Scalable Kernel Inverse Optimization
Authors: Youyuan Long
Tolga Ok
Pedro Zattoni Scroccaro
Peyman Mohajerin Esfahani
Venue: NeurIPS 2024 poster (2024)
Keywords: Optimization
Imitation Learning
Inverse Optimization
Primary area: optimization
TL;DR: Proposing a kernel-based Inverse Optimization model with a scalable algorithm designed for imitation learning tasks.

Abstract: Inverse Optimization (IO) is a framework for learning the unknown objective function of an expert decision-maker from a past dataset.
In this paper, we extend the hypothesis class of IO objective functions to a reproducing kernel Hilbert space (RKHS), thereby enhancing feature representation to an infinite-dimensional space.
We demonstrate that a variant of the representer theorem holds for a specific training loss, allowing the reformulation of the problem as a finite-dimensional convex optimization program.
To address scalability issues commonly associated with kernel methods, we propose the Sequential Selection Optimization (SSO) algorithm to efficiently train the proposed Kernel Inverse Optimization (KIO) model.
Finally, we validate the generalization capabilities of the proposed KIO model and the effectiveness of the SSO algorithm through learning-from-demonstration tasks on the MuJoCo benchmark.

tasks on Scalable Kernel Inverse Optimization
Youyuan Long
Tolga Ok
Delft Center for Systems Control
Delft Center for Systems and Control Delft University of Technology
Delft University of Technology
The Netherlands Longyouyuann422gmal.ccom
The Netherlands
T. Www.kkkeett..... .nl
Pedro Zattoni Scroccaro
Peyman Mohajerin Esfahani
Delft Center for Systems and Control
Delft Center for Systems and Control
Delft University of Technology
Delft University of Technology
The Netherlands
The Netherlands
P. /attooniScrocaarotuudl...tt
P. MohajeeiinEsfaaami studelft.
Abstract
Inverse Optimization (IO) is a framework for learning the unknown objective function of expert decision-man makke from a past dataset. In this paper, extend the hypothesis class of IO objective functions to a reproducing kernel Hilbert space (RKHS), thereby enhancing feature representation an infinite-dimensional space. We demonstrate that a variant of the representer theorem holds for specific training loss, allowing the reformulation of the problem a finiteddimensional convex optimization program. To address scalability issues commonly associated with kernel methods, we propose the Sequential Selection Optimization (SSO) algorithm to efficiently train the proposed Kernel Inverse Optimization (KIO) model. Finally, we validate the generalization capabilities of the proposed KIO model and the effectiveness of the SSO algorithm through the MuJoCo benchmark.
Introduction
Inverse Optimization (IO) is distinct from traditional optimization problems, where we typically seek the optimal decision variables by optimizing objective function over set of constraints. In contrast, inverse optimization works "in reverse" by inferring the optimization objective given the optimal solution. The inherent assumption in IO is that agent generates its decision by solving an optimization problem. The assumed optimization problem called the Forward Optimization Problem (FOP), which parametric in the exogenous signal & with the corresponding optimal solution û. Therefore, 10 aims to deduce the objective function of the FOP from a dataset of exogenous signal and decision pairs, {(S,, û,) ) In this work, we assume the constraints are known priori. Consequently, we can leverage the FOP derived from the expert's 's dataset by solving it to mimic the expert's behavior when encountering new exogenous signals. 10 has garnered widespread attention within several fields, giving rise to numerous studies encompassing both theoretical and applied research. Application domains include vehicle routing [11, 31, 41], transportation system modeling [29, 8], portfolio optimization [25, 39, 23], power systems [9, 16, 32]. electric vehicle charging problems [17], network design [15], healthcare problems [5], as well as controller design [2, 14]. For more detailed discussion on different applications of 1O, we refer the readers to the recent survey paper [10].
10 can be categorized into classic 10 and data-driven I0. In classic I0, only a single signal- -eeas--eeision pair considered, where the decision assumed be the optimal solution of the FOP (i.e., there
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

no noise), and different classes of FOPs have been studied, such as linear conic problems [1, 36, 20]. However, in real-world applications, there are usually many observations of signal-decisinn pairs, and due to the presence of noise, it is usually unreasonable to assume that all observed decisions are optimal w.r.t single, true data-generating FOP. Additionally, for complex tasks, the chosen FOP may only approximate the task, not allowing for perfect replication of the observed behavior from the expert. These cases are referred to as data-driven IO problems. In such scenarios, a loss function is usually used to compute the discrepancy between observed data and the decision generated by the learned FOP. Examples of loss functions include the 2-norm distance loss [4], suboptimality loss [25], variational inequality loss [8], KKT loss [21]. and augmented suboptimality loss [40].
In data-driven IO, the objective function of the FOP is typically non-linear with respect to an exogenous signal 8. Hence, classical methods that learn an FOP based on linear function classes may oversimplify the problem and lead to suboptimal solutions. One effective approach for addressing the expressibility issue in data-driven IO problems is the introduction of kernel methods. These methods have been extensively studied within the context of IO [35, 8] and have shown promising results for scaling IO to address practical problems. The application of kernel methods in IO allows for the exploration of a broader class of optimization problems, thereby enhancing the model's ability to generalize from observed decisions to unseen situations. Specifically, using kernelized approach facilitates the embedding of decision data into a richer feature space, enabling the deduction of an FOP that not only fits the training data but also exhibits strong generalization capabilities.
Contributions. We list the contributions of this work as follows:
(1) Kernelized 10 Formulation: We propose novel Kernel Inverse Optimization (KIO) model based on suboptimality loss [25]. The proposed approach leverages kernel methods to enable IO models to operate outperform existing imitation learning (IL) algorithms on complex continuous control tasks in low-data regimes.
feature spaces, which allows KIO to
(2) Sequential Selection Optimization Algorithm:
To address the quadratic computational complexity of the proposed KIO model, we introduce the Sequential Selection Optimization (SSO) algorithm inspired by coordinate descent style updates. This algorithm selectively optimizes components of the decision variable, greatly enhancing efficiency and scalability while provably converging to the same solution of our proposed KIO model.
(3) Open Source Code: source implementation of the proposed KIO model and the SSO algorithm, along with the source code of the experiments in Github'.
To foster reproducibility and further research, we provide an open-
IR denotes the space of n-dimensional non-negative vectors. The identity square matrix with dimension n is denoted by I,, For a symmetric matrix Q. the inequality Q 2 0 (respectively,
Notation
Q > 0) means that Q is positive semi remi--efinite (respectively, positive definite). The trace of matrix Q is denoted as Tr(Q). Given a vector x E R", we use the shorthand notation || == x Qr. Symmetric block matrices are described by the upper diagonal elements while the lower diagonal
elements are replaced by "*" The Frobenius norm of matrix Q is denoted as ||Q|| The notation Qij represents the element in the i-th row and j-th column of the matrix Q The Fuclidean inner product of and y is denoted x y.
2 Preliminaries
2.1 Inverse Optimization
In general, to solve data-driven IO problem, we need to design two components: the Forward Opti- mization Problem (FOP) and the loss function. Specifically, the FOP corresponds to the optimization problem we aim to "fit" to the observed dataset D { { where each input-output pair (&,, û.) E x IR.. In this paper, we use the "hat" notation (e.g.. 8) to denote objects that depend on the dataset. Our goal is to find a parameter vector 0 O such that
FOP(e, $) min Fo(5, u), wEU(3)
(1)

