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
$\sigma^2$. The codewords satisfy an average power constraint $P$, resulting in
the signal-to-noise ratio (SNR) of $P/\sigma^2$. 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.

本研究考虑采用模拟编码方案进行空基础凸优化问题的求解，结果表明，在低信噪比情况下，简单的缩放传输模拟编码方案可以达到低信噪比情况下最优的收敛速度，并且提出了一种 “量化和调制” 的方案，几乎在所有的信噪比情况下都达到了最优的收敛速度。