We study semi-supervised stochastic optimization when labeled data is scarce but predictions from pre-trained models are available. PPI and SVRG both reduce variance through control variates -- PPI uses predictions, SVRG uses reference gradients. We show they are mathematically equivalent and develop PPI-SVRG, which combines both. Our convergence bound decomposes into the standard SVRG rate plus an error floor from prediction uncertainty. The rate depends only on loss geometry; predictions affect only the neighborhood size. When predictions are perfect, we recover SVRG exactly. When predictions degrade, convergence remains stable but reaches a larger neighborhood. Experiments confirm the theory: PPI-SVRG reduces MSE by 43--52\% under label scarcity on mean estimation benchmarks and improves test accuracy by 2.7--2.9 percentage points on MNIST with only 10\% labeled data.

翻译：本研究探讨了在标注数据稀缺但可获得预训练模型预测结果的场景下的半监督随机优化问题。预测驱动推断（PPI）与随机方差缩减梯度（SVRG）均通过控制变量法降低方差——PPI利用模型预测值，SVRG则采用参考梯度。我们证明二者在数学上等价，并提出了融合两者的PPI-SVRG算法。其收敛界可分解为标准SVRG收敛率与预测不确定性导致的误差基底之和。该收敛率仅取决于损失函数的几何特性，而预测质量仅影响最终邻域大小。当预测完全准确时，算法退化为标准SVRG；当预测质量下降时，算法仍保持稳定收敛，但会抵达更大的邻域。实验验证了理论结果：在均值估计基准测试中，PPI-SVRG在标注稀缺条件下将均方误差降低43–52%；在仅使用10%标注数据的MNIST数据集上，测试准确率提升2.7–2.9个百分点。