In Section 3, we verify that this geometry is linearly represented in the residual stream of transformers.

Humans can length-generalize in integer addition because they understand the essential principle of the task. Nevertheless, itisobserved that Transformers typically learn to solve addition only up to the training sequence length (Lee et al., 2024), which is different from thetruearithmetic algorithm thathumans "implement".

Large language models (LLMs) process and predict sequences containing text to answer questions, and address tasks including document summarization, providing recommendations, writing software and solving quantitative problems. We provide a mathematical framework for LLMs by describing the encoding of text sequences into sequences of tokens, defining the architecture for next-token prediction models, explaining how these models are learned from data, and demonstrating how they are deployed to address a variety of tasks. The mathematical sophistication required to understand this material is not high, and relies on straightforward ideas from information theory, probability and optimization. Nonetheless, the combination of ideas resting on these different components from the mathematical sciences yields a complex algorithmic structure; and this algorithmic structure has demonstrated remarkable empirical successes. The mathematical framework established here provides a platform from which it is possible to formulate and address questions concerning the accuracy, efficiency and robustness of the algorithms that constitute LLMs. The framework also suggests directions for development of modified and new methodologies.

We study the fractal structure of language, aiming to provide a precise formalism for quantifying properties that may have been previously suspected but not formally shown. We establish that language is: (1) self-similar, exhibiting complexities at all levels of granularity, with no particular characteristic context length, and (2) long-range dependent (LRD), with a Hurst parameter of approximately 0.7.Based on these findings, we argue that short-term patterns/dependencies in language, such as in paragraphs, mirror the patterns/dependencies over larger scopes, like entire documents. This may shed some light on how next-token prediction can capture the structure of text across multiple levels of granularity, from words and clauses to broader contexts and intents. In addition, we carry out an extensive analysis across different domains and architectures, showing that fractal parameters are robust.Finally, we demonstrate that the tiny variations in fractal parameters seen across LLMs improve upon perplexity-based bits-per-byte (BPB) in predicting their downstream performance. We hope these findings offer a fresh perspective on language and the mechanisms underlying the success of LLMs.

What computational structure are we building into large language models when we train them on next-token prediction? Here, we present evidence that this structure is given by the meta-dynamics of belief updating over hidden states of the data-generating process. Leveraging the theory of optimal prediction, we anticipate and then find that belief states are linearly represented in the residual stream of transformers, even in cases where the predicted belief state geometry has highly nontrivial fractal structure. We investigate cases where the belief state geometry is represented in the final residual stream or distributed across the residual streams of multiple layers, providing a framework to explain these observations. Furthermore we demonstrate that the inferred belief states contain information about the entire future, beyond the local next-token prediction that the transformers are explicitly trained on. Our work provides a general framework connecting the structure of training data to the geometric structure of activations inside transformers.

Transformers replace recurrence with a memory that grows with sequence length and self-attention that enables ad-hoc look ups over past tokens. Consequently, they lack an inherent incentive to compress history into compact latent states with consistent transition rules. This often leads to learning solutions that generalize poorly. We introduce Next-Latent Prediction (NextLat), which extends standard next-token training with self-supervised predictions in the latent space. Specifically, NextLat trains a transformer to learn latent representations that are predictive of its next latent state given the next output token. Theoretically, we show that these latents provably converge to belief states, compressed information of the history necessary to predict the future. This simple auxiliary objective also injects a recurrent inductive bias into transformers, while leaving their architecture, parallel training, and inference unchanged. NextLat effectively encourages the transformer to form compact internal world models with its own belief states and transition dynamics -- a crucial property absent in standard next-token prediction transformers. Empirically, across benchmarks targeting core sequence modeling competencies -- world modeling, reasoning, planning, and language modeling -- NextLat demonstrates significant gains over standard next-token training in downstream accuracy, representation compression, and lookahead planning. NextLat stands as a simple and efficient paradigm for shaping transformer representations toward stronger generalization.

Recent advances in reasoning domains with neural networks have primarily been enabled by a training recipe that optimizes Large Language Models, previously trained to predict the next-token in a sequence, with reinforcement learning algorithms. We introduce a framework to study the success of this paradigm, and we theoretically expose the optimization mechanisms by which reinforcement learning improves over next-token prediction in this setting. We study learning from mixture distributions of short and long ``chain-of-thought'' sequences encoding a single task. In particular, when the task consists of predicting the parity of $d$ bits and long sequences are rare, we show how reinforcement learning after next-token prediction enables autoregressive transformers to generalize, whereas mere next-token prediction requires extreme statistical or computational resources to do so. We further explain how reinforcement learning leverages increased test-time computation, manifested in longer responses, to facilitate this learning process. In a simplified setting, we theoretically prove that autoregressive linear models following this training recipe can efficiently learn to predict the parity of $d$ bits as long as the proportion of long demonstrations in the data mix is not exponentially small in the input dimension $d$. Finally, we demonstrate these same phenomena in other settings, including the post-training of Llama-series models on mixture variations of common mathematical reasoning benchmarks.

The rapid development research of Large Language Models (LLMs) based on transformer architectures raises key challenges, one of them being the task of distinguishing between human-written text and LLM-generated text. As LLM-generated textual content, becomes increasingly complex over time, and resembles human writing, traditional detection methods are proving less effective, especially as the number and diversity of LLMs continue to grow with new models and versions being released at a rapid pace. This study proposes VietBinoculars, an adaptation of the Binoculars method with optimized global thresholds, to enhance the detection of Vietnamese LLM-generated text. We have constructed new Vietnamese AI-generated datasets to determine the optimal thresholds for VietBinoculars and to enable benchmarking. The results from our experiments show results show that VietBinoculars achieves over 99\% in all two domains of accuracy, F1-score, and AUC on multiple out-of-domain datasets. It outperforms the original Binoculars model, traditional detection methods, and other state-of-the-art approaches, including commercial tools such as ZeroGPT and DetectGPT, especially under specially modified prompting strategies.