained from the Perspective API. The dataset contains approximately 21K toxic prompts with toxicity scores greater than 0.5 and 77K non-toxic prompts. Following Zhao et al. [Zhao et al. , 2026 ] , we train toxicity steering methods using the Jigsaw Unintended Bias in Toxicity Classification dataset 7 7 7 https://www.kaggle.com/c/jigsaw-unintended-bias-in-toxicity-classification and evaluate them on RealToxicityPrompts. Specifically, we construct a balanced Jigsaw training set of 10K sentences by sampling according to toxicity scores, with 5K toxic and 5K benign examples. For evaluation, we select 500 toxic prompts from RealToxicityPrompts and use them as inputs to the LLMs. This setup allows us to assess whether steering methods reduce toxic generations while preserving response quality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GSM8K [Cobbe et al. , 2021 ] contains approximately 8.5K grade-school math word problems paired with natural-language solutions. The dataset contains roughly 7.5K training examples and 1K test examples. We use the original test split for out-of-distribution evaluation. All methods are trained on TruthfulQA as described above and then evaluated on the 1K GSM8K test examples in a 5-shot setting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MMLU [Hendrycks et al. , 2020 ] is a large-scale multiple-choice benchmark covering a broad range of academic and professional subjects. It contains approximately 100K training examples and 14K test examples. We use the default test split for out-of-distribution evaluation and perform 5-shot generation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.3 Metrics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Truthfulness (%). Truthfulness is evaluated on TruthfulQA Lin et al. [ 2021 ] using a publicly available fine-tuned LLaMA2-7B judge 8 8 8 https://huggingface.co/allenai/truthfulqa-truth-judge-llama2-7B . The judge predicts a binary truthfulness label for each generated answer using prompts of the form:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We report the percentage of generations classified as truthful.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Informativeness (%). Informativeness is evaluated on TruthfulQA Lin et al. [ 2021 ] using a publicly available fine-tuned LLaMA2-7B judge 9 9 9 https://huggingface.co/allenai/truthfulqa-info-judge-llama2-7B . The judge predicts whether a response is informative or helpful using prompts of the form:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We report the percentage of generations classified as informative.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Truthfulness × \\\\times Informativeness (%). Following prior work, we report the percentage of responses that are both truthful and informative. For each prompt, this metric is computed as the product of the binary truthfulness and informativeness labels, and the final score is averaged over all prompts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Win Rate (%). We adopt the reward-model win-rate metric used in RewardBench [Lambert et al. , 2025 ] and ODESteer [Zhao et al. , 2026 ] . Given a set of prompts { x i } i = 1 N \\\\{x_{i}\\\\}_{i=1}^{N} and two candidate systems A A and B B , let s i A s_{i}^{A} and s i B s_{i}^{B} denote their reward-model scores under the same reward model. The win rate of A A over B B is defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where 𝟙 ( ⋅ ) \\\\mathbbm{1}(\\\\cdot) is the indicator function. A value of 0.5 0.5 indicates parity with system B B , values greater than 0.5 0.5 indicate that A A outperforms B B , and ties contribute 0.5 0.5 by convention.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RM mean {}_{\\\\text{mean}} and RM 90 . For UltraFeedback, we use an LLM-as-a-judge reward model to score helpfulness. Specifically, we use Skywork/Skywork-Reward-V2-Llama-3.1-8B 10 10 10 https://huggingface.co/Skywork/Skywork-Reward-V2-Llama-3.1-8B . We report RM mean {}_{\\\\text{mean}} , the average reward score across examples and three random seeds, and RM 90 , the 90th percentile reward score.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Toxicity Score. Toxicity is evaluated using the Perspective API 11 11 11 https://perspectiveapi.com/ . The toxicity score is a probability-like score indicating the likelihood that a reader would perceive the generated text as toxic. Lower values therefore indicate less toxic generations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Perplexity. We evaluate generation fluency using perplexity under GPT-2-XL Radford et al. [ 2019 ] . Given a generated response y = ( y 1 , … , y T ) y=(y_{1},\\\\ldots,y_{T}) , GPT-2-XL assigns a next-token negative log-likelihood\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The perplexity of y y is then, PPL ( y ) = exp ( ℒ ( y ) ) . \\\\mathrm{PPL}(y)=\\\\exp\\\\left(\\\\mathcal{L}(y)\\\\right). Lower perplexity indicates that the generated text is more fluent under GPT-2-XL.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dist- n n . We measure lexical diversity using Dist- n n for n ∈ { 1 , 2 , 3 } n\\\\in\\\\{1,2,3\\\\} . For a generated response y y , let 𝒢 n ( y ) \\\\mathcal{G}_{n}(y) denote the multiset of all word-level n n -grams in y y , obtained by whitespace tokenization. Dist- n n is defined as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If the response contains fewer than n n words, we set Dist - n ( y ) = 0 \\\\mathrm{Dist}\\\\text{-}n(y)=0 . Higher Dist- n n indicates greater lexical diversity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.4 Implementation Details\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All our experiments are conducted on a machine equipped with 1x H100 80GB. We extend ODESteer codebase 12 12 12 https://github.com/ZhaoHongjue/odesteer to implement our steering methods and baselines. Cobras uses the following default values: steering strength of 0.65 0.65 , abstention K K of 32 32 , abstention percentile of 98 98 , geodesic Euler steps K K = 10.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.1 Effect of steering strength T T on Cobras\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 shows that steering strength has a non-monotonic effect. On TruthfulQA, moderate T T generally improves True × \\\\times Info, with peaks around T = 0.6 T=0.6 for LLaMA3.1-8B, T = 0.7 T=0.7 for Mistral-7B, and T = 0.4 T=0.4 for Qwen2.5-7B; larger values often degrade performance, suggesting over-steering. Falcon-7B is flatter and weaker overall, with a sharp drop at high T T . On UltraFeedback, the trend is more model-dependent: LLaMA3.1-8B benefits from moderate T T , Falcon-7B is relatively stable, while Mistral-7B and Qwen2.5-7B degrade as T T increases. Overall, these results support using moderate steering, typically T ≈ 0.5 T\\\\approx 0.5 – 0.7 0.7 for TruthfulQA and lower-to-moderate values for UltraFeedback.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.2 Diversity on Detoxification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 shows the generation diversity of steering methods on the detoxification task using Dist- n n metrics. Overall, Cobras preserves lexical diversity close to the original model across all four backbones. The gap is small for Mistral-7B and LLaMA3.1-8B, where Dist-2 and Dist-3 remain nearly unchanged compared to the original model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also observe that some baselines reduce diversity more noticeably. For example, CAA lowers diversity on Falcon-7B and Qwen2.5-7B, while SphericalSteer shows a larger drop on Qwen2.5-7B. In contrast, Cobras maintains high Dist-2 and Dist-3 scores, suggesting that its detoxification effect does not come from generating overly repetitive or generic responses.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.3 Impact of geodesic Euler steps K K\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 shows the effect of increasing the number of geodesic Euler steps K K in our proposed Cobras steering method on TruthfulQA. Overall, increasing K K improves the True × \\\\times Info score across all models, although the trend differs slightly by model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For Falcon-7B, larger K K increases informativeness, but also reduces the True score, leading to only a modest gain in True × \\\\times Info. For Mistral-7B, performance is relatively stable across different values of K K , with a small improvement at K = 10 K=10 . LLaMA3.1-8B benefits the most from increasing K K , where True × \\\\times Info improves consistently from 64.6 64.6 at K = 1 K=1 to 68.5 68.5 at K = 10 K=10 . These results suggest that using more geodesic Euler steps generally improves steering quality, with K = 10 K=10 giving the best overall performance in our experiments.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"D.4 Performance vs Steering Strength for baselines\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To select the steering strength for each baseline, we sweep over model-specific ranges: T = 20 T=20 – 23 23 for Falcon-7B, T = 2 T=2 – 5 5 for Mistral-7B, T = 3 T=3 – 6 6 for LLaMA3.1-8B, and T = 13 T=13 – 16 16 for Qwen2.5-7B. Figure 4 shows the performance of RepE, ITI, CAA, and ODESteer on TruthfulQA and UltraFeedback. For baselines without a comparable steering-strength parameter, we use the best values reported in their papers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results show that the best steering strength depends on both the model and the method. In general, some methods improve TruthfulQA performance at moderate values of T T , but larger steering strengths can hurt UltraFeedback performance. This trend is especially visible for ODESteer on Mistral-7B and LLaMA3.1-8B, where increasing T T leads to a large drop in UltraFeedback win rate. CAA is more stable on UltraFeedback and often improves with larger T T , while ITI tends to change more gradually across the sweep.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To analyze the quality of the steering, we show random samples obtained from UltraFeedback and TruthfulQA on steering methods.\"\n }\n]","isConversational":false}}},{"rowIdx":396,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.11954"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Graph-Constrained Policy Learning for Extreme Clinical Code Prediction"},"authors":{"kind":"list like","value":["Amritpal Singh","Sebastian Torres","Khawar Shakeel","Syed Ahmad Chan Bukhari"],"string":"[\n \"Amritpal Singh\",\n \"Sebastian Torres\",\n \"Khawar Shakeel\",\n \"Syed Ahmad Chan Bukhari\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.11954v1"},"published":{"kind":"timestamp","value":"2026-07-11T00:00:00","string":"2026-07-11T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"I Introduction"},{"type":"para","text":"Clinical code prediction maps unstructured documentation to standardized diagnosis codes used for reimbursement, quality measurement, cohort discovery, epidemiology, and risk adjustment. In U.S. inpatient care, diagnoses are recorded using ICD-10-CM, a deeply hierarchical system that refines broad chapters into sections, subsections, categories, subcategories, and billable leaf codes."},{"type":"para","text":"Automated ICD coding from discharge summaries is challenging because it combines long-document understanding with extreme multi-label prediction. Notes may contain several thousand tokens, while the label space is large, sparse, and heavily imbalanced: common conditions appear frequently, but many clinically important leaf codes occur rarely. Prior work has explored linear classifiers, CNNs, RNNs, label-wise attention, transformer encoders, graph-enhanced architectures, and retrieval-augmented systems [ 1 , 2 , 3 , 4 , 5 ] ; however, most methods still treat coding as flat multi-label classification over leaf codes."},{"type":"para","text":"This paper casts ICD-10-CM code prediction as graph-constrained traversal, formalized as a finite-horizon MDP over the pruned ICD-10-CM hierarchy (Section IV-A ). A single decoder-only language model descends the hierarchy level by level. At each depth, a lightweight environment exposes the valid children of the current frontier through tool calls; the model selects the children supported by the note, and the descent continues until billable leaves are reached. Hard validity constraints restrict each selection to the candidate set C t C_{t} , so structurally invalid codes receive zero probability by construction. Instead of scoring the full label space, O ( | L | ) O(|L|) , for each note, the traversal policy reduces inference to O ( ∑ ℓ | C ℓ | ) O(\\sum_{\\ell}|C_{\\ell}|) , where | C ℓ | ≪ | L | |C_{\\ell}|\\ll|L| after pruning to relevant branches. Extreme multi-label classification therefore becomes a short sequence of sparse, graph-constrained subset decisions. As Table I details, our system combines hierarchy-guided inference, hard structural constraints, a trained policy, multi-turn context across depths, and formal MDP grounding; prior systems omit at least one of these properties."},{"type":"para","text":"On MIMIC-IV discharge summaries, the trained traversal policy significantly outperforms three strong flat baselines, CAML [ 1 ] , LAAT [ 3 ] , and PLM-ICD [ 14 ] , on both a curated 50-code subset and the full 15,761-code ICD-10-CM space. On the full label space, SFT-1 + exceeds the strongest flat baseline, LAAT, by + 0.044 +0.044 micro-F1 and + 0.157 +0.157 macro-F1. CAML and PLM-ICD collapse to 0.002 macro-F1, indicating limited ability to discriminate among 15,761 candidates without structural guidance. These results show that graph-constrained decomposition partially insulates against the rare-code bottleneck."},{"type":"para","text":"We further ask whether additional architectural or algorithmic complexity is warranted once graph-constrained traversal is introduced. We conduct a controlled factorial study on the same MIMIC-IV data, isolating architecture, training algorithm, and annotation budget: one shared traversal policy versus a three-specialist cascade, supervised fine-tuning versus GRPO reinforcement learning, and 2,000 versus 5,000 gold trajectories. Three findings hold at both label-space scales. First, the shared policy matches the cascade on accuracy while avoiding the cascade’s context-window overflow failure on 28–32% of full-space test notes. Second, additional supervised trajectories are the only intervention that reliably improves performance, with gains roughly doubling from the 50-code to the full-space setting. Third, GRPO does not improve over supervised continuation under matched data. Together, these findings support a simple practical conclusion: use one shared traversal policy and invest in supervised trajectories rather than cascaded architectures or reinforcement learning."},{"type":"heading","level":1,"text":"II Related Work"},{"type":"heading","level":2,"text":"II-A Hierarchical and Graph-Aware Coding"},{"type":"para","text":"Because ICD is hierarchical, several models incorporate code structure. HyperCore represents ICD codes with hyperbolic and co-graph embeddings to exploit hierarchical relations and co-occurrence patterns [ 4 ] . Later graph-based and knowledge-guided models [ 15 , 17 , 18 ] use code descriptions, ancestor relations, and label graphs to improve rare-code generalization. These methods show that ICD structure carries predictive signal, but most use the hierarchy as an auxiliary representation or regularizer rather than traversing it directly. In Section V , we show that direct graph-constrained traversal yields larger gains as the label space grows."},{"type":"heading","level":2,"text":"II-B Clinical Foundation Models and LLM Coding"},{"type":"para","text":"Clinical language models such as ClinicalBERT [ 12 ] and BioBERT [ 13 ] showed that biomedical pretraining improves representation learning for healthcare NLP. For ICD coding, transformer encoders [ 14 ] and generative prompting have been proposed as alternatives to CNN/RNN pipelines, but long discharge notes and large code vocabularies remain limiting factors. Prompt-only LLM coding also raises reliability concerns [ 21 ] , including nonexistent codes, mixed ICD revisions, or plausible but invalid descendants. Closest to our setting, Boyle et al. [ 22 ] use the ICD hierarchy to search sparsely rather than score the full vocabulary, but rely on prompting alone. We instead bind generation to an explicit candidate set from the current ICD-10-CM graph and train the traversal policy on oracle demonstrations."},{"type":"heading","level":2,"text":"II-C Positioning"},{"type":"para","text":"Table I situates our approach against representative prior systems along five axes that together define the design space for hierarchical ICD coding."},{"type":"para","text":"Our formulation combines all five properties: the ICD-10-CM graph is traversed at inference time with hard validity constraints, the traversal policy is trained directly on oracle demonstrations, decisions at each depth condition on the full accumulated history, and the procedure is grounded in a formal MDP (Section IV-A )."},{"type":"heading","level":1,"text":"III Data and Task Construction"},{"type":"heading","level":2,"text":"III-A Source and Cohort Construction"},{"type":"para","text":"Following the MIMIC-IV coding setup of Boyle et al. [ 22 ] , we construct our dataset from MIMIC-IV [ 8 ] and MIMIC-IV-Note [ 9 ] , joining admissions, diagnosis codes, and discharge notes by admission identifier. We retain admissions with non-null discharge notes coded exclusively under ICD-10, discarding any admission containing at least one ICD-9 code to avoid mixing code schemas. This yields 122,281 admissions, each represented as a discharge note and gold diagnosis code set."},{"type":"para","text":"We then validate diagnosis codes against the official active ICD-10-CM code list (April 2026 release, 74,719 codes), retaining only admissions whose gold codes are present in this reference set. This removes admissions with deprecated, malformed, or non-billable codes, leaving 122,197 notes. This validated cohort serves as the base population for both label spaces and downstream splits. All data access, processing, and model training were conducted within an AWS SageMaker instance operating under a Business Associate Agreement (BAA) with Amazon, in compliance with MIMIC-IV data use requirements."},{"type":"heading","level":2,"text":"III-B Label Space and Code Graph"},{"type":"para","text":"Similar to Boyle et al. [ 22 ] , we define two label spaces over the validated cohort: all , containing every observed code with 15,761 leaves, and frequent , containing the 50 most common codes. These provide controlled settings at different points on the difficulty–realism trade-off. For each setting, we build a pruned ICD-10-CM graph retaining gold codes and their ancestors up to the chapter level, while discarding unreferenced branches (Table II )."},{"type":"heading","level":2,"text":"III-C Splits and Statistics"},{"type":"para","text":"For each label space, we sample 6,500 notes from the validated cohort and partition them into four disjoint splits: 2,000 for supervised fine-tuning, 3,000 for continuation training, 500 for validation, and 1,000 for testing. The continuation split is used for both additional supervised fine-tuning and GRPO reinforcement learning, supporting the factorial design in Section IV-E . Splitting is performed at the note level with seed 42 so that no admission appears in more than one split. Test and validation sets are held fixed across both label spaces and all training arms."},{"type":"para","text":"Table II reports summary statistics. Notes in the all setting carry more gold codes on average than in the frequent setting because frequent includes only diagnoses falling within the top-50 codes. Because ICD coding comparisons are sensitive to cohort construction, split strategy, and code-version handling [ 10 , 20 ] , we fix note-level splits, evaluate both label spaces, and hold test and validation sets constant across all arms."},{"type":"heading","level":2,"text":"III-D Gold Trajectory Construction"},{"type":"para","text":"Each training example is a multi-turn gold trajectory demonstrating the correct graph traversal for a note. We expand each gold ICD-10-CM code to its full ancestor path, from chapter root to billable leaf, and replay the resulting paths through the traversal environment. At each turn, the environment presents candidate nodes and the oracle selects exactly those on a gold ancestor path. Leaf codes are recorded automatically upon expansion, and the trajectory ends with an ANSWER turn listing all recorded leaf codes."},{"type":"para","text":"This yields one deterministic sequence per note, with assistant tool calls and environment responses tracing every gold code from root to leaf. During tokenization, environment-response tokens are masked from the training loss, so the model is supervised only on its own decisions: which nodes to select and when to answer."},{"type":"heading","level":1,"text":"IV Methodology"},{"type":"heading","level":2,"text":"IV-A Traversal as a Markov Decision Process"},{"type":"para","text":"We formalize ICD-10-CM code prediction as a finite-horizon MDP over the pruned ICD-10-CM graph. 𝒢 = ( 𝒱 , ℰ ) \\mathcal{G}=(\\mathcal{V},\\mathcal{E}) , where 𝒱 \\mathcal{V} is the set of code-graph nodes and ℰ \\mathcal{E} the directed parent → \\rightarrow child edges."},{"type":"para","text":"State. A state s t = ( x , F t , R t ) s_{t}=(x,F_{t},R_{t}) comprises the discharge note x x (fixed throughout the episode), the frontier F t ⊆ 𝒱 F_{t}\\subseteq\\mathcal{V} of nodes pending expansion, and the set R t R_{t} of billable leaf codes recorded so far. The initial state is s 0 = ( x , { root } , ∅ ) s_{0}=(x,\\{\\textsc{root}\\},\\emptyset) ."},{"type":"para","text":"Action. At each step the policy first expands every node in F t F_{t} , revealing the candidate set"},{"type":"para","text":"The action a t ⊆ C t a_{t}\\subseteq C_{t} is the subset of children the note supports. The constraint a t ⊆ C t a_{t}\\subseteq C_{t} is enforced by the environment, so structurally invalid selections receive zero probability by construction rather than by penalty."},{"type":"para","text":"Transition. The transition is deterministic. Any selected node that is a billable leaf is added to R t R_{t} ; the remainder form the next frontier:"},{"type":"para","text":"The episode terminates when F t + 1 = ∅ F_{t+1}=\\emptyset ."},{"type":"para","text":"Reward. For supervised fine-tuning the reward is implicit: the policy is trained to imitate oracle trajectories constructed from the gold code sets (Section III-D ). For reinforcement learning (GRPO-1, Section IV-E ), we define an explicit terminal reward"},{"type":"para","text":"where C ∗ C^{*} is the gold code set and path-F1 awards partial credit for correct ancestor paths (Equation 1 ). Rewards are zero for all non-terminal transitions; only the terminal state receives the reward defined above."},{"type":"para","text":"Policy. The policy π θ ( a t ∣ s t ) \\pi_{\\theta}(a_{t}\\mid s_{t}) is parameterized by a single QLoRA adapter over a decoder-only backbone. Because the full traversal history ( a 0 , C 0 , … , a t − 1 , C t − 1 ) (a_{0},C_{0},\\ldots,a_{t-1},C_{t-1}) is retained in the context window, the Markov property holds with respect to the model’s input: each action conditions on the note, the complete descent so far, and the current candidate set."},{"type":"para","text":"The horizon is bounded by the depth of the deepest selected path in 𝒢 \\mathcal{G} (typically 5–7 expand-and-select rounds). The effective action space at each step satisfies | C t | ≪ | 𝒱 | |C_{t}|\\ll|\\mathcal{V}| , reducing the combinatorial multi-label problem to a short sequence of sparse, graph-constrained subset selections. Section IV-B describes the concrete implementation of this MDP as a tool-calling environment."},{"type":"heading","level":2,"text":"IV-B Traversal Environment"},{"type":"para","text":"We implement the MDP defined in Section IV-A as a guided descent through the pruned ICD-10-CM graph, mediated by a lightweight environment that maintains a traversal agenda and exposes two tools and a terminal action:"},{"type":"para","text":"expand_level() pops all currently pending nodes from the agenda and reveals their children, grouped by parent. If a pending node is a billable leaf (a code with no descendants in the pruned ICD-10-CM graph) it is recorded automatically as a final answer rather than expanded, since there is nothing below it to drill into. When the agenda is empty, the environment returns DONE and lists all recorded codes."},{"type":"para","text":"select(children=[...]) accepts a list of child IDs drawn exclusively from the most recent expand_level() output and queues them as pending for the next expansion. The environment validates every ID against the set of children just presented; invalid IDs are rejected, which realizes the constrained-decoding principle of restricting generation to a valid candidate set [ 19 ] at the level of tool arguments. An empty list is permitted, allowing the model to skip an entire level when no candidates are supported by the note."},{"type":"para","text":"ANSWER: code1, code2, ... terminates the trajectory and emits all recorded leaf codes as the model’s final prediction."},{"type":"para","text":"The environment enforces breadth-first traversal: each expand_level() call processes every pending node at the same hierarchy depth in a single turn, so a complete traversal spans 5-7 depth levels (approximately 12–14 model turns) regardless of how many codes the note carries (Figure 1 )."},{"type":"para","text":"The model interacts with the environment through Qwen3’s function-calling interface. The rollout driver parses each <tool_call> , executes the requested tool, and returns the result as a tool-response turn. During training, model-generated tokens are trainable and environment-injected tokens are masked; during evaluation, active rollouts are batched by wave for tractability at 1,000+ notes."},{"type":"heading","level":2,"text":"IV-C Shared Traversal Policy (SFT-1)"},{"type":"para","text":"A single traversal policy makes all selection decisions across hierarchy depths within one continuous context window, conditioning each level on the accumulated traversal history. Prior LLM traversal of ICD hierarchies [ 22 ] relies on prompting alone; we train the policy directly on oracle demonstrations. We fine-tune on the 2,000-note SFT split using the gold trajectories. Each trajectory is replayed through the rollout driver to produce the trainable/masked split described in Section IV-B . All fine-tuning uses QLoRA [ 7 ] with rank r = 16 r{=}16 on a 4-bit quantized base model; continuation arms warm-start from the preceding adapter."},{"type":"heading","level":2,"text":"IV-D Cascade Baseline (SFT-3)"},{"type":"para","text":"The cascade decomposes the same traversal into three independent single-turn classification problems, one per hierarchy depth: a chapter specialist, a subsection specialist, and a code specialist. Each specialist receives the discharge note and a flat list of candidates at its assigned level, and responds with a single line ( SELECTED: id1, id2, ... ) choosing every candidate the note supports. There is no tool-call protocol, no multi-turn interaction, and no shared context across levels. At inference time the three are executed sequentially, so the chapter specialist’s selections determine which candidates the subsection specialist sees, and in turn which billable codes are presented to the code specialist. All three share a common system prompt (Supplementary Materials, Appendix A)."},{"type":"para","text":"The candidate lists at the first two levels (L1: chapters, L2: subsections) are modest at both label-space scales, but the third level (L3: billable codes) presents all billable-leaf descendants of selected subsections in one batched prompt. This makes the cascade efficient, requiring three model calls per note, but creates a scaling limitation. Under the all label space, the L3 prompt often exceeds the 32,000-token cap used for a single 46GB GPU. During training, over-cap L3 examples are dropped rather than truncated; during evaluation, the candidate tail is truncated, imposing a hard recall ceiling on 28% of SFT-3 test notes and 32% of SFT-3 + test notes. The shared traversal policy has 0% truncation at the same scale."},{"type":"para","text":"We train the three specialists on the same 2,000-note SFT split used for SFT-1, with each note re-expressed as three single-turn examples (one per level), yielding 6,000 training examples total."},{"type":"heading","level":2,"text":"IV-E Continuation and RL Arms"},{"type":"para","text":"The shared traversal policy and cascade baseline each use 2,000 notes for initial training. To isolate the effects of data, architecture, and training algorithm, we define three continuation arms on the same 3,000-note continuation split."},{"type":"para","text":"SFT-1 + warm-starts from SFT-1 and continues supervised fine-tuning on 3,000 additional gold trajectories constructed as in Sections III-D and IV-B ."},{"type":"para","text":"SFT-3 + warm-starts each SFT-3 specialist and continues supervised fine-tuning on the same continuation split, re-expressed as per-level single-turn examples."},{"type":"para","text":"GRPO-1 warm-starts from SFT-1 and applies Group Relative Policy Optimization on the same 3,000 notes. The model generates live rollouts against the environment, and each rollout is scored with the terminal reward from Section IV-A . We run GRPO-1 on the frequent label space only; the all variant was omitted because training time scales to approximately 2–3 GPU-days and the frequent result already shows no benefit over supervised continuation."},{"type":"heading","level":2,"text":"IV-F Evaluation Protocol"},{"type":"para","text":"All arms are evaluated on the same held-out test set (1,000 notes) and validation set (500 notes), shared across both label spaces and all training arms (Section III-C ). Evaluation rollouts are executed through the batched rollout driver (Section IV-B ) under greedy decoding (temperature 0) for determinism and paired comparability."},{"type":"para","text":"Zero-shot floors. To anchor the fine-tuning results, we evaluate two zero-shot baselines using the base model base model (Qwen3-4B-Instruct-2507 [ 11 ] ) with no task-specific training. The traversal-format base (Base in Table III ) receives the same system prompt, tool schema, and discharge note as SFT-1, scoring near-zero micro-F1 on both label spaces (0.001), reflecting protocol-following failure rather than lack of clinical knowledge. The cascade base (Base-3) evaluates the same untrained model under the single-turn SELECTED: format of SFT-3, clearing the floor meaningfully (0.229 on frequent , 0.058 on all ). The gap between the two floors quantifies the protocol-following overhead that the gold trajectories (Section III-D ) are designed to close."},{"type":"para","text":"Metrics. We report micro-averaged precision, recall, and F1 (pooled across all notes and codes, weighting each prediction equally) and macro-averaged precision, recall, and F1 (averaged across the codes present in the gold set, weighting rare and common codes equally). Micro-F1 is the headline metric; macro-F1 surfaces rare-code performance, which dominates the all label space. We additionally report path-F1, a hierarchical variant that awards partial credit for predictions landing in the correct region of the ICD-10-CM hierarchy rather than requiring an exact leaf match. For a code c c , let A ( c ) = { c } ∪ anc ( c ) A(c)=\\{c\\}\\cup\\mathrm{anc}(c) denote its augmentation with all ancestors along the path toward the root (the root itself excluded), and extend this to a set of codes by A ( C ) = ⋃ c ∈ C A ( c ) A(C)=\\bigcup_{c\\in C}A(c) . Writing C ^ i \\hat{C}_{i} and C i C_{i} for the predicted and gold codes of note i i , path-F1 replaces exact-match set membership with membership over the augmented sets:"},{"type":"para","text":"with path-F1 their harmonic mean. Equation 1 gives the micro variant (sums pooled over notes); the macro variant averages the per-code F1 over the augmented gold codes. Because a mispredicted leaf still shares ancestors with the true code, this rewards a prediction that lands on the correct branch over one that is entirely unrelated."},{"type":"para","text":"Statistical testing. We use a paired bootstrap to assess differences between arms. Because micro-F1 and macro-F1 are pooled statistics that do not decompose into per-note scores, a paired t-test is not applicable. For each of 5,000 resamples we draw 1,000 test notes with replacement and recompute both arms’ pooled metrics on the identical resampled set. We report the mean difference, 95% confidence interval, and two-sided p-value (values below 1 / 5 , 000 1/5{,}000 reported as p < 0.001 p<0.001 ). We report uncorrected p-values for approximately fifteen planned comparisons. 1 1 1 The large effects (supervised data) survive Bonferroni correction. The small architecture deltas that reach nominal significance ( p = 0.002 p=0.002 – 0.026 0.026 ) would not all survive correction at α = 0.05 / 15 ≈ 0.003 \\alpha=0.05/15\\approx 0.003 , and we interpret them accordingly."},{"type":"para","text":"We additionally report per-level selection accuracy (chapter, subsection, billable code) to diagnose where recall is lost, and for the cascade under the all label space, the fraction of test notes affected by L3 candidate-list truncation (Section IV-D )."},{"type":"heading","level":1,"text":"V Results"},{"type":"heading","level":2,"text":"V-A Overall Performance"},{"type":"para","text":"Table III reports micro- and macro-averaged precision, recall, and F1 for all arms on the held-out test set ( n = 1 , 000 n{=}1{,}000 ) at both label-space scales. The results are summarized below."},{"type":"para","text":"The traversal policy outperforms flat baselines, with the margin growing at scale. Three flat multi-label baselines anchor these results externally. CAML and PLM-ICD both reach competitive micro-F1 on frequent (0.668 and 0.691) but collapse to near-identical scores on all (0.194 and 0.196), with macro-F1 falling to 0.002 for both—indicating that neither convolutional attention nor pretrained transformer encoders can discriminate among 15,761 candidates without structural guidance. Only LAAT survives at scale (0.483 on all ), and SFT-1 trained on just 2,000 trajectories already matches it (0.478 vs. 0.483). SFT-1 + significantly outperforms all three baselines at both scales (Table IV ), with the margin widening sharply on all : + 0.044 +0.044 micro-F1 over LAAT and + 0.157 +0.157 macro-F1, confirming that the traversal formulation’s advantage is concentrated where flat scoring is weakest. Notably, PLM-ICD’s pretrained encoder offers no advantage over CAML’s CNN at the full scale, suggesting that the bottleneck is the flat output layer rather than the representation - with only 2,000 training notes the vast majority of the 15,761 output neurons see fewer than five positive examples."},{"type":"para","text":"More supervised data helps, and the effect grows with scale. The + 3 , 000 +3{,}000 -note continuation (SFT-1 + over SFT-1) is the largest gain in the table: + 0.022 +0.022 micro-F1 on frequent ( 0.687 → 0.709 0.687\\rightarrow 0.709 ) and + 0.049 +0.049 on all ( 0.478 → 0.527 0.478\\rightarrow 0.527 ). The lift roughly doubles from the 50-code to the full label space. SFT-1 + is the best arm in both settings, and the same pattern holds for the cascade (SFT-3 + over SFT-3 by comparable margins). Macro-F1 gains track micro, so the improvement is not confined to high-frequency codes."},{"type":"para","text":"GRPO does not improve over supervised continuation. GRPO-1 (0.688 micro-F1) ties its SFT-1 warm-start (0.687) and falls below SFT-1 + (0.709) on both micro and macro. Training directly on the coding metric via RL on 3,000 notes offers no benefit over simply continuing supervised fine-tuning on the same data from the same initialization."},{"type":"para","text":"Neither architecture wins consistently. The cascade (SFT-3, SFT-3 + ) lands within roughly one micro-F1 point of the corresponding shared-policy arm (SFT-1, SFT-1 + ) at both scales. On all , SFT-3 + scores 0.513 vs. SFT-1 + ’s 0.527 on micro but 0.260 vs. 0.256 on macro; the deltas flip sign across metrics and variants. Statistical tests follow in Section V-B . Where the cascade does lose is structurally: 28–32% of test notes suffer L3 candidate-list truncation under the all label space (Section IV-D ), a constraint the shared policy never encounters."},{"type":"heading","level":2,"text":"V-B Statistical Comparisons"},{"type":"para","text":"Table IV reports paired-bootstrap estimates (5,000 resamples) of the difference in pooled micro- and macro-F1 between matched arms, with 95% confidence intervals and two-sided p-values. All external comparisons are significant ( p < 0.001 p<0.001 ). The margins are modest on frequent ( + 0.014 +0.014 to + 0.042 +0.042 micro-F1) but large on all ( + 0.044 +0.044 to + 0.333 +0.333 micro-F1), confirming that the traversal formulation’s advantage grows with label-space scale. On frequent , SFT-1 without continuation data already matches LAAT ( p = 0.08 p=0.08 , n.s.)."},{"type":"para","text":"Among the internal comparisons, the data effect is the only robust finding: SFT-1 + over SFT-1 is significant on all four tests ( p < 0.001 p<0.001 ), with the lift doubling from frequent to all . GRPO-1 ties SFT-1 (micro p = 0.892 p=0.892 ) and falls significantly below SFT-1 + ( p < 0.001 p<0.001 ). Architecture deltas are small ( ≤ 0.014 \\leq 0.014 ), inconsistent in sign, and would not survive Bonferroni correction."},{"type":"heading","level":2,"text":"V-C Hierarchical Recall by Depth"},{"type":"para","text":"To diagnose where in the hierarchy codes are lost, we compute micro-pooled recall at each depth of the ICD-10-CM tree: at depth d d , each predicted and gold leaf code is collapsed to its depth- d d ancestor, and recall is computed over the resulting sets. 2 2 2 Codes whose paths are shorter than d d do not contribute at that depth, so deeper columns reflect only codes that actually reach that level of the hierarchy. The denominators therefore shrink at deeper levels. Figure 3 plots this for the best shared-policy and cascade arms at both label-space scales."},{"type":"para","text":"Recall degrades monotonically with depth, from 0.80-0.83 at the chapter level to 0.70 (frequent) and 0.49 (all) at the billable leaf. The frequent → \\rightarrow all gap widens at each depth, confirming that rare-code discrimination at the bottom of the tree drives the macro-F1 collapse. The cascade tracks the shared policy at shallow depths but diverges where L3 truncation (32% of test notes) imposes a hard recall ceiling the shared policy never encounters."},{"type":"heading","level":1,"text":"VI Discussion"},{"type":"heading","level":2,"text":"VI-A The Rare-Code Bottleneck"},{"type":"para","text":"The macro-F1 collapse from frequent to all ( 0.651 → 0.256 0.651\\rightarrow 0.256 for the best arm) is the starkest result in the study. Micro-F1 also drops ( 0.709 → 0.527 0.709\\rightarrow 0.527 ), but the far larger macro gap shows the degradation is concentrated in the long tail of rare codes rather than spread uniformly. The flat baselines degrade far more sharply, confirming that the traversal formulation’s hierarchical decomposition partially insulates against the rare-code problem without solving it. The per-depth analysis (Figure 3 ) corroborates this: chapter-level recall is comparable across scales, but the gap widens at each depth, largest at the billable leaf where rare codes dominate."},{"type":"heading","level":2,"text":"VI-B Supervision Is the Bottleneck"},{"type":"para","text":"Of the three variables we isolate, only additional supervised data reliably improves performance (Table IV ). Neither RL nor architectural specialization produces a comparable or even consistent gain. This is worth stating plainly because the natural design impulse points the other way: once a hierarchy is available, it is tempting to assign specialist models to each level and to replace hand-built trajectories with learned rewards. Our factorial study tests both assumptions and finds neither justified at this scale. Our results suggest that, at least for extraction-bound graph-constrained coding at this model scale, the complexity does not pay off. GRPO ties its supervised warm-start and loses to supervised continuation on the same data; the cascade matches the shared policy but never beats it. The bottleneck is not how the model is trained or how the traversal is decomposed, but how many gold demonstrations it sees."},{"type":"para","text":"This conclusion is scoped to the regime we test (a single 4B model, one reward design, 2,000–5,000 notes), and RL or specialization may yet help at larger scales or with richer reward shaping. What our study contributes is a controlled baseline: any added machinery should be shown to beat simply adding more supervised data at matched compute."},{"type":"heading","level":2,"text":"VI-C Consolidation Scales Where the Cascade Cannot"},{"type":"para","text":"On headline accuracy the shared traversal policy and the three-specialist cascade are statistically indistinguishable at both label-space scales. The consolidation argument therefore rests not on accuracy but on deployment simplicity (one adapter, no inter-stage orchestration) and a structural scaling advantage."},{"type":"para","text":"The cascade hits a scaling wall at the full code space: its batched L3 prompt forces every billable-leaf descendant of the selected subsections into one context window, overflowing it for 28-32% of test notes and making those codes unselectable regardless of model quality. A larger memory budget would raise this ceiling, but the pressure is structural - the single-shot design’s prompt cost grows with label-space size, whereas the shared policy’s per-turn fan-out distributes the same load across turns (0% truncation)."},{"type":"heading","level":2,"text":"VI-D Limitations"},{"type":"para","text":"All results are from a single 4B-parameter model (Qwen3-4B-Instruct-2507); RL and architectural specialization may behave differently at larger scales or with other model families. The GRPO result is based on a single reward design, one hyperparameter configuration, and one warm-start on the frequent label space only; the full-scale RL cell is argued but not directly measured. All data comes from MIMIC-IV, a single US academic medical center, and the gold labels are billed administrative codes known to be incomplete, so measured precision is a lower bound when the model assigns a clinically valid code that was not billed. The 32,000-token context cap driving the cascade’s 28-32% truncation rate is a memory budget constraint (single 46GB GPU), not an architectural limit of the model. Finally, our training budget of 5,000 notes is small relative to the 122,197 available; the data-scaling claim rests on just two training budgets (2,000 and 5,000 notes) and we do not determine where the lift saturates."},{"type":"heading","level":1,"text":"VII Conclusion"},{"type":"para","text":"We formulated ICD-10-CM code prediction as a single graph-constrained traversal policy that significantly outperforms flat multi-label baselines at both label-space scales (by + 0.044 +0.044 micro-F1 and + 0.157 +0.157 macro-F1 over the strongest (LAAT) on the full 15,761-code space) and conducted a controlled study isolating architecture, training algorithm, and data. Across both scales, only additional supervised data reliably helped, RL offered no benefit over supervised continuation, and one shared policy matched the three-specialist cascade while avoiding its context-window truncation. Practitioners should therefore deploy one model, not three, and invest in annotation rather than architectural or algorithmic complexity."},{"type":"para","text":"Several directions remain open. The data-scaling curve has only two budgets (2,000 and 5,000 notes); larger annotation budgets and targeted sampling of rare-code-rich notes could determine where the supervised lift saturates. Sampling-based decoding (self-consistency, best-of- n n ) and per-code threshold tuning may improve rare-code recall, which the macro-F1 collapse identifies as the primary bottleneck. RL deserves re-examination with richer reward shaping and at larger model scales before a general negative conclusion is warranted. Finally, validation on a second institution and coding system would test whether these findings generalize beyond a single US academic center."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"I Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Clinical code prediction maps unstructured documentation to standardized diagnosis codes used for reimbursement, quality measurement, cohort discovery, epidemiology, and risk adjustment. In U.S. inpatient care, diagnoses are recorded using ICD-10-CM, a deeply hierarchical system that refines broad chapters into sections, subsections, categories, subcategories, and billable leaf codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Automated ICD coding from discharge summaries is challenging because it combines long-document understanding with extreme multi-label prediction. Notes may contain several thousand tokens, while the label space is large, sparse, and heavily imbalanced: common conditions appear frequently, but many clinically important leaf codes occur rarely. Prior work has explored linear classifiers, CNNs, RNNs, label-wise attention, transformer encoders, graph-enhanced architectures, and retrieval-augmented systems [ 1 , 2 , 3 , 4 , 5 ] ; however, most methods still treat coding as flat multi-label classification over leaf codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This paper casts ICD-10-CM code prediction as graph-constrained traversal, formalized as a finite-horizon MDP over the pruned ICD-10-CM hierarchy (Section IV-A ). A single decoder-only language model descends the hierarchy level by level. At each depth, a lightweight environment exposes the valid children of the current frontier through tool calls; the model selects the children supported by the note, and the descent continues until billable leaves are reached. Hard validity constraints restrict each selection to the candidate set C t C_{t} , so structurally invalid codes receive zero probability by construction. Instead of scoring the full label space, O ( | L | ) O(|L|) , for each note, the traversal policy reduces inference to O ( ∑ ℓ | C ℓ | ) O(\\\\sum_{\\\\ell}|C_{\\\\ell}|) , where | C ℓ | ≪ | L | |C_{\\\\ell}|\\\\ll|L| after pruning to relevant branches. Extreme multi-label classification therefore becomes a short sequence of sparse, graph-constrained subset decisions. As Table I details, our system combines hierarchy-guided inference, hard structural constraints, a trained policy, multi-turn context across depths, and formal MDP grounding; prior systems omit at least one of these properties.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On MIMIC-IV discharge summaries, the trained traversal policy significantly outperforms three strong flat baselines, CAML [ 1 ] , LAAT [ 3 ] , and PLM-ICD [ 14 ] , on both a curated 50-code subset and the full 15,761-code ICD-10-CM space. On the full label space, SFT-1 + exceeds the strongest flat baseline, LAAT, by + 0.044 +0.044 micro-F1 and + 0.157 +0.157 macro-F1. CAML and PLM-ICD collapse to 0.002 macro-F1, indicating limited ability to discriminate among 15,761 candidates without structural guidance. These results show that graph-constrained decomposition partially insulates against the rare-code bottleneck.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We further ask whether additional architectural or algorithmic complexity is warranted once graph-constrained traversal is introduced. We conduct a controlled factorial study on the same MIMIC-IV data, isolating architecture, training algorithm, and annotation budget: one shared traversal policy versus a three-specialist cascade, supervised fine-tuning versus GRPO reinforcement learning, and 2,000 versus 5,000 gold trajectories. Three findings hold at both label-space scales. First, the shared policy matches the cascade on accuracy while avoiding the cascade’s context-window overflow failure on 28–32% of full-space test notes. Second, additional supervised trajectories are the only intervention that reliably improves performance, with gains roughly doubling from the 50-code to the full-space setting. Third, GRPO does not improve over supervised continuation under matched data. Together, these findings support a simple practical conclusion: use one shared traversal policy and invest in supervised trajectories rather than cascaded architectures or reinforcement learning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"II Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-A Hierarchical and Graph-Aware Coding\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because ICD is hierarchical, several models incorporate code structure. HyperCore represents ICD codes with hyperbolic and co-graph embeddings to exploit hierarchical relations and co-occurrence patterns [ 4 ] . Later graph-based and knowledge-guided models [ 15 , 17 , 18 ] use code descriptions, ancestor relations, and label graphs to improve rare-code generalization. These methods show that ICD structure carries predictive signal, but most use the hierarchy as an auxiliary representation or regularizer rather than traversing it directly. In Section V , we show that direct graph-constrained traversal yields larger gains as the label space grows.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-B Clinical Foundation Models and LLM Coding\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Clinical language models such as ClinicalBERT [ 12 ] and BioBERT [ 13 ] showed that biomedical pretraining improves representation learning for healthcare NLP. For ICD coding, transformer encoders [ 14 ] and generative prompting have been proposed as alternatives to CNN/RNN pipelines, but long discharge notes and large code vocabularies remain limiting factors. Prompt-only LLM coding also raises reliability concerns [ 21 ] , including nonexistent codes, mixed ICD revisions, or plausible but invalid descendants. Closest to our setting, Boyle et al. [ 22 ] use the ICD hierarchy to search sparsely rather than score the full vocabulary, but rely on prompting alone. We instead bind generation to an explicit candidate set from the current ICD-10-CM graph and train the traversal policy on oracle demonstrations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"II-C Positioning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table I situates our approach against representative prior systems along five axes that together define the design space for hierarchical ICD coding.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our formulation combines all five properties: the ICD-10-CM graph is traversed at inference time with hard validity constraints, the traversal policy is trained directly on oracle demonstrations, decisions at each depth condition on the full accumulated history, and the procedure is grounded in a formal MDP (Section IV-A ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"III Data and Task Construction\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-A Source and Cohort Construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following the MIMIC-IV coding setup of Boyle et al. [ 22 ] , we construct our dataset from MIMIC-IV [ 8 ] and MIMIC-IV-Note [ 9 ] , joining admissions, diagnosis codes, and discharge notes by admission identifier. We retain admissions with non-null discharge notes coded exclusively under ICD-10, discarding any admission containing at least one ICD-9 code to avoid mixing code schemas. This yields 122,281 admissions, each represented as a discharge note and gold diagnosis code set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We then validate diagnosis codes against the official active ICD-10-CM code list (April 2026 release, 74,719 codes), retaining only admissions whose gold codes are present in this reference set. This removes admissions with deprecated, malformed, or non-billable codes, leaving 122,197 notes. This validated cohort serves as the base population for both label spaces and downstream splits. All data access, processing, and model training were conducted within an AWS SageMaker instance operating under a Business Associate Agreement (BAA) with Amazon, in compliance with MIMIC-IV data use requirements.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-B Label Space and Code Graph\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Similar to Boyle et al. [ 22 ] , we define two label spaces over the validated cohort: all , containing every observed code with 15,761 leaves, and frequent , containing the 50 most common codes. These provide controlled settings at different points on the difficulty–realism trade-off. For each setting, we build a pruned ICD-10-CM graph retaining gold codes and their ancestors up to the chapter level, while discarding unreferenced branches (Table II ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-C Splits and Statistics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each label space, we sample 6,500 notes from the validated cohort and partition them into four disjoint splits: 2,000 for supervised fine-tuning, 3,000 for continuation training, 500 for validation, and 1,000 for testing. The continuation split is used for both additional supervised fine-tuning and GRPO reinforcement learning, supporting the factorial design in Section IV-E . Splitting is performed at the note level with seed 42 so that no admission appears in more than one split. Test and validation sets are held fixed across both label spaces and all training arms.