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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.