固定维度下随机梯度预言机的紧采样复杂度
Tight Sampling Complexity with Stochastic Gradient Oracles in Fixed Dimensions
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中文总结 AI 辅助
本文针对固定维度下光滑强对数凹分布，利用随机梯度预言机，证明了达到给定TV精度的采样复杂度下界与上界，并给出同时关于条件数和精度的紧界，且适应无噪声情形。
中文摘要 AI 辅助
我们研究了在任意固定欧几里得维度中，对光滑强对数凹分布进行采样的随机梯度查询复杂度。势函数为$\mu$-强凸且$L$-光滑，其未知众数位于以原点为中心、半径为$\mu^{-1/2}$的球内。我们可访问无偏随机预言机，其方差至多为$\sigma^2$。对于每个$\sigma^2\ge0$和全变差（TV）精度$0<\varepsilon\le1/10$，我们证明了从目标分布采样达到$\epsilon$-TV距离内的分布的紧复杂度为\\[ N^\star_{\text{TV}}=\Theta\\!\left(\log(1+\kappa)+ \frac{\sigma^2}{\mu\epsilon}\right), \\] 其中$\kappa:=\frac L\mu$为条件数。注意，该复杂度界对条件数$\kappa$和精度$\epsilon$同时是紧的。此外，我们的紧复杂度界适应于无噪声情形$\sigma=0$，此时为$ N^\star_{\text{TV}}=\Theta\\!\left(\log(1+\kappa)\right)$。
英文摘要
We establish the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in every fixed dimension $d\geq1$. For $μ$-strongly convex, $L$-smooth potentials with unknown minimizers $x_f^\star$ in the ball $\mathbb{B}(0,μ^{-1/2})$, and unbiased gradient oracles with variance at most $σ^2$, the minimax worst-case expected query complexity is $Θ\left(\log(1+κ)+\frac{σ^2}{μ\varepsilon}\right)$, jointly optimal for the condition number $κ:=L/μ$, variance $σ^2\ge 0$, and TV accuracy $0<\varepsilon\leq 1/10$. In the noiseless setting $σ=0$, this tight complexity $Θ\left(\log(1+κ)\right)$ is independent of $\varepsilon$. Moreover, a gradient-only sampler generates an exact sample with ${O}(\log(1+κ))$ worst-case expected queries. Without previous ball $\mathbb{B}(0,μ^{-1/2})$, exact sampling from any initial point $x_0$ can be implemented with expected cost ${O}\left(\log(1+κ)+\log(1+\sqrtμ\|x_0-x_f^\star\|)\right)$, without knowing the initial distance. We also show that dependence on initial distance is generally unavoidable.
发表机构
- Shanghai University of Finance and Economics（上海财经大学）
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