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table II reports summary statistics. Notes in the all setting carry more gold codes on average than in the frequent setting because frequent includes only diagnoses falling within the top-50 codes. Because ICD coding comparisons are sensitive to cohort construction, split strategy, and code-version handling [ 10 , 20 ] , we fix note-level splits, evaluate both label spaces, and hold test and validation sets constant across all arms.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"III-D Gold Trajectory Construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each training example is a multi-turn gold trajectory demonstrating the correct graph traversal for a note. We expand each gold ICD-10-CM code to its full ancestor path, from chapter root to billable leaf, and replay the resulting paths through the traversal environment. At each turn, the environment presents candidate nodes and the oracle selects exactly those on a gold ancestor path. Leaf codes are recorded automatically upon expansion, and the trajectory ends with an ANSWER turn listing all recorded leaf codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This yields one deterministic sequence per note, with assistant tool calls and environment responses tracing every gold code from root to leaf. During tokenization, environment-response tokens are masked from the training loss, so the model is supervised only on its own decisions: which nodes to select and when to answer.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"IV Methodology\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-A Traversal as a Markov Decision Process\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We formalize ICD-10-CM code prediction as a finite-horizon MDP over the pruned ICD-10-CM graph. 𝒢 = ( 𝒱 , ℰ ) \\\\mathcal{G}=(\\\\mathcal{V},\\\\mathcal{E}) , where 𝒱 \\\\mathcal{V} is the set of code-graph nodes and ℰ \\\\mathcal{E} the directed parent → \\\\rightarrow child edges.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"State. A state s t = ( x , F t , R t ) s_{t}=(x,F_{t},R_{t}) comprises the discharge note x x (fixed throughout the episode), the frontier F t ⊆ 𝒱 F_{t}\\\\subseteq\\\\mathcal{V} of nodes pending expansion, and the set R t R_{t} of billable leaf codes recorded so far. The initial state is s 0 = ( x , { root } , ∅ ) s_{0}=(x,\\\\{\\\\textsc{root}\\\\},\\\\emptyset) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Action. At each step the policy first expands every node in F t F_{t} , revealing the candidate set\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The action a t ⊆ C t a_{t}\\\\subseteq C_{t} is the subset of children the note supports. The constraint a t ⊆ C t a_{t}\\\\subseteq C_{t} is enforced by the environment, so structurally invalid selections receive zero probability by construction rather than by penalty.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Transition. The transition is deterministic. Any selected node that is a billable leaf is added to R t R_{t} ; the remainder form the next frontier:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The episode terminates when F t + 1 = ∅ F_{t+1}=\\\\emptyset .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reward. For supervised fine-tuning the reward is implicit: the policy is trained to imitate oracle trajectories constructed from the gold code sets (Section III-D ). For reinforcement learning (GRPO-1, Section IV-E ), we define an explicit terminal reward\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where C ∗ C^{*} is the gold code set and path-F1 awards partial credit for correct ancestor paths (Equation 1 ). Rewards are zero for all non-terminal transitions; only the terminal state receives the reward defined above.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Policy. The policy π θ ( a t ∣ s t ) \\\\pi_{\\\\theta}(a_{t}\\\\mid s_{t}) is parameterized by a single QLoRA adapter over a decoder-only backbone. Because the full traversal history ( a 0 , C 0 , … , a t − 1 , C t − 1 ) (a_{0},C_{0},\\\\ldots,a_{t-1},C_{t-1}) is retained in the context window, the Markov property holds with respect to the model’s input: each action conditions on the note, the complete descent so far, and the current candidate set.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The horizon is bounded by the depth of the deepest selected path in 𝒢 \\\\mathcal{G} (typically 5–7 expand-and-select rounds). The effective action space at each step satisfies | C t | ≪ | 𝒱 | |C_{t}|\\\\ll|\\\\mathcal{V}| , reducing the combinatorial multi-label problem to a short sequence of sparse, graph-constrained subset selections. Section IV-B describes the concrete implementation of this MDP as a tool-calling environment.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-B Traversal Environment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We implement the MDP defined in Section IV-A as a guided descent through the pruned ICD-10-CM graph, mediated by a lightweight environment that maintains a traversal agenda and exposes two tools and a terminal action:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"expand_level() pops all currently pending nodes from the agenda and reveals their children, grouped by parent. If a pending node is a billable leaf (a code with no descendants in the pruned ICD-10-CM graph) it is recorded automatically as a final answer rather than expanded, since there is nothing below it to drill into. When the agenda is empty, the environment returns DONE and lists all recorded codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"select(children=[...]) accepts a list of child IDs drawn exclusively from the most recent expand_level() output and queues them as pending for the next expansion. The environment validates every ID against the set of children just presented; invalid IDs are rejected, which realizes the constrained-decoding principle of restricting generation to a valid candidate set [ 19 ] at the level of tool arguments. An empty list is permitted, allowing the model to skip an entire level when no candidates are supported by the note.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ANSWER: code1, code2, ... terminates the trajectory and emits all recorded leaf codes as the model’s final prediction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The environment enforces breadth-first traversal: each expand_level() call processes every pending node at the same hierarchy depth in a single turn, so a complete traversal spans 5-7 depth levels (approximately 12–14 model turns) regardless of how many codes the note carries (Figure 1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model interacts with the environment through Qwen3’s function-calling interface. The rollout driver parses each <tool_call> , executes the requested tool, and returns the result as a tool-response turn. During training, model-generated tokens are trainable and environment-injected tokens are masked; during evaluation, active rollouts are batched by wave for tractability at 1,000+ notes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-C Shared Traversal Policy (SFT-1)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A single traversal policy makes all selection decisions across hierarchy depths within one continuous context window, conditioning each level on the accumulated traversal history. Prior LLM traversal of ICD hierarchies [ 22 ] relies on prompting alone; we train the policy directly on oracle demonstrations. We fine-tune on the 2,000-note SFT split using the gold trajectories. Each trajectory is replayed through the rollout driver to produce the trainable/masked split described in Section IV-B . All fine-tuning uses QLoRA [ 7 ] with rank r = 16 r{=}16 on a 4-bit quantized base model; continuation arms warm-start from the preceding adapter.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-D Cascade Baseline (SFT-3)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cascade decomposes the same traversal into three independent single-turn classification problems, one per hierarchy depth: a chapter specialist, a subsection specialist, and a code specialist. Each specialist receives the discharge note and a flat list of candidates at its assigned level, and responds with a single line ( SELECTED: id1, id2, ... ) choosing every candidate the note supports. There is no tool-call protocol, no multi-turn interaction, and no shared context across levels. At inference time the three are executed sequentially, so the chapter specialist’s selections determine which candidates the subsection specialist sees, and in turn which billable codes are presented to the code specialist. All three share a common system prompt (Supplementary Materials, Appendix A).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The candidate lists at the first two levels (L1: chapters, L2: subsections) are modest at both label-space scales, but the third level (L3: billable codes) presents all billable-leaf descendants of selected subsections in one batched prompt. This makes the cascade efficient, requiring three model calls per note, but creates a scaling limitation. Under the all label space, the L3 prompt often exceeds the 32,000-token cap used for a single 46GB GPU. During training, over-cap L3 examples are dropped rather than truncated; during evaluation, the candidate tail is truncated, imposing a hard recall ceiling on 28% of SFT-3 test notes and 32% of SFT-3 + test notes. The shared traversal policy has 0% truncation at the same scale.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train the three specialists on the same 2,000-note SFT split used for SFT-1, with each note re-expressed as three single-turn examples (one per level), yielding 6,000 training examples total.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-E Continuation and RL Arms\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The shared traversal policy and cascade baseline each use 2,000 notes for initial training. To isolate the effects of data, architecture, and training algorithm, we define three continuation arms on the same 3,000-note continuation split.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SFT-1 + warm-starts from SFT-1 and continues supervised fine-tuning on 3,000 additional gold trajectories constructed as in Sections III-D and IV-B .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SFT-3 + warm-starts each SFT-3 specialist and continues supervised fine-tuning on the same continuation split, re-expressed as per-level single-turn examples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GRPO-1 warm-starts from SFT-1 and applies Group Relative Policy Optimization on the same 3,000 notes. The model generates live rollouts against the environment, and each rollout is scored with the terminal reward from Section IV-A . We run GRPO-1 on the frequent label space only; the all variant was omitted because training time scales to approximately 2–3 GPU-days and the frequent result already shows no benefit over supervised continuation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"IV-F Evaluation Protocol\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All arms are evaluated on the same held-out test set (1,000 notes) and validation set (500 notes), shared across both label spaces and all training arms (Section III-C ). Evaluation rollouts are executed through the batched rollout driver (Section IV-B ) under greedy decoding (temperature 0) for determinism and paired comparability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Zero-shot floors. To anchor the fine-tuning results, we evaluate two zero-shot baselines using the base model base model (Qwen3-4B-Instruct-2507 [ 11 ] ) with no task-specific training. The traversal-format base (Base in Table III ) receives the same system prompt, tool schema, and discharge note as SFT-1, scoring near-zero micro-F1 on both label spaces (0.001), reflecting protocol-following failure rather than lack of clinical knowledge. The cascade base (Base-3) evaluates the same untrained model under the single-turn SELECTED: format of SFT-3, clearing the floor meaningfully (0.229 on frequent , 0.058 on all ). The gap between the two floors quantifies the protocol-following overhead that the gold trajectories (Section III-D ) are designed to close.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Metrics. We report micro-averaged precision, recall, and F1 (pooled across all notes and codes, weighting each prediction equally) and macro-averaged precision, recall, and F1 (averaged across the codes present in the gold set, weighting rare and common codes equally). Micro-F1 is the headline metric; macro-F1 surfaces rare-code performance, which dominates the all label space. We additionally report path-F1, a hierarchical variant that awards partial credit for predictions landing in the correct region of the ICD-10-CM hierarchy rather than requiring an exact leaf match. For a code c c , let A ( c ) = { c } ∪ anc ( c ) A(c)=\\\\{c\\\\}\\\\cup\\\\mathrm{anc}(c) denote its augmentation with all ancestors along the path toward the root (the root itself excluded), and extend this to a set of codes by A ( C ) = ⋃ c ∈ C A ( c ) A(C)=\\\\bigcup_{c\\\\in C}A(c) . Writing C ^ i \\\\hat{C}_{i} and C i C_{i} for the predicted and gold codes of note i i , path-F1 replaces exact-match set membership with membership over the augmented sets:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with path-F1 their harmonic mean. Equation 1 gives the micro variant (sums pooled over notes); the macro variant averages the per-code F1 over the augmented gold codes. Because a mispredicted leaf still shares ancestors with the true code, this rewards a prediction that lands on the correct branch over one that is entirely unrelated.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Statistical testing. We use a paired bootstrap to assess differences between arms. Because micro-F1 and macro-F1 are pooled statistics that do not decompose into per-note scores, a paired t-test is not applicable. For each of 5,000 resamples we draw 1,000 test notes with replacement and recompute both arms’ pooled metrics on the identical resampled set. We report the mean difference, 95% confidence interval, and two-sided p-value (values below 1 / 5 , 000 1/5{,}000 reported as p < 0.001 p<0.001 ). We report uncorrected p-values for approximately fifteen planned comparisons. 1 1 1 The large effects (supervised data) survive Bonferroni correction. The small architecture deltas that reach nominal significance ( p = 0.002 p=0.002 – 0.026 0.026 ) would not all survive correction at α = 0.05 / 15 ≈ 0.003 \\\\alpha=0.05/15\\\\approx 0.003 , and we interpret them accordingly.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We additionally report per-level selection accuracy (chapter, subsection, billable code) to diagnose where recall is lost, and for the cascade under the all label space, the fraction of test notes affected by L3 candidate-list truncation (Section IV-D ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"V Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-A Overall Performance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table III reports micro- and macro-averaged precision, recall, and F1 for all arms on the held-out test set ( n = 1 , 000 n{=}1{,}000 ) at both label-space scales. The results are summarized below.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The traversal policy outperforms flat baselines, with the margin growing at scale. Three flat multi-label baselines anchor these results externally. CAML and PLM-ICD both reach competitive micro-F1 on frequent (0.668 and 0.691) but collapse to near-identical scores on all (0.194 and 0.196), with macro-F1 falling to 0.002 for both—indicating that neither convolutional attention nor pretrained transformer encoders can discriminate among 15,761 candidates without structural guidance. Only LAAT survives at scale (0.483 on all ), and SFT-1 trained on just 2,000 trajectories already matches it (0.478 vs. 0.483). SFT-1 + significantly outperforms all three baselines at both scales (Table IV ), with the margin widening sharply on all : + 0.044 +0.044 micro-F1 over LAAT and + 0.157 +0.157 macro-F1, confirming that the traversal formulation’s advantage is concentrated where flat scoring is weakest. Notably, PLM-ICD’s pretrained encoder offers no advantage over CAML’s CNN at the full scale, suggesting that the bottleneck is the flat output layer rather than the representation - with only 2,000 training notes the vast majority of the 15,761 output neurons see fewer than five positive examples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"More supervised data helps, and the effect grows with scale. The + 3 , 000 +3{,}000 -note continuation (SFT-1 + over SFT-1) is the largest gain in the table: + 0.022 +0.022 micro-F1 on frequent ( 0.687 → 0.709 0.687\\\\rightarrow 0.709 ) and + 0.049 +0.049 on all ( 0.478 → 0.527 0.478\\\\rightarrow 0.527 ). The lift roughly doubles from the 50-code to the full label space. SFT-1 + is the best arm in both settings, and the same pattern holds for the cascade (SFT-3 + over SFT-3 by comparable margins). Macro-F1 gains track micro, so the improvement is not confined to high-frequency codes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GRPO does not improve over supervised continuation. GRPO-1 (0.688 micro-F1) ties its SFT-1 warm-start (0.687) and falls below SFT-1 + (0.709) on both micro and macro. Training directly on the coding metric via RL on 3,000 notes offers no benefit over simply continuing supervised fine-tuning on the same data from the same initialization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neither architecture wins consistently. The cascade (SFT-3, SFT-3 + ) lands within roughly one micro-F1 point of the corresponding shared-policy arm (SFT-1, SFT-1 + ) at both scales. On all , SFT-3 + scores 0.513 vs. SFT-1 + ’s 0.527 on micro but 0.260 vs. 0.256 on macro; the deltas flip sign across metrics and variants. Statistical tests follow in Section V-B . Where the cascade does lose is structurally: 28–32% of test notes suffer L3 candidate-list truncation under the all label space (Section IV-D ), a constraint the shared policy never encounters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-B Statistical Comparisons\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table IV reports paired-bootstrap estimates (5,000 resamples) of the difference in pooled micro- and macro-F1 between matched arms, with 95% confidence intervals and two-sided p-values. All external comparisons are significant ( p < 0.001 p<0.001 ). The margins are modest on frequent ( + 0.014 +0.014 to + 0.042 +0.042 micro-F1) but large on all ( + 0.044 +0.044 to + 0.333 +0.333 micro-F1), confirming that the traversal formulation’s advantage grows with label-space scale. On frequent , SFT-1 without continuation data already matches LAAT ( p = 0.08 p=0.08 , n.s.).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Among the internal comparisons, the data effect is the only robust finding: SFT-1 + over SFT-1 is significant on all four tests ( p < 0.001 p<0.001 ), with the lift doubling from frequent to all . GRPO-1 ties SFT-1 (micro p = 0.892 p=0.892 ) and falls significantly below SFT-1 + ( p < 0.001 p<0.001 ). Architecture deltas are small ( ≤ 0.014 \\\\leq 0.014 ), inconsistent in sign, and would not survive Bonferroni correction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"V-C Hierarchical Recall by Depth\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To diagnose where in the hierarchy codes are lost, we compute micro-pooled recall at each depth of the ICD-10-CM tree: at depth d d , each predicted and gold leaf code is collapsed to its depth- d d ancestor, and recall is computed over the resulting sets. 2 2 2 Codes whose paths are shorter than d d do not contribute at that depth, so deeper columns reflect only codes that actually reach that level of the hierarchy. The denominators therefore shrink at deeper levels. Figure 3 plots this for the best shared-policy and cascade arms at both label-space scales.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recall degrades monotonically with depth, from 0.80-0.83 at the chapter level to 0.70 (frequent) and 0.49 (all) at the billable leaf. The frequent → \\\\rightarrow all gap widens at each depth, confirming that rare-code discrimination at the bottom of the tree drives the macro-F1 collapse. The cascade tracks the shared policy at shallow depths but diverges where L3 truncation (32% of test notes) imposes a hard recall ceiling the shared policy never encounters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"VI Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-A The Rare-Code Bottleneck\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The macro-F1 collapse from frequent to all ( 0.651 → 0.256 0.651\\\\rightarrow 0.256 for the best arm) is the starkest result in the study. Micro-F1 also drops ( 0.709 → 0.527 0.709\\\\rightarrow 0.527 ), but the far larger macro gap shows the degradation is concentrated in the long tail of rare codes rather than spread uniformly. The flat baselines degrade far more sharply, confirming that the traversal formulation’s hierarchical decomposition partially insulates against the rare-code problem without solving it. The per-depth analysis (Figure 3 ) corroborates this: chapter-level recall is comparable across scales, but the gap widens at each depth, largest at the billable leaf where rare codes dominate.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-B Supervision Is the Bottleneck\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Of the three variables we isolate, only additional supervised data reliably improves performance (Table IV ). Neither RL nor architectural specialization produces a comparable or even consistent gain. This is worth stating plainly because the natural design impulse points the other way: once a hierarchy is available, it is tempting to assign specialist models to each level and to replace hand-built trajectories with learned rewards. Our factorial study tests both assumptions and finds neither justified at this scale. Our results suggest that, at least for extraction-bound graph-constrained coding at this model scale, the complexity does not pay off. GRPO ties its supervised warm-start and loses to supervised continuation on the same data; the cascade matches the shared policy but never beats it. The bottleneck is not how the model is trained or how the traversal is decomposed, but how many gold demonstrations it sees.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This conclusion is scoped to the regime we test (a single 4B model, one reward design, 2,000–5,000 notes), and RL or specialization may yet help at larger scales or with richer reward shaping. What our study contributes is a controlled baseline: any added machinery should be shown to beat simply adding more supervised data at matched compute.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-C Consolidation Scales Where the Cascade Cannot\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On headline accuracy the shared traversal policy and the three-specialist cascade are statistically indistinguishable at both label-space scales. The consolidation argument therefore rests not on accuracy but on deployment simplicity (one adapter, no inter-stage orchestration) and a structural scaling advantage.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cascade hits a scaling wall at the full code space: its batched L3 prompt forces every billable-leaf descendant of the selected subsections into one context window, overflowing it for 28-32% of test notes and making those codes unselectable regardless of model quality. A larger memory budget would raise this ceiling, but the pressure is structural - the single-shot design’s prompt cost grows with label-space size, whereas the shared policy’s per-turn fan-out distributes the same load across turns (0% truncation).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"VI-D Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All results are from a single 4B-parameter model (Qwen3-4B-Instruct-2507); RL and architectural specialization may behave differently at larger scales or with other model families. The GRPO result is based on a single reward design, one hyperparameter configuration, and one warm-start on the frequent label space only; the full-scale RL cell is argued but not directly measured. All data comes from MIMIC-IV, a single US academic medical center, and the gold labels are billed administrative codes known to be incomplete, so measured precision is a lower bound when the model assigns a clinically valid code that was not billed. The 32,000-token context cap driving the cascade’s 28-32% truncation rate is a memory budget constraint (single 46GB GPU), not an architectural limit of the model. Finally, our training budget of 5,000 notes is small relative to the 122,197 available; the data-scaling claim rests on just two training budgets (2,000 and 5,000 notes) and we do not determine where the lift saturates.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"VII Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We formulated ICD-10-CM code prediction as a single graph-constrained traversal policy that significantly outperforms flat multi-label baselines at both label-space scales (by + 0.044 +0.044 micro-F1 and + 0.157 +0.157 macro-F1 over the strongest (LAAT) on the full 15,761-code space) and conducted a controlled study isolating architecture, training algorithm, and data. Across both scales, only additional supervised data reliably helped, RL offered no benefit over supervised continuation, and one shared policy matched the three-specialist cascade while avoiding its context-window truncation. Practitioners should therefore deploy one model, not three, and invest in annotation rather than architectural or algorithmic complexity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several directions remain open. The data-scaling curve has only two budgets (2,000 and 5,000 notes); larger annotation budgets and targeted sampling of rare-code-rich notes could determine where the supervised lift saturates. Sampling-based decoding (self-consistency, best-of- n n ) and per-code threshold tuning may improve rare-code recall, which the macro-F1 collapse identifies as the primary bottleneck. RL deserves re-examination with richer reward shaping and at larger model scales before a general negative conclusion is warranted. Finally, validation on a second institution and coding system would test whether these findings generalize beyond a single US academic center.\"\n }\n]","isConversational":false}}},{"rowIdx":397,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.11953"},"doi":{"kind":"null"},"version":{"kind":"number","value":2,"string":"2"},"title":{"kind":"string","value":"When Does Reward Teach State? A Hidden-Automaton Instrument and the Group-Language Boundary"},"authors":{"kind":"list like","value":["Jim Allchin"],"string":"[\n \"Jim Allchin\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.11953v2"},"published":{"kind":"timestamp","value":"2026-07-11T00:00:00","string":"2026-07-11T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Many agent tasks are secretly finite-state: onboarding and support workflows, tool-use and API-call sequences, compliance and triage procedures, multi-step form repair. An agent can appear competent during training yet fail when the workflow is longer, noisier, or differently distributed—a symptom of having learned a reward-seeking shortcut rather than the task’s underlying state machine. This is the goal-misgeneralization worry (Langosco et al., 2022 ; Shah et al., 2022 ) : capabilities that transfer, but attached to a proxy objective. Prior demonstrations of it are behavioral and define the “true goal” informally; here the true latent state is an exactly known object we can both normalize against and read out, so the worry becomes measurable."},{"type":"para","text":"A clarifying point about what is hidden. The agent observes the entire emitted symbol prefix, and the latent dfa state is a deterministic function of that prefix. So this is not belief-tracking in the POMDP sense (no irreducible uncertainty to integrate); it is a computation/memory problem—whether the agent maintains a known-computable running function of its fully observed history. That places the testbed alongside the emergent-world-representation line (probing a sequence model for a latent state it was never shown; § 2 ) and connects to the counting/length-generalization literature: the running function is exactly what recurrent nets compute inductively and attention-only models struggle to."},{"type":"para","text":"We turn formal-language recognition into a control problem. Instead of asking whether a model can classify a string, we ask whether an agent can steer a hidden automaton toward an accepting state, and whether in doing so it comes to represent the automaton’s latent state. The key move is that the automaton is fully known to the experimenter: we can (a) compute the optimal achievable return exactly, normalizing raw reward, and (b) read out the true hidden state at every step to probe the agent’s representation. Reward success and latent-state learning therefore become two separately measurable axes."},{"type":"para","text":"Thesis. Reward success and latent-state learning are distinct, separately measurable quantities, and whether they couple is governed by three axes the instrument turns independently (Figure 1 ): (i) optimizer strength; (ii) task structure—above all whether the automaton is a group-language (permutation) automaton, which a credit-assignment analysis and a held-out test establish as a high-precision, one-directional warning signal for the perception gap (a controllable-stabilizer/leverage refinement sharpens this in-sample but does not generalize); and (iii) how much observations reveal about the latent state. Weak RL leaves them decoupled; a stronger optimizer couples them except for the group-language class; and a label-free signal recovers state only as observations become informative. The contribution is the white-box, exactly-probeable instrument that lets us measure where on these axes a task sits rather than conjecture it. The instrument is also self-correcting: a pre-registered control overturned our initial reading in § 5.2 ."},{"type":"para","text":"Contributions. (1) A measurement methodology for “did the agent learn the task state, or a reward shortcut?”: express the task as a hidden FSM, yielding for free an exact oracle for optimal/random/full-control achievable return (hence oracle- normalized reward), the exact latent state at every step for linear probing, and an off-policy supervised capacity upper bound. The recipe applies to any FSM-approximable task; hidden partial-control dfa s are its cleanest instance. (2) A benchmark instance ( PartialControlDFAEnv ) with a taxonomy separating diagnostic automata from trivial ones. (3) An optimizer-strength characterization of reward ↔ \\leftrightarrow state coupling, established by elimination and a pre-registered, multi-seed A2C-vs-PPO control that overturned our first reading. (4) An auxiliary-signal analysis distinguishing a privileged state auxiliary from a label-free observation auxiliary. (5) A pre-training structural warning signal for the perception gap (§ 5.5 ): permutation (group-language) structure , computable from δ \\delta alone, which a credit-assignment analysis links to a starved learning signal (Appendix B ), tying the RL finding to the algebraic theory of automata, and which validates held-out on 153 153 capacity-controlled fresh automata as a one-directional, high-precision indicator (precision 0.86 0.86 ; permutation accounts for 89 89 of 109 109 perception gaps); a controllable-stabilizer/leverage refinement sharpens it in-sample only, and the prescribed fix closes the gap causally (Appendix D ). Two conclusions strictly require the exact probe and could not be drawn from reward + + oracle alone : the perception gap and the planning gap. A preliminary cross-scale port of the full instrument to a no-CoT Qwen2.5-3B/7B LLM agent—on which the structure and optimizer-strength axes appear, though the clean perception gap does not yet—is reported as a feasibility check in Appendix E and is not part of the main claims."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"para","text":"Reward machines and automaton-structured RL expose the automaton behind a (possibly non-Markov) reward and let RL exploit it (Toro Icarte et al., 2018 ) ; a parallel line infers the latent automaton, either to plan with (Toro Icarte et al., 2023 ) or to model history-dependent dynamics as a regular decision process via a finite transducer, with PAC guarantees (Brafman & De Giacomo, 2019 ) . We invert all of these : we hide the automaton and ask whether sparse-reward RL discovers its latent state as a byproduct of maximizing reward, using the known automaton purely as a measurement instrument—and our motivation is the dual risk that the agent instead latches a reward-correlated shortcut, the control analog of causal confusion in imitation (de Haan et al., 2019 ) ."},{"type":"para","text":"Transformers, RNNs and counting. Counting tasks such as parity separate architectures: RNNs count inductively, whereas attention-only encoders ( ac 0 -bounded under hard attention) need positional tricks and tend not to length-generalize (Strobl et al., 2024 ; Chang & Bisk, 2024 ; X. Huang et al., 2025 ) ; this Chomsky-hierarchy view of architecture-vs-task is mapped at scale by Delétang et al. ( 2023 ) and for transformers on regular languages by Bhattamishra et al. ( 2020 ) , with finite-precision RNN counting power characterized by Weiss et al. ( 2018 ) . Closest to our probe, Michalenko et al. ( 2019 ) decode RNN hidden states against minimal-DFA states—the recurrent-net analog of our latent-state probe, but for supervised language recognition rather than sparse-reward control. Most directly, Liu et al. ( 2023 ) show low-depth transformers can simulate any finite automaton, with constant-depth shortcuts for solvable semiautomata via Krohn–Rhodes, yet—crucially for us—these shortcuts are brittle and degrade as supervision gets sparser , while an LSTM stays at 100 % 100\\% on the same group automata. Our testbed reproduces and extends this separation inside an RL control loop with a hidden, sparsely-rewarded state, and gives it a normalized, probeable form. A precision point: the transformer literature draws two boundaries—the length-generalization boundary (roughly non-star-free) and the constant-depth simulation boundary (solvable vs. non-solvable). Our hard tasks (cyclic mod_k counters) are solvable but non-star-free ; we therefore sit on the star-free / length-generalization boundary, where the sparse-supervision regime of Liu et al. ( 2023 ) agrees with us, not on the solvable/non-solvable boundary. We connect this to the classical syntactic-monoid theory (Schützenberger, 1965 ; Krohn & Rhodes, 1965 ) ; star-free is a useful approximation, not exact (transformers length-generalize on some non-star-free languages such as ( a a ) ∗ (aa)^{\\ast} (X. Huang et al., 2025 ) ), consistent with our finding that mod_2 ( ≡ \\equiv last bit) is easy. 1 1 1 Our “controllable stabilizer” (a controllable identity/hold action) is a control-theoretic notion, distinct from the group-theoretic algebraic structure that Y. Huang et al. ( 2025 ) show governs length generalization of chain-of-thought; we use the term in the control sense throughout."},{"type":"para","text":"Emergent world representations and probing. A model trained only to predict the next token can represent the latent state of the generating process—Othello-GPT encodes the board, linearly decodable and causally tied to predictions (Li et al., 2023 ; Nanda et al., 2023 ) , with parallels in chess and program semantics. This is our closest methodological relative; we differ in three ways: those models are trained by supervised next-token prediction on expert play , whereas we study sparse-reward RL under partial control ; we supply an exact oracle and a known finite probe target, so world-model criteria can be checked exactly; and we deliberately drive reward and state apart . We adopt the probing safeguards this literature established—control tasks and selectivity (Hewitt & Liang, 2019 ; Alain & Bengio, 2016 ) —and run an interventional probe."},{"type":"para","text":"State abstraction and auxiliary tasks. Bisimulation/behavioral abstraction and predictive state representations—which, like us, define state through observable consequences rather than a posited hidden variable (Littman et al., 2001 ) —formalize “the state that matters” (Gelada et al., 2019 ; Zhang et al., 2021 ) ; unlike PSRs, our dfa state is known exactly and used for measurement, not learned as the agent’s state. Recurrent model-free RL is a strong POMDP baseline (Ni et al., 2022 ) . Auxiliary self-supervised objectives are standard under sparse reward (Jaderberg et al., 2017 ) . Our auxiliary predicts the exact latent dfa state, so an independent probe can verify it installs the intended representation. Reward hacking benchmarks measure exploit rates behaviorally (Pan et al., 2022 ; Skalse et al., 2022 ) ; we provide a white-box, exactly-probeable substrate where “did it learn the task state” is a measurable quantity, complementary to them."},{"type":"heading","level":1,"text":"3 The Hidden-Automaton Control Testbed"},{"type":"para","text":"Partial-control dfa environment. A task is a dfa M = ( Q , Σ , δ , q 0 , F ) M=(Q,\\Sigma,\\delta,q_{0},F) . An episode of length T T evolves q t + 1 = δ ( q t , a t ) q_{t+1}=\\delta(q_{t},a_{t}) . At each step, with probability p control p_{\\mathrm{control}} the agent chooses the symbol a t a_{t} ; otherwise the environment samples a t a_{t} uniformly. The agent observes the emitted symbol stream, whether the step is controllable, and time remaining—but never the hidden state q t q_{t} . Reward is sparse and terminal: r = 1 r=1 if q T ∈ F q_{T}\\in F else 0 . Full control trivializes most tasks (just emit the accepting word); partial control forces the agent to track the current hidden state and spend limited interventions wisely."},{"type":"para","text":"Exact oracle. Because M M and the control distribution are known, the optimal acceptance probability is computed by backward induction over (state, steps-remaining):"},{"type":"para","text":"We report oracle-normalized success = (empirical) / V ( q 0 , T ) =\\text{(empirical)}/V(q_{0},T) ( 1.0 = 1.0= optimal), comparable across tasks, lengths and control probabilities, plus random and full-control ceilings."},{"type":"heading","level":1,"text":"4 Diagnostic Instruments"},{"type":"para","text":"Linear dfa -state probe (on-policy): fit a linear classifier from the policy’s pre-head features to the true q t q_{t} on the agent’s own rollouts; report the held-out gap over the majority-class baseline, with balanced accuracy and a control-task/selectivity check (Hewitt & Liang, 2019 ) so a positive gap reflects representation, not probe capacity or visitation skew. Interventional probe (causal): edit the activation along the decoded-state direction and check whether the greedy action changes as δ \\delta predicts—the test that distinguishes a representation the policy consults from one merely present. Supervised capacity (off-policy): train each encoder with full supervision to predict q t q_{t} from random play—the architecture’s capacity , decoupled from the optimizer. The on-policy probe conflates “the representation tracks state” with “the agent keeps the automaton in a small set of states”; we therefore report both."},{"type":"heading","level":1,"text":"5 Experiments"},{"type":"heading","level":2,"text":"5.1 The oracle selects diagnostic tasks; the interface, not just architecture, governs capacity"},{"type":"para","text":"The exact oracle (§ 3 ) immediately separates diagnostic from trivial tasks: contains_0010 is non-diagnostic (random reaches optimal), while parity and mod_k have optimal flat in length , so any degradation at longer horizons is pure failure-to-track. For the supervised capacity bound (predict q t q_{t} off-policy), a full-prefix GRU represents counting perfectly and length-generalizes to 4 × 4\\times training length (OOD accuracy 1.00 1.00 ), whereas a window-MLP is window-bounded (chance OOD) and a plain transformer fails to length-generalize parity ( 0.55 0.55 , chance) despite full context—consistent with transformer-counting theory (§ 2 ). Every shortfall below is therefore a learning effect, not a representational one."},{"type":"heading","level":2,"text":"5.2 The reward/state decoupling is a shortcut fixed point: a pre-registered control"},{"type":"para","text":"Under deliberately weak on-policy RL (A2C, i.e. REINFORCE with a value baseline), the probe is at chance for every architecture—including the GRU we proved can represent these tasks—while reward is achievable: a strong reward/state decoupling . It is tempting to conclude “sparse-reward RL cannot install latent state.” We pre-registered a control to test exactly this (Appendix A ): recurrent PPO + + GAE + + advantage normalization vs. A2C on { \\{ window-MLP, GRU } × { \\}\\times\\{ parity, mod_3 } \\} , p control = 0.5 p_{\\mathrm{control}}{=}0.5 , 5 seeds, with a 2 × 2 2\\times 2 (optimizer × \\times credit-scheme) de-confound. The control overturns that claim. PPO raises reward and drags the probe up, most clearly for the GRU (probe gap + 0.01 → + 0.20 +0.01\\!\\to\\!+0.20 on parity, + 0.02 → + 0.21 +0.02\\!\\to\\!+0.21 on mod_3; OOD reward 0.50 → 0.75 0.50\\!\\to\\!0.75 on mod_3). At a fixed budget the 2 × 2 2\\times 2 shows the optimizer (PPO machinery) is the dominant driver—at fixed credit scheme, A2C → \\to PPO moves the probe gap far more than broadcast-vs-discounted does at fixed optimizer, cleanly so on parity (Table 1 ; budget is the other lever, below). Credit is not irrelevant, though: on mod_3 discounting lifts A2C ( + 0.02 → + 0.15 +0.02\\to+0.15 ) and, under PPO, broadcast edges out discounted return-to-go ( + 0.34 +0.34 vs. + 0.21 +0.21 ); the claim is optimizer- dominance , not credit-irrelevance. The surviving, sharper claims: coupling is governed by shortcut-escape —set by optimizer, budget, and structure; even with PPO recovery is partial and high-variance (gaps ∼ + 0.20 \\sim{+}0.20 vs. the supervised-capacity ceiling ≈ + 0.48 {\\approx}{+}0.48 / + 0.60 {+}0.60 —the representational bound, distinct from the lower privileged-auxiliary ceiling reached by the shaping ladder below); and the recurrence advantage surfaces in RL only under PPO (GRU + + PPO length-generalizes mod_3 OOD 0.75 ± .20 0.75{\\pm}.20 ; window-MLP 0.55 ± .02 0.55{\\pm}.02 ; transformer 0.49 ± .04 0.49{\\pm}.04 ). We report this correction as evidence the instrument is self-correcting."},{"type":"para","text":"The chance probe is a reward-shortcut fixed point—escapable or not. A natural worry is that the probe sits at chance only because A2C was under-run. Running A2C for 5 × 5\\times the budget ( 30 30 k iters, 10 10 seeds) shows the answer is task-dependent , and that is the finding. On mod_3 the extra budget escapes the shortcut: reward climbs 0.51 → 0.95 0.51\\!\\to\\!0.95 and the probe rises with it, + 0.02 → + 0.34 +0.02\\!\\to\\!+0.34 (near the + 0.36 +0.36 ceiling)—here the chance probe was under-training. On parity A2C stays locked in: reward is flat at 0.67 0.67 and the probe at chance ( + 0.02 +0.02 ) all the way to 30 30 k—here the reward-shortcut is a stable attractor , a persistent perception gap that more of the same optimizer never escapes. So the coupling is set by whether the agent escapes the reward-shortcut fixed point : a stronger optimizer (PPO) escapes it faster (the 2 × 2 2\\times 2 , Table 1 , is a fixed-budget slice), more budget escapes it on escapable tasks, and task structure governs whether it is escapable at all— parity ’s shortcut is stabilized by exactly the credit starvation that Appendix B derives as a fixed point of the learning dynamics, not an impossibility . Oracle-potential shaping—a separate change to the reward signal , not the optimizer ( F = γ Φ ( q t + 1 ) − Φ ( q t ) F{=}\\gamma\\Phi(q_{t+1}){-}\\Phi(q_{t}) , Φ \\Phi the oracle value, 5 seeds)—escapes it directly, reaching + 0.30 +0.30 on parity and + 0.35 +0.35 on mod_3, the privileged-auxiliary ceiling ( + 0.28 +0.28 / + 0.36 +0.36 ). And on a genuinely hard counter the recovered fraction stays bounded well below the ceiling at a shared budget (§ 5.4 : PPO/ceiling falls to 0.31 0.31 at k = 11 k{=}11 ), the signature of a fixed point rather than a mere budget wall."},{"type":"heading","level":2,"text":"5.3 Auxiliaries, and the observation-informativeness axis"},{"type":"para","text":"Under the competent optimizer (GRU + + PPO, 5 seeds) a privileged state auxiliary (predict true q t q_{t} ; an upper bound, not deployable) is nearly inert at low complexity—because PPO already recovers the state—but its benefit grows exactly where the base learner fails (under weak A2C, and at high complexity, § 5.4 ). A label-free auxiliary (predict the next emitted symbol) is vacuous when emissions are state-independent—it adds nothing beyond the base learner, so the probe reflects sparse PPO alone: near chance ( + 0.03 +0.03 ) where PPO is, and PPO’s own + 0.14 +0.14 on mod_5 at β = 0 \\beta{=}0 (Table 4 )—and it can even degrade reward, because next-symbol prediction then has no gradient toward q t q_{t} . That vacuity is a feature : it isolates “state revealed only through outcome reward.” Adding an observation channel of tunable informativeness β \\beta (a state coloring leaked with probability β \\beta ) turns this into a controlled axis: on mod_5 —where sparse PPO alone struggles—the label-free auxiliary flips from vacuous to fully effective as soon as observations carry any state , saturating already by β = 0.10 \\beta{=}0.10 (probe gap + 0.47 +0.47 , reward 0.93 0.93 , 5 seeds). A deployable label-free signal recovers latent state, and rescues reward, the moment observations carry any state —a sharp threshold, not a graded response."},{"type":"heading","level":2,"text":"5.4 Scaling and the easy/hard boundary"},{"type":"para","text":"Sweeping the mod_k family ( k ∈ { 2 , 3 , 5 , 7 , 9 , 11 } k\\in\\{2,3,5,7,9,11\\} , GRU + + PPO, 5 seeds; supervised capacity perfect at every k k ), the fraction of the achievable latent state that sparse-reward PPO recovers shrinks from ≈ 1 {\\approx}1 at k = 2 , 3 k{=}2,3 to ≈ 0.31 {\\approx}0.31 at k = 11 k{=}11 , while the privileged auxiliary’s reward benefit grows from ≈ 0 {\\approx}0 to + 0.25 +0.25 – 0.43 0.43 (Table 2 ). As a task becomes more stateful, agents trained on sparse outcome reward recover a shrinking fraction of its latent state, while the value of a state-tracking signal grows."},{"type":"para","text":"We then use the instrument as a falsification engine , authoring workflow shapes that each embody a candidate explanation for difficulty. Sparse PPO solves all of: reach-commit API stages; a long-range latch (a bit set once and held—so long-range distance is not the culprit); a threshold quota; and a terminal-invariant reconciliation (additive with a hold action, hence steerable to zero). The acute reward-vs-state gap and large auxiliary benefit appear only in hard-to-steer abstract counters. This falsifies, in turn, three intuitive predictors of difficulty—state count, long-range distance, and terminal-invariance alone—and isolates the real one: terminal-exact accumulation under poorly-controllable (mixing) dynamics ."},{"type":"para","text":"The instrument also separates two failure modes. ends_with_1000 shows a planning gap : the probe is high ( + 0.62 +0.62 ) yet reward is only 0.41 0.41 —the agent represents the state yet does not convert it to reward—categorically different from the perception gap ( mined_ean : reward gap 0.82 0.82 , probe at chance + 0.03 +0.03 ). This is the one place the exact probe is indispensable, and we show it rather than assert it. A reward-only auditor sees only the behavioral signals—the oracle-normalized reward gap and the OOD reward drop—and these are nearly uncorrelated with the recoverable state across the suite (Spearman with the probe gap − 0.29 -0.29 and − 0.25 -0.25 ). Among the nine tasks where the agent is visibly suboptimal (reward gap ≥ 0.25 {\\geq}0.25 ), the behavioral signature is near-identical yet the probe spans 0.03 0.03 ( mined_ean , perception) to 0.65 0.65 ( ends_with_101 , planning): behavior cannot tell a representation that is absent from one that is present-but-unexploited , though the two demand opposite fixes (Appendix C , Fig. 2 ). The probe makes the distinction a measurement."},{"type":"heading","level":2,"text":"5.5 A pre-training structural warning signal for the perception gap"},{"type":"para","text":"The verbal boundary has an exact algebraic form computable from ( δ , p control ) (\\delta,p_{\\mathrm{control}}) before any training —which is what lets the instrument predict rather than merely report. Part 1 (must-accumulate): call M M a permutation automaton if on its reachable states every symbol induces a bijection q ↦ δ ( q , a ) q\\mapsto\\delta(q,a) ; then no input ever merges states, the latent state never forgets, and the task recognizes a group language (non-star-free) (Schützenberger, 1965 ; Krohn & Rhodes, 1965 ) . When some symbol collapses states (synchronizing), old symbols are forgotten and the state is a function of recent context—the regime a recurrent policy tracks locally. This separates the suite parameter-free and explains the k = 2 k{=}2 vs. k ≥ 3 k{\\geq}3 result: mod_2 ≡ \\equiv last bit (synchronizing, easy) while mod_k k ≥ 3 \\texttt{mod\\_k}{}_{k\\geq 3} (odd k k , where doubling is invertible) is a genuine group counter that must accumulate ; no appeal to state count is needed ( parity , 2 states, is hard; no_three_zeros , 4 states, is easy). This permutation property is necessary but not sufficient : it is the must-accumulate half, and a low- k k counter like mod_3 is a permutation automaton yet still learnable because partial control buys enough leverage (Part 2). Part 2 (poorly-controllable): group structure alone is insufficient— workflow_reconcile is group-like yet learnable because it has a controllable hold (an identity transition that freezes the state once steered to accept). A scalar proxy is the control leverage L = ( V opt − V random ) / ( V full − V random ) L=(V_{\\mathrm{opt}}-V_{\\mathrm{random}})/(V_{\\mathrm{full}}-V_{\\mathrm{random}}) , read from the oracle. The precise predictor is the conjunction : group/permutation structure and low control leverage ."},{"type":"para","text":"In-sample, the predictor is exact. Treating it as a δ \\delta -only scorer— s = L s=L for permutation automata, s = + ∞ s={+}\\infty otherwise—against the measured labels (hard ⇔ \\Leftrightarrow privileged-aux reward benefit ≥ 0.20 {\\geq}0.20 ), it perfectly ranks the 15 designed tasks (ROC-AUC 1.00 1.00 ; every hard task L ≤ 0.50 L\\leq 0.50 , every easy one L ≥ 0.519 L\\geq 0.519 ), survives leave-one-out cross-validation 14 / 15 14/15 (lone error workflow_reconcile at the boundary), and within the designed group tasks (mined pair excluded) tracks hardness monotonically (Spearman ( L , aux benefit ) = − 0.83 (L,\\text{aux benefit})=-0.83 ). The mechanism is argued , not merely observed (Appendix B ): for a permutation automaton, evaluated at a state-blind policy such as initialization , the per-step credit-assignment signal decays geometrically in the uncontrolled steps remaining (to zero after one step for parity) unless a controllable stabilizer or a synchronizing/absorbing distinction keeps it Θ ( 1 ) \\Theta(1) —so sparse RL has no Θ ( 1 ) \\Theta(1) gradient with which to begin tracking the state. This conjunction is what the predictor encodes, and the exact oracle confirms the decay empirically."