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Lossy Source Coding with Focal Loss

Alex Dytso, Martina Cardone

TL;DR

This work introduces focal loss as a distortion measure for lossy source coding with soft reconstruction, unifying ideas from machine learning with rate-distortion theory. It derives single-shot converse and achievability bounds and proves that the asymptotic distortion-rate function $\mathsf{D}(\mathsf{R};\gamma)$ coincides with the log-loss case, i.e., $[H(X) - \mathsf{R}]^+$, while demonstrating that the focal term influences finite-blocklength performance. The analysis relies on an entropy-like quantity $H_\gamma(P_X)$ and a maximum-entropy bound $h_\gamma(|\mathcal{X}|)$, and introduces an auxiliary distribution $F_X$ to optimize the achievability bound, with the optimal choice asymptotically being $F_X = R_X$. Numerical examples with small alphabets and a Binomial source illustrate non-asymptotic differences induced by the focal term and highlight potential privacy implications for negative $\gamma$. Overall, the results provide a principled framework for focal-loss-based distortion in rate-distortion theory and point to future work on non-asymptotic tightening, excess-distortion analysis, multiterminal extensions, and privacy-oriented applications.

Abstract

Focal loss has recently gained significant popularity, particularly in tasks like object detection where it helps to address class imbalance by focusing more on hard-to-classify examples. This work proposes the focal loss as a distortion measure for lossy source coding. The paper provides single-shot converse and achievability bounds. These bounds are then used to characterize the distortion-rate trade-off in the infinite blocklength, which is shown to be the same as that for the log loss case. In the non-asymptotic case, the difference between focal loss and log loss is illustrated through a series of simulations.

Lossy Source Coding with Focal Loss

TL;DR

This work introduces focal loss as a distortion measure for lossy source coding with soft reconstruction, unifying ideas from machine learning with rate-distortion theory. It derives single-shot converse and achievability bounds and proves that the asymptotic distortion-rate function coincides with the log-loss case, i.e., , while demonstrating that the focal term influences finite-blocklength performance. The analysis relies on an entropy-like quantity and a maximum-entropy bound , and introduces an auxiliary distribution to optimize the achievability bound, with the optimal choice asymptotically being . Numerical examples with small alphabets and a Binomial source illustrate non-asymptotic differences induced by the focal term and highlight potential privacy implications for negative . Overall, the results provide a principled framework for focal-loss-based distortion in rate-distortion theory and point to future work on non-asymptotic tightening, excess-distortion analysis, multiterminal extensions, and privacy-oriented applications.

Abstract

Focal loss has recently gained significant popularity, particularly in tasks like object detection where it helps to address class imbalance by focusing more on hard-to-classify examples. This work proposes the focal loss as a distortion measure for lossy source coding. The paper provides single-shot converse and achievability bounds. These bounds are then used to characterize the distortion-rate trade-off in the infinite blocklength, which is shown to be the same as that for the log loss case. In the non-asymptotic case, the difference between focal loss and log loss is illustrated through a series of simulations.
Paper Structure (13 sections, 7 theorems, 49 equations, 4 figures)

This paper contains 13 sections, 7 theorems, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Paper Contributions and Outline
  4. Definitions and Single-Shot Preliminaries
  5. Converse
  6. Auxiliary Results
  7. Converse Bound
  8. Achievability
  9. Asymptotics, Comparisons, and Evaluations
  10. Asymptotics
  11. Examples
  12. Conclusion
  13. Proof of Proposition \ref{['prop:max_entropy']}

Key Result

Proposition 1

The quantity $H_\gamma(P_X)$ in eq:EntropyTerm satisfies the following two basic properties:

Figures (4)

  • Figure 1: Examples of $h_\gamma(| \mathcal{X}|)$ in \ref{['eq:FirstClaimPropMaxEntr']}.
  • Figure 2: Exact $d^*(2; \gamma)$ for Example \ref{['examp:M2X3']}, where $R_{X_1} = [1/3, 1/3,1/3]$ and $R_{X_2} = [2/3, 1/4, 1/12]$.
  • Figure 3: Converse and achievability bounds for the binomial source in Example \ref{['examp:binomial']} for different values of $\gamma$.
  • Figure 4: Bounds for the binomial source in Example \ref{['examp:binomial']}, $M\!=\!8$.

Theorems & Definitions (13)

  • Definition 1: Focal loss distortion
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Example 1
  • ...and 3 more