Title: Inverse Optimization Latent Variable Models for Learning Costs Applied to Route Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: inverse optimization

latent space
Abstract: Learning representations for solutions of constrained optimization problems (COPs) with unknown cost functions is challenging, as models like (Variational) Autoencoders struggle to enforce constraints when decoding structured outputs. We propose an Inverse Optimization Latent Variable Model (IO-LVM) that learns a latent space of COP cost functions from observed solutions and reconstructs feasible outputs by solving a COP with a solver in the loop. Our approach leverages estimated gradients of a Fenchel-Young loss through a non-differentiable deterministic solver to shape the latent space. Unlike standard Inverse Optimization or Inverse Reinforcement Learning methods, which typically recover a single or context-specific cost function, IO-LVM captures a distribution over cost functions, enabling the identification of diverse solution behaviors arising from different agents or conditions not available during the training process. We validate our method on real-world datasets of ship and taxi routes, as well as paths in synthetic graphs, demonstrating its ability to reconstruct paths and cycles, predict their distributions, and yield interpretable latent representations.

Inverse Optimization Latent Variable Models for Learning Costs Applied to Route Problems

Abstract

Learning representations for solutions of constrained optimization problems (COPs) with unknown cost functions is challenging, as models like (Variational) Autoencoders struggle to enforce constraints when decoding structured outputs. We propose an Inverse Optimization Latent Variable Model (IO-LVM) that learns a latent space of COP cost functions from observed solutions and reconstructs feasible outputs by solving a COP with a solver in the loop. Our approach leverages estimated gradients of a Fenchel-Young loss through a non-differentiable deterministic solver to shape the latent space. Unlike standard Inverse Optimization or Inverse Reinforcement Learning methods, which typically recover a single or context-specific cost function, IO-LVM captures a distribution over cost functions, enabling the identification of diverse solution behaviors arising from different agents or conditions not available during the training process. We validate our method on real-world datasets of ship and taxi routes, as well as paths in synthetic graphs, demonstrating its ability to reconstruct paths and cycles, predict their distributions, and yield interpretable latent representations. The code is available at https://github.com/AlanLahoud/IO-LVM

1 Introduction

When learning latent generative representations, it is often necessary for inferred samples to satisfy specific constraints, such as forming paths in a graph between designated start and target nodes, consistent with the feasible set of an associated constrained optimization problem (COP). A major challenge is learning such models when the feasible set of solutions is discrete, as the gradients of these solutions with respect to the model parameters are zero almost everywhere, and therefore non-informative [1]. In this paper, we pose this problem as learning a latent cost representation of the COP from observed solutions.

State-of-the-art approaches for recovering underlying cost functions of COPs from observed solutions, e.g., structured decisions performed by agents, primarily address the non-informative gradient problem by either smoothing solver operations [21], interpolating COP solutions [28], perturbing the COP cost [7], or relaxing the COP [39] to match statistics of the observed behavior (i.e., solutions). However, these methods assume a single underlying cost, making them unable to directly learn from data of multiple different agents with different underlying cost functions. To correctly learn from real world data containing behavior from different agents, they require supervision through agent labels.

In this paper, we introduce IO-LVM, a novel approach for learning latent representations of COP costs that can recover observed COP solutions, specifically for route problems in graphs. Our approach

does not assume a single underlying COP cost, allowing it to learn effectively even when multiple agents or context are represented in the data without labels. Similar to a Variational Autoencoder (VAE) [18], we use amortized inference and map into a meaningful and interpretable low-dimensional latent cost space. In contrast of ordinary V AEs, we guarantee that samples fulfill requirements of the feasible set (e.g., connected paths) by using a black-box COP solver in the generative step. To address non-informative gradient challenge posed by discrete solutions, we employ a gradient estimation technique based on perturbing the input of the black-box solver and the Fenchel-Young loss [7, 8].

Applied to path and route data, we can use IO-LVM to, e.g., generate new paths by selecting a latent cost and providing source and target nodes to the black-box solver, which ensures validity of the path. This allows us to infer how different agents would navigate between new source and target nodes. As the IO-LVM learns costs instead of the shape / geometry of solutions, a low-dimensional latent representation is often sufficient, offering easy interpretation and analysis such as clustering similar COP costs, denoising observed paths by finding a small number of representative paths, and detecting outliers and deviating behavior.

We state our contributions as follows: i) We introduce IO-LVM, a method that combines variational approximation techniques with COP solver gradient estimation to learn latent representations for the underlying costs of COPs based on observed decisions; ii) IO-LVM naturally constructs a meaningful and interpretable latent space, allowing for the reconstruction of observed path distributions without making assumptions about the distribution of inferred paths. Notably, the ability to recover distinct (e.g., multimodal) representations for the underlying costs enables the modeling of different agents making decisions; iii) We demonstrate the versatility of IO-LVM using both synthetic and real-world datasets, highlighting its potential for path analysis tasks such as naturally separating underlying costs into meaningful groups, reconstructing or denoising observed paths, and predicting paths for unseen start and target nodes. Our aim is not only to provide quantitative results but also to offer insights through visualizations of routes and latent variables.

1.1 Related Work

Obtaining useful gradients through optimizers for end-to-end learning is challenging and previous works focussed on convex solvers [3, 2] and linear or quadratic programs [11, 36]. However, these methods are mainly limited to continuous COP formulations and are difficult to extend to combinatorial problems such as the graph-based problems in our work with the non-informative gradients. More related to our work are efforts to differentiate through dynamic programming algorithms, such as dynamic programming differentiability [22], or more specifically, a differentiable version of the Floyd-Warshall algorithm to learn from observed paths in graphs [21]. Different to our work, their approach struggles with scalability as graph size increases.

Learning cost parameters from observed solutions is also done by Inverse Optimization [4, 32, 33], Inverse Reinforcement Learning, and Inverse Path Planning [37, 21]. Here, cost parameters are either global [21], linear [23, 38, 39, 24] or non-linear [13, 37, 12], and often learned with end-to-end gradient estimation by exploiting insight in the underlying optimization process or in some cases assuming a black box optimizer [29, 7]. In the later case, the Fenchel-Young loss, also used in this paper, has been demonstrated as a suitable way to match inferred and observed paths within a smooth and convex space [7]. Different to us, these methods typically assume a single underlying cost function or condition the cost on a given context, which may not capture the diversity of agent behaviors present in real-world scenarios.