---

replicates the data as closely as possible by minimizing a loss function, akin to classical empirical risk minimization problems. In this work, we focus on lifting the learning problem based on quadratic FOPs. These FOPs include linear constraints and continuous decision variables, as defined by
(2) where 0 : (Owm: Oen), Oun [R Onn E Rmxx, M()) Rmxn, ((s) E RT and á S R. is the feature function that maps & to higher-dimensional feature space to enhance the model's capacity. To simplify notation, we omit the explicit dependence of M and W on 8, denoting them as M and W, respectively.
Fo(8, := u Ou 2x)(8) Oxu and U(s) = {u E R" M(a)u S W()))
To learn 0, we solve regularized loss minimization problem using the Suboptimality Loss [25]
min KR(O) AE
+
Σ max (Fo(Se, ū.)) Fo u.)): EEE(()))
N
where θ = (0= (9 um: 0..) P I, R(0) 10 1/++ 11...11 1 and & is positive regulariza- tion parameter. The constraint On. = I prevents the trivial solution Oun = 0.n = and guarantees that the resulting FOP is convex optimization problem. Moreover, since Fo is linear in 0 the optimization program (3) is convex w.r.t. 9, and it can be reformulated from the "minimax" form single minimization problem. This reformulation based on dualizing the inner maximization problems of (3) and combining the resulting minimization problems.
Proposition (LMI reformulation [2]). For the hypothesis function and feasible set in (2), the optimization program (3) is equivalent to
N
min 8,4..
KR(O) + N
Σ
Po(Si, (+) + 7 +W )x)
s.t.
0 = (Oue Osu), Ovn 2 In. X T24, 76 ER
NSAA N
Ouu
MT A + 28 ($:)
20
N.S.N
%
2.2 The Kernel Method
The kernel method is powerful technique used in machine learning and statistics that exploits the structure of data embedded in higher-dimensional space. The kernel method has found numerous applications, including Support Vector Machines [12], Kernel Principal Component Analysis [33], and Kernel Linear Discriminant Analysis [6]. The fundamental idea behind the kernel method is to implicitly map input data into a higher- mation, thus enabling algorithms dinhesroimnsii space without explicitly computing the transfor- capture complex patterns and non-lineet relationships without heavy computational burden [35]. The kernel method generalizes the hypothesis of the optimization problem to a nonlinear function class based on a Reproducing Kernel Hilbert Space (RKHS) H. In this work, we primarily focus on lifting the original parametric optimization problem of the
form min (((fo(8)) 8,, (.)) ; |[]]rr) lee
where fo S - TR. such that fo... (s) T u == o(s) Osnn,
following Definition (2). The lifted problem is then defined as min ((((())) 8, a)) 1111lx))
JEH
with f in an RKHS H. By lifting the function class to H. we effectively optimize over nonlinear hypotheses.
In a vector-valued RKHS H. equipped with the inner product (,, HH as defined in [24], given a proper kernel function K S x S -> " that is symmetric and positive definite, Moore-Aree Aronszajn's reproducing kernels theory implies that there exists a unique RKHS with the reproducing properties induced by matrix-valued K, such that Vff E H fs) u (f, K(, s)u)H with linear operator K(+&)) R" + H. Furthermore, the Riesz representation theorem states that for every & E S and H for all f E H. Recall that the lifted optimization
R there exists unique function K(- problem is formed over all nonlinear hypotheses via f E H. However, as a result of the reproducing property [34], we can write the lifted optimization problem in the adfnite-dimensionall inner product space. Hence, the resulting optimization problem has the form
min& JEH
((, w: 81, a:) )
1-1 111fll1)
(5)
3

---

In what follows, we show that the solution of the optimization problem (5) exists and is finite when the problem is built over finite dataset DD of size N. We leverage the Representer theorem [35], which for an arbitrary loss function, as (5), states that the solution admits the kernel representation of the form
over (s) = Σ ($;; s)a; with a, E R" This result effectively suggests that optimizing RKHS has sparse solution over the linear hypotheses.
Kernel Inverse Optimization
In this section, we extend the Inverse Optimization (IO) model proposed by [2] to incorporate kernel methods. We consider hypothesis class of the form in Equation (1). However, kernelizing such hypotheses is not straightforward. Instead, we argue that by kernelizing the optimization problem associated with the loss function described in Equation (3), we can obtain a forward optimization problem (FOP) that minimizes the kernelized objective F(s, u)-
In Theorem 1, we dualize the problem in (3) and show that the optimal solution for 0% when plugged into Bw 6), admits an affine function of N coefficients w.r.t 8. The resulting FOP, obtained by Theorem 1, takes form consistent with the representer theorem when the loss function and kernel are defined appropriately, as discussed later in this section.
Theorem (Kernel reformulation). The Lagrangian dual of the optimization program (4) is
min P.A.D.
Σ
Aj
- P
- Tr(P)
4k
N
F
R5 83)
2T;
2r,
N
N
(6)
s.t.
= 0, A, 2 0, T R.
Vi S N
MM 00
14 S N
N
A
r
1 W N,
0
IN
where n(;;, 8,) = ((a) 6(3,) is the scaler-valued kernel function. The primal variables Bwn and Osu can be recovered using
A. 1) and Oan = Σ (b3))
N
1 Onn 2k
21
(7)
N
N
Proof. See Appendix A.I.
Notice that the complexity of the optimization program (6) does not depend on the dimensionality of the feature vector d(::). Consequently, this allows us to kernels generated from infinite- dimensional feature spaces, e.g., the Gaussian (a.k.a. radial basis function) kernel k(,,, 8,) = exp(->/l6; - ,)))). Program (6) a convex optimization problem and can be solved using off-the- shelf solvers, such MOSEK [3]. Once solved, we can recover the optimal primal variables 0* and 8r, from the optimal dual variables A; I.; and p* using (7). Notice that the dimensionality of 0r., depends é, meaning it can be anfinite-dimensionn
matrix. However, ultimate goal is to learn an FOP that replicates the behavior observed in the data. By combining (7) with (1), we have that, for the signal (Snew, Mnew; W,ew), the resulting FOP
min Man Woo
p
Σ
Q.
i N
A; ) u Σ 2=1
2 82)
21
4g N
(8)
2h
which again does not depend on the dimensionality of Φ. but only on the kernel function However, a key difference between solving the 10 problem using the kernel reformulation of Theorem I and other classical 10 approaches (c.g.. [25, 40]) is that the resulting FOP (8) explicitly depends on the

---

entire training dataset D. These models are sometimes called nonparametric models [26], indicating that the number of parameters of the model (in our case, P At, and I for all i S N) depends on the size of the training dataset.
Remark (A potential variant of representer theorem). In the primal problem (4), a regularized empirical risk loss is optimized over a set of constraints. In the learned objective function of the FOP (8), the term related to the features 8; (the linear coefficient of the optimizer u) can represented as a finite linear combination of kernel products evaluated on the input points in the training set data, i.e., -=+)==E= K(β,, -)x; with E R". A similar forward optimization problem can be obtained using a loss function defined in a vector-valued RKHS H with a corresponding kernel function K for functions f S R", as discussed in Section 2.2, via
N
(x D) k|ll Ix +
Σ
(F(Ss, Uug Oun f) - F(5s, u.; Ow: f)};
max mEU((A)
where F(s, (e,uu:0u, u f)= u u- + ((, s)u) and K(s, E Rd R" is set to be diagonal matrix, with diagonal entries corresponding to the same scalar kernel K, such that: K K(s,s") s')s k(s,8') for j (I........ Based on the representer theorem, the solution the optimization
problem min (uff D) admits the form (*) = Ei K(s, (s), = Σ k(ss, 3:) This FEH
result indicates that the learned FOP (8) exhibits characteristics consistent with the representer theorem, implying a potential variant the context of inverse optimization.
In the class of cost functions studied in this paper (2), Oun can be interpreted as a matrix that penalizes the components of the decision vector u. However, many problems, it is assumed that the expert generating the data equally penalizes each dimension of u, or equivalently, uses auu I,. This assumption holds, for instance, the Gymnasium MuJoCo environments [38], where the reward settings for all tasks apply the same penalty to each dimension of the decision vector. Intuitively, this means that the expert trained under such reward settings aims to reduce the magnitude of the decision vector uniformly across all dimensions, rather than favoring any specific dimension. Therefore, assuming a. = In as prior knowledge can reduce model complexity and lead to faster training, without degrading model performance.
Corollary 1 (Kernel reformulation for onn I,). The Lagrangian dual of the optimization program (4) with am = I, is
④4
T
01
min AgPa ViES
x(8 85)
21:;
21'
Σ
Tr(A,)
N
j=1
s.t.
A 2 0, r2 E RD
EES
(9)
Wi
TM 2
WES
N
[A r
0
Vie S.
IN
((:)) ())()))(())))) ((83) is the kernel function and S {1,..., N). The primal variable 0... can be recovered using (7).
where k(B;, 8j)
The proof of Corollary is the same as the proof of Theorem under the assumption that 6... = In- and is therefore omitted here. represents an index set, where all decision variables whose indices belong to S will be optimized, while decision variables whose indices do not belong to S retain their original values and are treated as constants. Corollary 1, all variables will be optimized, so S is the set comprising natural numbers from 1 to N. The concept of the index set S' is introduced to make Problem (9) compatible with the sub-optimization problems based on the coordinate descent method outlined in Section 4.
In the following section, we will focus on algorithms to solve Problem (9). However, all ideas also apply to the general problem (6).
5

