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Zeroth-order methods are extensively used in machine learning applications where gradients are infeasible or expensive to compute, such as black-box attacks, reinforcement learning, and language model fine-tuning. Existing optimization theory focuses on convergence to an arbitrary stationary point, but less is known about the implicit regularization that provides a fine-grained characterization of which particular solutions are reached. This paper shows that zeroth-order optimization with the standard two-point estimator favors solutions with small trace of Hessian, a measure widely used to distinguish between sharp and flat minima. The authors provide convergence rates of zeroth-order optimization to approximate flat minima for convex and sufficiently smooth functions, defining flat minima as minimizers that achieve the smallest trace of Hessian among all optimal solutions. Experiments on binary classification tasks with convex losses and language model fine-tuning support the theoretical findings. 

Although the dynamics of collisions between a molecule and a solid surface are ultimately quantum mechanical, decohering effects owing to the large number of interacting degrees of freedom typically obscure the wavelike nature of these events. However, a partial decoupling of internal molecular motion from external degrees of freedom can reveal striking interference effects despite significant momentum exchange between the molecule and the bath of surface vibrations. We report state-prepared and state-resolved measurements of methane scattering from a room-temperature gold surface that demonstrate total destructive interference between molecular states related by a reflection symmetry operation. High-contrast interference effects prevail for all processes investigated, including vibrationally excited and vibrationally inelastic collisions. The results demonstrate the distinctly quantum mechanical effect of discrete symmetries in molecular collision dynamics. 

Understanding the petrological and geochemical processes shaping the Moho transition zone (MTZ) is crucial for advancing our knowledge of thermal and chemical exchanges between the oceanic crust and the residual upper mantle. In this study, we systematically investigate the MTZ outcropped within the Zedong ophiolite, located in the eastern part of the Yarlung-Tsangpo Suture Zone (YTSZ), with the aim of at reconstructing the magmatic processes responsible for generating the petrological Moho. The Zedong MTZ comprises a sequence of dunite, wehrlite, pyroxenite, and gabbro, with frequent occurrences of clinopyroxene-rich lithologies. Cyclicity within the MTZ sequences is characterized by the recurrence of olivine-rich intervals and the presence of zig-zag patterns in both major and trace elements of clinopyroxenes. Zircon Usingle bondPb dating on the Zedong gabbros supports the coeval formation of the Zedong ophiolite with other YTSZ ophiolites. Clinopyroxene in the Zedong MTZ follows a differentiation sequence characterized by an increase in contents of Al2O3 and TiO2, coupled with a decrease in Mg#. This differentiation sequence along with frequent occurrences of amphibole suggest the evolution of a primitive hydrous melt depleted in Al2O3, TiO2, and Na2O. The depleted Ndsingle bondHf isotopes and rare earth element patterns of the MTZ rocks indicate that their parental magmas originated from fluid-enhanced re-melting of a previously depleted mantle. Additionally, we proposed that the initiation of a new subduction zone results in the re-melting of the mantle peridotite, leading to the formation of primitive hydrous basaltic melts. The variable lithologies observed in the Zedong MTZ arise from fractional crystallization and repeated replenishment of hydrous melts.