},{"type":"para","text":"Held-out, permutation structure predicts the perception gap. The clean in-sample gap invites the worry that it is fit to a curated family, so we ran the full instrument on 153 fresh random dfa s (stratified to populate both predictor classes; predicted from δ \\delta before any RL; GRU + + PPO, the original 33 33 at 5 5 seeds and 120 120 additional at 3 3 seeds, Appendix D ). The predictor targets the perception gap specifically—latent state that is representable yet not linearly recoverable from the trained agent—so we score it against linear state recoverability directly: a perception gap is linear probe recovery < 0.5 <0.5 when the state is in fact recoverable . (This is the direct form of the in-sample aux-benefit criterion: a large privileged-aux reward benefit means the state was missing; here we read the missing state off the probe directly.) The probe’s train/test split is by episode , not by row, so autocorrelated within-episode activations cannot leak across it; an earlier row-level split gives the same headline (re-scoring moves 2 2 of 153 153 automata and leaves precision at 0.86 0.86 ). We label on recoverability rather than on “reward present” because in this stratified random population the two mostly co-occur—reward is usually depressed for a gap as well (median normalized reward 0.45 0.45 across the 109 109 gaps; only 2 2 clear 0.85 0.85 )—so the held-out claim is specifically state-blindness , while the clean high-reward-yet-no-state case is the one exhibited on the curated suite ( mined_ean , above). The capacity confound is controlled directly: off-policy supervised probes saturate (val and OOD accuracy ≈ 1.0 {\\approx}1.0 ) for the automata, so a low on-policy recovery is a genuine learning failure, not a representational limit (no mined_ean -style artifact). A second validity check controls for control-relevance : an optimal controller need not distinguish dfa states that share an optimal value and optimal-action set at every remaining horizon, so failing to decode the full state need not be a control-relevant failure. Re-scoring recovery against this oracle quotient—which is nontrivial for 28 28 of the 153 153 automata (it merges, e.g., mined_ean ’s 20 20 states into 10 10 )—leaves the classification identical : every one of the 109 109 gaps also fails to recover the coarser control-relevant state (precision 0.86 0.86 , 0 automata reclassify), so these are genuine state-blindness, not agents that quietly learned a minimal control abstraction. The Appendix B direction transfers as a one-directional indicator: when an automaton is permutation it is a perception gap with precision 0.86 0.86 ( 89 / 103 89/103 ), and 89 89 of the 109 109 perception gaps are permutation (recall 0.82 0.82 ) —so group structure is a high-precision warning signal for state-blindness on automata it never saw, not a guarantee of safety for non-group tasks (below). The original 33 33 reproduce exactly ( 19 / 23 19/23 , precision 0.83 0.83 ); the 4.6 × 4.6\\times larger held-out set leaves the precision essentially unchanged, so the signal is not a small-sample artifact (Wilson 95 % 95\\% CI [ 0.78 , 0.92 ] [0.78,0.92] ), and it is insensitive to the recovery threshold ( 0.86 0.86 at rec < 0.5 <0.5 ; 0.84 0.84 and 0.88 0.88 at 0.4 0.4 and 0.6 0.6 ). Two caveats, both measured. First, the converse does not hold: non-permutation automata are heterogeneous (of 50 50 , 20 20 perception / 14 14 planning / 16 16 solved), so structure flags that a perception gap is likely, not that a non-group task is safe. Second, the leverage threshold does not transfer (the conjunction’s recall collapses; the scalar L L over-credits steerability), so we demote L L to a within-group severity proxy (the in-sample Spearman − 0.83 -0.83 ) and drop the two-sided classifier. The finding is not architecture-specific : on the original 33 33 it replicates on an LSTM (precision 0.78 0.78 , recall 0.78 0.78 ; 22 22 of the 24 24 GRU perception gaps are also LSTM perception gaps), with that architecture’s capacity equally saturated. The surviving claim is the well-supported one: permutation/group structure is a high-precision, mechanistically-grounded, held-out-validated predictor of the perception gap."},{"type":"para","text":"What generalizes cleanly is the fix . Even where prediction is imperfect, the predictor’s two named escape routes work as interventions on a known-hard task (Appendix D , Table 4 ): on mod_5 , latchability (adding a controllable hold, still a permutation automaton) lifts reward 0.50 → 0.89 0.50{\\to}0.89 and the probe + 0.17 → + 0.27 +0.17{\\to}+0.27 while cutting seed variance, and observability (the § 5.3 emission channel at β ≥ 0.1 \\beta{\\geq}0.1 ) lifts them to 0.93 0.93 and + 0.47 +0.47 . The audit-and-fix loop—make the latent state latchable or observable —is thus a demonstrated causal intervention needing only the oracle and probe, not a perfect a-priori predictor."},{"type":"heading","level":2,"text":"5.6 Non-synthetic instances mined from real logic and logs"},{"type":"para","text":"To test whether the failure mode arises in structure we did not author, we mine hidden dfa s from real, deployed logic via RPNI (Oncina & García, 1992 ; Angluin, 1987 ) (passive automaton learning) and run the identical instrument. We deliberately target checksum/divisibility code (ubiquitous, self-labeling, and structurally in our hard regime)—so this validates the pipeline and shows the hard structure exists in real code , not that it is common. mined_div7 (binary-value-divisible-by-7, recovered exactly, capacity 1.0 1.0 ) reproduces the full mod_7 gap: sparse PPO is partial ( 0.65 ± .27 0.65{\\pm}.27 , probe + 0.30 +0.30 ) while both the privileged and a deployable label-free ( β = 1 \\beta{=}1 ) auxiliary rescue it to optimal—the entire three-axis story on logic we recovered rather than authored. Conversely, all seven real process logs we mined (van Dongen, 2012 ; 2015 ; 2017 ; 2019 ; Mannhardt, 2017 ; de Leoni & Mannhardt, 2015 ; Mannhardt, 2016 ) (procurement, clinical, two loan-application, building-permit, billing, and fines processes) are non-permutation : five mine to 2-state reach-commit automata and two are decide-late automata, and all are learnable. They lie outside the high-precision perception-gap class (permutation structure), and empirically none exhibits the gap. Real FSM structure thus spans both regimes—real checksums are permutation (perception-hard), real workflows are non-permutation and here learnable—so the perception gap is present in real, deployed logic but is not the common case among the logs we surveyed."},{"type":"heading","level":1,"text":"6 Discussion, Limitations, and Conclusion"},{"type":"para","text":"The testbed converts a vague worry into a measurement. State-learning is gated not by the observation (once the full prefix is provided) nor by capacity (the GRU provably suffices), but by credit assignment and reward design : reward and latent-state learning couple only weakly and unreliably under sparse reward, the coupling strengthens with optimizer quality, and it degrades as the task grows more accumulation-heavy. Where the state is learned it is genuinely used : editing the hidden along the decoded-state direction shifts the greedy action to the δ \\delta -predicted one far more than a norm-matched random direction (across the seven policies where an auxiliary installed the state, 0.32 0.32 – 0.68 0.68 vs. 0.00 0.00 – 0.21 0.21 ). The practical reading is more reassuring than a doom result: any FSM-approximable workflow can be given an exact oracle and a state probe and audited white-box; permutation (group-language) structure is a reliable red flag for the perception gap (held-out precision 0.86 0.86 on 153 153 fresh automata, permutation accounting for 89 89 of 109 109 gaps, capacity controlled, § 5.5 ); and the fix falls out and is causally confirmed (make the latent state observable or latchable and the gap closes). The audit, not a perfect a-priori predictor, is the deployable artifact—and all seven real logs we mined are learnable."},{"type":"para","text":"Towards LLM agents (preliminary). As an external-validity check we ported the whole instrument to a real LLM agent (Qwen2.5-3B/7B) and pre-registered the test (Appendix E ). Two of the three axes reproduce—structure as a representational counting-horizon limit, and optimizer strength as the predicted weak-optimizer outcome—but it does not yet produce a clean LLM-scale perception gap (no optimizer we ran lifts reward above random on a representable group counter within budget, the accumulation difficulty of § 5.4 at scale). We frame this as a preliminary instrumentation check and release the instrument for the stronger-optimizer experiment."},{"type":"para","text":"Limitations. (i) Scale: the synthetic suite is small ( ≤ 11 \\leq 11 states, mostly binary); mined instances reach 20 states / 10-symbol alphabets, and the oracle/probe handle far larger, but larger automata remain future work. (ii) The reward/state decoupling behaves as a learning fixed point , not a fixed property (§ 5.2 ): a stronger optimizer, more budget, or richer credit escapes it (partial and seed-variable), and its persistence is task-dependent—it survives 5 × 5\\times the A2C budget on parity but not on mod_3 . (iii) The structural signal is mechanistically grounded (Appendix B ), but a held-out test on 153 capacity-controlled fresh dfa s (Appendix D ) shows only the one-directional claim generalizes: permutation structure is a high-precision warning signal for the perception gap (precision 0.86 0.86 ; 89 / 109 89/109 gaps permutation), but the converse fails (non-permutation automata are heterogeneous) and the leverage threshold does not transfer (we demote L L to a within-group severity proxy). The GRU result holds across 153 153 fresh automata, and on the original 33 33 replicates on an LSTM, with capacity controlled; a sharper predictor of which permutation automata are hardest, and of the non-group perception gaps, remains open, as does scaling beyond small automata; the causal fix, by contrast, transfers (Appendix D ). (iv) The hard regime is shown on non-synthetic logic, but the clean hard mined instance is a 7-state checksum we targeted ; “real workflows tend to be learnable” rests on seven logs. (v) § 5.3 ’s β = 0 \\beta{=}0 vacuity cell is 3-seed (its β ≥ 0.1 \\beta{\\geq}0.1 saturation cells are 5-seed). A hard instance mined from a richer real source with a saturated capacity bound is our top reach goal."},{"type":"para","text":"Conclusion. Hidden partial-control dfa s are a cheap, exactly-probeable laboratory for stateful-agent reliability. Reward success and latent-state learning are distinct quantities whose coupling is governed by three measurable axes—optimizer strength, task structure (group-language structure as a high-precision, held-out-validated, one-directional warning signal for the perception gap, with a credit-assignment mechanism behind it), and observation-informativeness—and the instrument separates a perception gap from a planning gap. The reusable contribution is the recipe—express a task as a hidden FSM and obtain an exact oracle plus exact latent-state probes—for asking, of any agent, whether it learned the task or only the reward."},{"type":"heading","level":1,"text":"Reproducibility statement"},{"type":"para","text":"The environment, exact oracle, probes, supervised capacity bound, and all training code are released (anonymized). Headline tables (§ 5.2 –§ 5.4 ) are 5-seed with 95% bootstrap CIs; the β \\beta grid is 3-seed at its coarse points ( β ∈ { 0 , 0.25 , 0.5 , 1 } \\beta\\in\\{0,0.25,0.5,1\\} ) and 5-seed at the transition points ( β ∈ { 0.10 , 0.15 , 0.20 } \\beta\\in\\{0.10,0.15,0.20\\} on mod_5 , including the β = 0.10 \\beta{=}0.10 saturation cell); the held-out validation (Appendix D ) is GRU over 153 153 fresh automata (the original 33 33 at 5 5 seeds, 120 120 additional at 3 3 seeds), with an LSTM cross-check on the original 33 33 (3-seed) and a per-automaton supervised-capacity control. Every linear state probe in the paper uses a train/test split by episode (no autocorrelated within-episode rows leak across the split); the released probe outputs also record the earlier row-level split, which shifts individual probe gaps by ≤ 0.02 {\\leq}0.02 (within their CIs) and leaves the held-out precision at 0.86 0.86 . The pre-registration of the § 5.2 control is in Appendix A . Full per-task grids, the mined-automaton pipeline, and selectivity analyses ship as released code and run logs. The LLM-agent instrument (NL environment, residual-stream probe, frozen and supervised capacity bounds, RL loop) and its pre-registration (Appendix E ) are released. The structural predictor, its in-sample ROC/leave-one-out evaluation, and its predictions over the random- dfa family are reproduced by a standalone, training-free script ( predictor_eval.py : δ ↦ \\delta\\mapsto hard/easy); the held-out scores (§ 5.5 , Appendix D ) are recomputed from the released run trees by analyze_randval.py ; the credit-assignment quantities of Appendix B are computed exactly from the oracle."},{"type":"heading","level":1,"text":"Use of large language models"},{"type":"para","text":"An LLM-based assistant helped implement the code, run the experimental pipeline, and draft parts of the text, working at every stage under the authors’ direction. The authors reviewed and verified the methodology, the results, and the writing, and accept full responsibility for all claims and content in this paper."},{"type":"heading","level":1,"text":"A Pre-registration of the optimizer-vs-reward-channel control (§ 5.2 )"},{"type":"para","text":"The § 5.2 correction rests on a control we fixed before running it, recorded here so the reversal reads as principled rather than post-hoc. Under weak on-policy RL the probe sat at chance for every architecture; we registered two competing explanations and a design to separate them."},{"type":"para","text":"Registered hypotheses. H opt (optimizer): the chance probe is an artifact of a weak optimizer; a stronger optimizer (recurrent PPO + + GAE + + advantage normalization) will raise the probe and reward together. H signal (reward channel): the chance probe is intrinsic to sparse terminal reward; the credit-assignment scheme (broadcast return vs. discounted reward-to-go), not the optimizer machinery, is what matters."},{"type":"para","text":"Registered design (fixed before seeing PPO results). A 2 × 2 2\\times 2 grid, optimizer { \\{ A2C, PPO } \\} × \\times credit-scheme { \\{ broadcast ( γ = 1 \\gamma{=}1 ), discounted-RTG } \\} , crossed with { \\{ window-MLP, GRU } × { \\}\\times\\{ parity , mod_3 } \\} , at p control = 0.5 p_{\\mathrm{control}}{=}0.5 , 5 seeds, identical network sizes, horizon, and evaluation protocol across cells; the GRU’s off-policy supervised capacity ( = 1.0 {=}1.0 ) was established first, so any cell’s shortfall is a learning, not a representational, effect."},{"type":"para","text":"Registered decision rule. Support H opt if the PPO cells raise the probe gap substantially over A2C holding the credit scheme fixed ; support H signal if the discounted-RTG cells do so holding the optimizer fixed . Outcome (Table 1 ): the optimizer axis moved the probe (A2C → {\\to} PPO at fixed credit scheme), the credit-scheme axis was secondary—supporting H opt and forcing us to overturn the too-strong reading that sparse-reward RL cannot install latent state. Recovery is nonetheless partial and seed-variable, the surviving (weaker) claim in § 5.2 ."},{"type":"heading","level":1,"text":"B Why permutation automata starve credit assignment"},{"type":"para","text":"This appendix gives the mechanism behind the § 5.5 boundary as two propositions under explicit assumptions , and confirms them empirically with the exact oracle; it is a mechanistic account, not a claim that sparse-reward RL provably cannot learn these tasks. Assumptions. An episode has horizon T T ; at each step the action is the agent’s with probability c = p control c=p_{\\mathrm{control}} and a uniform random symbol otherwise; the reward is the sparse terminal indicator R = 𝟏 [ q T ∈ F ] R=\\mathbf{1}[q_{T}\\in F] ; we use the broadcast ( γ = 1 \\gamma{=}1 ) estimator ∇ θ J = 𝔼 [ R ∑ t ∈ C ∇ θ log π θ ( a t ∣ h t ) ] \\nabla_{\\theta}J=\\mathbb{E}\\big[R\\sum_{t\\in C}\\nabla_{\\theta}\\log\\pi_{\\theta}(a_{t}\\mid h_{t})\\big] , where C C is the set of controllable steps. Crucially the policy π θ ( a t ∣ h t ) \\pi_{\\theta}(a_{t}\\mid h_{t}) conditions on the observed prefix h t h_{t} only (it cannot see future symbols). Proposition 1 holds for any such policy; Proposition 2 additionally assumes the policy is state-blind after t t —its controllable actions at steps u > t u>t do not depend on the random symbols drawn at the intervening uncontrolled steps. This holds at initialization and for any policy that does not yet track the latent state, which is exactly the regime in which learning must begin ; we show below that relaxing it requires the state-tracking the sparse signal has to install."},{"type":"para","text":"Define the state-influence of step t t as the spread of the eventual acceptance probability over the latent state just after the step,"},{"type":"para","text":"where the probability is over the (random and controllable) symbols of the remaining s s steps under the rollout policy. Δ t \\Delta_{t} upper-bounds how much any decision at step t t can change the return."},{"type":"para","text":"Proposition 1 (credit is bounded by state-influence). The contribution of step t t to the policy gradient obeys ∥ 𝔼 [ R ∇ θ log π θ ( a t ∣ h t ) ] ∥ ≤ Δ t ⋅ 𝔼 ∥ ∇ θ log π θ ( a t ∣ h t ) ∥ . \\big\\|\\mathbb{E}[\\,R\\,\\nabla_{\\theta}\\log\\pi_{\\theta}(a_{t}\\mid h_{t})\\,]\\big\\|\\leq\\Delta_{t}\\cdot\\mathbb{E}\\,\\|\\nabla_{\\theta}\\log\\pi_{\\theta}(a_{t}\\mid h_{t})\\|. In particular, if the acceptance probability does not depend on the symbol chosen at step t t , that step contributes exactly zero."},{"type":"para","text":"Proof. Condition on h t h_{t} . By the score-function identity 𝔼 a t [ ∇ θ log π θ ( a t ∣ h t ) g ( h t ) ] = 0 \\mathbb{E}_{a_{t}}[\\nabla_{\\theta}\\log\\pi_{\\theta}(a_{t}\\mid h_{t})\\,g(h_{t})]=0 for any g g not depending on a t a_{t} . Writing Pr [ R = 1 ∣ h t , a t ] = ρ ¯ ( h t ) + ( Pr [ R = 1 ∣ h t , a t ] − ρ ¯ ( h t ) ) \\Pr[R{=}1\\mid h_{t},a_{t}]=\\bar{\\rho}(h_{t})+\\big(\\Pr[R{=}1\\mid h_{t},a_{t}]-\\bar{\\rho}(h_{t})\\big) for any baseline ρ ¯ ( h t ) \\bar{\\rho}(h_{t}) , the first term integrates to zero and the residual is bounded in absolute value by the spread of Pr [ R = 1 ∣ h t , a t ] \\Pr[R{=}1\\mid h_{t},a_{t}] over a t a_{t} , which is at most Δ t \\Delta_{t} since each action only moves q t q_{t} within Q Q . Cauchy–Schwarz gives the stated bound. □ \\square"},{"type":"para","text":"Proposition 2 (geometric credit decay at a state-blind policy). Let M M be a permutation automaton and let P P be the transition operator induced by a uniform random symbol, with Dobrushin contraction coefficient λ < 1 \\lambda<1 on the reachable orbit (equivalently a spectral gap 1 − λ > 0 1-\\lambda>0 ). Under the state-blind assumption above, if at least k k of the s s steps after t t are uncontrolled, then Δ t ≤ C λ k \\Delta_{t}\\leq C\\,\\lambda^{k} for a constant C C depending only on M M . For parity ( ℤ 2 \\mathbb{Z}_{2} , λ = 0 \\lambda=0 ) a single uncontrolled step gives Δ t = 0 \\Delta_{t}=0 exactly."},{"type":"para","text":"Proof sketch. Under the state-blind assumption the action distribution at each controllable step u > t u>t is independent of the random symbols drawn after t t , so ρ s \\rho_{s} propagates backwards by alternately applying the random-symbol operator P P (uncontrolled steps) and a fixed action-mixture operator P π P_{\\pi} (controllable steps). Both are convex combinations of permutation matrices, hence doubly stochastic and non-expansive in the span seminorm; P P contracts the span by its Dobrushin coefficient λ \\lambda per application. After k k uncontrolled steps the span—and therefore Δ t \\Delta_{t} —is at most λ k \\lambda^{k} times its value at step T T . For parity, one uniform symbol makes q T q_{T} uniform regardless of q t q_{t} , so Δ t = 0 \\Delta_{t}=0 . □ \\square"},{"type":"para","text":"Why state-blindness, and why it cannot simply be dropped. For an arbitrary history-dependent policy the bound can fail: a policy that records an observed random symbol b b and replays it at a later controllable step cancels its effect ( q T = q t ⊕ a t ⊕ b ⊕ b = q t ⊕ a t q_{T}=q_{t}\\oplus a_{t}\\oplus b\\oplus b=q_{t}\\oplus a_{t} for parity), keeping Δ t = Θ ( 1 ) \\Delta_{t}=\\Theta(1) . But that cancellation requires the agent to already track and reproduce the latent symbol history—the very state-tracking the sparse reward must first install. At a state-blind policy (initialization) the assumption holds, so by Proposition 1 the intermediate-step credit is O ( λ ( 1 − c ) ( T − t ) ) O(\\lambda^{(1-c)(T-t)}) and there is no Θ ( 1 ) \\Theta(1) gradient with which to acquire that skill. The barrier is thus a fixed point of the learning dynamics , not an information-theoretic impossibility—consistent with these tasks becoming learnable once a controllable stabilizer or a warm start supplies the missing signal."},{"type":"para","text":"Consequences. Each subsequent step is uncontrolled independently with probability 1 − c 1-c , so in expectation step t t has Θ ( ( 1 − c ) s ) \\Theta((1-c)s) uncontrolled steps after it and its credit is O ( λ ( 1 − c ) ( T − t ) ) O(\\lambda^{(1-c)(T-t)}) —exponentially small in the steps-to-go for every intermediate step, with Θ ( 1 ) \\Theta(1) credit surviving only in the last few steps. A permutation automaton therefore receives no Θ ( 1 ) \\Theta(1) signal to represent or use the latent state at intermediate times ; the only escapes are (i) a controllable stabilizer —a hold action that removes that step from the mixing dynamics, letting the agent freeze the state and keep Δ t = Θ ( 1 ) \\Delta_{t}=\\Theta(1) (this is exactly workflow_reconcile , which is a permutation automaton yet learnable)—or (ii) a synchronizing/absorbing distinction, where P P is not mixing and the span never contracts to zero (this is no_three_zeros , workflow_api ). This is the § 5.5 predictor, derived rather than observed."},{"type":"para","text":"Empirical confirmation. Computing Δ t \\Delta_{t} exactly from the oracle under a uniform (state-blind) rollout—the regime of Proposition 2—(spread of ρ s \\rho_{s} over latent state vs. number of intervening random steps) matches the theory to the digit: parity collapses to Δ = 0 \\Delta=0 after a single random step; mod_k decays geometrically ( Δ ≈ 2 − r \\Delta\\approx 2^{-r} : 0.50 , 0.25 , 0.125 , 0.031 , … 0.50,0.25,0.125,0.031,\\dots after r = 1 , 2 , 3 , 5 r=1,2,3,5 ); workflow_api stays at Δ ≡ 1.0 \\Delta\\equiv 1.0 (absorbing accept/reject) and no_three_zeros stays high ( Δ ≈ 0.30 \\Delta\\approx 0.30 out to 16 16 steps); workflow_reconcile decays like a counter under random play but its hold action lets the policy avoid the decay. The vanishing-credit regime and the predictor’s hard class coincide exactly."},{"type":"heading","level":1,"text":"C Behavioral diagnostics cannot localize the failure"},{"type":"para","text":"Figure 2 substantiates the § 5.4 claim that an exact instrument sees what a behavior-only diagnostic cannot. Plotting every RL-evaluated task by its reward gap ( 1 − 1- val n , the strongest purely behavioral signal) against its probe gap, tasks at the same—large—reward gap span the entire probe range. A reward-only auditor would group mined_ean (perception gap, probe + 0.03 +0.03 ) with ends_with_1000 (planning gap, probe + 0.62 +0.62 ) as “the agent is failing,” yet they are opposite failures with opposite fixes. Across the suite the reward gap and OOD reward drop correlate only weakly with the probe (Spearman − 0.29 -0.29 , − 0.25 -0.25 ), confirming that no behavioral summary recovers the perception/planning distinction the probe makes directly."},{"type":"heading","level":1,"text":"D Held-out validation of the predictor, and the causal fix"},{"type":"para","text":"Fresh-DFA validation. To test whether the § 5.5 predictor generalizes beyond the curated suite, we generated fresh random dfa s ( 3 3 – 7 7 states, 2 2 – 3 3 symbols), oracle-filtered to be diagnostic, and stratified to populate both predictor classes (the predicted-hard class—permutation with low leverage—is ≈ 1 % {\\approx}1\\% of uniform-random diagnostic dfa s, so a natural sample would need thousands). We report 153 153 such automata: an initial 33 33 ( 5 5 seeds) and a 120 120 -automaton extension ( 3 3 seeds) generated from a separate seed, 4.6 × 4.6\\times larger so the held-out estimate is not small-sample. The predictor was computed from δ \\delta alone before any RL; we then ran the full instrument (GRU + + PPO, 6000 6000 iters). We control capacity directly : an off-policy supervised probe saturates (val and OOD accuracy ≈ 1.0 {\\approx}1.0 ) on the automata, so a low on-policy probe is a genuine perception gap (state recoverable but not recovered), not a representational limit. We therefore label each automaton perception -hard if probe recovery < 0.5 <0.5 , planning -hard if the state is recovered but reward is suboptimal (val < n 0.85 {}_{n}<0.85 ), else solved (Table 3 ). The Props 1–2 direction transfers: of the 103 103 permutation automata, 89 89 are perception gaps (precision 0.86 0.86 ), and 89 89 of all 109 109 perception gaps are permutation (recall 0.82 0.82 ) —permutation structure is a high-precision, one-directional signature of the perception gap on automata it never saw. The original 33 33 reproduce exactly ( 19 / 23 19/23 , precision 0.83 0.83 ); the larger set leaves precision essentially unchanged. Two caveats. The converse fails : non-permutation automata are heterogeneous ( 20 20 perception, 14 14 planning, 16 16 solved of 50 50 ), so group structure flags that state-blindness is likely, not that a non-group task is safe. We could not isolate a clean structural property shared by these non-group perception gaps (diameter, minimum image size, and mixing rate do not separate them), and notably they are largely not explained by the credit-vanishing mechanism (their acceptance-influence Δ \\Delta persists rather than decaying)—a mix of reward shortcuts (high reward, state unrepresented) and optimization failures. A structural account of perception gaps outside the group class is the one genuine open hole in the predictor story. And the leverage threshold does not transfer (the conjunction’s recall collapses, the scalar L L over-crediting steerability—it waves through most of the easy_perm stratum, which are predominantly perception gaps), so we demote L L to a within-group severity proxy. (On the original 33 33 , the result replicates on an LSTM—precision 0.78 0.78 /recall 0.78 0.78 , 22 / 24 22/24 GRU perception gaps shared, capacity equally saturated—so it is not a recurrence-cell artifact. A sharper predictor of which permutation automata are hardest, and of the non-group perception gaps, is open.)"},{"type":"para","text":"The causal fix transfers. Where prediction is imperfect, the prescribed design levers still work as interventions (Table 4 ). On mod_5 (predicted and measured hard), adding a controllable hold ( latchability : mod_5_hold , which raises L L from 0.433 0.433 to 0.523 0.523 and remains a permutation automaton) nearly doubles reward and roughly doubles recovered state, while sharply cutting seed variance; supplying an informative emission channel ( observability , the § 5.3 β \\beta lever) recovers the state and reward to near-optimal. Both are pre-registered as the predictor’s two named escape routes, and both close the gap."},{"type":"heading","level":1,"text":"E An LLM-agent instrument: a preliminary feasibility port"},{"type":"para","text":"The paper’s motivation is stateful agent reliability, so we ported the entire instrument to a real LLM agent and pre-registered the test (hypotheses and decision thresholds accompany the release). We render each step of PartialControlDFAEnv as natural-language tool I/O with the alphabet relabeled to arbitrary tokens (so a pretrained model cannot pattern-match “this is mod- k k ”) and the dfa rule kept out of the prompt; the agent emits one action token per controllable step with no chain of thought (so an activation probe is not undercut by state offloaded into emitted reasoning). The oracle transfers unchanged; we probe the residual stream at the action token with the same linear probe, selectivity control, and capacity bounds. We ran this on Qwen2.5-3B/-7B-Instruct on a single GPU (released code). The structure and optimizer-strength axes appear on a real LLM; the clean perception gap does not yet —this is a feasibility demonstration that the instrument transfers, not a scaling result."},{"type":"para","text":"Axis (ii), structure, replicates as a representational limit. A frozen pretrained model does not linearly encode the latent state (probe at chance for 3B and 7B on mod_5 , gap 0.03 0.03 ); a privileged supervised fine-tune (LoRA to predict q t q_{t} , the LLM analog of our capacity bound) installs it, but only at short counting horizons (Table 5 ): parity capacity decays 0.84 → 0.63 → 0.51 0.84\\to 0.63\\to 0.51 as the horizon grows L = 8 → 12 → 16 L{=}8\\to 12\\to 16 , while at L = 8 L{=}8 even mod_5 is representable ( 0.73 0.73 ). The limit is the length of the count , not the state-count—the star-free/length-generalization boundary of § 2 (§ 5.2 ’s transformer result), now as a representational limit on a 3–7B transformer."},{"type":"para","text":"Axis (i), optimizer strength, replicates—and the weak-optimizer outcome was predicted . On a representable task ( mod_5 @ L 8 @L8 , capacity 0.73 0.73 , oracle random 0.20 0.20 /optimal 0.55 0.55 ), our REINFORCE-level loop (group-normalized return, no value baseline or GAE) does not raise reward and destabilizes (reward drifts below random as the policy leaves the base model). This is not a surprise but a consistency check : it is exactly the synthetic weak-optimizer (A2C) regime of § 5.2 , which also failed to install state and which the stronger PPO + + GAE partially repaired. The reason we hit it first is compute, not design—a PPO + + GAE optimizer for an LLM (a value head and GAE over the agent’s decisions) is heavier than the group-baseline loop. We therefore also implemented that optimizer—a scalar value head, GAE over the agent’s ordered decisions, and a clipped surrogate with multiple epochs—and confirmed it works: on a reach-commit control ( workflow_api ) it raises reward stably , where the REINFORCE loop stagnated. On mod_5 @ L 8 @L8 , however, it still does not learn: conservative settings hold reward at the random baseline and aggressive settings destabilize it, with the state installed in neither case (frozen and post-PPO probe at chance, gap 0.04 0.04 ; reward ≈ \\approx random, 0.39 0.39 normalized; capacity 0.73 0.73 ). This mirrors the synthetic finding : synthetically PPO recovers reach-commit/local tasks but only partially recovers accumulation counters, and there only with ∼ 30 × {\\sim}30\\times the updates; at LLM scale the competent optimizer likewise learns the local task and not the accumulation counter within budget. The optimizer-strength ordering (REINFORCE degrades < < PPO stable-but-no-traction) and the accumulation-is-hard axis (§ 5.4 ) both reproduce."},{"type":"para","text":"Scope of the one thing we do not get. A single clean perception gap at LLM scale—reward clearly above random with the probe at chance and capacity saturated—never materializes, because no optimizer we ran lifts reward above random on a representable group counter within budget: the very accumulation credit-assignment difficulty the paper isolates blocks it at scale too. The blocker is thus optimization (the hardness of accumulation under sparse reward), not representation, so the open corner is a stronger / longer-trained LLM RL optimizer, not a larger model. We report this straight, and release the instrument (NL environment, exact oracle, residual-stream probe with selectivity, frozen and supervised capacity bounds, and both RL loops) so that corner can be filled directly."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Many agent tasks are secretly finite-state: onboarding and support workflows, tool-use and API-call sequences, compliance and triage procedures, multi-step form repair. An agent can appear competent during training yet fail when the workflow is longer, noisier, or differently distributed—a symptom of having learned a reward-seeking shortcut rather than the task’s underlying state machine. This is the goal-misgeneralization worry (Langosco et al., 2022 ; Shah et al., 2022 ) : capabilities that transfer, but attached to a proxy objective. Prior demonstrations of it are behavioral and define the “true goal” informally; here the true latent state is an exactly known object we can both normalize against and read out, so the worry becomes measurable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A clarifying point about what is hidden. The agent observes the entire emitted symbol prefix, and the latent dfa state is a deterministic function of that prefix. So this is not belief-tracking in the POMDP sense (no irreducible uncertainty to integrate); it is a computation/memory problem—whether the agent maintains a known-computable running function of its fully observed history. That places the testbed alongside the emergent-world-representation line (probing a sequence model for a latent state it was never shown; § 2 ) and connects to the counting/length-generalization literature: the running function is exactly what recurrent nets compute inductively and attention-only models struggle to.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We turn formal-language recognition into a control problem. Instead of asking whether a model can classify a string, we ask whether an agent can steer a hidden automaton toward an accepting state, and whether in doing so it comes to represent the automaton’s latent state. The key move is that the automaton is fully known to the experimenter: we can (a) compute the optimal achievable return exactly, normalizing raw reward, and (b) read out the true hidden state at every step to probe the agent’s representation. Reward success and latent-state learning therefore become two separately measurable axes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thesis. Reward success and latent-state learning are distinct, separately measurable quantities, and whether they couple is governed by three axes the instrument turns independently (Figure 1 ): (i) optimizer strength; (ii) task structure—above all whether the automaton is a group-language (permutation) automaton, which a credit-assignment analysis and a held-out test establish as a high-precision, one-directional warning signal for the perception gap (a controllable-stabilizer/leverage refinement sharpens this in-sample but does not generalize); and (iii) how much observations reveal about the latent state. Weak RL leaves them decoupled; a stronger optimizer couples them except for the group-language class; and a label-free signal recovers state only as observations become informative. The contribution is the white-box, exactly-probeable instrument that lets us measure where on these axes a task sits rather than conjecture it. The instrument is also self-correcting: a pre-registered control overturned our initial reading in § 5.2 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Contributions. (1) A measurement methodology for “did the agent learn the task state, or a reward shortcut?”: express the task as a hidden FSM, yielding for free an exact oracle for optimal/random/full-control achievable return (hence oracle- normalized reward), the exact latent state at every step for linear probing, and an off-policy supervised capacity upper bound. The recipe applies to any FSM-approximable task; hidden partial-control dfa s are its cleanest instance. (2) A benchmark instance ( PartialControlDFAEnv ) with a taxonomy separating diagnostic automata from trivial ones. (3) An optimizer-strength characterization of reward ↔ \\\\leftrightarrow state coupling, established by elimination and a pre-registered, multi-seed A2C-vs-PPO control that overturned our first reading. (4) An auxiliary-signal analysis distinguishing a privileged state auxiliary from a label-free observation auxiliary. (5) A pre-training structural warning signal for the perception gap (§ 5.5 ): permutation (group-language) structure , computable from δ \\\\delta alone, which a credit-assignment analysis links to a starved learning signal (Appendix B ), tying the RL finding to the algebraic theory of automata, and which validates held-out on 153 153 capacity-controlled fresh automata as a one-directional, high-precision indicator (precision 0.86 0.86 ; permutation accounts for 89 89 of 109 109 perception gaps); a controllable-stabilizer/leverage refinement sharpens it in-sample only, and the prescribed fix closes the gap causally (Appendix D ). Two conclusions strictly require the exact probe and could not be drawn from reward + + oracle alone : the perception gap and the planning gap. A preliminary cross-scale port of the full instrument to a no-CoT Qwen2.5-3B/7B LLM agent—on which the structure and optimizer-strength axes appear, though the clean perception gap does not yet—is reported as a feasibility check in Appendix E and is not part of the main claims.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reward machines and automaton-structured RL expose the automaton behind a (possibly non-Markov) reward and let RL exploit it (Toro Icarte et al., 2018 ) ; a parallel line infers the latent automaton, either to plan with (Toro Icarte et al., 2023 ) or to model history-dependent dynamics as a regular decision process via a finite transducer, with PAC guarantees (Brafman & De Giacomo, 2019 ) . We invert all of these : we hide the automaton and ask whether sparse-reward RL discovers its latent state as a byproduct of maximizing reward, using the known automaton purely as a measurement instrument—and our motivation is the dual risk that the agent instead latches a reward-correlated shortcut, the control analog of causal confusion in imitation (de Haan et al., 2019 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Transformers, RNNs and counting. Counting tasks such as parity separate architectures: RNNs count inductively, whereas attention-only encoders ( ac 0 -bounded under hard attention) need positional tricks and tend not to length-generalize (Strobl et al., 2024 ; Chang & Bisk, 2024 ; X. Huang et al., 2025 ) ; this Chomsky-hierarchy view of architecture-vs-task is mapped at scale by Delétang et al. ( 2023 ) and for transformers on regular languages by Bhattamishra et al. ( 2020 ) , with finite-precision RNN counting power characterized by Weiss et al. ( 2018 ) . Closest to our probe, Michalenko et al. ( 2019 ) decode RNN hidden states against minimal-DFA states—the recurrent-net analog of our latent-state probe, but for supervised language recognition rather than sparse-reward control. Most directly, Liu et al. ( 2023 ) show low-depth transformers can simulate any finite automaton, with constant-depth shortcuts for solvable semiautomata via Krohn–Rhodes, yet—crucially for us—these shortcuts are brittle and degrade as supervision gets sparser , while an LSTM stays at 100 % 100\\\\% on the same group automata. Our testbed reproduces and extends this separation inside an RL control loop with a hidden, sparsely-rewarded state, and gives it a normalized, probeable form. A precision point: the transformer literature draws two boundaries—the length-generalization boundary (roughly non-star-free) and the constant-depth simulation boundary (solvable vs. non-solvable). Our hard tasks (cyclic mod_k counters) are solvable but non-star-free ; we therefore sit on the star-free / length-generalization boundary, where the sparse-supervision regime of Liu et al. ( 2023 ) agrees with us, not on the solvable/non-solvable boundary. We connect this to the classical syntactic-monoid theory (Schützenberger, 1965 ; Krohn & Rhodes, 1965 ) ; star-free is a useful approximation, not exact (transformers length-generalize on some non-star-free languages such as ( a a ) ∗ (aa)^{\\\\ast} (X. Huang et al., 2025 ) ), consistent with our finding that mod_2 ( ≡ \\\\equiv last bit) is easy. 1 1 1 Our “controllable stabilizer” (a controllable identity/hold action) is a control-theoretic notion, distinct from the group-theoretic algebraic structure that Y. Huang et al. ( 2025 ) show governs length generalization of chain-of-thought; we use the term in the control sense throughout.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Emergent world representations and probing. A model trained only to predict the next token can represent the latent state of the generating process—Othello-GPT encodes the board, linearly decodable and causally tied to predictions (Li et al., 2023 ; Nanda et al., 2023 ) , with parallels in chess and program semantics. This is our closest methodological relative; we differ in three ways: those models are trained by supervised next-token prediction on expert play , whereas we study sparse-reward RL under partial control ; we supply an exact oracle and a known finite probe target, so world-model criteria can be checked exactly; and we deliberately drive reward and state apart . We adopt the probing safeguards this literature established—control tasks and selectivity (Hewitt & Liang, 2019 ; Alain & Bengio, 2016 ) —and run an interventional probe.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"State abstraction and auxiliary tasks. Bisimulation/behavioral abstraction and predictive state representations—which, like us, define state through observable consequences rather than a posited hidden variable (Littman et al., 2001 ) —formalize “the state that matters” (Gelada et al., 2019 ; Zhang et al., 2021 ) ; unlike PSRs, our dfa state is known exactly and used for measurement, not learned as the agent’s state. Recurrent model-free RL is a strong POMDP baseline (Ni et al., 2022 ) . Auxiliary self-supervised objectives are standard under sparse reward (Jaderberg et al., 2017 ) . Our auxiliary predicts the exact latent dfa state, so an independent probe can verify it installs the intended representation. Reward hacking benchmarks measure exploit rates behaviorally (Pan et al., 2022 ; Skalse et al., 2022 ) ; we provide a white-box, exactly-probeable substrate where “did it learn the task state” is a measurable quantity, complementary to them.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The Hidden-Automaton Control Testbed\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Partial-control dfa environment. A task is a dfa M = ( Q , Σ , δ , q 0 , F ) M=(Q,\\\\Sigma,\\\\delta,q_{0},F) . An episode of length T T evolves q t + 1 = δ ( q t , a t ) q_{t+1}=\\\\delta(q_{t},a_{t}) . At each step, with probability p control p_{\\\\mathrm{control}} the agent chooses the symbol a t a_{t} ; otherwise the environment samples a t a_{t} uniformly. The agent observes the emitted symbol stream, whether the step is controllable, and time remaining—but never the hidden state q t q_{t} . Reward is sparse and terminal: r = 1 r=1 if q T ∈ F q_{T}\\\\in F else 0 . Full control trivializes most tasks (just emit the accepting word); partial control forces the agent to track the current hidden state and spend limited interventions wisely.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Exact oracle. Because M M and the control distribution are known, the optimal acceptance probability is computed by backward induction over (state, steps-remaining):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We report oracle-normalized success = (empirical) / V ( q 0 , T ) =\\\\text{(empirical)}/V(q_{0},T) ( 1.0 = 1.0= optimal), comparable across tasks, lengths and control probabilities, plus random and full-control ceilings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Diagnostic Instruments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Linear dfa -state probe (on-policy): fit a linear classifier from the policy’s pre-head features to the true q t q_{t} on the agent’s own rollouts; report the held-out gap over the majority-class baseline, with balanced accuracy and a control-task/selectivity check (Hewitt & Liang, 2019 ) so a positive gap reflects representation, not probe capacity or visitation skew. Interventional probe (causal): edit the activation along the decoded-state direction and check whether the greedy action changes as δ \\\\delta predicts—the test that distinguishes a representation the policy consults from one merely present. Supervised capacity (off-policy): train each encoder with full supervision to predict q t q_{t} from random play—the architecture’s capacity , decoupled from the optimizer. The on-policy probe conflates “the representation tracks state” with “the agent keeps the automaton in a small set of states”; we therefore report both.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 The oracle selects diagnostic tasks; the interface, not just architecture, governs capacity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The exact oracle (§ 3 ) immediately separates diagnostic from trivial tasks: contains_0010 is non-diagnostic (random reaches optimal), while parity and mod_k have optimal flat in length , so any degradation at longer horizons is pure failure-to-track. For the supervised capacity bound (predict q t q_{t} off-policy), a full-prefix GRU represents counting perfectly and length-generalizes to 4 × 4\\\\times training length (OOD accuracy 1.00 1.00 ), whereas a window-MLP is window-bounded (chance OOD) and a plain transformer fails to length-generalize parity ( 0.55 0.55 , chance) despite full context—consistent with transformer-counting theory (§ 2 ). Every shortfall below is therefore a learning effect, not a representational one.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 The reward/state decoupling is a shortcut fixed point: a pre-registered control\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under deliberately weak on-policy RL (A2C, i.e. REINFORCE with a value baseline), the probe is at chance for every architecture—including the GRU we proved can represent these tasks—while reward is achievable: a strong reward/state decoupling . It is tempting to conclude “sparse-reward RL cannot install latent state.” We pre-registered a control to test exactly this (Appendix A ): recurrent PPO + + GAE + + advantage normalization vs. A2C on { \\\\{ window-MLP, GRU } × { \\\\}\\\\times\\\\{ parity, mod_3 } \\\\} , p control = 0.5 p_{\\\\mathrm{control}}{=}0.5 , 5 seeds, with a 2 × 2 2\\\\times 2 (optimizer × \\\\times credit-scheme) de-confound. The control overturns that claim. PPO raises reward and drags the probe up, most clearly for the GRU (probe gap + 0.01 → + 0.20 +0.01\\\\!\\\\to\\\\!+0.20 on parity, + 0.02 → + 0.21 +0.02\\\\!\\\\to\\\\!+0.21 on mod_3; OOD reward 0.50 → 0.75 0.50\\\\!\\\\to\\\\!0.75 on mod_3). At a fixed budget the 2 × 2 2\\\\times 2 shows the optimizer (PPO machinery) is the dominant driver—at fixed credit scheme, A2C → \\\\to PPO moves the probe gap far more than broadcast-vs-discounted does at fixed optimizer, cleanly so on parity (Table 1 ; budget is the other lever, below). Credit is not irrelevant, though: on mod_3 discounting lifts A2C ( + 0.02 → + 0.15 +0.02\\\\to+0.15 ) and, under PPO, broadcast edges out discounted return-to-go ( + 0.34 +0.34 vs. + 0.21 +0.21 ); the claim is optimizer- dominance , not credit-irrelevance. The surviving, sharper claims: coupling is governed by shortcut-escape —set by optimizer, budget, and structure; even with PPO recovery is partial and high-variance (gaps ∼ + 0.20 \\\\sim{+}0.20 vs. the supervised-capacity ceiling ≈ + 0.48 {\\\\approx}{+}0.48 / + 0.60 {+}0.60 —the representational bound, distinct from the lower privileged-auxiliary ceiling reached by the shaping ladder below); and the recurrence advantage surfaces in RL only under PPO (GRU + + PPO length-generalizes mod_3 OOD 0.75 ± .20 0.75{\\\\pm}.20 ; window-MLP 0.55 ± .02 0.55{\\\\pm}.02 ; transformer 0.49 ± .04 0.49{\\\\pm}.04 ). We report this correction as evidence the instrument is self-correcting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The chance probe is a reward-shortcut fixed point—escapable or not. A natural worry is that the probe sits at chance only because A2C was under-run. Running A2C for 5 × 5\\\\times the budget ( 30 30 k iters, 10 10 seeds) shows the answer is task-dependent , and that is the finding. On mod_3 the extra budget escapes the shortcut: reward climbs 0.51 → 0.95 0.51\\\\!\\\\to\\\\!0.95 and the probe rises with it, + 0.02 → + 0.34 +0.02\\\\!\\\\to\\\\!+0.34 (near the + 0.36 +0.36 ceiling)—here the chance probe was under-training. On parity A2C stays locked in: reward is flat at 0.67 0.67 and the probe at chance ( + 0.02 +0.02 ) all the way to 30 30 k—here the reward-shortcut is a stable attractor , a persistent perception gap that more of the same optimizer never escapes. So the coupling is set by whether the agent escapes the reward-shortcut fixed point : a stronger optimizer (PPO) escapes it faster (the 2 × 2 2\\\\times 2 , Table 1 , is a fixed-budget slice), more budget escapes it on escapable tasks, and task structure governs whether it is escapable at all— parity ’s shortcut is stabilized by exactly the credit starvation that Appendix B derives as a fixed point of the learning dynamics, not an impossibility . Oracle-potential shaping—a separate change to the reward signal , not the optimizer ( F = γ Φ ( q t + 1 ) − Φ ( q t ) F{=}\\\\gamma\\\\Phi(q_{t+1}){-}\\\\Phi(q_{t}) , Φ \\\\Phi the oracle value, 5 seeds)—escapes it directly, reaching + 0.30 +0.30 on parity and + 0.35 +0.35 on mod_3, the privileged-auxiliary ceiling ( + 0.28 +0.28 / + 0.36 +0.36 ). And on a genuinely hard counter the recovered fraction stays bounded well below the ceiling at a shared budget (§ 5.4 : PPO/ceiling falls to 0.31 0.31 at k = 11 k{=}11 ), the signature of a fixed point rather than a mere budget wall.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Auxiliaries, and the observation-informativeness axis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under the competent optimizer (GRU + + PPO, 5 seeds) a privileged state auxiliary (predict true q t q_{t} ; an upper bound, not deployable) is nearly inert at low complexity—because PPO already recovers the state—but its benefit grows exactly where the base learner fails (under weak A2C, and at high complexity, § 5.4 ). A label-free auxiliary (predict the next emitted symbol) is vacuous when emissions are state-independent—it adds nothing beyond the base learner, so the probe reflects sparse PPO alone: near chance ( + 0.03 +0.03 ) where PPO is, and PPO’s own + 0.14 +0.14 on mod_5 at β = 0 \\\\beta{=}0 (Table 4 )—and it can even degrade reward, because next-symbol prediction then has no gradient toward q t q_{t} . That vacuity is a feature : it isolates “state revealed only through outcome reward.” Adding an observation channel of tunable informativeness β \\\\beta (a state coloring leaked with probability β \\\\beta ) turns this into a controlled axis: on mod_5 —where sparse PPO alone struggles—the label-free auxiliary flips from vacuous to fully effective as soon as observations carry any state , saturating already by β = 0.10 \\\\beta{=}0.10 (probe gap + 0.47 +0.47 , reward 0.93 0.93 , 5 seeds). A deployable label-free signal recovers latent state, and rescues reward, the moment observations carry any state —a sharp threshold, not a graded response.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Scaling and the easy/hard boundary\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sweeping the mod_k family ( k ∈ { 2 , 3 , 5 , 7 , 9 , 11 } k\\\\in\\\\{2,3,5,7,9,11\\\\} , GRU + + PPO, 5 seeds; supervised capacity perfect at every k k ), the fraction of the achievable latent state that sparse-reward PPO recovers shrinks from ≈ 1 {\\\\approx}1 at k = 2 , 3 k{=}2,3 to ≈ 0.31 {\\\\approx}0.31 at k = 11 k{=}11 , while the privileged auxiliary’s reward benefit grows from ≈ 0 {\\\\approx}0 to + 0.25 +0.25 – 0.43 0.43 (Table 2 ). As a task becomes more stateful, agents trained on sparse outcome reward recover a shrinking fraction of its latent state, while the value of a state-tracking signal grows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We then use the instrument as a falsification engine , authoring workflow shapes that each embody a candidate explanation for difficulty. Sparse PPO solves all of: reach-commit API stages; a long-range latch (a bit set once and held—so long-range distance is not the culprit); a threshold quota; and a terminal-invariant reconciliation (additive with a hold action, hence steerable to zero). The acute reward-vs-state gap and large auxiliary benefit appear only in hard-to-steer abstract counters. This falsifies, in turn, three intuitive predictors of difficulty—state count, long-range distance, and terminal-invariance alone—and isolates the real one: terminal-exact accumulation under poorly-controllable (mixing) dynamics .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The instrument also separates two failure modes. ends_with_1000 shows a planning gap : the probe is high ( + 0.62 +0.62 ) yet reward is only 0.41 0.41 —the agent represents the state yet does not convert it to reward—categorically different from the perception gap ( mined_ean : reward gap 0.82 0.82 , probe at chance + 0.03 +0.03 ). This is the one place the exact probe is indispensable, and we show it rather than assert it. A reward-only auditor sees only the behavioral signals—the oracle-normalized reward gap and the OOD reward drop—and these are nearly uncorrelated with the recoverable state across the suite (Spearman with the probe gap − 0.29 -0.29 and − 0.25 -0.25 ). Among the nine tasks where the agent is visibly suboptimal (reward gap ≥ 0.25 {\\\\geq}0.25 ), the behavioral signature is near-identical yet the probe spans 0.03 0.03 ( mined_ean , perception) to 0.65 0.65 ( ends_with_101 , planning): behavior cannot tell a representation that is absent from one that is present-but-unexploited , though the two demand opposite fixes (Appendix C , Fig. 2 ). The probe makes the distinction a measurement.