---

Sequential Selection Optimization
Solving the kernel IO problem (9) involves optimizing a Semidefinite Program (SDP), which can become prohibitively costly if the number of semidefinite constraints and optimization variables grows too large. In our case, the size of the SDP grows quadratically with the size of the training dataset N.. For instance, in our experiments, solving (9) with N 20000 using CVXPY [13] requires up to 256GB of memory. Therefore, in this section, we propose a coordinate descent-type algorithm find approximate solution (9) by iteratively optimizing only a subset of the coordinates each iteration, keeping all other coordinates fixed. We define a pair of variables A₃ and T2 as the i-th coordinate, denoted as {A,, In Problem (9), each coordinate decoupled in the constraints. which enables the use of the coordinate descent framework here. We call this method Sequential Selection Optimization (SSO), and present Algorithm 1.
Algorithm 1 Sequential Selection Optimization (SSO)
1: Initialize 1.
2: for tort= 1,..., T do
3: Select a batch of p coordinates S {a:]<= where at E (1,...,N}
4: Update (Aa, Ia, 1 based on (9) with S
5:
end for
Here, we explain each step of Algorithm 1: (i) Initialization of the optimization variables. In general, the variables are usually initialized randomly or set to 1. These methods are simple but may not provide a good initial guess. (ii) Selection of a batch of p coordinates. The most straightforward approach to selecting p coordinates is to choose them eyclically. Alternatively, we can select coordinates at random at each iteration (not necessarily with equal probability). Lastly, we can choose coordinates greedily, selecting the components corresponding to the greatest descent or those with the largest gradient subgradient the current iteration [37]. (iii) Solving the KIO subproblem to update the selected coordinates. The mathematical expression of the subproblem is Problem (9) with S. where S' is set containing the indices of the coordinates that need to be updated. Note that the coordinates whose indices & S remain fixed. Therefore, the number of quadratic terms in (9) scales with | rather than N
Next, we propose two heuristics to accelerate the convergence speed of the SSO algorithm: a heuristic method for choosing which coordinates (line 3 of Algorithm 1) to optimize, and a warm-up trick to improve the initialization of the optimization variables (line of Algorithm 1).
4.1 Heuristic for choosing coordinates
At each iteration of Algorithm 1, intuitively, the largest improvement will be made by updating the "least optimal" set of p variables. One way evaluate their degree of suboptimality is to choose the variables with the most significant violation of the Karush- Kuhn-Tucker (KKT) conditions of the primal version of Program (9) (i.e., (4) with an I,), inspired by the Sequential Minimal Optimization (SMO) method from [30].
Proposition 2. For optimal decision variables of Problem (9), the coordinate {Ai, L'j that satisfies
W N
2M,1, > 0,
(10)
should also satisfy
A
T3
T
20 ($:)) ||28 (3x) 113
1) = 0.
Tr
(11)
aN
Proof. See Appendix A.2.
Proposition is based on KKT conditions. condition
Based on Condition (11), we can define the KKT violation
[
P
In
su ($,)
kkt. ekt_violator (i) = Tr
(12)
|
( (1)
IN

---

Using the violation condition (12), we can establish the following heuristic method to construct the set S in Algorithm I: given the current values of [A,, [.] at iteration we choose the p coordinates that satisfy Condition (10) with the maximum KKT violation (12). In practice, we additionally select some random coordinates to update at each iteration, ensuring that coordinates have the chance to be updated, including those that initially do not meet the criteria specified in Condition (10) and would otherwise never be updated. This random selection is inspired by the proven convergence of coordinate descent algorithms with uniformly random and cyclic coordinate selection [27, 7].
4.2 Warm-up trick for improved initialization
Another component of A Algorithm that may have a significant practical impact is how the optimiza- tion variables (A,, F are initialized. A poor initial guess (c.g., A, r = 0) can lead to slow solver convergence or even result in numerical instability. Here, we propose simple warm-up trick that leads to a better initialization of the optimization variables. First, we divide the original dataset D into n non-overlapping sub-datasets DI,..., D, and solve the n small problems (9) for each of these sub-datasets (N; [P:] and S is the set of indices of all the data in D:). We then concatenate the optimal solutions of these n solved small problems to form an initial guess. Even when the original problem (9) is intractable due to large training dataset (i.e., large N), each subproblem remains tractable for a small enough batch size N,, and its solutions are still feasible with respect to (9).
5
Numerical Experiments
5.1 Performance Evaluation
In this evaluation, KIO is implemented in its simplified version (9), incorporating a Gaussian kernel, and tested on continuous control datasets from the D4RL benchmark [18]. The model is trained using the SSO Algorithm 1. In each task, the model's performance is assessed over 100 test episodes, and the score* for KIO is the average score across these 100 episodes. The parentheses following KIO scores indicate the amount of data used.
For comparison, four additional agents are selected for this experiment. 10 is the inverse optimization model without the kernel method, introduced in Proposition To illustrate the effect of the kernel method, both the KIO and 10 models are trained on identical datasets. The scores of two behavior cloning agents, BC(TD3+BC) [19] and BC(CQL) [22], are taken from two offline reinforcement learning algorithms in which the entire dataset of 1 million samples was used for training. These papers implemented their respective behavior cloning agents using D4RL datasets, serving as base- lines to compare against their proposed offline reinforcement learning algorithms. In these studies, BC(TD3+BC) and BC(CQL) were evaluated over 10 seeds and seeds, respectively. The Teacher is the agent responsible for generating the dataset and serves as the target for imitation learning in this experiment.
Table 1: Performance of KIO, 10, two Behavior Cloning (BC) agents, and the Teacher agent on MuJoCo tasks from the D4RL benchmark on the normalized return metric. The numbers in parentheses represent the amount of data used by KIO and I0, and the score for KIO in each task is the average score over 100 episodes.
Task
KIO
10
BC[TD3+BOJ19] BC(CQL)
Teacher
Hopper-expert
109.9 (5k)
31.8
111.5
109.0
108.5
Hopper-medium
50.2 (5k)
20.6
30.0
29.0
47.2
Walker2d-expent
108.5 (10k)
0.9
56.0
125.7
107.1
Walker2d-medium
74.6 (5k)
0.0
11.4
6.6
68.1
Halfcheetab-expee exppet
84.4 (10k)
-1.7
105.2
107.0
88.1
Halfcheetah-meen mediuu
39.0 (5k)
-3.1
36.6
36.1
40.7
Evaluation for KIO. Table 1 presents the final experimental results, where KIO achieves com- petitive scores in four out of six tasks. In these tasks, except for a slightly lower score in the Halfchettah-expert task compared the teacher agent, KIO's scores are either close or exceed those of the teacher agent, indicating strong learning capabilities in complex control tasks. However,
'Reaadinng the definition of the score for one episode, we refer readers to the official documentation of Gymnasium [38] and the D4RL paper [18].
7

