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Fundamental Results on Asynchronous Parallel Optimization Algorithms

  • Author(s): Hannah, Robert Rafaeil
  • Advisor(s): Yin, Wotao
  • et al.
Abstract

In this thesis, we present a body of work on the performance and convergence properties of asynchronous-parallel algorithms completed over the course of my doctorate degree (Hannah, Feng, and Wotao Yin 2018; Hannah and Wotao Yin 2017b; T. Sun, Hannah, and Wotao Yin 2017; Hannah and Wotao Yin 2017a). Asynchronous algorithms eliminate the costly synchronization penalty of traditional synchronous-parallel algorithms. They do this by having computing nodes utilize the most recently available information to compute updates. However, it’s not immediately clear whether the trade-off of eliminating synchronization penalty at the cost of using outdated information is favorable.

We first give a comprehensive theoretical justification of the performance advantages of asynchronous algorithms, which we summarize as "Faster Iterations, Same Quality" (Hannah and Wotao Yin 2017a). Under a well-justified model, we show that asynchronous algorithms complete "Faster Iterations". Using renewal theory, we demonstrate how network delays, heterogeneous sub-problem difficulty and computing power greatly hinder synchronous algorithms, but have no impact on their asynchronous counterparts. We next prove the first exact convergence rate results for a variety of synchronous algorithms including synchronous ARock and synchronous randomized block coordinate descent (sync-RBCD). This allows us to make a fair comparison between these algorithms and their asynchronous counterparts. Finally, we show that a variety of asynchronous algorithms have a convergence rate that essentially matches the previously derived exact rates for synchronous counterparts so long as the delays are not too large. Hence asynchronous algorithms complete faster iteration that are of the "Same Quality" as synchronous algorithms. Therefore we conclude that a wide variety of asynchronous algorithms will always outcompete their synchronous counterparts if the delays are not too large, and especially at scale.

Next, we present the first asynchonous Nesterov-accelerated algorithm that attains a speedup: A2BCD (Hannah, Feng, and Wotao Yin 2018). We first prove that A2BCD attains NU_ACDM’s complexity to highest order. NU_ACDM is a state-of-the-art accelerated coordinate descent algorithm (Allen-Zhu, Qu, et al. 2016). Then we show that both A2BCD and NU_ACDM both have optimal complexity. Hence because A2BCD has faster iterations, and optimal complexity, it should be the fastest coordinate descent algorithm. We verify this with numerical experiments comparing A2BCD with NU_ACDM. We find that A2BCD is up to 4-5x faster than NU_ACDM, and hence conclude that our algorithm is the current fastest coordinate descent algorithm that exists. Finally, we derive a second-order ODE, which is the continuous-time limit of A2BCD. The ODE analysis motivates and clarifies our proof strategy.

Lastly, we present earlier foundational work that comprises the basis of the technical innovations that made the previous results possible (Hannah and Wotao Yin 2017b). We show that ARock and its many special cases may converge even under unbounded delays (both stochastic and deterministic). These results sidestep longstanding impossibility results derived in the 1980s by making slightly stronger assumptions. They were also an early demonstration of the power of meticulous Lyapunov-function construction techniques pioneered in this body of work.

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