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LEARN: An Invex Loss for Outlier Oblivious Robust Online Optimization

Adarsh Barik, Anand Krishna, Vincent Y. F. Tan

TL;DR

The paper tackles robust online optimization when an adversary can inject outliers in an unknown subset of rounds, even allowing unbounded domains and non-Lipschitz losses. It introduces the Learn loss $g(\theta) = -a\log(\exp(-f(\theta)/a) + b)$ to attenuate outlier effects and develops a robust online gradient method that guarantees sublinear clean dynamic regret, with matching lower-bound-like dependence on the outlier count $k$. A unified invex-loss analysis framework is proposed, enabling regret bounds for invex and related nonconvex losses, and extended to an expert-framework for unbounded domains, achieving $\mathcal{O}(\sqrt{V_T T} + V_T + \sqrt{T\log T} + k)$ regret in the unbounded setting. Empirical validation on online linear regression and online SVM demonstrates robust performance against outliers, supporting the theoretical claims and highlighting LEARN’s practical impact for robust, dynamic online learning.

Abstract

We study a robust online convex optimization framework, where an adversary can introduce outliers by corrupting loss functions in an arbitrary number of rounds k, unknown to the learner. Our focus is on a novel setting allowing unbounded domains and large gradients for the losses without relying on a Lipschitz assumption. We introduce the Log Exponential Adjusted Robust and iNvex (LEARN) loss, a non-convex (invex) robust loss function to mitigate the effects of outliers and develop a robust variant of the online gradient descent algorithm by leveraging the LEARN loss. We establish tight regret guarantees (up to constants), in a dynamic setting, with respect to the uncorrupted rounds and conduct experiments to validate our theory. Furthermore, we present a unified analysis framework for developing online optimization algorithms for non-convex (invex) losses, utilizing it to provide regret bounds with respect to the LEARN loss, which may be of independent interest.

LEARN: An Invex Loss for Outlier Oblivious Robust Online Optimization

TL;DR

The paper tackles robust online optimization when an adversary can inject outliers in an unknown subset of rounds, even allowing unbounded domains and non-Lipschitz losses. It introduces the Learn loss to attenuate outlier effects and develops a robust online gradient method that guarantees sublinear clean dynamic regret, with matching lower-bound-like dependence on the outlier count . A unified invex-loss analysis framework is proposed, enabling regret bounds for invex and related nonconvex losses, and extended to an expert-framework for unbounded domains, achieving regret in the unbounded setting. Empirical validation on online linear regression and online SVM demonstrates robust performance against outliers, supporting the theoretical claims and highlighting LEARN’s practical impact for robust, dynamic online learning.

Abstract

We study a robust online convex optimization framework, where an adversary can introduce outliers by corrupting loss functions in an arbitrary number of rounds k, unknown to the learner. Our focus is on a novel setting allowing unbounded domains and large gradients for the losses without relying on a Lipschitz assumption. We introduce the Log Exponential Adjusted Robust and iNvex (LEARN) loss, a non-convex (invex) robust loss function to mitigate the effects of outliers and develop a robust variant of the online gradient descent algorithm by leveraging the LEARN loss. We establish tight regret guarantees (up to constants), in a dynamic setting, with respect to the uncorrupted rounds and conduct experiments to validate our theory. Furthermore, we present a unified analysis framework for developing online optimization algorithms for non-convex (invex) losses, utilizing it to provide regret bounds with respect to the LEARN loss, which may be of independent interest.
Paper Structure (56 sections, 14 theorems, 119 equations, 6 figures, 1 table, 3 algorithms)

This paper contains 56 sections, 14 theorems, 119 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Outlier robustness
  3. Online convex optimization (OCO)
  4. OCO with robustness
  5. Main contributions
  6. Related works:
  7. Preliminaries and notation
  8. Learn loss
  9. Robust online convex optimization
  10. Regret analysis of the Learn loss
  11. Clean dynamic regret analysis
  12. Bounded domain
  13. Unbounded domain via an expert framework
  14. An overview of proof techniques
  15. Experimental validation
  16. ...and 41 more sections

Key Result

Lemma 3.1

The robust loss $g_t(\theta) \coloneqq g_t(s_t, \theta)$ constructed at each round $t\in [T]$ is a $\zeta_t$-invex function, i.e., for all $\vartheta_1, \vartheta_2 \in \mathbb{R}^d$, there exists a vector-valued function $\zeta_t:\mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^d$ such that Moreover, for actions $\theta_t$ generated in Algorithm alg:outlier robust OGD and $\omega_t^*$ as defined

Figures (6)

  • Figure 1: We compared Learn (Algorithm \ref{['alg:outlier robust OGD']}) and baselines (Section \ref{['sec: experimental validation main']}) on binary classification with hinge loss and sequential data. Figure \ref{['fig: svm clean decision']} shows the true decision boundary with no outliers and all the methods recover it exactly. Figure \ref{['fig: svm ktwothird decision']} shows how decision boundaries for baseline methods deviate from the ground truth in the presence of outliers while Learn remains robust to outliers.
  • Figure 2: Clean dynamic regret plots for different algorithms running on the Online Ridge Regression (Top row, Figures \ref{['fig: lin reg kcfr k0']}-\ref{['fig: lin reg kcfr k constant']}) and Online SVM (Bottom row, Figures \ref{['fig: svm kcfr k0']}-\ref{['fig: svm kcfr k constant']}).
  • Figure 3: The robust loss closely follows the square loss and flattens out as we move away from the minima.
  • Figure 4: Clean Regret for Online SVM in Presence of Varying Number of Outliers ($k$).
  • Figure 5: Visualization of Decision Boundary in Presence of Varying Number of Outliers ($k$).
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 2.1: Invex functions
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Theorem 3.5: Informal
  • proof
  • proof
  • Definition E.1: Invex functions
  • Remark E.2
  • ...and 24 more