A depth-recurrent transformer applies a weight-tied core a variable number of times, and prior work has shown that training with a randomized recursion count yields one checkpoint usable across a range of inference depths. We ask what such a model actually computes per token, and measure it directly. On a 135M-class model trained on FineWeb-Edu, the recurrent state converges to a per-token fixed point: mean successive-output KL divergence falls from 3.9e-1 at the second loop to 8.5e-6 by the sixteenth, and per-token state change decays in step. Crucially, this convergence is not uniform across tokens. The median token converges by loop six, while approximately 10 percent of tokens continue to update at the training-mean depth of eight, and mean convergence depth is ordered by token type (whitespace shallowest, content words deepest). This per-token variation is the central object of the paper. We show it is directly readable and that reading it outperforms learning to predict it: a training-free rule that halts each token once its output stabilizes attains uniform depth-8 quality at 4.94 average loops (a 38 percent reduction in average depth) and matches uniform depth across the average-depth range, whereas a linear router trained on convergence labels harvested from the same model requires nearly full depth and yields no reduction. The elasticity that makes this possible reproduces here as background (validation loss decreases monotonically from 3.80 at one loop to 3.20 at eight and remains stable to 32 loops). We report average depth as a FLOP proxy with a three-point wall-clock bracket rather than a realized speedup, make no FLOP-matched parity claim, and note that the allocation results are established at a single scale and seed. The complete study runs on a single RTX 4090 in approximately 100 GPU-hours.
Language model loss follows remarkably regular scaling laws over model and data size, yet it remains unclear why the aggregate loss should exhibit a power-law form. Existing explanations often attribute this regularity to a heavy-tailed spectrum of pattern difficulty in natural language, but this view has not been directly validated at token-level granularity in large-scale real-data training. We present a token-level framework that decomposes scaling laws into localized learning events of individual contextualized tokens. By fitting token loss trajectories with sigmoids, we show that token learning is concentrated in localized transitions, giving rise to a learning-time spectrum that dominates the scaling-law shape. Across more than one hundred pre-training runs on large and diverse real-language corpora with modern LLM architectures, scaling up to 6B parameters and 300B training tokens, the measured learning-time spectrum quantitatively reconstructs the validation loss derivative along the training-step $T$, data-scale $D$, and model-scale $M$ axes. We further show that the same signal is actionable: by reshaping the training distribution according to when tokens become learnable, we alter the optimization trajectory and achieve 11\% faster validation-loss reduction. These results provide direct empirical evidence that scaling laws are governed primarily by the distribution of token-level learning times, and that this distribution can be used not only to explain scaling behavior but also to improve training performance.
Large language models (LLMs) reach high accuracy in mathematical reasoning, but individual traces on the same problem diverge; some arrive at the correct answer while others fail. Prior work analyzes failure at the step, chunk, or sentence level, or at tokens where failure has already occurred. Neither identifies the precise token that triggers the shift toward failure. We introduce the cliff token, a token where the token-wise potential drops significantly under an adaptive threshold that scales with the local token-wise potential, based on a one-sided two-proportion z-test. Across seven models and three mathematical reasoning benchmarks (GSM1K, MATH500, AIME 2025), cliff tokens act as failure triggers; deleting the first cliff token and resampling recovers pass@64 to 1.0, while keeping it limits recovery to between 0.71 and 1.00. We further introduce a cliff taxonomy of deterministic, uncertain, and sampled-off cliffs, defined by greedy choice and token entropy. Each type has distinct probabilistic characteristics, and the taxonomy generalizes across model scales. Finally, we validate the taxonomy via single-token preference optimization at cliff positions (Cliff-DPO). Trained on GSM8K, Cliff-DPO improves accuracy across benchmarks by up to +6.6. Optimizing at uncertain and sampled-off cliffs improves reasoning, while deterministic cliffs do not.
Hybrid language models that mix attention and recurrent layers have shown promise: theoretically, recurrent layers ameliorate the limitations of pure transformers on state tracking, and empirically, hybrids can outperform pure transformers in loss and downstream evaluations \citep{waleffe2024empirical,merrill2026olmohybrid}. Yet it remains unclear which data or capabilities drive these gains, and to what degree they reflect the theoretical advantages motivating hybrid models. We address this question using the open weights from Olmo 3 \citep{olmo2025olmo3} and Olmo Hybrid \citep{merrill2026olmohybrid}: we compare the loss of a matched transformer and hybrid at the same target tokens under the same prefixes, stratifying the results by natural token tags, copy features, delimiter structure, and controlled synthetic probes. The hybrid has lower loss on most tag families, but the gains are not uniform: they are largest for open-class content words and smaller for many closed-class function words. Across prose, code, and markup, the hybrid's loss advantage is larger on opening delimiters than on the corresponding closing delimiters, and nearly vanishes on repeated $n$-grams. Synthetic probes show the same split: the hybrid is favored on pronoun-memory and entity-tracking tasks, whereas the transformer is favored on bracket-matching tasks that require choosing closing delimiters. These patterns suggest that the recurrent layers in hybrids improve predictions that leverage the semantic state of a document, whereas attention helps on tokens predictable by $n$-gram copying or syntactic bracket matching. We conclude with proof-of-concept filtered evaluations showing how token-level decompositions can sharpen pretraining diagnostics for hybrid architectures.
Sayantani Ghosh, Rajashik Datta, Amit Kumar Das +1cs.LG
Transformer-based language models such as BERT having 110M+ parameters have revolutionized natural language understanding, yet their internal mechanisms remain largely opaque to researchers and practitioners. Traditional attention-based interpretability methods often emphasize structurally important but semantically weak tokens such as punctuation marks rather than meaningful semantic relationships. This work introduces a lightweight and model-agnostic framework for quantifying token-level representational importance using hidden-state activation strengths at Layer 8 of BERT. The proposed Activation Flow Network (AFN) framework computes Token Activation Strength using the L2 norm of Layer-8 hidden representations, enabling direct ranking of semantically salient tokens. The study further introduces a threshold-based activation bucket formulation that partitions tokens into HIGH-activation and LOW-activation groups using an empirical upper-quartile activation boundary. Experimental observations demonstrate that semantically meaningful content words consistently occupy the HIGH-activation bucket and dominate representational activation shifts, while structurally supportive tokens contribute comparatively less. The results suggest that Layer 8 acts as a critical semantic consolidation zone balancing structural and semantic information processing. By revealing how activation magnitudes concentrate around semantically informative tokens, this work provides an interpretable and computationally efficient alternative to attentioncentric analysis, contributing toward transforming BERT from a "black box" into a more transparent "glass box" model for natural language understanding.