Shubham Jha

Shubham K Jha, Prathamesh Mayekar, Himanshu Tyagi

We consider over-the-air convex optimization on a $d-$dimensional space where coded gradients are sent over an additive Gaussian noise channel with variance $σ^2$. The codewords satisfy an average power constraint $P$, resulting in the signal-to-noise ratio (SNR) of $P/σ^2$. We derive bounds for the convergence rates for over-the-air optimization. Our first result is a lower bound for the convergence rate showing that any code must slowdown the convergence rate by a factor of roughly $\sqrt{d/\log(1+\mathtt{SNR})}$. Next, we consider a popular class of schemes called $analog$ $coding$, where a linear function of the gradient is sent. We show that a simple scaled transmission analog coding scheme results in a slowdown in convergence rate by a factor of $\sqrt{d(1+1/\mathtt{SNR})}$. This matches the previous lower bound up to constant factors for low SNR, making the scaled transmission scheme optimal at low SNR. However, we show that this slowdown is necessary for any analog coding scheme. In particular, a slowdown in convergence by a factor of $\sqrt{d}$ for analog coding remains even when SNR tends to infinity. Remarkably, we present a simple quantize-and-modulate scheme that uses $Amplitude$ $Shift$ $Keying$ and almost attains the optimal convergence rate at all SNRs.

Prathamesh Mayekar, Shubham Jha, Ananda Theertha Suresh et al.

Communication efficient distributed mean estimation is an important primitive that arises in many distributed learning and optimization scenarios such as federated learning. Without any probabilistic assumptions on the underlying data, we study the problem of distributed mean estimation where the server has access to side information. We propose \emph{Wyner-Ziv estimators}, which are communication and computationally efficient and near-optimal when an upper bound for the distance between the side information and the data is known. As a corollary, we also show that our algorithms provide efficient schemes for the classic Wyner-Ziv problem in information theory. In a different direction, when there is no knowledge assumed about the distance between side information and the data, we present an alternative Wyner-Ziv estimator that uses correlated sampling. This latter setting offers {\em universal recovery guarantees}, and perhaps will be of interest in practice when the number of users is large and keeping track of the distances between the data and the side information may not be possible. With this mean estimator at our disposal, we revisit basic problems in decentralized optimization and compression where our Wyner-Ziv estimator yields algorithms with almost optimal performance. First, we consider the problem of communication constrained distributed optimization and provide an algorithm which attains the optimal convergence rate by exploiting the fact that the gradient estimates are close to each other. Specifically, the gradient compression scheme in our algorithm first uses half of the parties to form side information and then uses our Wyner-Ziv estimator to compress the remaining half of the gradient estimates.