Dissertation
A finite-sample, finite-feature error bound and optimized adaptive sampling method for learning trajectories
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2026
DOI:
https://doi.org/10.17918/00011407
Abstract
This thesis develops a sampling framework for improving regression and dynamical systems learning when training data are expensive or unevenly informative. The main idea is that finite-sample error should not only be controlled by the total number of samples, but also by the distribution of these samples. Using replace-one stability arguments, we derive error bounds that separate local contributions from different regions, leading to an allocation principle that assigns more samples to regions with larger contributions to prediction error. Adaptive Random Fourier Features (ARFF) provide the main motivating application. Because ARFF models are mathematically tractable shallow random-feature models, the general sampling framework can be specialized to obtain an error decomposition with distinct finite-feature and finite-sample terms. In particular, the resulting bounds exhibit the expected 1/K + 1/N structure, where K is the number of random features and N is the number of training samples. This makes ARFF a useful setting in which to connect approximation theory, stability, and targeted data allocation. The thesis then extends the sampling framework to dynamical systems by viewing trajectory data as samples from a one-step flow map. In this setting, local one-step errors may be amplified under rollout, so sampling must account for both approximation difficulty and long-time predictive behavior. We introduce empirical loss-preimage sampling that uses current-model error to identify regions where additional samples are most useful. Numerical experiments demonstrate that targeted sampling can outperform uniform sampling, especially in regions that are dynamically important by uniform data collection.
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Details
- Title
- A finite-sample, finite-feature error bound and optimized adaptive sampling method for learning trajectories
- Creators
- Hunter R. Wages
- Contributors
- Gideon Simpson (Advisor)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University
- Number of pages
- xiii, 142 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 991022189169204721