canonical setting

Stochastic and finite-sum oracles

$V=m^{-1}\sum_{i=1}^m f_i$ or unbiased noisy gradients; oracle cost counts component or stochastic queries.

Updated Aug 6, 2026

Upper and lower bounds

Comparison table

5 scoped results. Tildes suppress logarithmic factors.

Best-known and comparison results for Stochastic and finite-sum oracles
Result	Algorithm or model	Complexity	Guarantee	Oracle / start	Assumptions and notes	Review	Sources
best upper	High-accuracy stochastic-gradient sampler	$\widetilde O((\kappa\sqrt d+\sigma_\psi^2/\alpha)\operatorname{polylog}(1/\varepsilon))$	$\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon$	Unbiased sub-exponential stochastic gradientKnown mode / paper initialization	$\alpha I\preceq\nabla^2V\preceq\beta I$ and uniform Orlicz-tail bound.Checked headline for light-tailed stochastic gradients.	Checkedpublished	Chen et al. (COLT 2026); Chewi Theorem 10.1.3 ↗
best lower	Bounded-variance stochastic-gradient lower bound	$\Omega(\sigma^2/(\alpha\varepsilon))$	TV accuracy proportional to $\varepsilon$	Unbiased bounded-variance stochastic gradientOne-dimensional Gaussian testing instance	Independent oracle noise in the stated minimax model.Shows light tails are necessary for polylogarithmic accuracy dependence.	Checkedpublished	Chen et al. (COLT 2026); Chewi Theorem 10.1.2 ↗
upper	Variance-reduced high-accuracy sampler	$\widetilde O((m+\kappa\sqrt{dm})\operatorname{polylog}(1/\varepsilon))$	High-accuracy sampling	Individual component value/gradientPaper initialization	Each $f_i$ is $\beta$ -smooth and $V$ is $\alpha$ -strongly convex.Counts individual component queries, not full gradients.	Checkedpublished	Chen et al. (COLT 2026); Chewi §10.1.2 ↗
upper	Finite-sum RM-ULMC	$\widetilde O(m+\kappa^2+\kappa^{4/3}d^{1/3}m^{2/3}/\varepsilon^{2/3})$	Low-accuracy sampling	Individual component gradientPaper initialization	Finite-sum strong convexity and component smoothness.Low-accuracy row; compare separately from polylogarithmic accuracy.	Checkedmonograph	Chewi (2026), §10.1.2 ↗
lower	Finite-sum zeroth-order lower bound	$\widetilde\Theta(L^2/\alpha)$	Constant sampling accuracy	Individual component value queriesUnit-ball support	$d\ll L^2/\alpha$ ; Lipschitz component functions.Different oracle and support model from the stochastic-gradient row.	Checkedmonograph	Chewi (2026), Chapter 9 bibliographical notes ↗

Scope

This table separates light-tailed stochastic gradients, bounded-variance gradients, finite sums, and zeroth-order component queries. These are different oracle models; the rows are grouped here because they measure the price of replacing a full exact gradient.

Main separation

Sub-exponential noise permits polylogarithmic accuracy dependence. Bounded variance alone forces inverse-accuracy dependence.