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 + 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/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.

中文翻译：

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无线优化的基本限制：模拟方案是否最佳？

我们考虑在广告维度空间上进行空中凸优化，其中编码梯度通过方差为 \sigma^2 的加性高斯噪声通道发送。码字满足平均功率约束 P，导致信噪比 (SNR) 为 P/\sigma^2。我们推导出空中优化收敛速度的界限。我们的第一个结果是收敛速度的下限，表明任何代码都必须将收敛速度减慢大约 \sqrt{d/log(1 + SNR)} 的因子。接下来，我们考虑一类流行的称为模拟编码的方案，其中发送梯度的线性函数。我们表明，一个简单的缩放传输模拟编码方案会导致收敛速度减慢 \sqrt{d(1 + 1/SNR)} 的因子。这与之前的下限相匹配，直到低 SNR 的常数因子，使缩放传输方案在低 SNR 下最佳。然而，我们表明这种减速对于任何模拟编码方案都是必要的。特别是，即使当 SNR 趋于无穷大时，模拟编码的收敛速度仍会减慢 \sqrt{d} 倍。值得注意的是，我们提出了一种简单的量化和调制方案，该方案使用幅移键控，并且几乎在所有 SNR 下都能达到最佳收敛速度。