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Successive Refinement in Large-Scale Computation: Advancing Model Inference Applications

Homa Esfahanizadeh, Alejandro Cohen, Shlomo Shamai, Muriel Medard

TL;DR

The paper addresses deadline-constrained, large-scale computation by introducing layered-resolution (successive refinement) to deliver intermediate results earlier and adaptively upgrade only as needed. It formulates a formal problem with $R$ resolution upgrades and derives layering strategies for both linear and nonlinear functions, including a linear matrix product and deep neural networks with piecewise-linear activations. The authors apply the approach to streaming distributed matrix multiplication and ML classification with adaptive resolution, showing significant reductions in early-resolution delays and high likelihood of meeting deadlines, while preserving final accuracy close to one-shot outcomes. The results demonstrate practical gains in deadline-based systems, adaptability, and resource efficiency, with potential extensions to broader ML architectures and training-time adaptation.

Abstract

Modern computationally-intensive applications often operate under time constraints, necessitating acceleration methods and distribution of computational workloads across multiple entities. However, the outcome is either achieved within the desired timeline or not, and in the latter case, valuable resources are wasted. In this paper, we introduce solutions for layered-resolution computation. These solutions allow lower-resolution results to be obtained at an earlier stage than the final result. This innovation notably enhances the deadline-based systems, as if a computational job is terminated due to time constraints, an approximate version of the final result can still be generated. Moreover, in certain operational regimes, a high-resolution result might be unnecessary, because the low-resolution result may already deviate significantly from the decision threshold, for example in AI-based decision-making systems. Therefore, operators can decide whether higher resolution is needed or not based on intermediate results, enabling computations with adaptive resolution. We present our framework for two critical and computationally demanding jobs: distributed matrix multiplication (linear) and model inference in machine learning (nonlinear). Our theoretical and empirical results demonstrate that the execution delay for the first resolution is significantly shorter than that for the final resolution, while maintaining overall complexity comparable to the conventional one-shot approach. Our experiments further illustrate how the layering feature increases the likelihood of meeting deadlines and enables adaptability and transparency in massive, large-scale computations.

Successive Refinement in Large-Scale Computation: Advancing Model Inference Applications

TL;DR

The paper addresses deadline-constrained, large-scale computation by introducing layered-resolution (successive refinement) to deliver intermediate results earlier and adaptively upgrade only as needed. It formulates a formal problem with resolution upgrades and derives layering strategies for both linear and nonlinear functions, including a linear matrix product and deep neural networks with piecewise-linear activations. The authors apply the approach to streaming distributed matrix multiplication and ML classification with adaptive resolution, showing significant reductions in early-resolution delays and high likelihood of meeting deadlines, while preserving final accuracy close to one-shot outcomes. The results demonstrate practical gains in deadline-based systems, adaptability, and resource efficiency, with potential extensions to broader ML architectures and training-time adaptation.

Abstract

Modern computationally-intensive applications often operate under time constraints, necessitating acceleration methods and distribution of computational workloads across multiple entities. However, the outcome is either achieved within the desired timeline or not, and in the latter case, valuable resources are wasted. In this paper, we introduce solutions for layered-resolution computation. These solutions allow lower-resolution results to be obtained at an earlier stage than the final result. This innovation notably enhances the deadline-based systems, as if a computational job is terminated due to time constraints, an approximate version of the final result can still be generated. Moreover, in certain operational regimes, a high-resolution result might be unnecessary, because the low-resolution result may already deviate significantly from the decision threshold, for example in AI-based decision-making systems. Therefore, operators can decide whether higher resolution is needed or not based on intermediate results, enabling computations with adaptive resolution. We present our framework for two critical and computationally demanding jobs: distributed matrix multiplication (linear) and model inference in machine learning (nonlinear). Our theoretical and empirical results demonstrate that the execution delay for the first resolution is significantly shorter than that for the final resolution, while maintaining overall complexity comparable to the conventional one-shot approach. Our experiments further illustrate how the layering feature increases the likelihood of meeting deadlines and enables adaptability and transparency in massive, large-scale computations.
Paper Structure (14 sections, 8 theorems, 62 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 14 sections, 8 theorems, 62 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Designing Layering Strategies
  4. Layered Evaluation of Linear Functions
  5. Layered Evaluation of nonlinear Functions
  6. Stream Distributed Matrix Multiplication
  7. System Design
  8. Simulation Results
  9. Related Work
  10. ML Classification with Adaptive Resolution
  11. System Design
  12. Simulation Results
  13. Related Work
  14. Discussion and Future Work

Key Result

Theorem 1

For the layering strategy in Definition def:layering_linear, we have where $\delta(r)=\delta+(R-r)2^{P_{\gamma_r(1)}+Q_{\gamma_r(2)}}$ is non-increasing in $r\in\{1,\dots,R\}$ and $\delta=2^{P_0+Q_d}+2^{P_d+Q_0}+2^{P_d+Q_d}$.

Figures (8)

  • Figure 1: Architecture of a deep neural network with $L$ hidden layers. The $i$-th hidden layer $H^{(i)}$ is identified via a linear transformation $\{W_{i-1},B_{i-1}\}$, followed by a point-wise nonlinear function $\sigma_{i}$; except for the last layer that is followed by a nonlinear function $\sigma'$ that is not necessarily point-wise. Each edge is associated with a weight that is described by the linear transformation.
  • Figure 2: Examples of point-wise activation functions: (a) piece-wise linear functions Relu $\sigma(a)=\max(a,0)$ and Leaky Relu $\sigma(a)=\max(a,\beta a)$, $\beta=0.05$ here; (b) Sigmoid $\sigma(a)=1/(1+\exp(-a))$ and its piece-wise linear approximation (PLA).
  • Figure 3: Stream distributed computation with successive refinement: A subset of task results are sufficient to obtain the first resolution result, and successively, the resolution is improved upon collecting more task results.
  • Figure 4: (left) Distribution and (right) success rate, for the execution delay of $R=4$ layers of resolution, based on $1000$ jobs.
  • Figure 5: Several randomly-selected samples in MNIST dataset.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Definition 1: Partitioning Vector
  • Example 1
  • Definition 2
  • Example 2
  • Definition 3
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Lemma 1
  • ...and 8 more