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Distributionally Robust Degree Optimization for BATS Codes

Hoover H. F. Yin, Jie Wang, Sherman S. M. Chow

TL;DR

The paper tackles the instability of BATS code rates arising from misestimating the destination rank distribution $\mathbf{h}$ when optimizing the degree distribution $\boldsymbol{\Psi}$. It introduces a distributionally robust optimization (DRO) framework that constrains optimization to a probability ball around the empirical rank distribution $\hat{\mathbf{h}}$, and analyzes both $1$-Wasserstein and total-variation metrics to provide out-of-sample guarantees. Through dual reformulations, the authors obtain tractable DRO programs that maximize the worst-case rate while preserving decodability constraints across discretized $x\in\mathcal{X}$. Numerical results show that Wasserstein DRO delivers a close-to-optimal rate with minimal rate fluctuations, outperforming non-DRO and other robust strategies, while TV-DRO may require larger samples to achieve similar performance. The work suggests that Wasserstein DRO offers a practical and effective route to robust BATS code deployment in uncertain wireless environments.

Abstract

Batched sparse (BATS) code is a network coding solution for multi-hop wireless networks with packet loss. Achieving a close-to-optimal rate relies on an optimal degree distribution. Technical challenges arise from the sensitivity of this distribution to the often empirically obtained rank distribution at the destination node. Specifically, if the empirical distribution overestimates the channel, BATS codes experience a significant rate degradation, leading to unstable rates across different runs and hence unpredictable transmission costs. Confronting this unresolved obstacle, we introduce a formulation for distributionally robust optimization in degree optimization. Deploying the resulting degree distribution resolves the instability of empirical rank distributions, ensuring a close-to-optimal rate, and unleashing the potential of applying BATS codes in real-world scenarios.

Distributionally Robust Degree Optimization for BATS Codes

TL;DR

The paper tackles the instability of BATS code rates arising from misestimating the destination rank distribution when optimizing the degree distribution . It introduces a distributionally robust optimization (DRO) framework that constrains optimization to a probability ball around the empirical rank distribution , and analyzes both -Wasserstein and total-variation metrics to provide out-of-sample guarantees. Through dual reformulations, the authors obtain tractable DRO programs that maximize the worst-case rate while preserving decodability constraints across discretized . Numerical results show that Wasserstein DRO delivers a close-to-optimal rate with minimal rate fluctuations, outperforming non-DRO and other robust strategies, while TV-DRO may require larger samples to achieve similar performance. The work suggests that Wasserstein DRO offers a practical and effective route to robust BATS code deployment in uncertain wireless environments.

Abstract

Batched sparse (BATS) code is a network coding solution for multi-hop wireless networks with packet loss. Achieving a close-to-optimal rate relies on an optimal degree distribution. Technical challenges arise from the sensitivity of this distribution to the often empirically obtained rank distribution at the destination node. Specifically, if the empirical distribution overestimates the channel, BATS codes experience a significant rate degradation, leading to unstable rates across different runs and hence unpredictable transmission costs. Confronting this unresolved obstacle, we introduce a formulation for distributionally robust optimization in degree optimization. Deploying the resulting degree distribution resolves the instability of empirical rank distributions, ensuring a close-to-optimal rate, and unleashing the potential of applying BATS codes in real-world scenarios.
Paper Structure (21 sections, 2 theorems, 32 equations, 2 figures)

This paper contains 21 sections, 2 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. BATS Codes and the Degree Distribution Issue
  3. Related Works
  4. Our Contribution
  5. BATS Codes
  6. Notations
  7. Encoding
  8. Recoding and Ranks of Batches
  9. Decoding
  10. Degree Optimization Framework
  11. Distributionally Robust Optimization (DRO)
  12. Choice of Probability Metric
  13. Wasserstein DRO
  14. Total Variation DRO
  15. Numerical Evaluations
  16. ...and 6 more sections

Key Result

Theorem 1

The Wasserstein DRO formulation for degree optimization is equivalent to

Figures (2)

  • Figure 1: Illustrations of significant rate degradation when overestimating the channel and its impact on rate fluctuation.
  • Figure 2: The $1$st and $3$rd quantiles of rate for different approaches: Approaches in literature and both DROs can significantly reduce the fluctuation of rates. The Wasserstein DRO has a close-to-optimal rate, where its plot almost collides with the optimal curve. Also, $98\%$ of the rate is upper-bounded by the expected rank (or capacity).

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2