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Revisiting Online Learning Approach to Inverse Linear Optimization: A Fenchel$-$Young Loss Perspective and Gap-Dependent Regret Analysis

Shinsaku Sakaue, Han Bao, Taira Tsuchiya

TL;DR

We study online learning for inverse linear optimization, aiming to recover an unknown linear objective $c^*$ from sequential observations. By casting the suboptimality and estimate losses as Fenchel--Young losses, we show the total loss equals the linearized Fenchel--Young regret, enabling standard online-to-batch transfers and offline guarantees without requiring agent optimality. The paper provides an $O(\sqrt{T})$ online bound for the total loss in general settings and, under a $\Delta$-gap condition on the agent's decision problems, a horizon-free bound of $O(1/\Delta^2)$, exploiting problem structure to surpass typical online-learning rates. This Fenchel--Young loss perspective clarifies the theoretical underpinnings of inverse linear optimization, yields horizon-independent guarantees in structured cases, and suggests avenues for extending surrogate losses beyond Fenchel--Young constructions.

Abstract

This paper revisits the online learning approach to inverse linear optimization studied by Bärmann et al. (2017), where the goal is to infer an unknown linear objective function of an agent from sequential observations of the agent's input-output pairs. First, we provide a simple understanding of the online learning approach through its connection to online convex optimization of \emph{Fenchel--Young losses}. As a byproduct, we present an offline guarantee on the \emph{suboptimality loss}, which measures how well predicted objectives explain the agent's choices, without assuming the optimality of the agent's choices. Second, assuming that there is a gap between optimal and suboptimal objective values in the agent's decision problems, we obtain an upper bound independent of the time horizon $T$ on the sum of suboptimality and \emph{estimate losses}, where the latter measures the quality of solutions recommended by predicted objectives. Interestingly, our gap-dependent analysis achieves a faster rate than the standard $O(\sqrt{T})$ regret bound by exploiting structures specific to inverse linear optimization, even though neither the loss functions nor their domains enjoy desirable properties, such as strong convexity.

Revisiting Online Learning Approach to Inverse Linear Optimization: A Fenchel$-$Young Loss Perspective and Gap-Dependent Regret Analysis

TL;DR

We study online learning for inverse linear optimization, aiming to recover an unknown linear objective from sequential observations. By casting the suboptimality and estimate losses as Fenchel--Young losses, we show the total loss equals the linearized Fenchel--Young regret, enabling standard online-to-batch transfers and offline guarantees without requiring agent optimality. The paper provides an online bound for the total loss in general settings and, under a -gap condition on the agent's decision problems, a horizon-free bound of , exploiting problem structure to surpass typical online-learning rates. This Fenchel--Young loss perspective clarifies the theoretical underpinnings of inverse linear optimization, yields horizon-independent guarantees in structured cases, and suggests avenues for extending surrogate losses beyond Fenchel--Young constructions.

Abstract

This paper revisits the online learning approach to inverse linear optimization studied by Bärmann et al. (2017), where the goal is to infer an unknown linear objective function of an agent from sequential observations of the agent's input-output pairs. First, we provide a simple understanding of the online learning approach through its connection to online convex optimization of \emph{Fenchel--Young losses}. As a byproduct, we present an offline guarantee on the \emph{suboptimality loss}, which measures how well predicted objectives explain the agent's choices, without assuming the optimality of the agent's choices. Second, assuming that there is a gap between optimal and suboptimal objective values in the agent's decision problems, we obtain an upper bound independent of the time horizon on the sum of suboptimality and \emph{estimate losses}, where the latter measures the quality of solutions recommended by predicted objectives. Interestingly, our gap-dependent analysis achieves a faster rate than the standard regret bound by exploiting structures specific to inverse linear optimization, even though neither the loss functions nor their domains enjoy desirable properties, such as strong convexity.
Paper Structure (20 sections, 8 theorems, 39 equations, 1 figure, 1 algorithm)

This paper contains 20 sections, 8 theorems, 39 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. Problem Setting
  6. Fenchel--Young Loss
  7. Fenchel--Young-Loss Perspective
  8. $\ell^\mathrm{sub}$ is a Fenchel--Young Loss
  9. $\ell^\mathrm{sub} + \ell^\mathrm{est} =$ the Linearized Regret of a Fenchel--Young Loss
  10. Discussion on a Difference from the Previous Understanding
  11. Theoretical Guarantees for General Cases
  12. Online Guarantee on $\ell^\mathrm{sub} + \ell^\mathrm{est}$
  13. Offline Guarantee on $\ell^\mathrm{sub}$
  14. Gap-Dependent Bound on Total Loss
  15. Proof of Theorem \ref{['thm:fast-rate']}
  16. ...and 5 more sections

Key Result

Proposition 2.2

For any $x \in \mathbb{R}^n$, the Fenchel--Young loss $L_\Omega(\cdot; x)$ is non-negative and convex. For any $\hat{c} \in \mathbb{R}^n$, if $\hat{x} \in \mathop{\mathrm{arg\,max}}\nolimits_{x' \in \mathbb{R}^n}{\langle\hat{c}, x'\rangle - \Omega(x')}$ exists, then the residual vector is a subgradi

Figures (1)

  • Figure 1: An illustration of the gap condition. The gray area shows polyhedral feasible region $X_t$. The vertex $x$ is the unique optimal solution for $c^*$ if $c^*$ lies in the interior of the normal cone $N(x)$, shown in blue. The $\Delta$-gap condition requires that the cosine of the angle between $c^*$ and $x - \hat{x}$ is at least $\Delta/\|c^*\|$ for every $\hat{x} \in X_t$; this is true if $c^*$ (the head of the blue arrow) is distant from the boundary of $N(x)$ (the blue lines) by at least $\Delta$.

Theorems & Definitions (17)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 4.1: cf. orabona2023modern
  • Corollary 4.2
  • Theorem 4.3
  • ...and 7 more