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2025/Lightspeed Geometric Dataset Distance via Sliced Optimal Transport/cb12e187-6831-4d4c-be93-32292d5f41a2_origin.pdf

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2025/Lightspeed Geometric Dataset Distance via Sliced Optimal Transport/full.md

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+# Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
+
+Khai Nguyen\*1 Hai Nguyen\*2 Tuan Pham' Nhat Ho
+
+# Abstract
+
+We introduce sliced optimal transport dataset distance (s-OTDD), a model-agnostic, embedding-agnostic approach for dataset comparison that requires no training, is robust to variations in the number of classes, and can handle disjoint label sets. The core innovation is Moment Transform Projection (MTP), which maps a label, represented as a distribution over features, to a real number. Using MTP, we derive a data point projection that transforms datasets into one-dimensional distributions. The s-OTDD is defined as the expected Wasserstein distance between the projected distributions, with respect to random projection parameters. Leveraging the closed form solution of one-dimensional optimal transport, s-OTDD achieves (near-)linear computational complexity in the number of data points and feature dimensions and is independent of the number of classes. With its geometrically meaningful projection, s-OTDD strongly correlates with the optimal transport dataset distance while being more efficient than existing dataset discrepancy measures. Moreover, it correlates well with the performance gap in transfer learning and classification accuracy in data augmentation.
+
+# 1. Introduction
+
+Dataset distances provide a powerful framework for comparing datasets based on their underlying structures, distributions, or content. These measures are essential in applications where understanding the relationships between datasets drives decision-making, such as assessing data quality, detecting distributional shifts, or quantifying biases. They play a critical role in machine learning workflows, enabling tasks like domain adaptation, transfer learning,
+
+*Equal contribution  $^{1}$ Department of Statistics and Data Sciences, University of Texas at Austin, Texas, USA  $^{2}$ Qualcomm AI Research, Qualcomm Vietnam Company Limited. Correspondence to: Khai Nguyen <khainb@utexas.edu>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+continual learning, and fairness evaluation. Additionally, dataset distances are valuable in emerging areas such as synthetic data evaluation, 3D shape comparison, and federated learning, where comparing heterogeneous data distributions is fundamental. By capturing meaningful similarities and differences between datasets, these measures facilitate data-driven insights, enhance model robustness, and support novel applications across diverse fields.
+
+A common approach to comparing datasets relies on proxies, such as analyzing the learning curves of a predefined model (Leite & Brazdil, 2005; Gao & Chaudhari, 2021) or examining its optimal parameters (Achille et al., 2019; Khodak et al., 2019) on a given task. Another strategy involves making strong assumptions about the similarity or co-occurrence of labels between datasets (Tran et al., 2019). However, these methods often lack theoretical guarantees, are heavily dependent on the choice of the probe model, and require training the model to completion (e.g., to identify optimal parameters) for each dataset under comparison. To address limitations of previous approaches, model-agnostic approaches are developed. These methods often assess task similarity based on the similarity between joint or conditional input-output distributions, occasionally incorporating the loss function as well. Principled notions of domain discrepancy (Ben-David et al., 2006; Mansour et al., 2009) provide a more rigorous foundation but are frequently impractical due to computational intractability or poor scalability to large datasets.
+
+More recently, approaches based on Optimal Transport (Villani, 2008; Peyre & Cuturi, 2020) (OT) have demonstrated promise in modeling dataset similarities (Alvarez-Melis & Fusi, 2020; Tan et al., 2021). Authors in (Alvarez-Melis & Fusi, 2020) introduced optimal transport dataset distance (OTDD) which is based on a hierarchical OT framework drawing from (Yurochkin et al., 2019). In this approach, an inner OT problem calculates the label distance between the class-conditional distributions of two supervised learning tasks. This label distance is then integrated into the transportation cost of an outer OT problem, yielding a dataset distance that accounts for both sample and label discrepancies, and can handle disjoint sets of labels from different datasets. Meanwhile, authors in (Tan et al., 2021) treat the optimal transport plan between the input distributions of datasets as a joint probability distribution and leverage con
+
+ditional entropy to quantify the difference between datasets.
+
+A key limitation of OT-based approaches is their high computational complexity. These methods require pairwise OT (or entropy-regularized OT) computations across datasets and classes. This process can be prohibitively expensive, especially for applications that demand frequent dataset similarity evaluations. Solving optimal transport incurs a time complexity of  $\mathcal{O}(n^3\log n)$  (Peyre & Cuturi, 2020) and a space complexity of  $\mathcal{O}(n^2)$ , where  $n$  is the maximum number of realizations in the two input distributions. Using entropic regularization (Cuturi, 2013) for approximate solutions reduces the complexity to  $\mathcal{O}(n^2\log n / \epsilon^3)$  (Altschuler et al., 2017), with  $\epsilon$  denoting the accuracy level. To enhance efficiency, (Liu et al., 2025) proposes using multidimensional scaling (MDS) (Cox & Cox, 2000) to embed class labels as vectors while preserving the Wasserstein distance geometry for label distances. The authors then leverage a reference distribution to form Wasserstein embeddings of datasets (Wang et al., 2013; Kolouri et al., 2021), termed Wasserstein Task Embedding (WTE). Although WTE is more efficient than OTDD, performing MDS and Wasserstein embedding remains costly due to the need for solving optimal transport. Additionally, WTE is not a valid metric.
+
+Sliced optimal transport or sliced Wasserstein (SW) distance (Bonneel et al., 2015; Rabin et al., 2012) is a well-known approach to obtain scalable optimal transport distance between distributions. The SW relies on random projections to exploit the closed-form of Wasserstein distance in one-dimension with  $\mathcal{O}(n\log n)$  in time complexity and  $\mathcal{O}(n)$  in space complexity. The challenge in using SW as a dataset distance is that carrying out projection for class labels is non-trivial. Authors in (Bonet et al., 2024) bypass the problem by using MDS to embed class labels into vectors on a chosen manifold, then utilize SW on product of manifolds i.e., Cartan-Hadamard Sliced-Wasserstein (CHSW), as the dataset distance. Nevertheless, conducting MDS as WTE leads to extra computation which has quadratic computation in terms of the number of classes that is undesirable when having a large number of classes.
+
+Our goal is to develop a model-agnostic sliced optimal transport distance for comparing datasets, which is also embedding-agnostic, requiring no additional preprocessing. Furthermore, we aim for the distance to achieve (near-)linear computational complexity with respect to the number of data points and feature dimensions, while being robust to the number of classes in the datasets. To this end, we propose a novel approach for projecting a data point-comprising a feature and a label-onto a real number. This projection includes an innovative label (class) projection, which maps a label, represented as a conditional distribution over features (as in OTDD), to a scalar.
+
+Contribution. In summary, our contributions are three-fold:
+
+1. We propose Moment Transform Projection (MTP), which maps a label, a distribution over high-dimensional features, to a single scalar. MTP first projects the distribution of interest onto one dimension using a feature projection, then computes a scaled moment to convert the projected one-dimensional distribution into a scalar. We prove that MTP is injective under certain regularity assumptions of the underlying dataset distributions. Using MTP, we derive data point projection that combines the label projections from multiple MTPs with a feature projection to obtain a one-dimensional representation of a data point. By connecting data point projection to the hierarchical hybrid projection approach (Nguyen & Ho, 2024), we show that data point projection is injective, given the injectivity of the MTP.
+
+2. We propose the sliced optimal transport dataset distance (s-OTDD), the first formal dataset distance based on sliced optimal transport. The s-OTDD is defined as the expected Wasserstein distance between the projected one-dimensional distributions of two input datasets, where the projections are determined by random parameters. We prove that the s-OTDD is a valid distance on the space of distributions over the product of the feature space and the space of all distributions on the feature space. Moreover, we discuss in detail the computational properties of the s-OTDD including approximation error, computational complexities, and other computational benefits.
+
+3. We show that the s-OTDD exhibits a strong correlation with the OTDD when comparing subsets from MNIST and CIFAR10, while being faster than existing competitors. Moreover, we observe that it correlates well with the performance gap in transfer learning across various datasets and modalities, including NIST datasets for images, as well as AG News, DBPedia, Yelp Reviews, Amazon Reviews, and Yahoo Answers for text. Finally, we also find that the s-OTDD correlates well with classification accuracy in data augmentation on CIFAR10 and Tiny-ImageNet.
+
+Organization. We begin by reviewing preliminaries on optimal transport dataset distance in Section 2. Next, we present the proposed s-OTDD and its key contributions in Section 3. Experiments demonstrating the favorable performance of the s-OTDD are provided in Section 4. Finally, we conclude in Section 5 and defer proofs of key results and additional materials to the Appendices.
+
+Notations. We denote  $\mathcal{P}(\mathcal{X})$  as the set of all distributions on the set  $\mathcal{X}$ . We denote  $\mathbb{R}^d$  as the set of real numbers in  $d > 0$  dimensions,  $\mathbb{N} = 1,2,\dots ,\infty$  as the set of natural numbers, and  $\mathbb{S}^{d - 1}$  as the unit hypersphere in  $d > 1$  dimensions. For a dataset  $\mathcal{D} = (x_{1},y_{1}),\ldots ,(x_{n},y_{n})$ , we define its empirical distribution as  $P_{\mathcal{D}} = \frac{1}{n}\sum_{i = 1}^{n}\delta_{(x_i,y_i)}$ . We denote  $[[n]]$  as the set  $\{1,\dots ,n\}$ . For any two sequences  $a_{n}$  and  $b_{n}$ , the notation  $a_{n} = \mathcal{O}(b_{n})$  means that  $a_{n}\leq C b_{n}$  for all  $n\geq 1$  for a constant  $C$ . Lastly,  $\lambda!$  denotes the factorial of  $\lambda >0$ .
+
+# 2. Preliminaries
+
+In this section, we review the definition of Wasserstein distance, sliced Wasserstein distance, and their computational properties, and restate the optimal transport dataset distance.