Autoencoders [17] and Variational Autoencoders (VAEs) [18] are often used to learn lowerdimensional representations of data and there have been attempts to simplify solving COPs by learning better representations of the feasible solutions [6]. However, we observe in our work that VAEs do not reliable generate feasible solutions in complex route problems like TSP and shortest path. In contrast to V AE, IO-LVM uses a COP solver in the decoding step and thereby guarantees that generated samples are feasible solutions.

2 Preliminaries

Below, we recap the Evidence Lower Bound (ELBO) for deep latent variable models and introduce Fenchel-Young losses which are both fundamental for our approach.

Evidence Lower Bound (ELBO). The objective in latent variable models is to identify the latent variables z that best explain the observed data x . However, it is well-known that directly computing the posterior P ( z | x ) is generally intractable. To address this, a variational distribution q φ ( z | x ) is introduced and learned with a lower bound objective (ELBO) [18, 31]. The ELBO makes a trade-off between accurately reconstructing the input data (the expected log-likelihood) using a model p θ ( x | z ) and adhering to the prior distribution P ( z ) for the latent variables, i.e.,

where D KLis Kullback-Leibler (KL) divergence and β is a balance factor [16, 9]. In our approach detailed in Sec. 3, we also use a learned q φ (encoder), but replace the usual reconstruction loss with a Fenchel-Young loss (see below).

Fenchel-Young Losses Fenchel-Young losses are a versatile class of loss functions derived from convex duality theory, specifically leveraging Fenchel conjugates [8, 7, 5]. These losses generalize several common loss functions used in structured prediction tasks, such as cross-entropy and hinge losses, by formulating the learning problem as a regularized prediction task [8]. Fenchel-Young losses are defined as l FY ( y , x ) = Ω FC ( y ) + Ω( x ) -〈 y , x 〉 , where Ω( x ) is a chosen regularization function, Ω FC ( y ) is its conjugate, and 〈 ., . 〉 represents the inner product between two vectors. In our context, x represents a decision vector (e.g., a structured output), and y a cost vector; the loss measures how suboptimal x is under the cost structure implied by y , and is minimized when x is the optimal decision under y and regularizer Ω .

3 Inverse Optimization Latent Variable Model

In this section, we introduce the notations and problem definition, present IO-LVM in a general COP setting (Sec. 3.1), and discuss assumptions for path and cycle applications (Sec. 3.2).

Notation and Problem Definition. Our dataset D = { ( x i , p i ) } N i =1 consists of structured decision vectors x i ∈ X in a constrained space X , e.g., connected paths performed by agents in a graph, and corresponding problem requirements p i ∈ P , e.g., start and target nodes for the path. We denote by ω a black-box solver for the COP that takes cost vectors y i and problem requirements p i to output an optimal COP solution ˆ x i = ω ( y i , p i ) , where the COP solved by ω is defined as argmin x ∈X ( p i ) 〈 y i , x 〉 which, although formulated with linear costs, can represent a wide variety of COPs, specially combinatorial ones [20]. The main goal is to model a low-dimensional representation of COP costs y i ∈ Y that leads to the observed decision vectors from D . Concretely, we aim to estimate the posterior distribution P ( z | x ) , where z ∈ Z ⊆ R k is a latent vector in a space of dimension k .

3.1 IO-LVM Description and Learning

We learn a latent representation Z with amortized inference [19] using a nonlinear mapping q φ to map samples x i to the latent space Z , and then reconstruct them back to the constrained space X , similar to a V AE. Different from V AE models, where reconstruction is done by a decoder network, our reconstruction is non-trivial due to the constraints on the COP solution space X . E.g., X contains valid paths between specific nodes in a graph. To achieve this, we define our reconstruction as a composition of functions g θ : Z → Y and ω : Y × P → X , where the former is a nonlinear map parameterized by θ , and the latter is a solver that is potentially non-differentiable. This sequence of transformations is visualized in Fig. 1.

To learn our IO-LVM, we adapt the VAE's ELBO objective by changing the reconstruction loss (first term) of Eq. (1). We introduce the solver ω and a suitable distance measure d in X resulting in the term E q φ ( z | x ) [ d ( x , ω ( y θ , p )) ] , where y θ := g θ ( z ) . In contrast to V AE models, the black box solver ω in our reconstruction generally prohibits end-to-end learning with a common loss such as the Mean Squared Error. For this reason, we use the Fenchel-Young loss for d by inducing perturbations in the input space of the COP. Consequently, our loss function is defined as:

By choosing Ω (in l FY) to be the conjugate of E glyph[epsilon1] [min x ∈X 〈 y + glyph[epsilon1] , x 〉 ] , we rewrite (brief derivation in Appendix A) l FY as

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The second term corresponds to the expected optimal cost under random perturbations of the cost vector y . In Proposition 1 (see below), we justify this choice in the context of our problem setting by showing that, under mild conditions on the perturbation distribution E , this formulation ensures that no solution is a priori favoured by the noise or problem structure, and solutions with the same cost impact gradients with the same probability. Specifically, we assume that the components of glyph[epsilon1] ∼ E are independent, zero-mean, and identically distributed with finite variance, and that E is closed under convolution, e.g., Gaussian noise. In this case, the Fenchel-Young loss gradient with respect to y -elements is analytically computed as ∇ y l glyph[epsilon1] FY ( y , x ) = x -ˆ x glyph[epsilon1] , where ˆ x glyph[epsilon1] = ω ( y + glyph[epsilon1] ) , minimizing the Fenchel-Young loss if and only if ˆ x glyph[epsilon1] = x [7]. This allows us to obtain gradient estimates w.r.t. the weights θ , as the chain of gradients in the reconstruction block can now be written as

where ˆ x θ glyph[epsilon1] = ω ( y θ + glyph[epsilon1] ) . Estimating the gradients in Eq. (4) is generally done in a Monte Carlo fashion, which is expensive due to the need of running the solver ω several times. To avoid this, we use the property of expectation linearity to rewrite the reconstruction loss in Eq. (3) as

highlighting that the estimator is unbiased with a double expectation. This result allows us to use a stochastic gradient descent (SGD) method to learn θ and φ , as described in Alg. 1. By leveraging SGD, the solver runs once (rather than several times in a Monte Carlo fashion) per data sample during training, reducing the computational cost per iteration.

