Neural scaling laws establish a predictable relationship between model performance and data or compute, offering crucial guidance for resource allocation in new domains and tasks. Yet such laws are most needed precisely where they are hardest to obtain: fitting one for a new model and task pair demands expensive sweeps that typically exhaust the very compute budget the law is meant to economize. We study meta-scaling laws: relations that characterize when neural scaling behavior is preserved and predict how it changes under data transformations, enabling forecasts across domains when the shift can be approximated by a compatible transformation. We address this by identifying the right invariants: scaling laws are preserved under bijective (information-preserving) transformations of the data and modified in predictable, information-theoretically grounded ways under non-bijective transformations that lower its information resolution $ρ$: a single axis along which a law fit in one domain can be transported to another. We validate this across language, vision, and speech, and demonstrate two cross-domain applications: predicting scaling for language models trained on electronic health records from laws fit on general text, and predicting time-series classification scaling under varying levels of noise injection, recovering the data-scaling exponents to within $3\%$ error.
We study risk control for online policy learning, where an agent adaptively collects data and aims to learn a new policy whose risk stays below a tolerance threshold at every round with high probability. Such policy updates and data collection induce feedback covariate shift (FCS): observed data depend on history through evolving policies and the data-collection process. FCS brings two substantive problems for risk control: 1) efficient risk evaluation for each candidate policy and 2) valid policy selection with high-probability risk control. To address these, we propose a permutation-based weighted risk evaluator with weights efficiently approximated by the Sinkhorn algorithm and select policies whose estimated risk upper confidence bounds (UCB) fall below the target threshold, with the UCB obtained through a proposed Gaussian bootstrap method. Theoretically, we prove that our risk evaluator has a no-larger variance upper bound than that of the standard importance-weighted evaluator. Our policy control method also yields an asymptotically valid simultaneous risk bound and provides asymptotic high-probability risk control for the selected policy. Empirically, we evaluate our method on the online contextual bandit problem, safe question answering benchmark, and Medical Claim Filtering tasks. Our method maintains the target risk level, achieves lower risk than baseline methods, has lower variance in risk estimation, and achieves reliable policy selection with high-probability risk control. Further ablation studies support our theoretical claims.