Proving the Limited Scalability of Centralized Distributed Optimization via a New Lower Bound Construction

We consider centralized distributed optimization in the classical federated learning setup, where $n$ workers jointly find an $\varepsilon$-stationary point of an $L$-smooth, $d$-dimensional nonconvex function $f$, having access only to unbiased stochastic gradients with variance $\sigma^2$. Each worker requires at most $h$ seconds to compute a stochastic gradient, and the communication times from the server to the workers and from the workers to the server are $\tau_{s}$ and $\tau_{w}$ seconds per coordinate, respectively. One of the main motivations for distributed optimization is to achieve scalability with respect to $n$. For instance, it is well known that the distributed version of SGD has a variance-dependent runtime term $\frac{h \sigma^2 L \Delta}{n \varepsilon^2},$ which improves with the number of workers $n,$ where $\Delta = f(x^0) - f^*,$ and $x^0 \in R^d$ is the starting point. Similarly, using unbiased sparsification compressors, it is possible to reduce both the variance-dependent runtime term and the communication runtime term. However, once we account for the communication from the server to the workers $\tau_{s}$, we prove that it becomes infeasible to design a method using unbiased random sparsification compressors that scales both the server-side communication runtime term $\tau_{s} d \frac{L \Delta}{\varepsilon}$ and the variance-dependent runtime term $\frac{h \sigma^2 L \Delta}{\varepsilon^2},$ better than poly-logarithmically in $n$, even in the homogeneous (i.i.d.) case, where all workers access the same distribution. To establish this result, we construct a new "worst-case" function and develop a new lower bound framework that reduces the analysis to the concentration of a random sum, for which we prove a concentration bound. These results reveal fundamental limitations in scaling distributed optimization, even under the homogeneous assumption.