Title: Online Inverse Linear Optimization: Efficient Logarithmic-Regret Algorithm, Robustness to Suboptimality, and Lower Bound
Venue: NeurIPS 2025
Accepted: yes
Keywords: inverse optimization

online learning
Abstract: In online inverse linear optimization, a learner observes time-varying sets of feasible actions and an agent's optimal actions, selected by solving linear optimization over the feasible actions. The learner sequentially makes predictions of the agent's true linear objective function, and their quality is measured by the *regret*, the cumulative gap between optimal objective values and those achieved by following the learner's predictions. A seminal work by Bärmann et al. (2017) obtained a regret bound of $O(\sqrt{T})$, where $T$ is the time horizon. Subsequently, the regret bound has been improved to $O(n^4 \ln T)$ by Besbes et al. (2021, 2025) and to $O(n \ln T)$ by Gollapudi et al. (2021), where $n$ is the dimension of the ambient space of objective vectors. However, these logarithmic-regret methods are highly inefficient when $T$ is large, as they need to maintain regions specified by $O(T)$ constraints, which represent possible locations of the true objective vector. In this paper, we present the first logarithmic-regret method whose per-round complexity is independent of $T$; indeed, it achieves the best-known bound of $O(n \ln T)$. Our method is strikingly simple: it applies the online Newton step (ONS) to appropriate exp-concave loss functions. Moreover, for the case where the agent's actions are possibly suboptimal, we establish a regret bound of $O(n\ln T + \sqrt{\Delta_T n\ln T})$, where $\Delta_T$ is the cumulative suboptimality of the agent's actions. This bound is achieved by using MetaGrad, which runs ONS with $\Theta(\ln T)$ different learning rates in parallel. We also present a lower bound of $\Omega(n)$, showing that the $O(n\ln T)$ bound is tight up to an $O(\ln T)$ factor.

Online Inverse Linear Optimization: Efficient Logarithmic-Regret Algorithm, Robustness to Suboptimality, and Lower Bound

Abstract

In online inverse linear optimization, a learner observes time-varying sets of feasible actions and an agent's optimal actions, selected by solving linear optimization over the feasible actions. The learner sequentially makes predictions of the agent's true linear objective function, and their quality is measured by the regret , the cumulative gap between optimal objective values and those achieved by following the learner's predictions. A seminal work by Bärmann et al. (2017) obtained a regret bound of O ( √ T ) , where T is the time horizon. Subsequently, the regret bound has been improved to O ( n 4 ln T ) by Besbes et al. (2021, 2025) and to O ( n ln T ) by Gollapudi et al. (2021), where n is the dimension of the ambient space of objective vectors. However, these logarithmic-regret methods are highly inefficient when T is large, as they need to maintain regions specified by O ( T ) constraints, which represent possible locations of the true objective vector. In this paper, we present the first logarithmic-regret method whose per-round complexity is independent of T ; indeed, it achieves the best-known bound of O ( n ln T ) . Our method is strikingly simple: it applies the online Newton step (ONS) to appropriate exp-concave loss functions. Moreover, for the case where the agent's actions are possibly suboptimal, we establish a regret bound of O ( n ln T + √ ∆ T n ln T ) , where ∆ T is the cumulative suboptimality of the agent's actions. This bound is achieved by using MetaGrad, which runs ONS with Θ(ln T ) different learning rates in parallel. We also present a lower bound of Ω( n ) , showing that the O ( n ln T ) bound is tight up to an O (ln T ) factor.

1 Introduction

Optimization problems serve as forward models of various processes and systems, ranging from human decision-making to natural phenomena. In real-world applications, the true objective function of such models is rarely known a priori. This motivates the problem of inferring the objective function from observed optimal solutions, or inverse optimization . Early work in this area emerged from geophysics, aiming at estimating subsurface structure from seismic wave data [11, 53]. Subsequently, inverse optimization has been extensively studied [2, 13, 14, 28], applied across various domains, such as

Taira Tsuchiya

Taihei Oki

transportation [6], power systems [9], and healthcare [12], and have laid the foundation for various machine learning methods, including inverse reinforcement learning [44] and contrastive learning [50].

This study focuses on an elementary yet fundamental case where the objective function of forward optimization is linear. We consider an agent who repeatedly selects an action from a set of feasible actions by solving forward linear optimization. 1 Let n be a positive integer and R n the ambient space where forward optimization is defined. For t = 1 , . . . , T , given a set X t ⊆ R n of feasible actions, the agent selects an action x t ∈ X t that maximizes x ↦→⟨ c ∗ , x ⟩ over X t , where c ∗ ∈ R n is the agent's internal objective vector and ⟨· , ·⟩ denotes the standard inner product on R n . We want to infer c ∗ from observations consisting of the feasible sets and the agent's optimal actions, i.e., { ( X t , x t ) } T t =1 .

For this problem, Bärmann et al. [4, 5] have shown that online learning methods are effective for inferring the agent's underlying objective vector c ∗ . Consider a learner who aims to infer c ∗ . For t = 1 , . . . , T , the learner makes a prediction ˆ c t of c ∗ based on the past observations { ( X i , x i ) } t -1 i =1 and receives ( X t , x t ) as feedback. Let ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ represent an optimal action induced by the learner's t th prediction. The regret of choosing ˆ x t instead of x t is defined as ∑ T t =1 ⟨ c ∗ , x t -ˆ x t ⟩ . 2 Their idea is to regard R n ∋ c ↦→⟨ c, ˆ x t -x t ⟩ as a cost function and apply online learning methods, such as the online gradient descent (OGD). Then, ∑ T t =1 ⟨ c ∗ , x t -ˆ x t ⟩ ≤ ∑ T t =1 ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ = O ( √ T ) follows from the standard guarantee of online learning methods. As such, online learning methods with sublinear regret bounds can make the average regret converge to zero as T →∞ .

While the regret bound of O ( √ T ) is optimal in general online linear optimization (e.g., Hazan [26, Section 3.2]), the above online inverse linear optimization has special problem structures that could allow for better regret bounds; intuitively, feedback ( X t , x t ) is more informative about c ∗ , which defines the regret, due to the optimality of x t ∈ X t for c ∗ . Besbes et al. [7, 8] indeed showed that a logarithmic regret bound of O ( n 4 ln T ) is possible, and Gollapudi et al. [25] further improved the bound to O ( n ln T ) . 3 While these methods significantly improve the dependence on T in the regret bounds, their per-round computation cost is prohibitively high when T is large. Specifically, these methods iteratively update (appropriately inflated) regions that indicate possible locations of the true objective vector c ∗ and set prediction ˆ c t to the 'center' of the regions (the circumcenter in Besbes et al. [7, 8] and the centroid in Gollapudi et al. [25]). Since those regions are represented by O ( T ) constraints, their per-round complexity grows polynomially in T , at least in a straightforward implementation. Indeed, Besbes et al. [7, 8] and Gollapudi et al. [25] only claim that their methods run in poly( n, T ) time. This is in stark contrast to the earlier online-learning approach of Bärmann et al. [4, 5], whose per-round complexity is independent of T ; however, its O ( √ T ) -regret bound is much worse in terms of T . Is it then possible to design a logarithmic-regret method whose per-round complexity is independent of T ?

1.1 Our contributions

In this paper, we first present an O ( n ln T ) -regret method whose per-round complexity is independent of T (Theorem 3.1), answering the above question affirmatively. Table 1 summarizes the comparisons of our result with prior work. Our method is very simple: we apply the online Newton step (ONS) [27] to appropriately designed exp-concave loss functions. We believe this simplicity is a strength of our method, which makes it accessible to a wider audience and easier to implement.

We then address more realistic situations where the agent's actions can be suboptimal. We establish a regret bound of O ( n ln T + √ ∆ T n ln T ) , where ∆ T denotes the cumulative suboptimality of the agent's actions over T rounds (Theorem 4.1). We also apply this result to the offline setting via the online-to-batch conversion (Corollary 4.2). This bound is achieved by applying MetaGrad [55, 56], a universal online learning method that runs ONS with Θ(ln T ) different learning rates in parallel, to the suboptimality loss [43], a loss function commonly used in inverse optimization. While universal online learning is originally intended to adapt to unknown types of loss functions, our result shows that it is useful for adapting to unknown suboptimality levels in online inverse linear optimization. At

a high level, our important contribution lies in uncovering the deeper connection between inverse optimization and online learning, thereby enabling the former to leverage the powerful toolkit of the latter.

Finally, we present a regret lower bound of Ω( n ) (Theorem 5.1). Thus, the upper bound of O ( n ln T ) achieved by the method of Gollapudi et al. [25] and ours is tight up to an O (ln T ) factor. While the proof idea is somewhat straightforward, this lower bound clarifies the optimal dependence on n in the regret bound, thereby resolving a question raised in Besbes et al. [8, Section 7].

1.2 Related work

Classic studies on inverse optimization explored formulations for identifying parameters of forward optimization from a single observation [2, 29]. Recently, data-driven inverse optimization, which is intended to infer parameters of forward optimization from multiple noisy (possibly suboptimal) observations, has drawn significant interest [3, 6, 10, 35, 39, 42, 43, 52, 63]. This body of work has addressed offline settings with other criteria than the regret, which we formally define in (2). The suboptimality loss was introduced by Mohajerin Esfahani et al. [43] in this context.

