Underpinning the success of deep learning is effective regularizations that allow a variety of priors in data to be modeled.

Recent studies have shown that proximal gradient (PG) method and accelerated gradient method (APG) with restarting can enjoy a linear convergence under a weaker condition than strong convexity, namely a quadratic growth condition (QGC). However, the faster convergence of restarting APG method relies on the potentially unknown constant in QGC to appropriately restart APG, which restricts its applicability.

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Optimization problems over discrete or quantized variables are very challenging in general due to the combinatorial nature of their search space. Piecewise-affine regularization (PAR) provides a flexible modeling and computational framework for quantization based on continuous optimization. In this work, we focus on the setting of supervised learning and investigate the theoretical foundations of PAR from optimization and statistical perspectives. First, we show that in the overparameterized regime, where the number of parameters exceeds the number of samples, every critical point of the PAR-regularized loss function exhibits a high degree of quantization. Second, we derive closed-form proximal mappings for various (convex, quasi-convex, and non-convex) PARs and show how to solve PAR-regularized problems using the proximal gradient method, its accelerated variant, and the Alternating Direction Method of Multipliers. Third, we study statistical guarantees of PAR-regularized linear regression problems; specifically, we can approximate classical formulations of $\ell_1$-, squared $\ell_2$-, and nonconvex regularizations using PAR and obtain similar statistical guarantees with quantized solutions.

Lastly, we demonstrate the superiority of stochastic proximal methods compared to subgradient-based approaches via extensive experiments.

Lastly, we demonstrate the superiority of stochastic proximal methods compared to subgradient-based approaches via extensive experiments.