+
+Wasserstein Distance. The Wasserstein- $p$  distance (Villani, 2008; Peyre & Cuturi, 2020)  $(p \geq 1)$  between two distributions  $\mu \in \mathcal{P}(\mathcal{X})$  and  $\nu \in \mathcal{P}(\mathcal{X})$ , where  $\mathcal{X}$  are subsets of  $\mathbb{R}^d$  and has a ground metric  $d_{\mathcal{X}}: \mathcal{X} \times \mathcal{X} \to \mathbb{R}^+$ , is defined as:
+
+$$
+\mathbf {W} _ {p} ^ {p} (\mu , \nu) := \inf  _ {\pi \in \Pi (\mu , \nu)} \int_ {\mathcal {X} \times \mathcal {X}} d _ {\mathcal {X}} (x, x ^ {\prime}) ^ {p} d \pi (x, x ^ {\prime}), \tag {1}
+$$
+
+where  $\Pi (\mu ,\nu)$  is the set of all possible transportation plan i.e., all joint distributions  $\pi (x,x^{\prime})$  such that  $\pi (x,\mathcal{X}) = \mu (x)$  and  $\pi (\mathcal{X},x^{\prime}) = \nu (x^{\prime})$ . When  $\mu$  and  $\nu$  are discrete i.e.,  $\mu = \sum_{i = 1}^{n}\alpha_{i}\delta_{x_{i}}$  and  $\nu = \sum_{j = 1}^{m}\beta_{j}\delta_{x_{j}^{\prime}}$  with  $\sum_{i = 1}^{n}\alpha_{i} = \sum_{j = 1}^{m}\beta_{j} = 1$  ( $\alpha_{i} > 0\forall i$ , and  $\beta_{j} > 0\forall j$ ), we have:
+
+$$
+\mathrm {W} _ {p} ^ {p} (\mu , \nu) := \min  _ {\gamma \in \Gamma (\boldsymbol {\alpha}, \boldsymbol {\beta})} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \gamma_ {i j} d _ {\mathcal {X}} ^ {p} \left(x _ {i}, x _ {j} ^ {\prime}\right), \tag {2}
+$$
+
+where  $\Gamma (\alpha ,\beta) = \{\gamma \in \mathbb{R}_+^{n\times m}\mid \gamma \mathbf{1} = \beta ,\gamma^\top \mathbf{1} = \alpha \}$ ,  $\pmb {\alpha} = (\alpha_{1},\dots ,\alpha_{n})$ , and  $\beta = (\beta_{1},\ldots ,\beta_{n})$ . The time complexity and the space complexity of the Wasserstein distance is  $\mathcal{O}(n^3\log n)$  (Peyre & Cuturi, 2020) and  $\mathcal{O}(n^2)$  in turn which are very expensive. Using entropic regularization reduces the complexity to  $\mathcal{O}(n^2\log n / \epsilon^3)$  with  $\epsilon$  denoting the accuracy level. To avoid such quadratic complexity, sliced Wasserstein is proposed as an alternative solution.
+
+Sliced Wasserstein distance. The sliced Wasserstein (SW) distance is motivated from the closed-form solution of the one-dimensional Wasserstein distance with ground metric  $d_{\mathcal{X}}(x,x') = h(x - x')$  with  $\mathcal{X} \subset \mathbb{R}$  and  $h$  is a strictly convex function:
+
+$$
+\mathrm {W} _ {p} ^ {p} (\mu , \nu) = \int_ {0} ^ {1} d _ {\mathcal {X}} \left(F _ {\mu} ^ {- 1} (z), F _ {\nu} ^ {- 1} (z)\right) ^ {p} d z, \tag {3}
+$$
+
+where  $F_{\mu}^{-1}$  and  $F_{\nu}^{-1}$  are inverse CDF of  $\mu$  and  $\nu$  respectively. When  $\mu$  and  $\nu$  are discrete with at most  $n$  supports, the time complexity and the space complexity for computing the closed-form are  $\mathcal{O}(n\log n)$  (Peyre & Cuturi, 2020) and  $\mathcal{O}(n)$  respectively. To exploit the closed-form, the SW distance relies on random projections. For  $p\geq 1$ , the SW distance (Bonneel et al., 2015) of  $p$ -th order between two distributions  $\mu \in \mathcal{P}(\mathcal{X})$  and  $\nu \in \mathcal{P}(\mathcal{X})$  with  $\mathcal{X}\subset \mathbb{R}^d$  is defined as follow:
+
+$$
+\mathbf {S W} _ {p} ^ {p} (\mu , \nu) = \mathbb {E} _ {\theta \sim \mathcal {U} (\mathbb {S} ^ {d - 1})} \left[ \mathbf {W} _ {p} ^ {p} \left(\mathcal {R} _ {\theta} \sharp \mu , \mathcal {R} _ {\theta} \sharp \nu\right) \right], \tag {4}
+$$
+
+where  $\mathcal{R}_{\theta}\sharp \mu$  and  $\mathcal{R}_{\theta}\sharp \nu$  are the one-dimensional pushforward distributions of  $\mu$  and  $\nu$  through the function  $\mathcal{R}_{\theta}(x) = \theta^{\top}x$  (derived from Radon Transform (RT) (Heggason, 2011)), and  $\mathcal{U}(\mathbb{S}^{d - 1})$  is the uniform distribution over
+
+the unit hypersphere in  $d$  dimensions. In addition to the computational benefit, SW has been widely known for its low sample complexity (Manole et al., 2022; Nadjahi et al., 2020; Nietert et al., 2022; Boedihardjo, 2025).
+
+Optimal Transport dataset distance. We are given two datasets  $\mathcal{D}_1 = \{(x_i, y_i)\}_{i=1}^n$  and  $\mathcal{D}_2 = \{(x_j', y_j')\}_{j=1}^m$ , where  $(x_i, y_i), (x_j', y_j') \in \mathcal{X} \times \mathcal{Y}, \forall i \in [n]$ ,  $\forall j \in [[m]]$  with  $\mathcal{X}$  is the space of features and  $\mathcal{Y}$  is the space of labels. We assume that we know the support distance on the space of features  $\mathcal{X}$  i.e.,  $d_{\mathcal{X}}: \mathcal{X} \times \mathcal{X} \to \mathbb{R}^+$  e.g., Euclidean distance. In contrast to the mild assumption of having  $d_{\mathcal{X}}$ , we rarely know the distance on the space of labels  $\mathcal{Y}$  e.g., we cannot tell if the label "dog" is closer to the label "cat" than to the label "bird". As a solution, authors in (Alvarez-Melis & Fusi, 2020) propose to map a label into a distribution over features, then use distances between distributions as the proxy for the distance between labels. In particular, for the first dataset, we assume that  $(x_1, y_1), \ldots, (x_n, y_n) \sim q(X, Y)$  where  $q(X, Y)$  is an unknown joint distribution. After that, we can define  $q_y(X) = q(X|Y = y)$  as the conditional distribution over features given the label  $y$ . Similarly, for the second dataset, we can obtain  $q_{y'}(X')$  for a label  $y'$ . Therefore, the distance between labels can be defined as:
+
+$$
+d _ {\mathcal {Y}} \left(y, y ^ {\prime}\right) = \mathbb {D} \left(q _ {y} (X), q _ {y ^ {\prime}} \left(X ^ {\prime}\right)\right), \tag {5}
+$$
+
+where  $\mathbb{D}$  is a distance between two distributions. In practice, we observe  $q_{y}(X)$  and  $q_{y^{\prime}}(X^{\prime})$  in empirical forms, hence, optimal transport distances naturally serve as a measure for  $\mathbb{D}$ . Authors in (Alvarez-Melis & Fusi, 2020) suggest using the Wasserstein distance i.e.,  $d_{\mathcal{Y}}(y,y^{\prime}) = W_{p}(q_{y}(X),q_{y^{\prime}}(X^{\prime}))$ , where we abuse the notation of the Wasserstein distance between two pdfs as the Wasserstein distance between the two distributions. Due to the expensive computational complexities of the Wasserstein distance, authors in (Alvarez-Melis & Fusi, 2020) propose to approximate two distributions by two multivariate Gaussians, then use the closed-form solution of Wasserstein-2 distance as the label distance. Nevertheless, the approximation can only capture the first two moments of the two distributions, hence, the resulting comparison might not be accurate. After having the label distance, the optimal transport dataset distance can be defined as:
+
+$$
+\begin{array}{l} O T D D (\mathcal {D} _ {1}, \mathcal {D} _ {2}) \\ = \min  _ {\gamma \in \Gamma \left(\frac {1}{n}, \frac {1}{m}\right)} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \gamma_ {i j} d \left(\left(x _ {i}, y _ {i}\right), \left(x _ {j} ^ {\prime}, y _ {j} ^ {\prime}\right)\right), \tag {6} \\ \end{array}
+$$
+
+which is the Wasserstein distance with the ground metric:  $d^{p}((x_{i},y_{i}),(x_{j}^{\prime},y_{j}^{\prime})) = d_{\mathcal{X}}^{p}(x_{i},x_{j}^{\prime}) + d_{\mathcal{Y}}^{p}(y_{i},y_{j}^{\prime})$ , which is the combination of the feature distance and the label distance. As long as the feature distance and the label distance are valid metrics, the OTTD is a valid metric on the product space of  $\mathcal{X}\times \mathcal{P}(\mathcal{X})$  (Alvarez-Melis & Fusi, 2020).
+
+# 3. Sliced Optimal Transport Dataset Distances
+
+While the OTDD is a natural distance between two datasets, it inherits the computational challenges of the Wasserstein distance. Let  $n$  denote the maximum number of data points in the two datasets,  $c$  the maximum number of classes, and  $n_{\mathrm{max}}$  the maximum number of samples per class. The time complexity of OTDD is  $\mathcal{O}(n^3\log n + c^2 (n_{\mathrm{max}}^3\log n_{\mathrm{max}} + d))$ , and the memory complexity is  $\mathcal{O}(n^2 +c^2)$ . When using a Gaussian approximation for the label distribution, the time complexity becomes  $\mathcal{O}(n^3\log n + c^2 (d^3 +n_{\mathrm{max}}d^2))$ . As a result, OTDD may not be scalable for large datasets. To address this, we aim to develop a sliced optimal transport version of OTDD that leverages the one-dimensional closed-form of the Wasserstein distance. This requires developing a novel approach to project a data point onto a single scalar.
+
+# 3.1. Label Projection
+
+As mentioned in Section 2, a label is represented as a distribution over the feature space  $\mathcal{X}$ . Given a label  $y$ , we would like to map the label distribution  $q_{y}$  to a scalar. To achieve our goal, there are two steps: projecting  $q_{y}$  to one-dimension through feature projection, and transforming the one-dimensional projection of  $q_{y}$  into a scalar.
