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DFORD: Directional Feedback based Online Ordinal Regression Learning

Naresh Manwani, M Elamparithy, Tanish Taneja

TL;DR

This work introduces directional feedback as a weak supervision signal for online ordinal regression, formulating the DFORD framework with both linear and kernel variants. It develops unbiased estimators and an exploration-exploitation scheme to learn from left/right feedback while preserving threshold ordering in expectation. The linear version achieves an $O(\log T)$ regret bound, and the kernel variant employs truncation to manage memory. Empirical results on real and synthetic data show DFORD attaining performance comparable to, and sometimes surpassing, full-information baselines, validating directional feedback as an effective online learning signal for ordinal tasks.

Abstract

In this paper, we introduce directional feedback in the ordinal regression setting, in which the learner receives feedback on whether the predicted label is on the left or the right side of the actual label. This is a weak supervision setting for ordinal regression compared to the full information setting, where the learner can access the labels. We propose an online algorithm for ordinal regression using directional feedback. The proposed algorithm uses an exploration-exploitation scheme to learn from directional feedback efficiently. Furthermore, we introduce its kernel-based variant to learn non-linear ordinal regression models in an online setting. We use a truncation trick to make the kernel implementation more memory efficient. The proposed algorithm maintains the ordering of the thresholds in the expected sense. Moreover, it achieves the expected regret of $\mathcal{O}(\log T)$. We compare our approach with a full information and a weakly supervised algorithm for ordinal regression on synthetic and real-world datasets. The proposed approach, which learns using directional feedback, performs comparably (sometimes better) to its full information counterpart.

DFORD: Directional Feedback based Online Ordinal Regression Learning

TL;DR

This work introduces directional feedback as a weak supervision signal for online ordinal regression, formulating the DFORD framework with both linear and kernel variants. It develops unbiased estimators and an exploration-exploitation scheme to learn from left/right feedback while preserving threshold ordering in expectation. The linear version achieves an regret bound, and the kernel variant employs truncation to manage memory. Empirical results on real and synthetic data show DFORD attaining performance comparable to, and sometimes surpassing, full-information baselines, validating directional feedback as an effective online learning signal for ordinal tasks.

Abstract

In this paper, we introduce directional feedback in the ordinal regression setting, in which the learner receives feedback on whether the predicted label is on the left or the right side of the actual label. This is a weak supervision setting for ordinal regression compared to the full information setting, where the learner can access the labels. We propose an online algorithm for ordinal regression using directional feedback. The proposed algorithm uses an exploration-exploitation scheme to learn from directional feedback efficiently. Furthermore, we introduce its kernel-based variant to learn non-linear ordinal regression models in an online setting. We use a truncation trick to make the kernel implementation more memory efficient. The proposed algorithm maintains the ordering of the thresholds in the expected sense. Moreover, it achieves the expected regret of . We compare our approach with a full information and a weakly supervised algorithm for ordinal regression on synthetic and real-world datasets. The proposed approach, which learns using directional feedback, performs comparably (sometimes better) to its full information counterpart.
Paper Structure (48 sections, 11 theorems, 65 equations, 4 figures, 2 tables, 4 algorithms)

This paper contains 48 sections, 11 theorems, 65 equations, 4 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Ordinal Regression
  4. Ordinal Regression Using Interval Labels
  5. Preliminaries
  6. Ordinal Regression
  7. Linear Ordinal Regression Model
  8. Kernel Ordinal Regression
  9. Loss Function for Ordinal Regression
  10. Online Learning Setting
  11. Interval Label For Ordinal Regression
  12. Directional Feedback and Label Uncertainty
  13. Efficient Exploration of Labels Under Directional Feedback
  14. Unbiased Estimators of Labels $z_i^t$
  15. Unbiased Estimator of $\tau_i^t=z_i^t \mathbb{I}[z_i^t(\mathbf{w}^t.\mathbf{x}^t - \theta_i^t)<0]$:
  16. ...and 33 more sections

Key Result

Theorem 1

$\tilde{\tau}_i^t$ is an unbiased estimator of $\tau_i^t$. Thus, for $i=1\ldots K$, we have $\mathbb{E}[\tilde{\tau}_i^t]=\tau_i^t$.

Figures (4)

  • Figure 1: Example of label probability distribution for $K = 7$ and $\gamma=0.7$. (a) Case 1: $\hat{y} = 5$ and (b) Case 2: $\hat{y} = 2$.
  • Figure 2: Hyperparameter Search ($\lambda$ and $\gamma$) for all datasets
  • Figure 3: Searching for truncation parameter $(\delta)$ for Abalone and Synthetic dataset
  • Figure 4: Average Mean Absolute Error (MAE) curves for various datasets.

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Theorem 3: Regret Bound for DFORD-Kernel
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • ...and 1 more