TY - GEN

T1 - Differentially Quantized Gradient Descent

AU - Lin, Chung-Yi

AU - Kostina, Victoria

AU - Hassibi, Babak

N1 - KAUST Repository Item: Exported on 2021-09-07
Acknowledgements: This work was supported in part by the National Science Foundation (NSF) under grants CCF-1751356, CCF-1956386, CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASAs Jet Propulsion Laboratory through the President and Directors Fund, and by King Abdullah University of Science and Technolog
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2021/7/12

Y1 - 2021/7/12

N2 - 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 only information that the server knows about the problem instance is what it receives from the worker via a rate-limited noiseless communication channel. We introduce the technique we call differential quantization (DQ) that compensates past quantization errors to make the descent trajectory of a quantized algorithm follow 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 convergence rate of $\max\{\sigma_{\text{GD}}, \rho_{n}2^{-R}\}$ , where $\sigma_{\text{GD}}$ is the convergence rate of unquantized gradient descent (GD), $\rho_{n}$ is the covering efficiency of the quantizer, and $R$ is the bitrate per problem dimension $n$ . Thus at any $R\geq\log_{2}\rho_{n}/\sigma_{\text{GD}}$ , the convergence rate 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 GD-like quantized algorithm can converge faster than $\max\{\sigma_{\text{GD}}, 2^{-R}\}$ . Since quantizers exist with $\rho_{n}\rightarrow 1$ as $n\rightarrow\infty$ (Rogers, 1963), this means that DQ-GD is asymptotically optimal. In contrast, naively quantized GD where the worker directly quantizes the gradient attains only $\sigma_{\text{GD}}+\rho_{n}2^{-R}$ . The technique 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 convergence rate obtained by the differentially quantized algorithm compared to its unquantized counterpart. Experimental results on both simulated and realworld least-squares problems validate our theoretical analysis.

AB - 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 only information that the server knows about the problem instance is what it receives from the worker via a rate-limited noiseless communication channel. We introduce the technique we call differential quantization (DQ) that compensates past quantization errors to make the descent trajectory of a quantized algorithm follow 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 convergence rate of $\max\{\sigma_{\text{GD}}, \rho_{n}2^{-R}\}$ , where $\sigma_{\text{GD}}$ is the convergence rate of unquantized gradient descent (GD), $\rho_{n}$ is the covering efficiency of the quantizer, and $R$ is the bitrate per problem dimension $n$ . Thus at any $R\geq\log_{2}\rho_{n}/\sigma_{\text{GD}}$ , the convergence rate 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 GD-like quantized algorithm can converge faster than $\max\{\sigma_{\text{GD}}, 2^{-R}\}$ . Since quantizers exist with $\rho_{n}\rightarrow 1$ as $n\rightarrow\infty$ (Rogers, 1963), this means that DQ-GD is asymptotically optimal. In contrast, naively quantized GD where the worker directly quantizes the gradient attains only $\sigma_{\text{GD}}+\rho_{n}2^{-R}$ . The technique 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 convergence rate obtained by the differentially quantized algorithm compared to its unquantized counterpart. Experimental results on both simulated and realworld least-squares problems validate our theoretical analysis.

UR - http://hdl.handle.net/10754/670945

UR - https://ieeexplore.ieee.org/document/9518254/

U2 - 10.1109/isit45174.2021.9518254

DO - 10.1109/isit45174.2021.9518254

M3 - Conference contribution

BT - 2021 IEEE International Symposium on Information Theory (ISIT)

PB - IEEE

ER -