The line of work by Bärmann et al. [4, 5], Besbes et al. [7, 8], and Gollapudi et al. [25], mentioned in Section 1, is the most relevant to our work; we present the detailed comparisons with them in Appendix A. It is worth mentioning here that Gollapudi et al. [25] obtained an exp( O ( n ln n )) -regret bound, in addition to the O ( n ln T ) bound in Table 1; therefore, it is possible to achieve a finite regret bound, although the dependence on n is exponential. Recently, Sakaue et al. [49] obtained a finite regret bound by assuming a gap between the optimal and suboptimal objective values. Unlike their work, we do not require such gap assumptions. Online inverse linear optimization can also be viewed as a variant of stochastic linear bandits [1, 18], where noisy objective values are given as feedback, instead of optimal actions. Intuitively, the optimal-action feedback is more informative and allows for the O ( n ln T ) regret upper bound, while there is a lower bound of Ω( n √ T ) in stochastic linear bandits [18, Theorem 3]. Online-learning approaches to other related settings have also been studied [20, 30, 59]; see Besbes et al. [8, Section 1.2] for an extensive discussion on the relation to these studies. Additionally, Chen and Kılınç-Karzan [15] and Sun et al. [51] studied online-learning methods for related settings with different criteria.

ONS [27] is a well-known online convex optimization (OCO) method that achieves a logarithmic regret bound for exp-concave loss functions. While ONS requires the prior knowledge of the expconcavity, universal online learning methods, including MetaGrad, can automatically adapt to the unknown curvatures of loss functions, such as the strong convexity and exp-concavity [55, 56, 58, 64]. Our strategy for achieving robustness to suboptimal feedback is to combine the regret bound of MetaGrad (Proposition 2.6) with the self-bounding technique (see Section 4 for details), which is widely adopted in the online learning literature [23, 60, 66].

Contextual search [38, 48] is a related problem of inferring the value of ⟨ c ∗ , x t ⟩ for an underlying vector c ∗ given context vectors x t . The method of Gollapudi et al. [25] is based on techniques developed in this context. Robustness to corrupted feedback is also studied in contextual search [36, 46, 47].

However, note that the problem setting is different from ours. Also, the regret in contextual search is defined with optimal choices even under corrupted feedback, and the regret bounds scale linearly with the cumulative corruption level. By contrast, our regret is defined with the agent's possibly suboptimal actions and our regret bound grows only at the rate of √ ∆ T for the cumulative suboptimality ∆ T .

Improving the per-round complexity is crucial. This topic has gathered particular attention in online portfolio selection [17, 34, 57]. There exists a trade-off between the per-round complexity and regret bounds among known methods for this problem, and advancing this frontier is recognized as important research [32, 54, 65]. When it comes to online inverse linear optimization, logarithmic regret bounds had only been achieved by the somewhat inefficient methods of Besbes et al. [7, 8] and Gollapudi et al. [25], while the efficient online-learning approach of Bärmann et al. [4, 5] only enjoys the O ( √ T ) -regret bound. This background highlights the significance of our efficient O ( n ln T ) -regret method, which realizes the benefits of both approaches that previously existed in a trade-off relationship.

2 Preliminaries

2.1 Problem setting

Weconsider an online learning setting with two players, a learner and an agent . The agent sequentially solves linear optimization problems of the following form for t = 1 , . . . , T :

where c ∗ is the agent's objective vector, which is unknown to the learner. Every feasible set X t ⊆ R n is non-empty and compact, and the agent's action x t always belongs to X t . Weassume that the agent's action is optimal for (1), i.e., x t ∈ arg max x ∈ X t ⟨ c ∗ , x ⟩ , except in Section 4, where we discuss the case where x t can be suboptimal. The set, X t , is not necessarily convex; we only assume access to an oracle that returns an optimal solution x ∈ arg max x ′ ∈ X t ⟨ c, x ′ ⟩ for any c ∈ R n . If X t is a polyhedron, any solver for linear programs (LPs) of the form (1) can serve as the oracle. Even if (1) is, for example, an integer LP, we may use empirically efficient solvers, such as Gurobi, to obtain an optimal solution.

The learner sequentially makes a prediction of c ∗ for t = 1 , . . . , T . Let Θ ⊆ R n denote a set of linear objective vectors, from which the learner picks predictions. We assume that Θ is a closed convex set and that the true objective vector c ∗ is contained in Θ . For t = 1 , . . . , T , the learner outputs a prediction ˆ c t of c ∗ based on past observations { ( X i , x i ) } t -1 i =1 and then receives ( X t , x t ) as feedback from the agent. Let ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ denote an optimal action induced by the learner's t th prediction. 4 We consider the following two measures of the quality of predictions ˆ c 1 , . . . , ˆ c T ∈ Θ :

Following prior work [7, 8, 25], we call R c ∗ T the regret , which is the cumulative gap between the optimal objective values and the objective values achieved by following the learner's predictions. Note that we have ⟨ c ∗ , x t -ˆ x t ⟩ ≥ 0 as long as x t is optimal for c ∗ . While the regret is a natural performance measure, the second one, ˜ R c ∗ T , in (2) is convenient when considering the online-learning approach [4, 5]. We always have R c ∗ T ≤ ˜ R c ∗ T since the additional term consisting of ⟨ ˆ c t , ˆ x t -x t ⟩ is non-negative due to the optimality of ˆ x t for ˆ c t ; intuitively, this term quantifies how well ˆ c t explains the agent's choice x t . Our upper bounds in Theorems 3.1 and 4.1 apply to ˜ R c ∗ T , and our lower bound in Theorem 5.1 applies to R c ∗ T .

Remark 2.1. The problem setting of Besbes et al. [7, 8] involves context functions and initial knowledge sets , which might make their setting appear more general than ours. However, it is not difficult to confirm that our methods are applicable to their setting. See Appendix A for details.

2.2 Boundedness assumptions and suboptimality loss

We introduce the following bounds on the sizes of X t and Θ .

Assumption 2.2. The ℓ 2 -diameter of Θ is bounded by D > 0 , i.e., max { ∥ c -c ′ ∥ 2 : c, c ′ ∈ Θ } ≤ D . Similarly, the ℓ 2 -diameter of X t is bounded by K > 0 for t = 1 , . . . , T . Furthermore, there exists B > 0 satisfying the following condition:

Assuming bounds on the diameters is common in the previous studies [4, 5, 7, 8, 25]. We additionally introduce B > 0 to measure the sizes of X t and Θ taking their mutual relationship into account. Note that the choice of B = DK is always valid due to the Cauchy-Schwarz inequality. This quantity is inspired by a semi-norm of gradients used in Van Erven et al. [56] and enables sharper analysis than that conducted by simply setting B = DK .

We also define the suboptimality loss for later use.

Definition 2.3. For t = 1 , . . . , T , for any action set X t and the agent's possibly suboptimal action x t , the suboptimality loss is defined by ℓ t ( c ) := max x ∈ X t ⟨ c, x ⟩ - ⟨ c, x t ⟩ for all c ∈ Θ .

That is, ℓ t ( c ) is the suboptimality of x t ∈ X t for c . Mohajerin Esfahani et al. [43] introduced this as a loss function that enjoys desirable computational properties in the context of inverse optimization. Specifically, the suboptimality loss is convex, and there is a convenient expression of a subgradient.

Proposition 2.4 (cf. Bärmann et al. [4, Proposition 3.1]) . The suboptimality loss, ℓ t : Θ → R , is convex. Moreover, for any ˆ c t ∈ Θ and ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ , it holds that ˆ x t -x t ∈ ∂ℓ t (ˆ c t ) .

Confirming these properties is not difficult: the convexity is due to the fact that ℓ t is the pointwise maximum of linear functions c ↦→⟨ c, x ⟩-⟨ c, x t ⟩ , and the subgradient expression is a consequence of Danskin's theorem [19] (or one can directly prove this as in Bärmann et al. [4, Proposition 3.1]). It is worth mentioning that, as pointed out by Sakaue et al. [49], ˜ R c ∗ T appears as the linearized upper bound on the regret with respect to the suboptimality loss, i.e., ∑ T t =1 ( ℓ t (ˆ c t ) -ℓ t ( c ∗ )) ≤ ∑ T t =1 ⟨ ˆ c t -c ∗ , g t ⟩ = ˜ R c ∗ T , where g t = ˆ x t -x t ∈ ∂ℓ t (ˆ c t ) . This enables the online-to-batch conversion for the suboptimality loss, as discussed in Section 4.1. Additionally, we have ˜ R c ∗ T = R c ∗ T + ∑ T t =1 ℓ t (ˆ c t ) in (2).

2.3 ONS and MetaGrad

Webriefly describe ONS and MetaGrad, based on Hazan [26, Section 4.4] and Van Erven et al. [56], to aid understanding of our methods. Appendix B shows the details for completeness. Readers who wish to proceed directly to our results may skip this subsection, taking Propositions 2.5 and 2.6 as given.

For convenience, we first state a specific form of ONS's O ( n ln T ) regret bound, which is later used in MetaGrad and in our analysis. See Algorithm 1 in Appendix B.1 for the pseudocode of ONS.

Proposition 2.5. Let W ⊆ R n be a closed convex set whose ℓ 2 -diameter is at most W > 0 . Let w 1 , . . . , w T and g 1 , . . . , g T be vectors in R n satisfying the following conditions for some G,H > 0 :

Take any η ∈ ( 0 , 1 5 H ] and define loss functions f η t : W → R for t = 1 , . . . , T as follows:

Let w η 1 , . . . , w η T ∈ W be the outputs of ONS applied to f η 1 , . . . , f η T . Then, for any u ∈ W , it holds that

Next, we describe MetaGrad (see Algorithm 2 in Appendix B.3), which we apply to the following general OCO problem on a closed convex set, W ⊆ R n . For t = 1 , . . . , T , we select w t ∈ W based on information obtained up to the end of round t -1 ; then, we incur f t ( w t ) and observe a subgradient, g t ∈ ∂f t ( w t ) , where f t : W → R denotes the t th convex loss function. We assume that W and g t for t = 1 , . . . , T satisfy the conditions in (3). Our goal is to make the regret with respect to f t , i.e., ∑ T t =1 ( f t ( w t ) -f t ( u )) , as small as possible for any comparator u ∈ W .