Algorithm details. The algorithm details the steps in the training process using an encoder h φ to model q φ ( z | x ) , and a mapping g θ . Note that in step 1, the problem requirement sample can be leveraged into the encoder as additional information. Step 2 samples the latent value using the VAE re-parametrization trick [19]. In step 3, we include a transformation Φ to ensure that costs y θ fits to COP input space Y . It is common that some COPs require their cost elements to be positive, for instance, which is generally fixed by ordinary activation functions Φ . In step 4, we compute the COP solution given an inferred and

Algorithm 1 One epoch of IO-LVM training process.

1: Components:

2: - Encoder h φ ; Decoder g θ .

3: Input: Dataset D = { ( x i , p i ) } N i =1

4: Output: Trained model parameters

5: for each sample ( x , p ) ∈ D do

6: Step 1: Encoding: ( µ, σ ) = h φ ( x , p ) .

7: Step 2: Sample z : z = µ + σ · glyph[epsilon1] , glyph[epsilon1] ∼ N .

8: Step 3: Map z to COP cost space: y θ = Φ( g θ ( z )) .

9: Step 4: Solve the COP using ω , p and the inferred cost y θ : ˆ x θ glyph[epsilon1] = ω ( y θ + glyph[epsilon1] , p ) , where glyph[epsilon1] ∼ N .

10: Step 5: Compute the loss as in Eq. (2).

11: Step 6: Update the encoder and decoder parameters ( φ , θ ) allowed by Eq. (4).

12: end for

perturbed cost. In step 6, the back-propagation is allowed due to the gradient estimator described in Eq. (4), bridging the gap of the non-differentiable COP solver.

3.2 IO-LVM Assumptions and Applications

We assume that the observed structured decision samples in D presented in Sec. 3 are optimal COP solutions, e.g., the taxi drivers solve a shortest path problem under their own underlying cost values. Therefore, the variations in the observed structured decision samples arises from differences in

valuations of COP costs. Below, we describe how we model the two problem domains used in experiments in Sec. 4.

glyph[negationslash]

Path Planning. For a fixed directed graph with edge set E , we can model a set of paths as a set of binary vectors X ⊆ { 0 , 1 } | E | corresponding to edge usage. Here, D contains samples of paths ´ in the graph. In this case, a common path requirement is { p = ( s, t ) | s, t ∈ V, s = t } , defining start and target nodes of those paths. In this scenario, we assume there is an underlying set of edges costs for each data sample such that a Shortest Path Problem (SPP) solver (e.g., Dijkstra) ω recovers the observed paths.

Hamiltonian Cycles. For a fixed, either directed or undirected graph, with edge set E , we can model a set of cycles also as a set of binary vectors X ⊆ { 0 , 1 } | E | corresponding to edge usage. Here, D contains samples of Hamiltonian cycles in the graph. We assume that there is an underlying set of edge costs for each data sample such that a Traveling Salesman Problem (TSP) solver recovers the observed cycle. Although path requirements could be added, we consider a null set in our experiments.

Proposition 1. Suppose all feasible routes x ∈ X have equal length, i.e., the same number of edges. Let glyph[epsilon1] ∈ R | E | be a random perturbation vector with independent, zero-mean components, each with identical variance σ 2 . Then, the perturbed cost vector y + glyph[epsilon1] induces a distribution over X in which all solution costs have equal expected value and equal variance, preventing bias toward specific solution in X after perturbation, and paths with the same cost impact gradients with the same probability.

This condition holds exactly for Hamiltonian cycles, where all feasible solutions have the same length by definition, and serves as a reasonable approximation in our real-world path planning applications, where the graph is embedded in a space with relatively uniform node spacing and no significant shortcuts. Therefore, the corresponding Fenchel-Young loss chosen in 5, defined via the additive perturbation, is a good choice for both experimental settings. When this is not the case, alternative perturbation methods would be required. A proof for Proposition 1 is provided in Appendix B.

Note that in both applications, as in general discrete problems , the COP can be formulated with a linear objective [20]: ˆ x ∈ argmin x ∈X 〈 y , x 〉 , fulfilling the requirement for the gradient estimator. Importantly, as in other works in structured prediction [7, 28, 21], IO-LVM is limited to problems for which fast solvers ω exist, since the solvers are the computational bottleneck of the training process.

3.3 IO-LVM Inference Tasks

Once the IO-LVM training process is complete, we can perform inference tasks in the above applications. IO-LVM allows reconstructing paths from parts of the low-dimensional latent space using again, as in the training step, a composition of the learned decoder and the solver, i.e., Φ( g θ ) and ω ( . ) , allowing us to observe different patterns reconstructed in the path space. By fitting a kernel density estimation (KDE) into the low-dimensional learned latent space, we can sample latent values and generate paths as mentioned above, leading to an inferred path distribution.

Selecting a proper β during the training process is not only useful to mitigate the issue of posterior collapse or avoid overfitting as in β -VAEs [34], but in our case can also be useful to denoise a test path at inference time, i.e., remove uncommon patters in the observed test path by encoding it through the learned h ( φ ) and decoding it using Φ( g θ ) and ω ( . ) . In a similar direction, similarity metrics between a test path x and inferred paths from IO-LVM can also be used to detect if x is an outlier.

4 Experiments

The experiments focus on route problems in graphs using ω Dijkstra and a TSP solver for path planning and Hamiltonian cycles assumptions, respectively, as described in Sec. 3.2. Details of the IO-LVM training process are provided in Appendix F. We use four different datasets to evaluate IO-LVM in tasks like route reconstruction, path distribution prediction, facilitation to latent space analysis, and its potential for unsupervised learning tasks such as anomaly detection and denoising. The datasets used in the experiments are presented below, while further details are described in Appendix C.

A) Synthetic Waxman Random Graph. We generate a Waxman graph [35] with 700 nodes and 7230 edges; with three different cost functions for the edges costs y , all of them on the basis of a

nonlinear function from unobserved features. We increased the costs of southern edges for agent 1 and of northern edges for agent 3, while agent 2 was unbiased in terms of south/north edges (in Fig. 2 and 5 it is possible to observe those preferences). We solved the SPP using Dijsktra for each agent cost multiple times on the basis of the generated edges costs plus a Gaussian noise, i.e., ω ( y + glyph[epsilon1] ) , to generate multiple paths to D . These paths are generated in two manners, one set with a single source-target pair ( Single S&T ), and another set with multiple pairs ( Mult. S&T ). In both cases, 6000 paths were generated, which 5000 were used for training.