+
+Feature Projection. Since a label is treated as a distribution over the feature space, we can project a label to an one-dimensional distribution through a feature projection i.e., a mapping from  $\mathcal{FP}_{\theta}:\mathcal{X}\to \mathbb{R}$  with the projection parameter  $\theta$  belongs to a projection space  $\Theta$ . We can choose any feature projection methods based on the prior knowledge of the feature space  $\mathcal{X}$  e.g., Euclidean space (Bonneel et al., 2015), images (Nguyen & Ho, 2022), functions (Garrett et al., 2024), spherical space (Tran et al., 2024; Quellmalz et al., 2023; 2024), hyperbolic space (Bonet et al., 2023), manifolds (Bonet et al., 2024; Nguyen & Mueller, 2024), and so on. It is worth noting that a feature projection is injective if  $\mathcal{FP}_{\theta}\sharp \mu_1 = \mathcal{FP}_{\theta}\sharp \mu_2$  for all  $\theta \in \Theta$  then  $\mu_{1} = \mu_{2}$  for any  $\mu_{1},\mu_{2}\in \mathcal{P}(\mathcal{X})$ . The injectivity of the projection is vital to preserve the distributional metricity.
+
+Scaled Moment. We now discuss how to map a one-dimensional distribution into a scalar. More importantly, we want the transformation to be injective. To design such transformation, we rely on scaled moments.
+
+Definition 1. Given a distribution  $\mu \in \mathcal{P}(\mathbb{R})$  with the density function  $f_{\mu}$  and a set  $\Lambda \subset \mathbb{N}$  such that  $\int_{\mathbb{R}}\frac{x^{\lambda}}{\lambda!} f_{\mu}(x)dx < \infty$  for all  $\lambda \in \Lambda$ , the  $\lambda$ -th scaled moment of  $\mu$  is defined as:
+
+$$
+\mathcal {S M} _ {\lambda} (\mu) = \int_ {\mathbb {R}} \frac {x ^ {\lambda}}{\lambda !} f _ {\mu} (x) d x \tag {7}
+$$
+
+The difference between the scaled moment and the conventional moment is that the scaled moment is scaled by the factorial function of the order  $\lambda$ . The scaling is for avoiding
+
+exploding in value of high moments.
+
+Moment Transform Projection. After the feature projection, we obtain an one-dimensional projection of the label distribution  $\mathcal{FP}_{\theta} \sharp q_{y}$ . We then can obtain the scaled moment of  $\mathcal{FP}_{\theta} \sharp q_{y}$  as the final desired scalar. Before giving the formal definition, we first state the following assumption.
+
+Assumption 1 (Existence of projected scaled moments). A distribution  $\mu \in \mathcal{P}(\mathbb{R}^d)$  with the density function  $f_{\mu}$  has all projected  $\lambda$ -th scaled moments ( $\lambda \in \Lambda \subset \mathbb{N}$ ) given a feature projection  $\mathcal{FP}_{\theta}$  if  $\int_{\mathbb{R}^d} \frac{(\mathcal{FP}_{\theta}(x))^{\lambda}}{\lambda!} f_{\mu}(x) dx < \infty$  for all  $\theta \in \mathbb{S}^{d-1}$  and  $\lambda \in \Lambda$ .
+
+Definition 2. Given a feature projection  $\mathcal{FP}_{\theta}:\mathcal{X}\to \mathbb{R}$  and a set  $\Lambda \subset \mathbb{N}$ , a distribution  $\mu \in \mathcal{P}(\mathbb{R}^d)$  ( $d > 1$ ) that satisfies Assumption 1, the Moment Transform projection  $\mathcal{MTP}_{\lambda ,\theta}:\mathcal{P}(\mathbb{R}^d)\to \mathbb{R}$ , with  $\theta \in \mathbb{S}^{d - 1}$  and  $\lambda \in \Lambda$ , is defined as follows:
+
+$$
+\begin{array}{l} \mathcal {M T P} _ {\lambda , \theta} (\mu) = \mathcal {S M} _ {\lambda} \left(\mathcal {F P} _ {\theta} \sharp \mu\right) \\ = \int_ {\mathbb {R} ^ {d}} \frac {(\mathcal {F P} _ {\theta} (x)) ^ {\lambda}}{\lambda !} f _ {\mu} (x) d x. \tag {8} \\ \end{array}
+$$
+
+When  $\mu$  is an empirical distribution i.e.,  $\mu = \frac{1}{n}\sum_{i=1}^{n}\delta_{x_i}$ , the MTP projection of  $\mu$  given  $\lambda$  and a feature projection  $\mathcal{FP}_{\theta}$  is  $\mathcal{MTP}_{\lambda,\theta} = \frac{1}{n}\sum_{i=1}^{n}\frac{(\mathcal{FP}_{\theta}(x))^{\lambda}}{\lambda!}$ . We now discuss the distributionally injectivity of the MTP.
+
+Proposition 1. For  $\mu, \nu \in \mathcal{P}(\mathbb{R}^d)$  and a injective feature projection  $\mathcal{FP}_{\theta}$  and a set  $\Lambda \subset \mathbb{N}$ ,  $\mathcal{MTP}_{\lambda,\theta}(\mu) = \mathcal{MTP}_{\lambda,\theta}(\nu)$  for all  $\theta \in \mathbb{S}^{d-1}$  and  $\lambda \in \Lambda$  implies  $\mu = \nu$  if either following conditions hold:
+
+(1)  $\mu$  and  $\nu$  satisfies Assumption 1 with an infinite set  $\Lambda = \mathbb{N}$  and moment generating functions (MGFs) of  $\mathcal{FP}_{\theta} \sharp \mu$  and  $\mathcal{FP}_{\theta} \sharp \nu$  exist for all  $\theta \in \Theta$ .  
+(2)  $\mu$  and  $\nu$  satisfies Assumption 1 with an finite set  $\Lambda \subset \mathbb{N}$ , and for all  $\theta \in \Theta$ , the projected Hankel matrices
+
+$$
+A _ {\theta , \mu} = \left[ \begin{array}{c c c c} m _ {\theta , \mu , 0} & m _ {\theta , \mu , 1} & \dots & m _ {\theta , \mu , \lambda_ {\max}} \\ m _ {\theta , \mu , 1} & m _ {\theta , \mu , 2} & \dots & m _ {\theta , \mu , 0} \\ \vdots & \vdots & \ddots & \vdots \\ m _ {\theta , \mu , \lambda_ {\max}} & m _ {\theta , \mu , 0} & \dots & m _ {\theta , \mu , \lambda_ {\max} - 1} \end{array} \right]
+$$
+
+with  $m_{\theta,\mu,\lambda}$  is  $\lambda$ -th moment of  $\mathcal{FP}_{\theta} \sharp \mu$  (similar with  $m_{\theta,\nu,\lambda}$ ) are positive definite and  $|m_{\theta,\mu,\lambda}| < CD^{\lambda} \lambda!$  and  $|m_{\theta,\nu,\lambda}| < CD^{\lambda} \lambda!$  for some constants  $C$  and  $D$  for all  $\lambda \in \Lambda$ .
+
+The proof of Proposition 1 is given in Appendix A.1 and is based on the Hamburger moment problem (Reed & Simon, 1975; Chihara, 2011).
+
+# 3.2. Data Point Projection
+
+With the proposed label projection, we can propose data point projection. We recall that a data point is a pair of a
+
+feature and a label or equivalently a pair of a feature and a distribution over features, which is denoted as  $(x,q_y)\in \mathcal{X}\times \mathcal{P}(\mathcal{X})$ . For the feature domain, as discussed, we can use any feature projections based on the prior knowledge of the feature space, denoted as  $\mathcal{FP}_{\theta}:\mathcal{X}\to \mathbb{R}$ . For the label, we can use MTP defined in the Section 3.1. Now, we need to combine the outputs of the feature projection and the label projection to obtain the final scalar.
+
+Definition 3. Given a data point  $(x, q_y) \in \mathcal{X} \times \mathcal{P}(\mathcal{X})$  and  $k \geq 1$ , the data point projection can be defined as follows:
+
+$$
+\begin{array}{l} \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} (x, q _ {y}) = \psi^ {(1)} \mathcal {F P} _ {\theta} (x) \\ + \sum_ {i = 1} ^ {k} \psi^ {(i + 1)} \mathcal {M T P} _ {\lambda^ {(i)}, \phi} (q _ {y}), \tag {9} \\ \end{array}
+$$
+
+where  $\psi = (\psi^{(1)},\psi^{(2)},\ldots ,\psi^{(k + 1)})\in \mathbb{S}^k,\theta \in \Theta ,\lambda = (\lambda^{(1)},\dots ,\lambda^{(k)})\in \Lambda^k,\phi \in \Phi$  with  $\Theta$  and  $\Phi$  are projection space of the feature projections.
+
+Given a dataset  $\mathcal{D} = \{(x_1, q_{y_1}), \ldots, (x_1, q_{y_n})\}$ , we have the projected distribution of the corresponding empirical distribution through the data point projection is  $\mathcal{DP}_{\psi, \theta, \lambda, \phi}^k P_{\mathcal{D}} = \frac{1}{n} \sum_{i=1}^{n} \delta_{\mathcal{DP}_{\psi, \theta, \lambda, \phi}^k(x_i, q_{y_i})}$ . The proposed data point projection combines the outputs of  $k \geq 1$  MTPs to obtain the final projection value. It follows the principle of hierarchical hybrid projection in (Nguyen & Ho, 2024) to retain the overall injectivity.
+
+Corollary 1. The data point projection is injective when the feature projection and the MTP is injective i.e., for  $\mu, \nu \in \mathcal{X} \times \mathcal{P}(\mathcal{X})$  if  $\mathcal{D}\mathcal{P}_{\psi, \theta, \lambda, \phi}^{k} \sharp \mu = \mathcal{D}\mathcal{P}_{\psi, \theta, \lambda, \phi}^{k} \sharp \nu$  for all  $\psi \in \mathbb{S}, \theta \in \Theta, \lambda \in \Lambda \subset \mathbb{N}, \phi \in \Phi$ .
+
+The Corollary 1 follows the fact that the data point projection is a composition of injective projections (Nguyen & Ho, 2024) i.e., the Radon Transform projection and the MTP.
+
+It is worth noting that we can use the same value for  $\theta$  and  $\phi$  when we use a single feature projection to project feature and to construct label projections. This approach not only reduces memory consumption for storing projection parameters but also saves computation since we need to project features only once.
+
+# 3.3. Sliced Optimal Transport Dataset Distance
+
+With projections of data points, we can now introduce the sliced optimal transport dataset distance (s-OTDD).