MetaGrad maintains η -experts , each of whom is associated with one of Θ(ln T ) different learning rates η ∈ ( 0 , 1 5 H ] . Each η -expert applies ONS to loss functions f η t of the form (4), where w t ∈ W

is the t th output of MetaGrad and g t ∈ ∂f t ( w t ) is given as feedback. In each round t , given the outputs w η t of η -experts (which are computed based on information up to round t -1 ), MetaGrad computes w t ∈ W by aggregating them via the exponentially weighted average (EWA).

For any comparator u ∈ W , define ˜ R u T := ∑ T t =1 ⟨ w t -u, g t ⟩ and V u T := ∑ T t =1 ⟨ w t -u, g t ⟩ 2 . Since all functions f t are convex, the regret with respect to f t , or ∑ T t =1 ( f t ( w t ) -f t ( u )) , is bounded by ˜ R u T from above. Furthermore, from the definition of f η t , we can decompose ˜ R u T as follows:

which simultaneously holds for all η > 0 . The first summation on the right-hand side, i.e., the regret of EWA compared to w η t , is indeed as small as O (ln ln T ) , while Proposition 2.5 ensures that the second summation is O ( n ln T ) . Thus, the right-hand side is O ( n ln T η + ηV u T ) . If we knew the true V u T value, we could choose η ≃ √ n ln T/V u T to achieve O (√ n ln T · V u T ) . This might seem impossible as we do not know any of u , g t , and w t beforehand. However, we can show that at least one of Θ(ln T ) values of η leads to almost the same regret, eschewing the need for knowing V u T . Formally, MetaGrad achieves the following regret bound (cf. Van Erven et al. [56, Corollary 8]). 5

Proposition 2.6. Let W ⊆ R n be given as in Proposition 2.5. Let w 1 , . . . , w T ∈ W be the outputs of MetaGrad applied to convex loss functions f 1 , . . . , f T : W → R . Assume that for every t = 1 , . . . , T , subgradient g t ∈ ∂f t ( w t ) satisfies the conditions (3) in Proposition 2.5. Then, it holds that

We outline how this result applies to exp-concave losses. Taking W , G , and H to be constants and ignoring the additive term of O ( n ln( T/n )) for simplicity, we have ˜ R u T = O ( √ n ln T · V u T ) . If all f t are α -exp-concave for some α ≤ 1 / ( GW ) , then f t ( w t ) -f t ( u ) ≤ ⟨ w t -u, g t ⟩ -α 2 ⟨ w t -u, g t ⟩ 2 holds (e.g., Hazan [26, Lemma 4.3]). Summing this over t and using Proposition 2.6 yield

where the last inequality is due to √ ax -bx ≤ a 4 b for any a ≥ 0 , b > 0 , and x ≥ 0 . Remarkably, MetaGrad achieves the O ( n α ln T ) regret bound without prior knowledge of α , whereas ONS achieves this regret bound by using the α value. Furthermore, even when some f t are not exp-concave, MetaGrad still enjoys a regret bound of O ( √ T ln ln T ) [56, Corollary 8]. As such, MetaGrad can automatically adapt to the unknown curvature of loss functions (at the cost of the negligible ln ln T factor), which is the key feature of universal online learning methods.

3 An efficient O ( n ln T ) -regret method based on ONS

This section presents an efficient logarithmic-regret method for online inverse linear optimization. Our method is remarkably simple: we apply ONS to exp-concave loss functions defined similarly to the η -experts' losses (4) used in MetaGrad. The proof is very short given the ONS's regret bound in Proposition 2.5. Despite this simplicity, we can achieve the regret bound of O ( n ln T ) , which matches the best-known regret upper bound of Gollapudi et al. [25], with far lower per-round complexity.

Theorem 3.1. Assume that for every t = 1 , . . . , T , action x t ∈ X t is optimal for c ∗ ∈ Θ . Let ˆ c 1 , . . . , ˆ c T ∈ Θ be the outputs of ONS applied to loss functions defined as follows for t = 1 , . . . , T :

where ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ and we set η = 1 5 B . 6 Then, for R c ∗ T and ˜ R c ∗ T in (2) , it holds that

Proof. Consider using Proposition 2.5 in the current setting with W = Θ , w η t = w t = ˆ c t , g t = ˆ x t -x t , u = c ∗ , W = D , G = K , and H = B . Since the optimality of x t and ˆ x t for c ∗ and ˆ c t , respectively, ensures ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ ≥ 0 , we have ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ 2 ≤ B ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ due to Assumption 2.2. Therefore, ˜ R c ∗ T = ∑ T t =1 ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ and V c ∗ T := ∑ T t =1 ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ 2 satisfy V c ∗ T ≤ B ˜ R c ∗ T . By using this and Proposition 2.5 with η = 1 5 B , for some constant C ONS > 0 , it holds that

and rearranging the terms yields ˜ R c ∗ T = O ( Bn ln ( DKT Bn )) . 7 This also applies to R c ∗ T ≤ ˜ R c ∗ T .

Time complexity. We discuss the time complexity of the method. Let τ solve be the time for solving linear optimization to find ˆ x t and τ G-proj the time for the generalized projection onto Θ used in ONS (see Appendix B.1). In each round t , we compute ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ in τ solve time; after that, the ONS update takes O ( n 2 + τ G-proj ) time. Therefore, it runs in O ( τ solve + n 2 + τ G-proj ) time per round, which is independent of T . If problem (1) is an LP, τ solve equals the time for solving the LP (cf. Cohen et al. [16] and Jiang et al. [33]). Also, τ G-proj is often affordable as Θ is usually specified by the learner and hence has a simple structure. For example, if Θ is the unit Euclidean ball, the generalized projection can be computed in O ( n 3 ) time by singular value decomposition (e.g., Mhammedi et al. [41, Section 4.1]). We may also use the quasi-Newton-type method for further efficiency [40].

4 Robustness to suboptimal feedback with MetaGrad

In practice, assuming that the agent's actions are always optimal is unrealistic. This section discusses how to handle suboptimal feedback effectively. Here, we let x t ∈ X t denote an arbitrary action taken by the agent, which the learner observes. Now that x t may have nothing to do with c ∗ , we can no longer ensure meaningful bounds on the regret that compares ˆ x t with optimal actions. For example, if revealed actions x t remain all zeros for t = 1 , . . . , T , we can learn nothing about c ∗ , and hence the regret that compares ˆ x t with optimal actions grows linearly in T in the worst case. Considering this issue, we highlight that the regret, R c ∗ T = ∑ T t =1 ⟨ c ∗ , x t -ˆ x t ⟩ , used here is defined with the agent's possibly suboptimal actions x t , not with those optimal for c ∗ . Small upper bounds on this regret ensure that, if the agent's actions x t are nearly optimal for c ∗ , so are ˆ x t . This regret still satisfies R c ∗ T ≤ ˜ R c ∗ T = ∑ T t =1 ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ since ˆ x t is optimal for ˆ c t . Additionally, recall that the suboptimality loss, ℓ t , in Definition 2.3 can be defined for any action x t ∈ X t and that ℓ t ( c ∗ ) = max x ∈ X t ⟨ c ∗ , x ⟩ - ⟨ c ∗ , x t ⟩ ≥ 0 indicates the suboptimality of x t for c ∗ . Below, we use ∆ T := ∑ T t =1 ℓ t ( c ∗ ) to denote the cumulative suboptimality of the agent's actions x t .

In this setting, it is not difficult to show that ONS used in Theorem 3.1 enjoys a regret bound that scales linearly with ∆ T . However, the linear dependence on ∆ T is not satisfactory, as it results in a regret bound of O ( T ) even for small suboptimality that persists across all rounds. The following theorem ensures that by applying MetaGrad to the suboptimality losses, we can obtain a regret bound that scales with √ ∆ T .

Theorem 4.1. Let ˆ c 1 , . . . , ˆ c T ∈ Θ be the outputs of MetaGrad applied to the suboptimality losses, ℓ 1 , . . . , ℓ T , given in Definition 2.3. Let ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ for t = 1 , . . . , T . Then, it holds that

Proof. Similar to the proof of Theorem 3.1, we apply Proposition 2.6 with W = Θ , w t = ˆ c t , g t = ˆ x t -x t , u = c ∗ , W = D , G = K , and H = B ; in addition, g t = ˆ x t -x t ∈ ∂ℓ t (ˆ c t ) holds due

to Proposition 2.4. Thus, Proposition 2.6 ensures the following bound for some constant C MG > 0 : 8

where ˜ R c ∗ T = ∑ T t =1 ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ and V c ∗ T = ∑ T t =1 ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ 2 . In contrast to the case of Theorem 3.1, ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ 2 ≤ B ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ is not ensured since ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ can be negative due to the suboptimality of x t . Instead, we will show that the following inequality holds:

If ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ ≥ 0 , (7) is immediate from ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ 2 ≤ B ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ and ℓ t ( c ∗ ) ≥ 0 . If ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ < 0 , ⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ 2 ≤ B ( -⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ ) holds. In addition, we have ℓ t ( c ∗ ) = max x ∈ X t ⟨ c ∗ , x ⟩-⟨ c ∗ , x t ⟩ ≥ ⟨ c ∗ , ˆ x t -x t ⟩ ≥ ⟨ c ∗ , ˆ x t -x t ⟩-⟨ ˆ c t , ˆ x t -x t ⟩ = -⟨ ˆ c t -c ∗ , ˆ x t -x t ⟩ , where the second inequality follows from ⟨ ˆ c t , ˆ x t -x t ⟩ ≥ 0 . Multiplying both sides by 2 yields

Thus, (7) holds in any case, and hence V c ∗ T ≤ B ˜ R c ∗ T +2 B ∆ T . Substituting this into (6), we obtain

We assume ˜ R c ∗ T > 0 ; otherwise, the trivial bound of ˜ R c ∗ T ≤ 0 holds. By the subadditivity of x ↦→ √ x for x ≥ 0 , we have ˜ R c ∗ T ≤ √ a ˜ R c ∗ T + b , where a = C 2 MG Bn ln ( DKT Bn ) and b = √ 2 a ∆ T + a C MG . Since x ≤ √ ax + b implies x = 4 3 x -x 3 ≤ 4 3 ( √ ax + b ) -x 3 = -1 3 ( √ x -2 √ a ) 2 + 4 3 ( a + b ) ≤ 4 3 ( a + b ) for any a, b, x ≥ 0 , we obtain ˜ R c ∗ T ≤ 4 3 ( a + b ) = O ( Bn ln ( DKT Bn ) + √ ∆ T Bn ln ( DKT Bn ) ) .