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.5 A pre-training structural warning signal for the perception gap\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The verbal boundary has an exact algebraic form computable from ( δ , p control ) (\\\\delta,p_{\\\\mathrm{control}}) before any training —which is what lets the instrument predict rather than merely report. Part 1 (must-accumulate): call M M a permutation automaton if on its reachable states every symbol induces a bijection q ↦ δ ( q , a ) q\\\\mapsto\\\\delta(q,a) ; then no input ever merges states, the latent state never forgets, and the task recognizes a group language (non-star-free) (Schützenberger, 1965 ; Krohn & Rhodes, 1965 ) . When some symbol collapses states (synchronizing), old symbols are forgotten and the state is a function of recent context—the regime a recurrent policy tracks locally. This separates the suite parameter-free and explains the k = 2 k{=}2 vs. k ≥ 3 k{\\\\geq}3 result: mod_2 ≡ \\\\equiv last bit (synchronizing, easy) while mod_k k ≥ 3 \\\\texttt{mod\\\\_k}{}_{k\\\\geq 3} (odd k k , where doubling is invertible) is a genuine group counter that must accumulate ; no appeal to state count is needed ( parity , 2 states, is hard; no_three_zeros , 4 states, is easy). This permutation property is necessary but not sufficient : it is the must-accumulate half, and a low- k k counter like mod_3 is a permutation automaton yet still learnable because partial control buys enough leverage (Part 2). Part 2 (poorly-controllable): group structure alone is insufficient— workflow_reconcile is group-like yet learnable because it has a controllable hold (an identity transition that freezes the state once steered to accept). A scalar proxy is the control leverage L = ( V opt − V random ) / ( V full − V random ) L=(V_{\\\\mathrm{opt}}-V_{\\\\mathrm{random}})/(V_{\\\\mathrm{full}}-V_{\\\\mathrm{random}}) , read from the oracle. The precise predictor is the conjunction : group/permutation structure and low control leverage .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In-sample, the predictor is exact. Treating it as a δ \\\\delta -only scorer— s = L s=L for permutation automata, s = + ∞ s={+}\\\\infty otherwise—against the measured labels (hard ⇔ \\\\Leftrightarrow privileged-aux reward benefit ≥ 0.20 {\\\\geq}0.20 ), it perfectly ranks the 15 designed tasks (ROC-AUC 1.00 1.00 ; every hard task L ≤ 0.50 L\\\\leq 0.50 , every easy one L ≥ 0.519 L\\\\geq 0.519 ), survives leave-one-out cross-validation 14 / 15 14/15 (lone error workflow_reconcile at the boundary), and within the designed group tasks (mined pair excluded) tracks hardness monotonically (Spearman ( L , aux benefit ) = − 0.83 (L,\\\\text{aux benefit})=-0.83 ). The mechanism is argued , not merely observed (Appendix B ): for a permutation automaton, evaluated at a state-blind policy such as initialization , the per-step credit-assignment signal decays geometrically in the uncontrolled steps remaining (to zero after one step for parity) unless a controllable stabilizer or a synchronizing/absorbing distinction keeps it Θ ( 1 ) \\\\Theta(1) —so sparse RL has no Θ ( 1 ) \\\\Theta(1) gradient with which to begin tracking the state. This conjunction is what the predictor encodes, and the exact oracle confirms the decay empirically.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Held-out, permutation structure predicts the perception gap. The clean in-sample gap invites the worry that it is fit to a curated family, so we ran the full instrument on 153 fresh random dfa s (stratified to populate both predictor classes; predicted from δ \\\\delta before any RL; GRU + + PPO, the original 33 33 at 5 5 seeds and 120 120 additional at 3 3 seeds, Appendix D ). The predictor targets the perception gap specifically—latent state that is representable yet not linearly recoverable from the trained agent—so we score it against linear state recoverability directly: a perception gap is linear probe recovery < 0.5 <0.5 when the state is in fact recoverable . (This is the direct form of the in-sample aux-benefit criterion: a large privileged-aux reward benefit means the state was missing; here we read the missing state off the probe directly.) The probe’s train/test split is by episode , not by row, so autocorrelated within-episode activations cannot leak across it; an earlier row-level split gives the same headline (re-scoring moves 2 2 of 153 153 automata and leaves precision at 0.86 0.86 ). We label on recoverability rather than on “reward present” because in this stratified random population the two mostly co-occur—reward is usually depressed for a gap as well (median normalized reward 0.45 0.45 across the 109 109 gaps; only 2 2 clear 0.85 0.85 )—so the held-out claim is specifically state-blindness , while the clean high-reward-yet-no-state case is the one exhibited on the curated suite ( mined_ean , above). The capacity confound is controlled directly: off-policy supervised probes saturate (val and OOD accuracy ≈ 1.0 {\\\\approx}1.0 ) for the automata, so a low on-policy recovery is a genuine learning failure, not a representational limit (no mined_ean -style artifact). A second validity check controls for control-relevance : an optimal controller need not distinguish dfa states that share an optimal value and optimal-action set at every remaining horizon, so failing to decode the full state need not be a control-relevant failure. Re-scoring recovery against this oracle quotient—which is nontrivial for 28 28 of the 153 153 automata (it merges, e.g., mined_ean ’s 20 20 states into 10 10 )—leaves the classification identical : every one of the 109 109 gaps also fails to recover the coarser control-relevant state (precision 0.86 0.86 , 0 automata reclassify), so these are genuine state-blindness, not agents that quietly learned a minimal control abstraction. The Appendix B direction transfers as a one-directional indicator: when an automaton is permutation it is a perception gap with precision 0.86 0.86 ( 89 / 103 89/103 ), and 89 89 of the 109 109 perception gaps are permutation (recall 0.82 0.82 ) —so group structure is a high-precision warning signal for state-blindness on automata it never saw, not a guarantee of safety for non-group tasks (below). The original 33 33 reproduce exactly ( 19 / 23 19/23 , precision 0.83 0.83 ); the 4.6 × 4.6\\\\times larger held-out set leaves the precision essentially unchanged, so the signal is not a small-sample artifact (Wilson 95 % 95\\\\% CI [ 0.78 , 0.92 ] [0.78,0.92] ), and it is insensitive to the recovery threshold ( 0.86 0.86 at rec < 0.5 <0.5 ; 0.84 0.84 and 0.88 0.88 at 0.4 0.4 and 0.6 0.6 ). Two caveats, both measured. First, the converse does not hold: non-permutation automata are heterogeneous (of 50 50 , 20 20 perception / 14 14 planning / 16 16 solved), so structure flags that a perception gap is likely, not that a non-group task is safe. Second, the leverage threshold does not transfer (the conjunction’s recall collapses; the scalar L L over-credits steerability), so we demote L L to a within-group severity proxy (the in-sample Spearman − 0.83 -0.83 ) and drop the two-sided classifier. The finding is not architecture-specific : on the original 33 33 it replicates on an LSTM (precision 0.78 0.78 , recall 0.78 0.78 ; 22 22 of the 24 24 GRU perception gaps are also LSTM perception gaps), with that architecture’s capacity equally saturated. The surviving claim is the well-supported one: permutation/group structure is a high-precision, mechanistically-grounded, held-out-validated predictor of the perception gap.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"What generalizes cleanly is the fix . Even where prediction is imperfect, the predictor’s two named escape routes work as interventions on a known-hard task (Appendix D , Table 4 ): on mod_5 , latchability (adding a controllable hold, still a permutation automaton) lifts reward 0.50 → 0.89 0.50{\\\\to}0.89 and the probe + 0.17 → + 0.27 +0.17{\\\\to}+0.27 while cutting seed variance, and observability (the § 5.3 emission channel at β ≥ 0.1 \\\\beta{\\\\geq}0.1 ) lifts them to 0.93 0.93 and + 0.47 +0.47 . The audit-and-fix loop—make the latent state latchable or observable —is thus a demonstrated causal intervention needing only the oracle and probe, not a perfect a-priori predictor.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.6 Non-synthetic instances mined from real logic and logs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test whether the failure mode arises in structure we did not author, we mine hidden dfa s from real, deployed logic via RPNI (Oncina & García, 1992 ; Angluin, 1987 ) (passive automaton learning) and run the identical instrument. We deliberately target checksum/divisibility code (ubiquitous, self-labeling, and structurally in our hard regime)—so this validates the pipeline and shows the hard structure exists in real code , not that it is common. mined_div7 (binary-value-divisible-by-7, recovered exactly, capacity 1.0 1.0 ) reproduces the full mod_7 gap: sparse PPO is partial ( 0.65 ± .27 0.65{\\\\pm}.27 , probe + 0.30 +0.30 ) while both the privileged and a deployable label-free ( β = 1 \\\\beta{=}1 ) auxiliary rescue it to optimal—the entire three-axis story on logic we recovered rather than authored. Conversely, all seven real process logs we mined (van Dongen, 2012 ; 2015 ; 2017 ; 2019 ; Mannhardt, 2017 ; de Leoni & Mannhardt, 2015 ; Mannhardt, 2016 ) (procurement, clinical, two loan-application, building-permit, billing, and fines processes) are non-permutation : five mine to 2-state reach-commit automata and two are decide-late automata, and all are learnable. They lie outside the high-precision perception-gap class (permutation structure), and empirically none exhibits the gap. Real FSM structure thus spans both regimes—real checksums are permutation (perception-hard), real workflows are non-permutation and here learnable—so the perception gap is present in real, deployed logic but is not the common case among the logs we surveyed.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Discussion, Limitations, and Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The testbed converts a vague worry into a measurement. State-learning is gated not by the observation (once the full prefix is provided) nor by capacity (the GRU provably suffices), but by credit assignment and reward design : reward and latent-state learning couple only weakly and unreliably under sparse reward, the coupling strengthens with optimizer quality, and it degrades as the task grows more accumulation-heavy. Where the state is learned it is genuinely used : editing the hidden along the decoded-state direction shifts the greedy action to the δ \\\\delta -predicted one far more than a norm-matched random direction (across the seven policies where an auxiliary installed the state, 0.32 0.32 – 0.68 0.68 vs. 0.00 0.00 – 0.21 0.21 ). The practical reading is more reassuring than a doom result: any FSM-approximable workflow can be given an exact oracle and a state probe and audited white-box; permutation (group-language) structure is a reliable red flag for the perception gap (held-out precision 0.86 0.86 on 153 153 fresh automata, permutation accounting for 89 89 of 109 109 gaps, capacity controlled, § 5.5 ); and the fix falls out and is causally confirmed (make the latent state observable or latchable and the gap closes). The audit, not a perfect a-priori predictor, is the deployable artifact—and all seven real logs we mined are learnable.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Towards LLM agents (preliminary). As an external-validity check we ported the whole instrument to a real LLM agent (Qwen2.5-3B/7B) and pre-registered the test (Appendix E ). Two of the three axes reproduce—structure as a representational counting-horizon limit, and optimizer strength as the predicted weak-optimizer outcome—but it does not yet produce a clean LLM-scale perception gap (no optimizer we ran lifts reward above random on a representable group counter within budget, the accumulation difficulty of § 5.4 at scale). We frame this as a preliminary instrumentation check and release the instrument for the stronger-optimizer experiment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Limitations. (i) Scale: the synthetic suite is small ( ≤ 11 \\\\leq 11 states, mostly binary); mined instances reach 20 states / 10-symbol alphabets, and the oracle/probe handle far larger, but larger automata remain future work. (ii) The reward/state decoupling behaves as a learning fixed point , not a fixed property (§ 5.2 ): a stronger optimizer, more budget, or richer credit escapes it (partial and seed-variable), and its persistence is task-dependent—it survives 5 × 5\\\\times the A2C budget on parity but not on mod_3 . (iii) The structural signal is mechanistically grounded (Appendix B ), but a held-out test on 153 capacity-controlled fresh dfa s (Appendix D ) shows only the one-directional claim generalizes: permutation structure is a high-precision warning signal for the perception gap (precision 0.86 0.86 ; 89 / 109 89/109 gaps permutation), but the converse fails (non-permutation automata are heterogeneous) and the leverage threshold does not transfer (we demote L L to a within-group severity proxy). The GRU result holds across 153 153 fresh automata, and on the original 33 33 replicates on an LSTM, with capacity controlled; a sharper predictor of which permutation automata are hardest, and of the non-group perception gaps, remains open, as does scaling beyond small automata; the causal fix, by contrast, transfers (Appendix D ). (iv) The hard regime is shown on non-synthetic logic, but the clean hard mined instance is a 7-state checksum we targeted ; “real workflows tend to be learnable” rests on seven logs. (v) § 5.3 ’s β = 0 \\\\beta{=}0 vacuity cell is 3-seed (its β ≥ 0.1 \\\\beta{\\\\geq}0.1 saturation cells are 5-seed). A hard instance mined from a richer real source with a saturated capacity bound is our top reach goal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Conclusion. Hidden partial-control dfa s are a cheap, exactly-probeable laboratory for stateful-agent reliability. Reward success and latent-state learning are distinct quantities whose coupling is governed by three measurable axes—optimizer strength, task structure (group-language structure as a high-precision, held-out-validated, one-directional warning signal for the perception gap, with a credit-assignment mechanism behind it), and observation-informativeness—and the instrument separates a perception gap from a planning gap. The reusable contribution is the recipe—express a task as a hidden FSM and obtain an exact oracle plus exact latent-state probes—for asking, of any agent, whether it learned the task or only the reward.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Reproducibility statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The environment, exact oracle, probes, supervised capacity bound, and all training code are released (anonymized). Headline tables (§ 5.2 –§ 5.4 ) are 5-seed with 95% bootstrap CIs; the β \\\\beta grid is 3-seed at its coarse points ( β ∈ { 0 , 0.25 , 0.5 , 1 } \\\\beta\\\\in\\\\{0,0.25,0.5,1\\\\} ) and 5-seed at the transition points ( β ∈ { 0.10 , 0.15 , 0.20 } \\\\beta\\\\in\\\\{0.10,0.15,0.20\\\\} on mod_5 , including the β = 0.10 \\\\beta{=}0.10 saturation cell); the held-out validation (Appendix D ) is GRU over 153 153 fresh automata (the original 33 33 at 5 5 seeds, 120 120 additional at 3 3 seeds), with an LSTM cross-check on the original 33 33 (3-seed) and a per-automaton supervised-capacity control. Every linear state probe in the paper uses a train/test split by episode (no autocorrelated within-episode rows leak across the split); the released probe outputs also record the earlier row-level split, which shifts individual probe gaps by ≤ 0.02 {\\\\leq}0.02 (within their CIs) and leaves the held-out precision at 0.86 0.86 . The pre-registration of the § 5.2 control is in Appendix A . Full per-task grids, the mined-automaton pipeline, and selectivity analyses ship as released code and run logs. The LLM-agent instrument (NL environment, residual-stream probe, frozen and supervised capacity bounds, RL loop) and its pre-registration (Appendix E ) are released. The structural predictor, its in-sample ROC/leave-one-out evaluation, and its predictions over the random- dfa family are reproduced by a standalone, training-free script ( predictor_eval.py : δ ↦ \\\\delta\\\\mapsto hard/easy); the held-out scores (§ 5.5 , Appendix D ) are recomputed from the released run trees by analyze_randval.py ; the credit-assignment quantities of Appendix B are computed exactly from the oracle.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Use of large language models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An LLM-based assistant helped implement the code, run the experimental pipeline, and draft parts of the text, working at every stage under the authors’ direction. The authors reviewed and verified the methodology, the results, and the writing, and accept full responsibility for all claims and content in this paper.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"A Pre-registration of the optimizer-vs-reward-channel control (§ 5.2 )\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The § 5.2 correction rests on a control we fixed before running it, recorded here so the reversal reads as principled rather than post-hoc. Under weak on-policy RL the probe sat at chance for every architecture; we registered two competing explanations and a design to separate them.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Registered hypotheses. H opt (optimizer): the chance probe is an artifact of a weak optimizer; a stronger optimizer (recurrent PPO + + GAE + + advantage normalization) will raise the probe and reward together. H signal (reward channel): the chance probe is intrinsic to sparse terminal reward; the credit-assignment scheme (broadcast return vs. discounted reward-to-go), not the optimizer machinery, is what matters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Registered design (fixed before seeing PPO results). A 2 × 2 2\\\\times 2 grid, optimizer { \\\\{ A2C, PPO } \\\\} × \\\\times credit-scheme { \\\\{ broadcast ( γ = 1 \\\\gamma{=}1 ), discounted-RTG } \\\\} , crossed with { \\\\{ window-MLP, GRU } × { \\\\}\\\\times\\\\{ parity , mod_3 } \\\\} , at p control = 0.5 p_{\\\\mathrm{control}}{=}0.5 , 5 seeds, identical network sizes, horizon, and evaluation protocol across cells; the GRU’s off-policy supervised capacity ( = 1.0 {=}1.0 ) was established first, so any cell’s shortfall is a learning, not a representational, effect.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Registered decision rule. Support H opt if the PPO cells raise the probe gap substantially over A2C holding the credit scheme fixed ; support H signal if the discounted-RTG cells do so holding the optimizer fixed . Outcome (Table 1 ): the optimizer axis moved the probe (A2C → {\\\\to} PPO at fixed credit scheme), the credit-scheme axis was secondary—supporting H opt and forcing us to overturn the too-strong reading that sparse-reward RL cannot install latent state. Recovery is nonetheless partial and seed-variable, the surviving (weaker) claim in § 5.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"B Why permutation automata starve credit assignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This appendix gives the mechanism behind the § 5.5 boundary as two propositions under explicit assumptions , and confirms them empirically with the exact oracle; it is a mechanistic account, not a claim that sparse-reward RL provably cannot learn these tasks. Assumptions. An episode has horizon T T ; at each step the action is the agent’s with probability c = p control c=p_{\\\\mathrm{control}} and a uniform random symbol otherwise; the reward is the sparse terminal indicator R = 𝟏 [ q T ∈ F ] R=\\\\mathbf{1}[q_{T}\\\\in F] ; we use the broadcast ( γ = 1 \\\\gamma{=}1 ) estimator ∇ θ J = 𝔼 [ R ∑ t ∈ C ∇ θ log π θ ( a t ∣ h t ) ] \\\\nabla_{\\\\theta}J=\\\\mathbb{E}\\\\big[R\\\\sum_{t\\\\in C}\\\\nabla_{\\\\theta}\\\\log\\\\pi_{\\\\theta}(a_{t}\\\\mid h_{t})\\\\big] , where C C is the set of controllable steps. Crucially the policy π θ ( a t ∣ h t ) \\\\pi_{\\\\theta}(a_{t}\\\\mid h_{t}) conditions on the observed prefix h t h_{t} only (it cannot see future symbols). Proposition 1 holds for any such policy; Proposition 2 additionally assumes the policy is state-blind after t t —its controllable actions at steps u > t u>t do not depend on the random symbols drawn at the intervening uncontrolled steps. This holds at initialization and for any policy that does not yet track the latent state, which is exactly the regime in which learning must begin ; we show below that relaxing it requires the state-tracking the sparse signal has to install.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Define the state-influence of step t t as the spread of the eventual acceptance probability over the latent state just after the step,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the probability is over the (random and controllable) symbols of the remaining s s steps under the rollout policy. Δ t \\\\Delta_{t} upper-bounds how much any decision at step t t can change the return.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proposition 1 (credit is bounded by state-influence). The contribution of step t t to the policy gradient obeys ∥ 𝔼 [ R ∇ θ log π θ ( a t ∣ h t ) ] ∥ ≤ Δ t ⋅ 𝔼 ∥ ∇ θ log π θ ( a t ∣ h t ) ∥ . \\\\big\\\\|\\\\mathbb{E}[\\\\,R\\\\,\\\\nabla_{\\\\theta}\\\\log\\\\pi_{\\\\theta}(a_{t}\\\\mid h_{t})\\\\,]\\\\big\\\\|\\\\leq\\\\Delta_{t}\\\\cdot\\\\mathbb{E}\\\\,\\\\|\\\\nabla_{\\\\theta}\\\\log\\\\pi_{\\\\theta}(a_{t}\\\\mid h_{t})\\\\|. In particular, if the acceptance probability does not depend on the symbol chosen at step t t , that step contributes exactly zero.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proof. Condition on h t h_{t} . By the score-function identity 𝔼 a t [ ∇ θ log π θ ( a t ∣ h t ) g ( h t ) ] = 0 \\\\mathbb{E}_{a_{t}}[\\\\nabla_{\\\\theta}\\\\log\\\\pi_{\\\\theta}(a_{t}\\\\mid h_{t})\\\\,g(h_{t})]=0 for any g g not depending on a t a_{t} . Writing Pr [ R = 1 ∣ h t , a t ] = ρ ¯ ( h t ) + ( Pr [ R = 1 ∣ h t , a t ] − ρ ¯ ( h t ) ) \\\\Pr[R{=}1\\\\mid h_{t},a_{t}]=\\\\bar{\\\\rho}(h_{t})+\\\\big(\\\\Pr[R{=}1\\\\mid h_{t},a_{t}]-\\\\bar{\\\\rho}(h_{t})\\\\big) for any baseline ρ ¯ ( h t ) \\\\bar{\\\\rho}(h_{t}) , the first term integrates to zero and the residual is bounded in absolute value by the spread of Pr [ R = 1 ∣ h t , a t ] \\\\Pr[R{=}1\\\\mid h_{t},a_{t}] over a t a_{t} , which is at most Δ t \\\\Delta_{t} since each action only moves q t q_{t} within Q Q . Cauchy–Schwarz gives the stated bound. □ \\\\square\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proposition 2 (geometric credit decay at a state-blind policy). Let M M be a permutation automaton and let P P be the transition operator induced by a uniform random symbol, with Dobrushin contraction coefficient λ < 1 \\\\lambda<1 on the reachable orbit (equivalently a spectral gap 1 − λ > 0 1-\\\\lambda>0 ). Under the state-blind assumption above, if at least k k of the s s steps after t t are uncontrolled, then Δ t ≤ C λ k \\\\Delta_{t}\\\\leq C\\\\,\\\\lambda^{k} for a constant C C depending only on M M . For parity ( ℤ 2 \\\\mathbb{Z}_{2} , λ = 0 \\\\lambda=0 ) a single uncontrolled step gives Δ t = 0 \\\\Delta_{t}=0 exactly.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proof sketch. Under the state-blind assumption the action distribution at each controllable step u > t u>t is independent of the random symbols drawn after t t , so ρ s \\\\rho_{s} propagates backwards by alternately applying the random-symbol operator P P (uncontrolled steps) and a fixed action-mixture operator P π P_{\\\\pi} (controllable steps). Both are convex combinations of permutation matrices, hence doubly stochastic and non-expansive in the span seminorm; P P contracts the span by its Dobrushin coefficient λ \\\\lambda per application. After k k uncontrolled steps the span—and therefore Δ t \\\\Delta_{t} —is at most λ k \\\\lambda^{k} times its value at step T T . For parity, one uniform symbol makes q T q_{T} uniform regardless of q t q_{t} , so Δ t = 0 \\\\Delta_{t}=0 . □ \\\\square\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why state-blindness, and why it cannot simply be dropped. For an arbitrary history-dependent policy the bound can fail: a policy that records an observed random symbol b b and replays it at a later controllable step cancels its effect ( q T = q t ⊕ a t ⊕ b ⊕ b = q t ⊕ a t q_{T}=q_{t}\\\\oplus a_{t}\\\\oplus b\\\\oplus b=q_{t}\\\\oplus a_{t} for parity), keeping Δ t = Θ ( 1 ) \\\\Delta_{t}=\\\\Theta(1) . But that cancellation requires the agent to already track and reproduce the latent symbol history—the very state-tracking the sparse reward must first install. At a state-blind policy (initialization) the assumption holds, so by Proposition 1 the intermediate-step credit is O ( λ ( 1 − c ) ( T − t ) ) O(\\\\lambda^{(1-c)(T-t)}) and there is no Θ ( 1 ) \\\\Theta(1) gradient with which to acquire that skill. The barrier is thus a fixed point of the learning dynamics , not an information-theoretic impossibility—consistent with these tasks becoming learnable once a controllable stabilizer or a warm start supplies the missing signal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consequences. Each subsequent step is uncontrolled independently with probability 1 − c 1-c , so in expectation step t t has Θ ( ( 1 − c ) s ) \\\\Theta((1-c)s) uncontrolled steps after it and its credit is O ( λ ( 1 − c ) ( T − t ) ) O(\\\\lambda^{(1-c)(T-t)}) —exponentially small in the steps-to-go for every intermediate step, with Θ ( 1 ) \\\\Theta(1) credit surviving only in the last few steps. A permutation automaton therefore receives no Θ ( 1 ) \\\\Theta(1) signal to represent or use the latent state at intermediate times ; the only escapes are (i) a controllable stabilizer —a hold action that removes that step from the mixing dynamics, letting the agent freeze the state and keep Δ t = Θ ( 1 ) \\\\Delta_{t}=\\\\Theta(1) (this is exactly workflow_reconcile , which is a permutation automaton yet learnable)—or (ii) a synchronizing/absorbing distinction, where P P is not mixing and the span never contracts to zero (this is no_three_zeros , workflow_api ). This is the § 5.5 predictor, derived rather than observed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Empirical confirmation. Computing Δ t \\\\Delta_{t} exactly from the oracle under a uniform (state-blind) rollout—the regime of Proposition 2—(spread of ρ s \\\\rho_{s} over latent state vs. number of intervening random steps) matches the theory to the digit: parity collapses to Δ = 0 \\\\Delta=0 after a single random step; mod_k decays geometrically ( Δ ≈ 2 − r \\\\Delta\\\\approx 2^{-r} : 0.50 , 0.25 , 0.125 , 0.031 , … 0.50,0.25,0.125,0.031,\\\\dots after r = 1 , 2 , 3 , 5 r=1,2,3,5 ); workflow_api stays at Δ ≡ 1.0 \\\\Delta\\\\equiv 1.0 (absorbing accept/reject) and no_three_zeros stays high ( Δ ≈ 0.30 \\\\Delta\\\\approx 0.30 out to 16 16 steps); workflow_reconcile decays like a counter under random play but its hold action lets the policy avoid the decay. The vanishing-credit regime and the predictor’s hard class coincide exactly.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"C Behavioral diagnostics cannot localize the failure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 substantiates the § 5.4 claim that an exact instrument sees what a behavior-only diagnostic cannot. Plotting every RL-evaluated task by its reward gap ( 1 − 1- val n , the strongest purely behavioral signal) against its probe gap, tasks at the same—large—reward gap span the entire probe range. A reward-only auditor would group mined_ean (perception gap, probe + 0.03 +0.03 ) with ends_with_1000 (planning gap, probe + 0.62 +0.62 ) as “the agent is failing,” yet they are opposite failures with opposite fixes. Across the suite the reward gap and OOD reward drop correlate only weakly with the probe (Spearman − 0.29 -0.29 , − 0.25 -0.25 ), confirming that no behavioral summary recovers the perception/planning distinction the probe makes directly.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"D Held-out validation of the predictor, and the causal fix\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fresh-DFA validation. To test whether the § 5.5 predictor generalizes beyond the curated suite, we generated fresh random dfa s ( 3 3 – 7 7 states, 2 2 – 3 3 symbols), oracle-filtered to be diagnostic, and stratified to populate both predictor classes (the predicted-hard class—permutation with low leverage—is ≈ 1 % {\\\\approx}1\\\\% of uniform-random diagnostic dfa s, so a natural sample would need thousands). We report 153 153 such automata: an initial 33 33 ( 5 5 seeds) and a 120 120 -automaton extension ( 3 3 seeds) generated from a separate seed, 4.6 × 4.6\\\\times larger so the held-out estimate is not small-sample. The predictor was computed from δ \\\\delta alone before any RL; we then ran the full instrument (GRU + + PPO, 6000 6000 iters). We control capacity directly : an off-policy supervised probe saturates (val and OOD accuracy ≈ 1.0 {\\\\approx}1.0 ) on the automata, so a low on-policy probe is a genuine perception gap (state recoverable but not recovered), not a representational limit. We therefore label each automaton perception -hard if probe recovery < 0.5 <0.5 , planning -hard if the state is recovered but reward is suboptimal (val < n 0.85 {}_{n}<0.85 ), else solved (Table 3 ). The Props 1–2 direction transfers: of the 103 103 permutation automata, 89 89 are perception gaps (precision 0.86 0.86 ), and 89 89 of all 109 109 perception gaps are permutation (recall 0.82 0.82 ) —permutation structure is a high-precision, one-directional signature of the perception gap on automata it never saw. The original 33 33 reproduce exactly ( 19 / 23 19/23 , precision 0.83 0.83 ); the larger set leaves precision essentially unchanged. Two caveats. The converse fails : non-permutation automata are heterogeneous ( 20 20 perception, 14 14 planning, 16 16 solved of 50 50 ), so group structure flags that state-blindness is likely, not that a non-group task is safe. We could not isolate a clean structural property shared by these non-group perception gaps (diameter, minimum image size, and mixing rate do not separate them), and notably they are largely not explained by the credit-vanishing mechanism (their acceptance-influence Δ \\\\Delta persists rather than decaying)—a mix of reward shortcuts (high reward, state unrepresented) and optimization failures. A structural account of perception gaps outside the group class is the one genuine open hole in the predictor story. And the leverage threshold does not transfer (the conjunction’s recall collapses, the scalar L L over-crediting steerability—it waves through most of the easy_perm stratum, which are predominantly perception gaps), so we demote L L to a within-group severity proxy. (On the original 33 33 , the result replicates on an LSTM—precision 0.78 0.78 /recall 0.78 0.78 , 22 / 24 22/24 GRU perception gaps shared, capacity equally saturated—so it is not a recurrence-cell artifact. A sharper predictor of which permutation automata are hardest, and of the non-group perception gaps, is open.)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The causal fix transfers. Where prediction is imperfect, the prescribed design levers still work as interventions (Table 4 ). On mod_5 (predicted and measured hard), adding a controllable hold ( latchability : mod_5_hold , which raises L L from 0.433 0.433 to 0.523 0.523 and remains a permutation automaton) nearly doubles reward and roughly doubles recovered state, while sharply cutting seed variance; supplying an informative emission channel ( observability , the § 5.3 β \\\\beta lever) recovers the state and reward to near-optimal. Both are pre-registered as the predictor’s two named escape routes, and both close the gap.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"E An LLM-agent instrument: a preliminary feasibility port\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The paper’s motivation is stateful agent reliability, so we ported the entire instrument to a real LLM agent and pre-registered the test (hypotheses and decision thresholds accompany the release). We render each step of PartialControlDFAEnv as natural-language tool I/O with the alphabet relabeled to arbitrary tokens (so a pretrained model cannot pattern-match “this is mod- k k ”) and the dfa rule kept out of the prompt; the agent emits one action token per controllable step with no chain of thought (so an activation probe is not undercut by state offloaded into emitted reasoning). The oracle transfers unchanged; we probe the residual stream at the action token with the same linear probe, selectivity control, and capacity bounds. We ran this on Qwen2.5-3B/-7B-Instruct on a single GPU (released code). The structure and optimizer-strength axes appear on a real LLM; the clean perception gap does not yet —this is a feasibility demonstration that the instrument transfers, not a scaling result.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Axis (ii), structure, replicates as a representational limit. A frozen pretrained model does not linearly encode the latent state (probe at chance for 3B and 7B on mod_5 , gap 0.03 0.03 ); a privileged supervised fine-tune (LoRA to predict q t q_{t} , the LLM analog of our capacity bound) installs it, but only at short counting horizons (Table 5 ): parity capacity decays 0.84 → 0.63 → 0.51 0.84\\\\to 0.63\\\\to 0.51 as the horizon grows L = 8 → 12 → 16 L{=}8\\\\to 12\\\\to 16 , while at L = 8 L{=}8 even mod_5 is representable ( 0.73 0.73 ). The limit is the length of the count , not the state-count—the star-free/length-generalization boundary of § 2 (§ 5.2 ’s transformer result), now as a representational limit on a 3–7B transformer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Axis (i), optimizer strength, replicates—and the weak-optimizer outcome was predicted . On a representable task ( mod_5 @ L 8 @L8 , capacity 0.73 0.73 , oracle random 0.20 0.20 /optimal 0.55 0.55 ), our REINFORCE-level loop (group-normalized return, no value baseline or GAE) does not raise reward and destabilizes (reward drifts below random as the policy leaves the base model). This is not a surprise but a consistency check : it is exactly the synthetic weak-optimizer (A2C) regime of § 5.2 , which also failed to install state and which the stronger PPO + + GAE partially repaired. The reason we hit it first is compute, not design—a PPO + + GAE optimizer for an LLM (a value head and GAE over the agent’s decisions) is heavier than the group-baseline loop. We therefore also implemented that optimizer—a scalar value head, GAE over the agent’s ordered decisions, and a clipped surrogate with multiple epochs—and confirmed it works: on a reach-commit control ( workflow_api ) it raises reward stably , where the REINFORCE loop stagnated. On mod_5 @ L 8 @L8 , however, it still does not learn: conservative settings hold reward at the random baseline and aggressive settings destabilize it, with the state installed in neither case (frozen and post-PPO probe at chance, gap 0.04 0.04 ; reward ≈ \\\\approx random, 0.39 0.39 normalized; capacity 0.73 0.73 ). This mirrors the synthetic finding : synthetically PPO recovers reach-commit/local tasks but only partially recovers accumulation counters, and there only with ∼ 30 × {\\\\sim}30\\\\times the updates; at LLM scale the competent optimizer likewise learns the local task and not the accumulation counter within budget. The optimizer-strength ordering (REINFORCE degrades < < PPO stable-but-no-traction) and the accumulation-is-hard axis (§ 5.4 ) both reproduce.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Scope of the one thing we do not get. A single clean perception gap at LLM scale—reward clearly above random with the probe at chance and capacity saturated—never materializes, because no optimizer we ran lifts reward above random on a representable group counter within budget: the very accumulation credit-assignment difficulty the paper isolates blocks it at scale too. The blocker is thus optimization (the hardness of accumulation under sparse reward), not representation, so the open corner is a stronger / longer-trained LLM RL optimizer, not a larger model. We report this straight, and release the instrument (NL environment, exact oracle, residual-stream probe with selectivity, frozen and supervised capacity bounds, and both RL loops) so that corner can be filled directly.\"\n }\n]","isConversational":false}}},{"rowIdx":398,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.11952"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Scalable Optimal Transport Algorithm for Network Alignment"},"authors":{"kind":"list like","value":["Elaheh Hassani","Durga Mandarapu","Qi Yu","Hanghang Tong","Ariful Azad"],"string":"[\n \"Elaheh Hassani\",\n \"Durga Mandarapu\",\n \"Qi Yu\",\n \"Hanghang Tong\",\n \"Ariful Azad\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.11952v1"},"published":{"kind":"timestamp","value":"2026-07-11T00:00:00","string":"2026-07-11T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"Code Availability: The source code, data, and other artifacts are available at https://github.com/elawh1/FastAlign ."},{"type":"heading","level":1,"text":"1. Introduction"},{"type":"para","text":"The goal of network alignment is to identify node correspondences between two networks. It is a fundamental primitive for integrating graph-structured data across sources in many applications, including social network analysis (Cao and Yu, 2016 ) , fraud detection (Du et al. , 2021 ) , and knowledge graphs (Wang et al. , 2018 ) . For example, aligning users across social platforms supports cross-platform recommendations; matching entities across transaction networks helps detect suspicious activities; aligning entities across incomplete knowledge graphs enables the construction of unified knowledge bases. As modern networks grow in size and dynamic networks require repeated realignments, it is imperative that alignment algorithms be both accurate and fast."},{"type":"para","text":"Existing network alignment methods exhibit a tradeoff between scalability and accuracy (Figure 1 ). Consistency-based methods use a fast linear transformation between two networks, although their accuracy suffers because they do not fully capture global structure (Zhang and Tong, 2016 ) . Embedding-based methods improve accuracy by learning node representations, at the cost of higher runtime from deep representation learning (Yan et al. , 2021 ; Zhang et al. , 2020 , 2021 ) . Optimal transport (OT)-based methods achieve state-of-the-art (SOTA) empirical accuracy by optimizing a transport objective that combines node similarity, structural consistency, neighborhood consistency, and anchor supervision (Zeng et al. , 2023 ; Yu et al. , 2025b ; Zeng et al. , 2024 ; Tang et al. , 2023 ) . This accuracy comes with its own scalability limitation as OT-based methods repeatedly construct and update dense cross-network cost and alignment matrices over several iterations, making them expensive in both runtime and memory."},{"type":"para","text":"To address this scalability bottleneck without sacrificing alignment accuracy, we present FastAlign , a scalable sparsity-aware framework for OT-based network alignment. Rather than introducing a new alignment model, FastAlign preserves the original OT formulation and reinterprets its computation as a set of recurring mixed sparse-dense operations. This reinterpretation exposes where graph sparsity can be exploited and where dense cross-network operations must be optimized for memory efficiency. FastAlign realizes this decomposition using sparsity-aware graph operations and memory-efficient kernel fusion to reduce memory traffic and avoid unnecessary intermediate materialization. As a result, FastAlign accelerates and scales OT-based network alignment, making it practical at large graph sizes."},{"type":"para","text":"Building on this reinterpretation, FastAlign combines four optimizations. First, it uses sparsity-aware computations to replace all dense matrix computations involving graph structures. Second, it introduces a custom sparse-dense matrix multiplication (SpMM) algorithm for multiplying a sparse matrix by a wide dense matrix, as arises in OT-based alignment, unlike generic SpMM settings where the dense matrix is tall and skinny. Third, it uses domain-specific kernel fusion to combine chains of memory-bound operations to reduce intermediate materialization and memory traffic. Fourth, its GPU implementation keeps data resident on device and reuses dense matrices across stages to avoid unnecessary data movement and tiles kernels for optimized performance."},{"type":"para","text":"We implement FastAlign on CPU and GPU and evaluate it on a range of real-world and synthetic network alignment datasets. Our results show that FastAlign achieves comparable alignment quality to SOTA OT-based methods while substantially reducing end-to-end runtime."},{"type":"para","text":"In summary, this paper makes the following contributions:"},{"type":"para","text":"We present FastAlign , a scalable sparsity-aware framework for OT-based network alignment that preserves the alignment accuracy while improving runtime and memory efficiency."},{"type":"para","text":"We provide a computational reinterpretation of OT-based network alignment using mixed sparse-dense operations."},{"type":"para","text":"We develop sparsity-aware graph computation, including a custom SpMM kernel and the fusion of adjacent memory-bound operations, to accelerate OT-based network alignment."},{"type":"para","text":"With extensive experiments, we demonstrate that FastAlign achieves 3.89 × 3.89\\times – 9.45 × 9.45\\times speedup on CPU and 2.24 × 2.24\\times – 32.54 × 32.54\\times speedup on GPU relative to SOTA network alignment algorithms."},{"type":"heading","level":1,"text":"2. Background"},{"type":"para","text":"Let 𝐆 1 = ( 𝐕 1 , 𝐄 1 , 𝐀 1 ) \\mathbf{G}_{1}=(\\mathbf{V}_{1},\\mathbf{E}_{1},\\mathbf{A}_{1}) and 𝐆 2 = ( 𝐕 2 , 𝐄 2 , 𝐀 2 ) \\mathbf{G}_{2}=(\\mathbf{V}_{2},\\mathbf{E}_{2},\\mathbf{A}_{2}) denote two networks, with | 𝐕 1 | = n 1 |\\mathbf{V}_{1}|=n_{1} , | 𝐕 2 | = n 2 |\\mathbf{V}_{2}|=n_{2} , and adjacency matrices 𝐀 1 ∈ ℝ n 1 × n 1 \\mathbf{A}_{1}\\in\\mathbb{R}^{n_{1}\\times n_{1}} and 𝐀 2 ∈ ℝ n 2 × n 2 \\mathbf{A}_{2}\\in\\mathbb{R}^{n_{2}\\times n_{2}} . Nodes optionally carry attributes encoded as 𝐗 1 ∈ ℝ n 1 × d \\mathbf{X}_{1}{\\in}\\mathbb{R}^{n_{1}\\times d} and 𝐗 2 ∈ ℝ n 2 × d \\mathbf{X}_{2}\\in\\mathbb{R}^{n_{2}\\times d} . We are further given a set of anchor pairs ℒ ⊆ 𝐕 1 × 𝐕 2 \\mathcal{L}\\subseteq\\mathbf{V}_{1}\\times\\mathbf{V}_{2} , where each pair denotes a known correspondence between the two networks. Network alignment computes a soft alignment matrix 𝐒 ∈ ℝ n 1 × n 2 \\mathbf{S}\\in\\mathbb{R}^{n_{1}\\times n_{2}} , where 𝐒 i j \\mathbf{S}_{ij} scores likelihood of node i ∈ 𝐕 1 i\\in\\mathbf{V}_{1} corresponding to node j ∈ 𝐕 2 j\\in\\mathbf{V}_{2} . Figure 2 illustrates network alignment problem. Existing approaches can be categorized as consistency-based, embedding-based, or OT-based (Zhang and Tong, 2020 ) . Of these, OT-based approaches consistently achieve SOTA performance (Yu et al. , 2025a ) , our algorithm builds on this paradigm. So, we detail OT-based methods in this section and defer the rest to Section 5 ."},{"type":"heading","level":2,"text":"2.1. Optimal transport for network alignment"},{"type":"para","text":"Optimal transport (OT) maps one discrete distribution to another by minimizing a total transport cost, known as the Wasserstein distance (Peyré and Cuturi, 2019 ) . OT-based network alignment treats the node sets 𝐕 1 \\mathbf{V}_{1} and 𝐕 2 \\mathbf{V}_{2} as discrete distributions with uniform mass vectors 𝝁 = 1 n 1 𝟏 n 1 \\bm{\\mu}=\\tfrac{1}{n_{1}}\\mathbf{1}_{n_{1}} and 𝝂 = 1 n 2 𝟏 n 2 \\bm{\\nu}=\\tfrac{1}{n_{2}}\\mathbf{1}_{n_{2}} . The soft alignment matrix 𝐒 \\mathbf{S} corresponds to the OT transport plan, whose entry 𝐒 i j \\mathbf{S}_{ij} encodes the amount of mass routed from node i ∈ 𝐕 1 i\\in\\mathbf{V}_{1} to node j ∈ 𝐕 2 j\\in\\mathbf{V}_{2} and is interpreted as the strength of their correspondence. Given a cost matrix 𝐂 ∈ ℝ n 1 × n 2 \\mathbf{C}\\in\\mathbb{R}^{n_{1}\\times n_{2}} that encodes the penalty for aligning each pair of nodes, the optimal alignment is obtained by solving"},{"type":"para","text":"where ⟨ ⋅ ⟩ \\langle\\cdot\\rangle denotes the Frobenius inner product, Π ( 𝝁 , 𝝂 ) = { 𝐒 ∈ ℝ ≥ 0 n 1 × n 2 : 𝐒𝟏 = 𝝁 , 𝐒 ⊤ 𝟏 = 𝝂 } \\Pi(\\bm{\\mu},\\bm{\\nu})=\\{\\mathbf{S}\\in\\mathbb{R}_{\\geq 0}^{n_{1}\\times n_{2}}:\\mathbf{S}\\mathbf{1}=\\bm{\\mu},\\;\\mathbf{S}^{\\top}\\mathbf{1}=\\bm{\\nu}\\} is the set of feasible soft alignments, and ε > 0 \\varepsilon>0 controls the strength of the entropic regularizer H ( 𝐒 ) H(\\mathbf{S}) that makes the problem strictly convex and efficiently solvable by the Sinkhorn algorithm (Cuturi, 2013 ) . The objective in Eq. 1 is shaped by two design choices: how the cost matrix 𝐂 \\mathbf{C} is constructed and how the resulting optimization is solved. We discuss each in turn."},{"type":"heading","level":2,"text":"2.2. Decomposition of the Cost Matrix"},{"type":"para","text":"The design of the cost matrix 𝐂 \\mathbf{C} is what most directly shapes alignment quality. A well-designed cost captures three consistency principles together with prior anchor knowledge (Zeng et al. , 2023 ) . Node consistency ( 𝐂 node \\mathbf{C}_{\\text{node}} ) requires matched nodes to have similar attributes and similar structural positions within their respective networks. Edge consistency ( 𝐂 edge \\mathbf{C}_{\\text{edge}} ) requires matched node pairs to preserve their connecting edges. Neighborhood consistency ( 𝐂 nbr \\mathbf{C}_{\\text{nbr}} ) requires matched nodes to have matched neighborhoods, so that neighbors in one graph map to neighbors in the other. Finally, the cost should reflect prior anchor preferences ( 𝐂 anc \\mathbf{C}_{\\text{anc}} ), biasing the alignment toward known correspondences (Zeng et al. , 2023 ) . The overall cost matrix combines these terms additively, where λ e , λ n , λ a ≥ 0 \\lambda_{e},\\lambda_{n},\\lambda_{a}\\geq 0 weigh the contribution of each term:"},{"type":"heading","level":2,"text":"2.3. The OT solver"},{"type":"para","text":"Given the current cost matrix 𝐂 \\mathbf{C} and the predefined marginal vectors 𝝁 , 𝝂 \\bm{\\mu},\\bm{\\nu} , the Sinkhorn algorithm computes the alignment matrix 𝐒 \\mathbf{S} by iteratively rescaling its rows and columns to satisfy the marginal constraints while keeping the alignment cost low. The algorithm maintains two dual vectors 𝐚 \\mathbf{a} and 𝐛 \\mathbf{b} and alternately updates them in the log domain (for numerical stability):"},{"type":"para","text":"where LSE is the log-sum-exp reduction. On convergence, the alignment matrix is recovered as"},{"type":"para","text":"Eq. 4 yields the optimal alignment for the current 𝐂 \\mathbf{C} . Because some components of 𝐂 \\mathbf{C} themselves depend on 𝐒 \\mathbf{S} , the cost is then updated using the newly computed alignment and the Sinkhorn solve is repeated. This alternation realizes the constrained proximal point method (Xie et al. , 2020 ) , which solves the problem as a sequence of fixed-cost OT subproblems with guaranteed convergence."},{"type":"heading","level":1,"text":"3. Scaling OT-Based Network Alignment"},{"type":"para","text":"In this section, we introduce FastAlign , a scalable and fast network alignment framework. FastAlign achieves its speedups through a deliberate two-stage pipeline. First, it recasts OT-based alignment as a sequence of linear-algebraic operations on matrices with explicit shapes and sparsity. Second, it accelerates these kernels using custom sparse algorithms, kernel fusion, and hardware-mapped parallel implementations across CPUs and GPUs. This two-stage framework is necessary because the initial linear-algebraic decomposition exposes optimization pathways that are usually hidden by conventional OT formulations described in Section 2 ."},{"type":"heading","level":2,"text":"3.1. Foundation of FastAlign : Linear-algebraic Decomposition of Network Alignment"},{"type":"para","text":"To decompose the OT-based algorithm in linear-algebraic form, we first observe that the computations described in Section 2 can be organized in three phases: (a) Phase 1 normalizes the adjacency matrices and computes the alignment-independent cost components 𝐂 node \\mathbf{C}_{\\text{node}} and 𝐂 anc \\mathbf{C}_{\\text{anc}} ; (b) Phase 2 computes the alignment-dependent components 𝐂 edge \\mathbf{C}_{\\text{edge}} and 𝐂 nbr \\mathbf{C}_{\\text{nbr}} ; and (c) Phase 3 invokes an OT solver to produce the alignment from the current cost matrix. These three phases of FastAlign are described in Algorithm 1 and Figure 3 ."},{"type":"para","text":"Algorithm 1 takes two networks with node features and a set of anchor alignments and returns the alignment matrix 𝐒 \\mathbf{S} . After normalizing the adjacency matrices into 𝐀 ¯ 1 , 𝐀 ¯ 2 \\bar{\\mathbf{A}}_{1},\\bar{\\mathbf{A}}_{2} (Line 3 ) and representing the anchors as one-hot encodings 𝐑 1 ( 0 ) ∈ ℝ n 1 × | ℒ | \\mathbf{R}_{1}^{(0)}\\in\\mathbb{R}^{n_{1}\\times|\\mathcal{L}|} and 𝐑 2 ( 0 ) ∈ ℝ n 2 × | ℒ | \\mathbf{R}_{2}^{(0)}\\in\\mathbb{R}^{n_{2}\\times|\\mathcal{L}|} , the algorithm proceeds in three phases:"},{"type":"para","text":"Phase 1: Computing 𝐂 node \\mathbf{C}_{\\text{node}} (line 3-5). The node-consistency cost is built in three steps. First, repeated multiplication ( 1 − β ) 𝐀 ¯ i 𝐑 i + β 𝐑 i ( 0 ) (1-\\beta)\\,\\bar{\\mathbf{A}}_{i}\\mathbf{R}_{i}+\\beta\\,\\mathbf{R}_{i}^{(0)} produces a personalized-PageRank-style positional encoding of each node with respect to the anchor set; the l l -th column of 𝐑 i \\mathbf{R}_{i} encodes the position of every node relative to the l l -th anchor (Line 6 ). Second, Line 7 forms the initial 𝐂 node \\mathbf{C}_{\\text{node}} from these encodings via two matrix multiplications. Finally, the cost is iteratively propagated across networks via the triple product 𝐀 ¯ 1 𝐂 node 𝐀 ¯ 2 \\bar{\\mathbf{A}}_{1}\\mathbf{C}_{\\text{node}}\\bar{\\mathbf{A}}_{2} , yielding the final 𝐂 node \\mathbf{C}_{\\text{node}} that remains fixed for the rest of the algorithm. Computing 𝐂 node \\mathbf{C}_{\\text{node}} thus reduces to six matrix multiplications of distinct shapes: some involve sparse operands, some are repeated, and some occur in tandem with others. We exploit these patterns in Section 3.4 ."},{"type":"para","text":"Phase 1: Computing 𝐂 anc \\mathbf{C}_{\\text{anc}} (line 9 ). The anchor-preference cost encodes prior supervision directly from the known correspondences. OneHot ( ℒ ) \\textsc{OneHot}(\\mathcal{L}) places a one at each anchor pair ( i , j ) ∈ ℒ (i,j)\\in\\mathcal{L} and zero elsewhere. Each entry of 𝐂 anc \\mathbf{C}_{\\text{anc}} is ≈ 0 \\approx 0 on anchor pairs and − log ( ε ) -\\log(\\varepsilon) otherwise, biasing transport toward known correspondences."},{"type":"para","text":"Phase 2: Computing 𝐂 edge \\mathbf{C}_{\\text{edge}} (line 7 and 10). The edge-consistency cost is built in two steps. First, we form intra-network dissimilarity matrices that capture how neighbors within each network relate to one another with respect to the anchor set: 𝐂 1 = ( 𝐑 1 𝐑 1 ⊤ ) ⊙ 𝐀 ¯ 1 \\mathbf{C}_{1}=(\\mathbf{R}_{1}\\mathbf{R}_{1}^{\\top})\\odot\\bar{\\mathbf{A}}_{1} and analogously for 𝐂 2 \\mathbf{C}_{2} (Line 10 ). This triple product, called sampled dense–dense matrix multiplication (SDDMM), is less expensive than a full dense product because only the entries selected by 𝐀 ¯ 1 \\bar{\\mathbf{A}}_{1} are computed. Since 𝐂 1 \\mathbf{C}_{1} and 𝐂 2 \\mathbf{C}_{2} do not depend on 𝐒 \\mathbf{S} , they are precomputed in Phase 1. 𝐂 edge \\mathbf{C}_{\\text{edge}} is computed by transporting 𝐂 1 \\mathbf{C}_{1} and 𝐂 2 \\mathbf{C}_{2} across the networks via the triple product 𝐂 1 𝐒𝐂 2 ⊤ \\mathbf{C}_{1}\\mathbf{S}\\mathbf{C}_{2}^{\\top} (Line 14 )."},{"type":"para","text":"Phase 2: Computing 𝐂 nbr \\mathbf{C}_{\\text{nbr}} (line 15 ). The neighborhood-consistency cost ensures that for any node pair ( i , j ) (i,j) , the alignment score 𝐒 i j \\mathbf{S}_{ij} is consistent with the alignment scores of i i ’s and j j ’s neighbors. It is computed by propagating the current alignment across the two networks via the triple product 𝐂 nbr = 𝐀 ¯ 1 ⊤ 𝐒 𝐀 ¯ 2 \\mathbf{C}_{\\text{nbr}}=\\bar{\\mathbf{A}}_{1}^{\\top}\\mathbf{S}\\bar{\\mathbf{A}}_{2} ."},{"type":"para","text":"Phase 2: Computing 𝐂 \\mathbf{C} (line 12). After each cost term is computed, we add the matrices to obtain the final cost matrix for the current iteration. We also add the proximal term λ p log ( S ) \\lambda_{p}\\log(S) , which couples the cost to the current alignment."},{"type":"para","text":"Phase 3: Computing 𝐒 \\mathbf{S} (line 13-17). We solve the entropic OT of Section 2.3 by Sinkhorn (Line 18 -15). First, the dual vectors 𝐚 \\mathbf{a} and 𝐛 \\mathbf{b} are refined by k alternating SoftMin updates. Each SoftMin is a log-sum-exp reduction over the cost matrix: 𝐚 \\mathbf{a} reduces 𝐂 \\mathbf{C} over its rows (Line 14), and 𝐛 \\mathbf{b} reduces 𝐂 \\mathbf{C} over its columns, equivalently a row reduction over 𝐂 T \\mathbf{C}^{T} (Line 15). Second, the alignment is updated from the dual vectors 𝐚 \\mathbf{a} and 𝐛 \\mathbf{b} (Line 21 ), where the outer products 𝟏 𝐚 ⊤ \\mathbf{1}\\mathbf{a}^{\\top} and 𝐛𝟏 ⊤ \\mathbf{b}\\mathbf{1}^{\\top} broadcast the dual vectors into n 1 × n 2 n_{1}\\times n_{2} matrices, combined with 𝐂 \\mathbf{C} , and exponentiated element-wise to produce 𝐒 \\mathbf{S} ."