+
+Definition 4. Let  $\mathcal{D}_1$  and  $\mathcal{D}_2$  be the two given datasets,  $P_{\mathcal{D}_1}$  and  $P_{\mathcal{D}_2}$  be corresponding empirical distributions of  $\mathcal{D}_1$  and  $\mathcal{D}_2$  respectively, the sliced optimal transport dataset distance (s-OTDD) of order  $p > 0$  is defined as follows:
+
+$$
+\begin{array}{l} s - O T D D _ {p} ^ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}\right) \\ = \mathbb {E} \left[ \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right], \tag {10} \\ \end{array}
+$$
+
+where the expectation is with respect to the random projection parameters  $(\psi, \theta, \lambda, \phi) \sim \mathcal{U}(\mathbb{S}) \otimes \mathcal{U}(\mathbb{S}^{d-1}) \otimes \sigma(\Lambda^k) \otimes \mathcal{U}(\Phi)$  with  $\phi \in \Phi$ , and  $\sigma(\Lambda^k)$  is the uniform distribution on  $\Lambda^k$  when  $\Lambda$  is finite or the product of zero-truncated Poisson distributions (Cohen, 1960) when  $\Lambda$  is infinite.
+
+Proposition 2. The sliced optimal transport dataset distance  $s\text{-OTDD}_p(\mathcal{D}_1,\mathcal{D}_2)$  defines a valid metric on  $\mathcal{P}(\mathcal{X}\times \mathcal{P}(\mathcal{X}))$  the space of distributions over feature and label-distribution pairs if the data point projection is injective.
+
+Proof of Proposition 2 is given in Appendix A.2. The proposition strengthens the geometrical benefits of the s-OTDD.
+
+Numerical Approximation. The expectation in s-OTDD is intractable, hence, numerical approximation such as Monte Carlo integration must be used. In particular, we sample  $(\psi_{1},\theta_{1},\lambda_{1},\phi_{1}),\ldots ,(\psi_{L},\theta_{L},\lambda_{L},\phi_{L})\sim \mathcal{U}(\mathbb{S})\otimes$ $\mathcal{U}(\mathbb{S}^{d - 1})\otimes \sigma (\Lambda^k)\otimes \mathcal{U}(\Phi)$  with the number of projections  $L > 0$  , then we form the following estimation:
+
+$$
+\begin{array}{l} \mathrm {s -} \widehat {\mathrm {O T D D}} _ {p} ^ {p} (\mathcal {D} _ {1}, \mathcal {D} _ {2}; L) = \\ \frac {1}{L} \sum_ {l = 1} ^ {L} \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right). \tag {11} \\ \end{array}
+$$
+
+In practice, we do not know the number of moments. Hence, we can assume it to be infinite and use the product of zero-truncated Poisson distributions (Cohen, 1960) with some rate hyperparameters as  $\sigma(\Lambda^k)$ . Another approach is to cap the number of moments to be  $\lambda_{max} \geq 1$  i.e.,  $\Lambda = \{1,2,\dots,\lambda_{max}\}$ , then use the uniform distribution  $\mathcal{U}(\Lambda^k)$  as  $\sigma(\Lambda^k)$ . We now discuss the approximation error of the s-OTDD when using the Monte Caro estimation.
+
+Proposition 3. For any  $p \geq 1$ , and two datasets  $\mathcal{D}_1$  and  $\mathcal{D}_2$ , we have the following approximation error:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left| s \widehat {- O T D D} _ {p} ^ {p} (\mathcal {D} _ {1}, \mathcal {D} _ {2}; L) - s \text {-} O T D D _ {p} ^ {p} (\mathcal {D} _ {1}, \mathcal {D} _ {2}) \right| \right] \leq \\ \frac {1}{\sqrt {L}} \operatorname {V a r} \left[ W _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right], \tag {12} \\ \end{array}
+$$
+
+where the variance is with respect to  $(\psi, \theta, \lambda, \phi) \sim \mathcal{U}(\mathbb{S}) \otimes \mathcal{U}(\mathbb{S}^{d-1}) \otimes \sigma(\Lambda^k) \otimes \mathcal{U}(\Phi)$ .
+
+The proof of Proposition 3 is given in Appendix A.3. We can see that the approximation error reduces fast when increasing the number of projections  $L$  i.e.,  $\mathcal{O}(L^{-1/2})$ . It is worth noting that variance reduction can also be used (Nguyen & Ho, 2023; Nguyen et al., 2024; Leluc et al., 2024). We refer the reader to Algorithm 1 in Appendix B for a detailed computational algorithm.
+
+Computational Complexities. Using a feature projection with linear complexity in the number of dimensions  $d$  (e.g., Radon transform, convolution, etc), the time complexity of s-OTDD is  $\mathcal{O}(L(n\log n + dn))$  and the space complexity is
+
+![](images/1f05a3d81123b452db3301a27118a67f4717fce91875de486f5edc8ca867be38.jpg)
+
+![](images/9d14a84f80b4f76f02e24ca8b4361823cbea0a9671b53ec1dc0436ce32b3ca42.jpg)
+
+![](images/1d3eb7414c66466dabe9531b2367d466c323c7c5f344ab826c9e9e8ca4c8736b.jpg)
+
+![](images/07292f4af890af65b678b51e4b507ed2018b320486c3d715351321cfcdc20111.jpg)
+
+![](images/bfe3c366156d2ec4011083c0744e4b4b0a7c06a56879a9983030523a9a7881e2.jpg)  
+Figure 1: The figure shows distance correlation with OTDD (Exact) of OTDD (Gaussian approximation), WTE, CHSW, and s-OTDD.
+
+![](images/4d691a8a3c6b11564b7bcdea27c28849dd35906f1c406d6c9b26b2dcf0e1e8ae.jpg)
+
+![](images/7d3b32efc5411c8d7027da77c80d92471858360b5aa0fb038eca3fe3332f3c34.jpg)
+
+![](images/72c6d24c2e9bddcd2295e48c79c3dff336fa9a92e9fa834198e8ba2a792adccd.jpg)
+
+$\mathcal{O}(L(d + n))$ , where  $n$  is the number of data points and  $L$  is the number of projections. Interestingly, both time complexity and memory complexity do not depend on the number of classes  $c$  and the maximum class size  $n_{max}$  (good for imbalanced data). In a greater detail, the time complexity for applying projection for the class  $i$  is  $\mathcal{O}(Ln_i d)$  where  $n_i$  is the size of the class  $i$ . As a result, the total time complexity for projection is  $\mathcal{O}\left(L \sum_{i=1}^{k} n_i d\right) = \mathcal{O}(Lnd)$ . After having the projections, solving one-dimensional Wasserstein distances costs  $\mathcal{O}(Ln \log n)$  in time. For space complexity,  $\mathcal{O}(L(d + n))$  is for storing projection parameters and projections. It is worth noting that the majority of computation of s-OTDD is for projecting and computing inverse CDFs of projected distributions of datasets, and such computation can be done independently for each dataset. After that, s-OTDD value can be obtained with linear complexity with the precomputed projected inverse CDFs. Therefore, s-OTDD is suitable for distributed and federated settings where datasets can be stored across machines.
+
+# 4. Experiments
+
+In Section 4.1, we conduct analysis for the s-OTDD in comparing subsets of the MNIST dataset (LeCun et al., 1998) and the CIFAR10 dataset (Krizhevsky et al., 2009). In particular, we analyze the correlations, which Spearman's rank correlation denoted by  $\rho$  and Pearson correlation denoted by  $r$ , of the s-OTDD with OTDD. While prior work (Alvarez-Melis & Fusi, 2020) reports only Spearman's rank correlation, we also compute Pearson correlation for a more comprehensive analysis, though Spearman's correlation remains our primary metric to maintain comparability with existing
+
+bases. Furthermore, we compare the computational time of the proposed s-OTDD with existing dataset distance metrics and investigate its dependency on the number of projections and on the number of moments. In Section 4.2, we evaluate the correlations of the s-OTDD with the performance gap in transfer learning on image NIST datasets (Deng, 2012) and a diverse set of text datasets (Zhang et al., 2015) including AG News, DBPedia, Yelp Reviews (with both 5-way classification and binary polarity labels), Amazon Reviews (with both 5-way classification and binary polarity labels), and Yahoo Answers. Also, to demonstrate the robustness and scability of proposed method, we empirically validate the correlations between s-OTDD distance and transferability on large-scale dataset, which is Split Tiny-ImageNet (Le & Yang, 2015) at  $224 \times 224$  resolution. Finally, we test the performance of the s-OTDD with the OTDD as the reference in distance-driven data augmentation for image classification on CIFAR10 (Krizhevsky et al., 2009) dataset and Tiny-ImageNet (Le & Yang, 2015) dataset in Section 4.3. When dealing with the high dimensional datasets, i.e. CIFAR10 dataset and Tiny-ImageNet dataset, we use the convolution feature projection (Nguyen & Ho, 2022) for the s-OTDD while we use the Radon Transform projection for the NIST datasets and text feature datasets. We also provide experiments to compare the convolution feature projection and the Radon transform projection on the NIST datasets in Figure 12 in Appendix B. In general, the convolution-based projections require fewer projections yet exhibit stronger positive correlations between distance and adaptation performance. For all experiments, we use s-OTDD with  $k = 5$  and  $\sigma(\Lambda^k)$  be the product of  $k$  Truncated Poisson distributions which have the corresponding
+
+rate parameters be  $1,\dots,5$  in turn.
+
+# 4.1. Empirical Analysis of Sliced Optimal Transport Dataset Distance
+
+Distance Correlation. We treat the OTDD (Exact) as the desired dataset distance. After that, we compare the proposed s-OTDD with OTDD (Gaussian approximation), WTE, and CHSW in terms of correlations with the OTDD. We randomly split MNIST and CIFAR10 to create subdataset pairs, each ranging in size from 5,000 to 10,000. For each split, we plot the results of each method and OTDD (Exact), and report their correlations. The results are fully presented in Figure 9 and Figure 10 in Appendix B. The process is repeated 10 times and is summarized in Figure 1. From the figure, we observe that the s-OTDD (10,000 projections) has significant correlations with the OTDD (Exact) in both datasets. Moreover, s-OTDD shows comparable performance with OTDD (Gaussian approximation) and WTE while being computationally faster than them (see the next analysis).
