Yifan Bai (白一凡)

Typical dimension reduction techniques for nonoverlapping sparse optimization involve screening or sieving strategies based on a dual certificate derived from the first-order optimality condition, approximating the gradients or exploiting certain inherent low-dimensional structure of the sparse solution. In comparison, dimension reduction rules for overlapping group sparsity are generally less developed because the subgradient structure is more complex, making the link between sparsity pattern and the dual variable indirect due to the non-separability. In this work, we propose new dual certificates for overlapping group sparsity and a novel adaptive scheme for identifying the support of the overlapping group LASSO. We demonstrate how this scheme can be integrated into and significantly accelerate existing algorithms, including Primal-Dual splitting method, alternating direction method of multipliers and a recently developed variable projection scheme based on over-parameterization. We provide convergence analysis of the method and verify its practical effectiveness through experiments on standard datasets.

Iterative algorithms are fundamental tools for approximating fixed-points of nonexpansive operators in real Hilbert spaces. Among them, Krasnosel’skiĭ–Mann iteration and Halpern iteration are two widely used schemes. In this work, we formalize the convergence of these two fixed-point algorithms in the interactive theorem prover Lean4 based on type dependent theory. To this end, weak convergence and topological properties in the infinite-dimensional real Hilbert space are formalized. Definition and properties of nonexpansive operators are also provided. As a useful tool in convex analysis, we then formalize the Fejér monotone sequence. Building on these foundations, we verify the convergence of both the iteration schemes. Our formalization provides reusable components for machine-checked convergence analysis of fixed-point iterations and theories of convex analysis in real Hilbert spaces. Our code is available at https://github.com/TTony2019/fixed-point-iterations-in-lean.

As large language models (LLMs) are increasingly deployed for software engineering, constructing high-quality benchmarks is crucial for evaluating not just the functional correctness, but also the formal verifiability of generated code. However, existing benchmarks are limited by the quantity and quality of positive and negative test cases, leading to an overestimation of model capabilities in generating specifications and implementations. To address this, we propose VeriScale, a novel framework driven by the adversarial implementations. It consists of two stages: test-suite expansion to construct diverse and challenging test cases, and test-suite reduction to distill them into compact yet discriminative suites. While VeriScale is general, we instantiate it on Verina to construct VerinaPlus, which expands the original test suites by over 83\times, and VerinaLite, a lightweight 14\times variant. Our experiments across eight state-of-the-art LLMs demonstrate that VerinaPlus exposes substantial model weaknesses hidden by the original benchmark, evidenced by sharp score drops on both SpecGen and CodeGen tasks, whereas VerinaLite maintains this discriminative power at a fraction of the evaluation cost. The enhanced benchmarks and source code are publicly available at https://github.com/XiaoyangLiu-sjtu/VeriScale.