},{"type":"heading","level":2,"text":"3.2. Optimization opportunities"},{"type":"para","text":"The decomposition in Algorithm 1 exposes the computation as a pattern of recurring dense and sparse linear-algebra operations rather than as isolated steps, which lets us reason about costs and bottlenecks directly. Assuming n 1 ≈ n 2 ≈ n n_{1}\\approx n_{2}\\approx n (Zeng et al. , 2023 ; Yu et al. , 2025b ) , Phase 1 incurs 𝒪 ( n 3 ) \\mathcal{O}(n^{3}) work across multiple matrix multiplications; Phase 2 updates the 𝐒 \\mathbf{S} -dependent terms in 𝒪 ( n 3 ) \\mathcal{O}(n^{3}) and produces a dense 𝒪 ( n 2 ) \\mathcal{O}(n^{2}) matrix in each outer iteration; Phase 3 runs Sinkhorn at 𝒪 ( n 2 ) \\mathcal{O}(n^{2}) per iteration. Memory traffic is a second bottleneck: the dense 𝒪 ( n 2 ) \\mathcal{O}(n^{2}) cost and alignment matrices are streamed repeatedly, rescaled row- and column-wise each Sinkhorn iteration, and re-accessed during every cost update. Profiling PARROT, a representative OT-based method, confirms that runtime is spread across all three phases: fixed-cost computation 40 – 69 % 40\\text{--}69\\% , cost update 15 – 36 % 15\\text{--}36\\% , and Sinkhorn 14 – 23 % 14\\text{--}23\\% ."},{"type":"para","text":"Because the three phases share a small set of recurring primitives, accelerating those primitives accelerates the pipeline end-to-end. But the primitives are diverse: the algorithm calls more than ten distinct matrix multiplications differing in operand sparsity, shape, and composition with adjacent operations. 𝐂 node \\mathbf{C}_{\\text{node}} requires dense, sparse, and sparse-dense-sparse triple products; 𝐂 edge \\mathbf{C}_{\\text{edge}} requires dense-dense-sparse (SDDMM) and sparse-dense-sparse triple products; the OT solver requires matrix–vector products interleaved with elementwise operations. Off-the-shelf libraries such as Intel MKL deliver suboptimal performance across this mix. We identify three opportunities for closing the gap: (a) develop efficient sparse computations to reduce the runtime, (b) fuse a sequence of steps into a combined operation to reduce memory traffic, and (c) develop hardware-mapped parallelization for modern GPUs."},{"type":"heading","level":2,"text":"3.3. Sparsity-aware Network Alignment"},{"type":"para","text":"Since real-world networks are sparse, representing graphs as sparse matrices exposes asymptotically more efficient computations that typically translate into faster runtimes in practice. FastAlign represents the input graphs as sparse matrices and replaces every dense graph computation with a sparse kernel. The positional encoding (Line 3) and 𝐂 n o d e \\mathbf{C}_{node} propagation (Line 5) multiply a sparse adjacency matrix with a dense matrix, which we cast as an SpMM. Similarly, the sparse-dense-sparse triple product (Line 5) is factored as two successive SpMMs. The same pattern recurs in the S-dependent costs 𝐂 e d g e \\mathbf{C}_{edge} (Line 10) and C n b r C_{nbr} (Line 11), which are also computed as two SpMMs. We compute the intra-network dissimilarities (Line 7) using SDDMM that evaluates only the masked entries rather than materializing the full dense product."},{"type":"para","text":"Computational Efficiency. These sparse substitutions change the asymptotic cost of the dominant operations. In the propagation and transport triple products (Lines 5, 10, 11), replacing GEMM with SpMM reduces computational complexity from 𝒪 ( n 3 ) \\mathcal{O}(n^{3}) to 𝒪 ( nnz ⋅ n ) \\mathcal{O}(\\textit{nnz}\\cdot n) , where n n is the number of nodes and nnz is the number of nonzeros (edges) in the sparse matrix. For the intra-network dissimilarity (Line 7), each of the nnz masked entries is an inner product of length | ℒ | |\\mathcal{L}| , reducing the cost from 𝒪 ( n 2 ⋅ | ℒ | ) \\mathcal{O}(n^{2}\\cdot|\\mathcal{L}|) to 𝒪 ( nnz ⋅ | ℒ | ) \\mathcal{O}(\\textit{nnz}\\cdot|\\mathcal{L}|) ."},{"type":"para","text":"Memory Reduction. Utilizing sparse matrix representations scales down graph storage overhead from 𝒪 ( n 2 ) \\mathcal{O}(n^{2}) to 𝒪 ( nnz ) \\mathcal{O}(\\textit{nnz}) . We further observe that the alignment matrix is often sparse in practice because each node in one network typically has high alignment probability with only a small subset of nodes in the other network. (Tang et al. , 2023 ; Yu et al. , 2026 ) Thus, reducing the numerical precision from double to single precision does not noticeably affect alignment accuracy. This also reduces the memory footprint and enables FastAlign to scale to larger graphs."},{"type":"heading","level":2,"text":"3.4. Custom SpMM for wide dense cost matrix"},{"type":"para","text":"Several of the most expensive steps in Algorithm 1 share a common matrix-product shape. A dense cost matrix is multiplied on both sides by a sparse adjacency-structured matrix. An example is the 𝐂 node \\mathbf{C}_{\\text{node}} propagation 𝐀 ¯ 1 𝐂 𝐀 ¯ 2 ⊤ \\bar{\\mathbf{A}}_{1}\\mathbf{C}\\,\\bar{\\mathbf{A}}_{2}^{\\top} (Line 8 ), where 𝐂 \\mathbf{C} is dense matrix of size n 1 × n 2 n_{1}\\times n_{2} and the outer matrices are sparse matrices 𝐀 ¯ 1 ∈ ℝ n 1 × n 1 \\bar{\\mathbf{A}}_{1}\\in\\mathbb{R}^{n_{1}\\times n_{1}} and 𝐀 ¯ 2 ∈ ℝ n 2 × n 2 \\bar{\\mathbf{A}}_{2}\\in\\mathbb{R}^{n_{2}\\times n_{2}} . The same sparse-dense-sparse structure recurs in the edge and neighborhood consistency terms 𝐂 edge = 𝐂 1 𝐒𝐂 2 ⊤ \\mathbf{C}_{\\text{edge}}=\\mathbf{C}_{1}\\mathbf{S}\\mathbf{C}_{2}^{\\top} (Line 14 ) and 𝐂 nbr = 𝐀 ¯ 1 ⊤ 𝐒 𝐀 ¯ 2 \\mathbf{C}_{\\text{nbr}}=\\bar{\\mathbf{A}}_{1}^{\\top}\\mathbf{S}\\bar{\\mathbf{A}}_{2} (Line 15 ); there the outer factors 𝐂 1 , 𝐂 2 \\mathbf{C}_{1},\\mathbf{C}_{2} are also sparse, with the same sparsity pattern as the adjacency matrix. Computing each term requires two SpMMs that share the same structure, a sparse n × n n\\times n matrix multiplied by a dense matrix with Θ ( n ) \\Theta(n) columns. We call this primitive SpMM on a wide dense matrix . Here wide means not tall-skinny : the number of columns in the dense matrix is comparable to or larger than its rows. In our algorithm, this column count grows with the graph. Even in the position-aware encoding (Line 6 ), which is another SpMM, 𝐑 k \\mathbf{R}_{k} has width ≈ 0.2 n \\approx 0.2\\,n , and it grows without bound as the graphs grow. This is the opposite of the regime that off-the-shelf libraries such as Intel MKL is optimized for, where the dense matrix has a small, often fixed column dimension d d . To see why that mismatch hurts us, we first review how a generic SpMM is computed and why it is efficient in its intended regime."},{"type":"para","text":"Why generic SpMM underperforms. A general SpMM (Yang et al. , 2018 ; Qian et al. , 2026 ; Huang et al. , 2020 ) computes 𝐌 = 𝐀𝐁 \\mathbf{M}=\\mathbf{A}\\mathbf{B} where 𝐀 \\mathbf{A} is the sparse n × n n\\times n matrix, and B and M are dense n × d n\\times d matrices, where d ≪ n d\\ll n . This kernel computes one output row at a time 𝐌 [ i , : ] = ∑ k : A i k ≠ 0 A i k 𝐁 [ k , : ] \\mathbf{M}[i,:]=\\sum_{k\\,:\\,A_{ik}\\neq 0}A_{ik}\\mathbf{B}[k,:] , where k k runs only over the columns in which row i i of 𝐀 \\mathbf{A} has a nonzero, A i k A_{ik} is that nonzero value, 𝐁 [ k , : ] \\mathbf{B}[k,:] is the k k -th row of the dense input matrix, and 𝐌 [ i , : ] \\mathbf{M}[i,:] is the output row being accumulated. These kernels are tuned for tall-skinny 𝐁 \\mathbf{B} ( d ≪ n ) (d\\ll n) where a row 𝐁 [ k , : ] \\mathbf{B}[k,:] is short, stays cache-resident, and is reused across every nonzero that references column k k of 𝐀 \\mathbf{A} . This setting is the most common application of SpMM when used in iterative solvers and graph neural networks."},{"type":"para","text":"In FastAlign , 𝐁 \\mathbf{\\mathbf{B}} is wide, not tall-skinny. A row 𝐁 [ k , : ] \\mathbf{B}[k,:] is now θ ( n ) \\theta(n) , so it may no longer stay cache-resident and cannot be reused across the nonzeros that reference column k k of 𝐀 \\mathbf{A} , which limits data reuse in FastAlign . In the worst case, each nonzero from the sparse matrix fetches its corresponding dense row from memory, giving 𝒪 ( nnz ⋅ n ) \\mathcal{O}(\\textit{nnz}\\cdot n) memory traffic. Since SpMM is a memory-bound operation, the standard SpMM libraries (e.g., Intel MKL) hurt the performance of FastAlign ."},{"type":"para","text":"The fix: custom SpMM kernel. We design this kernel with a column blocking model to improve cache residency. We partition dense matrices 𝐁 \\mathbf{B} and 𝐌 \\mathbf{M} into Q = ⌈ n 2 / b c ⌉ Q=\\lceil n_{2}/b_{c}\\rceil narrow blocks of b c b_{c} contiguous columns and process one block at a time. To compute block q q of output, whose columns start at j q = ( q − 1 ) b c j_{q}=(q-1)b_{c} , we fill it one output row at a time: 𝐌 [ i , j q : j q + b c ] = ∑ k : 𝐀 i k ≠ 0 𝐀 i k 𝐁 [ k , j q : j q + b c ] \\mathbf{M}[i,j_{q}:j_{q}+b_{c}]=\\sum_{k\\,:\\,\\mathbf{A}_{ik}\\neq 0}\\mathbf{A}_{ik}\\mathbf{B}[k,j_{q}:j_{q}+b_{c}] . Since each memory access within each column block computation is limited to a narrow partition of dense matrix, the B partition needs to reside in cache, reduced to n 1 × b c n_{1}\\times b_{c} instead of n 1 × n 2 n_{1}\\times n_{2} that does not fit into L2/L3. This will increase cache hits when consecutive rows share similar columns of nonzeros."},{"type":"para","text":"Figure 4 traces one block: To fill the b c b_{c} output block 𝐌 [ k , j q : j q + b c ] \\mathbf{M}[k,j_{q}:j_{q}+b_{c}] , the kernel iterates over all rows of 𝐀 \\mathbf{A} accumulating the 𝐀 i k 𝐁 [ k , j q : j q + b c ] \\mathbf{A}_{ik}\\mathbf{B}[k,j_{q}:j_{q}+b_{c}] for each nonzero. Since each memory access within a column-block computation is limited to a narrow partition of the dense matrix, the resident 𝐁 \\mathbf{B} partition shrinks to n 1 × b c n_{1}\\times b_{c} , which fits into L2/L3, instead of the full n 1 × n 2 n_{1}\\times n_{2} , which does not."},{"type":"para","text":"Payoff. Blocking leaves the arithmetic and total access count unchanged at 𝒪 ( nnz ⋅ n ) \\mathcal{O}(\\textit{nnz}\\cdot n) . It changes which level of memory is accessed. By keeping each 𝐁 \\mathbf{B} block cache-resident, it reduces the DRAM traffic for reading 𝐁 \\mathbf{B} , converting memory-bound stream into a cache-resident reuse pattern. For a SpMM with the structure of our application, this improved cache reuse accelerates the SpMM computations throughout our model."},{"type":"heading","level":2,"text":"3.5. Domain-specific Kernel Fusion"},{"type":"para","text":"According to Algorithm 1 , the cost matrix 𝐂 \\mathbf{C} assembly (Line 12), and Sinkhorn dual vector update (Lines 14,15, Eq. 3 ) are written as compact algebraic expressions; however, a literal computation allocates and populates a n 1 × n 2 n_{1}\\times n_{2} intermediate matrix: each log, scale, and add streams a full dense intermediate to memory and back. Since these matrices are large, the memory traffic dominates the computation even though every individual operation is cheap. We merge each algebraic chain into a single path over the output/input matrices. We observed such chains that occur and fused them."},{"type":"para","text":"Cost matrix 𝐂 \\mathbf{C} computation. Cost computation (Line 12) combines four dense terms by scaling, log, and addition: 𝐂 ← 𝐂 node + λ e 𝐂 edge + λ n 𝐂 nbr + λ a 𝐂 anc \\mathbf{C}\\leftarrow\\mathbf{C}_{\\text{node}}+\\lambda_{e}\\mathbf{C}_{\\text{edge}}+\\lambda_{n}\\mathbf{C}_{\\text{nbr}}+\\lambda_{a}\\mathbf{C}_{\\text{anc}} . Naively computing this results in materializing at least 6 separate dense n 1 × n 2 n_{1}\\times n_{2} matrices. The fused kernel instead visits each entry ( i , j ) (i,j) once, loads the five contributing values, applies the scale/log/add, and writes 𝐂 i j \\mathbf{C}_{ij} in one pass, without any intermediates."},{"type":"para","text":"Sinkhorn dual updates. Each dual update is a log-sum-exp reduction over the cost matrix (Eq. 3 ): the row reduction that updates a and the column reduction that updates b. Evaluated naively, each update shifts, exponentiates, and reduces the cost matrix, then applies a log and a final shift which materializes a full n 1 × n 2 n_{1}\\times n_{2} intermediate matrix at each step. We instead design a single fused kernel that applies element-wise shift and exponential inside the reduction, in one streaming pass with a max-shift for stability, so the n 1 × n 2 n_{1}\\times n_{2} intermediate is not materialized. The same kernel serves both updates: applied to 𝐂 \\mathbf{C} to refresh 𝐚 \\mathbf{a} and to 𝐂 T \\mathbf{C}^{T} to refresh 𝐛 \\mathbf{b} ."},{"type":"heading","level":2,"text":"3.6. GPU-aware Optimizations"},{"type":"para","text":"The computational reformulation of FastAlign naturally maps to GPU architectures, as the performance-critical matrix operations reduce to sparse and dense linear algebra kernels. Our GPU implementation builds on the sparsity-aware formulation (Section 3.3 ) and domain-specific kernel fusion (Section 3.5 ) using cuSPARSE for sparse matrix operations and cuBLAS for dense linear algebra. Rather than using these libraries out of the box, we tune their performance by selecting memory layouts and SpMM algorithms to match the structure of each operation in the alignment pipeline."},{"type":"para","text":"Minimizing latency. Unlike the CPU implementation, the GPU implementation must carefully minimize both device-side memory movement and host–device transfers to reduce latency. Cost computation involves sparse–dense–sparse matrix products, which we break down into two SpMM calls with transposing the intermediate dense matrices. As explicitly materializing transposed matrices is expensive, we use transpose views to avoid additional memory allocation. To minimize host–device communication, all graphs, cost matrices, and alignment matrices remain resident in GPU memory, while intermediate buffers are reused throughout the alignment."},{"type":"para","text":"GPU-specific kernel fusion. The Sinkhorn dual updates assign one thread block per column and execute two streaming passes: the first reduces the column minimum using warp-level shuffles ( __shfl_down_sync ), while the second accumulates the shifted exponentials. The same kernel serves the column reduction for updating 𝐛 \\mathbf{b} by operating on a transpose view of 𝐂 \\mathbf{C} , produced by a tiled transpose kernel that stages 32 × 32 32\\times 32 tiles in shared memory with + 1 +1 column padding to eliminate bank conflicts, achieving coalesced reads and writes in a single pass."},{"type":"para","text":"Parallel streams and reusing SpMM plans. To improve GPU utilization, FastAlign issues independent kernels for separate graphs across parallel, non-blocking CUDA streams. For example, the row-normalization kernels of the two graph intra-distance matrices (Line 10 of Algorithm 1 ) run concurrently on separate streams rather than sequentially. Furthermore, standard cuSPARSE SpMM calls incur per-invocation overhead for planning, handle setup, and buffer allocation. Because these calls repeat within a loop, FastAlign pre-computes the plan once and reuses it across iterations. This amortizes the initialization overhead, ensuring subsequent calls incur only raw execution costs."},{"type":"heading","level":1,"text":"4. Evaluation"},{"type":"para","text":"We evaluate FastAlign to answer the following research questions:"},{"type":"para","text":"How accurate is FastAlign compared to SOTA network alignment algorithms?"},{"type":"para","text":"How does FastAlign accelerate alignment on real networks?"},{"type":"para","text":"How does FastAlign scale with increasing network size?"},{"type":"para","text":"How much does each of our techniques contribute to the overall speedups achieved by FastAlign ?"},{"type":"heading","level":2,"text":"4.1. Experimental Setup"},{"type":"para","text":"For experiments, we use an AMD EPYC 7763 CPU with 64 cores and 512 GB of memory. For GPU experiments, we use an NVIDIA A100 GPU with 40 GB of memory and CUDA 12.8. All methods are evaluated in a semi-supervised setting, where 20% of the ground-truth node pairs are used as prior knowledge, i.e., anchor links. We report runtime and accuracy metrics averaged over five runs. Runtime measures the end-to-end alignment computation time after the input graphs are read, including all steps required to compute the final alignment matrix. In our experiments, FastAlign exhibits convergence behavior similar to the OT baseline, PARROT."},{"type":"heading","level":3,"text":"4.1.1. Metrics"},{"type":"para","text":"We evaluate the effectiveness of FastAlign compared to the baselines using the following two accuracy metrics. For each test node x ∈ G 1 x\\in G_{1} , let x ′ ∈ G 2 x^{\\prime}\\in G_{2} denote its ground-truth match. We rank all candidate nodes in G 2 G_{2} by their alignment score with x x . A prediction is counted as a hit at K K if x ′ x^{\\prime} appears among the top- K K ranked candidates. For a test set with n n node pairs, Hits @ K = # hits @ K n \\mathrm{Hits@}K=\\frac{\\#\\mathrm{hits@}K}{n} . Mean Reciprocal Rank (MRR) is measured as the average inverse rank of the ground-truth match: MRR = 1 n ∑ ( x , x ′ ) ∈ 𝒮 test 1 rank ( x , x ′ ) \\mathrm{MRR}=\\frac{1}{n}\\sum_{(x,x^{\\prime})\\in\\mathcal{S}_{\\mathrm{test}}}\\frac{1}{\\operatorname{rank}(x,x^{\\prime})}"},{"type":"heading","level":3,"text":"4.1.2. Baselines"},{"type":"para","text":"Following the benchmark study (Yu et al. , 2025a ) , we select six top-accuracy baselines that cover the three main categories of network alignment methods. FINAL (Zhang and Tong, 2016 ) is consistency-based; BRIGHT (Yan et al. , 2021 ) and NetTrans (Zhang et al. , 2020 ) are embedding-based; and JOENA (Yu et al. , 2025b ) , PARROT (Zeng et al. , 2023 ) , and SLOTAlign (Tang et al. , 2023 ) are OT-based. PARROT is the closest baseline to FastAlign , since our framework preserves the same OT-based alignment formulation. We also evaluated NeXtAlign (Zhang et al. , 2021 ) , but it timed out on most datasets and is omitted."},{"type":"heading","level":3,"text":"4.1.3. Datasets"},{"type":"para","text":"Table 1 summarizes the real-world datasets we used to evaluate FastAlign against the baselines. For scalability experiments, we generated synthetic graphs using the Erdős–Rényi model with an average node degree of 10 10 (Erdős et al. , 1960 ) ."},{"type":"heading","level":2,"text":"4.2. Accuracy Results"},{"type":"para","text":"Table 2 compares the Hits@ 10 10 and MRR of FastAlign against all baselines on the real-world datasets. We report GPU results in this table and observe similar accuracy trends on CPU. CPU accuracy results are omitted because several baselines time out on larger datasets, as discussed in the next Subsection. FastAlign and other OT-based methods achieve substantially higher accuracy than consistency-based and embedding-based methods. Compared with state-of-the-art OT-based baselines, FastAlign achieves nearly identical or better accuracy on most datasets, showing that its optimizations do not degrade alignment quality and preserve original OT alignment objective. We attribute the slightly lower accuracy on DBP 15 15 K to floating-point roundoff inconsistencies, especially from fused kernels as they change the order of numerical operations."},{"type":"heading","level":2,"text":"4.3. Performance Analysis"},{"type":"para","text":"Figure 5 compares the runtime of FastAlign against all baselines on real-world datasets using CPU and GPU implementations, respectively. On CPU, missing bars indicate that the corresponding baseline timed out on that dataset, whereas on GPU, all baselines completed. FastAlign achieves a 3.89 × 3.89\\times – 9.45 × 9.45\\times speedup on CPU and a 2.24 × 2.24\\times – 32.54 × 32.54\\times speedup on GPU compared to PARROT, the fastest baseline. Across all baselines, the speedup ranges from 3.89 × 3.89\\times to 193.90 × 193.90\\times on CPU and from 2.24 × 2.24\\times to 1321.85 × 1321.85\\times on GPU, with the largest speedups achieved over slow baselines."},{"type":"para","text":"These significantly high speedups demonstrate the effectiveness of FastAlign ’s computational reinterpretation and optimizations discussed in Section 3 . In general, the speedups increase with dataset size. For datasets of similar size, the speedups follow the sparsity trend: higher sparsity leads to less data movement through memory and therefore larger speedups. GPU speedups are higher than CPU speedups primarily because kernel fusion and GPU-specific optimizations have a stronger impact on GPU execution. In particular, kernel fusion improves data locality and reuse, reducing memory access latency and contributing to the superior performance of FastAlign on GPU."},{"type":"heading","level":2,"text":"4.4. Scalability Analysis"},{"type":"para","text":"Figure 6 a shows the runtime scalability comparison of FastAlign for synthetic datasets against the baselines that did not time out on graphs with 20 20 K nodes. FastAlign scales more efficiently than all baselines, with speedups increasing as graph size grows — from 0.82 × 0.82\\times at 10K nodes to 9.41 × 9.41\\times at 110K nodes over the fastest baseline. The overall speedups are in the range 0.82 × 0.82\\times – 14.49 × 14.49\\times ."},{"type":"para","text":"FastAlign scales to graphs with up to 110 110 K nodes, while all baselines except BRIGHT run out of memory at smaller graph sizes. The maximum problem size supported by FastAlign is determined by its peak memory usage during Sinkhorn optimization. The peak live state uses nine dense ( n 1 × n 2 ) (n_{1}\\times n_{2}) matrices and two sparse graph matrices. Using single precision (Section 3.3 ), the dense matrices require ( 4 ⋅ 9 ⋅ n 1 n 2 = 36 n 1 n 2 ) (4\\cdot 9\\cdot n_{1}n_{2}=36n_{1}n_{2}) bytes, which bounds the implementation to approximately 110 110 K nodes on a CPU with 512 512 GB memory. As long as they fit in memory, FastAlign aligns the networks."},{"type":"para","text":"Figure 6 b presents the scalability comparison on GPU using an NVIDIA A100 (80GB variant). FastAlign and BRIGHT scale to 50K-node graphs, while FINAL supports up to 35K nodes, PARROT , JOENA , and SLOTAlign up to 30K, and NetTrans only up to 10K. Although the y-axis is shown in log scale to accommodate the wide range of runtimes, the scalability trend on GPU mirrors the CPU results: FastAlign ’s speedups increase with graph size. From 10K to 50K nodes, it achieves speedups in the range of 4.16 × 4.16\\times – 86.88 × 86.88\\times over BRIGHT , the fastest baseline. Across all baselines and graph sizes, the speedups are in the range of 4.16 × 4.16\\times – 2525.88 × 2525.88\\times ."},{"type":"heading","level":2,"text":"4.5. Ablation studies"},{"type":"para","text":"Figure 7 shows the runtime contribution of each FastAlign optimization: sparsity-aware computation, kernel fusion, and custom SpMM. The number on each bar reports the speedup of that artifact over PARROT. To isolate algorithmic gains from platform effects, we include a straightforward reimplementation of the baseline, PARROT in CPP and CUDA as an intermediate baseline. Both CPP and CUDA baselines are only marginally faster than the original PyTorch implementation of PARROT, with the speedups diminishing as graph size grows, confirming that naive reimplementations with a platform change yield no meaningful performance benefit. On smaller networks, we observe a modest speedup from framework conversion alone. However, on larger datasets with at least 10 10 K nodes, the overhead of PyTorch over CPP/CUDA becomes negligible and the C++/CUDA baselines are no longer faster, confirming that FastAlign ’s speedups are driven by its algorithmic optimizations rather than the platform change."},{"type":"para","text":"On CPU, sparsity alone improves runtime by 0.8 × 0.8\\times – 2.6 × 2.6\\times , and adding fusion compounds these gains to 2.2 × 2.2\\times – 4.6 × 4.6\\times , with custom SpMM further increasing speedups to 2.9 × 2.9\\times – 6.6 × 6.6\\times . On GPU, sparsity contributes 2.6 × 2.6\\times – 4.4 × 4.4\\times to the speedup, and adding fusion brings the overall speedup to 2.2 × 2.2\\times – 32.5 × 32.5\\times . Across both CPU and GPU, all optimizations contribute to performance, with fusion consistently compounding the benefits of sparsity."},{"type":"heading","level":1,"text":"5. Related Work"},{"type":"para","text":"In this section, we review the related work, which can be categorized into two groups: network alignment and optimal transport."},{"type":"heading","level":3,"text":"5.0.1. Network Alignment"},{"type":"para","text":"Research on network alignment has largely developed along two directions: consistency-based and embedding-based approaches. The first direction is based on the idea that aligned nodes should have structurally and semantically similar positions, so that the connectivity and attributes of a node’s neighbors carry over from one network to the other. IsoRank (Singh et al. , 2008 ) propagates pairwise similarities on the product graph, FINAL (Zhang and Tong, 2016 ) extends this with attribute and edge consistency for attributed graphs, and BIG-ALIGN (Koutra et al. , 2013 ) recovers a near-permutation between adjacency matrices via a Frobenius objective. Computationally, these methods iterate over an 𝒪 ( n 2 ) \\mathcal{O}(n^{2}) product or similarity space that dominates cost at scale, motivating coarsening-based designs like MOANA (Zhang et al. , 2019 ) . In terms of accuracy, however, they capture limited global graph geometry, and the consistency premise weakens when the two networks differ substantially (Zhang et al. , 2021 ) ."},{"type":"para","text":"The second learns node embeddings that encode each network’s topology while pulling anchor pairs together: IONE (Liu et al. , 2016 ) models directional follower/followee context, REGAL (Heimann et al. , 2018 ) factorizes a cross-network similarity matrix, CrossMNA (Chu et al. , 2019 ) fuses intra-/inter-/global features, and NetTrans (Zhang et al. , 2020 ) , BRIGHT (Yan et al. , 2021 ) , and NeXtAlign (Zhang et al. , 2021 ) further improve robustness through learned transformations, restart-based positional features, and consistency-aware sampling, respectively. Computationally, the cost shifts to training: repeated gradient updates over deep encoders, often with pair sampling, are expensive and lack convergence guarantees. In accuracy, they model cross-network relations only indirectly through anchor paths and are sensitive to sampling noise and embedding-space distortion (Yu et al. , 2025b ; Zhang et al. , 2021 ) ."},{"type":"heading","level":3,"text":"5.0.2. Optimal Transport"},{"type":"para","text":"Optimal transport (OT) has recently been applied to network alignment, representing each graph as a distribution over its nodes and seeking the minimum-cost transport between them (Chen et al. , 2020 ) . Methods differ mainly in how the transport cost is defined. Early designs transport spectral node embeddings (EMD (Nikolentzos et al. , 2017 ) ) or represent graphs through filtered graph signals and Gaussian graph kernels (FGOT (Maretic et al. , 2022 ) , GOT (Petric Maretic et al. , 2019 ) ), while later costs jointly encode node and edge correspondence via the Wasserstein and Gromov–Wasserstein (GW) distances (Xu et al. , 2019 ; Chen et al. , 2020 ) . A prominent line aligns graphs through Gromov–Wasserstein structural discrepancy (S-GWL (Xu et al. , 2019 ) , GraphOTC (O’Connor et al. , 2021 ) ), while GOAT (Saad-Eldin et al. , 2021 ) accelerates it via the Sinkhorn algorithm. More recent work learns the cost instead of fixing it: SLOTAlign (Tang et al. , 2023 ) encodes a GW objective with a parameter-free GNN, CombAlign (Chen et al. , 2025 ) ensembles embedding- and OT-based alignments, and JOENA (Yu et al. , 2025b ) jointly learns embeddings and transport. PARROT (Zeng et al. , 2023 ) builds random-walk topology and consistency regularization into the OT cost and solves it with a constrained proximal point method."},{"type":"para","text":"While OT-based methods achieve higher accuracy, they remain the most computationally demanding category. Their cost stems from repeatedly updating dense 𝒪 ( n 2 ) \\mathcal{O}(n^{2}) matrices, and while proximal solvers (Yu et al. , 2025b ) reduce iteration counts, graph sparsity and sparse-linear-algebra primitives remain underexploited. As a result, even the most scalable OT aligners are confined to moderate-sized graphs, a gap our work targets by retaining the accuracy of regularized OT while improving runtime and scalability through a computation-optimized algorithm."},{"type":"heading","level":1,"text":"6. Conclusion"},{"type":"para","text":"In this paper, we scale OT-based network alignment to large graphs while preserving its accuracy. Instead of designing a new alignment model, we present FastAlign , which keeps the original OT formulation and treats its computation as a small set of recurring sparse-dense operations. FastAlign exploits graph sparsity to skip work over nonexistent edges, fuses memory-bound kernels to cut redundant memory traffic, adds a custom SpMM kernel for the wide dense matrices in OT alignment, and uses GPU-aware optimizations to reuse data on device. Experiments on real and synthetic datasets show that FastAlign matches the accuracy of SOTA OT-based methods while running up to 9.45 × 9.45\\times faster on CPU and up to 32.54 × 32.54\\times faster on GPU than the strongest OT baseline."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"Code Availability: The source code, data, and other artifacts are available at https://github.com/elawh1/FastAlign .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1. Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The goal of network alignment is to identify node correspondences between two networks. It is a fundamental primitive for integrating graph-structured data across sources in many applications, including social network analysis (Cao and Yu, 2016 ) , fraud detection (Du et al. , 2021 ) , and knowledge graphs (Wang et al. , 2018 ) . For example, aligning users across social platforms supports cross-platform recommendations; matching entities across transaction networks helps detect suspicious activities; aligning entities across incomplete knowledge graphs enables the construction of unified knowledge bases. As modern networks grow in size and dynamic networks require repeated realignments, it is imperative that alignment algorithms be both accurate and fast.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Existing network alignment methods exhibit a tradeoff between scalability and accuracy (Figure 1 ). Consistency-based methods use a fast linear transformation between two networks, although their accuracy suffers because they do not fully capture global structure (Zhang and Tong, 2016 ) . Embedding-based methods improve accuracy by learning node representations, at the cost of higher runtime from deep representation learning (Yan et al. , 2021 ; Zhang et al. , 2020 , 2021 ) . Optimal transport (OT)-based methods achieve state-of-the-art (SOTA) empirical accuracy by optimizing a transport objective that combines node similarity, structural consistency, neighborhood consistency, and anchor supervision (Zeng et al. , 2023 ; Yu et al. , 2025b ; Zeng et al. , 2024 ; Tang et al. , 2023 ) . This accuracy comes with its own scalability limitation as OT-based methods repeatedly construct and update dense cross-network cost and alignment matrices over several iterations, making them expensive in both runtime and memory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address this scalability bottleneck without sacrificing alignment accuracy, we present FastAlign , a scalable sparsity-aware framework for OT-based network alignment. Rather than introducing a new alignment model, FastAlign preserves the original OT formulation and reinterprets its computation as a set of recurring mixed sparse-dense operations. This reinterpretation exposes where graph sparsity can be exploited and where dense cross-network operations must be optimized for memory efficiency. FastAlign realizes this decomposition using sparsity-aware graph operations and memory-efficient kernel fusion to reduce memory traffic and avoid unnecessary intermediate materialization. As a result, FastAlign accelerates and scales OT-based network alignment, making it practical at large graph sizes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Building on this reinterpretation, FastAlign combines four optimizations. First, it uses sparsity-aware computations to replace all dense matrix computations involving graph structures. Second, it introduces a custom sparse-dense matrix multiplication (SpMM) algorithm for multiplying a sparse matrix by a wide dense matrix, as arises in OT-based alignment, unlike generic SpMM settings where the dense matrix is tall and skinny. Third, it uses domain-specific kernel fusion to combine chains of memory-bound operations to reduce intermediate materialization and memory traffic. Fourth, its GPU implementation keeps data resident on device and reuses dense matrices across stages to avoid unnecessary data movement and tiles kernels for optimized performance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We implement FastAlign on CPU and GPU and evaluate it on a range of real-world and synthetic network alignment datasets. Our results show that FastAlign achieves comparable alignment quality to SOTA OT-based methods while substantially reducing end-to-end runtime.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In summary, this paper makes the following contributions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We present FastAlign , a scalable sparsity-aware framework for OT-based network alignment that preserves the alignment accuracy while improving runtime and memory efficiency.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We provide a computational reinterpretation of OT-based network alignment using mixed sparse-dense operations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We develop sparsity-aware graph computation, including a custom SpMM kernel and the fusion of adjacent memory-bound operations, to accelerate OT-based network alignment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With extensive experiments, we demonstrate that FastAlign achieves 3.89 × 3.89\\\\times – 9.45 × 9.45\\\\times speedup on CPU and 2.24 × 2.24\\\\times – 32.54 × 32.54\\\\times speedup on GPU relative to SOTA network alignment algorithms.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2. Background\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let 𝐆 1 = ( 𝐕 1 , 𝐄 1 , 𝐀 1 ) \\\\mathbf{G}_{1}=(\\\\mathbf{V}_{1},\\\\mathbf{E}_{1},\\\\mathbf{A}_{1}) and 𝐆 2 = ( 𝐕 2 , 𝐄 2 , 𝐀 2 ) \\\\mathbf{G}_{2}=(\\\\mathbf{V}_{2},\\\\mathbf{E}_{2},\\\\mathbf{A}_{2}) denote two networks, with | 𝐕 1 | = n 1 |\\\\mathbf{V}_{1}|=n_{1} , | 𝐕 2 | = n 2 |\\\\mathbf{V}_{2}|=n_{2} , and adjacency matrices 𝐀 1 ∈ ℝ n 1 × n 1 \\\\mathbf{A}_{1}\\\\in\\\\mathbb{R}^{n_{1}\\\\times n_{1}} and 𝐀 2 ∈ ℝ n 2 × n 2 \\\\mathbf{A}_{2}\\\\in\\\\mathbb{R}^{n_{2}\\\\times n_{2}} . Nodes optionally carry attributes encoded as 𝐗 1 ∈ ℝ n 1 × d \\\\mathbf{X}_{1}{\\\\in}\\\\mathbb{R}^{n_{1}\\\\times d} and 𝐗 2 ∈ ℝ n 2 × d \\\\mathbf{X}_{2}\\\\in\\\\mathbb{R}^{n_{2}\\\\times d} . We are further given a set of anchor pairs ℒ ⊆ 𝐕 1 × 𝐕 2 \\\\mathcal{L}\\\\subseteq\\\\mathbf{V}_{1}\\\\times\\\\mathbf{V}_{2} , where each pair denotes a known correspondence between the two networks. Network alignment computes a soft alignment matrix 𝐒 ∈ ℝ n 1 × n 2 \\\\mathbf{S}\\\\in\\\\mathbb{R}^{n_{1}\\\\times n_{2}} , where 𝐒 i j \\\\mathbf{S}_{ij} scores likelihood of node i ∈ 𝐕 1 i\\\\in\\\\mathbf{V}_{1} corresponding to node j ∈ 𝐕 2 j\\\\in\\\\mathbf{V}_{2} . Figure 2 illustrates network alignment problem. Existing approaches can be categorized as consistency-based, embedding-based, or OT-based (Zhang and Tong, 2020 ) . Of these, OT-based approaches consistently achieve SOTA performance (Yu et al. , 2025a ) , our algorithm builds on this paradigm. So, we detail OT-based methods in this section and defer the rest to Section 5 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1. Optimal transport for network alignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optimal transport (OT) maps one discrete distribution to another by minimizing a total transport cost, known as the Wasserstein distance (Peyré and Cuturi, 2019 ) . OT-based network alignment treats the node sets 𝐕 1 \\\\mathbf{V}_{1} and 𝐕 2 \\\\mathbf{V}_{2} as discrete distributions with uniform mass vectors 𝝁 = 1 n 1 𝟏 n 1 \\\\bm{\\\\mu}=\\\\tfrac{1}{n_{1}}\\\\mathbf{1}_{n_{1}} and 𝝂 = 1 n 2 𝟏 n 2 \\\\bm{\\\\nu}=\\\\tfrac{1}{n_{2}}\\\\mathbf{1}_{n_{2}} . The soft alignment matrix 𝐒 \\\\mathbf{S} corresponds to the OT transport plan, whose entry 𝐒 i j \\\\mathbf{S}_{ij} encodes the amount of mass routed from node i ∈ 𝐕 1 i\\\\in\\\\mathbf{V}_{1} to node j ∈ 𝐕 2 j\\\\in\\\\mathbf{V}_{2} and is interpreted as the strength of their correspondence. Given a cost matrix 𝐂 ∈ ℝ n 1 × n 2 \\\\mathbf{C}\\\\in\\\\mathbb{R}^{n_{1}\\\\times n_{2}} that encodes the penalty for aligning each pair of nodes, the optimal alignment is obtained by solving\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ⟨ ⋅ ⟩ \\\\langle\\\\cdot\\\\rangle denotes the Frobenius inner product, Π ( 𝝁 , 𝝂 ) = { 𝐒 ∈ ℝ ≥ 0 n 1 × n 2 : 𝐒𝟏 = 𝝁 , 𝐒 ⊤ 𝟏 = 𝝂 } \\\\Pi(\\\\bm{\\\\mu},\\\\bm{\\\\nu})=\\\\{\\\\mathbf{S}\\\\in\\\\mathbb{R}_{\\\\geq 0}^{n_{1}\\\\times n_{2}}:\\\\mathbf{S}\\\\mathbf{1}=\\\\bm{\\\\mu},\\\\;\\\\mathbf{S}^{\\\\top}\\\\mathbf{1}=\\\\bm{\\\\nu}\\\\} is the set of feasible soft alignments, and ε > 0 \\\\varepsilon>0 controls the strength of the entropic regularizer H ( 𝐒 ) H(\\\\mathbf{S}) that makes the problem strictly convex and efficiently solvable by the Sinkhorn algorithm (Cuturi, 2013 ) . The objective in Eq. 1 is shaped by two design choices: how the cost matrix 𝐂 \\\\mathbf{C} is constructed and how the resulting optimization is solved. We discuss each in turn.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2. Decomposition of the Cost Matrix\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The design of the cost matrix 𝐂 \\\\mathbf{C} is what most directly shapes alignment quality. A well-designed cost captures three consistency principles together with prior anchor knowledge (Zeng et al. , 2023 ) . Node consistency ( 𝐂 node \\\\mathbf{C}_{\\\\text{node}} ) requires matched nodes to have similar attributes and similar structural positions within their respective networks. Edge consistency ( 𝐂 edge \\\\mathbf{C}_{\\\\text{edge}} ) requires matched node pairs to preserve their connecting edges. Neighborhood consistency ( 𝐂 nbr \\\\mathbf{C}_{\\\\text{nbr}} ) requires matched nodes to have matched neighborhoods, so that neighbors in one graph map to neighbors in the other. Finally, the cost should reflect prior anchor preferences ( 𝐂 anc \\\\mathbf{C}_{\\\\text{anc}} ), biasing the alignment toward known correspondences (Zeng et al. , 2023 ) . The overall cost matrix combines these terms additively, where λ e , λ n , λ a ≥ 0 \\\\lambda_{e},\\\\lambda_{n},\\\\lambda_{a}\\\\geq 0 weigh the contribution of each term:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3. The OT solver\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given the current cost matrix 𝐂 \\\\mathbf{C} and the predefined marginal vectors 𝝁 , 𝝂 \\\\bm{\\\\mu},\\\\bm{\\\\nu} , the Sinkhorn algorithm computes the alignment matrix 𝐒 \\\\mathbf{S} by iteratively rescaling its rows and columns to satisfy the marginal constraints while keeping the alignment cost low. The algorithm maintains two dual vectors 𝐚 \\\\mathbf{a} and 𝐛 \\\\mathbf{b} and alternately updates them in the log domain (for numerical stability):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where LSE is the log-sum-exp reduction. On convergence, the alignment matrix is recovered as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Eq. 4 yields the optimal alignment for the current 𝐂 \\\\mathbf{C} . Because some components of 𝐂 \\\\mathbf{C} themselves depend on 𝐒 \\\\mathbf{S} , the cost is then updated using the newly computed alignment and the Sinkhorn solve is repeated. This alternation realizes the constrained proximal point method (Xie et al. , 2020 ) , which solves the problem as a sequence of fixed-cost OT subproblems with guaranteed convergence.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3. Scaling OT-Based Network Alignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we introduce FastAlign , a scalable and fast network alignment framework. FastAlign achieves its speedups through a deliberate two-stage pipeline. First, it recasts OT-based alignment as a sequence of linear-algebraic operations on matrices with explicit shapes and sparsity. Second, it accelerates these kernels using custom sparse algorithms, kernel fusion, and hardware-mapped parallel implementations across CPUs and GPUs. This two-stage framework is necessary because the initial linear-algebraic decomposition exposes optimization pathways that are usually hidden by conventional OT formulations described in Section 2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1. Foundation of FastAlign : Linear-algebraic Decomposition of Network Alignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To decompose the OT-based algorithm in linear-algebraic form, we first observe that the computations described in Section 2 can be organized in three phases: (a) Phase 1 normalizes the adjacency matrices and computes the alignment-independent cost components 𝐂 node \\\\mathbf{C}_{\\\\text{node}} and 𝐂 anc \\\\mathbf{C}_{\\\\text{anc}} ; (b) Phase 2 computes the alignment-dependent components 𝐂 edge \\\\mathbf{C}_{\\\\text{edge}} and 𝐂 nbr \\\\mathbf{C}_{\\\\text{nbr}} ; and (c) Phase 3 invokes an OT solver to produce the alignment from the current cost matrix. These three phases of FastAlign are described in Algorithm 1 and Figure 3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm 1 takes two networks with node features and a set of anchor alignments and returns the alignment matrix 𝐒 \\\\mathbf{S} . After normalizing the adjacency matrices into 𝐀 ¯ 1 , 𝐀 ¯ 2 \\\\bar{\\\\mathbf{A}}_{1},\\\\bar{\\\\mathbf{A}}_{2} (Line 3 ) and representing the anchors as one-hot encodings 𝐑 1 ( 0 ) ∈ ℝ n 1 × | ℒ | \\\\mathbf{R}_{1}^{(0)}\\\\in\\\\mathbb{R}^{n_{1}\\\\times|\\\\mathcal{L}|} and 𝐑 2 ( 0 ) ∈ ℝ n 2 × | ℒ | \\\\mathbf{R}_{2}^{(0)}\\\\in\\\\mathbb{R}^{n_{2}\\\\times|\\\\mathcal{L}|} , the algorithm proceeds in three phases:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase 1: Computing 𝐂 node \\\\mathbf{C}_{\\\\text{node}} (line 3-5). The node-consistency cost is built in three steps. First, repeated multiplication ( 1 − β ) 𝐀 ¯ i 𝐑 i + β 𝐑 i ( 0 ) (1-\\\\beta)\\\\,\\\\bar{\\\\mathbf{A}}_{i}\\\\mathbf{R}_{i}+\\\\beta\\\\,\\\\mathbf{R}_{i}^{(0)} produces a personalized-PageRank-style positional encoding of each node with respect to the anchor set; the l l -th column of 𝐑 i \\\\mathbf{R}_{i} encodes the position of every node relative to the l l -th anchor (Line 6 ). Second, Line 7 forms the initial 𝐂 node \\\\mathbf{C}_{\\\\text{node}} from these encodings via two matrix multiplications. Finally, the cost is iteratively propagated across networks via the triple product 𝐀 ¯ 1 𝐂 node 𝐀 ¯ 2 \\\\bar{\\\\mathbf{A}}_{1}\\\\mathbf{C}_{\\\\text{node}}\\\\bar{\\\\mathbf{A}}_{2} , yielding the final 𝐂 node \\\\mathbf{C}_{\\\\text{node}} that remains fixed for the rest of the algorithm. Computing 𝐂 node \\\\mathbf{C}_{\\\\text{node}} thus reduces to six matrix multiplications of distinct shapes: some involve sparse operands, some are repeated, and some occur in tandem with others. We exploit these patterns in Section 3.4 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase 1: Computing 𝐂 anc \\\\mathbf{C}_{\\\\text{anc}} (line 9 ). The anchor-preference cost encodes prior supervision directly from the known correspondences. OneHot ( ℒ ) \\\\textsc{OneHot}(\\\\mathcal{L}) places a one at each anchor pair ( i , j ) ∈ ℒ (i,j)\\\\in\\\\mathcal{L} and zero elsewhere. Each entry of 𝐂 anc \\\\mathbf{C}_{\\\\text{anc}} is ≈ 0 \\\\approx 0 on anchor pairs and − log ( ε ) -\\\\log(\\\\varepsilon) otherwise, biasing transport toward known correspondences.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase 2: Computing 𝐂 edge \\\\mathbf{C}_{\\\\text{edge}} (line 7 and 10). The edge-consistency cost is built in two steps. First, we form intra-network dissimilarity matrices that capture how neighbors within each network relate to one another with respect to the anchor set: 𝐂 1 = ( 𝐑 1 𝐑 1 ⊤ ) ⊙ 𝐀 ¯ 1 \\\\mathbf{C}_{1}=(\\\\mathbf{R}_{1}\\\\mathbf{R}_{1}^{\\\\top})\\\\odot\\\\bar{\\\\mathbf{A}}_{1} and analogously for 𝐂 2 \\\\mathbf{C}_{2} (Line 10 ). This triple product, called sampled dense–dense matrix multiplication (SDDMM), is less expensive than a full dense product because only the entries selected by 𝐀 ¯ 1 \\\\bar{\\\\mathbf{A}}_{1} are computed. Since 𝐂 1 \\\\mathbf{C}_{1} and 𝐂 2 \\\\mathbf{C}_{2} do not depend on 𝐒 \\\\mathbf{S} , they are precomputed in Phase 1. 𝐂 edge \\\\mathbf{C}_{\\\\text{edge}} is computed by transporting 𝐂 1 \\\\mathbf{C}_{1} and 𝐂 2 \\\\mathbf{C}_{2} across the networks via the triple product 𝐂 1 𝐒𝐂 2 ⊤ \\\\mathbf{C}_{1}\\\\mathbf{S}\\\\mathbf{C}_{2}^{\\\\top} (Line 14 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase 2: Computing 𝐂 nbr \\\\mathbf{C}_{\\\\text{nbr}} (line 15 ). The neighborhood-consistency cost ensures that for any node pair ( i , j ) (i,j) , the alignment score 𝐒 i j \\\\mathbf{S}_{ij} is consistent with the alignment scores of i i ’s and j j ’s neighbors. It is computed by propagating the current alignment across the two networks via the triple product 𝐂 nbr = 𝐀 ¯ 1 ⊤ 𝐒 𝐀 ¯ 2 \\\\mathbf{C}_{\\\\text{nbr}}=\\\\bar{\\\\mathbf{A}}_{1}^{\\\\top}\\\\mathbf{S}\\\\bar{\\\\mathbf{A}}_{2} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase 2: Computing 𝐂 \\\\mathbf{C} (line 12). After each cost term is computed, we add the matrices to obtain the final cost matrix for the current iteration. We also add the proximal term λ p log ( S ) \\\\lambda_{p}\\\\log(S) , which couples the cost to the current alignment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Phase 3: Computing 𝐒 \\\\mathbf{S} (line 13-17). We solve the entropic OT of Section 2.3 by Sinkhorn (Line 18 -15). First, the dual vectors 𝐚 \\\\mathbf{a} and 𝐛 \\\\mathbf{b} are refined by k alternating SoftMin updates. Each SoftMin is a log-sum-exp reduction over the cost matrix: 𝐚 \\\\mathbf{a} reduces 𝐂 \\\\mathbf{C} over its rows (Line 14), and 𝐛 \\\\mathbf{b} reduces 𝐂 \\\\mathbf{C} over its columns, equivalently a row reduction over 𝐂 T \\\\mathbf{C}^{T} (Line 15). Second, the alignment is updated from the dual vectors 𝐚 \\\\mathbf{a} and 𝐛 \\\\mathbf{b} (Line 21 ), where the outer products 𝟏 𝐚 ⊤ \\\\mathbf{1}\\\\mathbf{a}^{\\\\top} and 𝐛𝟏 ⊤ \\\\mathbf{b}\\\\mathbf{1}^{\\\\top} broadcast the dual vectors into n 1 × n 2 n_{1}\\\\times n_{2} matrices, combined with 𝐂 \\\\mathbf{C} , and exponentiated element-wise to produce 𝐒 \\\\mathbf{S} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2. Optimization opportunities\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The decomposition in Algorithm 1 exposes the computation as a pattern of recurring dense and sparse linear-algebra operations rather than as isolated steps, which lets us reason about costs and bottlenecks directly. Assuming n 1 ≈ n 2 ≈ n n_{1}\\\\approx n_{2}\\\\approx n (Zeng et al. , 2023 ; Yu et al. , 2025b ) , Phase 1 incurs 𝒪 ( n 3 ) \\\\mathcal{O}(n^{3}) work across multiple matrix multiplications; Phase 2 updates the 𝐒 \\\\mathbf{S} -dependent terms in 𝒪 ( n 3 ) \\\\mathcal{O}(n^{3}) and produces a dense 𝒪 ( n 2 ) \\\\mathcal{O}(n^{2}) matrix in each outer iteration; Phase 3 runs Sinkhorn at 𝒪 ( n 2 ) \\\\mathcal{O}(n^{2}) per iteration. Memory traffic is a second bottleneck: the dense 𝒪 ( n 2 ) \\\\mathcal{O}(n^{2}) cost and alignment matrices are streamed repeatedly, rescaled row- and column-wise each Sinkhorn iteration, and re-accessed during every cost update. Profiling PARROT, a representative OT-based method, confirms that runtime is spread across all three phases: fixed-cost computation 40 – 69 % 40\\\\text{--}69\\\\% , cost update 15 – 36 % 15\\\\text{--}36\\\\% , and Sinkhorn 14 – 23 % 14\\\\text{--}23\\\\% .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because the three phases share a small set of recurring primitives, accelerating those primitives accelerates the pipeline end-to-end. But the primitives are diverse: the algorithm calls more than ten distinct matrix multiplications differing in operand sparsity, shape, and composition with adjacent operations. 𝐂 node \\\\mathbf{C}_{\\\\text{node}} requires dense, sparse, and sparse-dense-sparse triple products; 𝐂 edge \\\\mathbf{C}_{\\\\text{edge}} requires dense-dense-sparse (SDDMM) and sparse-dense-sparse triple products; the OT solver requires matrix–vector products interleaved with elementwise operations. Off-the-shelf libraries such as Intel MKL deliver suboptimal performance across this mix. We identify three opportunities for closing the gap: (a) develop efficient sparse computations to reduce the runtime, (b) fuse a sequence of steps into a combined operation to reduce memory traffic, and (c) develop hardware-mapped parallelization for modern GPUs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3. Sparsity-aware Network Alignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since real-world networks are sparse, representing graphs as sparse matrices exposes asymptotically more efficient computations that typically translate into faster runtimes in practice. FastAlign represents the input graphs as sparse matrices and replaces every dense graph computation with a sparse kernel. The positional encoding (Line 3) and 𝐂 n o d e \\\\mathbf{C}_{node} propagation (Line 5) multiply a sparse adjacency matrix with a dense matrix, which we cast as an SpMM. Similarly, the sparse-dense-sparse triple product (Line 5) is factored as two successive SpMMs. The same pattern recurs in the S-dependent costs 𝐂 e d g e \\\\mathbf{C}_{edge} (Line 10) and C n b r C_{nbr} (Line 11), which are also computed as two SpMMs. We compute the intra-network dissimilarities (Line 7) using SDDMM that evaluates only the masked entries rather than materializing the full dense product.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Computational Efficiency. These sparse substitutions change the asymptotic cost of the dominant operations. In the propagation and transport triple products (Lines 5, 10, 11), replacing GEMM with SpMM reduces computational complexity from 𝒪 ( n 3 ) \\\\mathcal{O}(n^{3}) to 𝒪 ( nnz ⋅ n ) \\\\mathcal{O}(\\\\textit{nnz}\\\\cdot n) , where n n is the number of nodes and nnz is the number of nonzeros (edges) in the sparse matrix. For the intra-network dissimilarity (Line 7), each of the nnz masked entries is an inner product of length | ℒ | |\\\\mathcal{L}| , reducing the cost from 𝒪 ( n 2 ⋅ | ℒ | ) \\\\mathcal{O}(n^{2}\\\\cdot|\\\\mathcal{L}|) to 𝒪 ( nnz ⋅ | ℒ | ) \\\\mathcal{O}(\\\\textit{nnz}\\\\cdot|\\\\mathcal{L}|) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Memory Reduction. Utilizing sparse matrix representations scales down graph storage overhead from 𝒪 ( n 2 ) \\\\mathcal{O}(n^{2}) to 𝒪 ( nnz ) \\\\mathcal{O}(\\\\textit{nnz}) . We further observe that the alignment matrix is often sparse in practice because each node in one network typically has high alignment probability with only a small subset of nodes in the other network. (Tang et al. , 2023 ; Yu et al. , 2026 ) Thus, reducing the numerical precision from double to single precision does not noticeably affect alignment accuracy. This also reduces the memory footprint and enables FastAlign to scale to larger graphs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4. Custom SpMM for wide dense cost matrix\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Several of the most expensive steps in Algorithm 1 share a common matrix-product shape. A dense cost matrix is multiplied on both sides by a sparse adjacency-structured matrix. An example is the 𝐂 node \\\\mathbf{C}_{\\\\text{node}} propagation 𝐀 ¯ 1 𝐂 𝐀 ¯ 2 ⊤ \\\\bar{\\\\mathbf{A}}_{1}\\\\mathbf{C}\\\\,\\\\bar{\\\\mathbf{A}}_{2}^{\\\\top} (Line 8 ), where 𝐂 \\\\mathbf{C} is dense matrix of size n 1 × n 2 n_{1}\\\\times n_{2} and the outer matrices are sparse matrices 𝐀 ¯ 1 ∈ ℝ n 1 × n 1 \\\\bar{\\\\mathbf{A}}_{1}\\\\in\\\\mathbb{R}^{n_{1}\\\\times n_{1}} and 𝐀 ¯ 2 ∈ ℝ n 2 × n 2 \\\\bar{\\\\mathbf{A}}_{2}\\\\in\\\\mathbb{R}^{n_{2}\\\\times n_{2}} . The same sparse-dense-sparse structure recurs in the edge and neighborhood consistency terms 𝐂 edge = 𝐂 1 𝐒𝐂 2 ⊤ \\\\mathbf{C}_{\\\\text{edge}}=\\\\mathbf{C}_{1}\\\\mathbf{S}\\\\mathbf{C}_{2}^{\\\\top} (Line 14 ) and 𝐂 nbr = 𝐀 ¯ 1 ⊤ 𝐒 𝐀 ¯ 2 \\\\mathbf{C}_{\\\\text{nbr}}=\\\\bar{\\\\mathbf{A}}_{1}^{\\\\top}\\\\mathbf{S}\\\\bar{\\\\mathbf{A}}_{2} (Line 15 ); there the outer factors 𝐂 1 , 𝐂 2 \\\\mathbf{C}_{1},\\\\mathbf{C}_{2} are also sparse, with the same sparsity pattern as the adjacency matrix. Computing each term requires two SpMMs that share the same structure, a sparse n × n n\\\\times n matrix multiplied by a dense matrix with Θ ( n ) \\\\Theta(n) columns. We call this primitive SpMM on a wide dense matrix . Here wide means not tall-skinny : the number of columns in the dense matrix is comparable to or larger than its rows. In our algorithm, this column count grows with the graph. Even in the position-aware encoding (Line 6 ), which is another SpMM, 𝐑 k \\\\mathbf{R}_{k} has width ≈ 0.2 n \\\\approx 0.2\\\\,n , and it grows without bound as the graphs grow. This is the opposite of the regime that off-the-shelf libraries such as Intel MKL is optimized for, where the dense matrix has a small, often fixed column dimension d d . To see why that mismatch hurts us, we first review how a generic SpMM is computed and why it is efficient in its intended regime.