Moreau-Yosida未校正朗之万采样的活动迹复杂度界

中文摘要 AI 辅助

我们研究针对非光滑复合目标的Moreau-Yosida未校正朗之万算法（MYULA），该目标为π(dx)∝exp{-f(x)-g(x)}dx，其中x∈ℝᵈ，f是具有L_f-利普希茨梯度的m-强凸函数，g是凸且G-利普希茨的函数。设g_λ为g的Moreau包络，π_λ为对应的平滑目标，a_λ=trH_λ，H_λ是g_λ的几乎处处/弱海森矩阵。我们证明，MYULA的主导离散化误差由参考活动迹B_ref控制，B_ref是从π_λ开始的一次MYULA更新的热子步中a_λ的平均值，而非全局曲率界d/λ。若M_λ是a_λ的几乎处处上界，则忽略对数因子，迭代次数N满足N≲(1/m)[L_f + (τ_f+G²+B_ref)/ε_alg² + M_λ/ε_alg]，其中τ_f:=supₓtr∇²f(x)，可确保√m·W₂(μ_N,π_λ)≤ε_alg，μ_N是第N次迭代的分布，W₂是二次瓦瑟斯坦距离。我们还证明了Moreau偏差界√m·W₂(π_λ,π)≤G²λ/4，因此选择λ≍ε/G²可得到针对π的端到端保证。通用估计B_ref≤d/λ给出了对精度的Õ(ε⁻³)依赖；对于本文考虑的结构化分段线性、套索型、组和全变分惩罚，曲率-管估计使B_ref与λ无关，对同一经典MYULA核给出Õ(ε⁻²)依赖。

英文摘要

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.

发表机构

• School of Mathematical Sciences, Peking University（北京大学数学科学学院）

机构由 AI 辅助整理，请以论文原文为准。