Ships Dataset. Using Automatic Identification System (AIS) data from the Danish Maritime Authority [10], we use ship locations collected during three months. We project the locations to a grid graph with 2513 nodes and 8924 edges, resulting in a set of 2500 paths with different start and target nodes, where 2,000 were used for training.

Taxi Dataset. We use the Cabspotting dataset [27], which contains real-world taxi trajectories in the city of San Francisco, following a preprocessing approach to build a graph based on the available trajectories [21]. Unlike the original setup, we increase the number of nodes and edges to 1125 and 8022 to better reflect the realism of the actual trajectories, resulting in 101344 trajectories.

TSPLIB. From TSPLIB95 [30], we use burma14 (14 nodes, 91 edges) and bayg29 (29 nodes, 406 edges) graphs. Actual (underlying) edges costs y are generated using a nonlinear function incorporating unobserved features and Euclidean distances between nodes as offset. We create two versions for each graph, one using 3 unobserved features and another using 50 unobserved features. 3000 actual Hamiltonian Cycles (2400 for training) are generated with a TSP solver ω using the underlying costs y as input.

4.1 Qualitative Latent Space Analysis

In this section, we analyze the learned latent space Z . The goal is to observe that paths generated by similar costs (e.g., coming from the same agent or similar context) are encoded close to each other in latent space after IO-LVM training. We also analyze how structure in the latent space influences reconstruction by sampling latent values and observing corresponding inferred routes.

Additional results of latent space analysis using the Taxi and TSPLIB datasets are reported in Appendix D.

Synthetic Waxman Paths. Fig. 2 shows the learned twodimensional latent space for the Mult S&T dataset. The colors represent the agent that executed each path. Importantly, the agent identity was not provided during training. We observe that IO-LVM successfully disentangles the factors associated with the costs of the three different agents. For example, Agent 1 (red) follows several distinct paths with different start and target nodes, yet these paths are consistently mapped to a common region in the latent space, reflecting their shared underlying transition costs, that is, to avoid southern edges when possible. In contrast, the VAE latent space fails to exhibit this structure: paths performed by the red agent are spread in

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different locations of the latent space. This suggests that V AEs are unable to disentangle the transition

cost structure inherent in the data. Illustrations of path reconstruction from different regions of the latent space are presented in Appendix D.

Ship Paths. The top-left chart in Fig. 3 illustrates the 2D latent space (3D latent space results are seen in Fig. 12 in the Appendix). Each hexagon in the right chart of Fig. 3 corresponds to a subspace of the latent space. For each hexagon, the average of the ships' width is plotted in color. Larger ships are less frequently found in the top-right corner of the latent space, leading to a low average ship width in that region. This is another example that IO-LVM was capable to capture unobserved factors within the latent space, i.e., the ship width information (provided by AIS) was not used during the training process.

We also analyze the reconstruction process through this dataset by sampling 20 latent values from Gaussians in the latent space and then computing the respective edges costs and paths. As seen in the right part of Fig. 3, neighbor latent values share a high number of edges in the graph (e.g., many path samples are the same). Moreover, an interesting pattern emerges: some regions of the latent space containing wider ships avoid the Oresund Strait when traveling from the east to the north part of Denmark even though it is the shortest path in terms of euclidean distance, as observed in the second column of the figure where ships prefer going through the Great Belt.

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4.2 Validating IO-LVM Reconstruction Quality

We evaluate reconstruction performance by comparing if the inferred output perfectly match the observed path (input to the encoder) with low latent dimensions on the TSPLIB datasets. Note that the task of reconstructing a perfect Hamiltonian cycle is challenging, since there are 13! ≈ 6 ∗ 10 9 possible Hamiltonian cycles for the burma14 dataset and 28! ≈ 3 ∗ 10 29 for the bayg29 . Results reported in Table 1 demonstrates that IO-LVM consistently outperforms VAE due to

its structured reconstruction process. As a naive reference, we include the ω ( y E ) , a non-learning baseline where the TSP solution is computed using edge costs defined as the Euclidean distances between node positions. Although VAEs demonstrate to have good performance when the reconstruction is measured by edge matching instead of full reconstruction, as show in Table 4 (Appendix E), they do not guarantee feasible outputs with respect to the COP, leading to worse results in a full match comparison. Also, results in Table 2 shows that even by increasing the latent dimension of

VAEs, they are still far from the reconstruction power of IO-LVMs for the most challenging dataset ( bayg29 with 50 features). The gap is even higher when the number of training samples are smaller, as also shown in Table 2. Additional results on the variation of the number of latent dimensions are provided in Appendix E.

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Fig. 4 illustrates some reconstructed samples of IO-LVM against VAE , where the decoder outputs paths as binary edge usage indicators (i.e., probabilities converted to binary). In the figure, thick edges illustrate the reconstructed paths for the first three test samples of each dataset ( burma14 and bayg29 created with 50 features). It can be seen from the figure that, different from V AEs, IO-LVM ensures that the output forms a valid Hamiltonian cycle due to the inclusion of a TSP solver in its processing loop. This happens even when the reconstruction is not fully correct (e.g., most right graph in in Fig. 4).

Effect of varying β : Reconstruction versus Denoising We analyze the effect of varying β on two metrics in the Single S&T synthetic Waxman dataset: i) the number of distinct paths reconstructed by the decoder using the test dataset; ii) and the Intersection over Union (IoU) metric between observed and inferred edges usage during training. Figure 5 shows that as β increases, the number of distinct paths decreases, indicating a denoising effect due to the diminished influence of the reconstruction loss. This results in the decoder reducing diversity of generated paths due to the posterior collapse. The IoU decreases with larger β , also reflecting a reduction in reconstruction power. In Figure 14 (Appendix E), we show a projection of noisy real-world taxi paths into similar but expected paths by reconstructing them using IO-LVM.