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why generic SpMM underperforms. A general SpMM (Yang et al. , 2018 ; Qian et al. , 2026 ; Huang et al. , 2020 ) computes 𝐌 = 𝐀𝐁 \\\\mathbf{M}=\\\\mathbf{A}\\\\mathbf{B} where 𝐀 \\\\mathbf{A} is the sparse n × n n\\\\times n matrix, and B and M are dense n × d n\\\\times d matrices, where d ≪ n d\\\\ll n . This kernel computes one output row at a time 𝐌 [ i , : ] = ∑ k : A i k ≠ 0 A i k 𝐁 [ k , : ] \\\\mathbf{M}[i,:]=\\\\sum_{k\\\\,:\\\\,A_{ik}\\\\neq 0}A_{ik}\\\\mathbf{B}[k,:] , where k k runs only over the columns in which row i i of 𝐀 \\\\mathbf{A} has a nonzero, A i k A_{ik} is that nonzero value, 𝐁 [ k , : ] \\\\mathbf{B}[k,:] is the k k -th row of the dense input matrix, and 𝐌 [ i , : ] \\\\mathbf{M}[i,:] is the output row being accumulated. These kernels are tuned for tall-skinny 𝐁 \\\\mathbf{B} ( d ≪ n ) (d\\\\ll n) where a row 𝐁 [ k , : ] \\\\mathbf{B}[k,:] is short, stays cache-resident, and is reused across every nonzero that references column k k of 𝐀 \\\\mathbf{A} . This setting is the most common application of SpMM when used in iterative solvers and graph neural networks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In FastAlign , 𝐁 \\\\mathbf{\\\\mathbf{B}} is wide, not tall-skinny. A row 𝐁 [ k , : ] \\\\mathbf{B}[k,:] is now θ ( n ) \\\\theta(n) , so it may no longer stay cache-resident and cannot be reused across the nonzeros that reference column k k of 𝐀 \\\\mathbf{A} , which limits data reuse in FastAlign . In the worst case, each nonzero from the sparse matrix fetches its corresponding dense row from memory, giving 𝒪 ( nnz ⋅ n ) \\\\mathcal{O}(\\\\textit{nnz}\\\\cdot n) memory traffic. Since SpMM is a memory-bound operation, the standard SpMM libraries (e.g., Intel MKL) hurt the performance of FastAlign .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The fix: custom SpMM kernel. We design this kernel with a column blocking model to improve cache residency. We partition dense matrices 𝐁 \\\\mathbf{B} and 𝐌 \\\\mathbf{M} into Q = ⌈ n 2 / b c ⌉ Q=\\\\lceil n_{2}/b_{c}\\\\rceil narrow blocks of b c b_{c} contiguous columns and process one block at a time. To compute block q q of output, whose columns start at j q = ( q − 1 ) b c j_{q}=(q-1)b_{c} , we fill it one output row at a time: 𝐌 [ i , j q : j q + b c ] = ∑ k : 𝐀 i k ≠ 0 𝐀 i k 𝐁 [ k , j q : j q + b c ] \\\\mathbf{M}[i,j_{q}:j_{q}+b_{c}]=\\\\sum_{k\\\\,:\\\\,\\\\mathbf{A}_{ik}\\\\neq 0}\\\\mathbf{A}_{ik}\\\\mathbf{B}[k,j_{q}:j_{q}+b_{c}] . Since each memory access within each column block computation is limited to a narrow partition of dense matrix, the B partition needs to reside in cache, reduced to n 1 × b c n_{1}\\\\times b_{c} instead of n 1 × n 2 n_{1}\\\\times n_{2} that does not fit into L2/L3. This will increase cache hits when consecutive rows share similar columns of nonzeros.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 traces one block: To fill the b c b_{c} output block 𝐌 [ k , j q : j q + b c ] \\\\mathbf{M}[k,j_{q}:j_{q}+b_{c}] , the kernel iterates over all rows of 𝐀 \\\\mathbf{A} accumulating the 𝐀 i k 𝐁 [ k , j q : j q + b c ] \\\\mathbf{A}_{ik}\\\\mathbf{B}[k,j_{q}:j_{q}+b_{c}] for each nonzero. Since each memory access within a column-block computation is limited to a narrow partition of the dense matrix, the resident 𝐁 \\\\mathbf{B} partition shrinks to n 1 × b c n_{1}\\\\times b_{c} , which fits into L2/L3, instead of the full n 1 × n 2 n_{1}\\\\times n_{2} , which does not.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Payoff. Blocking leaves the arithmetic and total access count unchanged at 𝒪 ( nnz ⋅ n ) \\\\mathcal{O}(\\\\textit{nnz}\\\\cdot n) . It changes which level of memory is accessed. By keeping each 𝐁 \\\\mathbf{B} block cache-resident, it reduces the DRAM traffic for reading 𝐁 \\\\mathbf{B} , converting memory-bound stream into a cache-resident reuse pattern. For a SpMM with the structure of our application, this improved cache reuse accelerates the SpMM computations throughout our model.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5. Domain-specific Kernel Fusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"According to Algorithm 1 , the cost matrix 𝐂 \\\\mathbf{C} assembly (Line 12), and Sinkhorn dual vector update (Lines 14,15, Eq. 3 ) are written as compact algebraic expressions; however, a literal computation allocates and populates a n 1 × n 2 n_{1}\\\\times n_{2} intermediate matrix: each log, scale, and add streams a full dense intermediate to memory and back. Since these matrices are large, the memory traffic dominates the computation even though every individual operation is cheap. We merge each algebraic chain into a single path over the output/input matrices. We observed such chains that occur and fused them.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cost matrix 𝐂 \\\\mathbf{C} computation. Cost computation (Line 12) combines four dense terms by scaling, log, and addition: 𝐂 ← 𝐂 node + λ e 𝐂 edge + λ n 𝐂 nbr + λ a 𝐂 anc \\\\mathbf{C}\\\\leftarrow\\\\mathbf{C}_{\\\\text{node}}+\\\\lambda_{e}\\\\mathbf{C}_{\\\\text{edge}}+\\\\lambda_{n}\\\\mathbf{C}_{\\\\text{nbr}}+\\\\lambda_{a}\\\\mathbf{C}_{\\\\text{anc}} . Naively computing this results in materializing at least 6 separate dense n 1 × n 2 n_{1}\\\\times n_{2} matrices. The fused kernel instead visits each entry ( i , j ) (i,j) once, loads the five contributing values, applies the scale/log/add, and writes 𝐂 i j \\\\mathbf{C}_{ij} in one pass, without any intermediates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sinkhorn dual updates. Each dual update is a log-sum-exp reduction over the cost matrix (Eq. 3 ): the row reduction that updates a and the column reduction that updates b. Evaluated naively, each update shifts, exponentiates, and reduces the cost matrix, then applies a log and a final shift which materializes a full n 1 × n 2 n_{1}\\\\times n_{2} intermediate matrix at each step. We instead design a single fused kernel that applies element-wise shift and exponential inside the reduction, in one streaming pass with a max-shift for stability, so the n 1 × n 2 n_{1}\\\\times n_{2} intermediate is not materialized. The same kernel serves both updates: applied to 𝐂 \\\\mathbf{C} to refresh 𝐚 \\\\mathbf{a} and to 𝐂 T \\\\mathbf{C}^{T} to refresh 𝐛 \\\\mathbf{b} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.6. GPU-aware Optimizations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The computational reformulation of FastAlign naturally maps to GPU architectures, as the performance-critical matrix operations reduce to sparse and dense linear algebra kernels. Our GPU implementation builds on the sparsity-aware formulation (Section 3.3 ) and domain-specific kernel fusion (Section 3.5 ) using cuSPARSE for sparse matrix operations and cuBLAS for dense linear algebra. Rather than using these libraries out of the box, we tune their performance by selecting memory layouts and SpMM algorithms to match the structure of each operation in the alignment pipeline.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Minimizing latency. Unlike the CPU implementation, the GPU implementation must carefully minimize both device-side memory movement and host–device transfers to reduce latency. Cost computation involves sparse–dense–sparse matrix products, which we break down into two SpMM calls with transposing the intermediate dense matrices. As explicitly materializing transposed matrices is expensive, we use transpose views to avoid additional memory allocation. To minimize host–device communication, all graphs, cost matrices, and alignment matrices remain resident in GPU memory, while intermediate buffers are reused throughout the alignment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GPU-specific kernel fusion. The Sinkhorn dual updates assign one thread block per column and execute two streaming passes: the first reduces the column minimum using warp-level shuffles ( __shfl_down_sync ), while the second accumulates the shifted exponentials. The same kernel serves the column reduction for updating 𝐛 \\\\mathbf{b} by operating on a transpose view of 𝐂 \\\\mathbf{C} , produced by a tiled transpose kernel that stages 32 × 32 32\\\\times 32 tiles in shared memory with + 1 +1 column padding to eliminate bank conflicts, achieving coalesced reads and writes in a single pass.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Parallel streams and reusing SpMM plans. To improve GPU utilization, FastAlign issues independent kernels for separate graphs across parallel, non-blocking CUDA streams. For example, the row-normalization kernels of the two graph intra-distance matrices (Line 10 of Algorithm 1 ) run concurrently on separate streams rather than sequentially. Furthermore, standard cuSPARSE SpMM calls incur per-invocation overhead for planning, handle setup, and buffer allocation. Because these calls repeat within a loop, FastAlign pre-computes the plan once and reuses it across iterations. This amortizes the initialization overhead, ensuring subsequent calls incur only raw execution costs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4. Evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate FastAlign to answer the following research questions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"How accurate is FastAlign compared to SOTA network alignment algorithms?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"How does FastAlign accelerate alignment on real networks?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"How does FastAlign scale with increasing network size?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"How much does each of our techniques contribute to the overall speedups achieved by FastAlign ?\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1. Experimental Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For experiments, we use an AMD EPYC 7763 CPU with 64 cores and 512 GB of memory. For GPU experiments, we use an NVIDIA A100 GPU with 40 GB of memory and CUDA 12.8. All methods are evaluated in a semi-supervised setting, where 20% of the ground-truth node pairs are used as prior knowledge, i.e., anchor links. We report runtime and accuracy metrics averaged over five runs. Runtime measures the end-to-end alignment computation time after the input graphs are read, including all steps required to compute the final alignment matrix. In our experiments, FastAlign exhibits convergence behavior similar to the OT baseline, PARROT.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1.1. Metrics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate the effectiveness of FastAlign compared to the baselines using the following two accuracy metrics. For each test node x ∈ G 1 x\\\\in G_{1} , let x ′ ∈ G 2 x^{\\\\prime}\\\\in G_{2} denote its ground-truth match. We rank all candidate nodes in G 2 G_{2} by their alignment score with x x . A prediction is counted as a hit at K K if x ′ x^{\\\\prime} appears among the top- K K ranked candidates. For a test set with n n node pairs, Hits @ K = # hits @ K n \\\\mathrm{Hits@}K=\\\\frac{\\\\#\\\\mathrm{hits@}K}{n} . Mean Reciprocal Rank (MRR) is measured as the average inverse rank of the ground-truth match: MRR = 1 n ∑ ( x , x ′ ) ∈ 𝒮 test 1 rank ( x , x ′ ) \\\\mathrm{MRR}=\\\\frac{1}{n}\\\\sum_{(x,x^{\\\\prime})\\\\in\\\\mathcal{S}_{\\\\mathrm{test}}}\\\\frac{1}{\\\\operatorname{rank}(x,x^{\\\\prime})}\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1.2. Baselines\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Following the benchmark study (Yu et al. , 2025a ) , we select six top-accuracy baselines that cover the three main categories of network alignment methods. FINAL (Zhang and Tong, 2016 ) is consistency-based; BRIGHT (Yan et al. , 2021 ) and NetTrans (Zhang et al. , 2020 ) are embedding-based; and JOENA (Yu et al. , 2025b ) , PARROT (Zeng et al. , 2023 ) , and SLOTAlign (Tang et al. , 2023 ) are OT-based. PARROT is the closest baseline to FastAlign , since our framework preserves the same OT-based alignment formulation. We also evaluated NeXtAlign (Zhang et al. , 2021 ) , but it timed out on most datasets and is omitted.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"4.1.3. Datasets\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 summarizes the real-world datasets we used to evaluate FastAlign against the baselines. For scalability experiments, we generated synthetic graphs using the Erdős–Rényi model with an average node degree of 10 10 (Erdős et al. , 1960 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2. Accuracy Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 compares the Hits@ 10 10 and MRR of FastAlign against all baselines on the real-world datasets. We report GPU results in this table and observe similar accuracy trends on CPU. CPU accuracy results are omitted because several baselines time out on larger datasets, as discussed in the next Subsection. FastAlign and other OT-based methods achieve substantially higher accuracy than consistency-based and embedding-based methods. Compared with state-of-the-art OT-based baselines, FastAlign achieves nearly identical or better accuracy on most datasets, showing that its optimizations do not degrade alignment quality and preserve original OT alignment objective. We attribute the slightly lower accuracy on DBP 15 15 K to floating-point roundoff inconsistencies, especially from fused kernels as they change the order of numerical operations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3. Performance Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 5 compares the runtime of FastAlign against all baselines on real-world datasets using CPU and GPU implementations, respectively. On CPU, missing bars indicate that the corresponding baseline timed out on that dataset, whereas on GPU, all baselines completed. FastAlign achieves a 3.89 × 3.89\\\\times – 9.45 × 9.45\\\\times speedup on CPU and a 2.24 × 2.24\\\\times – 32.54 × 32.54\\\\times speedup on GPU compared to PARROT, the fastest baseline. Across all baselines, the speedup ranges from 3.89 × 3.89\\\\times to 193.90 × 193.90\\\\times on CPU and from 2.24 × 2.24\\\\times to 1321.85 × 1321.85\\\\times on GPU, with the largest speedups achieved over slow baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These significantly high speedups demonstrate the effectiveness of FastAlign ’s computational reinterpretation and optimizations discussed in Section 3 . In general, the speedups increase with dataset size. For datasets of similar size, the speedups follow the sparsity trend: higher sparsity leads to less data movement through memory and therefore larger speedups. GPU speedups are higher than CPU speedups primarily because kernel fusion and GPU-specific optimizations have a stronger impact on GPU execution. In particular, kernel fusion improves data locality and reuse, reducing memory access latency and contributing to the superior performance of FastAlign on GPU.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4. Scalability Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 6 a shows the runtime scalability comparison of FastAlign for synthetic datasets against the baselines that did not time out on graphs with 20 20 K nodes. FastAlign scales more efficiently than all baselines, with speedups increasing as graph size grows — from 0.82 × 0.82\\\\times at 10K nodes to 9.41 × 9.41\\\\times at 110K nodes over the fastest baseline. The overall speedups are in the range 0.82 × 0.82\\\\times – 14.49 × 14.49\\\\times .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FastAlign scales to graphs with up to 110 110 K nodes, while all baselines except BRIGHT run out of memory at smaller graph sizes. The maximum problem size supported by FastAlign is determined by its peak memory usage during Sinkhorn optimization. The peak live state uses nine dense ( n 1 × n 2 ) (n_{1}\\\\times n_{2}) matrices and two sparse graph matrices. Using single precision (Section 3.3 ), the dense matrices require ( 4 ⋅ 9 ⋅ n 1 n 2 = 36 n 1 n 2 ) (4\\\\cdot 9\\\\cdot n_{1}n_{2}=36n_{1}n_{2}) bytes, which bounds the implementation to approximately 110 110 K nodes on a CPU with 512 512 GB memory. As long as they fit in memory, FastAlign aligns the networks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 6 b presents the scalability comparison on GPU using an NVIDIA A100 (80GB variant). FastAlign and BRIGHT scale to 50K-node graphs, while FINAL supports up to 35K nodes, PARROT , JOENA , and SLOTAlign up to 30K, and NetTrans only up to 10K. Although the y-axis is shown in log scale to accommodate the wide range of runtimes, the scalability trend on GPU mirrors the CPU results: FastAlign ’s speedups increase with graph size. From 10K to 50K nodes, it achieves speedups in the range of 4.16 × 4.16\\\\times – 86.88 × 86.88\\\\times over BRIGHT , the fastest baseline. Across all baselines and graph sizes, the speedups are in the range of 4.16 × 4.16\\\\times – 2525.88 × 2525.88\\\\times .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5. Ablation studies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 7 shows the runtime contribution of each FastAlign optimization: sparsity-aware computation, kernel fusion, and custom SpMM. The number on each bar reports the speedup of that artifact over PARROT. To isolate algorithmic gains from platform effects, we include a straightforward reimplementation of the baseline, PARROT in CPP and CUDA as an intermediate baseline. Both CPP and CUDA baselines are only marginally faster than the original PyTorch implementation of PARROT, with the speedups diminishing as graph size grows, confirming that naive reimplementations with a platform change yield no meaningful performance benefit. On smaller networks, we observe a modest speedup from framework conversion alone. However, on larger datasets with at least 10 10 K nodes, the overhead of PyTorch over CPP/CUDA becomes negligible and the C++/CUDA baselines are no longer faster, confirming that FastAlign ’s speedups are driven by its algorithmic optimizations rather than the platform change.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On CPU, sparsity alone improves runtime by 0.8 × 0.8\\\\times – 2.6 × 2.6\\\\times , and adding fusion compounds these gains to 2.2 × 2.2\\\\times – 4.6 × 4.6\\\\times , with custom SpMM further increasing speedups to 2.9 × 2.9\\\\times – 6.6 × 6.6\\\\times . On GPU, sparsity contributes 2.6 × 2.6\\\\times – 4.4 × 4.4\\\\times to the speedup, and adding fusion brings the overall speedup to 2.2 × 2.2\\\\times – 32.5 × 32.5\\\\times . Across both CPU and GPU, all optimizations contribute to performance, with fusion consistently compounding the benefits of sparsity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5. Related Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this section, we review the related work, which can be categorized into two groups: network alignment and optimal transport.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.0.1. Network Alignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Research on network alignment has largely developed along two directions: consistency-based and embedding-based approaches. The first direction is based on the idea that aligned nodes should have structurally and semantically similar positions, so that the connectivity and attributes of a node’s neighbors carry over from one network to the other. IsoRank (Singh et al. , 2008 ) propagates pairwise similarities on the product graph, FINAL (Zhang and Tong, 2016 ) extends this with attribute and edge consistency for attributed graphs, and BIG-ALIGN (Koutra et al. , 2013 ) recovers a near-permutation between adjacency matrices via a Frobenius objective. Computationally, these methods iterate over an 𝒪 ( n 2 ) \\\\mathcal{O}(n^{2}) product or similarity space that dominates cost at scale, motivating coarsening-based designs like MOANA (Zhang et al. , 2019 ) . In terms of accuracy, however, they capture limited global graph geometry, and the consistency premise weakens when the two networks differ substantially (Zhang et al. , 2021 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The second learns node embeddings that encode each network’s topology while pulling anchor pairs together: IONE (Liu et al. , 2016 ) models directional follower/followee context, REGAL (Heimann et al. , 2018 ) factorizes a cross-network similarity matrix, CrossMNA (Chu et al. , 2019 ) fuses intra-/inter-/global features, and NetTrans (Zhang et al. , 2020 ) , BRIGHT (Yan et al. , 2021 ) , and NeXtAlign (Zhang et al. , 2021 ) further improve robustness through learned transformations, restart-based positional features, and consistency-aware sampling, respectively. Computationally, the cost shifts to training: repeated gradient updates over deep encoders, often with pair sampling, are expensive and lack convergence guarantees. In accuracy, they model cross-network relations only indirectly through anchor paths and are sensitive to sampling noise and embedding-space distortion (Yu et al. , 2025b ; Zhang et al. , 2021 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"5.0.2. Optimal Transport\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Optimal transport (OT) has recently been applied to network alignment, representing each graph as a distribution over its nodes and seeking the minimum-cost transport between them (Chen et al. , 2020 ) . Methods differ mainly in how the transport cost is defined. Early designs transport spectral node embeddings (EMD (Nikolentzos et al. , 2017 ) ) or represent graphs through filtered graph signals and Gaussian graph kernels (FGOT (Maretic et al. , 2022 ) , GOT (Petric Maretic et al. , 2019 ) ), while later costs jointly encode node and edge correspondence via the Wasserstein and Gromov–Wasserstein (GW) distances (Xu et al. , 2019 ; Chen et al. , 2020 ) . A prominent line aligns graphs through Gromov–Wasserstein structural discrepancy (S-GWL (Xu et al. , 2019 ) , GraphOTC (O’Connor et al. , 2021 ) ), while GOAT (Saad-Eldin et al. , 2021 ) accelerates it via the Sinkhorn algorithm. More recent work learns the cost instead of fixing it: SLOTAlign (Tang et al. , 2023 ) encodes a GW objective with a parameter-free GNN, CombAlign (Chen et al. , 2025 ) ensembles embedding- and OT-based alignments, and JOENA (Yu et al. , 2025b ) jointly learns embeddings and transport. PARROT (Zeng et al. , 2023 ) builds random-walk topology and consistency regularization into the OT cost and solves it with a constrained proximal point method.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"While OT-based methods achieve higher accuracy, they remain the most computationally demanding category. Their cost stems from repeatedly updating dense 𝒪 ( n 2 ) \\\\mathcal{O}(n^{2}) matrices, and while proximal solvers (Yu et al. , 2025b ) reduce iteration counts, graph sparsity and sparse-linear-algebra primitives remain underexploited. As a result, even the most scalable OT aligners are confined to moderate-sized graphs, a gap our work targets by retaining the accuracy of regularized OT while improving runtime and scalability through a computation-optimized algorithm.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6. Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, we scale OT-based network alignment to large graphs while preserving its accuracy. Instead of designing a new alignment model, we present FastAlign , which keeps the original OT formulation and treats its computation as a small set of recurring sparse-dense operations. FastAlign exploits graph sparsity to skip work over nonexistent edges, fuses memory-bound kernels to cut redundant memory traffic, adds a custom SpMM kernel for the wide dense matrices in OT alignment, and uses GPU-aware optimizations to reuse data on device. Experiments on real and synthetic datasets show that FastAlign matches the accuracy of SOTA OT-based methods while running up to 9.45 × 9.45\\\\times faster on CPU and up to 32.54 × 32.54\\\\times faster on GPU than the strongest OT baseline.\"\n }\n]","isConversational":false}}},{"rowIdx":399,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.10441"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Context by Distinct Information: An Auditable Dirichlet-Process Working Memory for Long, Redundant Context Streams"},"authors":{"kind":"list like","value":["Siddharth Pal","Viktoria Rojkova"],"string":"[\n \"Siddharth Pal\",\n \"Viktoria Rojkova\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.10441v1"},"published":{"kind":"timestamp","value":"2026-07-11T00:00:00","string":"2026-07-11T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 The problem: context is metered in tokens, but information is not"},{"type":"para","text":"Context engineering is the construction and maintenance of the information exposed to a model at inference time. A context stream may contain natural-language turns, document passages, tool outputs, program state, observations, structured records, or mixtures of them. The stream can be long, hierarchical, and multi-rate: a slow-moving fact or entity can remain relevant across thousands of fast local updates. It is also usually redundant. A name recurs across a conversation, a retrieved passage restates an earlier fact, the same tool emits repeated status records, a code or identifier appears hundreds of times, and a log template fires thousands of times."},{"type":"para","text":"The standard memory choices nevertheless scale with tokens. A fixed-state recurrent or state-space model processes the stream in linear time and constant-size inference memory, but folds all prior information into a bounded state. Its content-addressed recall degrades once the number of independently retrievable items exceeds the state capacity, the regime isolated theoretically by copying separations and empirically by recall benchmarks for efficient sequence models (Jelassi et al., 2024 ; Arora et al., 2023 ) . Full attention removes that cap by retaining a key–value entry for every token, but the cache grows with the stream and the compute of a dense read grows quadratically during training. Fixed windows and eviction caches bound the cost, but they decide what survives by recency, fixed roles, or attention statistics rather than by whether the information is already represented."},{"type":"para","text":"These are three answers to the same question: what is the unit of context? If the unit is a token, then repeated evidence is charged repeatedly. If the unit is a distinct item represented in the model’s key space, a long redundant stream can be carried by a much smaller working memory. This paper studies the latter design and, more importantly, the boundary around it. Not every task needs the same past. The experiments separate three regimes:"},{"type":"para","text":"Recall-carried information : a later query depends on a particular earlier item, including a rare one. A content-addressed distinct-item cache is the natural memory."},{"type":"para","text":"Summary-carried information : the target depends on a long-range aggregate or smooth history. A recurrent or state-space path is the natural memory."},{"type":"para","text":"Locality-carried information : the target depends mainly on the immediate past. A recency window is the natural memory."},{"type":"para","text":"The contribution is not a claim that novelty is universally superior. It is a context decomposition that separates recall-carried, summary-carried, and locality-carried information and assigns each to a distinct memory structure, together with trained end-to-end evidence locating where each one pays and where it does not, so that the novelty-gated cache is placed as one component of a working memory rather than a universal compressor."},{"type":"heading","level":1,"text":"2 Theoretical anchor: remembering distinct items, not tokens"},{"type":"para","text":"The companion mechanism paper (Pal and Rojkova, 2026 ) places an allocate-on-novelty cache between fixed-state recurrence and full attention. We use that result as the theoretical anchor and summarize only the parts needed to make the present paper self-contained."},{"type":"heading","level":2,"text":"2.1 The Dirichlet-process allocation rule as a memory operator"},{"type":"para","text":"Let the cache contain occupied slots i i , each with key κ i \\kappa_{i} , value ν i \\nu_{i} , usage count, and provenance. For an incoming key–value pair ( k t , v t ) (k_{t},v_{t}) , novelty is"},{"type":"para","text":"where the experiments use cosine similarity. The allocation rule is"},{"type":"para","text":"This is the DP-means rule, the small-variance maximum-a-posteriori limit of a Dirichlet-process mixture (Kulis and Jordan, 2012 ) , used here not for offline latent-variable inference but as the write operator of the working memory. Repeats merge into the slot they already spawned; genuinely new items open a slot. On streams generated from a finite set, the slot count therefore saturates near the number of distinct items. A query q q reads over occupied slots only,"},{"type":"para","text":"so the read cost follows the number of slots rather than the number of observed tokens."},{"type":"para","text":"The companion paper establishes the mechanism-level claims that the present work relies on: at fourfold redundancy, the cache matches full-attention associative recall with a fourfold smaller cache; it dominates recency, heavy-hitter (Zhang et al., 2023 ) , sink-plus-window (Xiao et al., 2024 ) , and SnapKV-style (Li et al., 2024 ) fixed-budget policies on the recall–size frontier; beside a state-space backbone it answers both an item-recall query and a long-range aggregate at the lowest memory tested; and a minimal two-parameter novelty-threshold gate trained on task loss recovers the allocation rule exactly, whereas an over-parameterized saliency gate fails. The relevant inductive bias is novelty, not gate capacity (Pal and Rojkova, 2026 ) ."},{"type":"heading","level":2,"text":"2.2 Static and surprise-adaptive context"},{"type":"para","text":"The base cache uses a fixed threshold τ \\tau . The companion paper also defines a surprise-adaptive form for non-stationary streams. If o t ∈ { 0 , 1 } o_{t}\\in\\{0,1\\} records whether step t t opened a slot, an exponential moving average"},{"type":"para","text":"tracks the recent allocation rate. The effective budget is"},{"type":"para","text":"and least-used slots are evicted whenever the occupied count exceeds M t M_{t} . The cache heats during bursts of novel information and consolidates when the stream settles. At matched average cost it improves recall when the working set varies, but correctly offers no advantage when the demand is stationary (Pal and Rojkova, 2026 ) . In the trained windowed models below, an explicit adaptive schedule turns out to be redundant because the learned soft gate already adapts token by token; that negative result is part of the present paper."},{"type":"heading","level":2,"text":"2.3 Why this is a context-engineering primitive"},{"type":"para","text":"The cache changes context construction in three ways."},{"type":"para","text":"First, it changes the scaling variable from observed tokens to occupied novelty classes. The claim is operational, not philosophical: the cache size is the admitted-slot count, and all reported savings use that measured count."},{"type":"para","text":"Second, it makes the retained context inspectable when the frontend gives slots recognizable semantics. A slot can expose its first occurrence, merged occurrences, usage, key, value, and source offsets. An eviction cache can report what survived, but not why two repeated observations became one object; a fixed-state model cannot expose a discrete retained set at all. The auditability claim is therefore strongest in the structured streams below, where the slot table literally becomes a template catalog, formulary, code set, or place map. It is not guaranteed merely by using the gate: interpretability depends on the key representation and merge policy."},{"type":"para","text":"Third, it separates context construction from context reading. Allocation can be more expensive than appending every token, but repeated reads operate on the much smaller occupied set. That build-versus-read trade is appropriate when context is long-lived, queried repeatedly, or memory-bound, and inappropriate for a single read over a short sequence."},{"type":"heading","level":1,"text":"3 Position: context engineering already allocates on novelty by hand"},{"type":"para","text":"The mechanism is new as a learned key–value memory for a deep sequence model, but the allocation principle is not. Several mature systems already construct context by keeping a representative only when the incoming observation is sufficiently new."},{"type":"para","text":"Online log parsing. Drain, Spell, and related production parsers stream log lines, compare each line to templates retained so far, open a template when the line is dissimilar to all of them, and otherwise merge it into the nearest template while generalizing mismatched positions (Du and Li, 2016 ; He et al., 2017 ) . That is allocate-on-novelty with a domain-specific similarity and merge. The DP cache expresses the same loop as a differentiable memory whose threshold, similarity, and merge can in principle be learned."},{"type":"para","text":"Macro-tokenization. Long structured records are commonly aggregated into one token per claim, admission, processing step, or other schema-defined unit. A macro-token is a hand-authored slot: context is allocated per process unit instead of per raw event. The cache replaces the schema-fixed allocation with a content-driven one and can apply the same rule across levels of a hierarchy."},{"type":"para","text":"Keyframe selection. Visual SLAM systems in the ORB-SLAM line insert a keyframe when the current view is sufficiently novel against retained keyframes and close loops through nearest-neighbour lookup over that store (Gálvez-López and Tardós, 2012 ; Mur-Artal et al., 2015 ) . The memory grows with the explored environment rather than the length of the trajectory. Recent feed-forward streaming reconstruction instead compresses the past into fixed roles such as an anchor, a recency window, and a trajectory summary (GCT, 2026 ) ; this is well suited to locality-carried geometry but cannot answer a revisit query once the relevant place has left the fixed window."},{"type":"para","text":"These systems are not presented as baselines for a universal context engine. They establish that novelty allocation is already a useful context-construction pattern wherever repeated observations refer to a finite or slowly growing set of objects. The paper’s question is whether the same principle survives end-to-end representation learning and how it should be combined with recurrence and recency."},{"type":"heading","level":1,"text":"4 The working-memory component"},{"type":"heading","level":2,"text":"4.1 Backbone and reads"},{"type":"para","text":"The sequence model is a standard residual stack: token embedding, ℓ \\ell blocks, and a linear head. Within each block the mixing operator, or read, is one of:"},{"type":"para","text":"a diagonal state-space convolution, implemented as a learned per-channel decay applied causally, representing the fixed-state endpoint (Gu et al., 2022 ) ;"},{"type":"para","text":"full causal attention, representing the growing token-cache endpoint;"},{"type":"para","text":"novelty-gated causal attention, in which a token enters the key–value cache only if its key is dissimilar to every earlier admitted key, and later queries attend over admitted slots alone."},{"type":"para","text":"The admitted fraction is the number of cache slots divided by the number of input tokens and is measured during training. A coupled read adds a recurrent path beside the content-addressed path inside the block: the recurrence integrates; the cache recalls. We measure four coupled forms: state-space plus full attention, state-space plus novelty-gated cache (the deployment component), all three paths together, and a selective input-gated recurrence replacing the plain state-space path (Gu and Dao, 2023 ) ."},{"type":"para","text":"This architecture should be read as a context decomposition rather than a single selector. The recurrent path carries compressed history; the gated cache carries distinct addressable items; a windowed baseline carries immediate local context. The experiments determine which path matters for which target."},{"type":"heading","level":2,"text":"4.2 What counts as a context item"},{"type":"para","text":"The gate operates on keys, not on an externally guaranteed ontology. A context item is therefore the unit emitted by the frontend and separated by the learned key geometry. In the control studies it is a character token. In the structured studies it is an event with categorical and hierarchical features. In a future multi-turn or retrieval-augmented system it could be a turn, sentence, chunk, tool result, entity mention, or explicit fact, but those frontends are not evaluated here and no claim is made that cosine novelty alone will discover those units without representation learning and provenance-aware merging."},{"type":"para","text":"For structured events, the frontend encodes two fused streams: a categorical stream containing event type, code, and code system, and a numeric stream containing amounts and durations. Feature-wise modulation fuses the streams (Perez et al., 2018 ) . Explicit boundary tokens mark hierarchy levels such as line, claim, and admission and provide natural cache segmentation points. The claims next-code study uses the categorical stream with boundary tokens. The redesigned cost task adds the larger outpatient corpus and regularization; the original plan for the numeric fusion remains a broader frontend direction. The character-level control uses no domain frontend so the read comparison is clean."},{"type":"heading","level":2,"text":"4.3 Auditable working memory"},{"type":"para","text":"Each retained slot can be represented as"},{"type":"para","text":"The model still learns the key space, so a slot is not automatically a human concept. In the structured measurements, however, the slots align with explicit domain objects. The question “what context did the model retain?” can then be answered by the slot table itself. This is a separate design objective from cache reduction and motivates retaining provenance even when only the key and value are needed for prediction."},{"type":"heading","level":1,"text":"5 Mechanism evidence on real context streams"},{"type":"para","text":"Four public, ground-truthed streams test whether the distinct-item property exists outside the synthetic probe. The first three also appear in the companion mechanism paper; here they establish the kinds of context for which an inspectable slot table is natural."},{"type":"para","text":"On Loghub HDFS, whitespace-tokenized lines, positional match similarity, and wildcard generalization on merge turn the cache into an online log parser without further engineering. It recovers 15 templates against 14 ground-truth templates and groups lines at 0.89 accuracy, compressing 2,000 lines into 15 inspectable slots. This is not competitive with tuned production parsers on more diverse logs: on BGL, the bare token similarity reaches only 0.47, where engineered parsers do better. The point is that the cache recovers the parser family’s core allocation loop, not that it displaces mature systems (Zhu et al., 2023 ) ."},{"type":"para","text":"On the MIMIC-IV open demo, 18,087 prescription events yield 628 slots, one per distinct drug to within three, and nearest-slot reading recovers the administration route at 0.995, matching an attention memory that stores every event at 29 times the size (Johnson et al., 2023 ) . On DE-SynPUF, 150,000 inpatient diagnosis events yield 3,933 slots against 3,981 distinct codes and recall each code’s ICD-9 chapter at 0.989; a fixed-budget eviction cache at one quarter of the distinct count collapses to 0.30 (CMS, 2010 ) ."},{"type":"para","text":"The KITTI row makes the context interpretation particularly clear. Places are ten-metre occupied cells and the query is a revisit: a frame whose place was last seen at least 500 frames earlier. At a budget equal to the distinct-place count, the sliding window recalls none of the revisits on any of three sequences because every revisit exceeds the matched window. The heavy-hitter is unstable across sequences, from 0.34 to 1.00, because it retains popular places and drops the tail. Allocate-on-novelty recalls every revisit on every sequence while holding exactly one slot per place. The memory grows with the environment and remains flat along the trajectory, the invariance a mapping context should have (Geiger et al., 2012 ) ."},{"type":"para","text":"Across these domains the retained context becomes a concrete object: template catalog, formulary, code set, or place map. The companion paper adds two mechanism-level boundaries relevant here. First, the threshold does not require per-domain tuning across the recommendation, clinical, and claims streams: any value on a broad plateau yields the same distinct-item cache. Second, on bursty BGL alerts, a surprise-adaptive budget recalls 0.94 at 18 average slots where a matched fixed budget recalls 0.75; on nearly stationary MovieLens windows it ties the matched fixed budget, as it should (Pal and Rojkova, 2026 ) ."},{"type":"heading","level":1,"text":"6 A trained control study: summary plus distinct recall at half the cache"},{"type":"para","text":"Mechanism-level allocation does not establish what happens when the cache is embedded in a model trained end to end, where the representations themselves move under the gate. Character-level language modeling on enwik8 isolates the read without a domain frontend. We train four-layer residual stacks of width 256, sequence length 512, batch size 32, and 6,000 steps over the 90 MB training split, using identical schedules, and evaluate bits per character on held-out data (Mahoney, 2006 ) ."},{"type":"para","text":"Four findings build on one another."},{"type":"para","text":"First, novelty-gated attention reaches 2.073 versus 2.087 for full attention, a 0.014-bit margin within expected run-to-run variation at this scale and therefore treated as parity rather than a win. It does so while admitting 49% of tokens. The fixed-state path, with no item cache, trails both by about 0.11 bits per character. At this horizon, half the learned keys carry the content-addressed information the task uses."},{"type":"para","text":"Second, adding the state-space path to full attention reaches 1.932, improving by 0.155 bits over attention alone. The scale of this gain is an order of magnitude larger than the attention-versus-gated difference and is consistent with the two paths doing different work: the recurrence carries smooth positional and local structure, while attention carries addressable recall."},{"type":"para","text":"Third, composing those two results gives the proposed component. State-space plus gated cache reaches 1.877 at an attended fraction of 0.53. It is nominally 0.055 below state-space plus full attention, a single-seed margin we do not rely on. The central statement is that restricting the content-addressed path to the novelty-kept half of the tokens gives up nothing against coupling recurrence with full attention."},{"type":"para","text":"Fourth, retaining both full attention and the gated cache reaches 1.836. It saves no cache because full attention remains, so it is not a deployment candidate. Its role is evidential: if the gated cache were merely a cheaper approximation to attention, adding it beside full attention should contribute nothing. Instead it improves the coupled read by a further 0.096 bits. Organizing context by distinct content supplies an inductive view not automatically extracted by dense attention."},{"type":"para","text":"Three boundaries accompany the table. The models are small and far from the enwik8 state of the art; the comparison is internal to matched backbones, not a leaderboard claim. The headline entries are single-seed. And the saving is in cache and inference read cost, not in training clock time: the unoptimized novelty scan is quadratic and makes the gated read slower per step than attention (1,122 versus 882 seconds; 1,404 for the component)."},{"type":"para","text":"The seed boundary has been retired. Retraining the five principal reads over three seeds on Apple-silicon hardware gives: state-space 2.191 ± \\pm 0.016, full attention 2.029 ± \\pm 0.064, gated read 2.014 ± \\pm 0.065 at fraction 0.49, coupled read 1.933 ± \\pm 0.009, and component 1.922 ± \\pm 0.035 at fraction 0.51. Across ten contiguous validation blocks per seed, the gated read is at or below full attention in 29 of 30 seed-block pairs. The component is below full attention in all 30. Against the full-attention coupled read, the component wins all ten blocks on two seeds and loses all ten on the third. Thus the durable result remains parity with the full-attention coupling at half the attended tokens; any further margin is within seed variation."},{"type":"para","text":"A selective, input-gated recurrence changes the best text result. Paired with the gated cache it reaches 1.709 ± \\pm 0.008 bpc at an admitted fraction of 0.61 over three seeds, two tenths below the plain component with every seed separated, below the three-path read’s 1.836, and with the tightest spread in the study. On concatenated source code the same coupling reaches 1.394 ± \\pm 0.019 at fraction 0.50, four tenths below its cache-free ablation (1.811 ± \\pm 0.008) and about half a bit below full attention, whose seeds vary by ± \\pm 0.28. This is the strongest text coupling measured. It is also sharply domain-specific: on the claims task below it collapses, showing that the recurrent summary operator must match the stream even when the novelty cache remains useful."},{"type":"heading","level":1,"text":"7 Context scaling: the advantage grows with horizon"},{"type":"para","text":"The decisive question for long-context engineering is how the gap changes with horizon. We sweep sequence length over 256, 512, and 1,024 tokens, adjust batch size to 24, 16, and 8 to remain feasible, and train full attention, novelty-gated attention, and the component for 3,000 steps at each horizon. Within each horizon the reads share the same training budget, so those gaps are clean. Across horizons, the fixed step count means longer sequences receive a different optimization budget; absolute loss trends across horizons are not the claim. These runs are also shorter than the 6,000-step control above."},{"type":"para","text":"The component remains nearly flat as the horizon quadruples (2.42, 2.41, 2.39), while matched attention degrades (2.36, 2.46, 2.69). The gap moves from − 0.060 -0.060 at 256, through a favorable but non-claimable + 0.051 +0.051 within spread at 512, to + 0.300 +0.300 at 1,024 with cleanly separated spreads. The novelty-gated read tracks attention at each horizon, so the parity result is not confined to 512 tokens. The component’s admitted fraction falls from approximately 0.45 to 0.42 as context lengthens: the longer context accumulates more redundancy, and the memory saving grows where the performance advantage does."},{"type":"para","text":"The short-horizon reversal is part of the result. At 256 tokens the relevant context fits comfortably in attention and the recurrent path contributes no measurable benefit. The component is a long-context design, not a universal replacement for short attention."},{"type":"heading","level":1,"text":"8 Which context to keep depends on how the task uses the past"},{"type":"para","text":"A frozen-threshold sweep traces the trained cache-size frontier; a sliding-window read at matched budgets determines what that frontier means. The result is deliberately reported against a novelty-first interpretation."},{"type":"para","text":"Thinning helps both policies on short-horizon character prediction. The gated read improves monotonically as its admitted fraction falls from 0.75 to 0.20, from 2.350 to 2.305 bpc at 3,000 steps. The window improves as it narrows, and the smallest window is the best point in the sweep: 2.262 at a 0.12 fraction. Near a matched fraction of 0.2, the policies are at parity: 2.305 at 0.20 for novelty versus 2.312 at 0.22 for the window. The task is locality-carried, so recency is the correct context policy."},{"type":"para","text":"Training the deployed budget policies end to end reveals a second boundary. Sink-plus-window trains as stably as attention and the SnapKV-style read nearly so, but the heavy-hitter read trains poorly: 2.69 and 3.14 bpc at its two budgets versus 2.09 for attention. Selection by accumulated attention mass is noisy while the attention itself is untrained, and the hard selection feeds the noise back into learning. Heavy-hitter eviction is a plausible inference-time policy; it is a poor training-time primitive in this experiment. Novelty does not depend on the model’s own accumulated saliency statistics and is therefore better behaved during learning."},{"type":"para","text":"Two corpus controls narrow the interpretation further. Source code is substantially more compressible than prose by the attention baseline (1.62 versus 2.35 bpc), yet the learned gate settles near half on both (0.47 prose and 0.50 code). The admitted fraction does not measure corpus-level compressibility; it reflects local similarity in the learned key space. On code, the gated read has a small deficit to attention (1.640 versus 1.622), whereas on prose it has a small surplus. A selective recurrence without attention reaches 1.912 at the 6,000-step budget, within the coupled reads’ seed band, reinforcing that this short control corpus does not require strong item-addressed recall."},{"type":"para","text":"The sweep also asks which occurrence of a repeated key should be kept. Allocate-on-novelty keeps the first. Its mirror keeps only the latest occurrence of each group of similar earlier keys, matching the copying behavior associated with induction heads (Olsson et al., 2022 ) . Under the same protocol, keep-latest reaches 2.267 ± \\pm 0.002 bpc at an admitted fraction of 0.21 over three seeds. It beats every keep-first threshold point at less than half their fraction, beats the matched 0.22 sliding window (2.312), and is statistically tied with the narrowest window’s 2.262 at fraction 0.12. On source code it is the steadiest read measured, 1.731 ± \\pm 0.043 versus 1.94 ± \\pm 0.28 for full attention and keep-first at that budget."},{"type":"para","text":"This revises the trained read, not the mechanism. At the mechanism level a slot represents the whole cluster, and the companion paper’s merge study found the stored occurrence immaterial to the result. In a trained attention read, however, the retained value carries positional context, and the most recent occurrence can be the useful one. That result is again domain-dependent: on the thousand-event claims task, keep-latest scores 8.55 ± \\pm 0.17, behind keep-first at 8.48 ± \\pm 0.08, behind full attention, and well behind the component. Recency wins when the repeated occurrence’s latest local context matters; first-allocation wins when the slot is intended to represent a stable distinct entity."},{"type":"heading","level":1,"text":"9 A recall-carried long-context task: the crossover on real claims"},{"type":"para","text":"The control corpus identifies boundaries but does not force distinct-entity recall. The end-to-end structured task is next-code prediction on DE-SynPUF inpatient claims. For each member, claims are ordered by date and represented as a member boundary followed by, for each claim, a claim boundary and its ICD-9 diagnosis codes. The model predicts the next token. Evaluation uses bits per event and top-1 accuracy on held-out members, so generalization is across patients rather than within a patient’s history. The stream contains 576,885 events and a vocabulary of 4,096 codes and boundaries. Four reads are trained for 3,000 steps at three horizons, with the batch scaled to hold the token budget fixed at the two longer horizons."},{"type":"para","text":"At 256 and 512 events, all four reads tie within seed spread. At 1,024 events the ordering separates. The component reaches 8.085 ± \\pm 0.040 bits per event, 0.311 below full attention with non-overlapping spreads, while attending over 45% of events; the plain coupled read follows at 8.107. The crossover between 512 and 1,024 events matches the shape of the enwik8 horizon sweep. An event contains more semantic content than a character, and the member history relevant to the next diagnosis spans hundreds of events."},{"type":"para","text":"The matched-budget policies determine which context rule is correct. The sliding window that led the short character sweep trails the component by more than half a bit. The strongest deployed policy, the heavy hitter, trails by about a third. The rare and distinct codes matter, so popularity and recency are the wrong retention criteria. The fixed-state backbone at the short horizon scores 8.021, confirming the bounded-state floor. As in the context sweep, only within-horizon comparisons are claimed because the training step count is fixed across horizons."},{"type":"para","text":"The selective coupling that leads prose and code fails severely here. At the 1,024-event horizon, the selective backbone alone reaches 9.64 ± \\pm 0.07 and with the cache 9.77 ± \\pm 0.05, approximately 1.5 bits behind the plain component. The input-gated recurrence that suits smooth text streams fails on the spiky categorical stream, and the cache cannot rescue a failed summary path. There is no universal recurrent companion: selective summary wins text; plain linear summary wins claims; keep-latest wins text budgets and loses claims. What is stable across the three corpora is narrower: the best measured read on prose, code, and claims contains the novelty-gated cache, while the domain determines the other path."