Optimal discrepancy-principle safety factor for continuous-time SGD

Determine whether the discrepancy principle for the continuous-time stochastic-gradient-descent model analyzed in the paper is optimally stopped with safety factor equal to one, as is the case for most deterministic filter-based methods under white noise.

Background

The paper proves a high-probability minimax-rate guarantee for a discrepancy stopping rule with a safety factor required to exceed a constant κ₀ determined by the contraction argument. The authors note that this requirement may be conservative and discuss possible refinements of the step-size and safety-factor thresholds.

For deterministic filter-based regularization methods under white noise, the discrepancy principle is generally stopped at safety factor one to obtain optimality. The paper does not establish whether the same sharp safety-factor choice remains valid for the pathwise discrepancy principle applied to the continuous-time stochastic-gradient-descent dynamics with multiplicative fluctuations.