4.3 Inference Results

Path Distribution Prediction. To measure the prediction quality of the overall test paths distribution, we used a kernel density estimator (KDE) in the learned latent space to estimate its probability density function and sample latent values from it. We evaluate the quality of predicted path distributions using two metrics: Jensen-Shannon divergence ( D JS, lower is better) and root mean squared error (RMSE, lower is better), both computed on edge usage frequencies between predicted and ground truth paths given fixed start and target nodes. For the Ship and Taxi datasets, these nodes are selected as the most frequent pairs in the data to ensure consistency in comparisons at inference time. Path

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samples for the V AE and IO-LVM are obtained using KDE samples from their latent spaces, inferring their corresponding edges costs y θ , and solving Dijkstra on them, i.e., ω ( y θ ) . We also compare the results against two other baselines: the Perturbed Optimizer (PO) baseline [7], which estimates gradients for structured prediction without modeling a latent space. We adapt PO to our setting by removing contextual inputs and reintroducing the training-time noise glyph[epsilon1] at inference to sample paths as ω ( y θ + glyph[epsilon1] ) ; and a naive BO baseline, where we minimize RMSE ( x , ω ( y E + glyph[epsilon1] BO )) through Bayesian Optimization where the distribution variance from which glyph[epsilon1] is sampled is a single decision variable. Results in Table 3 show that IO-LVM outperforms VAE, which fails to model constraints under limited data, e.g., in the Ship dataset, the sparse path count relative to the graph size prevented convergence (NC). IO-LVM also outperforms PO, whose naive sampling, i.e., ω ( y θ + glyph[epsilon1] ) , fails to capture some characteristics of the true path distribution. The resulting inferred path distribution by IO-LVM of the Taxi dataset is illustrated with the blue paths of Figure 6 (top-left chart), and can be compared with the green observed paths in the test dataset with the same pairs of start and target nodes (top-mid chart).

Outlier Detection. We generate two path outliers in the taxi dataset as illustrated in the top-right chart of Fig. 6, one by using Dijsktra considering edge costs as Euclidean distances between nodes (purple path), and the other solving Dijkstra on learned costs but removing edges in latitude/longitude box, enforcing deviation (red path). To analyze if each of the observed paths (greens) and each of the outlier paths are actually outliers, we compute a low τ -th quantile Q τ (e.g., τ = 0 . 02 ) of a distance measured between this test path x and the path distribution using inferred samples ˆ x θ by IO-LVM, i.e., Q τ z ∼Z [ d ( x , ˆ x θ ( z ))] , a score reflecting how far each observed path lies from most of the paths in the inferred distribution. Although alternative distances d could be chosen, we compute d as the RMSE between the cost of those paths by sampling N edges from the learned latent space, i.e., d = RMSE ( [ 〈 x , y θ i 〉 ] N i =1 , [ 〈 ˆ x θ , y θ i 〉 ] N i =1 ) . This distance reflects how far two paths are from each other when projected on learned costs. A high quantile score Q τ suggests that the path is unlikely under the inferred path distribution and therefore anomalous. We compute this score Q τ for each path in the data

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Q=0.02

Outlier2

100

85.2

81.8

81.9

88.0

50

0

(green) and the created outliers (purple and red) for τ = 0 . 02 . Results showing higher scores for outlier paths are reported in the bottom graphs of Fig. 6, and could also indicate the existence of other potential outliers in the test dataset (green bars).

5 Conclusion

This paper proposed IO-LVM, a novel approach for learning latent representations of COP costs, specifically for paths and cycles in graphs. The method leverages amortized inference and integrates black-box solvers within a probabilistic framework, allowing for the modeling of multiple agents and diverse behaviors in graphs. By employing a Fenchel-Young loss with perturbed inputs, it overcomes the gradient challenges in optimizing COPs, ensuring feasible and interpretable path reconstructions. The learned latent space captures meaningful structures, highlighting the model's characteristic to distinct agent behaviors, while maintaining accurate path reconstruction and prediction. Our method description is valid for a general set of COPs if gradient estimation is available.

Acknowledgments and Disclosure of Funding

This work has been supported by the Industrial Graduate School Collaborative AI & Robotics funded by the Swedish Knowledge Foundation Dnr:20190128, and the Knut and Alice Wallenberg Foundation through Wallenberg AI, Autonomous Systems and Software Program (WASP).

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A Perturbed Fenchel-Young brief derivation

For a regularized maximization problem, the Fenchel-Young loss is defined in [8] as

where Ω is a convex regularized and Ω FC is its convex conjugate. By choosing the regularized Ω as the conjugate of the perturbed minimum cost, i.e., Ω FC = E glyph[epsilon1] [ min x ′ 〈 y + glyph[epsilon1] , x ′ 〉 ] , the loss is rewritten as

Since the only important terms for our gradient-based learning approach are the terms dependent on y , i.e., we are only interested in gradients with respect to y , and inverting the sign of the loss to be suitable to a minimization problem instead of a maximization problem, we simplify the loss for the gradient computation as:

B Perturbed Fenchel-Young for Graph-based Applications

Proof of Proposition 1. Let y ∈ R | E | be the edge cost vector, and let glyph[epsilon1] ∈ R | E | be a random vector with independent, zero-mean components added to each edge. Let ˜ y = y + glyph[epsilon1] denote the perturbed cost vector. For any path or cycle x ∈ X ⊆ { 0 , 1 } | E | , represented as an indicator vector over edges, the perturbed cost is given by 〈 ˜ y , x 〉 , representing the inner product between both variables.

We first show that the expected cost of a path remains unchanged under perturbation:

since each component of glyph[epsilon1] has zero mean and x is fixed.

Next, we compute the variance of the perturbed cost. The variance of the cost 〈 ˜ y , x 〉 = 〈 y + glyph[epsilon1] , x 〉 arises from the perturbation:

Assuming the components glyph[epsilon1] e are independent with identical variance σ 2 , we obtain:

where ‖ x ‖ 0 denotes the number of edges in the path x . Hence, if all x ∈ X have equal length, the variance is the same for all paths.

As a direct consequence, the additive perturbation model induces equal expected cost and variance across paths or cycles when their lengths are equal. This condition holds exactly in our Hamiltonian cycle experiment, where all feasible cycles visit each node once and thus have fixed length. In the path planning experiment, although path lengths may vary, the uniform spatial layout in our real-world datasets (ship and taxi) and lack of large shortcuts make it reasonable to assume that most feasible paths between start and target nodes have similar lengths, while too lengthy paths are extremely unlike to be optimal even with perturbation. Consequently, the induced variance does not vary much, validating the use of this perturbation model in both settings.