---

without the kernel method, the I0 model demonstrates weak learning capabilities, achieving low scores in the Hopper task and failing learn in the other two more challenging tasks. We argue that the weak performance of the IO model is due to the limitations of its hypothesis class, which lacks the richness needed to learn an effective policy for imitation learning tasks. This limitation arises from its reliance on predefined feature spaces, which may fail to capture the complexities of more sophisticated environments. All hyperparameters used in this experiment for KIO are listed in Appendix B.
Task
SCS
SSO
Hopper-expert Wolloe2
Hopper-medium Hallchestah-modlum
Obj Value
Score
Obj Value
Score
Hopper-expert
185.219
109.9
185.220
110.2
10
Hartchscaa-oooett
Hopper medium
218.761
50.2
218.761
51.8
2alker22-exxpett
140. 12.1
108.5
140.121
109.2
10
Walker2d medium
medium
151.117
74.6
151.117
74.9
Hallchettsh-epert
165.041
84.4
165.041
83.8
Halfchoctal--mcdimm
188. 184
39.0
188.184
39.7
10-1
Error
Table 2: Final Objective Function Value and Score (average return over 100 evaluations) for SCS [28] and SSO (20 iterations for all tasks) algorithms. The ultimate Objective Function Values of the two algo- rithms are nearly identical, yet across the majority of tasks, SSO achieves a slightly higher score compared to SCS.
10-2
10-3
10-4
10-5
Iteration
Figure 1: Convergence curves for SSO.
Evaluation for SSO. Table 2 presents the optimal objective function values and task scores obtained by the centralized algorithm, Splitting Conic Solver (SCS) [28], and the distributed algorithm, Sequential Selection Optimization (SSO), for solving Problem (9). In this experiment, SCS is employed to directly address the large-scale problem (9) with |S| = N. The corresponding solution is evaluated 100 times, and the average score is taken as the final result. Mcanwhile, SSO addresses the large-scale problem by solving a series of subproblems. After each iteration, the current solution is evaluated 100 times, and the average score is recorded. After 20 iterations, there are 20 corresponding scores, and we select the highest score along with the objective function value from the last iteration and report it. The results show that SSO and SCS yield nearly identical optimal objective function values. However, except for the Halfcheetah meeiiammmmmmmmmmmmmm task, SSO achieved higher scores across all other tasks. Figure 1 displays the convergence performance of the SSO algorithm across six distinct tasks, with the horizontal axis representing the number of iterations and the vertical axis representing the error between the current objective function value and the optimal objective function value (calculated by SCS) in Problem (9). The SSO algorithm demonstrates fast convergence rate. By the 10th iteration, the errors for all tasks are below 1. and by the 20th iteration, the errors have further diminished to approximately le-4 for all tasks.
In the previous evaluation of the SSO algorithm, we limited the maximum training data to 10k to ensure that we could directly solve Problem (9) without SSO. Thereby, we were able to compare the results with and without the SSO algorithm. However, to further verify the effectiveness of the SSO algorithm, we tested it on a new task (the medium-expert dataset) using 100k data points. At this scale, due to memory limitations, we were unable to solve Problem (9) directly without the SSO algorithm, thus the effectiveness of the SSO algorithm is inferred solely from its experimental performance. Table 3 presents the results of the KIO model optimized by the SSO algorithm. The results indicate that the KIO model achieves competitive results and scales effectively to larger data sizes. We list the hyperparameters used in this experiment in Appendix B.
5.2
Ablation Studies
We perform ablation studies to understand the contribution of each individual component: Heuristic Coordinates Selection (Section 4. and Warm-Up Trick (Section 4.2). Our results, presented in Figure 2, compare the performance of SSO with and without each component (all model hyperparameters remain unchanged as shown in Appendix B).
We use the Hopper task as the testing task, with the first 5000 data points from the D4RL Hopper-expert dataset as training data. The block coordinate for each iteration consists of 2500 coordinates ([S) 2500). When applying the Warm-Up Trick, we partition the data
8

---

Table Performance of KIO, two Behavior Cloning (BC) agents, and the Teacher agent on MuJoCo tasks from the D4RL benchmark on the normalized return metric. The numbers in parentheses represent the amount of data used by KIO, and the score for KIO each task is the average score over 100 episodes.
Task
KIO
BC(TD3+BC)
BC(CQL)
Teacher
Hopper-mediumg meddrr--mmiuu-eppertt
79.6 (100k)
89.6
111.9
64.8
100.1 (100k)
12.0
11.3
82.7
Halficherah-meddum-experr
46.4 (100k)
67.6
35.8
64.4
into two equal parts and solve two subproblems (9), each with |S| the computational time required for the Warm-Up Trick is approximately equal to the time needed for two iterations of the SSO algorithm.
2500. Therefore,
In Figure 2, we present the results of 20 iter-
-i- Activate Heuratic Naive
ations, with the vertical axis representing the
-q- HouHistiiic
Wormp
-4-- Houristic +W mlp
difference between the current objective func-
10
tion value and the optimal value in Problem (9).
10a
Houristic Adlivated
10a
Both the Heuristic Coordinates Selection, ab- breviated as Heuristic in Figure 2, and Warm- Up Trick significantly accelerate the algorithm. With the Warm-Up Trick, the initial objective function value is markedly reduced. Without the Warm-Up Trick, the Heuristic curve requires approximately 10 iterations to reach the initial values of the SSO curve, whereas the Warm- Up Trick requires only about the time of two iterations. The Heuristic Coordinates Selection results in rapid descent of the error curves. The WarmUp curve, however, becomes nearly flat af- ter a few iterations, until the 17th iteration when the Heuristic Coordinates Selection method is activated, causing rapid decrease in the curve.
10a
10
10
10-2
10-
10-3
10
10-5
10
15
20
Iteration
Figure 2: Convergence curves on the MuJoCo Hop- per task with the first 5k data points from the D4RL Hopper-expert dataset. The vertical axis represents the difference between the current objective func- tion value and the optimal value. Sequential Se- lection Optimization (orange) exhibits the fastest convergence rate.
Conclusion and Limitations
We introduced Kernel Inverse Optimization (KIO), an inverse optimization model leveraging kernel methods, along with its simplified variant and the theoretical derivations. Subsequently, we proposed the Sequential Selection Optimization (SSO) algorithm for training the KIO model, which addresses memory issues by decomposing the original problem into a series of subproblems. Our empirical results demonstrate that KIO exhibits strong learning capabilities in complex control tasks, while the SSO algorithm achieves rapid convergence to the optimal solution within a limited number of iterations.
One of the limitations of this model is the computational cost of adding a new data point. In that case, all training data are required compute the coefficients for the FOP problem (see FOP (8)) for that point. Thus, as the amount of training data grows, so does the computational cost. Another limitation is the absence theoretical analysis on the convergence rate of the SSO algorithm, which we leave for future research. Furthermore, in our numerical experiments in Section 5, the proposed KIO model, even with the SSO algorithm, required substantial memory resources up to 256 GB when using 100k data points. However, we believe that further optimization in implementation could reduce these memory requirements. Finally, our numerical experiments in Section 5, we observed that initialization strategies critically impact the performance of the SSO algorithm. Thus, exploring alternative initialization strategies beyond the one proposed in Section 4.2 presents a promising direction for future work.

