梯度流可优化性蕴含低温吉布斯测度的泛函不等式，实现高效采样。
Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials
- 从任意初值可优化，推出低溫吉布斯測度滿足Poincaré不等式
- 在温和条件下，采样复杂度為O(C'+1/β)，β≥Ω(d)
- 適用於新類型非對數凹密度，擴展了高效採樣範圍
本文證明：若任意初始點下函數F均能通過梯度流優化，則其低溫吉布斯測度μ_β = e^{-βF}/Z 滿足Poincaré不等式。在梯度流收斂率的弱正則性假設下，μ_β 的Poincaré常數為O(C'+1/β)，其中C'為F全局最小值鄰域上μ_β的常數，且β≥Ω(d)。在額外條件下，μ_β滿足對數-索博列夫不等式，常數為O(β max(S,1) max(C',1))，S為μ_β的二階矩。高層次意義上，從所有初始點可優化，意味著可從任意初值高效採樣。若僅需除某集合S外的所有初始點，則μ_β滿足弱Poincaré不等式，參數為(O(C'+1/β), O(μ_β(S)))，β=Ω(d)。這表明從「大多數」初始點可優化，即可從合適熱啟動中採樣。正則性假設極弱，因此我們可高效採樣多類新的自然且有趣的非對數凹密度，這是相對少有例子的重要設定。另一推論是，對滿足更輕鬆正則性條件（非光滑）的對數凹測度，獲得高效離散時間採樣結果，類似於Lehec (2023)。
原文摘要 · Abstract (English)
In this paper, we prove that optimizability of any function F using Gradient Flow from all initializations implies a Poincaré Inequality for Gibbs measures mu_{beta} = e^{-beta F}/Z at low temperature. In particular, under mild regularity assumptions on the convergence rate of Gradient Flow, we establish that mu_{beta} satisfies a Poincaré Inequality with constant O(C'+1/beta) for beta >= Omega(d), where C' is the Poincaré constant of mu_{beta} restricted to a neighborhood of the global minimizers of F. Under an additional mild condition on F, we show that mu_{beta} satisfies a Log-Sobolev Inequality with constant O(beta max(S, 1) max(C', 1)) where S denotes the second moment of mu_{beta}. Here asymptotic notation hides F-dependent parameters. At a high level, this establishes that optimizability via Gradient Flow from every initialization implies a Poincaré and Log-Sobolev Inequality for the low-temperature Gibbs measure, which in turn imply sampling from all initializations. Analogously, we establish that under the same assumptions, if F can be initialized from everywhere except some set S, then mu_{beta} satisfies a Weak Poincaré Inequality with parameters (O(C'+1/beta), O(mu_{beta}(S))) for β= Omega(d). At a high level, this shows while optimizability from 'most' initializations implies a Weak Poincaré Inequality, which in turn implies sampling from suitable warm starts. Our regularity assumptions are mild and as a consequence, we show we can efficiently sample from several new natural and interesting classes of non-log-concave densities, an important setting with relatively few examples. As another corollary, we obtain efficient discrete-time sampling results for log-concave measures satisfying milder regularity conditions than smoothness, similar to Lehec (2023).
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