C Datasets details

Synthetic Waxman Random Graph Wegenerate a Waxman graph [35] with 700 nodes ( α = 0 . 05 , β = 0 . 6 ), where the probability of an edge between two nodes u and v is given by P ( u, v ) = α · exp ( -d ( u,v ) β · d max ) , where we considered d ( u, v ) as the Euclidean distance between nodes u and v , and d max is the maximum distance between of two nodes, consequently ending up in 7230 edges. We create three edge cost sets to simulate three different agents performing decisions to go from start and end nodes. The edge costs are based on Euclidean distances, with higher costs for the southern edges

for agent 1, and higher costs in the northern for agent 3, while agent 2 is not biased by the edges position. For each agent, we add a random noise in the cost elements y so the generated paths can be different from each other even within the same agent. The "observed" paths are generated by running the Dijkstra on the noisy edge costs. Two sets of 6,000 observed paths are generated: one with a single source and target pair (Fig. 2, top-left) and another with multiple source-target pairs (Fig. 2, bottom-left). In each of these sets, 5,000 paths are used for training IO-LVM and the baselines, while 1,000 are used for evaluation purposes. Further details on cost generation are provided in the code.

Ships dataset We use the Automatic Identification System (AIS) data provided by the Danish Maritime Authority [10], considering latitude and longitude projected in a 2D space for simplicity. The analysis focuses on paths from the first week of the months January 2024, May 2024, and June 2024. Only paths that exceed a distance of 4 units (in latitude/longitude) in Euclidean space are included. A path is considered completed either when the ship speed approaches zero or when there is an abrupt change in its heading. In some cases, there are gaps in the latitude/longitude signals; when such jumps occur, we segment the data and treat them as separate paths. We created a grid graph with a distance of 0.09 units between adjacent nodes, focusing on the area where there are more route options to be taken, which in total led to 2513 nodes and 8924 edges. This resulted in approximately 2,500 ship paths, for which the first 2000 (in the order available from the data) is used for training. Note that this is the most sparse dataset, i.e. few paths in comparison with the graph size. This led, in our experiments, the V AE not being able to learn even after considerably more (e.g., 10x more) epochs than IO-LVM.

Taxi dataset The data and the preprocessing follows exactly what is done in [21] with their available preprocessing code, except for the fact that we increase the density of the grid, so that the resulting in a bigger graph, consisted of 1125 nodes and 8022 edges and 101344 trajectories. We split the train and test datasets by randomizing the taxi drivers (anonymized driver id), where 70% was used for training and 30% for test.

TSPLIB We use datasets from TSPLIB95 [30], a library of benchmark instances for the Traveling Salesman Problem (TSP) and related optimization problems. Specifically, we selected two graphs: burma14 , which consists of 14 nodes representing locations in Myanmar, forming a complete graph with 91 edges, and bayg29 , which consists of 29 nodes representing the coordinates of cities in Bavaria, Germany, forming a complete graph with 406 edges. To assign the actual edge costs y , we uses the Euclidean distance between nodes as an offset, and design a nonlinear function that incorporates unobserved features to calculate edge costs. We generate two datasets for each graph, one considering 3 unobserved features (less complex), and one considering 50 unobserved features (more complex). The observed paths are generated as ω ( y ) without noise, where ω represents a TSP search solver. In all experiments with TSPLIB, even with different training size, we always used the same 600 test samples for a fair evaluation and proper comparisons. For the main results 2400 samples were used for training, while we had other experiments (see tables in the main paper) where we also used 1000 and 10000 training samples to understand the role of the training size in the reconstruction results.

D Additional Results: Latent Space Analysis

Single S&T. We sample 20 latent values from Gaussians in the latent space and compare distribution of paths generated by the composition of the decoder and solver. Reconstructed synthetic paths from the Single S & T dataset are shown in the bottom graphs of Fig. 7. It is observed that points closer in the latent space share a high number of edges in the graph. Additionally, as the variance increases, the number of distinct reconstructed paths grows, indicating consistency in the learned latent space. e.g., difference between third and fourth columns in Fig. 7.

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Hamiltonian Cycles. We select nine samples from the test set of the burma14 graph, distributed across different regions of the latent space. They are organized into three groups of three samples each (see the top graph of Fig. 9). The corresponding paths that generated these latent values by the encoder are visualized in the bottom graph of Fig. 9.

Paths with more edges intersection tend to be closer to each other in the latent space. We generalize this analysis by computing the Euclidean distance between all pairs of latent values versus the Manhattan distance based on edge usage (path choice) between those paths. For each group of sample pairs with a specific Manhattan distance, we calculate the average Euclidean distance in the latent space. The results, presented in Fig. 8, reveal that samples that are closer in the latent space are also closer in terms of edge intersection.

E Additional Results: Inference

Taxi Dataset. We sample three different latent vectors from the learned latent space on the Taxi Dataset and compute three different set of learned edges costs, i.e., y θ = Φ( g θ ( z )) . Then, we use Dijkstra as ω ( y θ ) on those three set of edges to compute the shortest path given two set of nodes in the extremes of the graph on those learned edges. The resulting edges costs, normalized by the mean and standard deviation of a bigger sampling population of edges, and the resulting shortest paths, are illustrated in Figure 10.

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TSPLIB reconstruction The results in Table 4 are Recall of edges reconstruction in the TSPLIB datasets for both training and test path samples. IO-LVM is capable to reach almost 1.0 in all cases. VAEs can also do a good reconstruction when analyzing this metric. However, most of the mistakes in VAEs reconstruction leads to a non-structured output, i.e., a non Hamiltonian cycle, which can be observed with some reconstruction examples in Figure 11.

Varying the number of latent dimensions

In our experiments, we observed that for certain tasks, a very low number of latent dimensions was enough. For instance, in the Ship Dataset, attempting to add a third dimension to the latent space revealed that the second and third dimensions are highly correlated (Fig. 12), indicating that the third dimension is unnecessary. Conversely, in the Hamiltonian Cycles experiment, where 50 hidden features were used to generate edge costs with a complex relationship, increasing the latent dimensions proved beneficial in mitigating underfitting during the reconstruction process. This effect is illustrated in Fig. 13, which compares the performance of using 2 latent dimensions (left graphs) versus 10 latent dimensions (right graphs) for both the burma14 (top graphs) and bayg29 datasets. In the right-hand charts, we observe that using 10 latent dimensions achieves 100% Recall in the reconstructions for the training datasets and improves Recall for the test datasets compared to the

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E.1 Denoising Taxi paths

By encoding observed paths using the trained encoder of IO-LVM, and then decoding the corresponding latent value through the decoder + solver, it is possible to observe slight differences between the input and output. This is due to the well-balances β chosen in the training process to avoid overfitting, leading to a denoising feature, i.e., removing uncommon patterns in an observed path. Figure 14