If every x t is optimal, i.e., ∆ T = 0 , the bound recovers that in Theorem 3.1. Note that MetaGrad requires no prior knowledge of ∆ T ; it automatically achieves the bound that scales with √ ∆ T , analogous to the original bound in Proposition 2.6 that scales with √ V u T . Moreover, a refined version of MetaGrad [56] enables us to achieve a similar bound without prior knowledge of K , B , or T (see Appendix B.4). Universal online learning methods shine in such scenarios where adaptivity to unknown quantities is desired. Another noteworthy point is that the last part of the proof uses the self-bounding technique [23, 60, 66]. Specifically, we derived ˜ R c ∗ T ≲ a + b from ˜ R c ∗ T ≤ √ a ˜ R c ∗ T + b , where the latter means that ˜ R c ∗ T is upper bounded by a term of lower order in ˜ R c ∗ T itself, hence the name self-bounding. We expect that the combination of universal online learning methods and self-bounding, through relations like V c ∗ T ≲ ˜ R c ∗ T +∆ T used above, will be a useful technique for deriving meaningful guarantees in online inverse linear optimization.

Time complexity. The use of MetaGrad comes with a slight increase in time complexity. First, as with the case of ONS, ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ is computed in each round, taking τ solve time. Then, each η -expert performs the ONS update, taking O ( n 2 + τ G-proj ) time. Since MetaGrad maintains Θ(ln T ) distinct η values, the total per-round time complexity is O ( τ solve +( n 2 + τ G-proj ) ln T ) . If the O ( τ G-proj ln T ) factor is a bottleneck, we can use more efficient universal algorithms [41, 61] to reduce the number of projections from Θ(ln T ) to 1 . Moreover, the O ( n 2 ) factor can also be reduced by sketching techniques (see Van Erven et al. [56, Section 5]).

4.1 Online-to-batch conversion

We briefly discuss the implication of Theorem 4.1 in the offline setting, where feedback follows some underlying distribution. As noted in Section 2.2, the bound in Theorem 4.1 applies to the regret with respect to the suboptimality loss, ∑ T t =1 ( ℓ t (ˆ c t ) -ℓ t ( c ∗ )) , since it is bounded by ˜ R c ∗ T from above. Therefore, the standard online-to-batch conversion (e.g., Orabona [45, Theorem 3.1]) implies the following convergence of the average prediction in terms of the suboptimality loss.

Corollary 4.2. For any non-empty and compact X ⊆ R n , x ∈ X , and c ∈ Θ , define the corresponding suboptimality loss as ℓ X,x ( c ) := max x ′ ∈ X ⟨ c, x ′ ⟩ - ⟨ c, x ⟩ . Let ∆ > 0 and define X ∆ as the set of observations ( X,x ) with bounded suboptimality, ℓ X,x ( c ∗ ) ≤ ∆ . Assume that { ( X t , x t ) } T t =1 are drawn i.i.d. from some distribution on X ∆ (hence ∆ T ≤ ∆ T ). Let ˆ c 1 , . . . , ˆ c T ∈ Θ be the outputs of MetaGrad applied to the suboptimality losses ℓ t = ℓ X t ,x t for t = 1 , . . . , T . Then, it holds that

Bärmann et al. [4, Theorem 3.14] also obtained a similar offline guarantee via the online-to-batch conversion. Their convergence rate is O ( 1 √ T ) even when ∆ = 0 , whereas our Corollary 4.2 offers the faster rate of O ( ln T T ) if ∆ = 0 . It also applies to the case of ∆ > 0 , which is important in practice because stochastic feedback is rarely optimal at all times. We emphasize that if regret bounds scale linearly with ∆ T , the above online-to-batch conversion cannot ensure that the excess suboptimality loss (the left-hand side) converge to zero as T → 0 . This observation lends support to the importance of the √ ∆ T -dependent regret bound we established in Theorem 4.1.

5 Ω( n ) lower bound

We construct an instance where any online learner incurs an Ω( n ) regret, implying that the O ( n ln T ) upper bound is tight up to an O (ln T ) factor. More strongly, the following Theorem 5.1 shows that, for any B > 0 that gives the tight upper bound in Assumption 2.2, no learner can achieve a regret smaller than Bn 4 , which means that the Bn factor in our Theorem 3.1 is inevitable.

Theorem 5.1. Let n be a positive integer and Θ = [ -1 √ n , + 1 √ n ] n . For any T ≥ n , B > 0 , and the learner's outputs ˆ c 1 , . . . , ˆ c T ∈ Θ , there exist c ∗ ∈ Θ and X 1 , . . . , X T ⊆ R n such that

hold, where R c ∗ T = ∑ T t =1 ⟨ c ∗ , x t -ˆ x t ⟩ , x t ∈ arg max x ∈ X t ⟨ c ∗ , x ⟩ , ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ , and the expectation is taken over the learner's possible randomness.

̸

Proof. We focus on the first n rounds and show that any learner must incur Bn 4 in these rounds; in the remaining rounds, we may use any instance since the optimality of x t for c ∗ ensures ⟨ c ∗ , x t -ˆ x t ⟩ ≥ 0 . For t = 1 , . . . , n , let X t = { x ∈ R n : -B 4 √ n ≤ x ( t ) ≤ B 4 √ n, x ( i ) = 0 for i = t } , where x ( i ) denotes the i th element of x . That is, X t is the line segment on the t th axis from -B 4 √ n to B 4 √ n . Then, max { ⟨ c -c ′ , x -x ′ ⟩ : c, c ′ ∈ Θ , x, x ′ ∈ X t } = B holds for each t = 1 , . . . , n . Let c ∗ ∈ Θ be a random vector such that each entry is -1 √ n or 1 √ n with probability 1 2 , which is drawn independently of any other randomness. Then, the optimal action, x t ∈ X t , which is zero everywhere except that its t th coordinate equals c ∗ ( t ) | c ∗ ( t ) | · B 4 √ n , achieves ⟨ c ∗ , x t ⟩ = B 4 . Note that the learner's t th prediction ˆ c t is independent of c ∗ ( t ) since it depends only on past observations, { ( X i , x i ) } t -1 i =1 , which have no information about c ∗ ( t ) . Thus, ˆ x t ∈ arg max x ∈ X t ⟨ ˆ c t , x ⟩ is also independent of c ∗ ( t ) , and hence

where the expectation is taken over the randomness of c ∗ . This implies that any deterministic learner incurs Bn 4 in the first n rounds in expectation. Thanks to Yao's minimax principle [62], we can conclude that for any randomized learner, there exists c ∗ ∈ Θ such that E [ R c ∗ T ] ≥ Bn 4 holds.

In the above proof, we restricted X 1 , . . . , X T to line segments so that each x t ∈ arg max x ∈ X t ⟨ c ∗ , x ⟩ reveals nothing about c ∗ ( t +1) , . . . , c ∗ ( n ) . Whether a similar lower bound holds when all X t are fulldimensional remains an open question. Another side note is that the Ω( n ) lower bound does not contradict the O ( √ T ) upper bound of Bärmann et al. [4]. Their OGD-based method indeed achieves a regret bound of O ( DK √ T ) , where D and K are upper bounds on the ℓ 2 -diameters of Θ and X t , respectively. In the above proof, T ≥ n , D ≥ 1 , and K ≥ B 2 √ n hold, implying that their regret upper bound is DK √ T ≳ Bn . Hence, the Ω( n ) -lower bound and their O ( DK √ T ) -upper bound are compatible.

6 Conclusion and discussion

We have presented an efficient ONS-based method that achieves an O ( n ln T ) -regret bound for online inverse linear optimization. Then, we have extended the method to deal with suboptimal feedback based on MetaGrad, achieving an O ( n ln T + √ ∆ T n ln T ) -regret bound, where ∆ T is the cumulative suboptimality of the agent's actions. Finally, we have presented a lower bound of Ω( n ) , which shows that the O ( n ln T ) upper bound is tight up to an O (ln T ) factor. Regarding limitations, our work is restricted to the case where the agent's optimization problem is linear, as mentioned in Footnote 1; how to deal with non-linearity is an important direction for future work. In online portfolio selection, ONS is efficient but inferior to the universal portfolio algorithm regarding the dependence on the gradient norm [57]. Exploring possible similar relationships in online inverse linear optimization is left for future work. Last but not least, closing the O (ln T ) gap between the upper and lower bounds is an important open problem. Interestingly, if all X t are line segments as in Section 5 and the learner can observe X t in the beginning of round t , the algorithm of Gollapudi et al. [25, Theorem 5.2] offers a regret upper bound of O ( n 5 log 2 n ) , which is finite and polynomial in n . We also provide an additional discussion on a finite regret bound for the case of n = 2 in Appendix C.

