Isotropic Noise in Stochastic and Quantum Convex Optimization

Annie Marsden (Google Deepmind) · Aaron Sidford (Stanford) · Chenyi Zhang (Stanford University) · Liam O'Carroll (Stanford University)
algorithm improvementdimension-dependent ratesisotropic errorisotropic noiselipschitz convex functionlower boundspolylogarithmic factorsquantum algorithmquantum isotropifierquantum stochastic convex optimizationstate-of-the-art complexitystochastic gradient oraclesub-exponential noiseunbiased estimatevariance-bounded quantum sampling

We consider the problem of minimizing a $d$-dimensional Lipschitz convex function using a stochastic gradient oracle. We introduce and motivate a setting where the noise of the stochastic gradient is isotropic in that it is bounded in every direction with high probability. We then develop an algorithm for this setting which improves upon prior results by a factor of $d$ in certain regimes, and as a corollary, achieves a new state-of-the-art complexity for sub-exponential noise. We give matching lower bounds (up to polylogarithmic factors) for both results. Additionally, we develop an efficient quantum isotropifier, a quantum algorithm which converts a variance-bounded quantum sampling oracle into one that outputs an unbiased estimate with isotropic error. Combining our results, we obtain improved dimension-dependent rates for quantum stochastic convex optimization.