When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the

When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the

Learning distributions over permutations is a fundamental problem in machine learning, with applications in ranking, combinatorial optimization, structured prediction, and data association. Existing methods rely on mixtures of parametric families or neural networks with expensive variational inference procedures. In this work, we propose a novel approach that leverages alternative representations for permutations, including Lehmer codes, Fisher-Yates draws, and Insertion-Vectors. These representations form a bijection with the symmetric group, allowing for unconstrained learning using conventional deep learning techniques, and can represent any probability distribution over permutations. Our approach enables a trade-off between expressivity of the model family and computational requirements. In the least expressive and most computationally efficient case, our method subsumes previous families of well established probabilistic models over permutations, including Mallow's and the Repeated Insertion Model. Experiments indicate our method significantly outperforms current approaches on the jigsaw puzzle benchmark, a common task for permutation learning. However, we argue this benchmark is limited in its ability to assess learning probability distributions, as the target is a delta distribution (i.e., a single correct solution exists). We therefore propose two additional benchmarks: learning cyclic permutations and re-ranking movies based on user preference. We show that our method learns non-trivial distributions even in the least expressive mode, while traditional models fail to even generate valid permutations in this setting.

As post-training techniques evolve, large language models (LLMs) are increasingly augmented with structured multi-step reasoning abilities, often optimized through reinforcement learning. These reasoning-enhanced models outperform standard LLMs on complex tasks and now underpin many commercial LLM APIs. However, to protect proprietary behavior and reduce verbosity, providers typically conceal the reasoning traces while returning only the final answer. This opacity introduces a critical transparency gap: users are billed for invisible reasoning tokens, which often account for the majority of the cost, yet have no means to verify their authenticity. This opens the door to token count inflation, where providers may overreport token usage or inject synthetic, low-effort tokens to inflate charges. To address this issue, we propose CoIn, a verification framework that audits both the quantity and semantic validity of hidden tokens. CoIn constructs a verifiable hash tree from token embedding fingerprints to check token counts, and uses embedding-based relevance matching to detect fabricated reasoning content. Experiments demonstrate that CoIn, when deployed as a trusted third-party auditor, can effectively detect token count inflation with a success rate reaching up to 94.7%, showing the strong ability to restore billing transparency in opaque LLM services. The dataset and code are available at https://github.com/CASE-Lab-UMD/LLM-Auditing-CoIn.

Recent advances in large language models (LLMs) have popularized the chain-of-thought (CoT) paradigm, in which models produce explicit reasoning steps in natural language. Although this approach improves interpretability and facilitates external auditing, it may not represent the most computationally efficient method for internal reasoning. In contrast, human cognition relies on implicit mental representations that recall past sensory and episodic information without requiring complete verbalization. In this paper, we propose a framework that integrates implicit mental representations into the internal reasoning processes of LLMs. Preliminary experiments indicate that incorporating an Implicit Memory Module (IMM) into a simple GPT model yields a reduction of between 35% and 57% in final training loss compared to a regular GPT baseline. The addition of an explicit interpretability channel (e.g., a chain-of-thought decoder) is straightforward to implement within this approach. We outline theoretical foundations, propose technical mechanisms to scale the memory module, and discuss how these ideas may lead to more efficient and robust reasoning, with optional future extensions for explicit auditability.

When training deep neural networks, a model's generalization error is often observed to follow a power scaling law dependent both on the model size and the data size. Perhaps the best known example of such scaling laws are for transformer-based large language models, where networks with billions of parameters are trained on trillions of tokens of text. Yet, despite sustained widespread interest, a rigorous understanding of why transformer scaling laws exist is still missing. To answer this question, we establish novel statistical estimation and mathematical approximation theories for transformers when the input data are concentrated on a low-dimensional manifold. Our theory predicts a power law between the generalization error and both the training data size and the network size for transformers, where the power depends on the intrinsic dimension $d$ of the training data. Notably, the constructed model architecture is shallow, requiring only logarithmic depth in $d$. By leveraging low-dimensional data structures under a manifold hypothesis, we are able to explain transformer scaling laws in a way which respects the data geometry. Moreover, we test our theory with empirical observation by training LLMs on natural language datasets. We find the observed empirical data scaling laws closely agree with our theoretical predictions. Taken together, these results rigorously show the intrinsic dimension of data to be a crucial quantity affecting transformer scaling laws in both theory and practice.