TamedMALA: Critically Scaled Score Taming for Robust Langevin Sampling

The Metropolis-adjusted Langevin algorithm (MALA) can be highly efficient when the score provides a reliable local proposal direction, but excessively large scores can instead produce proposals that are almost always rejected. We introduce a proposal-scale-coupled family of score tamers, ${\mathcal{T}}_{\delta ,c,p,\gamma }(g)=\frac{g}{{\left(1+{\delta }^{\gamma }\Vert g{\Vert }^p/c^p\right)}^{1/p}},$ combined with an exact Metropolis–Hastings correction. For fixed-dimensional product blocks, we derive the leading Hastings expansion and show a scaling transition governed by the coupling exponent $\gamma$: for $\gamma <1$, taming worsens the proposal-variance scaling, whereas for $\gamma \ge 1$ the classical $d^{-1/3}$ MALA scaling is preserved. At the critical coupling $\gamma =1$, we obtain an explicit nonnegative asymptotic efficiency penalty on regular targets, clarifying that taming should act as a robustness safeguard rather than improve well-behaved MALA. We further show that $p=2$ uniquely balances saturated deterministic drift with proposal noise. The resulting coordinate-wise TamedMALA achieves essentially MALA-level stationary efficiency across six independent tune-and-evaluate repetitions, while maintaining reliable recovery from deliberately pathological score states where MALA exhibits rejection-induced sticking. A short-chain tuner selects the additional taming scale automatically with stable stationary performance.