Parameter estimation in generalized linear models, such as linear and logistic regression problems, is among the most fundamental and well-studied statistical optimization problems. It serves as the primary workhorse in statistical studies arising from a variety of disciplines, ranging from economics [Smi12], biology [VGSM05], and the social sciences [Gor10].

It is known that the minimum-mean-squared-error (MMSE) denoiser under Gaussian noise can be written as a proximal operator, which suffices for asymptotic convergence of plug-and-play (PnP) methods but does not reveal the structure of the induced regularizer or give convergence rates. We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive (to the best of our knowledge) the first sublinear convergence guarantee for PnP proximal gradient descent with an MMSE denoiser. We validate the theory with a one-dimensional synthetic study that recovers the implicit regularizer. We also validate the theory with imaging experiments (deblurring and computed tomography), which exhibit the predicted sublinear behavior.

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In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in\mathcal Y}\phi(x, y) - \max_{z\in\mathcal Z}\psi(x, z)]$, where both $\Phi(x) = \max_{y\in\mathcal Y}\phi(x, y)$ and $\Psi(x)=\max_{z\in\mathcal Z}\psi(x, z)$ are weakly convex functions, and $\phi(x, y), \psi(x, z)$ are strongly concave functions in terms of $y$ and $z$, respectively. It covers two families of problems that have been studied but are missing single-loop stochastic algorithms, i.e., difference of weakly convex functions and weakly convex strongly-concave min-max problems. We propose a stochastic Moreau envelope approximate gradient method dubbed SMAG, the first single-loop algorithm for solving these problems, and provide a state-of-the-art non-asymptotic convergence rate. The key idea of the design is to compute an approximate gradient of the Moreau envelopes of $\Phi, \Psi$ using only one step of stochastic gradient update of the primal and dual variables. Empirically, we conduct experiments on positive-unlabeled (PU) learning and partial area under ROC curve (pAUC) optimization with an adversarial fairness regularizer to validate the effectiveness of our proposed algorithms.

Both component functions are in expectation forms, i.e.,

In the limitβ, the measureµβ concentrates around solutions of the minimisation in eq.(24).

In the Appendix, we give proofs of all results from the main text. We say a function f: R Y! R is M -Lipschitz if for any y 2Y and ˆ y We can also define the Moreau envelope of a function f: R Y! R by The proof of all results in this section can be straightforwardly extended to these settings. Boyd et al. 2004; Bauschke, Combettes, et al. 2011; Rockafellar 1970), but is also useful and Interestingly, there is a similar equivalent characterization for Lipschitz functions as well. Finally, we show that any smooth loss is square-root-Lipschitz. Lipschitz losses is more general than the class of smooth losses studied in Srebro et al. 2010 .

In linear regression, interpolating the square loss is equivalent to interpolating many other losses (such as the absolute loss) on the training set. Similarly, in the context of linear classification, many works (Soudry et al. 2018; Ji and Telgarsky 2019; Muthukumar et al. 2021) have shown that optimizing