Acknowledgments and Disclosure of Funding

SS was supported by JST ERATO Grant Number JPMJER1903. TT was supported by JST ACT-X Grant Number JPMJAX210E and JSPS KAKENHI Grant Number JP24K23852. HB was supported by JST PRESTO Grant Number JPMJPR24K6. TO was supported by JST FOREST Grant Number JPMJFR232L and JSPS KAKENHI Grant Number JP22K17853.

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NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: See Abstract, Section 1, and Section 1.1.

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A Detailed comparisons with previous results

Below we compare our results with Bärmann et al. [4, 5], Besbes et al. [7, 8], and Gollapudi et al. [25].

Bärmann et al. [4, 5] used ˜ R c ∗ T as the performance measure, as with our Theorems 3.1 and 4.1, and provided two specific methods. The first one, based on the multiplicative weights update (MWU), is tailored for the case where Θ is the probability simplex, i.e., Θ = { c ∈ R n | c ≥ 0 , ∥ c ∥ 1 = 1 } . The authors assumed a bound of K ∞ > 0 on the ℓ ∞ -diameters of X t and obtained a regret bound of O ( K ∞ √ T ln n ) . The second one is based on the online gradient descent (OGD) and applies to general convex sets Θ . The authors assumed that the ℓ 2 -diameters of Θ and X t are bounded by D > 0 and K > 0 , respectively, and obtained a regret bound of O ( DK √ T ) . In the first case, our Theorem 3.1 with B = K ∞ , D = √ 2 , and K ≤ 2 √ nK ∞ offers a bound of O ( K ∞ n ln( T/ √ n )) ; in the second case, we obtain a bound of O ( DKn ln( T/n )) by setting B = DK . In both cases, our bounds improve the dependence on T from √ T to ln T , while scaled up by a factor of n , up to logarithmic terms. Regarding the computation time, their MWU and OGD methods run in O ( τ solve + τ E-proj + n ) time per round, where τ E-proj is the time for the Euclidean projection onto Θ , hence faster than our method. Also, suboptimal feedback is discussed in Bärmann et al. [4, Sections 3.1]. However, their bound does not achieve the logarithmic dependence on T even when ∆ T = 0 , unlike our Theorem 4.1.

Besbes et al. [7, 8] used R c ∗ T as the performance measure, which is upper bounded by ˜ R c ∗ T . They assumed that c ∗ lies in the unit Euclidean sphere and that the ℓ 2 -diameters of X t are at most one. Under these conditions, they obtained the first logarithmic regret bound of O ( n 4 ln T ) . By applying Theorem 3.1 to this case, we obtain a bound of O ( n ln( T/n )) , which is better than their bound by a factor of n 3 . As discussed in Section 1, their method relies on the idea of narrowing down regions represented with O ( T ) constraints, and hence it seems inefficient for large T ; indeed Besbes et al. [8, Theorem 4] only claims that the total time complexity is polynomial in n and T . Considering this, our ONS-based method is arguably much faster while achieving the better regret bound.

On the problem setting of Besbes et al. [7, 8]. As mentioned in Remark 2.1, the problem setting of Besbes et al. [7, 8] is seemingly different from ours. In their setting, in each round t , the learner first observes ( X t , f t ) , where f t : X t → R n is called a context function . Then, the learner chooses ˆ x t ∈ X t and receives an optimal action x t ∈ arg max x ∈ X t ⟨ c ∗ , f t ( x ) ⟩ as feedback. It is assumed that the learner can solve max x ∈ X t ⟨ c, f t ( x ) ⟩ for any c ∈ R n and that all f t are 1 -Lipschitz, i.e., ∥ f t ( x ) -f t ( x ′ ) ∥ 2 ≤ ∥ x -x ′ ∥ 2 for all x, x ′ ∈ X t . We note that our methods work in this setting, while the presence of f t might make their setting appear more general. Specifically, we redefine X t as the image of f t , i.e., { f t ( x ) : x ∈ X t } . Then, their assumption ensures that we can find f t (ˆ x t ) ∈ X t that maximizes X t ∋ ξ ↦→⟨ ˆ c t , ξ ⟩ , and the ℓ 2 -diameter of the newly defined X t is bounded by 1 due to the 1 -Lipschitzness of f t . Therefore, by defining g t = f t (ˆ x t ) -f t ( x t ) and applying it in Theorems 3.1 and 4.1, we recover the bounds therein on ∑ T t =1 ⟨ ˆ c t -c ∗ , f t (ˆ x t ) -f t ( x t ) ⟩ , with D , K , and B being constants. The bounds also apply to the regret, ∑ T t =1 ⟨ c ∗ , f t ( x t ) -f t (ˆ x t ) ⟩ , used in Besbes et al. [7, 8]. Additionally, Besbes et al. [7, 8] consider a (possibly non-convex) initial knowledge set C 0 ⊆ R n that contains c ∗ . We note, however, that they do not care about whether predictions ˆ c t lie in C 0 or not since the regret, their performance measure, does not explicitly involve ˆ c t . Indeed, predictions ˆ c t that appear in their method are chosen from ellipsoidal cones that properly contain C 0 in general. Therefore, our methods carried out on a convex set Θ ⊇ C 0 work similarly in their setting.

Gollapudi et al. [25] studied essentially the same problem as online inverse linear optimization under the name of contextual recommendation (where they and Besbes et al. [7, 8] appear to have been unaware of each other's work). As with Besbes et al. [7, 8], Gollapudi et al. [25] assumed that c ∗ and X 1 , . . . , X T lie in the unit Euclidean ball, denoted by B n . Similar to Besbes et al. [7, 8], their method maintains the region K t , which is the intersection of hyperplanes { c ∈ R d : ⟨ c -ˆ c s , x s -ˆ x s ⟩ ≥ 0 } for s = 1 , . . . , t -1 , and sets ˆ c t to the centroid of K t + 1 T B n , where + is the Minkowski sum. As regards the regret analysis, their key idea is to use the approximate Grünbaum theorem: whenever the learner incurs ⟨ c ∗ , x t -ˆ x t ⟩ ≥ 1 T , Vol ( K t + 1 T B n ) decreases by a constant factor, where Vol denotes the volume. Consequently, Vol ( K 1 + 1 T B n ) / Vol ( K T + 1 T B n ) ≲ T n implies the regret bound of R c ∗ T = ∑ t ⟨ c ∗ , x t -ˆ x t ⟩ = O ( n ln T ) . As such, the per-round complexity of their method also inherently depends on T , and Gollapudi et al. [25, Section 1.2] only claims the total time complexity of poly( n, T ) . In this setting, our ONS-based method achieves a regret bound of O ( n ln( T/n )) and is arguably more efficient since the per-round complexity is independent of T .

Algorithm 1 Online Newton Step

Estimating the per-round complexity of Gollapudi et al. [25]. As described above, the method of Gollapudi et al. [25] requires ˆ x t for each t , and hence the per-round complexity involves τ solve , the time to solve max x ∈ X t ⟨ ˆ c t , x ⟩ . Aside from this, its per-round complexity is dominated by the cost for computing the centroid of K t + 1 T B n , where K t is represented by O ( T ) hyperplanes. It is known that the problem of exactly computing the centroid is #P-hard in general, but we can approximate it via sampling with a membership oracle of K t + 1 T B n . To the best of our knowledge, computing a point that is ε -close to the centroid takes O ( n 4 /ε 2 ) membership queries, up to logarithmic factors [21, Theorem 5.7], and it is natural to set ε = 1 /T to make the approximation error negligible. Thus, it takes O ( n 4 T 2 ) membership queries. Regarding the complexity of the membership oracle, naively checking whether a given point satisfies all the O ( T ) linear constraints of K t takes O ( nT ) time. Handling the Minkowski sum with 1 T B would complicates the procedure, though it can be done in poly( n, T ) time by using, for example, Frank-Wolfe-type algorithms [22, 24, 31, 37]. For now, O ( nT ) would be a reasonable (optimistic) estimate of the complexity of the membership oracle. Consequently, the total per-round complexity of their method is estimated to be O ( τ solve + n 5 T 3 ) (or higher).

B Details of ONS and MetaGrad

We present the details of ONS and MetaGrad. The main purpose of this section is to provide simple descriptions and analyses of those algorithms, thereby assisting readers who are not familiar with them. As in Appendix B.4, we can also derive a regret bound of MetaGrad that yields a similar result to Theorem 4.1 directly from the results of Van Erven et al. [56].

First, we discuss the regret bound of ONS used by η -experts in MetaGrad, proving Proposition 2.5. Then, we establish the regret bound of MetaGrad in Proposition 2.6.

B.1 Regret bound of ONS

Let I n ∈ R n × n denote the identity matrix. For any A,B ∈ R n × n , A ⪰ B means that A -B is positive semidefinite. For positive semidefinite A ∈ R n × n , let ∥ x ∥ A = √ x ⊤ Ax for x ∈ R n . Let W ⊆ R n be a closed convex set. A function q : W → R is α -exp-concave for some α > 0 if W ∋ w ↦→ e -αq ( w ) is concave. For twice differentiable q , this is equivalent to ∇ 2 q ( w ) ⪰ α ∇ q ( w ) ∇ q ( w ) ⊤ . The following regret bound of ONS mostly comes from the standard analysis [26, Section 4.4], and hence readers familiar with it can skip the subsequent proof. The only modification lies in the use of β instead of Wλ (defined below), where β ≤ Wλ always holds and hence slightly tighter. This leads to the multiplicative factor of B , rather than DK , in Theorems 3.1 and 4.1.

