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Neural Quantile Optimization for Edge-Cloud Networking

Bin Du, He Zhang, Xiangle Cheng, Lei Zhang

TL;DR

This work addresses edge-cloud traffic scheduling under SD-WAN with $95^{\text{th}}$ percentile billing by recasting a constrained integer program as a series of differentiable continuous problems via the Gumbel-Softmax reparameterization. It introduces the Gumbel-Softmax Sampling Network (GSSN), a decoupled, unsupervised neural architecture that maps problem data to feasible traffic-allocation schemes, enabling fast, high-quality sampling and serving as effective warm-starts for traditional solvers. Empirical results show GSSN outperforms random sampling and provides superior warm-starts for Gurobi, with encouraging generalization to more time steps and users. The approach offers a scalable, practical pathway for real-time edge-cloud routing under complex percentile-based billing constraints, with potential extensions to larger and more dynamic networks.

Abstract

We seek the best traffic allocation scheme for the edge-cloud computing network that satisfies constraints and minimizes the cost based on burstable billing. First, for a fixed network topology, we formulate a family of integer programming problems with random parameters describing the various traffic demands. Then, to overcome the difficulty caused by the discrete feature of the problem, we generalize the Gumbel-softmax reparameterization method to induce an unconstrained continuous optimization problem as a regularized continuation of the discrete problem. Finally, we introduce the Gumbel-softmax sampling network to solve the optimization problems via unsupervised learning. The network structure reflects the edge-cloud computing topology and is trained to minimize the expectation of the cost function for unconstrained continuous optimization problems. The trained network works as an efficient traffic allocation scheme sampler, remarkably outperforming the random strategy in feasibility and cost function value. Besides testing the quality of the output allocation scheme, we examine the generalization property of the network by increasing the time steps and the number of users. We also feed the solution to existing integer optimization solvers as initial conditions and verify the warm-starts can accelerate the short-time iteration process. The framework is general with solid performance, and the decoupled feature of the random neural networks is adequate for practical implementations.

Neural Quantile Optimization for Edge-Cloud Networking

TL;DR

This work addresses edge-cloud traffic scheduling under SD-WAN with percentile billing by recasting a constrained integer program as a series of differentiable continuous problems via the Gumbel-Softmax reparameterization. It introduces the Gumbel-Softmax Sampling Network (GSSN), a decoupled, unsupervised neural architecture that maps problem data to feasible traffic-allocation schemes, enabling fast, high-quality sampling and serving as effective warm-starts for traditional solvers. Empirical results show GSSN outperforms random sampling and provides superior warm-starts for Gurobi, with encouraging generalization to more time steps and users. The approach offers a scalable, practical pathway for real-time edge-cloud routing under complex percentile-based billing constraints, with potential extensions to larger and more dynamic networks.

Abstract

We seek the best traffic allocation scheme for the edge-cloud computing network that satisfies constraints and minimizes the cost based on burstable billing. First, for a fixed network topology, we formulate a family of integer programming problems with random parameters describing the various traffic demands. Then, to overcome the difficulty caused by the discrete feature of the problem, we generalize the Gumbel-softmax reparameterization method to induce an unconstrained continuous optimization problem as a regularized continuation of the discrete problem. Finally, we introduce the Gumbel-softmax sampling network to solve the optimization problems via unsupervised learning. The network structure reflects the edge-cloud computing topology and is trained to minimize the expectation of the cost function for unconstrained continuous optimization problems. The trained network works as an efficient traffic allocation scheme sampler, remarkably outperforming the random strategy in feasibility and cost function value. Besides testing the quality of the output allocation scheme, we examine the generalization property of the network by increasing the time steps and the number of users. We also feed the solution to existing integer optimization solvers as initial conditions and verify the warm-starts can accelerate the short-time iteration process. The framework is general with solid performance, and the decoupled feature of the random neural networks is adequate for practical implementations.
Paper Structure (18 sections, 1 theorem, 29 equations, 11 figures, 5 tables, 2 algorithms)

This paper contains 18 sections, 1 theorem, 29 equations, 11 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Organization of the Paper
  3. The Gumbel-Softmax Reparameterization
  4. Problem Formulation
  5. Neural Network Architecture and Algorithm
  6. The Neural Network Input
  7. Neural Network Architecture
  8. How to Use GSSN
  9. Numerical Results
  10. Problem Settings
  11. Neural Network Training and the Sampling Distribution
  12. The Comparison of WarmUp Solutions
  13. Use Gurobi to Test the Solution Quality of Three Warmups
  14. Generalization Test of GSSN
  15. Conclusion and Discussion
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $X \sim \mathrm{Concrete}(\bm{\alpha}, \tau)$ with location parameter $\bm{\alpha} \in (0, +\infty)^{d}$ and temperature $\tau \in (0, +\infty)$, then for $k= 1,2,\dots, d$, we have

Figures (11)

  • Figure 1: The architecture of SD-WAN consists of business-grade IP VPN, broadband Internet, and wireless services.SD-WANdefine
  • Figure 2: The network topology of the SD-WAN. The "Internet", "MPLS-A", "MPLS-B", and "MPLS-C" are four different ISPs. MPLS is the abbreviation of multi-protocol label switching, which enables ISPs to build intelligent networks that deliver various services over a single infrastructure 2004Generalized. The four peering links between ISPs and Hub also contribute to the total cost based on their utilization.
  • Figure 3: The schematic of the network. The transformation context of input data between three encoders - the link encoder, the program encoder, and the Ranking AutoEncoder - is represented by the red dashed line box. The outputs of GSSN are obtained through multiple sampling of Gumbel-Softmax to derive the final solution probability distribution.
  • Figure 4: The averaged value of the soft-loss function of the training (blue curve) and testing problems (yellow curve). The soft-loss converges after 60 epochs. The training and testing errors behave similarly mainly because their problem data are sampled from the same distribution.
  • Figure 5: The histogram of the loss of the GSSN output. (We normalized the frequency so that the area of the bin corresponds to the probability, as do the rest of the histograms in the paper) The output is random since we implement the Gumbel-Softmax trick in sampling the traffic allocations. We compute the value of the objective function for each output and plot the corresponding histogram over 1000 samples. The dash-line shows the Gaussian distribution of the same mean and standard deviation.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Proposition 2.1
  • Definition 2.1
  • Remark 1
  • Remark 2
  • Remark 3