Raghav Thind, Youran Sun, Ling Liang, Haizhao Yang
原文摘要
When an optimization problem uses a surrogate constraint, a decision that is feasible for the surrogate need not be feasible for the true constraint. We propose Selected-Risk Calibration (SRC), a framework that calibrates a learned constraint against the optimizer's returned decisions for objectives drawn from a given distribution. SRC guarantees that, with probability at least $1-δ$, every margin it certifies on held-out objectives has selected risk, the probability that the returned decision is infeasible, of at most $γ$. We also show that the random-input coverage guarantee of split conformal prediction does not bound the selected risk, and that likelihood-ratio reweighting for feedback covariate shift does not apply when the learned constraint is active at the returned decision. Our experiments show that a conformal margin covering about $90\%$ of random inputs covered as few as $17\%$ of the returned decisions, and every certified margin had a violation rate below its target on unseen objectives. Moreover, at equal cost, SRC's correction had fewer or equal infeasible solutions to the baselines in 57 of 62 comparisons. The SRC framework applies to any setting in which a surrogate model chooses a decision whose feasibility can be verified for a sample of instances.
FARMS: A Verifiable Foundation for Automated Computational Science
Di Wu, Ling Liang, Haizhao Yang
AI 速读 基于摘要
作者提出 FARMS，将计算数学知识转化为可供 AI 解释与复用、并带有显式保证的“计算知识”基础。
cs.AI · cs.LO
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Automated computational science could accelerate discovery by selecting, adapting and developing numerical methods. This prospect requires mathematical knowledge that AI systems can interpret and reuse with explicit guarantees. Here we present FARMS, a faithful and efficient foundation that transforms computational mathematics into AI-ready computing knowledge. FARMS connects formal statements to source context, algorithmic meaning and quantitative guarantees through source preparation, reusable representations and semantic review. Across 171 targets in randomized numerical linear algebra and numerical partial differential equations, 157 were accepted by the pipeline, with mean estimated costs of US \$0.85-2.29 per target across sources. Lean-checked implications from explicit source premises connect selected statements to computational skills, illustrated by a heat-flow solve and wave-response compression. FARMS therefore supports an affordable, reusable knowledge foundation for trustworthy automated computational science.
Gromov--Wasserstein optimal transport (GWOT) aligns metric measure spaces by matching their within-domain relational structures, but large-scale GWOT remains challenging because its objective is nonconvex and projection onto the transport polytope is often solved only approximately in practice. This leads to a gap between practical projected-gradient implementations and convergence theory, which typically assumes exact projections. For squared-loss GWOT, we propose an inexact projected-gradient framework with a verifiable feasibility-residual-based inexact condition for the projection subproblem. This condition is directly computable and avoids unknown quantities such as the exact projection point. Under this implementable condition, we prove subsequential convergence to stationary points and, with a mild tolerance-decay condition, convergence of the whole sequence. The resulting method retains the simplicity and sparsity of projected-gradient schemes while providing rigorous convergence guarantees, turning projected-gradient methods into a principled and scalable approach for GWOT with provable reliability.