Of course, the count function should be fixed.

Natural languages exhibit striking regularities in their statistical structure, including notably the emergence of Zipf's and Heaps' laws. Despite this, it remains broadly unclear how these properties relate to the modern tokenisation schemes used in contemporary transformer models. In this note, we analyse the information content (as measured by the Shannon entropy) of various corpora under the assumption of a Zipfian frequency distribution, and derive a closed-form expression for the slot entropy expectation value. We then empirically investigate how byte--pair encoding (BPE) transforms corpus statistics, showing that recursive applications of BPE drive token frequencies toward a Zipfian power law while inducing a characteristic growth pattern in empirical entropy. Utilizing the ability of transformers to learn context dependent token probability distributions, we train language models on corpora tokenised at varying BPE depths, revealing that the model predictive entropies increasingly agree with Zipf-derived predictions as the BPE depth increases. Attention-based diagnostics further indicate that deeper tokenisation reduces local token dependencies, bringing the empirical distribution closer to the weakly dependent (near IID) regime. Together, these results clarify how BPE acts not only as a compression mechanism but also as a statistical transform that reconstructs key informational properties of natural language.

Personalized federated learning with differential privacy has been considered a feasible solution to address non-IID distribution of data and privacy leakage risks. However, current personalized federated learning methods suffer from inflexible personalization and convergence difficulties due to two main factors: 1) Firstly, we observe that the prevailing personalization methods mainly achieve this by personalizing a fixed portion of the model, which lacks flexibility.

A growing body of research runs human subject evaluations to study whether providing users with explanations of machine learning models can help them with practical real-world use cases. However, running user studies is challenging and costly, and consequently each study typically only evaluates a limited number of different settings, e.g., studies often only evaluate a few arbitrarily selected model explanation methods. To address these challenges and aid user study design, we introduce Simulated Evaluations (SimEvals). SimEvals involve training algorithmic agents that take as input the information content (such as model explanations) that would be presented to the user, to predict answers to the use case of interest. The algorithmic agent's test set accuracy provides a measure of the predictiveness of the information content for the downstream use case. We run a comprehensive evaluation on three real-world use cases (forward simulation, model debugging, and counterfactual reasoning) to demonstrate that SimEvals can effectively identify which explanation methods will help humans for each use case. These results provide evidence that \simevals{} can be used to efficiently screen an important set of user study design decisions, e.g., selecting which explanations should be presented to the user, before running a potentially costly user study.

Conventional wisdom in the age of LLMs dictates that solving IQ-test-like visual puzzles from the ARC-AGI-1 benchmark requires capabilities derived from massive pretraining. To counter this, we introduce CompressARC, a 76K parameter model without any pretraining that solves 20% of evaluation puzzles by minimizing the description length (MDL) of the target puzzle purely during inference time. The MDL endows CompressARC with extreme generalization abilities typically unheard of in deep learning. To our knowledge, CompressARC is the only deep learning method for ARC-AGI where training happens only on a single sample: the target inference puzzle itself, with the final solution information removed. Moreover, CompressARC does not train on the pre-provided ARC-AGI "training set". Under these extremely data-limited conditions, we do not ordinarily expect any puzzles to be solvable at all. Yet CompressARC still solves a diverse distribution of creative ARC-AGI puzzles, suggesting MDL to be an alternative feasible way to produce intelligence, besides conventional pretraining.

Deep temporal architectures such as TCNs achieve strong predictive performance on sequential data, yet theoretical understanding of their generalization remains limited. We address this gap through three contributions: introducing an evaluation methodology for temporal models, revealing surprising empirical phenomena about temporal dependence, and the first architecture-aware theoretical framework for dependent sequences. Fair-Comparison Methodology. We introduce evaluation protocols that fix effective sample size $N_{\text{eff}}$ to isolate temporal structure effects from information content. Empirical Findings. Applying this method reveals that under $N_{\text{eff}} = 2000$, strongly dependent sequences ($ρ= 0.8$) exhibit approx' $76\%$ smaller generalization gaps than weakly dependent ones ($ρ= 0.2$), challenging the conventional view that dependence universally impedes learning. However, observed convergence rates ($N_{\text{eff}}^{-1.21}$ to $N_{\text{eff}}^{-0.89}$) significantly exceed theoretical worst-case predictions ($N^{-0.5}$), revealing that temporal architectures exploit problem structure in ways current theory does not capture. Lastly, we develop the first architecture-aware generalization bounds for deep temporal models on exponentially $β$-mixing sequences. By embedding Golowich et al.'s i.i.d. class bound within a novel blocking scheme that partitions $N$ samples into approx' $B \approx N/\log N$ quasi-independent blocks, we establish polynomial sample complexity under convex Lipschitz losses. The framework achieves $\sqrt{D}$ depth scaling alongside the product of layer-wise norms $R = \prod_{\ell=1}^{D} M^{(\ell)}$, avoiding exponential dependence. While these bounds are conservative, they prove learnability and identify architectural scaling laws, providing worst-case baselines that highlight where future theory must improve.

This article trained a network for perceiving three-dimensional motion information of binocular vision target, which can provide real-time three-dimensional coordinate, velocity, and acceleration, and has a basic spatiotemporal perception capability. Understood the ability of neural networks to fit nonlinear problems from the perspective of PID. Considered a single-layer neural network as using a second-order difference equation and a nonlinearity to describe a local problem. Multilayer networks gradually transform the raw representation to the desired representation through multiple such combinations. Analysed some reference principles for designing neural networks. Designed a relatively small PID convolutional neural network, with a total of 17 layers and 413 thousand parameters. Implemented a simple but practical feature reuse method by concatenation and pooling. The network was trained and tested using the simulated randomly moving ball datasets, and the experimental results showed that the prediction accuracy was close to the upper limit that the input image resolution can represent. Analysed the experimental results and errors, as well as the existing shortcomings and possible directions for improvement. Finally, discussed the advantages of high-dimensional convolution in improving computational efficiency and feature space utilization. As well as the potential advantages of using PID information to implement memory and attention mechanisms.

We study how trajectory value depends on the learning algorithm in policy-gradient control. Using Trajectory Shapley in an uncertain LQR, we find a robust negative correlation between a trajectory's information content--Persistence of Excitation (PE)--and its marginal value under vanilla REINFORCE (e.g., r 0.38). We prove a variance-mediated mechanism: (i) for fixed energy, higher PE yields lower gradient variance; (ii) near saddle regions, higher variance increases the probability of escaping poor basins and thus raises marginal contribution. When the update is stabilized (state whitening or Fisher preconditioning), this variance channel is neutralized and information content dominates, flipping the correlation positive (e.g., r +0.29). Hence, trajectory value is algorithm-relative: it emerges from the interaction between data statistics and update dynamics. Experiments on LQR validate the two-step mechanism and the flip, and show that decision-aligned scores (Leave-One-Out) complement Shapley for pruning near the full set, while Shapley remains effective for identifying high-impact (and toxic) subsets.