},{"type":"heading","level":1,"text":"10 The rest of the program, measured"},{"type":"heading","level":2,"text":"10.1 Incremental cost"},{"type":"para","text":"At inference, the gate is incremental. The cache contains only admitted keys; each arriving key is compared with the occupied set; a query reads over the occupied set alone. Streaming 65,536 tokens of width 256 at eightfold redundancy on one CPU, the cache answers a query in 64 microseconds, while attention over the full stream takes 1,355 microseconds: a 21-fold read speedup with an eightfold smaller cache. The cost is allocation: 56 microseconds per incoming token versus 0.3 microseconds for attention’s append. This reproduces at scale the companion paper’s build-versus-read law. The cache is favorable when the retained context is queried repeatedly or memory is binding, not when a sequence is built once and read once."},{"type":"heading","level":2,"text":"10.2 Operational logs at full scale: one clean result and one wall"},{"type":"para","text":"The trained log study uses fully annotated Loghub-2.0 releases under a strict temporal split: the first 90% of each stream for training and a later contiguous slice for evaluation, three seeds per read. Copy-last-template accuracy is reported as the free floor supplied by a storm-dominated log."},{"type":"para","text":"On HDFS, 1.8 million lines and 37 templates, copy-last accuracy is 0.574. Every learned read is far below the uniform loss floor, approximately 0.98 bits per event versus 5.21, and above the copy floor at 0.66 accuracy. The gated read matches full attention at cache fraction 0.42. The component leads attention on all three seeds, 0.975 ± \\pm 0.008 versus 0.990 ± \\pm 0.011 bits, at fraction 0.39. The selective coupling ties the plain component at 0.974 ± \\pm 0.006 and fraction 0.40, showing that its failure on claims is not a universal failure on event data. Keep-latest defines an extreme-memory point: repeated storm lines make the latest duplicate sufficient, reducing the fraction to 0.03–0.04, roughly one tenth of the other reads, at a cost of about 0.09 bits (1.08 ± \\pm 0.06 versus 0.975). It is a frontier point, not the winner."},{"type":"para","text":"On BGL, 4.6 million lines and 320 templates, every read fails under the same protocol. Losses range from 9.5 to 12.5 bits versus an 8.3 uniform floor, accuracy lies far below the copy floor, and seed spreads reach a bit or more. The later slice changes regime: BGL’s alert-template distribution drifts over months. A model confidently wrong on the future slice has encountered distribution shift, not a memory-capacity limit. The trained log claim is therefore restricted to streams whose template distribution is sufficiently stationary across the split. Rolling refits or within-regime evaluation are required for drifting streams."},{"type":"heading","level":2,"text":"10.3 One stream, two targets: recall-carried versus summary-carried"},{"type":"para","text":"The original claims cost head predicted the payment decile of the arriving claim and produced validation losses above the ten-class uniform floor for every read, indicating memorization of training members. That experiment was withdrawn. The redesign adds outpatient claims, increasing the data by approximately twelvefold, plus dropout, weight decay, and best-checkpoint selection on a held-out slice."},{"type":"para","text":"The redesigned task clears the 3.322-bit uniform floor. Every read transfers at 3.268–3.276 bits per claim, with three-seed spreads of ± \\pm 0.003. The ordering, though modest, is diagnostic: the two recurrent-path reads tie at 3.268 for the summary alone and for the component, while full attention and the gated read tie at 3.276. The cache neither helps nor hurts; the recurrent summary carries the signal. The extensions obey the same split to the third decimal: selective recurrence and its coupling reach 3.269–3.270, and keep-latest joins the attention side at 3.278."},{"type":"para","text":"Thus the same underlying stream supports two different context requirements. Next-code prediction is recall-carried and favors the distinct-item cache. Cost forecasting is summary-carried and favors recurrence. Context policy cannot be selected from the corpus alone; it must be selected from the task’s dependency on the past."},{"type":"heading","level":2,"text":"10.4 The trained adaptive budget: a settled negative in this setting"},{"type":"para","text":"Inside a trained model, the surprise-modulated threshold ties the fixed gate to three decimal places on both standard and deliberately bursty enwik8 while admitting more of the stream. The first burst construction had phases longer than the training window, leaving each window quasi-stationary. A repaired test uses bursts four times shorter than the window so every window crosses several regime changes. The result remains a tie at added cost: 1.137 versus 1.137 bpc, admitted fraction 0.71 versus 0.49."},{"type":"para","text":"The interpretation is that the learned soft gate already adapts to content token by token. An explicit budget schedule layered on top is redundant inside these windowed trained models. The adaptive concentration remains useful at the mechanism level for truly streaming caches with phase-level demand swings, as shown by BGL alert storms in the companion study, but is not justified by the present trained backbones."},{"type":"heading","level":2,"text":"10.5 Architecture robustness"},{"type":"para","text":"Widening the four-layer backbone preserves the ordering. At width 384, the component reaches 1.839, the full-attention coupled read 1.849, and attention 1.945. At width 512, attention reaches 1.894, the component 1.823, and the selective coupling 1.686 at admitted fraction 0.63. The width axis is therefore covered at 256, 384, and 512 with the same qualitative result."},{"type":"para","text":"At six layers and width 256, the initial 5,000-step experiment inverts: attention and the gated read lead at 1.91–1.92 and the coupled reads trail by approximately 0.08. A plausible interpretation is that additional attention layers learn local composition that substitutes for the recurrent path while the coupled optimization cost remains. Doubling the training budget to 10,000 steps changes the conclusion: the component reaches 1.707 at fraction 0.59 versus 1.731 for attention, with blocked means 1.747 ± \\pm 0.034 versus 1.774 ± \\pm 0.035 in the same direction. The earlier inversion was an optimization artifact. At depth, the coupling requires more training budget."},{"type":"heading","level":1,"text":"11 Context-engineering consequences"},{"type":"para","text":"The experiments support a design rule more specific than generic prompt compression."},{"type":"heading","level":2,"text":"11.1 Three memory structures, three dependency types"},{"type":"para","text":"A practical long-context system should not force every kind of information through one memory policy."},{"type":"para","text":"This decomposition also explains why a single benchmark can be misleading. A locality-dominated corpus makes a window look universally sufficient; a recall probe makes novelty look universally superior; an aggregate target makes recurrence look sufficient. The architecture should be judged on a set of tasks that deliberately separates these dependencies."},{"type":"heading","level":2,"text":"11.2 Context objects, not merely compressed tokens"},{"type":"para","text":"The strongest use of the cache is not to replace a tokenizer with clustering. It is to maintain a set of addressable context objects whose size follows the distinct information active in the stream. In a general context engine, a slot should therefore carry more than a hidden key–value pair: source references, timestamps or offsets, merged evidence, confidence, and an update history. Those additions do not change the measured mechanism, but they turn its inspectable slot set into an auditable working memory."},{"type":"para","text":"The paper does not establish that arbitrary language turns will organize themselves into stable human-readable facts. The real-stream alignments arise because the frontends bind keys to explicit movies, templates, drugs, codes, and places. Applying the method to retrieval-augmented generation or multi-turn assistants requires a frontend and merge semantics that preserve propositions, entities, negation, temporal updates, and source authority. A novelty gate can decide that two representations are close; it cannot by itself decide whether one statement supersedes another or whether two similar passages conflict."},{"type":"heading","level":2,"text":"11.3 The relevant scaling law"},{"type":"para","text":"Let N N be observed context units and K K the number of distinct retained items. Full attention stores O ( N ) O(N) keys and reads O ( N ) O(N) per query. The present cache stores approximately O ( K ) O(K) and reads O ( K ) O(K) , while naive online allocation costs O ( N K ) O(NK) because each incoming key scans occupied slots. When redundancy is high, K ≪ N K\\ll N ; when every item is novel, K ≈ N K\\approx N and the cache offers no asymptotic saving. Approximate nearest-neighbour indexing, hierarchical slots, or batched allocation are engineering requirements for scaling the write path, not changes to the principle."},{"type":"para","text":"The measured incremental result makes the trade concrete: at N = 65 , 536 N=65{,}536 and eightfold redundancy, reading is 21 times faster and the cache is eight times smaller, while each write is much more expensive than attention’s append. The right deployment is persistent, repeatedly queried context, not transient one-shot prompting."},{"type":"heading","level":1,"text":"12 Scope"},{"type":"para","text":"All experiments use public data: Loghub, the MIMIC-IV open demo, CMS DE-SynPUF synthetic claims, KITTI, and enwik8. No proprietary stream, schema, or domain knowledge from an operational deployment enters the experiments. Mechanism-level studies use synthetic keys bound to real entity streams and train-free reads; they validate cache statistics and, for logs, one ground-truthed end task, but are not trained domain models. The trained control, claims task, and context-length sweep are replicated over three seeds, with paired per-block statistics for the control. The corpus-swap and some architecture-robustness measurements are single-seed where stated. Compared runs share hardware; the headline control table uses an NVIDIA T4, while seed replicates, context sweeps, claims, and mechanism studies use Apple M1 and M4 systems."},{"type":"para","text":"The evidence does not yet include long multi-turn conversations, retrieval-augmented generation, tool-using agents, or production context builders. The paper therefore does not claim that the current cosine gate and merge rule are sufficient for semantic facts in those systems. It establishes a lower-level result: a learned novelty-gated content-addressed path can retain distinct information at substantially below token-count cost, can be coupled with recurrent summaries without surrendering performance, and has measurable regimes where it is preferable to recency or fixed-budget eviction. Extending the frontend to proposition-level context, preserving contradictions and temporal updates, evaluating answer faithfulness and provenance, and optimizing the allocation scan are the next steps."},{"type":"para","text":"Within those limits, the central context-engineering claim is deliberate: context need not be a transcript, a fixed summary, or a window. It can be a structured working set whose size follows the distinct information the task may need to recall, while recurrence carries aggregates and recency carries local texture."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 The problem: context is metered in tokens, but information is not\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Context engineering is the construction and maintenance of the information exposed to a model at inference time. A context stream may contain natural-language turns, document passages, tool outputs, program state, observations, structured records, or mixtures of them. The stream can be long, hierarchical, and multi-rate: a slow-moving fact or entity can remain relevant across thousands of fast local updates. It is also usually redundant. A name recurs across a conversation, a retrieved passage restates an earlier fact, the same tool emits repeated status records, a code or identifier appears hundreds of times, and a log template fires thousands of times.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The standard memory choices nevertheless scale with tokens. A fixed-state recurrent or state-space model processes the stream in linear time and constant-size inference memory, but folds all prior information into a bounded state. Its content-addressed recall degrades once the number of independently retrievable items exceeds the state capacity, the regime isolated theoretically by copying separations and empirically by recall benchmarks for efficient sequence models (Jelassi et al., 2024 ; Arora et al., 2023 ) . Full attention removes that cap by retaining a key–value entry for every token, but the cache grows with the stream and the compute of a dense read grows quadratically during training. Fixed windows and eviction caches bound the cost, but they decide what survives by recency, fixed roles, or attention statistics rather than by whether the information is already represented.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These are three answers to the same question: what is the unit of context? If the unit is a token, then repeated evidence is charged repeatedly. If the unit is a distinct item represented in the model’s key space, a long redundant stream can be carried by a much smaller working memory. This paper studies the latter design and, more importantly, the boundary around it. Not every task needs the same past. The experiments separate three regimes:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recall-carried information : a later query depends on a particular earlier item, including a rare one. A content-addressed distinct-item cache is the natural memory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Summary-carried information : the target depends on a long-range aggregate or smooth history. A recurrent or state-space path is the natural memory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Locality-carried information : the target depends mainly on the immediate past. A recency window is the natural memory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The contribution is not a claim that novelty is universally superior. It is a context decomposition that separates recall-carried, summary-carried, and locality-carried information and assigns each to a distinct memory structure, together with trained end-to-end evidence locating where each one pays and where it does not, so that the novelty-gated cache is placed as one component of a working memory rather than a universal compressor.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Theoretical anchor: remembering distinct items, not tokens\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The companion mechanism paper (Pal and Rojkova, 2026 ) places an allocate-on-novelty cache between fixed-state recurrence and full attention. We use that result as the theoretical anchor and summarize only the parts needed to make the present paper self-contained.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 The Dirichlet-process allocation rule as a memory operator\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let the cache contain occupied slots i i , each with key κ i \\\\kappa_{i} , value ν i \\\\nu_{i} , usage count, and provenance. For an incoming key–value pair ( k t , v t ) (k_{t},v_{t}) , novelty is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where the experiments use cosine similarity. The allocation rule is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is the DP-means rule, the small-variance maximum-a-posteriori limit of a Dirichlet-process mixture (Kulis and Jordan, 2012 ) , used here not for offline latent-variable inference but as the write operator of the working memory. Repeats merge into the slot they already spawned; genuinely new items open a slot. On streams generated from a finite set, the slot count therefore saturates near the number of distinct items. A query q q reads over occupied slots only,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so the read cost follows the number of slots rather than the number of observed tokens.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The companion paper establishes the mechanism-level claims that the present work relies on: at fourfold redundancy, the cache matches full-attention associative recall with a fourfold smaller cache; it dominates recency, heavy-hitter (Zhang et al., 2023 ) , sink-plus-window (Xiao et al., 2024 ) , and SnapKV-style (Li et al., 2024 ) fixed-budget policies on the recall–size frontier; beside a state-space backbone it answers both an item-recall query and a long-range aggregate at the lowest memory tested; and a minimal two-parameter novelty-threshold gate trained on task loss recovers the allocation rule exactly, whereas an over-parameterized saliency gate fails. The relevant inductive bias is novelty, not gate capacity (Pal and Rojkova, 2026 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Static and surprise-adaptive context\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The base cache uses a fixed threshold τ \\\\tau . The companion paper also defines a surprise-adaptive form for non-stationary streams. If o t ∈ { 0 , 1 } o_{t}\\\\in\\\\{0,1\\\\} records whether step t t opened a slot, an exponential moving average\"\n },\n {\n \"type\": \"para\",\n \"text\": \"tracks the recent allocation rate. The effective budget is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and least-used slots are evicted whenever the occupied count exceeds M t M_{t} . The cache heats during bursts of novel information and consolidates when the stream settles. At matched average cost it improves recall when the working set varies, but correctly offers no advantage when the demand is stationary (Pal and Rojkova, 2026 ) . In the trained windowed models below, an explicit adaptive schedule turns out to be redundant because the learned soft gate already adapts token by token; that negative result is part of the present paper.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Why this is a context-engineering primitive\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The cache changes context construction in three ways.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, it changes the scaling variable from observed tokens to occupied novelty classes. The claim is operational, not philosophical: the cache size is the admitted-slot count, and all reported savings use that measured count.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, it makes the retained context inspectable when the frontend gives slots recognizable semantics. A slot can expose its first occurrence, merged occurrences, usage, key, value, and source offsets. An eviction cache can report what survived, but not why two repeated observations became one object; a fixed-state model cannot expose a discrete retained set at all. The auditability claim is therefore strongest in the structured streams below, where the slot table literally becomes a template catalog, formulary, code set, or place map. It is not guaranteed merely by using the gate: interpretability depends on the key representation and merge policy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Third, it separates context construction from context reading. Allocation can be more expensive than appending every token, but repeated reads operate on the much smaller occupied set. That build-versus-read trade is appropriate when context is long-lived, queried repeatedly, or memory-bound, and inappropriate for a single read over a short sequence.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Position: context engineering already allocates on novelty by hand\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mechanism is new as a learned key–value memory for a deep sequence model, but the allocation principle is not. Several mature systems already construct context by keeping a representative only when the incoming observation is sufficiently new.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Online log parsing. Drain, Spell, and related production parsers stream log lines, compare each line to templates retained so far, open a template when the line is dissimilar to all of them, and otherwise merge it into the nearest template while generalizing mismatched positions (Du and Li, 2016 ; He et al., 2017 ) . That is allocate-on-novelty with a domain-specific similarity and merge. The DP cache expresses the same loop as a differentiable memory whose threshold, similarity, and merge can in principle be learned.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Macro-tokenization. Long structured records are commonly aggregated into one token per claim, admission, processing step, or other schema-defined unit. A macro-token is a hand-authored slot: context is allocated per process unit instead of per raw event. The cache replaces the schema-fixed allocation with a content-driven one and can apply the same rule across levels of a hierarchy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Keyframe selection. Visual SLAM systems in the ORB-SLAM line insert a keyframe when the current view is sufficiently novel against retained keyframes and close loops through nearest-neighbour lookup over that store (Gálvez-López and Tardós, 2012 ; Mur-Artal et al., 2015 ) . The memory grows with the explored environment rather than the length of the trajectory. Recent feed-forward streaming reconstruction instead compresses the past into fixed roles such as an anchor, a recency window, and a trajectory summary (GCT, 2026 ) ; this is well suited to locality-carried geometry but cannot answer a revisit query once the relevant place has left the fixed window.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"These systems are not presented as baselines for a universal context engine. They establish that novelty allocation is already a useful context-construction pattern wherever repeated observations refer to a finite or slowly growing set of objects. The paper’s question is whether the same principle survives end-to-end representation learning and how it should be combined with recurrence and recency.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 The working-memory component\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Backbone and reads\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sequence model is a standard residual stack: token embedding, ℓ \\\\ell blocks, and a linear head. Within each block the mixing operator, or read, is one of:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"a diagonal state-space convolution, implemented as a learned per-channel decay applied causally, representing the fixed-state endpoint (Gu et al., 2022 ) ;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"full causal attention, representing the growing token-cache endpoint;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"novelty-gated causal attention, in which a token enters the key–value cache only if its key is dissimilar to every earlier admitted key, and later queries attend over admitted slots alone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The admitted fraction is the number of cache slots divided by the number of input tokens and is measured during training. A coupled read adds a recurrent path beside the content-addressed path inside the block: the recurrence integrates; the cache recalls. We measure four coupled forms: state-space plus full attention, state-space plus novelty-gated cache (the deployment component), all three paths together, and a selective input-gated recurrence replacing the plain state-space path (Gu and Dao, 2023 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This architecture should be read as a context decomposition rather than a single selector. The recurrent path carries compressed history; the gated cache carries distinct addressable items; a windowed baseline carries immediate local context. The experiments determine which path matters for which target.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 What counts as a context item\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The gate operates on keys, not on an externally guaranteed ontology. A context item is therefore the unit emitted by the frontend and separated by the learned key geometry. In the control studies it is a character token. In the structured studies it is an event with categorical and hierarchical features. In a future multi-turn or retrieval-augmented system it could be a turn, sentence, chunk, tool result, entity mention, or explicit fact, but those frontends are not evaluated here and no claim is made that cosine novelty alone will discover those units without representation learning and provenance-aware merging.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For structured events, the frontend encodes two fused streams: a categorical stream containing event type, code, and code system, and a numeric stream containing amounts and durations. Feature-wise modulation fuses the streams (Perez et al., 2018 ) . Explicit boundary tokens mark hierarchy levels such as line, claim, and admission and provide natural cache segmentation points. The claims next-code study uses the categorical stream with boundary tokens. The redesigned cost task adds the larger outpatient corpus and regularization; the original plan for the numeric fusion remains a broader frontend direction. The character-level control uses no domain frontend so the read comparison is clean.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Auditable working memory\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each retained slot can be represented as\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model still learns the key space, so a slot is not automatically a human concept. In the structured measurements, however, the slots align with explicit domain objects. The question “what context did the model retain?” can then be answered by the slot table itself. This is a separate design objective from cache reduction and motivates retaining provenance even when only the key and value are needed for prediction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Mechanism evidence on real context streams\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Four public, ground-truthed streams test whether the distinct-item property exists outside the synthetic probe. The first three also appear in the companion mechanism paper; here they establish the kinds of context for which an inspectable slot table is natural.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On Loghub HDFS, whitespace-tokenized lines, positional match similarity, and wildcard generalization on merge turn the cache into an online log parser without further engineering. It recovers 15 templates against 14 ground-truth templates and groups lines at 0.89 accuracy, compressing 2,000 lines into 15 inspectable slots. This is not competitive with tuned production parsers on more diverse logs: on BGL, the bare token similarity reaches only 0.47, where engineered parsers do better. The point is that the cache recovers the parser family’s core allocation loop, not that it displaces mature systems (Zhu et al., 2023 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the MIMIC-IV open demo, 18,087 prescription events yield 628 slots, one per distinct drug to within three, and nearest-slot reading recovers the administration route at 0.995, matching an attention memory that stores every event at 29 times the size (Johnson et al., 2023 ) . On DE-SynPUF, 150,000 inpatient diagnosis events yield 3,933 slots against 3,981 distinct codes and recall each code’s ICD-9 chapter at 0.989; a fixed-budget eviction cache at one quarter of the distinct count collapses to 0.30 (CMS, 2010 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The KITTI row makes the context interpretation particularly clear. Places are ten-metre occupied cells and the query is a revisit: a frame whose place was last seen at least 500 frames earlier. At a budget equal to the distinct-place count, the sliding window recalls none of the revisits on any of three sequences because every revisit exceeds the matched window. The heavy-hitter is unstable across sequences, from 0.34 to 1.00, because it retains popular places and drops the tail. Allocate-on-novelty recalls every revisit on every sequence while holding exactly one slot per place. The memory grows with the environment and remains flat along the trajectory, the invariance a mapping context should have (Geiger et al., 2012 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across these domains the retained context becomes a concrete object: template catalog, formulary, code set, or place map. The companion paper adds two mechanism-level boundaries relevant here. First, the threshold does not require per-domain tuning across the recommendation, clinical, and claims streams: any value on a broad plateau yields the same distinct-item cache. Second, on bursty BGL alerts, a surprise-adaptive budget recalls 0.94 at 18 average slots where a matched fixed budget recalls 0.75; on nearly stationary MovieLens windows it ties the matched fixed budget, as it should (Pal and Rojkova, 2026 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 A trained control study: summary plus distinct recall at half the cache\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Mechanism-level allocation does not establish what happens when the cache is embedded in a model trained end to end, where the representations themselves move under the gate. Character-level language modeling on enwik8 isolates the read without a domain frontend. We train four-layer residual stacks of width 256, sequence length 512, batch size 32, and 6,000 steps over the 90 MB training split, using identical schedules, and evaluate bits per character on held-out data (Mahoney, 2006 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Four findings build on one another.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, novelty-gated attention reaches 2.073 versus 2.087 for full attention, a 0.014-bit margin within expected run-to-run variation at this scale and therefore treated as parity rather than a win. It does so while admitting 49% of tokens. The fixed-state path, with no item cache, trails both by about 0.11 bits per character. At this horizon, half the learned keys carry the content-addressed information the task uses.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, adding the state-space path to full attention reaches 1.932, improving by 0.155 bits over attention alone. The scale of this gain is an order of magnitude larger than the attention-versus-gated difference and is consistent with the two paths doing different work: the recurrence carries smooth positional and local structure, while attention carries addressable recall.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Third, composing those two results gives the proposed component. State-space plus gated cache reaches 1.877 at an attended fraction of 0.53. It is nominally 0.055 below state-space plus full attention, a single-seed margin we do not rely on. The central statement is that restricting the content-addressed path to the novelty-kept half of the tokens gives up nothing against coupling recurrence with full attention.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fourth, retaining both full attention and the gated cache reaches 1.836. It saves no cache because full attention remains, so it is not a deployment candidate. Its role is evidential: if the gated cache were merely a cheaper approximation to attention, adding it beside full attention should contribute nothing. Instead it improves the coupled read by a further 0.096 bits. Organizing context by distinct content supplies an inductive view not automatically extracted by dense attention.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Three boundaries accompany the table. The models are small and far from the enwik8 state of the art; the comparison is internal to matched backbones, not a leaderboard claim. The headline entries are single-seed. And the saving is in cache and inference read cost, not in training clock time: the unoptimized novelty scan is quadratic and makes the gated read slower per step than attention (1,122 versus 882 seconds; 1,404 for the component).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The seed boundary has been retired. Retraining the five principal reads over three seeds on Apple-silicon hardware gives: state-space 2.191 ± \\\\pm 0.016, full attention 2.029 ± \\\\pm 0.064, gated read 2.014 ± \\\\pm 0.065 at fraction 0.49, coupled read 1.933 ± \\\\pm 0.009, and component 1.922 ± \\\\pm 0.035 at fraction 0.51. Across ten contiguous validation blocks per seed, the gated read is at or below full attention in 29 of 30 seed-block pairs. The component is below full attention in all 30. Against the full-attention coupled read, the component wins all ten blocks on two seeds and loses all ten on the third. Thus the durable result remains parity with the full-attention coupling at half the attended tokens; any further margin is within seed variation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A selective, input-gated recurrence changes the best text result. Paired with the gated cache it reaches 1.709 ± \\\\pm 0.008 bpc at an admitted fraction of 0.61 over three seeds, two tenths below the plain component with every seed separated, below the three-path read’s 1.836, and with the tightest spread in the study. On concatenated source code the same coupling reaches 1.394 ± \\\\pm 0.019 at fraction 0.50, four tenths below its cache-free ablation (1.811 ± \\\\pm 0.008) and about half a bit below full attention, whose seeds vary by ± \\\\pm 0.28. This is the strongest text coupling measured. It is also sharply domain-specific: on the claims task below it collapses, showing that the recurrent summary operator must match the stream even when the novelty cache remains useful.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Context scaling: the advantage grows with horizon\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The decisive question for long-context engineering is how the gap changes with horizon. We sweep sequence length over 256, 512, and 1,024 tokens, adjust batch size to 24, 16, and 8 to remain feasible, and train full attention, novelty-gated attention, and the component for 3,000 steps at each horizon. Within each horizon the reads share the same training budget, so those gaps are clean. Across horizons, the fixed step count means longer sequences receive a different optimization budget; absolute loss trends across horizons are not the claim. These runs are also shorter than the 6,000-step control above.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The component remains nearly flat as the horizon quadruples (2.42, 2.41, 2.39), while matched attention degrades (2.36, 2.46, 2.69). The gap moves from − 0.060 -0.060 at 256, through a favorable but non-claimable + 0.051 +0.051 within spread at 512, to + 0.300 +0.300 at 1,024 with cleanly separated spreads. The novelty-gated read tracks attention at each horizon, so the parity result is not confined to 512 tokens. The component’s admitted fraction falls from approximately 0.45 to 0.42 as context lengthens: the longer context accumulates more redundancy, and the memory saving grows where the performance advantage does.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The short-horizon reversal is part of the result. At 256 tokens the relevant context fits comfortably in attention and the recurrent path contributes no measurable benefit. The component is a long-context design, not a universal replacement for short attention.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8 Which context to keep depends on how the task uses the past\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A frozen-threshold sweep traces the trained cache-size frontier; a sliding-window read at matched budgets determines what that frontier means. The result is deliberately reported against a novelty-first interpretation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thinning helps both policies on short-horizon character prediction. The gated read improves monotonically as its admitted fraction falls from 0.75 to 0.20, from 2.350 to 2.305 bpc at 3,000 steps. The window improves as it narrows, and the smallest window is the best point in the sweep: 2.262 at a 0.12 fraction. Near a matched fraction of 0.2, the policies are at parity: 2.305 at 0.20 for novelty versus 2.312 at 0.22 for the window. The task is locality-carried, so recency is the correct context policy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training the deployed budget policies end to end reveals a second boundary. Sink-plus-window trains as stably as attention and the SnapKV-style read nearly so, but the heavy-hitter read trains poorly: 2.69 and 3.14 bpc at its two budgets versus 2.09 for attention. Selection by accumulated attention mass is noisy while the attention itself is untrained, and the hard selection feeds the noise back into learning. Heavy-hitter eviction is a plausible inference-time policy; it is a poor training-time primitive in this experiment. Novelty does not depend on the model’s own accumulated saliency statistics and is therefore better behaved during learning.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two corpus controls narrow the interpretation further. Source code is substantially more compressible than prose by the attention baseline (1.62 versus 2.35 bpc), yet the learned gate settles near half on both (0.47 prose and 0.50 code). The admitted fraction does not measure corpus-level compressibility; it reflects local similarity in the learned key space. On code, the gated read has a small deficit to attention (1.640 versus 1.622), whereas on prose it has a small surplus. A selective recurrence without attention reaches 1.912 at the 6,000-step budget, within the coupled reads’ seed band, reinforcing that this short control corpus does not require strong item-addressed recall.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The sweep also asks which occurrence of a repeated key should be kept. Allocate-on-novelty keeps the first. Its mirror keeps only the latest occurrence of each group of similar earlier keys, matching the copying behavior associated with induction heads (Olsson et al., 2022 ) . Under the same protocol, keep-latest reaches 2.267 ± \\\\pm 0.002 bpc at an admitted fraction of 0.21 over three seeds. It beats every keep-first threshold point at less than half their fraction, beats the matched 0.22 sliding window (2.312), and is statistically tied with the narrowest window’s 2.262 at fraction 0.12. On source code it is the steadiest read measured, 1.731 ± \\\\pm 0.043 versus 1.94 ± \\\\pm 0.28 for full attention and keep-first at that budget.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This revises the trained read, not the mechanism. At the mechanism level a slot represents the whole cluster, and the companion paper’s merge study found the stored occurrence immaterial to the result. In a trained attention read, however, the retained value carries positional context, and the most recent occurrence can be the useful one. That result is again domain-dependent: on the thousand-event claims task, keep-latest scores 8.55 ± \\\\pm 0.17, behind keep-first at 8.48 ± \\\\pm 0.08, behind full attention, and well behind the component. Recency wins when the repeated occurrence’s latest local context matters; first-allocation wins when the slot is intended to represent a stable distinct entity.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"9 A recall-carried long-context task: the crossover on real claims\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The control corpus identifies boundaries but does not force distinct-entity recall. The end-to-end structured task is next-code prediction on DE-SynPUF inpatient claims. For each member, claims are ordered by date and represented as a member boundary followed by, for each claim, a claim boundary and its ICD-9 diagnosis codes. The model predicts the next token. Evaluation uses bits per event and top-1 accuracy on held-out members, so generalization is across patients rather than within a patient’s history. The stream contains 576,885 events and a vocabulary of 4,096 codes and boundaries. Four reads are trained for 3,000 steps at three horizons, with the batch scaled to hold the token budget fixed at the two longer horizons.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At 256 and 512 events, all four reads tie within seed spread. At 1,024 events the ordering separates. The component reaches 8.085 ± \\\\pm 0.040 bits per event, 0.311 below full attention with non-overlapping spreads, while attending over 45% of events; the plain coupled read follows at 8.107. The crossover between 512 and 1,024 events matches the shape of the enwik8 horizon sweep. An event contains more semantic content than a character, and the member history relevant to the next diagnosis spans hundreds of events.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The matched-budget policies determine which context rule is correct. The sliding window that led the short character sweep trails the component by more than half a bit. The strongest deployed policy, the heavy hitter, trails by about a third. The rare and distinct codes matter, so popularity and recency are the wrong retention criteria. The fixed-state backbone at the short horizon scores 8.021, confirming the bounded-state floor. As in the context sweep, only within-horizon comparisons are claimed because the training step count is fixed across horizons.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selective coupling that leads prose and code fails severely here. At the 1,024-event horizon, the selective backbone alone reaches 9.64 ± \\\\pm 0.07 and with the cache 9.77 ± \\\\pm 0.05, approximately 1.5 bits behind the plain component. The input-gated recurrence that suits smooth text streams fails on the spiky categorical stream, and the cache cannot rescue a failed summary path. There is no universal recurrent companion: selective summary wins text; plain linear summary wins claims; keep-latest wins text budgets and loses claims. What is stable across the three corpora is narrower: the best measured read on prose, code, and claims contains the novelty-gated cache, while the domain determines the other path.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"10 The rest of the program, measured\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"10.1 Incremental cost\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At inference, the gate is incremental. The cache contains only admitted keys; each arriving key is compared with the occupied set; a query reads over the occupied set alone. Streaming 65,536 tokens of width 256 at eightfold redundancy on one CPU, the cache answers a query in 64 microseconds, while attention over the full stream takes 1,355 microseconds: a 21-fold read speedup with an eightfold smaller cache. The cost is allocation: 56 microseconds per incoming token versus 0.3 microseconds for attention’s append. This reproduces at scale the companion paper’s build-versus-read law. The cache is favorable when the retained context is queried repeatedly or memory is binding, not when a sequence is built once and read once.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"10.2 Operational logs at full scale: one clean result and one wall\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The trained log study uses fully annotated Loghub-2.0 releases under a strict temporal split: the first 90% of each stream for training and a later contiguous slice for evaluation, three seeds per read. Copy-last-template accuracy is reported as the free floor supplied by a storm-dominated log.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On HDFS, 1.8 million lines and 37 templates, copy-last accuracy is 0.574. Every learned read is far below the uniform loss floor, approximately 0.98 bits per event versus 5.21, and above the copy floor at 0.66 accuracy. The gated read matches full attention at cache fraction 0.42. The component leads attention on all three seeds, 0.975 ± \\\\pm 0.008 versus 0.990 ± \\\\pm 0.011 bits, at fraction 0.39. The selective coupling ties the plain component at 0.974 ± \\\\pm 0.006 and fraction 0.40, showing that its failure on claims is not a universal failure on event data. Keep-latest defines an extreme-memory point: repeated storm lines make the latest duplicate sufficient, reducing the fraction to 0.03–0.04, roughly one tenth of the other reads, at a cost of about 0.09 bits (1.08 ± \\\\pm 0.06 versus 0.975). It is a frontier point, not the winner.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On BGL, 4.6 million lines and 320 templates, every read fails under the same protocol. Losses range from 9.5 to 12.5 bits versus an 8.3 uniform floor, accuracy lies far below the copy floor, and seed spreads reach a bit or more. The later slice changes regime: BGL’s alert-template distribution drifts over months. A model confidently wrong on the future slice has encountered distribution shift, not a memory-capacity limit. The trained log claim is therefore restricted to streams whose template distribution is sufficiently stationary across the split. Rolling refits or within-regime evaluation are required for drifting streams.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"10.3 One stream, two targets: recall-carried versus summary-carried\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The original claims cost head predicted the payment decile of the arriving claim and produced validation losses above the ten-class uniform floor for every read, indicating memorization of training members. That experiment was withdrawn. The redesign adds outpatient claims, increasing the data by approximately twelvefold, plus dropout, weight decay, and best-checkpoint selection on a held-out slice.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The redesigned task clears the 3.322-bit uniform floor. Every read transfers at 3.268–3.276 bits per claim, with three-seed spreads of ± \\\\pm 0.003. The ordering, though modest, is diagnostic: the two recurrent-path reads tie at 3.268 for the summary alone and for the component, while full attention and the gated read tie at 3.276. The cache neither helps nor hurts; the recurrent summary carries the signal. The extensions obey the same split to the third decimal: selective recurrence and its coupling reach 3.269–3.270, and keep-latest joins the attention side at 3.278.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus the same underlying stream supports two different context requirements. Next-code prediction is recall-carried and favors the distinct-item cache. Cost forecasting is summary-carried and favors recurrence. Context policy cannot be selected from the corpus alone; it must be selected from the task’s dependency on the past.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"10.4 The trained adaptive budget: a settled negative in this setting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Inside a trained model, the surprise-modulated threshold ties the fixed gate to three decimal places on both standard and deliberately bursty enwik8 while admitting more of the stream. The first burst construction had phases longer than the training window, leaving each window quasi-stationary. A repaired test uses bursts four times shorter than the window so every window crosses several regime changes. The result remains a tie at added cost: 1.137 versus 1.137 bpc, admitted fraction 0.71 versus 0.49.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The interpretation is that the learned soft gate already adapts to content token by token. An explicit budget schedule layered on top is redundant inside these windowed trained models. The adaptive concentration remains useful at the mechanism level for truly streaming caches with phase-level demand swings, as shown by BGL alert storms in the companion study, but is not justified by the present trained backbones.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"10.5 Architecture robustness\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Widening the four-layer backbone preserves the ordering. At width 384, the component reaches 1.839, the full-attention coupled read 1.849, and attention 1.945. At width 512, attention reaches 1.894, the component 1.823, and the selective coupling 1.686 at admitted fraction 0.63. The width axis is therefore covered at 256, 384, and 512 with the same qualitative result.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At six layers and width 256, the initial 5,000-step experiment inverts: attention and the gated read lead at 1.91–1.92 and the coupled reads trail by approximately 0.08. A plausible interpretation is that additional attention layers learn local composition that substitutes for the recurrent path while the coupled optimization cost remains. Doubling the training budget to 10,000 steps changes the conclusion: the component reaches 1.707 at fraction 0.59 versus 1.731 for attention, with blocked means 1.747 ± \\\\pm 0.034 versus 1.774 ± \\\\pm 0.035 in the same direction. The earlier inversion was an optimization artifact. At depth, the coupling requires more training budget.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"11 Context-engineering consequences\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The experiments support a design rule more specific than generic prompt compression.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"11.1 Three memory structures, three dependency types\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A practical long-context system should not force every kind of information through one memory policy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This decomposition also explains why a single benchmark can be misleading. A locality-dominated corpus makes a window look universally sufficient; a recall probe makes novelty look universally superior; an aggregate target makes recurrence look sufficient. The architecture should be judged on a set of tasks that deliberately separates these dependencies.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"11.2 Context objects, not merely compressed tokens\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The strongest use of the cache is not to replace a tokenizer with clustering. It is to maintain a set of addressable context objects whose size follows the distinct information active in the stream. In a general context engine, a slot should therefore carry more than a hidden key–value pair: source references, timestamps or offsets, merged evidence, confidence, and an update history. Those additions do not change the measured mechanism, but they turn its inspectable slot set into an auditable working memory.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The paper does not establish that arbitrary language turns will organize themselves into stable human-readable facts. The real-stream alignments arise because the frontends bind keys to explicit movies, templates, drugs, codes, and places. Applying the method to retrieval-augmented generation or multi-turn assistants requires a frontend and merge semantics that preserve propositions, entities, negation, temporal updates, and source authority. A novelty gate can decide that two representations are close; it cannot by itself decide whether one statement supersedes another or whether two similar passages conflict.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"11.3 The relevant scaling law\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let N N be observed context units and K K the number of distinct retained items. Full attention stores O ( N ) O(N) keys and reads O ( N ) O(N) per query. The present cache stores approximately O ( K ) O(K) and reads O ( K ) O(K) , while naive online allocation costs O ( N K ) O(NK) because each incoming key scans occupied slots. When redundancy is high, K ≪ N K\\\\ll N ; when every item is novel, K ≈ N K\\\\approx N and the cache offers no asymptotic saving. Approximate nearest-neighbour indexing, hierarchical slots, or batched allocation are engineering requirements for scaling the write path, not changes to the principle.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The measured incremental result makes the trade concrete: at N = 65 , 536 N=65{,}536 and eightfold redundancy, reading is 21 times faster and the cache is eight times smaller, while each write is much more expensive than attention’s append. The right deployment is persistent, repeatedly queried context, not transient one-shot prompting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"12 Scope\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All experiments use public data: Loghub, the MIMIC-IV open demo, CMS DE-SynPUF synthetic claims, KITTI, and enwik8. No proprietary stream, schema, or domain knowledge from an operational deployment enters the experiments. Mechanism-level studies use synthetic keys bound to real entity streams and train-free reads; they validate cache statistics and, for logs, one ground-truthed end task, but are not trained domain models. The trained control, claims task, and context-length sweep are replicated over three seeds, with paired per-block statistics for the control. The corpus-swap and some architecture-robustness measurements are single-seed where stated. Compared runs share hardware; the headline control table uses an NVIDIA T4, while seed replicates, context sweeps, claims, and mechanism studies use Apple M1 and M4 systems.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The evidence does not yet include long multi-turn conversations, retrieval-augmented generation, tool-using agents, or production context builders. The paper therefore does not claim that the current cosine gate and merge rule are sufficient for semantic facts in those systems. It establishes a lower-level result: a learned novelty-gated content-addressed path can retain distinct information at substantially below token-count cost, can be coupled with recurrent summaries without surrendering performance, and has measurable regimes where it is preferable to recency or fixed-budget eviction. Extending the frontend to proposition-level context, preserving contradictions and temporal updates, evaluating answer faithfulness and provenance, and optimizing the allocation scan are the next steps.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Within those limits, the central context-engineering claim is deliberate: context need not be a transcript, a fixed summary, or a window. It can be a structured working set whose size follows the distinct information the task may need to recall, while recurrence carries aggregates and recency carries local texture.\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":3,"numItemsPerPage":100,"numTotalItems":1000,"offset":300,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NTkxODM3NSwianRpIjoiZTFmYWZjZjQtMjA2MS00NTQ0LWE0MjgtNjZjMGQ1MWUzMWM0Iiwic3ViIjoiL2RhdGFzZXRzL1ByaW1lSW50ZWxsZWN0L2Rlc2h1ZmZsZS1wYXBlcnMtdjEtY29ycHVzIiwiZXhwIjoxNzg1OTIxOTc1LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.Fbv8GFzRgwQF0zj5-RB1b6eTWmPpCffWMfrcur5kGk_68nCarblha7fKKLoZEz5xgrrOVi8WMsHCwp1mue37Bw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":1000,"numBytesParquet":49695546}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