---

Acknowledgments
This work is partially supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (TRUST:949796).
References
[1] Ravindra K. Ahuja and James B Orlin. Inverse optimization. Operations research, 49(5):771- 783, 2001.
[2] Syed Adnan Akhtar, Arman Sharifi Kolarijani, and Peyman Mohajerin Esfahani. Learning for control: An inverse optimization approach. IEEE Control Systems Letters, 6:187-192, 2021.
[3] Mosek ApS. Mosek optimization toolbox for matlab. User's Guide and Reference Manual, Version, 4:1, 2019.
[4] Anil Aswani, Zuo-Jun Shen, and Auyon Siddiq. Inverse optimization with noisy data. Opera- tions Research, 66(3):870 892, 2018.
[5] Turgay Ayer. Inverse optimization for assessing emerging technologies in breast cancer screen- ing. Annals of operations research, 230:57-85, 2015.
[6] Gaston Baudat and Fatiha Anouar. Generalized discriminant analysis using a kernel approach. Neural computation, 12(10):2385-240: 2000.
[7] Dimitri Bertsekas. Convex optimization algorithms. Athena Scientific, 2015.
[8] Dimitris Bertsimas, Vishal Gupta, and Ioannis Ch Paschalidis. Data-driven estimation in equilibrium using inverse optimization. Mathematical Programming, 153:595-66 633, 2015.
[9] John R Birge, Ali Hortaçsu, and Michael Pavlin. Inverse optimization for the recovery of market structure from market outcomes: application the miso electricity market. Operations Research, 65(4):837-855, 2017.
[10] Timothy CY Chan, Rafid Mahmood, and Ian Yihang Zhu. Inverse optimization: Theory and applications. Operations Research, 2023.
[11] Lu Chen, Yuyi Chen, and André Langevin. An inverse optimization approach for capacitated vehicle routing problem. European Journal of Operational Research, 295(3): 1087-1098, 2021.
[12] Corinna Cortes and Vladimir Vapnik. Support-vector networks. Machine learning, 20:273-297, 1995.
[13] Steven Diamond and Stephen Boyd. CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Rexearch, 17(83):1-5, 2016.
[14] Ioannis Dimanidis, Tolga Ok, and Peyman Mohajerin Esfahani. Offline reinforcement learning via inverse optimization. preprint available at https: htttps:///ooaaa/e http///maaaerinesaaaaataa www.aaai..n......aaa..aaa. io/
/abliccatiioos//arraa//////// Inw Inv.InI..p..... .df, 2024.
[15] András Faragó, Áron Szentesi, and Balázs Szviatovszki. Inverse optimization in high-speed networks. Discrete Applied Mathematics, 129(1):83-98, 2003.
[16] Ricardo Fernández- Blancoo Juan Miguel Morales, and Salvador Pineda. Forecasting the price-response of a pool of buildings via homothetic inverse optimization. Applied Energy, 290:1 16791, 2021.
[17] Ricardo Fernndez--Blamd Juan Miguel Morales, Salvador Pineda, and Álvaro Porras. Inverse optimization with kernel regression: Application to the power forecasting and bidding of fleet of electric vehicles. Computers & Operations Research, 134:105405, 2021.
[18] Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine. D4rl: Datasets for deep data-driven reinforcement learning, 2020.
10

---

[19] Scott Fujimoto and Shixiang Shane Gu. A minimalist approach to offline reinforcement learning. Advances neural information processing systems, 01220132-20443 2021.
[20] Garud Iyengar and Wanmo Kang. Inverse conic programming with applications. Operations Research Letters, 33(3):319-330 2005.
[21] Arezou Keshavarz, Yang Wang, and Stephen Boyd. Imputing a convex objective function. In
2011 IEEE international symposium intelligent control, pages 613-619. IEEE, 2011.
[22] Aviral Kumar, Aurick Zhou, George Tucker, and Sergey Levine. Conservative q-learning for offline reinforcement learning. Advances in Neural Information Processing Systems, 33:1179- 1191, 2020.
[23] Jonathan Yu-Meng Li. Inverse optimization of convex risk functions. Management Science, 67(11):713-7141, 2021.
[24] Charles Micchelli and Massimiliano Pontil. On learning vector-valued functions. Neural computation, 17(1):177-204, 2005.
[25] Peyman Mohajerin Esfahani and Daniel Kuhn. Data-driven distributionally robust optimization using the wasserstein metric: performance guarantees and tractable reformulations. Mathemati- cal Programming. 171(1-2):115-666 2018.
[26] Kevin P. Murphy. Probabilistic machine learning: an introduction. MIT press, 2022.
[27] Yu Nesterov. Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM Journal on Optimization, 22(2):341-305 2012.
[28] Brendan O'Donoghue, Eric Chu, Neal Parikh, and Stephen Boyd. Conic optimization via operator splitting and homogeneous self-dual embedding. Journal of Optimization Theory and Applications, 169(3):1022-168 June 2016.
[29] Michael Patriksson. The traffic assignment problem: models and methods. Courier Dover Publications, 2015.
[30] John Platt. Sequential minimal optimization: A fast algorithm for training support vector machines. Technical report, Microsoft, 1998.
[31] Mikael Rönnqvist, Gunnar Svenson, Patrik Flisberg, and Lars-Erik Jönsson. Calibrated route finder: Improving the safety, environmental consciousness, and cost effectiveness of truck routing in sweden. Interfaces, 47(5):372-395, 2017.
[32] Javier Saez-Gallego, Juan M Morales, Marco Zugno, and Henrik Madsen. A data-driven bidding model for a cluster of price-responsive consumers of electricity. IEEE Transactions on Power Systems, (1))50001150111 2016.
[33] Bernhard Schölkopf, Alexander Smola, and Klaus Klaue-Robert Müller. Kernel principal component analysis. In International conference on artificial neural networks, pages 583-588. Springer, 1997.
[34] Bernhard Scholkopf and Alexander J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA, USA, 2001.
[35] Soroosh Shafieezadeh-Abadee Daniel Kuhn, and Peyman Mohajerin Esfahani. Regularization via mass transportation. Journal of Machine Learning Research, 20(103): 2019.
[36] Zahed Shahmoradi and Taewoo Lee. Quantile inverse optimization: Improving stability in inverse linear programming. Operations research, 70(4))2538-2562, 2022.
[37] Hao-Jun Michael Shi, Shenyinying Tu, Yangyang Xu, and Wotao Yin. A primer coordinate descent algorithms. arXiv preprint arXiv: 2016.
[38] Mark Towers, Jordan K. Terry, Ariel Kwiatkowski, John U. Balis, Gianluca de Cola, Tristan Deleu, Manuel Goulão, Andreas Kallinteris, Arjun KG, Markus Krimmel, Rodrigo Perez- Vicente, Andrea Pierré, Sander Schulhoff, Jun Jet Tai, Andrew Tan Jin Shen, and Omar G. Younis. Gymnasium, March 2023.

---

[39] Shi Yu, Haoran Wang, and Chaosheng Dong, Learning risk preferences from investment portfo- lios using inverse optimization. Research International Business and Finance, 64:101879, 2023.
[40] Pedro Zattoni Scroccaro, Bilge Atasoy, and Peyman Mohajerin Esfahani. Learning in inverse optimization: Incenter cost, augmented suboptimality loss, and algorithms. Operations Research, 2024.
[41] Pedro Zattoni Scroccaro, Piet Beek, Peyman Mohajerin Esfahani, and Bilge Atasoy. Inverse optimization for routing problems. Transportation Science, 2024.
12

---

A Technical Proofs
A.1 Proof of Theorem
Ti ai
First, let P, X and Oun
be the Lagrange multiplier associated with the constraints Oun ? I.
M
20 ($.))
A E R4 and
h 0 respectively, where P, E Ruxx P, R.,
A Rd and 7/s E IR. Then define the Lagrangian function
N
L(Owu: Oen, Ao no Nr, Ar, Ts, a;) =k| Pp + k110su IP.. Tr(P(0wu 4n)
Σ
X x
N
Σ
U Qua + 22((8)) Tsu + 7/ + W,T.N. 1,
N
1m1
N
[A
r;]
Own
Mi A + 20 (00b((8)) 20000(6))
Tr
ai
%
1=1
(13)
and the Lagrange dual problem
max P.A.AATTT
inf
L(Oua: Peus dos Ve, P, 7á, 1,,
Oun 24.77
s.t.
p 0, X E R4. ViSN N
(14)
A
r
ž ViS
When the Lagrangian function (13) the point of the infimum with respect to Own Oen Ais 78. we have
aL
A1 A:))
80mm
2k0
un
Σ
+
Σ
As = Oun
p
i=1
f=1
0L
2k0 su +2
Σ
N
(b:)
0
2T
=0=
= = 00 Σ
N
(b$))
1
2r
00 sm
N
h
N
1=1
i=1
0L
W.
1, 2M,,, = = \; We
2M
N
N
0L 1
4N =0
0%
4N
Finally, substituting the expressions for Owns Owus Ao and a, into the Lagrange dual problem (14) and simplifying completes the proof.
A.2 Proof of Proposition?
Note that Problem (9) is the dual problem of Problem (4) with 6un In Here, enumerate the five KKT conditions that will be employed
N
a. 2M,,, Vi < N,
Omn =
((3))
21
(stationarity) (15a)
N
1=1
Wi
A,
(stationarity) (15b)
X A; =0 0, Vi < N,
(complementary slackness) (15c)
Mr
Pi 4N
In
MT
X
28 ($a)
Tr
= 0, Vi < N,
(complementary slackness) (15d)
7
X, E Rd. 14 < N.
(primal feasibility) (15e)
13

