On Stability and Generalization of Bilevel Optimization Problem

(Stochastic) bilevel optimization is a frequently encountered problem in machine learning with a wide range of applications such as meta-learning, hyper-parameter optimization, and reinforcement learning. Most of the existing studies on this problem only focused on analyzing the convergence or improving the convergence rate, while little effort has been devoted to understanding its generalization behaviors. In this paper, we conduct a thorough analysis on the generalization of first-order (gradient-based) methods for the bilevel optimization problem. We first establish a fundamental connection between algorithmic stability and generalization error in different forms and give a high probability generalization bound which improves the previous best one from $\bigO(\sqrt{n})$ to $\bigO(\log n)$, where $n$ is the sample size. We then provide the first stability bounds for the general case where both inner and outer level parameters are subject to continuous update, while existing work allows only the outer level parameter to be updated. Our analysis can be applied in various standard settings such as strongly-convex-strongly-convex (SC-SC), convex-convex (C-C), and nonconvex-nonconvex (NC-NC). Our analysis for the NC-NC setting can also be extended to a particular nonconvex-strongly-convex (NC-SC) setting that is commonly encountered in practice. Finally, we corroborate our theoretical analysis and demonstrate how iterations can affect the generalization error by experiments on meta-learning and hyper-parameter optimization.

Benchmarking features from different radiomics toolkits / toolboxes using Image Biomarkers Standardization Initiative

There is no consensus regarding the radiomic feature terminology, the underlying mathematics, or their implementation. This creates a scenario where features extracted using different toolboxes could not be used to build or validate the same model leading to a non-generalization of radiomic results. In this study, the image biomarker standardization initiative (IBSI) established phantom and benchmark values were used to compare the variation of the radiomic features while using 6 publicly available software programs and 1 in-house radiomics pipeline. All IBSI-standardized features (11 classes, 173 in total) were extracted. The relative differences between the extracted feature values from the different software and the IBSI benchmark values were calculated to measure the inter-software agreement. To better understand the variations, features are further grouped into 3 categories according to their properties: 1) morphology, 2) statistic/histogram and 3)texture features. While a good agreement was observed for a majority of radiomics features across the various programs, relatively poor agreement was observed for morphology features. Significant differences were also found in programs that use different gray level discretization approaches. Since these programs do not include all IBSI features, the level of quantitative assessment for each category was analyzed using Venn and the UpSet diagrams and also quantified using two ad hoc metrics. Morphology features earns lowest scores for both metrics, indicating that morphological features are not consistently evaluated among software programs. We conclude that radiomic features calculated using different software programs may not be identical and reliable. Further studies are needed to standardize the workflow of radiomic feature extraction.