source stringclasses 2
values | paper_id stringlengths 10 26 | doi stringlengths 15 34 ⌀ | version int64 1 6 | title stringlengths 22 201 | authors listlengths 1 43 | license stringclasses 2
values | source_url stringlengths 35 60 | published timestamp[s]date 2021-03-03 00:00:00 2026-07-16 00:00:00 | blocks listlengths 19 529 |
|---|---|---|---|---|---|---|---|---|---|
arxiv | 2607.12380 | null | 1 | SinAE: A Single-Architecture Flow-Matching Autoencoder for Cross-Domain Atomic Systems | [
"Yuxuan Ren",
"Fan Yang",
"Jianhua Yao",
"Yatao Bian"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12380v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "3D atomic structure generation underpins advances in drug discovery, materials design, and protein engineering. Despite differences in scale and domain conventions, the underlying data is the same: atoms ... |
arxiv | 2607.12304 | null | 1 | What Does a Temporal Benchmark Score Measure? Decomposing Channel Use in Video VLM Evaluation | [
"Farrukh Rahman"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12304v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "A score on a temporal video benchmark is meant to certify that a model understands time. But “temporal” hides two questions. The task question: does a question even require watching motion across frames, ... |
arxiv | 2607.12279 | null | 1 | A Shared Subcircuit Lets LLMs Count Down Across Tasks | [
"Jacob Dunefsky",
"Wes Gurnee",
"Emmanuel Ameisen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12279v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Mechanistic interpretability seeks to reverse-engineer complex neural networks, such as large language models (LLMs), in order to allow humans to reason about the algorithms which they implement. Just as ... |
arxiv | 2607.12267 | null | 1 | Track, Rank, Crack: Epistemic Working Memory Scales Multi-Hop Reasoning in Language Agents | [
"Ning Liu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12267v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Effective reasoning requires not only the ability to draw inferences but also the ability to manage what is known, what is uncertain, and what remains to be determined (Baddeley, 1992 ; 2000 ; Newell and ... |
arxiv | 2607.12266 | null | 1 | Saturation Makes Quantization Error Additive: A Coverage Model with a Certificate | [
"Joshua Hill"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12266v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Quantizing a transformer to low bit-widths is the dominant route to cheap inference, and the formats now being deployed quantize both weights and activations to 4 bits. What remains a design choice is mix... |
arxiv | 2607.12257 | null | 1 | On-Device Deep Research at 4B: Exposure Bounds Faithfulness, Retrieval Bounds Coverage | [
"Vinay Kumar Chaganti"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12257v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Deep research agents search sources, read them, and write a cited report. They are evaluated mainly as commercial products, compared on answer accuracy at frontier scale (Du et al., 2025 ; Xu et al., 2025... |
arxiv | 2607.12241 | null | 1 | Gradient-Free Topology Adaptation for Power Flow Surrogates via In-Context Whitening | [
"Ayushi Jolotia",
"Parikshit Pareek"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12241v1 | 2026-07-14T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The AC power flow (ACPF) problem is the core computation in grid operations: given generation and demand, it solves the remaining steady-state variables (e.g., bus voltages, generator reactive power, and ... |
arxiv | 2607.12215 | null | 1 | Fine-Tuned Multi-Agent Framework for Detecting OCEAN in Life Narratives | [
"Rasiq Hussain",
"Darshil Italiya",
"Joshua Oltmanns",
"Mehak Gupta"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12215v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "Fine-Tuned Multi-Agent Framework for Detecting OCEAN in Life Narratives"
},
{
"type": "para",
"text": "Rasiq Hussain 1 Darshil Italiya 1 Joshua Oltmanns 2 Mehak Gupta 1 1 Southern Methodist University 2 Washington University in St. Louis {rasiqh, ditaliya, mehakg}@smu.... |
arxiv | 2607.12204 | null | 1 | Forgetful Attention: A Trainable Support-Vector Memory with Certified Selection and Exact Unlearning | [
"Vishwajith Ramesh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12204v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Attention is increasingly understood not as a similarity heuristic but as a learning algorithm that runs at inference time. In this test-time-learning (TTL) view, a layer maintains a memory fit online to ... |
arxiv | 2607.12166 | null | 1 | From Geometric Recovery to Causal Validation: A Reproducible Audit of Sparse Autoencoder Features, from Superposition Geometry to Causal Inertness | [
"Mohamed Abdessalem Bal"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12166v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Mechanistic interpretability’s current toolkit rests on a specific pipeline: hypothesize that networks represent more features than they have dimensions by packing them into almost-orthogonal directions (... |
arxiv | 2607.12122 | null | 1 | An Agentic AI Scientific Community for Automated Neural Operator Discovery | [
"Luis Loo",
"Ulisses Braga-Neto"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12122v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Operator learning has become a central tool for building fast surrogates of partial differential equation (PDE) solvers kovachki2023neuraloperator . However, each operator family encodes a fixed inductive... |
arxiv | 2607.12104 | null | 1 | TraceSynth: Generating Production-Quality Kernel Traces with Constraint-Guided Diffusion Models | [
"Yuvraj Sehgal",
"Sneh Patel",
"Mahsa Panahandeh",
"Naser Ezzati-Jivan",
"Francois Tetreault"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12104v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Modern production systems increasingly rely on machine learning (ML) for operational intelligence, utilizing predictive models to forecast bottlenecks and automated analysis to reduce mean time to resolu... |
arxiv | 2607.12094 | null | 1 | Sparse Autoencoders for Interpretable Out-of-Distribution Detection | [
"Ayush Karmacharya",
"Luke Luschwitz",
"Lucia Romero",
"Yanan Niu",
"Joseph Campbell"
] | CC-BY-4.0 | https://arxiv.org/html/2607.12094v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "s> \\columncolor gray!15 c"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Machine learning algorithms are built on the assumption that data encountered at test-time comes from the same distribution as seen du... |
arxiv | 2607.11875 | null | 2 | Invariant Learning Dynamics of Transformers in Inductive Reasoning Tasks | [
"Tiberiu Musat",
"Tiago Pimentel",
"Nicolas Zucchet",
"Thomas Hofmann"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11875v2 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "While great progress has been made toward understanding the circuits (i.e. subnetworks, or weight structures) present in large language models [ 1 , 2 , 3 , 4 ] , we are still far from understanding the l... |
arxiv | 2607.11874 | null | 1 | A Minimalist Retargeting-Guided Reinforcement Learning Recipe for Dexterous Manipulation | [
"Yunhai Feng",
"Natalie Leung",
"Jiaxuan Wang",
"Lujie Yang",
"Haozhi Qi",
"Preston Culbertson"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11874v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "Keywords: Dexterous Manipulation, Motion Retargeting, Sim-to-Real"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Humanoid robots and anthropomorphic hands share the human body’s kinematic structure, opening t... |
arxiv | 2607.11873 | null | 1 | A Durability and Cross-Language Transfer Benchmark for a Validated Teaching-Feedback Classification Protocol | [
"Esteban U. Vega Barajas"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11873v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "A Durability and Cross-Language Transfer Benchmark for a Validated Teaching-Feedback Classification Protocol"
},
{
"type": "para",
"text": "Esteban U. Vega Barajas Universidad de Guadalajara"
},
{
"type": "para",
"text": "Keywords: teaching-evaluation feedb... |
arxiv | 2607.11826 | null | 1 | Transformer-Guided Swarm Intelligence for Frugal Neural Architecture Search | [
"Romain Amigon"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11826v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The design of neural networks has historically been dominated by an empirical approach, where hyperparameters and topology are chosen based on human experience. While Neural Architecture Search (NAS) has ... |
arxiv | 2607.11998 | null | 1 | HPC-Enabled Video-based Coastal Wave Parameter Estimation Using V-JEPA and Deep Spatiotemporal Learning | [
"Abubakar Hamisu Kamagata",
"Dharm Singh Jat",
"Attlee Munyaradzi Gamundani",
"Saravanakumar Paramasivam",
"Babangida Sani",
"Aliyu Zakariyya"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11998v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The measurement of waves at the coast is extremely important for the fields of coastal engineering, coastal erosion control, maritime navigation, disaster management, and climate change adaptation [ 17 ] ... |
arxiv | 2607.11796 | null | 1 | An Exact Instrument for State Usage in Selective State-Space Models, and the Input-Driven Migration It Reveals | [
"Raktim Bhattacharya"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11796v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Selective state-space models (SSMs) have made linear recurrences competitive with attention for long-sequence modeling ( 7 , 3 ) . A Mamba layer maintains, in each channel, a small linear state driven by ... |
arxiv | 2607.11752 | null | 1 | Paradoxes of Game Theoretic Equilibria and Price of Anarchy | [
"Georgios Piliouras",
"Ian Gemp",
"Siqi Liu",
"Luke Marris"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11752v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "A fundamental challenge in the study of multi-agent systems and economics is the development of target solution concepts that serve as both mathematically rigorous models of agent behavior and reliable p... |
arxiv | 2607.11751 | null | 1 | When Local Monitors Miss Compositional Harm: Diagnosing Distributed Backdoors in Multi-Agent Systems | [
"Yibo Hu",
"Ren Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11751v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Local monitors miss distributed backdoors because the attack is not local. In a multi-agent or tool-using LLM system, a planner assigns subtasks, agents write messages or call tools, and a final agent ass... |
arxiv | 2607.11734 | null | 1 | NeuralActuator: Neural Actuation Modeling for Robot Dynamics and External Force Perception | [
"Zhiyang Dou",
"John U. Onyemelukwe",
"Hangxing Zhang",
"Heng Zhang",
"Minghao Guo",
"Yunsheng Tian",
"Michal Piotr Lipiec",
"Joshua Jacob",
"Chao Liu",
"Peter Yichen Chen",
"Yuri Ivanov",
"Wojciech Matusik"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11734v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Differentiable simulators [ 24 , 36 , 13 , 22 , 15 ] enable efficient gradient-based optimization for robot control, with substantial advances in modeling rigid-body dynamics and recent extensions to soft... |
arxiv | 2607.11994 | null | 1 | Mitigating The Effect of Class Imbalance in Data with Hierarchical and Dependable Structure | [
"Bipin Chhetri",
"Deepika Giri",
"Avishek Kadel",
"Rabin Kumar Karki",
"Akbar Siami Namin"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11994v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Mitigating weaknesses has become increasingly necessary as software systems grow in complexity and adversaries adopt sophisticated attack techniques. Software and hardware weaknesses cataloged in the CWE ... |
arxiv | 2607.11725 | null | 1 | Time-Lag-Aware Deep Reinforcement Learning for Flexible Job-Shop Scheduling in PPVC Module Factories | [
"Ziheng Zhang",
"Wei Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11725v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Prefabricated Prefinished Volumetric Construction (PPVC) relocates structural work, mechanical, electrical, and plumbing (MEP) installation, and interior finishing from the construction site into a factor... |
arxiv | 2607.11656 | null | 2 | Imputation-free transformer learning enables robust Alzheimer's disease prediction and calibrated uncertainty quantification across heterogeneous clinical cohorts | [
"Christelle Schneuwly Diaz",
"Narmina Baghirova",
"Duy-Thanh Vu",
"Duy-Cat Can",
"Gilles Allali",
"Philippe Ryvlin",
"Oliver Y. Chén"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11656v2 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "1] \\orgdiv Platform of Bioinformatics, \\orgname Lausanne University Hospital"
},
{
"type": "para",
"text": "2] \\orgdiv Faculty of Biology and Medicine, \\orgname University of Lausanne"
},
{
"type": "para",
"text": "3] \\orgdiv Leenaards Memory Centre, \... |
arxiv | 2607.11653 | null | 1 | Bet on Features: Anytime-Valid and Feature-Aware Auditing of Conditional Quantile Forecasters | [
"Ivane Antonov",
"Sohom Mukherjee",
"Richard Pibernik",
"Yo Joong Choe"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11653v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "Abstract"
},
{
"type": "para",
"text": "Black-box conditional quantile forecasts are widely used for sequential decisions under asymmetric costs, such as inventory planning in supply chain management. Once deployed, such forecasters must be monitored continuously as da... |
arxiv | 2607.11632 | null | 1 | Reproducing human biases in route choice using large language models: Toward scalable behavioral modeling | [
"Jiangtao Han",
"Shoufeng Ma",
"Shuxian Xu",
"Geng Li",
"Shuai Ling",
"Ning Jia",
"Zhengbing He"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11632v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "In real-world decision-making, individuals do not always behave in a fully rational and objective manner. Instead, decisions are often shaped by subjective perceptions, resulting in systematic behavioral ... |
arxiv | 2607.11607 | null | 1 | Auditing the Risk Claims of Distributional Reinforcement Learning | [
"Hari Prasad"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11607v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "Distributional reinforcement learning (RL) replaces the scalar value function with a full distribution over returns (Bellemare, Dabney, and Munos 2017 ; Dabney et al. 2018b , a ) . The idea has been unusual... |
arxiv | 2607.11606 | null | 1 | Globally Consistent Coloring Schemes for Language Identification | [
"Moses Charikar",
"Jon Kleinberg",
"Chirag Pabbaraju"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11606v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Dramatic advances in the power of large language models have rewakened interest in abstract models of language learning. These models can be viewed as formalisms for language acquisition that operate at a... |
arxiv | 2607.11565 | null | 1 | Heuristic Learning for Active Flow Control Using Coding Agents | [
"Paul Garnier",
"Jonathan Viquerat",
"Elie Hachem"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11565v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "K eywords active flow control ⋅ \\cdot fluid dynamics ⋅ \\cdot heuristic learning ⋅ \\cdot coding agents ⋅ \\cdot reinforcement learning benchmarks ⋅ \\cdot interpretable control"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "pa... |
arxiv | 2607.11548 | null | 1 | Training-Free Off-Screen Player Imputation for Broadcast-Based Spatial Football Analytics | [
"Seongjin Choi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11548v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Video-based game state reconstruction (GSR) turns ordinary broadcast footage into player coordinates on a common pitch template, promising tracking-level analytics without stadium hardware [ 12 ] . But th... |
arxiv | 2607.11541 | null | 1 | Random Label Prediction Heads for Studying Memorization in Deep Neural Networks | [
"Marlon Becker",
"Jonas Konrad",
"Luis Garcia Rodriguez",
"Benjamin Risse"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11541v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern deep learning models are prone to overfitting due to their extreme over-parameterization (Nakkiran et al. , 2021 ) . A wide range of strategies have been proposed to mitigate this issue, including ... |
arxiv | 2607.11540 | null | 1 | Tropical Circuits with Scalar Multiplication Gates | [
"Christoph Hertrich",
"Moritz Stargalla"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11540v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Tropical circuits [ Juk23 ] , also known as max-plus circuits, are a variant of classical arithmetic circuits that use maximum and addition gates instead of addition and multiplication gates. Besides fund... |
arxiv | 2607.11530 | null | 1 | Learning Residual Kinematic Corrections for Continuous Neural Decoding via Reinforcement Learning | [
"Jiamian Li",
"Niall McShane",
"Attila Korik",
"Naomi du Bois",
"Karl McCreadie",
"Leen Jabban",
"Benjamin Metcalfe",
"Özgür Şimşek",
"Damien Coyle"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11530v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Brain-computer interfaces (BCIs) have demonstrated remarkable success across a wide range of domains, including cognitive and physical rehabilitation ( Millán et al. , 2010 ) ( Prasad et al. , 2010 ) ( Ce... |
arxiv | 2607.11510 | null | 1 | DAG-FM: A Foundation Model for Causal Discovery under Heterogeneous Causal Mechanisms | [
"Yikang Chen",
"Zhengkang Guan",
"Haoyuan Qian",
"Peng Cui",
"Yi Yang",
"Kun Kuang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11510v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Causal discovery aims to recover causal structures from data (Pearl, 2009 ) , playing a pivotal role in diverse fields such as bioinformatics (Zhang et al. , 2013 ) , epidemiology (Vandenbroucke et al. , ... |
arxiv | 2607.11508 | null | 1 | CDFM: Towards a General-Purpose Causal Discovery Foundation Model | [
"Jie Qiao",
"Ruichu Cai",
"Zijian Li",
"Weilin Chen",
"Pengfei Hua",
"Boyan Xu",
"Zhengming Chen",
"Zhifeng Hao",
"Peng Cui"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11508v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "1]School of Computer Science, Guangdong University of Technology, Guangzhou, China 2]College of Mathematics and Computer, Shantou University, Shantou, China 3]Department of Computer Science and Technology, Tsinghua University, Beijing, China \\authorcustomemails {qiaojie.chn,cuiru... |
arxiv | 2607.11497 | null | 1 | IG-GAN: A Generative Adversarial Network for Aerodynamic Data Generation Based on Intrinsic Geometry | [
"Ying Yan",
"Liwei Hu",
"Xiaoming Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11497v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "[inst1] organization=School of Information Science and Engineering, Hebei University of Science and Technology, city=Shijiazhuang, postcode=050091, country=China"
},
{
"type": "heading",
"level": 1,
"text": "1 INTRODUCTION"
},
{
"type": "para",
"text": ... |
arxiv | 2607.11493 | null | 1 | Agentic Skill Optimization over Lie Algebroids | [
"Sridhar Mahadevan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11493v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "K eywords Skill optimization ⋅ \\cdot Large Language Models ⋅ \\cdot Lie Algebroids ⋅ \\cdot Tangent Categories ⋅ \\cdot Infinitesimal causality"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Agentic skill op... |
arxiv | 2607.11445 | 10.1109/BioCAS67066.2025.00079 | 1 | Event-based Neural Decoding for Neuroprosthetic Motor Control | [
"Khaleelulla Khan Nazeer",
"Sirine Arfa",
"Matthias Jobst",
"Richard George",
"Christian Mayr"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11445v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "The development of intra-cortical Brain-Machine Interfaces has accelerated in recent years, aiming to restore functional independence to individuals affected by paralysis and neurodegenerative diseases. T... |
arxiv | 2607.11442 | null | 1 | Velocity Scheduled Flow Matching | [
"Vitalii Bondar"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11442v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "K eywords Flow matching ⋅ \\cdot Generative models ⋅ \\cdot Velocity profile ⋅ \\cdot Machine learning ⋅ \\cdot Image generation ⋅ \\cdot Ordinary differential equations ⋅ \\cdot Local truncation error ⋅ \\cdot Training-free adaptation."
},
{
"type": "heading",
"level"... |
arxiv | 2607.11429 | null | 1 | Physics-Aware Conditional SetGAN for Spatially Consistent Multi-User TR 38.901 Channel Generation | [
"Mauro Gonzalo Tarazona-Levano",
"David Lopez-Perez",
"Nicola Piovesan",
"David Gomez-Barquero"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11429v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Generating multi-user multiple-input multiple-output (MIMO) channel realizations is computationally expensive, especially when large numbers of user-equipment (UE) deployments and configurations must be e... |
arxiv | 2607.11388 | null | 1 | StructAgent: Harness Long-horizon Digital Agents with Unified Causal Structure | [
"Wenyi Wu",
"Sibo Zhu",
"Kun Zhou",
"Aayush Salvi",
"Zixuan Song",
"Biwei Huang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11388v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "Project Page Code"
},
{
"type": "para",
"text": "“The purpose of abstracting is not to be vague, but to create a new semantic level in which one can be absolutely precise.” — Edsger W. Dijkstra"
},
{
"type": "heading",
"level": 1,
"text": "1 Introductio... |
arxiv | 2607.11273 | null | 1 | Fixed-Protocol Amortized MPS Tomography with Conformalized Predictive Uncertainty | [
"Jian Xu",
"Delu Zeng",
"John Paisley",
"Qibin Zhao"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11273v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Quantum state tomography (QST)—estimating an unknown quantum state ρ \\rho from measurement data—is a cornerstone primitive for benchmarking and debugging quantum hardware. Its central difficulty is sampl... |
arxiv | 2607.11272 | null | 1 | Long-Memory Reservoir Computing for Data-Scarce Dengue Forecasting | [
"Rahul Goswami",
"Shinjini Paul",
"Palash Ghosh",
"Tanujit Chakraborty"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11272v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Dengue remains a major public health concern in many tropical and subtropical regions, where timely forecasting can support preparedness, resource allocation, vector-control planning, and early warning in... |
arxiv | 2607.13081 | null | 1 | SingGuard-NSFA: Extensible Guardrails for Agentic AI via Generative Reasoning and Real-Time Classification | [
" SingGuard Team"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13081v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "\\ul"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The rapid evolution of Large Language Models (LLMs) into autonomous agents marks a fundamental shift in how AI systems interact with the world. Modern agent... |
arxiv | 2607.11172 | null | 1 | STAMP: Provenance-Guided Credit Assignment for Deep Search Agents | [
"Ke Xu",
"Han Xu",
"Xinran Chen",
"Yuqian Wang",
"Zhixuan Li",
"Xiaojian Liu",
"Changwo Wu",
"Jianqiang Xia",
"Yuchen Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11172v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "STAMP: Provenance-Guided Credit Assignment for Deep Search Agents"
},
{
"type": "para",
"text": "Ke Xu 1,2,* , Han Xu 1,*,† , Xinran Chen 1 , Yuqian Wang 1 , Zhixuan Li 1 , Xiaojian Liu 1 , Changwo Wu 1 , Jianqiang Xia 1 , Yuchen Li 1 1 Baidu Inc., 2 Peking University ... |
arxiv | 2607.11146 | null | 1 | Rank-Conditioned Sample Reuse for the Plackett--Luce Best-of-$K$ Objective | [
"Melveena Jolly",
"Midhun Xavier"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11146v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning increasingly optimizes not the mean reward but a best-of- K K reward, the quantity a deployed system actually realizes when it samples K K candidates and keeps the winner. Two best-... |
arxiv | 2607.11116 | null | 1 | The Equilibrium Is the Initialization: Lazy Identity Collapse in Physics-Structured Deep Equilibrium Reasoning | [
"Joyjeet Singh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11116v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Implicit and recurrent depth is one of the field’s recurring hopes for reasoning: rather than fixing computation at a static layer count, let the model iterate — a deep equilibrium model (DEQ) solving for... |
arxiv | 2607.11095 | null | 1 | When cheap gradients fail: the measurement cost of attacking quantum classifiers | [
"Bacui Li",
"Chandra Thapa",
"Tansu Alpcan",
"Udaya Parampalli"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11095v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "Keywords : quantum machine learning, adversarial robustness, parameter-shift rule, shot noise, variational quantum circuits, quantum measurement, finite-shot gradient estimation"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "par... |
arxiv | 2607.11091 | null | 1 | Adapting Evidential Neural Networks to Test-Time Neighbor Fusion Improves Molecular Property Prediction | [
"Cameron Gruich",
"Weichi Yao",
"Yixin Wang",
"Bryan Goldsmith"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11091v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "† \\dagger Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA § \\S Michigan Institute for Data & AI in Society, University of Michigan, Ann Arbor, Michigan 48109-1042, USA ‡ \\ddagger Department of Statistics, University of Michigan, A... |
arxiv | 2607.11075 | null | 1 | Link Adaptation Using Joint-Thompson Sampling | [
"Vignatha Vinjam",
"Manjunath Kolavennu",
"Myna Vajha",
"Karthik Periyapattana Narayanaprasad"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11075v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Link adaptation (LA) is an essential MAC layer algorithm that selects link parameters such as Modulation and Coding Scheme (MCS), power control, number of layers, and precoders for dynamic wireless channe... |
arxiv | 2607.11976 | null | 2 | LiteTopK: Exploiting the Curse of Dimensionality for a Fused Indexer-TopK Kernel in Long-Context Sparse Attention | [
"Ziqi Yin",
"Jianyang Gao",
"Peiqi Yin",
"Jiangneng Li",
"Gao Cong"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11976v2 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Indexer-TopK, which computes scores under a specific scoring function and selects the per-row top- k k elements, has become a core primitive in modern machine learning systems, with applications in large ... |
arxiv | 2607.11052 | null | 1 | Domain-Aware Scaling Laws Uncover Data Synergy | [
"Kimia Hamidieh",
"Lester Mackey",
"David Alvarez-Melis"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11052v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recent improvements in Large Language Models (LLMs) are strongly shaped by their pretraining data distribution (Li et al., 2024a ; Soldaini et al., 2024 ) , yet prior work abstracts away composition and r... |
arxiv | 2607.11030 | 10.1145/3805712.3808434 | 1 | MMRM: A Multiplex Multimodal Representation Model for Product Ranking in E-commerce Search | [
"Zhen-Lin Chen",
"Maosen Sheng",
"Peng Lin",
"Jianmin Chen",
"Zhuojian Xiao",
"Dongyue Wang",
"Xiwei Zhao"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11030v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "by"
},
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "E-commerce search engines rely on ranking models to capture fine-grained user preferences across multiple objectives, such as Click-Through Rate (CTR), Add-t... |
arxiv | 2607.11022 | null | 1 | When the Reward Suite Is Leaky: A Preregistered Causal Contrast of Natural Verifier False Positives in RLVR | [
"Chuyifei Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11022v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning from verifiable rewards (RLVR) trains code models against test suites and treats a pass as ground truth (Lambert et al., 2024 ; DeepSeek-AI, 2025 ) . Deployed suites do not earn tha... |
arxiv | 2607.11020 | null | 2 | Can a Language Model Learn Facts Continually in Its Weights? | [
"Charles O'Neill"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11020v2 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "Can a Language Model Learn Facts Continually in Its Weights?"
},
{
"type": "para",
"text": "Charles O’Neill 1"
},
{
"type": "para",
"text": "1 Baseten"
},
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para"... |
arxiv | 2607.11013 | null | 1 | Overcoming Fourier Locking in Quantum Data Re-uploading Classifiers via Spectral Homotopy | [
"Spencer Topel"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11013v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "In current literature, parameterized quantum circuits utilizing data re-uploading (or DRU-PQCs) are often assumed to be advantageous due to their universal expressivity. However, as noted in recent theore... |
arxiv | 2607.11005 | null | 1 | Actor-Critic Learning for Extended Mean Field Control with Deterministic Policies | [
"Ziheng Cheng",
"Xin Guo",
"Huyên Pham",
"Yufei Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11005v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "heading",
"level": 4,
"text": "(Extended) MFC."
},
{
"type": "para",
"text": "(Extended) mean field control (MFC) provides a tractable framework for large population stochastic games, by optimizing the co... |
arxiv | 2607.10984 | null | 1 | EquiFusion: Kinematics-Agnostic Human Motion Prediction via Equivariant Latent Diffusion | [
"Cecilia Curreli",
"Florian Hofherr",
"Dominik Muhle",
"Abhishek Saroha",
"Riccardo Marin",
"Daniel Cremers"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10984v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Predicting future human motion from past observations is a core component of human intelligence and a prerequisite for mature spatial AI. Due to the inherent ambiguity of human intent, the field has shift... |
arxiv | 2607.10970 | null | 1 | Enhanced Byzantine-Robust Federated Learning Via Truncated-Quadratic Loss for Heterogeneous Data | [
"Zhi-Yong Wang",
"Hao Nan Sheng",
"Werner Stefan",
"Hing Cheung So",
"Linqi Song",
"Weitao Xu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10970v1 | 2026-07-13T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Federated learning (FL) has emerged as a popular paradigm to train large-scale deep learning models with data distributed among n n clients such as autonomous vehicles, wearable devices, smart robots and ... |
arxiv | 2607.10969 | null | 1 | CGS: Configurable Graph Summarization with Bounded Neighborhood Loss and Query Support | [
"Shubhadip Mitra",
"Sona Elza Simon",
"C Oswald",
"Arnab Bhattacharya",
"Arindam Pal"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10969v1 | 2026-07-13T00:00:00 | [
{
"type": "para",
"text": "by"
},
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "One of the key challenges in graph data mining is the ever increasing size of the graphs that prohibits its scalability in terms of storage, communication an... |
arxiv | 2607.10960 | null | 1 | Reinforcement Learning for Execution under Dynamic Fees in a Closed-Loop DEX Simulator | [
"Wen-Ting Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10960v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "Keywords: automated market makers; decentralized exchanges; dynamic fees; optimal execution; reinforcement learning; market simulation."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "In decentralized exchange... |
arxiv | 2607.10951 | null | 1 | Sticky Jump Diffusions: A Unifying View of Masked, Continuous, and Hybrid Diffusion | [
"Pascal Jutras-Dubé",
"Patrick Pynadath",
"Jeremy Lu",
"Yuan Gao",
"Ruqi Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10951v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Diffusion-based generative modeling has been extended from continuous data (Sohl-Dickstein et al. , 2015 ; Song and Ermon, 2019 ; Ho et al. , 2020 ; Song et al. , 2021 ; Karras et al. , 2022 ) to discrete... |
arxiv | 2607.13078 | null | 1 | Operational Evidence Gaps for LLMs in Fraud Detection and Trust-and-Safety Workflows | [
"Keyur Gabani"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13078v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Trust and safety is often a production-ML domain. Fraud scoring, scam interception, content moderation, and investigation support frequently operate under adversarial pressure, tight error budgets, and re... |
arxiv | 2607.10931 | 10.32470/gn6tuko | 1 | Fast Whole-Brain, Geometry-Aware Functional Alignment for Cross-Subject Decoding | [
"Pierre-Louis Barbarant",
"Florent Meyniel",
"Bertrand Thirion"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10931v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "10.32470/gn6tukoUniversité Paris-Saclay, Inria, CEA, Palaiseau 91120, France Cognitive Neuroimaging Unit, Institut National de la Santé et de la Recherche Médicale, Commissariat à l’Energie Atomique et aux énergies alternatives, Université Paris-Saclay, NeuroSpin center, 91191 Gif... |
arxiv | 2607.10923 | null | 1 | Infrared Organization and Critical Cognitive Field Formation in Transformer Dynamics | [
"Byung Gyu Chae"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10923v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I I. Introduction"
},
{
"type": "para",
"text": "Large language models based on the Transformer architecture have demonstrated remarkable emergent capabilities, including reasoning, abstraction, in-context learning, and long-range contextual memory [... |
arxiv | 2607.10915 | 10.1145/3705328.3759309 | 1 | Normative Alignment of Recommender Systems via Internal Label Shift | [
"Johannes Kruse",
"Kasper Lindskow",
"Michael Riis Andersen",
"Ryotaro Shimizu",
"Julian McAuley",
"Pierre-Alexandre Mattei",
"Jes Frellsen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10915v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Recommender systems play a central role in content delivery across a wide range of domains, including news, entertainment, and e-commerce (Jannach and Jugovac, 2019 ; Kruse et al . , 2023 ; Shimizu et al... |
arxiv | 2607.13075 | null | 1 | The Entanglement Wall: Activation-Space Probes as Risk Detectors, Not Context Adjudicators | [
"Dominik Schwarz"
] | CC-BY-4.0 | https://arxiv.org/html/2607.13075v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Context can change the meaning of an instruction without changing its words. “How do I kill this?” is benign when it refers to an operating-system process and dangerous when it refers to a person. A detec... |
arxiv | 2607.10910 | 10.1145/3805712.3809909 | 1 | ZoRRO: A Zero-Weight Personalized Recommender System for Scalable News Recommendation | [
"Johannes Kruse",
"Ryotaro Shimizu",
"Kasper Lindskow",
"Jon Tofteskov",
"Michael Riis Andersen",
"Julian McAuley",
"Jes Frellsen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10910v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "by"
},
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Unlike other domains, news platforms face unique challenges due to the constant influx of new articles and their short lifespan. This dynamic environment dem... |
arxiv | 2607.10887 | null | 1 | Transferable Implicit Solvent Machine Learning Potential for Drugs and Proteins Approaching Ab Initio Accuracy | [
"Jan Eckwert",
"Julija Zavadlav"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10887v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "[1,2] \\fnm Julija \\sur Zavadlav"
},
{
"type": "para",
"text": "1] \\orgdiv Multiscale Modeling of Fluid Materials, Department of Engineering Physics and Computation, TUM School of Engineering and Design and Department of Physics, TUM School of Natural Sciences, \\org... |
arxiv | 2607.10855 | null | 1 | Reliability Scaling Laws for Quantized Large Language Models | [
"Sirine Ayadi",
"Sándor Daróczi",
"Stephan Günnemann",
"Bertrand Charpentier"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10855v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "newfloatplacement \\undefine@key newfloatname \\undefine@key newfloatfileext \\undefine@key newfloatwithin"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) (Touvron et al., 2023 ; T... |
arxiv | 2607.10836 | null | 1 | Route, Communicate, and Reason: Gated Routing and Adaptive Depth for Efficient Multi-Agent Reasoning | [
"Sudipto Ghosh",
"Tanmoy Chakraborty"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10836v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "https://github.com/parmanu-lcs2/grade \\papertitle Route , Communicate , and Reason : Gated Routing and Adaptive Depth for Efficient Multi-Agent Reasoning \\papershorttitle Gated Routing and Adaptive Depth for Efficient Multi-Agent Reasoning \\papershortauthors S. Ghosh and T. Cha... |
arxiv | 2607.10804 | null | 1 | When does distribution shift break graph neural networks calibration? | [
"Abderaouf Bahi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10804v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Graph neural networks (GNNs) Bahi [ 2026 ] are now routinely deployed in settings where the graph seen at test time differs from the one used for training: social and financial networks evolve, sensor and... |
arxiv | 2607.10802 | null | 1 | Q-Learning Lab: Teaching Reinforcement Learning Through Learner-Generated Trace Analysis | [
"Ekkachai Jueng"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10802v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning (RL) has moved from a specialist topic to core curriculum: it underlies game-playing agents, robotics, recommendation, and—through reinforcement learning from human feedback—the ali... |
arxiv | 2607.10793 | null | 1 | Hierarchical Bayesian Quadrature | [
"Tim Weiland",
"Toni Karvonen",
"Philipp Hennig"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10793v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Many tasks in fields like scientific computing and probabilistic machine learning require the computation of a definite integral of a function f f over some domain Ω ⊂ ℝ d \\Omega\\subset\\mathbb{R}^{d} :... |
arxiv | 2607.10784 | null | 1 | LSTrans: Efficient Knowledge Transfer for Lightweight and Automated ECG Classification | [
"Yi Zhao",
"Jiajun Gao",
"Chenyang Xu",
"Yuxi Zhou",
"Hao Wang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10784v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Cardiovascular diseases (CVDs) remain a major cause of global mortality. This issue drives the demand for real-time cardiac monitoring. Electrocardiogram (ECG) analysis is the standard clinical tool to id... |
arxiv | 2607.10750 | null | 1 | Filtering Harmful Actions Isn't Enough: Phantom Transfer in Agentic SDF | [
"Chinmayi Dixit"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10750v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Synthetic data for training LLMs has become a widely used method due to the ease of creation and use. AI safety researchers use synthetic data to train model organisms to demonstrate behaviors, methods, a... |
arxiv | 2607.10740 | null | 1 | Multi-Scale Convolution with Optimal Transport Attention Effect on Multivariate Time Series | [
"HaoChong Fu",
"Jian Xu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10740v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Transformer [ 20 ] has achieved a success in natural language processing [ 2 ] and computer vision [ 4 ] , emerging as the dominant architecture for sequence modeling tasks. Inspired by this success, Tran... |
arxiv | 2607.11963 | null | 1 | Evaluating Reliability in Machine Learning Models for Early Chronic Kidney Disease Prediction: A Systematic Review of Data Leakage and Predictor Stability | [
"Mashrul Hossain",
"Nafesa Kibria",
"Fahim Shahriar"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11963v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "Evaluating Reliability in Machine Learning Models for Early"
},
{
"type": "para",
"text": "Chronic Kidney Disease Prediction: A Systematic Review of"
},
{
"type": "para",
"text": "Data Leakage and Predictor Stability"
},
{
"type": "para",
"text"... |
arxiv | 2607.10720 | 10.1145/3744256.3812595 | 1 | WattCouncil: Context-Aware Household Energy Scenario Generation With Governed LLMs | [
"Mohannad Takrouri",
"Nicolas M. Cuadrado A.",
"Martin Takáč"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10720v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Global energy consumption continues to grow alongside societal and economic development, intensifying efforts to improve energy efficiency and reduce carbon emissions to address climate change. At the sa... |
arxiv | 2607.10707 | null | 1 | MDQEC-QAS: Meta-Decoding for Quantum Error Correction with Hardware-Aware VQC Search and Confidence-Gated Recovery | [
"Prashant Kumar Choudhary",
"Nouhaila Innan",
"Muhammad Shafique",
"Rajeev Singh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10707v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "MDQEC-QAS: Meta-Decoding for Quantum Error Correction with Hardware-Aware VQC Search and Confidence-Gated Recovery"
},
{
"type": "para",
"text": "Prashant Kumar Choudhary 1 , Nouhaila Innan 2,3 , Muhammad Shafique 2,3 and Rajeev Singh 1"
},
{
"type": "para",
... |
arxiv | 2607.11962 | null | 1 | Contrastive Joint-Embedding Prediction for Representation Learning in Structural MRI | [
"Fabian Mager",
"Lars Kai Hansen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11962v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 4,
"text": "Self-Supervised Learning."
},
{
"type": "para",
"text": "Self-supervised learning (SSL) is a machine learning paradigm in which the model is trained without the need for labeled data. It leverages the data’s inherent structure to generate labels, mak... |
arxiv | 2607.10633 | null | 1 | Auditing Construct Overlap in Explainable Machine Learning: Evidence from Burnout-Depression Prediction Across Student Cohorts | [
"Alireza Dehghan",
"Negin Ashrafi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10633v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "Auditing Construct Overlap in Explainable Machine Learning: Evidence from Burnout-Depression Prediction Across Student Cohorts"
},
{
"type": "para",
"text": "Alireza Dehghan 1 Negin Ashrafi 2,*"
},
{
"type": "para",
"text": "1 Sharif University of Technolog... |
arxiv | 2607.10618 | null | 1 | Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularization | [
"Raziyeh Takbiri"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10618v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "Keywords: sparse demixing, nonlinear observations, non-convex regularization, folded-concave penalties, robust recovery, proximal alternating minimization, Kurdyka–Łojasiewicz."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para... |
arxiv | 2607.10611 | null | 2 | M+Adam: Low-Precision Training via Additive-Multiplicative Optimization | [
"Xiaoyuan Liang",
"Sebastian Loeschcke",
"Mads Toftrup",
"Anima Anandkumar"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10611v2 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Training large language models (LLMs) in full precision (FP32) incurs substantial memory, compute, and energy costs. Modern accelerators are optimized for reduced-precision formats such as FP8 and FP4, vi... |
arxiv | 2607.11958 | null | 1 | Constructed Reality, Contested Priors: Decoupling and the Architecture of Cognitive Relapse Under the Free Energy Principle | [
"MD Ibrahim Hossain Ridoy"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11958v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "ABSTRACT"
},
{
"type": "para",
"text": "Under the free energy principle, a predictive system does not observe reality directly; it maintains a generative model of the world and experiences that model’s best current hypothesis in place of direct sensory contact. This ra... |
arxiv | 2607.10605 | null | 1 | End-to-End Real-Time Drone-Based Person Detection Framework Using Deep Learning | [
"Payel Sarmah",
"Ayush Ranjan",
"Piyush Kaushik Bhattacharyya",
"Anil Kr. Shaw",
"Pradip Kr. Das"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10605v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Real-time person detection from aerial platforms has become an important component of modern intelligent surveillance systems. It is increasingly used in applications such as disaster management, building... |
arxiv | 2607.10593 | null | 1 | AutoNorm: Understanding Adaptive Normalization in Transformers through Differentiable Gating | [
"Piyush Kaushik Bhattacharyya",
"Divyanshu Rai",
"Swastik Singh",
"Kumar Aakash",
"Ayush Ranjan",
"Krutika Verma"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10593v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Normalization techniques have become fundamental to the stability and convergence of modern deep learning architectures. By normalizing feature statistics within each layer, techniques such as Batch Norma... |
arxiv | 2607.10582 | null | 1 | MemDecay: Region-Aware KV Cache Eviction for Efficient LLM Agent Inference | [
"Venkatesha Matam",
"Keon Kim"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10582v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) are increasingly used in long-horizon agentic workflows that combine multi-turn interaction, retrieval, tool use, and iterative planning. Unlike single-pass long-context tasks... |
arxiv | 2607.10571 | null | 1 | Learning from Local Walks on Dynamic Graphs with Bandit Feedback | [
"Sourav Chakraborty",
"Amit Kiran Rege",
"Claire Monteleoni",
"Lijun Chen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10571v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "In the stochastic multi-armed bandit (MAB) problem, a learner repeatedly chooses among several actions (interchangeably referred to as arms ) observes only the reward of the selected action, and seeks to ... |
arxiv | 2607.10569 | null | 1 | When Does Restricting a Coding Agent to execute_code Help? A Regime $\times$ Agent-Design Ablation | [
"Hong Yang",
"Qi Yu",
"Travis Desell"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10569v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Coding agents now ship with overlapping tool surfaces—Read, Grep, Glob, Edit, Write, and Bash on Claude Code; ACI primitives on SWE-agent; an MCP execute_code tool on emerging stacks—and the field has se... |
arxiv | 2607.10548 | null | 1 | Beyond Looking Up, Try Looking Around: Harmonizing Global Structure and Local Consistency in Optimal Transport for Short Text Clustering | [
"Zhihao Yao",
"Yuxuan Gu",
"Jixuan Yin",
"Bo Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10548v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "In the era of massive digital communication, short texts such as messages, queries, and comments are generated on an unprecedented scale. Effectively clustering and organizing these short texts is crucial... |
arxiv | 2607.10546 | null | 1 | LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation | [
"Jinyang Du",
"Hao Ma",
"Xiaohu Shi",
"Bo Yang",
"Yanchun Liang",
"Heow Pueh Lee",
"Chunguo Wu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10546v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "A central pursuit in modern computational science is bridging the gap between abundant empirical measurements and the underlying mathematical laws that dictate system dynamics [ 1 , 2 , 3 , 4 , 5 ] . Whil... |
arxiv | 2607.10540 | null | 1 | Representation Learning for Semiparametric Causal Mediation Analysis under No Essential Heterogeneity | [
"Roberto Faleh",
"Sofia Morelli",
"Holger Brandt"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10540v1 | 2026-07-12T00:00:00 | [
{
"type": "para",
"text": "Keywords: causal mediation, G-estimation, no essential heterogeneity, representation learning, structural mean models, TARNet."
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The primary goal of many randomize... |
arxiv | 2607.10534 | null | 1 | Cross-Layer Misalignment Detection in Agent Skills: A Progressive Loading-Aware Contrastive Learning Approach | [
"Chengjun Zhang",
"Yang Gao",
"Jianna Hur",
"Jingjing Zhang",
"Sagar Samtani"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10534v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Large Language Models (LLMs) are evolving from single-turn text generators into agentic Artificial Intelligence (AI) systems capable of reasoning, tool use, and interaction with external environments (Ya... |
arxiv | 2607.10517 | null | 1 | Conditional Optimal Bridge for Riemannian Activation Steering | [
"Seyed Arshan Dalili",
"Ajay Narayanan Sridhar",
"Vijaykrishnan Narayanan",
"Mehrdad Mahdavi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10517v1 | 2026-07-12T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Activation steering, also termed representation engineering, has emerged as a compelling alternative to fine-tuning for shaping the behavior of large language models (LLMs) at inference time (Rimsky et al... |
arxiv | 2607.11954 | null | 1 | Graph-Constrained Policy Learning for Extreme Clinical Code Prediction | [
"Amritpal Singh",
"Sebastian Torres",
"Khawar Shakeel",
"Syed Ahmad Chan Bukhari"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11954v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Clinical code prediction maps unstructured documentation to standardized diagnosis codes used for reimbursement, quality measurement, cohort discovery, epidemiology, and risk adjustment. In U.S. inpatient... |
arxiv | 2607.11953 | null | 2 | When Does Reward Teach State? A Hidden-Automaton Instrument and the Group-Language Boundary | [
"Jim Allchin"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11953v2 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Many agent tasks are secretly finite-state: onboarding and support workflows, tool-use and API-call sequences, compliance and triage procedures, multi-step form repair. An agent can appear competent durin... |
arxiv | 2607.11952 | null | 1 | Scalable Optimal Transport Algorithm for Network Alignment | [
"Elaheh Hassani",
"Durga Mandarapu",
"Qi Yu",
"Hanghang Tong",
"Ariful Azad"
] | CC-BY-4.0 | https://arxiv.org/html/2607.11952v1 | 2026-07-11T00:00:00 | [
{
"type": "para",
"text": "Code Availability: The source code, data, and other artifacts are available at https://github.com/elawh1/FastAlign ."
},
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "The goal of network alignment is to identif... |
arxiv | 2607.10441 | null | 1 | Context by Distinct Information: An Auditable Dirichlet-Process Working Memory for Long, Redundant Context Streams | [
"Siddharth Pal",
"Viktoria Rojkova"
] | CC-BY-4.0 | https://arxiv.org/html/2607.10441v1 | 2026-07-11T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 The problem: context is metered in tokens, but information is not"
},
{
"type": "para",
"text": "Context engineering is the construction and maintenance of the information exposed to a model at inference time. A context stream may contain natural-l... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.