We study the Moreau--Yosida unadjusted Langevin algorithm (MYULA) for the nonsmooth composite target \[ π(dx)\propto \exp\{-f(x)-g(x)\}\,dx, \qquad x\in\mathbb R^d, \] where \(f\) is \(m\)-strongly convex with \(L_f\)-Lipschitz gradient and \(g\) is convex and \(G\)-Lipschitz. Let \(g_λ\) be the Moreau envelope of \(g\), \(π_λ\) the corresponding smoothed target, and \(a_λ=\operatorname{tr}H_λ\), where \(H_λ\) is the a.e./weak Hessian of \(g_λ\). We show that the leading MYULA discretization error is controlled by the reference active trace \(B_{\mathrm{ref}}\), the average of \(a_λ\) along the heat substep of one MYULA update started from \(π_λ\), rather than by the global curvature bound \(d/λ\). If \(M_λ\) is an a.e. upper bound for \(a_λ\), then, up to logarithmic factors, \[ N \lesssim \frac{1}{m} \left[ L_f + \frac{ τ_f+G^2+B_{\mathrm{ref}} }{ \varepsilon_{\mathrm{alg}}^2 } + \frac{M_λ}{\varepsilon_{\mathrm{alg}}} \right], \qquad τ_f:= \sup_x\operatorname{tr}\nabla^2 f(x), \] iterations suffice to ensure \(\sqrt m\,W_2(μ_N,π_λ)\leq\varepsilon_{\mathrm{alg}}\), where \(μ_N\) is the law of the \(N\)-th iterate and \(W_2\) is the quadratic Wasserstein distance. We also prove the Moreau-bias bound \[ \sqrt m\,W_2(π_λ,π) \leq \frac{G^2λ}{4}. \] Thus, choosing \(λ\asymp\varepsilon/G^2\) gives an end-to-end guarantee for \(π\). The universal estimate \(B_{\mathrm{ref}}\leq d/λ\) yields \(\widetilde O(\varepsilon^{-3})\) accuracy dependence. For the structured piecewise-linear, lasso-type, group, and total-variation penalties considered here, curvature--tube estimates make \(B_{\mathrm{ref}}\) independent of \(λ\), yielding \(\widetilde O(\varepsilon^{-2})\) for the same classical MYULA kernel.
In this paper, we consider a class of multiblock nonconvex nonsmooth optimization problems, which covers many applications such as the analysis of pre-earthquake anomalies and machine learning. To solve this class of problems, we propose the inertial block proximal linearized method with two-phase adaptive momentum (IBPL$^+$-TP). Compared to the current methods, our method possesses three main advantages: (1) it introduces a two-phase adaptive momentum strategy to effectively update the extrapolation parameters, (2) it allows using two different extrapolation points to accelerate the convergence, (3) it allows the extrapolation parameters of these two extrapolation points to be independent of and unconstrained by all other parameters. While maintaining the above advantages, we prove that our method ensures the monotonic convergence of the objective function of this class of problems, and we also prove that the sequence generated by our method globally converges to a critical point, as well as establish the convergence rate of our method. To demonstrate the effectiveness of our method, we apply it to solve two nonconvex and nonsmooth machine learning problems, namely sparse nonnegative matrix factorization with $\ell_0$-constraints and sparse nonnegative CP decomposition with $\ell_0$-constraints. The numerical experimental results on solving these problems show that our method outperforms several state-of-the-art methods.
Adam is one of the most widely implemented and influential modern optimizers. Why is it effective across different optimization problems in practice? This question arguably lies at the center of the optimization community over the last decade and has motivated a substantial body of work aimed at understanding its convergence behavior. However, existing studies have mainly focused on the convergence rate of Adam in smooth nonconvex optimization, which unfortunately does not adequately capture practical settings, since many real-world problems are nonsmooth, such as those arising in training neural networks. Thus, these studies cannot fully explain the popularity and empirical success of Adam. Recently, an insightful and powerful framework called Online-to-Nonconvex Conversion has opened a new way to analyze Adam for nonsmooth nonconvex optimization. Unfortunately, prior works along this line share two common limitations. First, all of them ignore the important bias-correction term in the original Adam algorithm. Second and more importantly, many of them require extra operations that are not used in Adam, such as a clipping step. Therefore, the convergence guarantee for the original Adam method still remains unclear. In this work, we present the first finite-time analysis for the classical form of Adam, i.e., with the bias-correction step and without further algorithmic modifications, and prove that a randomly scaled learning rate ensures a convergence rate of $1/T^{\frac{2}{13}}$ for nonsmooth nonconvex optimization. Moreover, our result provably applies to the modern heavy-tailed noise regime, which is closer to practice. Interestingly, our theory is established under the parameter choice $β_1=β_2$, aligning with the recent empirical studies.