+
+Computational Time. We compare the computational time of the proposed s-OTDD with existing discussed dataset distances. We randomly split the MNIST dataset and the CIFAR10 dataset into two subdatasets which has the size varied from 1,000 to the full size of corresponding dataset. The total runtime includes all preprocessing steps, such as estimating the dataset's mean and covariance, as well as performing MDS for WTE and CHSW. We report the computational time in Figure 2. We observe that s-OTDD, with different numbers of projections, scales efficiently as the dataset size increases. Additionally, when moving from MNIST to CIFAR10 (which has a higher feature dimension), the computation time for s-OTDD does not increase significantly compared to OTDDs, WTE, and CHSW. In contrast to s-OTDD, other dataset distances struggle with relatively large datasets i.e., they are able to run for dataset sizes below 30,000 but then crash due to memory limitations (see Appendix C for the detail of the computational infrastructure). Thank to the computational benefits, s-OTDD successfully runs on large datasets, proving that it is more scalable than other existing dataset distances.
+
+Projection Analysis. In addition to investigating the effect of varying the number of projections on computational time in Figure 2, we assess the dependency of s-OTDD to the number of projections. Specifically, we vary the number of projections from 1,000 to 50,000 and examine datasets of different sizes: 1,000, 5,000, 10,000, 15,000, and 20,000, computing the correlations with the configuration using 50,000 projections in each case. The results are reported in Figure 3. From the figure, we see that the
+
+![](images/72a170787a7b9b3ecaee1de16561ae652fcbe4b3d8025104b974b41e29fba27a.jpg)  
+Figure 2: The figure shows computational time of OTDD (Exact), OTDD (Gaussian approx), WTE, CHSW (1,000, 5,000, 10,000 projections), and s-OTDD (1,000, 5,000, 10,000 projections) when varying size of two datasets.
+
+![](images/1b80544ab632c49ff93f23e89c956d75fb5ab8dd2d7ec2417b69319128fdb57c.jpg)
+
+![](images/3e46064c19b0124a441d0c0b3e5eb444db2fd52a208c675fe95d87d4140f0050.jpg)  
+Figure 3: The figure shows Pearson correlations of s-OTDD with s-OTDD (50,000 projections) when varying number of projections from 1,000 to 50,000 in MNIST dataset and CIFAR10 dataset. correlations grows very fast when increasing the number of projections, which strengthens the fast approximation rate in Proposition 3. Furthermore, to analyze how the number of projections influences the correlations with OTDD, we select a specific dataset size and compare the s-OTDD results using fewer projections (500, 1,000, 5,000, and 10,000) against the OTDD (Exact) baseline, with results presented in Figure 9 in Appendix B. The result indicates that increasing the number of projections enhances the alignment between s-OTDD and OTDD (Exact). We also conduct a similar projection analysis for CHSW in Figure 10 in Appendix B.
+
+![](images/47082d053d867f43c435e92473fc964dd54df2e34df19a3812b8230b19918956.jpg)
+
+Number of Moments Analysis. The number of moments for the label projection is denoted by  $k$  in the paper. It is noted that the higher  $k$ , the more moments information of the label distribution is gathered into the data point projection. Hence we want to have  $k$  as big as possible as long as it does not raises numerical issue i.e., overflow which might be due to the non-existence of the higher moments. We conduct an ablation study in which  $k$  is varied in Figure 7.
+
+# 4.2. Transfer Learning
+
+We apply the proposed method to transfer learning, following the OTDD framework. Transferability between source and target datasets is measured using the Performance Gap (PG), defined as the accuracy difference between fine-tuning on the full target dataset and using an adaptation method:
+
+$$
+P G \left(\mathcal {D} _ {S \rightarrow T}\right) = a c c u r a c y \left(\mathcal {D} _ {T}\right) - a c c u r a c y \left(\mathcal {D} _ {S \rightarrow T}\right)
+$$
+
+![](images/a9609149fa4d86363e416a471aa56179820e56708a635bbd207f49d3d06f1c11.jpg)  
+Figure 4: The figure shows correlations of OTDD (Exact) and s-OTDD (10,000 projections) with the performance gap when conducting transfer learning in *NIST datasets.
+
+![](images/34db029960a4d2505b99c9797382ea71ebbe00f9036996fb8e3f2e713740a3ba.jpg)
+
+![](images/2c36bdc8bbc5f5183cc16831b40b61bb2bc207338da1bc3536a8a106bb40eb2f.jpg)  
+Figure 5: The figure shows correlations of OTDD (Exact) and s-OTDD (10,000 projections) with the performance when conducting transfer learning in text datasets.
+
+![](images/4bdc390f3ec6c4bf5b6ff74b3b19bd75126896003e8964d48ffd9660fa0d5bb4.jpg)  
+Figure 6: The figure shows the Pearson and Spearman correlations between s-OTDD (500,000 projections) and the performance gain in the Split Tiny/ImageNet  $(224\times 224)$  experiment.
+
+PG serves a similar role to the Relative Drop metric in OTDD but offers a clearer view of adaptation transferability. A smaller PG indicates better adaptation for similar datasets, while a larger PG is expected for more distinct datasets.
+
+NIST Datasets. We use a simplified LeNet-5, freezing the convolutional layers while fine-tuning the fully connected ones. We utilize the entire dataset for s-OTDD computation, while all other methods are evaluated using a random sample of 10,000 data points. Figure 4 shows a positive correlations between s-OTDD with 10,000 projections and PG ( $\rho = 0.40$ ), which is roughly the same as OTDD (Exact). Additionally, OTDD (Gaussian approximation), CHSW, and WTE yield correlations of  $\rho = 0.40$ ,  $\rho = 0.16$ , and  $\rho = 0.37$ , respectively (see Figure 11 in Appendix B).
+
+Text Datasets. We limit the target dataset to 100 examples per class, fine-tune BERT on the source domain, then adapt and evaluate it on the target domain following OTDD settings. For distance calculations, we sample 5,000 points for OTDD (Exact) and 10,000 points for s-OTDD (10,000 projections). Sentences are embedded using BERT (base) (Li et al., 2022), and distances are computed on these embeddings. The results in Figure 5 show that although OTDD (Exact) outweighs a bit higher than s-OTDD on Spearman rank, but both score the same on Pearson  $(r = 0.48)$ . Crucially, s-OTDD performs the computation in roughly one-tenth the time, providing a substantial speed-efficiency advantage at the expense of a modest drop in Spearman.
+
+![](images/f62cffe7690623c884a99ccf417a8406f3db04f42149230a0bdf2181de0af3ba.jpg)
+
+Split Tiny-ImageNet. We randomly divide the Tiny ImageNet (Le & Yang, 2015) dataset into 10 disjoint tasks, each containing 20 classes. Each image is first rescaled to  $256 \times 256$  and then center-cropped to obtain a  $224 \times 224$  resolution. We initially train a ResNet-18 model on each task, serving as baseline, and subsequently freeze all layers except the final fully connected layer to fine-tune on the target task. We calculate the performance gain after transferability by  $Gain(\mathcal{D}_{S \rightarrow T}) = accuracy(\mathcal{D}_{S \rightarrow T}) - accuracy(\mathcal{D}_T)$ . For distance calculations, we draw 5,000 sample pairs and compute s-OTDD with 500,000 random projections. As shown in Figure 6, s-OTDD exhibits a strong correlation with the performance gain. In contrast, OTDD is infeasible at the  $224 \times 224$  resolution.
+
+From above experiments, we observe that not only s-OTDD achieve comparable performance but also more scalable and robust to existing dataset distances in transfer learning while it offers considerably better computational speed.
+
+# 4.3. Distance-Driven Data Augmentation
+
+As discussed in (Alvarez-Melis & Fusi, 2020), data augmentation enhances dataset quality and diversity but lacks clear guidelines for its effective implementation. A model-agnostic dataset metric can help compare and select the most suitable augmentation strategies. Intuitively, a smaller distance between the augmented source dataset and the target dataset positively correlates with higher accuracy. In this large-scale experiment, we use augmented Tiny-ImageNet as the source dataset, CIFAR10 as the target dataset, and ResNet-50 (He et al., 2016) as the classifier. Augmentations for Tiny-ImageNet include random variations in brightness, contrast, saturation (0.1-0.9), and hue (0-0.5). We evaluate the proposed method and OTDD (Exact), with results presented in Figure 8. For s-OTDD, we sample 50,000 data points per dataset and use 100,000 projections, while for OTDD (Exact), we sample 5,000 data points per dataset. The results are summarized in Figure 8.
+
+![](images/a3d25db9f1fbd986edb8bed455e835f110907b32b272608513d8614a05c1f3bc.jpg)
+
+![](images/039c6afe9c02bb8f1b943184b42dceb2a02d88e769d884ffa4be99b619fd3ed5.jpg)
+
+![](images/7bc0be40cdf56d2193959ac56f35609758e6544aa3d316c2040f14e415f568b1.jpg)
+
+![](images/17c34674c80eb06cbb56379a36677c66005d8634f17f1cf3a9d3d393847c3675.jpg)  
+Figure 7: Experiment when varying number of moments in s-OTDD from 1 to 6 in *NIST Adaptation Experiment.
+
+![](images/9e9cf5bfffe31b5178e2e84bb11a40e20d9566b5c916f415ec9b45f5b38437db.jpg)
+
+![](images/f1c773605719a8069fcfa1e43ceddc9f40471ad792e40ed0f531df9910b4b944.jpg)
+
+![](images/e674301ebab554f9964a7f6351a77d4ddc5839394fb4665c1d369dcb8f8832d7.jpg)  
+Figure 8: The figure shows correlations of OTDD (Exact) OTDD (Exact) and s-OTDD (100,000 projections) with the classification accuracy in data augmentation experiment.
+
+![](images/b657e7dd8841a36bcdc62cd07de009fd93ca9d012a2af1f404351c14b879efeb.jpg)
+
+Figure 8 shows that both OTDD (Exact) and our proposed s-OTDD method produce distances that are negatively correlated, which is natural, with test accuracy when transferring from augmented Tiny-ImageNet to CIFAR10. Despite processing ten times the dataset size compared to OTDD (Exact), s-OTDD is significantly faster, with a processing time in about  $53 \times 10^{3}$  seconds, whereas OTDD (Exact) takes over  $74 \times 10^{3}$  seconds, revealing a huge improvement in efficiency. Moreover, OTDD (Exact) achieves a Spearman correlations of  $-0.70$ , while s-OTDD attains a better correlations of  $-0.74$ . In other words, s-OTDD provides a reliable alternative distance to OTDD (Exact), achieving comparable performance while being faster.