Proposition B.1. Let W ⊆ R n be a closed convex set with the ℓ 2 -diameter of at most W > 0 . Assume that q 1 , . . . , q T : W → R are twice differentiable and α -exp-concave for some α > 0 . Additionally, assume that there exist β, λ > 0 such that max w ∈W ∣ ∣ ∇ q t ( w t ) ⊤ ( w -w t ) ∣ ∣ ≤ β and ∥∇ q t ( w t ) ∥ 2 ≤ λ hold. Let w 1 , . . . , w T ∈ W be the outputs of ONS (Algorithm 1). Then, for any u ∈ W , it holds that

where γ = 1 2 min { 1 β , α } is the parameter used in ONS.

Proof. We first give a useful inequality that follows from the α -exp-concavity. By the same analysis as the proof of Hazan [26, Lemma 4.3], for γ ≤ α 2 , we have

Note that we also have ∣ ∣ 2 γ ∇ q t ( w t ) ⊤ ( u -w t ) ∣ ∣ ≤ 2 γβ ≤ 1 . Since ln(1 -x ) ≤ -x -x 2 / 4 holds for x ≥ -1 , applying this with x = 2 γ ∇ q t ( w t ) ⊤ ( u -w t ) yields

We turn to the iterates of ONS. Since w t +1 is the projection of w t -1 γ A -1 t ∇ q t ( w t ) onto W with respect to the norm ∥·∥ A t , we have ∥ w t +1 -u ∥ 2 A t ≤ ∥ ∥ ∥ w t -1 γ A -1 t ∇ q t ( w t ) -u ∥ ∥ ∥ 2 A t for u ∈ W due to the Pythagorean theorem, hence

Rearranging the terms, we obtain

From A t = A t -1 + ∇ q t ( w t ) ∇ q t ( w t ) ⊤ , summing over t and ignoring γ 2 ( w T +1 -u ) ⊤ A T ( w T +1 -u ) ≥ 0 , we obtain

Since we have A 1 -∇ q 1 ( w 1 ) ∇ q 1 ( w 1 ) ⊤ = A 0 = εI n and ε = n W 2 γ 2 , the above inequality implies

The first term in the right-hand side is bounded as follows due to the celebrated elliptical potential lemma (e.g., Hazan [26, proof of Theorem 4.5]):

where we used det A 0 = ε n and det A T = det ( ∑ T t =1 ∇ q t ( w t ) ∇ q t ( w t ) ⊤ + εI n ) ≤ ( Tλ 2 + ε ) n , which follows from the fact that eigenvalues of ∑ T t =1 ∇ q t ( w t ) ∇ q t ( w t ) ⊤ are at most Tλ 2 . Combining (8), (9), and (10), we obtain

as desired.

B.2 Regret bound of η -expert

We now establish the regret bound of ONS in Proposition 2.5, which is used by η -experts in MetaGrad. Let η ∈ ( 0 , 1 5 H ] and consider applying ONS to the following loss functions, which are defined in (4):

As in Proposition 2.5, the ℓ 2 -diameter of W is at most W > 0 , and the following conditions hold:

Therefore, f η t satisfies the conditions in Proposition B.1 with α = 2 (1+2 ηH ) 2 , β = ηH +2 η 2 H 2 , and λ = η (1 + 2 ηH ) G . Since 1 α = 1 2 +2 ηH +2 η 2 H 2 ≥ β holds, we have γ = 1 2 min { 1 β , α } = α 2 . Thus, for any η ∈ ( 0 , 1 5 H ] , we have γ ∈ [ 25 49 , 1 ) ⊆ [ 1 2 , 1 ] and γλ = ηG 1+2 ηH ≤ G 7 H . Consequently, Proposition B.1 implies that for any u ∈ W , the regret of the η -expert's ONS is bounded as follows:

B.3 Regret bound of MetaGrad

We turn to MetaGrad applied to convex loss functions f 1 , . . . , f T : W → R . We here use w t ∈ W and g t ∈ ∂f t ( w t ) to denote the t th output of MetaGrad and a subgradient of f t at w t , respectively, for t = 1 , . . . , T . We assume that these satisfy the conditions in (3), as stated in Proposition 2.6.

Algorithm 2 describes the procedure of MetaGrad. Define η i = 2 -i 5 H for i = 0 , 1 , . . . , ⌈ 1 2 log 2 T ⌉ , called grid points , and let G ⊆ ( 0 , 1 5 H ] denote the set of all grid points. For each η ∈ G , η -expert runs ONS with loss functions f η 1 , . . . , f η T to compute w η 1 , . . . , w η T . In each round t , we obtain w t by

Algorithm 2 MetaGrad

2: for t = 1 , . . . , T :

4: Play w t = ∑ η ∈G ηp t w t ∑ η ∈G ηp η t .

3: Fetch w η t from η -experts for all η ∈ G .

η η

5: Observe g t ∈ ∂f t ( w t ) and send ( w t , g t ) to η -experts for all η ∈ G .

6: p η t +1 ← p η t exp( -f η t ( w η t )) /Z t for all η ∈ G , where Z t = ∑ η ∈G p η t exp( -f η t ( w η t )) .

aggregating the η -experts' outputs w η t based on the exponentially weighted average method (EWA). We set the prior as p η i 1 = C ( i +1)( i +2) for all η i ∈ G , where C = 1 + 1 1+ ⌈ 1 2 log 2 T ⌉ . Then, it is known that for every η ∈ G , the regret of EWA relative to the η -expert's choice w η t is bounded as follows:

where we used C ≥ 1 in the second inequality. We here omit the proof as it is completely the same as that of Van Erven and Koolen [55, Lemma 4] (see also Wang et al. [58, Lemma 1]).

We are ready to prove Proposition 2.6. Let V u T = ∑ T t =1 ⟨ w t -u, g t ⟩ 2 . By using f η t ( w t ) = 0 , (11), and (12), it holds that

for all η ∈ G . For brevity, let

If we knew V u T , we could set η to η ∗ := √ A V u T ≥ 1 5 H √ T to minimize the above regret bound, A η + ηV u T . Actually, we can do almost the same without knowing V u T thanks to the fact that the regret bound holds for all η ∈ G . If η ∗ ≤ 1 5 H , by construction we have a grid point η ∈ G such that η ∗ ∈ [ η 2 , η ] , hence

Otherwise, η ∗ = √ A V u T ≥ 1 5 H holds, which implies V u T ≤ 25 H 2 A . Thus, for η 0 = 1 5 H ∈ G , we have

Therefore, in any case, we have

obtaining the regret bound in Proposition 2.6.

Algorithm 3 O (1) -Regret Algorithm for n = 2 .

1: Set C 1 to S 1 .

2: for t = 1 , . . . , T :

3: Draw ˆ c t uniformly at random from C t .

4: Observe ( X t , x t ) .

5: C t +1 ←C t ∩ N t . ▷ N t is the normal cone.

B.4 Lipschitz adaptivity and anytime guarantee

Recent studies [41, 56] have shown that MetaGrad can be further made Lipschitz adaptive and agnostic to the number of rounds. Specifically, MetaGrad given in Van Erven et al. [56, Algorithms 1 and 2] works without knowing G , H , or T in advance, while using (a guess of) W . By expanding the proofs of Van Erven et al. [56, Theorem 7 and Corollary 8], we can confirm that the refined version of MetaGrad enjoys the following regret bound:

By using this in the proof of Theorem 4.1, we obtain

and the algorithm does not require knowing K , B , T , or ∆ T in advance.

C On removing the ln T factor: the case of n = 2

This section provides an additional discussion on closing the ln T gap in the upper and lower bounds on the regret. Specifically, focusing on the case of n = 2 , we provide a simple algorithm that achieves a regret bound of O (1) in expectation, removing the ln T factor. We also observe that extending the algorithm to general n ≥ 2 might be challenging. Note that Gollapudi et al. [25, Theorem 4.1] has already established a regret bound of exp( O ( n ln n )) as mentioned in Section 1.2, which implies an O (1) -regret bound for n = 2 . The purpose of this section is simply to stimulate discussions on closing the ln T gap by presenting another simple analysis. Below, let B n and S n -1 denote the unit Euclidean ball and sphere in R n , respectively, for any integer n > 1 .

C.1 An O (1) -regret method for n = 2

We focus on the case of n = 2 and present an algorithm that achieves a regret bound of O (1) in expectation. We assume that all x t ∈ X t are optimal for c ∗ for t = 1 , . . . , T . For simplicity, we additionally assume that all X t are contained in 1 2 B 2 and that c ∗ lies in S 1 . For any non-zero vectors c, c ′ ∈ R n , let θ ( c, c ′ ) denote the angle between the two vectors. The following lemma from Besbes et al. [8], which holds for general n ≥ 2 , is useful in the subsequent analysis.

Our algorithm, given in Algorithm 3, is a randomized variant of the one investigated by Besbes et al. [7, 8]. The procedure is intuitive: we maintain a set C t ⊆ S 1 that contains c ∗ , from which we draw ˆ c t uniformly at random, and update C t by excluding the area that is ensured not to contain c ∗ based on the t th feedback ( X t , x t ) . Formally, the last step takes the intersection of C t and the normal cone N t = { c ∈ R n : ⟨ c, x t -x ⟩ ≥ 0 , ∀ x ∈ X t } of X t at x t , which is a convex cone containing c ∗ . Therefore, every C t is a connected arc on S 1 and is non-empty due to c ∗ ∈ C t (see Figure 1).

Theorem C.2. For the above setting of n = 2 , Algorithm 3 achieves E [ R c ∗ T ] ≤ 2 π .

