Energy-Efficient Iterative Refinement Using Dynamic Precision

Abstract

Mixed precision is a promising approach to save energy in iterative refinement algorithms since it obtains speedup without necessitating additional cores and parallelisation. However, conventional mixed precision methods utilise statically defined precision in a loop, thus hindering further speed-up and energy savings. We overcome this problem by proposing novel methods which allow iterative refinement to utilise variable precision arithmetic dynamically in a loop (i.e. a trans-precision approach). Our methods restructure a numeric algorithm dynamically according to runtime numeric behaviour and remove unnecessary accuracy checks. We implemented our methods by extending one conventional mixed precision iterative refinement algorithm on an Intel Xeon E5-2650 2GHz core with MKL 2017 and XBLAS 1.0. Our dynamic precision approach demonstrates 2.0-2.6× speed-up and 1.8-2.4× energy savings compared to mixed precision iterative refinement when double precision solution accuracy is required for forward error and with matrix dimensions ranging from 4K to 32K.

AIR: Iterative refinement acceleration using arbitrary dynamic precision

“…In 2017, Carson and Higham discussed iterative refinements utilizing three types of precision arithmetic in terms of accuracy and run time [3]. In 2018, Lee et al proposed a Dynamic precision arithmetic Iterative Refinement (DynIR) that achieves double precision solution accuracy in the forward error, while minimizing run time further compared to statically defined mixed precision iterative refinements [18]. The iterative refinements considered in [3] selected precision statically, using three types of precision for steps 2, 3, and 4 respectively.…”

“…For example, for some applications requiring higher precision arithmetic [32], software-emulated precision arithmetic is required, resulting in significant slow-down. DynIR already improved ∼ 2× speedups over XMIR in [18] for large dense matrices of n = 16K by minimising the software emulated double-double precision arithmetic operations. DynIR associated with AIR can improve the speedups further by utilising the least sufficient software emulated arithmetic precision provided by AIR algorithm.…”

Half-Precision Floating-Point Formats for PageRank: Opportunities and Challenges

“…For instance, over 60 percent of energy consumed in a communication-bound graph algorithm such as PageRank (PR) is due to memory [5]. For iterative algorithms like PageRank, which are based on floating-point computations, reduced-precision and mixedprecision computation has been shown to reduce memory pressure [6,7]. For many applications, standard floating-point (FP) formats such as the IEEE-754 FP32 and FP64 formats are wider than necessary.…”

“…Mixed precision iterative algorithms have been proposed for linear solvers [6,18], where initial iterations calculate a first estimated solution at reduced precision, then iteratively improve the estimate using increasingly higher precision. Mixedprecision arithmetic could potentially incrementally increase the precision by adding 1 or 2 bits to the mantissa at a time [19].…”

Service Deployment Challenges in Cloud-to-Edge Continuum

“…Here, precision indicates the level of detail in the computation (e.g., the bit width of a number) while accuracy indicates the fidelity of the computed values to the exact values. Many computations inherently allow a reduction in accuracy [42,43] or even allow approximation [44,45].…”