In recent years there has been increasing interest in leveraging denoisers for solving general inverse problems. Two leading frameworks are regularization-by-denoising (RED) and plug-and-play priors (PnP) which incorporate explicit likelihood functions with priors induced by denoising algorithms. RED and PnP have shown state-of-the-art performance in diverse imaging tasks when powerful denoisersare used, such as convolutional neural networks (CNNs). However, the study of their convergence remains an active line of research. Recent works derive the convergence of RED and PnP methods by treating CNN denoisers as approximations for maximum a posteriori (MAP) or minimum mean square error (MMSE) estimators.

In recent years there has been increasing interest in leveraging denoisers for solving general inverse problems. Two leading frameworks are regularization-by-denoising (RED) and plug-and-play priors (PnP) which incorporate explicit likelihood functions with priors induced by denoising algorithms. RED and PnP have shown state-of-the-art performance in diverse imaging tasks when powerful denoisersare used, such as convolutional neural networks (CNNs). However, the study of their convergence remains an active line of research. Recent works derive the convergence of RED and PnP methods by treating CNN denoisers as approximations for maximum a posteriori (MAP) or minimum mean square error (MMSE) estimators.

Recent work by Woodworth et al. (2020) shows that the optimization dynamics of gradient descent for overparameterized problems can be viewed as low-dimensional dual dynamics induced by a mirror map, explaining the implicit regularization phenomenon from the mirror descent perspective. However, the methodology does not apply to algorithms where update directions deviate from true gradients, such as ADAM. We use the mirror descent framework to study the dynamics of smoothed sign descent with a stability constant $\varepsilon$ for regression problems. We propose a mirror map that establishes equivalence to dual dynamics under some assumptions. By studying dual dynamics, we characterize the convergent solution as an approximate KKT point of minimizing a Bregman divergence style function, and show the benefit of tuning the stability constant $\varepsilon$ to reduce the KKT error.

Deep neural networks have achieved remarkable empirical successes in the domains of computer vision, speech recognition, and natural language processing, sparking an interest in the theory behind their architectures and training. However, deep neural networks are often found to be highly overparameterized making them computationally expensive with large amounts of memory and computational power, which may take up to weeks on a modern multi-GPU server for large datasets such as ImageNet Deng et al. [7]. Hence, they are often unsuitable for smaller devices like embedded electronics, and there is a pressing demand for techniques to optimize models with reduced model size, faster inference and lower power consumption. Sparse networks, that is, neural networks in which a large subset of the model parameters are zero, have emerged as one of the leading approaches for reducing model parameter count. It has been shown empirically that deep neural networks can achieve state-of-the-art results under high levels of sparsity [17, 14, 22].