---

First, we choose coordinate i that satisfy Condition (10). Then based on KKT condition (15b), we have
A > 0.
(16)
Next, based on conditions in (15c) and (15c), can obtain
A 0.
(17)
Substituting the result (17) into Condition (15d) yields
A Tr
I3 IN
In 20
(0u = 0.
(18)
*
7
% is the decision variable of Problem (4) with Oun. = 4n Next, let's solve for its expression. By utilizing the Schur complement, can prove the following constraints equivalent
In M, X, 29.. ((())
= =0#
M X + 20 ((8:)
*
Therefore, Problem (4) with Onn In can be equivalently expressed as
min We s.t.
kll0.u su | + E/=
N
(2aTer ((5)) 1 W,T
A E R4, 74 E TR,
Vi
V N
(19)
Y M
20 <
Vi
Here, the variable "/ is highlighted. It can be easily proven that when 7 attain its optimal values, the equality the last constraint should hold: 7s the expression of is
M, A + 20 (8:)) Note that X (17), then
= 20 (3:
(20)
Substituting (20) into (18), one can obtain Condition (11).
B Hyperparameters of KIO
Table 4 provides the hyperparameters used the experiments in Section
Table 4: KIO Environment Specific Parameters. is the size the dataset, is the number of coordinates that updated in one iteration of SSO, k is the regularization coefficient, and scalar is the multiplier used in the implementation to avoid numerical instabilities.
Environment
Dataset
k
scalar
Hopper-expert
0:5k
le-6
200N
0.5N
Hopper-mediun
0:5k
1e-6
200N
0.5N
Hopper-mediung mepuu-
First 50k + Last 50k
le-6
200N
O.I.N
walker2d-expert
0:5k+10K 10k:
le-5
50N
0.5N
malder2d-medium
15k:20k
le-5
50N
0.5
walker2d-medium-on
mexeet expert Halfcheetaahexpper
First 50k + Last 50k
le-5
50N
O.LN
325k:330k+345k:3500
5e-6
50.V
0.5.N
Halfchectah-medheee mediiu
10k: 15k
5e-6
50.V
O.5.N
Halfcheetah
Halfchheeahhmmeduum-essyy medium expert
First 50k + Last 50k
1e-6
50N
0.IN
14

---

C Ablation Study on Kernel Function
Table 5: Performance of KIO on MuJoCo tasks from the D4RL benchmark on the normalized return metric. The scores in each task represent the average score over 100 episodes within the range of one standard deviation.
Task
RBF
Laplace
Linear
Hopper-medium
51 + 6.4
41 + 5.1
24 = 0.05
Hopper-expert
109.9 0.4
+ 26.7
28 0.5
Walker2d-medium
72 1
43 + 28.2
-0.19 = 0.006
Walker2d-expert
109.1 0.3
-0.02 0.1
halfcheetah-mediine
32.4 12.2
52 + 10.5
-0.8 = 0.6
Halfcheetah-expert
78.8 24.9
59.1 35.5
2.0 + 3.1
In addition to the studies presented in Section 5.2, we conducted experiments to evaluate our proposed model using different kernels commonly employed in machine learning: Gaussian (RBF), Laplacian, and linear kernels. The results for the RBF kernel Table 5 differ slightly from those Table 3 due to using different dataset partition for normalizing the states.
15

---

NeurIPS Paper Checklist
The checklist is designed encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
You should answer [Yes], [No], or [NA].
[NA] means either that the question is Not Applicable for that particular paper the relevant information is Not Available.
Please provide short (1-2 sentence) justification right after your answer (even for NA).
The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include (after eventual revisions) with the final version your paper, and its final version will published with the paper.
The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes] is generally preferable to "[No]" it is perfectly acceptable to answer "[No] provided proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset used"). In general, answering "[No] or "[NA] is not grounds for rejection. While the questions are phrased in binary way, we acknowledge that the true answer is often more nuanced, please just use your best judgment and write justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to question, in the justification please point to the section(s) where related material for the question can be found.
IMPORTANT, please:
Delete this instruction block, but keep the section heading "NeurIPS paper checklist",
Keep the checklist subsection headings, questions/answers and guidelines below.
Do not modify the questions and only use the provided macros for your answers.
1. Claims
Question:
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
The claims regarding extending the existing Inverse Optimization (IO) frame- work to kernel-methods is provided in Theorem I and in Equation (8). The claims regarding the effectiveness of the proposed Sequential Selection Optimization (SSO) and the general- ization capabilities of Kernel Inverse Optimization (KIO) justified in Section 5.
Justification:
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
16

---

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize other settings.
It is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
See the second paragraph of Section 6.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point any strong assumptions and how robust the results are violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic upproximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g. if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed penalize honesty concerning limitations.
the paper. The authors should use their best
3. Theory Assumptions and Proofs
For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Question:
Answer: [Yes]
The proof Theorem and Proposition are provided in Appendix A. and A.2 respectively. The proof of Corollary is the same the proof of Theorem with the assumption that Own = In, and therefore omitted.
Justification:
Guidelines:
17

---

The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification: We provided all hyperparameters for model training along with the original source code in Python.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution a novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how replicate the results, access to hosted model (e.g., in the of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend the nature of the contribution. For example
(a) If the contribution is primarily new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) If the contribution is primarily the architecture clearly and fully.
model architecture, the paper should describe
(c) If the contribution is new model (c.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
18

---

(d)
We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Question:
Answer: [Yes]
Justification: We provided the original source code along with documentation explaining how use the code.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.. http:///uuieeeeeeeeeuaiaiiiiioie
for more details.
While we encourage the release of code and data, we understand that this might be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https: ctáps.cc/publee httpl://ggg.dddeddddddduiiiieeee
/at////odeSSeeeeeeeeeeeeee ssi Policy) more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental Setting/Detalls
Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Question:
Answer: [Yes]
Justification: In Section 5, we provide the experimental details, while all hyperparameters are provided in Appendix B.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core the paper to level of detail that is necessary to appreciate the results and make sense of them.
19

---

The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment Statistical Significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [No]
Justification: The proposed algorithm and the evaluations in MuJoCo are all deterministic, hence, we did not provide error bars regarding the experimental results.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).
error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
We run our experiments on a machine with 8 CPU cores. However, the main computational bottleneck the memory, which is stated the conclusion section 6.
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, cloud provider, including relevant memory and storage.
20

---

The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't 't make it into the paper).
9. Code Of Ethics
Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics https: http://rrraa.cccpplll///tteeeeee
Question:
Answer: [Yes]
Justification:
Guidelines:
The answer NA means that the authors have not reviewed the NeurlPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. there special consid- eration due to laws or regulations in their jurisdiction).
10. Broader Impacts
Question:
Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (c.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks.
21

---

mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g.. pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: Our paper poses such risks and we do not release any model or dataset.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
If assets released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com datasets has curated licenses for some datasets. Their licensing guide can help determine the license dataset.
For existing datasets that re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information is not available online, the authors encouraged to reach out to the asset's ' creators.
22

