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Motivated by robust and quantile regression problems, we investigate the stochastic gradient descent (SGD) algorithm for minimizing an objective functionfthat is locally strongly convex with a sub--quadratic tail. This setting covers many widely used online statistical methods. We introduce a novel piecewise Lyapunov function that enables us to handle functionsfwith only first-order differentiability, which includes a wide range of popular loss functions such as Huber loss. Leveraging our proposed Lyapunov function, we derive finite-time moment bounds under general diminishing stepsizes, as well as constant stepsizes. We further establish the weak convergence, central limit theorem and bias characterization under constant stepsize, providing the first geometrical convergence result for sub--quadratic SGD. Our results have wide applications, especially in online statistical methods. In particular, we discuss two applications of our results. 1) Online robust regression: We consider a corrupted linear model with sub--exponential covariates and heavy--tailed noise. Our analysis provides convergence rates comparable to those for corrupted models with Gaussian covariates and noise. 2) Online quantile regression: Importantly, our results relax the common assumption in prior work that the conditional density is continuous and provide a more fine-grained analysis for the moment bounds. 

We report a novel ultra-thin metalens design based on photonic crystal slab (PCS) resonance modes. We experimentally verified with a metalens structure based on amorphous silicon on a quartz material platform by implementing the optical guided resonance on the PCS. The PCS metalens designs feature an ultra-thin device layer of about 160 nm at an operation wavelength of 940 nm. A full 2π transmission phase transition is realized by varying the air hole sizes at the design wavelength. Metalens devices with different phase change gradients were designed and fabricated to achieve different NAs. A maximum of 86.4% focusing efficiency is achieved. Imaging capabilities are characterized, and clear images are observed within the field of view. The PC resonance-based phase modulation design can be applied to optical beam manipulation, phase plate design, imaging, and laser beam formation applications.