势能上升法 IV：在 $\eta< 1$ 下对 Sherrington-Kirkpatrick 模型进行采样
Potential Hessian Ascent IV: Sampling the Sherrington-Kirkpatrick model at $β< 1$
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中文总结 AI 辅助
提出一种多项式时间算法，在 $\eta<1$ 下以 $o_n(1)$ 总变差误差采样 SK 模型吉布斯测度，结合算法随机定位与 Jarzynski 等式拒绝采样，并利用 TAP 自由能的局部正则性扩展了先前结果。
中文摘要 AI 辅助
我们给出一个多项式时间算法，用于在任意逆温度 $\eta < 1$ 下，以总变差距离（TVD）中 $o_n(1)$ 的误差从 Sherrington-Kirkpatrick（SK）模型的吉布斯测度中采样。该算法将算法随机定位（ASL）与通过 Jarzynski 等式（JE）在路径空间上的拒绝采样相结合。分析将作者先前 $\eta < 1/2$ 的结果 [ arXiv:2605.03718 ] 加以扩展，将随机微分方程（SDE）误差分析中的所有全局正则性要求替换为围绕可能轨迹的局部正则性。放松的正则性利用 Celentano 对 TAP 自由能局部强凸性的证明 [ arXiv:2208.09550 ] 得以确立。该分析利用了作者先前结果中发展的空腔插值理论和自由概率工具包，其中前者几乎逐字适用，而后者在辅以各种函数的 Lipschitz 和 $C^2$ 扩展后适用。ASL 和 JE 分析源于将 TAP 自由能用作随机定位吉布斯测度实际自由能的可高效计算的代理 [ § 3, arXiv:2605.03718 ]。我们给出一个“期望性质”列表，概括了 TAP 自由能所需的近似和正则性性质，将 [ § 2.5, arXiv:2605.03718 ] 中的性质放松为仅需局部正则性。这些通用期望性质可能适用于其他存在自由能代理的设置，在绕过通常基于函数不等式的途径的同时，给出算法采样保证。
英文摘要
We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington-Kirkpatrick (SK) model with $o_n(1)$ error in total-variation distance (TVD) at any inverse-temperature $β< 1$. The algorithm combines algorithmic stochastic localization (ASL) with rejection sampling over path-space via Jarzynski's equality (JE). The analysis extends the authors' prior $β< 1/2$ result [arXiv:2605.03718] by replacing all global regularity requirements in the stochastic differential equation (SDE) error analysis with local regularity around likely trajectories. The relaxed regularity is established using Celentano's proof of the local strong convexity of the TAP free energy [arXiv:2208.09550]. The analysis utilizes the cavity interpolation theory and free probability toolkit developed in the authors' previous result, where the former applies nearly verbatim and the latter applies supplemented with Lipschitz and $C^2$ extensions of various functions. The ASL and JE analysis arises from using the TAP free energy as an efficiently computable proxy for the actual free energy of the stochastically localized Gibbs measure [$ §$ 3, arXiv:2605.03718]. We give a list of \vocab{desiderata} encapsulating the approximation and regularity properties required of the TAP free energy, relaxing those of [$ §$ 2.5, arXiv:2605.03718] to only require local regularity. These generic desiderata are potentially applicable in other settings where a free energy surrogate exists, giving algorithmic sampling guarantees while bypassing the usual functional inequalities based approach.
发表机构
- Johns Hopkins University（约翰斯·霍普金斯大学）
- Colorado State University（科罗拉多州立大学）
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