论文标题
随机零订单梯度和Hessian估计器：降低方差和精制偏置界限
Stochastic Zeroth Order Gradient and Hessian Estimators: Variance Reduction and Refined Bias Bounds
论文作者
论文摘要
我们研究了$ \ mathbb {r}^n $中的随机零订单梯度和HESSIAN估计器。我们表明，通过沿随机正交方向进行有限差异，可以大大降低随机有限差估计器的方差。特别是，我们为平滑函数设计估计器，以便，如果人们使用$θ\ left（k \右）$随机方向从Stiefel的歧管歧管$ \ text {st}（n，k）$和有限差粒度$Δ$中，则渐变估计值的方差为$ \ nath Calcal calcal {o} {o}（o; \ frac {n} {k} - 1 \右） + \ left（\ frac {n^2} {k} - n \ right）δ^2 + \ frac {n^2Δ^4} {k} {k} {k} {k} {k} {k} {k} \ right）$ weled wewl of $ \ \ nester wewd wewl of $ \ \ n of $ \ n of。 \ frac {n^2} {k^2} -1 \ right） + \ left（\ frac {n^4} {k^2} - n^2 \ right）Δ^2 + \ frac {n^4Δ当$ k = n $时，差异会忽略不计。此外，我们为估计器提供了改善的偏置界限。平滑功能$ f $的梯度估计量的偏差是$ \ MATHCAL {O} \ left（δ^^2γ\ right）$，其中$δ$是有限的差异粒度，$γ$依赖于$ f $的高阶衍生物。经验观察证明了我们的结果。
We study stochastic zeroth order gradient and Hessian estimators for real-valued functions in $\mathbb{R}^n$. We show that, via taking finite difference along random orthogonal directions, the variance of the stochastic finite difference estimators can be significantly reduced. In particular, we design estimators for smooth functions such that, if one uses $ Θ\left( k \right) $ random directions sampled from the Stiefel's manifold $ \text{St} (n,k) $ and finite-difference granularity $δ$, the variance of the gradient estimator is bounded by $ \mathcal{O} \left( \left( \frac{n}{k} - 1 \right) + \left( \frac{n^2}{k} - n \right) δ^2 + \frac{ n^2 δ^4 }{ k } \right) $, and the variance of the Hessian estimator is bounded by $\mathcal{O} \left( \left( \frac{n^2}{k^2} - 1 \right) + \left( \frac{n^4}{k^2} - n^2 \right) δ^2 + \frac{n^4 δ^4 }{k^2} \right) $. When $k = n$, the variances become negligibly small. In addition, we provide improved bias bounds for the estimators. The bias of both gradient and Hessian estimators for smooth function $f$ is of order $\mathcal{O} \left( δ^2 Γ\right)$, where $δ$ is the finite-difference granularity, and $ Γ$ depends on high order derivatives of $f$. Our results are evidenced by empirical observations.
