At the same time, the field of dynamical systems benefited from deep learning and scaled to countless applications but does not allow parameter identification.

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Prior work suggests that human brain responses to language exhibit hierarchically organized "integration windows" that substantially constrain the

We consider the Continuum Bandit problem where the goal is to find the optimal action under an unknown reward function, with an additional monotonicity constraint (or, markdown constraint) that requires that the action sequence be non-increasing. This problem faithfully models a natural single-product dynamic pricing problem, called markdown pricing, where the objective is to adaptively reduce the price over a finite sales horizon to maximize expected revenues. Jia et al '21 and Chen '21 independently showed a tight $T^{3/4}$ regret bound over $T$ rounds under *minimal* assumptions of unimodality and Lipschitzness in the reward (or, revenue) function. This bound shows that the demand learning in markdown pricing is harder than unconstrained (i.e., without the monotonicity constraint) pricing under unknown demand which suffers regret only of the order of $T^{2/3}$ under the same assumptions (Kleinberg '04). However, in practice the demand functions are usually assumed to have certain functional forms (e.g.

Large Language Models (OpenAI et al., 2024; Team et al., 2025; DeepSeek-AI et al., 2025) based on the Transformer (Vaswani et al., 2023) architecture have achieved impressive results, approaching or exceeding human-level performance across multiple domains. Scaling laws (Hestness et al., 2017; Kaplan et al., 2020) are an established method for modeling the performance of these networks, enabling researchers to plan large-scale training runs based on curated sets of smaller experiments. Traditionally, these laws focus on predicting proxy metrics for model quality, such as pre-training log-perplexity. This has proven invaluable for optimizing training hyperparameters, like the optimal ratio of tokens to parameters. Another important direction in understanding the scaling of LLMs is tracking the behavior of more interpretable indicators of model capabilities, like accuracy on downstream benchmarks measuring the performance on general knowledge, reasoning, math and coding tasks. Despite early attempts to solve this problem (Grattafiori et al., 2024; Isik et al., 2025; Chen et al., 2025), scaling downstream metrics have been often referred to as noisy and unreliable (Schaeffer et al., 2025; Lourie et al., 2025). Current approaches to modeling the downstream performance performance of LLMs (Grattafiori et al., 2024; Chen et al., 2025; Bhagia et al., 2024) typically rely on a two-stage approach, where the training budget is first mapped to a proxy metric like mean log-probability of the correct answer, and then another dependence is established, mapping to benchmark accuracy. Work done as an intern at Apple.

Friction modeling plays a crucial role in achieving high-precision motion control in robotic operating systems. Traditional static friction models (such as the Stribeck model) are widely used due to their simple forms; however, they typically require predefined functional assumptions, which poses significant challenges when dealing with unknown functional structures. To address this issue, this paper proposes a physics-inspired machine learning approach based on the Kolmogorov-Arnold Network (KAN) for static friction modeling of robotic joints. The method integrates spline activation functions with a symbolic regression mechanism, enabling model simplification and physical expression extraction through pruning and attribute scoring, while maintaining both high prediction accuracy and interpretability. We first validate the method's capability to accurately identify key parameters under known functional models, and further demonstrate its robustness and generalization ability under conditions with unknown functional structures and noisy data. Experiments conducted on both synthetic data and real friction data collected from a six-degree-of-freedom industrial manipulator show that the proposed method achieves a coefficient of determination greater than 0.95 across various tasks and successfully extracts concise and physically meaningful friction expressions. This study provides a new perspective for interpretable and data-driven robotic friction modeling with promising engineering applicability. Introduction In robotic operating systems, friction plays a crucial role in determining motion control accuracy, particularly in high-precision, low-velocity, and force-controlled tasks, where its influence becomes markedly pronounced.