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F Implementation details

IO-LVM is trained according to Algorithm 1. We used PyTorch [26] for the implementation. We consider a learning rate of 0.00004 and a batch size of 250 for the Waxman synthetic dataset, the Ship dataset, while the Taxi dataset and the Hamiltonian Cycle experiment uses a learning rate of 0.0001 and a batch size of 200. COPs are solved in parallel for the batches. The neural network architectures do not have any special implementations. The encoder architecture consists of a neural network with 4 hidden layers, each containing 1000 neurons, with ReLU activation functions in the hidden layers. The decoder architecture is the same as the encoder, but with a Softplus activation function to ensure that all edge costs remain positive. The RMSProp optimizer is used for the Synthetic and Ship datasets, while the AdamW optimizer is used for the Hamiltonian Cycles experiment. The experiments were run on a CPU due to the bottleneck introduced by COP solvers. The processor model used was the 13th Gen Intel(R) Core(TM) i7-13700KF, which has 24 cores. We use Dijkstra from networkx library in python [15] for solving SPP: and Ortools routing python library for TSP solutions [14]. For all the experiments, we run with a fixed and high number of epochs. The most time consuming experiment is with the Taxi dataset dueto the number of samples. We observed that after 100 epochs (around 2 days to complete with our resources) was enough for the model to converge. The other experiments can be finished in a maximum of few hours with no more than 500 epochs (the higher the better). However, for the V AEs baseline, more epochs are generally needed (we observed convergence after 1200 epochs for the synthetic experiment, for example). Further details are found in the provided code at https://github.com/AlanLahoud/IO-LVM