+
+# 5. Conclusion
+
+We propose the sliced optimal transport dataset distance (s-OTDD), a versatile approach for comparing datasets that
+
+is model-agnostic, embedding-agnostic, training-free, and robust to variations in class count and disjoint label sets. At the heart of s-OTDD is the Moment Transform Projection (MTP), which encodes a label-represented as a feature distribution into a real number. This projection enables us to represent entire datasets as one-dimensional distributions by mapping individual data points accordingly. From theoretical aspects, we discuss theoretical properties of the s-OTDD including metricity properties and approximation rate. From the computational aspects, the proposed s-OTDD achieves (near-)linear computational complexity with respect to the number of data points and feature dimensions, while remaining independent of the number of classes. For experiments, we analyze the performance of s-OTDD by comparing its computational time with existing dataset distances, its correlations with OTDD, its dependence on the number of projections, and number of moments, using subsets of the MNIST and CIFAR10 datasets. Moreover, we evaluate the correlations between s-OTDD and transfer learning performance gaps on image NIST datasets, text datasets and large scale Split Tiny-[ImageNet (224x224 resolution). Finally, we test s-OTDD for data augmentation in image classification on CIFAR10 and Tiny-[ImageNet. Overall, s-OTDD is comparable to baseline while being significantly faster, more scalable, and more robust. Future works will focus on understanding the gradient flow (Alvarez-Melis & Fusi, 2021) of the s-OTDD and adapt the s-OTDD in continual learning applications (Lee et al., 2021; Ke et al., 2020; Yang & Cai, 2023; Goldfarb et al., 2024).
+
+# Impact Statement
+
+This paper presents work to advance the field of Machine Learning. There are potential societal consequences of our work, none of which we feel must be highlighted.
+
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+
+# Supplement to "Lightspeed Geometric Dataset Distance via Sliced Optimal Transport"
+
+We first give skipped proofs in the main text in Appendix A. We then present additional materials including algorithms and additional experiments in Appendix B. Finally, we report computational devices used for experiments in Appendix C.
+
+# A. Proofs
+
+# A.1. Proof of Proposition 1
+
+We can rewrite the MTP in Definition 2 as:
+
+$$
+\mathcal {M T P} _ {\lambda , \theta} (\mu) = \frac {\int_ {\mathbb {R} ^ {d}} (\mathcal {F P} _ {\theta} (x)) ^ {\lambda} f _ {\mu} (x) d x}{\lambda !} = \frac {\int_ {\mathbb {R} ^ {d}} t ^ {\lambda} f _ {\mathcal {F P} _ {\theta} \sharp \mu} (t) d t}{\lambda !},
+$$
+
+where  $f_{\mathcal{FP}_{\theta}\sharp \mu}$  is the density function of  $\mathcal{FP}_{\theta}\sharp \mu$ . Let  $T_{\mu,\theta}$  be the random variable of  $\mathcal{FP}_{\theta}\sharp \mu$ , we have:
+
+$$
+\mathcal {M T P} _ {\lambda , \theta} (\mu) = \frac {\mathbb {E} \left[ T _ {\mu , \theta} ^ {\lambda} \right]}{\lambda !}.
+$$
+
+For a given  $\theta$ , when  $\mathcal{MTP}_{\lambda,\theta}(\mu) = \mathcal{MTP}_{\lambda,\theta}(\nu)$  for all  $\lambda \in \Lambda$ , it implies  $\frac{\mathbb{E}[T_{\mu,\theta}^{\lambda}]}{\lambda!} = \frac{\mathbb{E}[T_{\nu,\theta}^{\lambda}]}{\lambda!}$  for all  $\lambda \in \Lambda$ .
+
+(1) When  $\Lambda$  is infinite i.e.,  $\Lambda = \mathbb{N}$ , we have  $\frac{\mathbb{E}[T_{\mu,\theta}^{\lambda}]}{\lambda!} = \frac{\mathbb{E}[T_{\nu,\theta}^{\lambda}]}{\lambda!}$  for all  $\lambda \subset \mathbb{N}$ . Therefore, we have:
+
+$$
+\sum_ {\lambda = 1} ^ {\infty} \frac {z ^ {\lambda} \mathbb {E} \left[ T _ {\mu , \theta} ^ {\lambda} \right]}{\lambda !} = \sum_ {\lambda = 1} ^ {\infty} \frac {z ^ {\lambda} \mathbb {E} \left[ T _ {\nu , \theta} ^ {\lambda} \right]}{\lambda !}.
+$$
+
+Since we assume that all projected scaled moment exists in Assumption 1 and the moment generating functions of  $\mathcal{FP}_{\theta}\sharp \mu$  and  $\mathcal{FP}_{\theta}\sharp \nu$  exist for all  $\theta \in \Theta$ , using Taylor's expansion, we have:
+
+$$
+\mathbb {E} [ e ^ {z T _ {\mu , \theta}} ] = \mathbb {E} [ e ^ {z T _ {\nu , \theta}} ],
+$$
+
+which means that moment generating function of  $\mathcal{FP}_{\theta}\sharp \mu$  equals moment generating function of  $\mathcal{FP}_{\theta}\sharp \nu$ . Therefore, we can conclude that  $\mathcal{FP}_{\theta}\sharp \mu = \mathcal{FP}_{\theta}\sharp \nu$  for all  $\theta \in \Theta$  due to the uniqueness of the moment generating function. Due to the assumption of injectivity of the feature projection, we obtain  $\mu = \nu$ . We complete the proof.
+
+(2) When  $\Lambda$  is finite i.e.,  $\Lambda = \{1, 2, \dots, \lambda_{max}\}$ , we assume that projected Hankel matrices
+
+$$
+A _ {\theta , \mu} = \left[ \begin{array}{c c c c} m _ {\theta , \mu , 0} & m _ {\theta , \mu , 1} & \ldots & m _ {\theta , \mu , \lambda_ {\max}} \\ m _ {\theta , \mu , 1} & m _ {\theta , \mu , 2} & \ldots & m _ {\theta , \mu , 0} \\ \vdots & \vdots & \ddots & \vdots \\ m _ {\theta , \mu , \lambda_ {\max}} & m _ {\theta , \mu , 0} & \ldots & m _ {\theta , \mu , \lambda_ {\max} - 1} \end{array} \right], A _ {\theta , \nu} = \left[ \begin{array}{c c c c} m _ {\theta , \nu , 0} & m _ {\theta , \nu , 1} & \ldots & m _ {\theta , \nu , \lambda_ {\max}} \\ m _ {\theta , \nu , 1} & m _ {\theta , \nu , 2} & \ldots & m _ {\theta , \nu , 0} \\ \vdots & \vdots & \ddots & \vdots \\ m _ {\theta , \nu , \lambda_ {\max}} & m _ {\theta , \nu , 0} & \ldots & m _ {\theta , \nu , \lambda_ {\max} - 1} \end{array} \right]
+$$
+
+, with  $m_{\theta,\mu,\lambda} = \int_{\mathbb{R}^d} (\mathcal{FP}_\theta(x))^{\lambda} f_\mu(x) dx$  and  $m_{\theta,\nu,\lambda} = \int_{\mathbb{R}^d} (\mathcal{FP}_\theta(x))^{\lambda} f_\nu(x) dx$ , are positive definite for all  $\theta \in \Theta$  and  $|m_{\theta,\mu,\lambda}| < CD^{\lambda} \lambda!$ , and  $|m_{\theta,\nu,\lambda}| < CD^{\lambda} \lambda!$  for some constants  $C$  and  $D$  for all  $\lambda \in \Lambda$ . From the Hamburger moment problem (Reed & Simon, 1975; Chihara, 2011), we know that moments uniquely define a distribution. Therefore, we have  $\mathcal{FP}_{\theta} \sharp \mu = \mathcal{FP}_{\theta} \sharp \nu$  for all  $\theta \in \Theta$ . Due to the assumption of injectivity of the feature projection, we obtain  $\mu = \nu$ . We complete the proof.
+
+# A.2. Proof of Proposition 2
+
+From Definition 4, we have:
+
+$$
+\mathrm {s - O T D D} _ {p} ^ {p} (\mathcal {D} _ {1}, \mathcal {D} _ {2}) = \mathbb {E} _ {(\psi , \theta , \lambda , \phi) \sim \mathcal {U} (\mathbb {S}) \otimes \mathcal {U} (\mathbb {S} ^ {d - 1}) \otimes \sigma (\Lambda^ {k}) \otimes \mathcal {U} (\Phi)} \left[ \mathrm {W} _ {p} ^ {p} \big (\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}} \big) \right].
+$$
+
+Since the Wasserstein distance is non-negative and symmetric (Peyre & Cuturi, 2020), the symmetry and non-negativity of the s-OTDD $_p^p$ ( $D_1, D_2$ ) follows directly from it. We now prove the triangle inequality of s-OTDD. Given any three datasets  $D_1, D_2$ , and  $D_3$ . We want to show that:
+
+$$
+\mathrm {s - O T D D} _ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}\right) \leq \mathrm {s - O T D D} _ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {3}\right) + \mathrm {s - O T D D} _ {p} \left(\mathcal {D} _ {3}, \mathcal {D} _ {2}\right).
