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Abstract We propose a new observable for the measurement of the forward–backward asymmetry$$(A_{FB})$$ in Drell–Yan lepton production. At hadron colliders, the$$A_{FB}$$ distribution is sensitive to both the electroweak (EW) fundamental parameter$$\sin ^{2} \theta _{W}$$ , the weak mixing angle, and the parton distribution functions (PDFs). Hence, the determination of$$\sin ^{2} \theta _{W}$$ and the updating of PDFs by directly using the same$$A_{FB}$$ spectrum are strongly correlated. This correlation would introduce large bias or uncertainty into both precise measurements of EW and PDF sectors. In this article, we show that the sensitivity of$$A_{FB}$$ on$$\sin ^{2} \theta _{W}$$ is dominated by its average value around theZpole region, while the shape (or gradient) of the$$A_{FB}$$ spectrum is insensitive to$$\sin ^{2} \theta _{W}$$ and contains important information on the PDF modeling. Accordingly, a new observable related to the gradient of the spectrum is introduced, and demonstrated to be able to significantly reduce the potential bias on the determination of$$\sin ^{2} \theta _{W}$$ when updating the PDFs using the same$$A_{FB}$$ data.
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Faster margin maximization rates for generic and adversarially robust optimization methods
Abstract First-order optimization methods tend to inherently favor certain solutions over others when minimizing an underdetermined training objective that has multiple global optima. This phenomenon, known asimplicit bias, plays a critical role in understanding the generalization capabilities of optimization algorithms. Recent research has revealed that in separable binary classification tasks gradient-descent-based methods exhibit an implicit bias for the$$\ell _2$$ -maximal margin classifier. Similarly, generic optimization methods, such as mirror descent and steepest descent, have been shown to converge to maximal margin classifiers defined by alternative geometries. While gradient-descent-based algorithms provably achievefastimplicit bias rates, corresponding rates in the literature for generic optimization methods are relatively slow. To address this limitation, we present a series of state-of-the-art implicit bias rates for mirror descent and steepest descent algorithms. Our primary technique involves transforming a generic optimization algorithm into an online optimization dynamic that solves a regularized bilinear game, providing a unified framework for analyzing the implicit bias of various optimization methods. Our accelerated rates are derived by leveraging the regret bounds of online learning algorithms within this game framework. We then show the flexibility of this framework by analyzing the implicit bias inadversarial training, and again obtain significantly improved convergence rates.
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- PAR ID:
- 10641843
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematical Programming
- ISSN:
- 0025-5610
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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