arxiv: 2605.08866 · v1 · submitted 2026-05-09 · 📊 stat.ML · cs.LG· math.OC

Recognition: 2 theorem links

· Lean Theorem

Tight Generalization Bounds for Noiseless Inverse Optimization

Hoomaan Maskan, Peyman Mohajerin Esfahani, Pouria Fatemi, Suvrit Sra

Pith reviewed 2026-05-12 01:40 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.OC

keywords inverse optimizationgeneralization boundsnoiseless settingregret boundsconsistent estimatorsinduced action setsample complexity

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The pith

Noiseless inverse optimization achieves a tight high-probability O(d/T) generalization bound on the induced action set.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that when demonstrations are produced exactly by optimizing an unknown ground-truth objective, any consistent estimator yields an induced action set whose error relative to the true optimal actions is bounded by O(d/T) with high probability. This rate cannot be improved in general for the estimators considered. Under added conditions that make the optimal action unique for each context, the bounds reach the rates known for best-arm identification in bandits. The analysis extends the same O(d/T) scaling to both instantaneous and cumulative regret, where the lower bound matches known upper bounds from adversarial environments. A parameter-free algorithm is given that recovers the objective with lower per-iteration cost than generic optimization solvers.

Core claim

In the noiseless setting of inverse optimization, demonstrations are produced exactly by optimizing an unknown ground-truth objective. For consistent estimators of the objective parameters, the induced action set generalizes with high probability at the rate O(d/T). This rate is tight across the considered estimators. The corresponding regret lower bounds match the upper bounds known for the adversarial setting, making the stochastic noiseless IO effectively adversarial for these estimators.

What carries the argument

The induced action set, the collection of actions that would be chosen by optimizing the estimated objective on new contexts, whose deviation from the true optimal actions is controlled by the generalization bound.

If this is right

The number of demonstrations required for small generalization error scales linearly with the dimension d.
Uniqueness of optimal actions upgrades the guarantees to match best-arm identification rates from the bandit literature.
Both instantaneous and cumulative regret inherit the tight O(d/T) scaling and the match to adversarial upper bounds.
Parameter-free algorithms recover the objective with lower per-iteration computational cost than standard solvers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The equivalence to adversarial regret suggests that techniques developed for robust adversarial inverse optimization may transfer directly to this stochastic noiseless case.
In domains that assume perfect rationality, such as certain economic or robotic planning settings, the results quantify the sample needs but also warn that performance remains vulnerable to worst-case data ordering.

Load-bearing premise

Demonstrations are generated exactly by optimizing a single ground-truth objective with no noise or suboptimality.

What would settle it

A family of noiseless demonstration sequences and a consistent estimator for which the probability that the induced action set deviates from the true one by more than c d/T remains bounded away from zero for large T and some fixed c.

Figures

Figures reproduced from arXiv: 2605.08866 by Hoomaan Maskan, Peyman Mohajerin Esfahani, Pouria Fatemi, Suvrit Sra.

**Figure 1.** Figure 1: (a) Relation between the set-level mismatch, the action-level mismatch, and the true suboptimality loss (regret), (b) Geometric intuition for tightness in the two-dimensional case θ = (1, θ2, θ3) for Theorem 3.1. Each demonstration induces an asymmetric strip constraint. Intersecting these constraints yields the consistency polytope (shaded) that contains θ ⋆ . The incenter estimator ˆθ in T has the maximu… view at source ↗

**Figure 2.** Figure 2: Generalization mismatch probability GET on fresh contexts versus T for d ∈ {5, 10, 20}. The depicted results are averaged over 10 runs with 90% confidence bands. The dashed line shows the theoretical upper bound from Theorem 2.1 with β = 0.1. The experiments confirm our theoretical findings. 6 Limitations and Concluding Remarks We established tight generalization guarantees for IO in the noiseless regime b… view at source ↗

read the original abstract

Inverse optimization (IO) seeks to infer the parameters of a decision-maker's objective from observed context--action data. We study noiseless IO, where demonstrations are generated by a ground-truth objective. We provide a high-probability ${O}(\frac{d}{T})$ generalization bound for the induced action set, where $d$ is the number of unknown parameters and $T$ is the size of the training dataset. We strengthen these guarantees under additional conditions that ensure uniqueness of the chosen action, bringing our IO guarantees in line with best-arm identification results in the bandit literature. We further show that the ${O}(\frac{d}{T})$ rate is tight over all consistent estimators considered here, and extend the result to both instantaneous and cumulative regret. Notably, the resulting regret lower bound matches the corresponding upper bounds in the adversarial setting, indicating that the stochastic IO setting is effectively adversarial for the class of estimators studied here. Finally, we propose a parameter-free algorithm with lower per-iteration complexity than generic solvers. Experiments validate the predicted rates and illustrate the tightness of our bounds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper pins down a tight O(d/T) high-probability bound on the recovered action set in noiseless inverse optimization and shows the regret lower bound matches adversarial rates.

read the letter

The core result is a high-probability O(d/T) generalization bound on the induced action set recovered from noiseless demonstrations, with a matching lower bound that shows the rate is tight for consistent estimators. The work also extends this to instantaneous and cumulative regret and notes that the lower bound aligns with known adversarial upper bounds, which suggests the stochastic noiseless setting is effectively adversarial for the estimators considered.

