Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling · arXivDesk
2608.13467 Aug 13, 2026
Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling Yuchen Xin, Zhihua Zhang
Abstract
We study the Moreau--Yosida unadjusted Langevin algorithm (MYULA) for the nonsmooth composite target π ( d x ) ∝ exp { − f ( x ) − g ( x ) } d x , x ∈ R d , π(dx)\propto \exp\{-f(x)-g(x)\}\,dx, \qquad x\in\mathbb R^d, π ( d x ) ∝ exp { − f ( x ) − g ( x )} d x , x ∈ R d ,
where
is
-strongly convex with
-Lipschitz gradient and
is convex and
-Lipschitz. Let
be the Moreau envelope of
,
the corresponding smoothed target, and
a λ = tr H λ a_λ=\operatorname{tr}H_λ a λ = tr H λ , where
is the a.e./weak Hessian of
. We show that the leading MYULA discretization error is controlled by the reference active trace
B r e f B_{\mathrm{ref}} B ref , the average of
along the heat substep of one MYULA update started from
, rather than by the global curvature bound
. If
is an a.e. upper bound for
, then, up to logarithmic factors,
N ≲ 1 m [ L f + τ f + G 2 + B r e f ε a l g 2 + M λ ε a l g ] , τ f : = sup x tr ∇ 2 f ( x ) , 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), N ≲ m 1 [ L f + ε alg 2 τ f + G 2 + B ref + ε alg M λ ] , τ f := x sup tr ∇ 2 f ( x ) , iterations suffice to ensure
m W 2 ( μ N , π λ ) ≤ ε a l g \sqrt m\,W_2(μ_N,π_λ)\leq\varepsilon_{\mathrm{alg}} m W 2 ( μ N , π λ ) ≤ ε alg , where
is the law of the
-th iterate and
is the quadratic Wasserstein distance. We also prove the Moreau-bias bound
m W 2 ( π λ , π ) ≤ G 2 λ 4 . \sqrt m\,W_2(π_λ,π) \leq \frac{G^2λ}{4}. m W 2 ( π λ , π ) ≤ 4 G 2 λ . Thus, choosing
λ ≍ ε / G 2 λ\asymp\varepsilon/G^2 λ ≍ ε / G 2 gives an end-to-end guarantee for
. The universal estimate
B r e f ≤ d / λ B_{\mathrm{ref}}\leq d/λ B ref ≤ d / λ yields
O ~ ( ε − 3 ) \widetilde O(\varepsilon^{-3}) O ( ε − 3 ) accuracy dependence. For the structured piecewise-linear, lasso-type, group, and total-variation penalties considered here, curvature--tube estimates make
B r e f B_{\mathrm{ref}} B ref independent of
, yielding
O ~ ( ε − 2 ) \widetilde O(\varepsilon^{-2}) O ( ε − 2 ) for the same classical MYULA kernel.
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