Extend quantized learning analysis to multi-pass SGD

Establish rigorous excess risk bounds for multi-pass stochastic gradient descent under the quantization framework introduced in this paper, where data features, labels, parameters, activations, and output gradients are quantized (via operators Q_d, Q_l, Q_p, Q_a, Q_o) for high-dimensional linear regression. The goal is to characterize the population risk of iterate-averaged multi-pass SGD with data reuse under these practical quantization constraints.

Background

The paper develops a theory of excess risk for one-pass, constant-stepsize SGD with iterate averaging under a comprehensive set of quantization operations applied to data, labels, parameters, activations, and gradients. The analysis yields bounds for general, additive, and multiplicative quantization, but it explicitly focuses on one-pass SGD.

In the conclusion, the authors state that extending their learning-theoretic analysis beyond the one-pass regime remains open. Multi-pass SGD introduces data reuse and different dynamics that are not addressed by the current framework, motivating a formal excess risk characterization under quantization.