Referee Report

2 major / 2 minor

Summary. The paper studies noiseless inverse optimization, deriving a high-probability O(d/T) generalization bound on the induced action set from context-action demonstrations generated by a ground-truth objective. It strengthens the bound to bandit-like rates under additional uniqueness conditions on the optimal action, proves that the O(d/T) rate is tight for all consistent estimators considered, extends the analysis to instantaneous and cumulative regret (with the lower bound matching adversarial upper bounds), and proposes a parameter-free algorithm with reduced per-iteration complexity. Experiments are used to validate the predicted scaling and illustrate tightness.

Significance. If the derivations hold, the work supplies the first tight, non-vacuous sample-complexity bounds for noiseless IO and explicitly connects the setting to best-arm identification in bandits. The observation that the stochastic IO problem is effectively adversarial for consistent estimators is a notable conceptual contribution. The parameter-free algorithm provides a concrete practical advance over generic solvers. Experiments confirming the O(d/T) scaling lend empirical support to the theory.

major comments (2)

The tightness claim is stated to hold over all consistent estimators; the precise definition of consistency and the exact estimator class must be stated explicitly (e.g., in the section containing the lower-bound construction) because this class is load-bearing for both the upper and lower bounds.
The regret lower bound is asserted to match adversarial upper bounds; the manuscript should include a direct side-by-side comparison of constants and the precise adversarial setting (e.g., the section on regret extensions) to substantiate the 'effectively adversarial' conclusion.

minor comments (2)

Notation for the induced action set and the recovered objective should be introduced with a dedicated definition block early in the paper to improve readability.
The abstract and introduction mention 'additional conditions ensuring uniqueness'; a brief discussion of how often these conditions hold in typical IO applications would help readers assess the strengthened guarantees.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The tightness claim is stated to hold over all consistent estimators; the precise definition of consistency and the exact estimator class must be stated explicitly (e.g., in the section containing the lower-bound construction) because this class is load-bearing for both the upper and lower bounds.

Authors: We agree that an explicit definition is needed to support the tightness claim. In the revised manuscript we will add, in the lower-bound section, a formal definition of consistency (estimators that recover the ground-truth objective almost surely as T tends to infinity under the noiseless model) together with the precise class of estimators to which the claim applies (the parameter-free algorithm and all generic convex solvers that return a feasible point in the dual polytope). This will make clear that the O(d/T) rate is tight precisely for this class. revision: yes
Referee: The regret lower bound is asserted to match adversarial upper bounds; the manuscript should include a direct side-by-side comparison of constants and the precise adversarial setting (e.g., the section on regret extensions) to substantiate the 'effectively adversarial' conclusion.

Authors: We will strengthen the presentation as suggested. In the revised regret-extensions section we will insert a direct side-by-side comparison (in text and a small table) of the leading constants in our stochastic lower bound and the matching adversarial upper bounds from the best-arm-identification literature, together with an explicit statement of the adversarial setting (an adversary that chooses contexts and feasible actions to force linear regret against any consistent estimator). This will substantiate the claim that the stochastic noiseless IO problem is effectively adversarial for the estimators studied. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation of the O(d/T) high-probability generalization bound on the induced action set proceeds directly from the noiseless model, the definition of the induced action set as actions optimal under the recovered objective, and standard concentration arguments over T samples with d parameters. Tightness follows from an independent lower-bound construction over a family of objectives with margin-separated optimal actions, without reducing to fitted quantities or self-referential definitions. Regret extensions and the parameter-free algorithm are likewise derived from the same primitives without circular steps. The result is self-contained against external benchmarks and does not rely on load-bearing self-citations or ansatzes smuggled from prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the noiseless data-generating process and uniqueness conditions; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Demonstrations are generated exactly by a ground-truth objective (noiseless setting)
Explicitly stated as the setting studied in the abstract.
domain assumption Additional conditions ensure uniqueness of the chosen action
Invoked to strengthen the O(d/T) bound to bandit-like guarantees.

pith-pipeline@v0.9.0 · 5500 in / 1370 out tokens · 52276 ms · 2026-05-12T01:40:42.737595+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We develop a scenario program view of noiseless IO, where each demonstration induces a random convex feasibility constraint on the parameter θ (Campi & Garatti, 2008). This yields a high-probability O(d/T) bound on set-level mismatch (Theorem 2.1).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We further show that the O(d/T) rate is tight over all consistent estimators... the resulting regret lower bound matches the corresponding upper bounds in the adversarial setting.

Reference graph

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