C

t

N

t

C

t

+1

c

∗

c

∗

ˆ

c

t

C

t

ε

Proof. For any connected arc C ⊆ S 1 , let A ( C ) ∈ [0 , 2 π ] denote its central angle, which equals its length. Fix C t . If ˆ c t ∈ C t ∩ int( N t ) , where int( · ) denotes the interior, ˆ x t = x t is the unique optimal solution for ˆ c t , hence ⟨ c ∗ , x t -ˆ x t ⟩ = 0 . Taking the expectation about the randomness of ˆ c t , we have

where we used Pr[ˆ c t ∈ C t \ int( N t )] = Pr[ˆ c t ∈ C t \ N t ] = A ( C t \ N t ) /A ( C t ) (since the boundary of N t has zero measure). If A ( C t ) ≥ π/ 2 , from ⟨ c ∗ , x t -ˆ x t ⟩ ≤ ∥ c ∗ ∥ 2 ∥ x t -ˆ x t ∥ 2 ≤ 1 , we have

If A ( C t ) < π/ 2 , Lemma C.1 and ˆ c t , c ∗ ∈ C t imply ⟨ c ∗ , x t -ˆ x t ⟩ ≤ sin θ ( c ∗ , ˆ c t ) ≤ sin A ( C t ) . Thus, by using 1 x sin x ≤ 1 ( x ∈ R ), we obtain

Therefore, we have E [ ⟨ c ∗ , x t -ˆ x t ⟩ ] ≤ A ( C t \ N t ) in any case. Consequently, we obtain

where the last inequality is due to C t +1 = C t ∩ N t , which implies C s ⊆ C t and C s ∩ ( C t \ N t ) = ∅ for any s > t , and hence no double counting occurs in the above summation.

C.2 Discussion on higher-dimensional cases

Algorithm 3 might appear applicable to general n ≥ 2 by replacing S 1 with S n -1 and defining A ( C t ) as the area of C t ⊆ S n -1 . However, this idea faces a challenge in bounding the regret when extending the above proof to general n ≥ 2 . 9

As suggested in the proof of Theorem C.2, bounding E [ ⟨ c ∗ , x t -ˆ x t ⟩ ] is trickier when A ( C t ) is small (cf. the case of A ( C t ) < π/ 2 ). Luckily, when n = 2 , we can bound it thanks to Lemma C.1 and sin θ ( c ∗ , ˆ c t ) ≤ sin A ( C t ) , where the latter roughly means the angle, θ ( c ∗ , ˆ c t ) , is bounded by the area, A ( C t ) , from above. Importantly, when n = 2 , both the central angle and the area of an arc are identified with the length of the arc, which is the key to establishing sin θ ( c ∗ , ˆ c t ) ≤ sin A ( C t ) . This is no longer true for n ≥ 3 . As in Figure 2, the area, A ( C t ) , can be arbitrarily small even if the angle

within there, or the maximum θ ( c ∗ , ˆ c t ) for c ∗ , ˆ c t ∈ C t , is large. 10 This is why the proof for the case of n = 2 does not directly extend to higher dimensions. We leave closing the O (ln T ) gap for n ≥ 3 as an important open problem for future research.

D Numerical experiments

We conducted numerical experiments to complement our theoretical results. Experiments were conducted on Google Colab equipped with an Intel ® Xeon ® CPU @ 2.20GHz, 12 GB RAM, running Ubuntu 22.04.4 LTS with Python 3.12.11. The code is available at https://github.com/ssakaue/ online-inverse-linear-optimization-code.

We use a setup based on the hard instance considered in our lower bound analysis (Section 5). Let c ∗ be a random vector with ∥ c ∗ ∥ 2 = 1 . The learner's prediction set Θ is the n -dimensional Euclidean unit ball. At each round t , we sample an endpoint v uniformly at random from the unit sphere in R n , and set X t = {-v, + v } . We report results for T = 10 , 000 rounds in dimensions n = 2 , 20 , and 200 . To mitigate randomness, we repeat each experiment 10 times independently and report mean and standard deviation.

Compared methods. We compare the following three methods:

ONS : our proposed method based on ONS,

OGD : the OGD-based method of Bärmann et al. [4],

CP : a cutting-plane style method inspired by Gollapudi et al. [25].

For CP, computing the centroid of the feasible region is #P-hard. Therefore, we adopt a randomized heuristic: we pre-sample n × 10 4 candidate points from the unit ball, eliminate those violating accumulated cuts, and approximate the centroid by averaging the remaining points.

Results. Table 2 summarizes the cumulative regret and runtime over T = 10 , 000 rounds (mean ± standard deviation across 10 trials).

n

= 20

52

07

= 2

±

2

.

±

±

Method

ONS

OGD

CP

7

8

2

.

.

.

n

81

46

77

4

0

.

31

33

.

55

.

±

0

.

80

±

43

0

.

50

.

68

829

.

91

±

287

.

00

n

= 200

46

.

19

67

515

.

.

±

0

.

88

41

50

±

±

1

.

36

43

.

75

When the dimension is small ( n = 2 ), CP achieves the lowest cumulative regret. However, its cumulative regret deteriorates significantly for n = 20 and n = 200 . This degradation stems from the limited number of pre-sampled candidate points: in principle, about T n samples would be required for an accurate approximation, which is infeasible even for moderate n . Moreover, although the above randomized centroid approximation substantially reduces computation by averaging over the

surviving candidates, it remains less scalable than OGD and ONS. These results highlight practical limitations of CP in moderate to high-dimensional settings.

In contrast, ONS consistently achieves low regret values across all dimensions while remaining computationally feasible. It outperforms OGD in terms of the regret and scales reasonably well with increasing n . Note that our ONS implementation uses a straightforward projection subroutine that repeatedly solves similar linear systems-an overhead that could be reduced by more sophisticated implementation techniques. Further speedups could also be achieved via quasi-Newton-type updates or sketching-based techniques, as discussed in Sections 3 and 4.

Taken together, these findings affirm that ONS provides a strong and scalable alternative to existing methods in online inverse linear optimization, especially when CP is computationally infeasible and OGD's regret performance is unsatisfactory.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of online inverse linear optimization. In this framework, at each round, the learner receives a feasible optimization set along with the optimal result of an oracle call (referred to as the agent). The learner aims to recover the underlying linear parameter from these observations by minimizing cumulative regret.\n\nPrevious work has shown that a black-box reduction to online convex optimization can be applied, yielding regret upper bounds of order $O(\\sqrt{T})$ or $O(n \\log T)$. However, the logarithmic regret rate was only achieved by methods with a per-round time complexity of $\\Omega(T^3)$, which is polynomial and thus impractical for large $T$.\n\nIn this work, the authors present a simple reduction that allows the use of Online Newton Step (ONS) with a loss function inspired by that of MetaGrad. This approach achieves a regret of order $O(n \\log T)$ while retaining the per-round computational complexity of ONS---i.e., constant per round, with the main computational bottleneck being the generalized projection step.\n\nMoreover, by applying MetaGrad instead of ONS---which automatically tunes the parameters of ONS---they show that their method remains robust even when the agent is suboptimal. Specifically, their regret scales as $O(\\sqrt{\\Delta_T})$ rather than linearly, where $\\Delta_T$ denotes the cumulative suboptimality of the agent.\n\nFinally, the authors provide an $\\Omega(n)$ lower bound (on a specific instance), demonstrating the optimality of their method up to a logarithmic factor in $T$.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper is well-written, and the analysis is, as the authors claim, relatively simple. It mainly involves defining an appropriate loss function and applying existing results from ONS and MetaGrad. This simplicity can be viewed as a strength (a straightforward method with easy implementation that significantly improves upon existing results) or as a weakness (the lack of novel technical contributions).\n\n\nWeaknesses:\n\nThe paper lacks numerical experiments. It would have been valuable to demonstrate, in practice, the benefits in terms of regret or computational cost using synthetic experiments. In particular, what is the effective cost of the generalized projection step in ONS when solving the optimization problem to sufficient accuracy for the regret guarantees? (It would be helpful to recall this in the paper.) If both algorithms are given similar time budgets (e.g., by controlling the precision with which the optimization problems are solved), would it be possible to compare with the regret bound of [25]?\n\nThere remains a gap between the upper and lower bounds: $\\Omega(n)$ versus $O(n \\log T)$. Moreover, constant regret might be achievable in general, although it is likely to be challenging. It would also be worthwhile to include in the comparison table the constant regret bound (albeit exponential in $n$) from [25]. Is it possible that an exponential dependence on $n$ is unavoidable for achieving constant regret?\n\nQUESTIONS:\n1) For the online-to-batch conversion, is it possible to obtain the result with high probability?\n\n2) Since MetaGrad simultaneously achieves both $DK\\sqrt{T}$ and $n \\log T$ regret bounds, it would be valuable to state this explicitly in your results. The $DK\\sqrt{T}$ bound may be preferable in regimes where $n$ is large.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-25"}

comments: SUMMARY:
This paper studies the problem of online inverse linear optimization. By applying the Online Newton Step (ONS) algorithm, the authors achieve logarithmic regret in time ($\log T$) with time independent per round complexity. The setting is further extended to account for suboptimal actions, and corresponding lower bounds are established.

STRENGTHS AND WEAKNESSES:
Strengths:  

- Per round complexity of different algorithms are compared and time independent per round complexity has been achieved. 

- Lower bound of regret in terms of dimensional dependence has been proved. 

Weaknesses: 

- It seems that there is a tradeoff between per round complexity and regret bound in Table 1. In the case of high dimensions ONS can be costly which can also be observed in $n^2$ term in per round complexity. 

