stat.MLOct 11, 2026

Notes on Uniform Policy Score Bounds and Lipschitz Continuity for Truncated Gaussian, Product Cauchy, Boltzmann, and Softmax Policies

Authors: Yanjie Dong

Organizations: Xi’an Jiaotong University

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Abstract

We establish uniform score bounds and Lipschitz continuity for truncated Gaussian, Cauchy, Boltzmann, and softmax policies under explicitly stated parameterizations.

Figures & tables

PolicyFixed structure / uniform boundRπR_{\pi}LπL_{\pi}
Truncated Gaussianμ=Wϕ\mu=W\phi , ∥ϕ∥2≤B\lVert\phi\rVert_{2}\leq B ; Σ\Sigma fixed; support diameter DDBD∥Σ−1∥opBD\lVert\Sigma^{-1}\rVert_{\mathrm{op}}B2D2∥Σ−1∥op 24\dfrac{B^{2}D^{2}\lVert\Sigma^{-1}\rVert_{\mathrm{op}}^{\,2}}{4}
Product Cauchyμ=Wϕ\mu=W\phi , ∥ϕ∥2≤B\lVert\phi\rVert_{2}\leq B ; scales γi>0\gamma_{i}>0B∑iγi−2B\sqrt{\sum_{i}\gamma_{i}^{-2}}2B2min⁡iγi2\dfrac{2B^{2}}{\min_{i}\gamma_{i}^{2}}
BoltzmannLinear preference ⟨W,F⟩F\langle W,F\rangle_{\mathrm{F}} ; ∥F∥∗≤B\lVert F\rVert_{*}\leq B ; finite positive carrier mass2Bτ\dfrac{2B}{\tau}B2τ2\dfrac{B^{2}}{\tau^{2}}
SoftmaxKK linear logits WϕW\phi ; ∥ϕ∥2≤B\lVert\phi\rVert_{2}\leq B ; finite action set2Bτ\dfrac{\sqrt{2}B}{\tau}B22τ2\dfrac{B^{2}}{2\tau^{2}}
Table 1: Uniform nuclear-score bounds and operator-to-nuclear score Lipschitz constants. Temperatures, scales, covariance, features, and supports are independent of the learned parameter. All constants are sufficient; only the specific sharpness statements in the text are claimed.
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