+$$
+
+From the triangle inequality of the Wasserstein distance, we have:
+
+$$
+\begin{array}{l} \mathrm {s - O T D D} _ {p} (\mathcal {D} _ {1}, \mathcal {D} _ {2}) = \left(\mathbb {E} _ {(\psi , \theta , \lambda , \phi) \sim \mathcal {U} (\mathbb {S}) \otimes \mathcal {U} (\mathbb {S} ^ {d - 1}) \otimes \sigma (\Lambda^ {k}) \otimes \mathcal {U} (\Phi)} \left[ \mathrm {W} _ {p} ^ {p} (\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}) \right]\right) ^ {\frac {1}{p}} \\ \leq \left(\mathbb {E} _ {(\psi , \theta , \lambda , \phi) \sim \mathcal {U} (\mathbb {S}) \otimes \mathcal {U} (\mathbb {S} ^ {d - 1}) \otimes \sigma (\Lambda^ {k}) \otimes \mathcal {U} (\Phi)} \left[ \left(\mathrm {W} _ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {3}}\right) \right. \right. \right. \\ \left. \left. + \mathrm {W} _ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {3}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right)\right) ^ {p} \right] \Bigg) ^ {\frac {1}{p}}. \\ \end{array}
+$$
+
+Using the Minkowski's inequality, we further have:
+
+$$
+\begin{array}{l} \left. \mathrm {s - O T D D} _ {p} (\mathcal {D} _ {1}, \mathcal {D} _ {2}) \leq \left(\mathbb {E} _ {(\psi , \theta , \lambda , \phi) \sim \mathcal {U} (\mathbb {S}) \otimes \mathcal {U} (\mathbb {S} ^ {d - 1}) \otimes \sigma (\Lambda^ {k}) \otimes \mathcal {U} (\Phi)} \left[ \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {3}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right]\right) ^ {\frac {1}{p}} \right. \\ \left. \right. + \left(\mathbb {E} _ {(\psi , \theta , \lambda , \phi) \sim \mathcal {U} (\mathbb {S}) \otimes \mathcal {U} (\mathbb {S} ^ {d - 1}) \otimes \sigma (\Lambda^ {k}) \otimes \mathcal {U} (\Phi)} \left[ \mathbf {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {3}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right)\right]\right) ^ {\frac {1}{p}} \\ = \mathrm {s} - \mathrm {O T D D} _ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {3}\right) + \mathrm {s} - \mathrm {O T D D} _ {p} \left(\mathcal {D} _ {3}, \mathcal {D} _ {2}\right), \\ \end{array}
+$$
+
+which completes the proof of the triangle inequality. For the identity of indiscernibles, when  $\mathcal{D}_1 = \mathcal{D}_2$ , we have directly that
+
+$$
+\begin{array}{l} \left. \mathrm {s - O T D D} _ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}\right) = \left(\mathbb {E} _ {\left(\psi , \theta , \lambda , \phi\right) \sim \mathcal {U} (\mathbb {S}) \otimes \mathcal {U} \left(\mathbb {S} ^ {d - 1}\right) \otimes \sigma \left(\Lambda^ {k}\right) \otimes \mathcal {U} (\Phi)} \left[ \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right]\right) ^ {\frac {1}{p}} \right. \\ = \left(\mathbb {E} _ {(\psi , \theta , \lambda , \phi) \sim \mathcal {U} (\mathbb {S}) \otimes \mathcal {U} (\mathbb {S} ^ {d - 1}) \otimes \sigma (\Lambda^ {k}) \otimes \mathcal {U} (\Phi)} [ 0 ]\right) ^ {\frac {1}{p}} = 0, \\ \end{array}
+$$
+
+due to the identity of indiscernibles of the Wasserstein distance when  $\mathcal{D}\mathcal{P}_{\psi ,\theta ,\lambda ,\phi}^{k}\sharp P_{\mathcal{D}_{1}} = \mathcal{D}\mathcal{P}_{\psi ,\theta ,\lambda ,\phi}^{k}\sharp P_{\mathcal{D}_{2}}$  . In addition, when s-OTDDp(D1,D2) = 0, it implies that Wp(DPk, Pd1, DPk, Pd2) = 0 for all ψ ∈ S, θ ∈ Θ, λ ∈ Λ, φ ∈ Φ. As a result, DPk, Pd1 = DPk, Pd2 for all ψ ∈ S, θ ∈ Θ, λ ∈ Λ, φ ∈ Φ. From the assumption of the injectivity of the data point projection, we obtain that PD1 = PD2 which implies D1 = D2. We conclude the proof here.
+
+# A.3. Proof of Proposition 3
+
+We recall the empirical approximation of the s-OTDD is:
+
+$$
+\widehat {\mathrm {s - O T D D}} _ {p} ^ {p} (\mathcal {D} _ {1}, \mathcal {D} _ {2}; L) = \frac {1}{L} \sum_ {l = 1} ^ {L} \mathrm {W} _ {p} ^ {p} (\mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {2}}).
+$$
+
+Using Holder's inequality, we have:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left| \widehat {\mathrm {s - O T D D}} _ {p} ^ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}; L\right) - \mathrm {s - O T D D} _ {p} ^ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}\right) \right| \right] \\ \leq \left(\mathbb {E} \left[ \left| \widehat {\mathrm {s - O T D D}} _ {p} ^ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}; L\right) - \mathrm {s - O T D D} _ {p} ^ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}\right) \right| ^ {2} \right]\right) ^ {\frac {1}{2}} \\ = \left( \right.\mathbb {E} \left[ \right.\left| \right. \frac {1}{L} \sum_ {l = 1} ^ {L} \mathbf {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) - \mathbb {E} \left[ \mathbf {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right)\right] ^ {2} \left. \right]\left. \right) ^ {\frac {1}{2}}. \\ \end{array}
+$$
+
+Since
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \frac {1}{L} \sum_ {l = 1} ^ {L} \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right] = \frac {1}{L} \sum_ {l = 1} ^ {L} \mathbb {E} \left[ \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {K} \sharp P _ {\mathcal {D} _ {2}}\right) \right] \\ = \mathbb {E} \left[ \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right], \\ \end{array}
+$$
+
+Algorithm 1 Computational Algorithm for s-OTDD  
+Input: Two datasets  $\mathcal{D}_1 = \{(x_1,y_1),\ldots ,(x_n,y_n)\}$  and  $\mathcal{D}_2 = \{(x_1',y_1'),\ldots ,(x_m',y_m')\}$ , label sets  $\mathcal{V}_1$  and  $\mathcal{V}_2$ , parameter  $p\geq 1$ , number of moments  $k$ , number of projections  $L$ , rate parameters  $r_1,\dots ,r_k > 0$ .  
+for  $l = 1$  to  $L$  do  
+Sample  $\theta_l\sim \mathcal{U}(\mathbb{S}^{d - 1})$   
+for  $i = 1$  to  $n$  do  
+Compute  $\mathcal{FP}_{\theta}(x_i)$   
+end for  
+for  $j = 1$  to  $m$  do  
+Compute  $\mathcal{FP}_{\theta}(x_j')$   
+end for  
+for  $j = 1$  to  $k$  do  
+Sample  $\lambda_l^{(j)}\sim \mathrm{TruncatedPoisson}(r_j)$   
+for  $a$  in  $\mathcal{V}_1$  do  
+Compute  $\mathcal{MTP}_{\lambda^{(i)},\theta}(q_a) = \frac{1}{n_a}\sum_{i = 1}^{n_a}\frac{\mathcal{FP}_{\theta}(x_i)^{\lambda^{(j)}}}{\lambda!}$  for  $n_a$  is the number of samples in class  $a$ .  
+end for  
+for  $a'$  in  $\mathcal{V}_2$  do  
+Compute  $\mathcal{MTP}_{\lambda^{(i)},\theta}(q_{a'}) = \frac{1}{m_{a'}}\sum_{i = 1}^{m_{s'}}\frac{\mathcal{FP}_{\theta}(x_i')^{\lambda^{(j)}}}{\lambda!}$  for  $m_{a'}$  is the number of samples in class  $a'$ .  
+end for  
+end for  
+Sample  $\psi_l\sim \mathcal{U}(\mathbb{S}^k)$   
+for  $i = 1$  to  $n$  do  
+ $\mathcal{DP}_{\psi ,\theta ,\lambda ,\theta}^k (x_i,q_{y_i}) = \psi^{(1)}\mathcal{FP}_{\theta}(x_i) + \sum_{i' = 1}^k\psi^{(k)}\mathcal{TP}_{\lambda^{(i')},\theta}(q_{y_i})$   
+end for  
+for  $i = j$  to  $m$  do  
+ $\mathcal{DP}_{\psi ,\theta ,\lambda ,\theta}^k (x_j',q_{y_j}') = \psi^{(1)}\mathcal{FP}_{\theta}(x_j') + \sum_{i' = 1}^k\psi^{(k)}\mathcal{TP}_{\lambda^{(i')},\theta}(q_{y_j}')$   
+end for  
+Compute  $w_l = W_p^p\left(\frac{1}{n}\sum_{i = 1}^{n}\delta_{\mathcal{DP}_{\psi ,\theta ,\lambda ,\theta}^k (x_i,q_{y_i}),\frac{1}{m}\sum_{j = 1}^{m}\delta_{\mathcal{DP}_{\psi ,\theta ,\lambda ,\theta}^k (x_j',q_{y_j}')}\right)$   
+end for  
+Return:  $\frac{1}{L}\sum_{l = 1}^{l}w_l$
+
+we have:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left| \widehat {\mathrm {s - O T D D}} _ {p} ^ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}; L\right) - \mathrm {s - O T D D} _ {p} ^ {p} \left(\mathcal {D} _ {1}, \mathcal {D} _ {2}\right) \right| \right] \\ \leq \left(V a r \left[ \frac {1}{L} \sum_ {l = 1} ^ {L} \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi_ {l}, \theta_ {l}, \lambda_ {l}, \phi_ {l}} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right]\right) ^ {\frac {1}{2}} \\ = \frac {1}{\sqrt {L}} \left(V a r \left[ \mathrm {W} _ {p} ^ {p} \left(\mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {1}}, \mathcal {D P} _ {\psi , \theta , \lambda , \phi} ^ {k} \sharp P _ {\mathcal {D} _ {2}}\right) \right]\right) ^ {\frac {1}{2}}, \\ \end{array}
+$$
+
+due to the i.i.d sampling of the projection parameters, which completes the proof.
+
+# B. Additional Materials
+
+Algorithms. We present the computational algorithm for s-OTDD in Algorithm 1. In the algorithm, we use the same feature projection for both the label projection (MTP) and the data point projection.
+
+Projection Analysis. As mentioned in the main text, we present the distance correlations with OTDD (Exact) of s-OTDD as a function of the number of projections in Figure 9 and similarly for CHSW in Figure 10. For s-OTDD, we observe that increasing the number of projections consistently improves the correlations for both MNIST and CIFAR10 datasets, indicating a clear trend. In contrast, the same does not hold true for CHSW, where the correlations do not show consistent
+
+![](images/eb89bd06cc3bb8083156f76b02e97629e3d60a223af2984ede5cf2794e0412ba.jpg)
+
+![](images/9047f46235acaed7d099fb58498e30fcf61fd282c92c059c5e42d6267459cc1d.jpg)
+
+![](images/51bc552126beaf81affdfadb04dfc9d74aabe9dee069adf1dc0bf62628f885bb.jpg)
+
+![](images/1c70e062dced7ab75be11c92072dc04812bc5c16191410fe98fd3d87505f112a.jpg)
+
+![](images/ea3aced44c2a90c0c80ec79ce56358838e8758addfca58a15c6bae6f7f24b0c9.jpg)  
+Figure 9: The figure shows distance correlations with OTDD (Exact) of s-OTDD when varying the number of projections.