Baselines. BOwas performed to optimize a single variable i.e., the variance of the perturbation. 100 calls for each run was performed by varying the initial state. PO was trained by setting a pre-defined glyph[epsilon1] (Synthetic and Ship Datasets), as in related works, or by leveraging the mean of trained latent values in IO-LVM (Taxi Dataset), since the training process of PO is equivalent to IO-LVM with a single point in the latent space. Further details are found in the provided code.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nSummary\nThis paper proposes a novel Inverse Optimization Latent Variable Model (IO-LVM) for learning implicit cost representations in constrained optimization problems (COPs) from observed optimal solutions. Unlike traditional Variational Autoencoders (VAEs), IO-LVM maps latent variables to cost function parameters and incorporates a black-box solver to ensure the generation of feasible, constraint-satisfying solutions. This design effectively addresses the challenges of non-differentiability and structural infeasibility common in structured output prediction. By leveraging Fenchel-Young losses and perturbed gradient estimators, the authors develop a principled training framework that enables unsupervised learning tasks such as path reconstruction, distribution prediction, interpretable latent space modeling, and anomaly detection.\n\nThe proposed approach is thoroughly evaluated on synthetic graphs, maritime shipping routes, taxi trajectories, and TSP benchmarks, covering multiple aspects such as reconstruction accuracy, distributional modeling, latent structure analysis, and inference performance. Comparative studies with representative baselines (VAE, Perturbed Optimizer, Bayesian Optimization) demonstrate the modeling capacity and practical potential of IO-LVM in unsupervised structured decision problems.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1. IO-LVM generates feasible solutions by mapping latent variables to cost vectors and invoking a black-box solver, which guarantees constraint satisfaction (e.g., valid paths or Hamiltonian cycles) and avoids the infeasible outputs often produced by VAE-based models.\n\n2. The learned latent space naturally captures behavior patterns of different agents or preferences, enabling interpretable clustering and path preference analysis.\n\n3. IO-LVM supports unsupervised inference tasks such as path denoising, anomaly detection, and distributional generation, filling a gap in current methods for unsupervised structured prediction.\n\nWeaknesses\n\n1. Despite adopting expectation linearization to approximate gradients, training still requires frequent calls to the black-box solver, which can be computationally intensive, especially on large-scale graphs.\n\n2. The paper focuses exclusively on the unsupervised setting and does not discuss how supervised methods might perform when agent labels or ground-truth cost functions are available.\n\n\nThe writing, theoretical formulation, and empirical performance of this work are all strong. The reviewer is generally supportive of acceptance. However, a discussion on the relationship and comparative boundaries between supervised and unsupervised methods would further strengthen the paper. The reviewer would be willing to raise the rating upon incorporation of this clarification in the revised version.\n\nQUESTIONS:\nThe manuscript is clearly written and technically sound. The reviewer has no additional questions at this time.\n\nLIMITATIONS:\nThis work primarily focuses on unsupervised cost inference in combinatorial optimization problems. Including a brief overview of recent advances in supervised CO solvers could further clarify the positioning and contribution of IO-LVM. Suggested references include (and potentially others) :\n\n[1] DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization\n\n[2] DISCO: Efficient Diffusion Solver for Large-Scale Combinatorial Optimization Problems\n\n[3] T2T: From Distribution Learning in Training to Gradient Search in Testing for Combinatorial Optimization", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a latent variable model to learn representations of (unknown) cost functions from examples of the form (problem requirements, structured decision vector). The model, dubbed Inverse Optimization Latent Variable Model (IO-LVM), operates as follows:\n* The pair is stochastically encoded to a latent variable, which is then decoded, similar to VAE\n* Unlike a VAE, the decoded variable is not directly a sample. It represents costs for a constrained optimization problem (COP). The costs are passed to a black-box COP solver (with some perturbation to improve the efficiency of gradient estimation). The use of a solver in the reconstruction guarantees that the generated samples satisfy the problem constraints.\n* A Fenchel-Young loss is used in place of the usual VAE reconstruction loss. This allows gradients to be estimated even though we cannot differentiate through the black-box solver.\n\nThe model is evaluated on two graph problems, routing and Hamiltonian cycles, over several datasets. Compared to traditional VAEs, IO-LVM is shown to produce better reconstructions and a more interpretable latent space. It is also demonstrated to be useful for anomaly detection.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n* The structure of the model guarantees that samples satisfy the constraints, which is a very useful property.\n* Experiments demonstrate both the quantitative and qualitative superiority of IO-LVM to VAEs (and other baselines) for the tasks considered.\n* The writing of the paper is generally clear and understandable.\n\nWeaknesses\n* The approach is limited to problems for which there is a suitable Fenchel-Young loss with analytic gradients. (Although, this covers a significant class of problems, e.g. those with linear objectives.) Experiments mostly only consider routing on graphs (also a bit about Hamiltonian cycles), which perhaps does not demonstrate the full utility of the model.\n* To get equation (3) from the definition of $l_{FY}$ and the chosen $\\Omega^{FC}$, one must argue that the $\\Omega$ term disappears. The disappearing term is not mentioned at all, which was confusing. This is not a terribly long derivation, so I think it should be included in the appendix (or cite an existing derivation).\n\nQUESTIONS:\nDo you have ideas for other problems to which IO-LVM could be applied? (No need to run additional experiments.)\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a new method called IO-LVM for tackling inverse optimization problems, i.e., the problem of learning unknown cost functions from the solutions of constrained optimization problems. The work combines two techniques: latent-variable variational encoder models and gradient estimation via French-Young loss, to deal with two issues (1) learning of the underlying distributional representations of the cost vectors, and (2) non-informative gradient in discrete solution spaces, respectively. They present their framework’s versatility in route problems.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n1. The paper tackles some well-motivated real issues in inverse optimization literature. The method IO-LVM presents a reasonable routine for solving these issues.\n2. The method allows some interesting inference analysis by exploring the learned cost distribution, as a consequence of adapting the VAE models.\n\n**Weaknesses**\nThe mathematical rigor needs some polishment in Section 3. This include undefined problem setups, undefined/inconsistent notations, etc.:\n\n  - The symbol $\\hat{x}\\_{\\epsilon}$ is never introduced. From Eq. (3) it looks like the gradient of the expectation term. Moreover, when you write $x=\\hat{x}\\_{\\epsilon}(\\theta)$,  $x$ are data while $\\hat{x}\\_{\\epsilon}$ is a function of the trainable parameters $\\theta$.  Thus, I believe it is more natural to flip the equation and write \"if and only if we set $\\theta$ such that $\\hat{x}\\_{\\epsilon}(\\theta) = x$\". \n\n  - In eq. (4), the LHS uses $y^\\theta$ (with no $z$), while the RHS inconsistently uses $g\\_{\\theta}(z)$ and introduces notation $\\hat{x}\\_{\\epsilon}^\\theta$ with a new index $\\theta$ for the first time.\n  - A formal and mathematical formulation of the original optimization problem should be presented. No *mathematical language* asserts that you have the assumption that the dataset is heterogeneous with different agents solving a same optimization problem that has a common constraint space yet differs in personalized linear cost. I can raise more questions about the problem setting if not specified. For example, following Figure 2, do you assume that conditional on an agent, and a given problem requirement (fixed origin-destination), the cost is determined or random? \n  - See first bullet in Questions.\n\nThese all impede a smooth reading and understanding of the methods.\n\n2. Weak theory with insufficient and vague explanation.\n  - Authors highlighted that the formulation ensures “equal variance and unbiasedness across the solution space” in Line 132, while in Proposition 1 the phrase used is “all solution costs have equal expected value and equal variance” What do these two phrases mean in mathematical language? The two expressions do not mean the same thing to me. \n  - Furthermore, in Line 141, it is not clear what it means that eq. (5) highlights “the estimator is unbiased with a double expectation”. You probably mean eq. (5) is an unbiased estimator of the original objective in eq. (2).  Moreover, in Line 190, I cannot clearly see how the FY loss with additive perturbation yields a good choice.\n\nQUESTIONS:\n1. It seems that the chosen FY loss is well-defined (as in your citation [7]) only when the COP is exactly a linear program with convex polytope constraint set. Otherwise, the linear operator (inner product) does not make sense, and the gradient may not be valid. I see no reason to define the FY loss this way if the objective of the COP for example is a general quadratic form of decisions $x^\\top Y x$ where $Y$ is the cost matrix. I also doubt if the gradient derivation that works for the convex constraint set assumption, can also work here when the decision set is discrete. Can you confirm if this is the case? If yes, then I believe you need to specify it for a rigorous presentation, and you should explicitly discuss that your methods do not work for the general COPs. \n2. I may not fully understand but the presented work does not perform or generalize inverse optimization; it effectively works on a related but not exactly same problem. I have no doubt It learns a latent representation of the cost function, and it is possible to sample a cost function out of the latent model. However, it is not clear if the framework can give the best prediction of the cost vector given a decision solution out of the COP, unless the black-box solver $\\omega$ represents a invertible map that can recover $y^\\theta$ from $\\hat{x}_\\epsilon^\\theta$. \n3. In Line 143, you mentioned that a benefit of your derived objective eq. (5) is that the solver runs once per data sample in SGD. I don’t see this as an advantage without any evidence. You need to compare with SGD that runs with solving multiple Monte-Carlo per data sample. It potentially yield better results as it offers a less noisy gradient that can lead to a steeper descent and require less iterations of training.\n\nLIMITATIONS:\n1. Following the first bullet in Questions, I believe the presented method is not sufficient for handling the general class of constrained optimization problems as claimed in Abstract and Intro.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

comments: SUMMARY:
The paper proposes IO-LVM, a generative model that learns latent cost representations of constrained optimization problems (COPs) from observed structured decisions, such as paths in graphs. By integrating variational inference with a black-box solver and using Fenchel-Young losses with perturbation-based gradient estimation, IO-LVM can generate feasible solutions and capture a distribution over cost functions, enabling modeling of diverse agent behaviors without labels. The method is evaluated on synthetic and real-world datasets, showing strong performance in reconstruction, latent space structure, and downstream tasks.

STRENGTHS AND WEAKNESSES:
Strengths

The proposed approach ensures constraint satisfaction via solver-in-the-loop decoding and handles multimodal behaviors by learning cost distributions. Theoretical grounding is solid, and experiments show clear advantages over VAEs and other baselines across multiple tasks. The learned latent space is interpretable and supports applications like denoising and anomaly detection.

Weaknesses

The method assumes optimal demonstrations and requires efficient solvers, limiting applicability to complex or noisy real-world tasks. Solver calls during training could be a bottleneck. Connections to more recent inverse RL or graph generative methods are not clearly stated.

QUESTIONS:
See weaknesses above as well as the following questions.

* How does the model perform under suboptimal or noisy data?  (can you at least provide a discussion for this?)
* Can it scale with approximate solvers or caching? 
* Is the latent space interpretable under interpolation? 
* Can the method generalize beyond routing tasks to other COPs like scheduling?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-03