Differentially Quantized Gradient Methods
Differentially Quantized Gradient Methods
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DOI：
10.1109/tit.2022.3171173
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发表时间：
2020-02
影响因子：
2.5
通讯作者：
Chung-Yi Lin;V. Kostina;B. Hassibi
中科院分区：
文献类型：
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作者：
Chung-Yi Lin;V. Kostina;B. Hassibi
请考虑以下分布式优化方案。工作人员可以访问用于计算梯度的训练数据，而服务器则根据其目标精度或延迟约束来决定何时停止迭代计算。服务器通过速率受限的无噪音通信通道从工作器接收有关问题实例的所有信息。我们引入了被称为差分量化(DQ)的原理，它规定补偿过去的量化误差，以将量化算法的下降轨迹引导到其未量化算法的下降轨迹。假设目标函数是光滑的、强凸的，我们证明了差分量化梯度下降(DQ-GD)的线性压缩因子为max-R，R是非量化梯度下降(GD)的压缩因子，是量化器的覆盖效率，R是每问题维比特率。因此，在任何$R\geq\log_{2}\rho_{n}/\sigma{\mathm{Gd}}$比特上，DQ-GD的压缩因子与未量化的GD的压缩因子相同，即不存在由于量化而造成的损耗。我们证明了在某一类中没有算法能比$\max\\sigma_{\mathm{Gd}}，2^{-R}\}$更快地收敛。由于量化器的存在使得$Rho_(N)to 1$作为$n\to Inty$(Rogers，1963)，这意味着DQ-GD是渐进最优的。相比之下，工作者直接量化梯度的天真量子化Gd仅达到$\sigma_{\mathm{Gd}}+\Rho_{n}2^{-R}$。差分量化原理继续适用于动量梯度法，如内斯特罗夫的加速梯度下降法和波利亚克的重球法。同样，对于这些算法，如果速率高于某一阈值，则差分量化算法获得的压缩因子与其未量化算法相比没有损失，而且，差分量化重球方法在所有(甚至是未量化的)梯度方法中获得了最优的收缩。对最小二乘问题的实验结果验证了我们的理论分析。
Consider the following distributed optimization scenario. A worker has access to training data that it uses to compute the gradients while a server decides when to stop iterative computation based on its target accuracy or delay constraints. The server receives all its information about the problem instance from the worker via a rate-limited noiseless communication channel. We introduce the principle we call differential quantization (DQ) that prescribes compensating the past quantization errors to direct the descent trajectory of a quantized algorithm towards that of its unquantized counterpart. Assuming that the objective function is smooth and strongly convex, we prove that differentially quantized gradient descent (DQ-GD) attains a linear contraction factor of $\max \{\sigma _{\mathrm {GD}}, \rho _{n} 2^{-R}\}$ , where $\sigma _{\mathrm {GD}}$ is the contraction factor of unquantized gradient descent (GD), $\rho _{n} \geq 1$ is the covering efficiency of the quantizer, and $R$ is the bitrate per problem dimension $n$ . Thus at any $R\geq \log _{2} \rho _{n} /\sigma _{\mathrm {GD}}$ bits, the contraction factor of DQ-GD is the same as that of unquantized GD, i.e., there is no loss due to quantization. We show a converse demonstrating that no algorithm within a certain class can converge faster than $\max \{\sigma _{\mathrm {GD}}, 2^{-R}\}$ . Since quantizers exist with $\rho _{n} \to 1$ as $n \to \infty $ (Rogers, 1963), this means that DQ-GD is asymptotically optimal. In contrast, naively quantized GD where the worker directly quantizes the gradient barely attains $\sigma _{\mathrm {GD}} + \rho _{n}2^{-R}$ . The principle of differential quantization continues to apply to gradient methods with momentum such as Nesterov’s accelerated gradient descent, and Polyak’s heavy ball method. For these algorithms as well, if the rate is above a certain threshold, there is no loss in contraction factor obtained by the differentially quantized algorithm compared to its unquantized counterpart, and furthermore, the differentially quantized heavy ball method attains the optimal contraction achievable among all (even unquantized) gradient methods. Experimental results on least-squares problems validate our theoretical analysis.