Extend the convergence analysis to active gradient clipping

Extend the convergence analysis of the Throttle algorithm for Byzantine-robust asynchronous stochastic gradient descent to the regime in which the clipping threshold is smaller than the uniform stochastic-gradient bound and therefore clips honest gradients, as in the experiments with clipping threshold \(\lambda_t=1\).

Background

The main convergence theorem analyzes Throttle under the choice $\lambda_t=G$, where the bounded-gradient assumption ensures that clipping acts as the identity on honest gradients. The experiments instead use a smaller threshold, $\lambda_t=1$, for which clipping is typically active. The paper explicitly identifies extending the theoretical analysis to this experimentally relevant clipping regime as unresolved.