- The reported overall complexity can be somewhat misleading. For example, $\tau_{proj}$ itself can depend on $n$, in which case the overall complexity may be well more expensive that just $n^2$.  

- While the paper motivates the results from a practical perspective, there is no numerical experiment to demonstrate the effectiveness of the proposed methods over state-of-the-art.

QUESTIONS:
Is the suboptimal feedback setting considered in the paper related in any way to the framework of online optimization with hints? If so, it would be helpful to clarify the connection and highlight any similarities or differences.

LIMITATIONS:
Adequately discussed.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper aims to solve the problem of online inverse linear optimization, where a learner tries to deduce a hidden linear objective function of an agent by observing the agent's chosen optimal actions over time.\nThe quality of the learner's predictions is measured by regret, which is the cumulative gap between optimal objective values and those achieved by following the learner's predictions.\nIn the literature, Besbes et al. 2021 and 2025 propose an algorithm that achieves a logarithmic regret bound, while it is not computationally efficient since the number of constraints grows as fast as a polynomial of time length $T$. The authors propose an efficient algorithm that achieves logarithmic regret $O(n \\ln(T)$ and has a per-round complexity independent of $T$.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper presents the first logarithmic-regret method whose per-round computational complexity is independent of the time horizon $T$. This is a significant advancement, as previous logarithmic-regret methods were highly inefficient for large time steps.\nAlso, the proposed method achieves the best-known logarithmic regret bound of $O(n\\ln (T))$\n\n2. The method is described as \"strikingly simple\" as it applies the Online Newton Step (ONS) to appropriate exp-concave loss functions. This simplicity is considered a strength, making it more accessible and easier to implement.\n\n3. Corollary 4.2 provides an online-to-batch conversion for offline settings. The comparison with [Bärmann et al., 2020] on offline guarantees notes that their prior work is $O(1/T)$ even when $\\Delta=0$, whereas this paper offers a faster $O(\\ln(T)/T)$ rate, highlighting an improvement.\n\nWeakness:\n\n1. The primary limitation explicitly stated by the authors is that their work is restricted to the case where the agent's underlying optimization problem is linear. Extending this to nonlinear settings is identified as an important direction for future work.\n\n2. While the lower bound of $\\Omega(n)$ and upper bound of $O(n\\ln(T))$ are established, there remains an $O(\\ln(T))$ gap between them that the authors identify as an important open problem.\n\nQUESTIONS:\n1. Line 290 - 296, you state that MetaGrad's per-round complexity is $O(\\tau_{\\text{solve}} + (n^2 + \\tau_{G-\\text{proj}}) \\ln(T)$ and that \"If the $\\tau_{G-\\text{proj}} \\ln(T)$ factor is a bottleneck, we can use more efficient universal algorithms [Mhammedi et al., 2019, Yang et al., 2024] to reduce the number of projections from $\\Theta(lnT)$ to $1$.\"  Could you briefly explain how these more efficient universal algorithms (e.g., [Mhammedi et al., 2019, Yang et al., 2024]) reduce the number of projections to 1 in the context of MetaGrad? And what is the factor or concern that prevents you from choosing those more advanced algorithms in the first place?\n\n2. Line 338 - 339, you mention, \"Whether a similar lower bound holds when all $X_t$ are full-dimensional remains an open question.\" Can you briefly speculate on what specific challenges arise in constructing a similar lower bound proof for the full-dimensional case compared to the line segments used in Theorem 5.1?\n\n3. Line 715 - 727, in Section C.2, you discuss the challenge of extending the $O(1)$ regret bound for $n=2$ to $n \\geq 3$ due to the area $A(C_t)$ becoming arbitrarily small while $\\theta(c^\\star, \\hat{c})$ does not. Could you elaborate more on the most promising direction you believe to close the gap for $n \\geq 3$?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis papers studies the problem online inverse linear optimization, proposing an algorithm with an improved regret bound of $O(n \\ln T)$  with a simple and relatively efficient algorithm. The main idea is the use of the Online Newton Step with a surrogate function proposed by the authors of the algorithm MetaGrad. Furthermore, the authors show that when the learner observes a suboptimal action instead of an optimal one as in the case of the original problem, using MetaGrad with $\\Theta(\\log T)$ experts leads to a similar regret bound with an extra penalty of $O(\\sqrt{\\Delta_T n \\ln T})$, where $\\Delta_T$ is the culmulative gap of the optimal payoffs and the payoffs of the observed actions. Finally, the authors also show a $\\Theta(n)$ lower bound on the regret of any (possibly randomized) algorithm for the problem, and show how one can get $O(1)$ regret for $n = 2$.\n\nSTRENGTHS AND WEAKNESSES:\nThe algorithm proposed to the problem is surprisingly simple and yields strong results for the problem. It is quite interesting to see such a simple algorithm given that the previous algorithms for the same (or similar problem) use more intricate algorithms based on cutting plane methods. Ultimately the regret bounds follow from results from MetaGrad (as the authors note, even Thm 3.1 is MetaGrad with only one expert), but this should not be seen as a negative point for this paper, since this application was not obvious at all in hindsight and the authors do shave off some constants in their analysis. The lower bound is also simple, so I feel the results overall are quite elegant.\n\nOne problem (although relatively minor) the paper has is with notation. It is quite heavy at some moments, and it can get quite confusing at some points. I know the MetaGrad paper and know that a lot of this notation is from the original MetaGrad paper, but this is not a good excuse to use the same notation in the main paper. \n\nAnother minor problem with presentation is that it was not clear to me when first reading the paper whether the constants the big-Oh notation was hiding in sec 1 could be hiding a dependency on $n$ (such as the B or D constants later on). Appendix A (which is quite well-written) clarifies this a lot, but it would be nice to say, for example, that the rates in the table in Sec 1 are for all sets being constrained to the unit ball for the sake of uniformity.\n\nQUESTIONS:\nHere are a few parts where notation is heavy and maybe a bit confusing:\n-  In Prop 2.5, do we need $w_i \\in \\mathcal{W}$? Or is it ok for them to be arbitrary vectors? Also, having $w_t$ as arbitrary vectors while $w_t^{\\eta}$ are iterates was quite confusing. It makes sense for MetaGrad, but at this point it was quite confusing;\n- For Theorem 4.1 we need the subgradients used by MetaGrad to be the specific gradients given by Prop 2.4, right? I don't think this is specified anywhere\nOther minor comments:\n- lines 49-50: \"(...) about $c^*$ defining the regret.\" not clear what you meant here\n- line 138: \"Alternately\" here is to mean that first the player places the prediction and only afterwards receives feedback, if I understood it correctly, but I don't think this is the correct way to say that.\n- Assumption 2.2: \"$\\ell_2$-diameter\" is slightly ambiguous since we don't know if you mean squared norm or not. If you have the space, I would add the mathematical formula to clear any possible confusions;\n\nLIMITATIONS:\nThe author's do a good job on placing their contributions in the literature and discussing limitations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of online inverse linear optimization, where the learner is given a sequence of feasible sets $X_t \\subseteq \\mathbb{R}^n$ and corresponding optimal decisions $x_t$, and seeks to recover the unknown linear cost vector $c^*$. The authors propose a learning algorithm based on the Online Newton Step (ONS) method applied to a specially constructed loss function. Assuming this loss is exp-concave, the algorithm achieves a regret bound of $O(n \\log T)$, with per-round computational complexity independent of the time horizon $T$. The algorithm is also extended to handle suboptimal actions using a MetaGrad-style ensemble. A lower bound of $\\Omega(n)$ is provided, establishing the optimality of the regret's dimensional dependence up to logarithmic factors.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n1. The paper studies a well-motivated and practically relevant problem in online learning and inverse optimization.\n2. The paper includes a matching lower bound, which strengthens the theoretical framing.\n\n### Weaknesses\n\n1. The central theoretical guarantee relies entirely on the assumption that the proposed loss function is exp-concave. However, the paper does not prove this property or provide sufficient conditions under which it holds. Since exp-concavity is a strong assumption that does not hold generically, the regret bound is not fully justified.\n2. Once exp-concavity is assumed, the regret analysis is a standard application of the ONS algorithm. The work does not introduce new techniques or insights beyond this reuse.\n3. The paper includes no experiments to demonstrate the method’s practical efficiency or its advantage over existing baselines such as constraint-accumulation methods. This significantly limits its potential impact.\n4. The regret bound hides constants involving quantities such as $B, D, K$, which are not analyzed or estimated. This limits the practical interpretability of the results.\n5. Although the paper focuses on linear objectives, the framing and claims suggest broader applicability. In reality, the approach is tailored to a narrow setting, and the key assumptions do not generalize easily.\n\nQUESTIONS:\n1. Can you provide a formal proof or structural conditions under which the proposed loss function is exp-concave? Without this, the regret bound in Theorem 3.1 is not well grounded.\n2. Have you considered implementing your algorithm and comparing its runtime and regret behavior to prior methods (e.g., Besbes et al., which accumulate constraints over time)? Even synthetic experiments could strengthen the paper.\n3. The regret bounds include constants tied to parameters like $B, D, K$. Can you give concrete estimates or examples that help interpret these constants in typical inverse optimization settings?\n4. This work is clearly focused on linear objectives. Section 6 mentions the potential for kernelization. Can your self-bounding analysis extend to non-linear objectives or convex settings? Is exp-concavity still plausible in those cases?\n5. The $\\log T$ gap between upper and lower bounds remains open. Is this tight, or can the upper bound be improved with further algorithmic refinements?\n\nLIMITATIONS:\nWhile the paper discusses the restriction to linear objectives and the unresolved $\\log T$ gap, it does not sufficiently highlight the reliance on the unproven exp-concavity assumption, which is critical to the validity of the main result. This should be addressed explicitly.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}