The development of score-based diffusion models is deeply rooted in the theory of stochastic processes.

Recently, there has been a marked increase in interest regarding the generalization capabilities of unregularized gradient-based learning methods.

Lipschitzness, which we make precise in Section 2 and in Assumption 3.1.

As a result, the demands for interval estimation, and consequently for its validity and precision, have experienced a sustained increase over time and are reflected in a number of recent studies. For example, in proteomics, confidence intervals are employed to assess the association between post-translational modifications and intrinsically disordered regions of proteins, validating hypotheses derived from predictive models and facilitating large-scale functional analyses (Tunyasuvunakool et al., 2021; Bludau et al., 2022). In genomic research, confidence intervals are leveraged to characterize the distribution of gene expression levels, enabling robust inferences about promoter sequence effects and genetic variability (Vaishnav et al., 2022). In the realm of environmental science, interval estimation can be used to monitor deforestation rates of forests, yielding uncertainty-aware insights critical for climate policy formulation (Bullock et al., 2020). As for social sciences, confidence intervals are utilized to evaluate relationships between socioeconomic factors, bolstering the robustness of conclusions drawn from census data (Ding et al., 2021).

Gradient Descent (GD) is a simple yet efficient baseline algorithm for solving nonconvex optimization problems.

This problem-dependent sample complexityresult is expressed in terms of the sub-optimality gapsof the state-action pairs that are visited during exploration.