---

13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/moodl as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification:
The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
23

---

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
24


Reviews:
[{"review_id": "kQs36lPH9I", "reviewer": "Reviewer_9Go2", "summary": "This paper concerns the topic of Inverse Optimization (IO) which deals with learning the objective function of the decision maker given previous data from an expert decision maker. By kernelizing the objective function, they extend the framework to features from a potentially infinite-dimensional space. The main contributions of this paper is a scalable SDP formulation of the kernel inverse optimization problem and a Sequential Selection Optimization algorithm that works by selecting a batch of variables at random or according to a KKT-based heuristic to optimizer over. The effectiveness of the SSO algorithm is shown using experimentally using imitation learning tasks from the MuJoCo benchmark.", "questions": "1. In Section 4.2, how does one concatenate the optimal solutions of the n solved small problems to form an initial guess? Have you evaluated different possible ways to form an initial guess? \n2. Have you considered evaluating your SSO algorithm on different types of kernel functions? \n3. For Table 1, you report only the amount of data the KIO+SSO method used but you didn't report how much data the BC(TD3+BC) and BC(CQL) offline reinforcement learning algorithms used. Is there a reason for this?\n4. For Table 1, why are the scores for IO so uniformly low compared to KIO? Is it possible that IO and KIO may perform similarly in simpler imitation learning tasks?\n5. For Table 2, it is quite interesting to see that SCS and SSO converged to more or less the same objective value and it would be interesting to see how many iterations (and time) it took for SCS to converge vs. how many iterations for SSO to converge. This could be better represented in a plot. The convergence could be bad in the beginning due to low quality of the initial guess and then improve as the algorithm converges. Its fairly obvious that the scores will be the same range for all trials if the objective value is the same, so it doesn't help to report objective value and score side-by-side.\n6. Have you considered reporting the speed of convergence for KIO + SSO method vs. BC(TD3+BC) and BC(CQL) offline reinforcement learning algorithms in terms of a plot of objective value vs. # of iterations or objective value vs. time?", "limitations": "The authors acknowledge the limitations of their work, which are (1) lack of convergence guarantees for Sequential Selection Optimization (SSO) algorithm for the given different variable selection strategies and (2) the large memory requirements.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The mains strength of this paper is the presentation of a scalable SDP formulation of the kernel optimization problem and a thorough experimental evaluation across a variety of MuJoCo benchmark tasks which includes an ablation study to evaluate different combinations of variable selection and warm-up strategies.", "weaknesses": "The main weakness is the lack of convergence guarantees for Sequential Selection Optimization (SSO) algorithm for the given different variable selection strategies. However, this is not a major drawback as the main contribution is a scalable SDP formulation for the kernel inverse optimization problem. The authors used Gaussian kernel for all experiments but it would be good to have more discussion about how type of kernel influences the performance of the SSO algorithm or quality of the solutions. There is also a lack of detail about how the baseline methods BC(TD3+BC) and BC(CQL) and Teacher were evaluated (e.g. hyperparameters, amount of data used and number of iterations to converge). There is a lack of convergence plots for SSO and other baselines in Table 1 and 2.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the comments and questions. Please find our response below.\n\n**In Section 4.2, how does one concatenate the optimal solutions of the n solved small problems to form an initial guess? Have you evaluated different possible ways to form an initial guess?**\n\t\nIn our approach, each subproblem independently solves the optimization problem (Problem 10 in the paper) using non-overlapping subsets of the dataset, $\\mathcal{D}_j \\subset \\mathcal{D}$. This process yields learned parameter values for $\\Lambda_i$ and $\\Gamma_i$, each associated with a data point in $\\mathcal{D}_j$. We then aggregate these parameter values to construct an initialization for the problem encompassing the complete dataset, $\\mathcal{D}$.\n\t\nWe appreciate your highlighting the significance of initialization, as our experiments confirmed that it critically impacts the performance of the SSO algorithm. In the ablation studies Section 5.2, we also experimented with zero initialization for the parameters, referred in Figure 2 as \"Heuristic\" without WarmUp. Compared to the initialization strategy proposed in Section 4.2 (that is, WarmUp), this method often resulted in slower convergence. Investigating different initialization strategies presents a promising direction for future work.\n\n**Have you considered evaluating your SSO algorithm on different types of kernel functions?**\n\t\nPlease refer to the global rebuttal for our extended analysis of different kernel functions.\n \n**For Table 1, you report only the amount of data the KIO+SSO method used but you didn't report how much data the BC(TD3+BC) and BC(CQL) offline reinforcement learning algorithms used. Is there a reason for this?**\n\n We agree with the suggestion of specifying the dataset size used by each algorithm in table 1. For clarity, we provide the information regarding the dataset sizes to the global rebuttal.\n \n**For Table 1, why are the scores for IO so uniformly low compared to KIO? Is it possible that IO and KIO may perform similarly in simpler imitation learning tasks?**\n\nIn Table 1, the scores for IO are low due to the complexity of the MuJoCo tasks. In other words, the hypothesis class used by standard IO techniques is not rich enough to learn an effective policy for such imitation learning tasks. This is due to its reliance on predefined feature spaces, which may not capture the intricacies of more sophisticated environments. In contrast, Kernel Inverse Optimization (KIO) inherently leverages the flexibility of kernel methods to operate in high-dimensional spaces, enabling it to adapt more effectively to the complexity of larger imitation learning problems. As our results demonstrate, KIO consistently outperforms traditional IO in these challenging settings by mitigating the need for manual feature engineering and allowing for more robust learning from data.\n    \t\nIndeed, in simpler imitation learning tasks, traditional IO can perform quite competitively, e.g., for the control tasks in reference [2] in the paper. For such tasks, KIO methods may not outperform traditional IO by a large margin, mainly due to the lack of necessity of complex policies to effectively tackle these simple tasks.\n\n**It would be interesting to see how many iterations (and time) it took for SCS to converge vs. how many iterations for SSO to converge.**\n\nIn our experiments, we employed the Splitting Conic Solver (SCS) as a black-box optimizer to solve Problem 10 using the 5k/10k data points in a single run. In contrast, the Sequential Selection Optimization (SSO) algorithm is an iterative algorithm, that runs SCS for the sub-problems defined over $\\mathcal{D}_j$, with samples selected based on the KKT-based heuristic outlined in Section 4.1. This is why we only reported the error-iteration curves for SSO across different tasks.\n\n**Its fairly obvious that the scores will be the same range for all trials if the objective value is the same, so it doesn't help to report objective value and score side-by-side.**\n\nWe reported the objective and evaluation scores side-by-side to illustrate that the solution found by SSO not only minimizes the error but also achieves high evaluation scores. This is crucial for assessing performance in imitation learning tasks due to the possibility of compounding errors in dynamical tasks, leading to low evaluation scores even when the training error (i.e., objective value) is low (e.g., see Ross et al. \"A Reduction of Imitation Learning and Structured Prediction to No-Regret Online Learning\", 2011).\n\n**Have you considered reporting the speed of convergence for KIO + SSO method vs. BC(TD3+BC) and BC(CQL)offline reinforcement learning algorithms in terms of a plot of objective value vs. # of iterations or objective value vs. time?**\n\t\nWe agree that the speed of convergence is an important factor to consider when comparing SSO with the benchmarks. However, since we used the final results reported in references [17] and [20] for the imitation learning algorithms (BC(TD3+BC) and BC(CQL)), we do not have access to the detailed convergence data needed to generate a plot of objective value versus iterations or time. To provide such a comparison, we would need to re-implement the algorithms and replicate the results from these papers, which is beyond the scope of our current study.\n\n- - -\n\nWe hope that we have addressed all the concerns raised. Please let us know if you have any further comments or questions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3DyznFC9Da", "reviewer": "Reviewer_tcjN", "summary": "This paper extend the hypothesis class of Inverse Optimization functions to RKHSs, there by the feature mappings could lie in a infinite dimensional space.\nThis paper also discuss about the scalability issue and then proposed a SSO algorithm to the proposed KIO model.", "questions": "None.", "limitations": "None.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Clarity: This paper is well-written and easy to follow.\n\nThis paper investigates the kernel method into the existing IO framework. This paper also provides\nthe SSO method for scalability, heuristic for choosing coordinates and warm-up trick. \nThis brings some new insights to the optimization community.", "weaknesses": "The choice of kernels is usually problem-dependent. It will be better to discuss some different\ntypes of kernels. Besides, the hyperparameter tuning of kernels should be reported.\n\nPlease report the standard deviation in the experiments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the comments. Please find our response below.\n\n**The choice of kernels is usually problem-dependent. It will be better to discuss some different types of kernels. Besides, the hyperparameter tuning of kernels should be reported.**\n\t\nPlease refer to the global rebuttal for our extended analysis of different kernel functions.\n\nBeyond the hyperparameters reported in Appendix Section B, we did not conduct an extensive hyperparameter search for kernel functions. \n\n**Please report the standard deviation in the experiments.**\n\nPlease refer to the global rebuttal for the extended results that include the standard deviations.\n\n- - -\n\nWe hope that we have addressed all the concerns raised. Please let us know if you have any further comments or questions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AQDv7dwjny", "reviewer": "Reviewer_o6JR", "summary": "The paper  presents an innovative approach to inverse optimization using kernel methods. Inverse optimization (IO) aims to learn the unknown objective function of an expert decision-maker from past data by determining the optimization goal given the optimal solution, which is the reverse of traditional optimization. The authors extend the hypothesis class of IO objective functions to a reproducing kernel Hilbert space (RKHS), thereby enhancing features to an infinite-dimensional space. They demonstrate a variant of the representer theorem for a specific training loss, reformulating the problem to a finite-dimensional convex optimization. To address scalability issues with kernel methods, the paper proposes the Sequential Selection Optimization (SSO) algorithm, which selectively optimizes components of the decision variable, improving efficiency and scalability while ensuring convergence to the same solution as the KIO model. The KIO model's generalization capabilities and the SSO algorithm's effectiveness are validated through learning-from-demonstration tasks within the MuJoCo benchmark.", "questions": "Inverse optimization methods require high-quality demonstration data to learn the target function. If there is noise or error in the demonstration data, what impact will this have on the effectiveness of the article? How to overcome this defect, because obtaining high-quality data may not be so easy", "limitations": "yes", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The inverse optimization is extended to the Reproducing Kernel Hilbert Space (RKHS), which improves the complexity and expressiveness of the objective function. This innovative method can handle more complex decision problems.\nThe sequential selection optimization (SSO) algorithm is proposed to solve the scalability problem of the kernel method. The SSO algorithm improves the training efficiency of the model and enables the KIO model to process large-scale data sets.\nThe article not only proves the effectiveness of the KIO model in theory, but also verifies its practical application effect through the learning demonstration task in the MuJoCo benchmark, demonstrating the generalization ability of the model.", "weaknesses": "The nonlinear and high-dimensional nature of kernel methods makes the models less interpretable. A clear understanding and interpretation of the optimization model may be required.\nThe performance of the KIO model depends on the selection of kernel functions and hyperparameters. Different kernel functions and hyperparameter settings can significantly affect the results and often need to be adjusted through methods such as cross-validation, which increases the complexity of model training. So I hope the author can add more details in this part", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the comments and questions. Please find our response below.\n\n**The nonlinear and high-dimensional nature of kernel methods makes the models less interpretable. A clear understanding and interpretation of the optimization model may be required.**\n\nWe agree with the reviewer that using kernel methods makes the model less interpretable, compared to a simple standard IO model. However, for complex learning tasks, in order to learn a high-performance policy, it is necessary to use richer hypothesis classes like the kernel-based IO model proposed in our paper. This can be seen in Table 1, where IO without kernels achieves low performance compared to KIO and the benchmarks BC(TD3+BC) and BC(CQL), which are Neural Network-based policies.\n\n**The performance of the KIO model depends on the selection of kernel functions and hyperparameters. Different kernel functions and hyperparameter settings can significantly affect the results and often need to be adjusted through methods such as cross-validation, which increases the complexity of model training. So I hope the author can add more details in this part.**\n\nPlease refer to the global rebuttal for our extended analysis of different kernel functions.\n\n**Inverse optimization methods require high-quality demonstration data to learn the target function. If there is noise or error in the demonstration data, what impact will this have on the effectiveness of the article? How to overcome this defect, because obtaining high-quality data may not be so easy**\n\nWe agree with the reviewer that the quality of the demonstration can have an impact on the quality of the learned policy. However, Inverse Optimization methods, in particular, the ones based on loss function can be quite robust to noisy data (e.g., reference [31]). In fact, the data we used in our experiments is noisy, due to internal noise during the action sampling phase of the expert agents. Moreover, we also show results when using data from \"medium\" quality agents, which further showcases the effectiveness of our KIO approach, even with medium-quality, noisy data. Nonetheless, we conducted an experiment using one of the datasets with two types of noise and reported the final scores. Please see the general rebuttal for the results.\n- - -\nWe hope that we have addressed all the concerns raised. Please let us know if you have any further comments or questions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all the reviewers for their time and insightful feedback. We have addressed the reviewer's questions by providing additional information and simulation results. In summary:


## Kernel comparison table

Based on feedback from all reviewers, we have extended our numerical results to evaluate our proposed model using different kernels commonly employed in machine learning, namely, the Gaussian (RBF), Laplacian, and linear kernels. These results are shown in the table below.

| Task                | RBF         | Laplace    | Linear       |
|--------------------|-------------|------------|--------------|
| hopper-medium      | 0.51±0.064  | 0.41±0.051 | 0.024±0.0005 |
| hopper-expert      | 1.099±0.004 | 0.71±0.267 | 0.028±0.005  |
| walker2d-medium    | 0.72±0.14   |0.43±0.282  |-0.0019±6e-5  |
| walker2d-expert    | 1.091±0.003 |1.031±0.226 |-0.0002±0.001 |
| halfcheetah-medium | 0.324±0.122 |0.052±0.105 | -0.008±0.006 |
| halfcheetah-expert | 0.788±0.249 |0.591±0.355 | 0.020±0.031  |

Please note that the results for the RBF kernel in the table above differ slightly from those in the manuscript. This variance is due to a different partition of the dataset used for normalizing the states.

## Noisy data table

In response to the question from reviewer o6JR, we conducted an additional experiment in the hopper-medium environment. We introduced two types of zero-mean noise to the actions in the dataset to assess the performance of KIO with the RBF kernel. The noise applied has a standard deviation that is a factor of the standard deviation of the actions in the dataset, as shown in the table below.

| Noise std ratio    | Gaussian    | Uniform    |
|--------------------|-------------|------------|
| 0.00               |  0.51±0.064 | 0.51±0.064 |
| 0.05               |  0.499±0.050| 0.457±0.063|
| 0.10               |  0.442±0.048| 0.372±0.026|

The results indicate that KIO with the RBF kernel is more robust to Gaussian noise than uniform noise with the same standard deviation.

## Dataset size

In Table 1, we have used the scores reported by the respective papers (references [17] and [20] in our paper) for each task. These works employ the entire dataset, which consists of 1 million samples per task, as described in the D4RL paper (reference [16] in our paper).

- - -
More details are given in the individual replies to each reviewer. We hope that we have adequately addressed all concerns raised by the reviewers, and we remain open to further suggestions that may enhance the quality of the manuscript.

Meta-review: The paper presents an SDP formulation for the the IO problem and proposes a SSO style algorithm for efficiently solving it. All the reviews seem to agree that the methodology (SDP formulation) is novel. Also, the empirical comparison with the baselines seems favourable. I will go with the recommendation by the reviewers of accepting the paper. It will be helpful if appropriate convergence results are cited for the coordinate descent algorithm, in the final version if accepted.
Decision: Accept (poster)