+
+![](images/a6af45e76737b5ea48e43050dec004e7427ed2d46faf11400add5528ea50cbaa.jpg)
+
+![](images/963330954a6bf155be00abffd2efe299de4550a886685093c5d5b48e5a1aaff6.jpg)
+
+![](images/ef0bdf42fe34b1a330704e0e79a83a7a088c986ceca2abc2f466c335e9906be0.jpg)
+
+![](images/805dd1731325e4e9b255ded4f8115a185ff73223615d8889d7af70bc3785c042.jpg)
+
+![](images/46cebb808fe5a5b4483a6765faba1a087de1f16a3d884d6103be13d8a8e16980.jpg)
+
+![](images/81a151d6128e5317b924e0d60198b60f6715269a6da1f9ea1086fe39f4493c0b.jpg)
+
+![](images/ed9a9d61b70bed9d9e62504d8d3895f4f36eb6a70f45a90b5096748a1066cb7c.jpg)
+
+![](images/0c01fb6d3bb505803bb172ae5d15d97c94108fa50410dd6d2a83b0c74af24987.jpg)  
+Figure 10: The figure shows distance correlation with OTDD (Exact) of CHSW when varying the number of projections.
+
+![](images/5835f19a91b7412fda30def1f2f0642915f5e873cd603e979806f8236e770c08.jpg)
+
+![](images/47d6ca69a65c1f95272ae2d902392b2deb5f523d223bf555f24125505b56e465.jpg)
+
+![](images/c0f62ddc411d2c212363fddc972b19304e9db8a0be9574da9f555b4ba54bb4f8.jpg)
+
+improvement as the number of projections increases. This important observation suggests that s-OTDD in its population form is highly correlated with OTDD (Exact), and therefore, reducing the Monte Carlo approximation error by increasing the number of projections results in a more accurate correlations with OTDD (Exact).
+
+Transfer Learning. We present the complete experimental results for transfer learning using the *NIST datasets in Figure 11. As highlighted in Section 4.2, the figure demonstrates clear positive correlations between s-OTDD with 10,000 projections and PG, with a Spearman's rank correlation coefficient of  $\rho = 0.42$ , which is identical to that of OTDD (Exact). For the other distances, we observe that OTDD (Gaussian approximation), CHSW, and WTE yield correlations of  $\rho = 0.44$ ,  $\rho = 0.15$ , and  $\rho = 0.43$ , respectively. These results reinforce the reliability of s-OTDD in capturing meaningful transfer learning behavior, comparable to OTDD (Exact) and other related metrics.
+
+![](images/2a5b9f67d34945b2a3be9fa5892df3e3b14270c830331c035690a99dfe2a17d5.jpg)  
+Figure 11: The figures show Pearson and Spearman correlation of OTDD (Exact), OTDD (Gaussian approximation), CHSW (10,000 projections), WTE, and s-OTDD (10,000 projections) with the performance gap when conducting transfer learning in *NIST datasets.
+
+![](images/7f22cba060f772586d8902305c7789c11e4898ba93d4c7f23b1a6f9b754cd029.jpg)
+
+![](images/a71b3d0bd59ad53deec782638d2b3e5903e52484fed49480018a4d14ece48887.jpg)
+
+![](images/aaa722116eb119ed3613d256cf13e581b966ce57fb35ba3cefaba31f7d8cadbf.jpg)
+
+![](images/64afbcaa7b71057e3841c17943c7f20ac8d42d2cd81f42eb7297c5e7cf1b8838.jpg)
+
+![](images/ff7018df60f1d459af5ea3fd308793a98bae327fb7091f3fd5353f8cfab0e214.jpg)
+
+![](images/a6a08cecf083648e6835255d4713e2689217c66bb9d1390efda023b76715a5bf.jpg)
+
+![](images/6818f45d6e4efec7bd307ae12fc2e97457b25c86afabaa5e15d4702db497c76f.jpg)
+
+![](images/b98c653c3aed32cbbeb77dd5eae2ba1d68c89d0683a13fba36d5fd445e30a527.jpg)
+
+![](images/a336a99fca4c14a67d43f739c3fbac7d5e886b3c8ec0d6121fc8eb5fcad83fc3.jpg)
+
+![](images/5c30d0efd37165dd525bf92a4b9621d09b4bb378a5e059a0e0d8de7afc59f838.jpg)  
+Figure 12: The figure compares two projection methods, linear projections (Radon Transform projection) and convolution projection, using the *NIST dataset experiments.
+
+![](images/858e5d1ba3dd03be30b79936587017628be25e20a7fbc9a7d25bec38effeb134.jpg)
+
+![](images/a723ef2a3440c7df4e0de2c877dfa2188e2d9d556af288d7b3f2b6caff7ef8e8.jpg)
+
+![](images/f586c24d273020e41771997554b6e5e74fbf79455fdf0af7fac1a502f1a1caaa.jpg)
+
+![](images/7a849d5c3f882a5c3ec9e1b2471c57e8dc1517a06404cfc2ef809f3a446be867.jpg)
+
+![](images/25a7b28d528061c5b3906045a1035ba840773c8f998febbdec83eaff75469557.jpg)  
+Figure 13: The figure shows Pearson and Spearman correlations of distance and Performance Gap in term of other metrics, i.e F1, Precision and Recall.
+
+![](images/544be12deb089016f460511991f887cdc4bc77d138c96799507bb84dfabaabe3.jpg)
+
+![](images/dcfc7a264e1e8db5160471a39826429c5a5812ae3297fc993244b99456fe3fa3.jpg)
+
+# C. Computational Devices
+
+For the runtime experiments and distance computations, we conducted tests using 8 CPU cores with 128GB of memory. For model training experiments, such as training BERT and ResNet, we used an NVIDIA A100 GPU.

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+# Limitations of Measure-First Protocols in Quantum Machine Learning
+
+Casper Gyurik $^{123}$  Riccardo Molteni $^{12}$  Vedran Dunjko $^{12}$
+
+# Abstract
+
+In recent times, there have been major developments in two distinct yet connected domains of quantum information. On the one hand, substantial progress has been made in so-called randomized measurement protocols. Here, a number of properties of unknown quantum states can be deduced from surprisingly few measurement outcomes, using schemes such as classical shadows. On the other hand, significant progress has been made in quantum machine learning. For example, exponential advantages have been proven when the data consists of quantum states and quantum algorithms can coherently measure multiple copies of input states. In this work, we aim to understand the implications and limitations of combining randomized measurement protocols with quantum machine learning, although the implications are broader. Specifically, we investigate quantum machine learning algorithms that, when dealing with quantum data, can either process it entirely using quantum methods or measure the input data through a fixed measurement scheme and utilize the resulting classical information. We prove limitations for the general class of quantum machine learning algorithms that use fixed measurement schemes on the input quantum states. Our results have several implications. From the perspective of randomized measurement procedures, we show limitations of measure-first protocols in the average case, improving on the state-of-the-art which only focuses on worst-case scenarios. Additionally, previous lower bounds were only known for physically unrealizable states. We improve upon this by employing quantum pseudosystems.
+
+dorandom functions to prove that a learning separation also exists when dealing with physically realizable states, which may be encountered in experiments. From a machine learning perspective, our results are crucial for defining a physically meaningful task that shows fully quantum machine learning processing is not only more efficient but also necessary for solving certain problems. The tasks at hand are also realistic, as the algorithms and proven separations hold when working with efficiently preparable states and remain robust in the presence of measurement and preparation errors.
+
+# 1. Introduction
+
+A central question in quantum machine learning revolves around understanding the various types of advantages one can achieve by exploiting quantum effects. Some of the most interesting scenarios arise when the dataset itself comprises quantum states, which can then be processed fully coherently, or through elaborate measurement strategies. In this context, exponential advantages have been identified when coherent measurements of multiple copies of a given quantum state are allowed (Chen et al., 2022b; Huang et al., 2022; 2020; King et al., 2025; Chen et al., 2024a;b). In a parallel related line, there have been significant breakthroughs in extracting useful classical information from quantum states using the versatile toolkit of randomized measurements (Elben et al., 2023). This toolkit includes the groundbreaking concept of classical shadows (Huang et al., 2020; 2022), which can extract an efficient classical description of quantum states that allows one to compute various physical properties. These two distinct research lines cast doubt on the advantages of quantum machine learning protocols that process quantum data directly compared to those that use a fixed measurement procedure, which can also allow for a coherent manipulation of quantum states before the fixed measurement, in order to extract valuable classical data<sup>1</sup>. In particular we address the question: is it possible
+
+that a "measure-first" protocol can be universally used as a substitute for any "fully-quantum" protocol in the general context of machine learning?
+
+We formally define what we mean by a "measure-first" or a "fully-quantum" protocol in Section 2 (see Definition 2.3 and Definition 2.5), and we provide an overview in Figure 1. Importantly, we focus on physically relevant scenarios where the quantum states in the dataset are efficiently preparable, rather than arbitrary unphysical states that do not allow a polynomial description in terms of quantum gates.
+
+Our main conceptual contribution is to resolve the above question negatively by presenting a concrete machine learning scenario that clearly exhibits the limitations of any measure-first protocol. In a machine learning setup, we construct a task that requires reproducing an undisclosed measurement from the data. This task shows an exponential difference in the number of data needed between fully-quantum protocols and measure-first protocols.
+
+Outside of machine learning, certain limitations of measure-first protocols are implied by other works in various tasks such as distributed sampling (Montanaro, 2019) and relational problems (Aaronson et al., 2024). However, these results primarily address separations in worst-case scenarios, which contrasts with the typical focus in machine learning settings where average-case performance is usually sufficient. In this paper, we advance these findings by identifying the limitations of measure-first protocols that need only achieve average-case correctness for a set of efficiently preparable quantum states, i.e. physical states. On a technical level, establishing limitations for measure-first protocols in a general quantum machine learning setting necessitates novel techniques beyond those utilized in previous studies. In particular, limitations for the task of predicting properties of quantum states were also recently derived for worst-case settings (Grier et al., 2024) exploiting lower bounds for classical one-way communication complexity. However, this approach inherently limits the focus to states that are not efficiently preparable. This limitation arises because, if states were efficiently preparable, there would be a concise classical description of their preparation procedure, enabling the transmission of valid classical messages to the recipient. Our main technical contribution lies in exploiting that, from the more general machine learning standpoint, it makes sense to constrain the computational power of the learning protocol. This allows us to additionally impose the requirement that states must be efficiently preparable and still achieve a separation by utilizing a novel construction that combines results on one-way communication complexity and pseudorandom quantum states (Brakerski & Shmueli, 2019). The task we consider is also experimentally robust, in the sense that we allow for errors in the preparation of the input quantum states and on the measurement outcomes which label them.
+
+![](images/75caa7c0a7bc85297134a82a9cf17619348f10ee73fa2c0906f1b9a4682049f4.jpg)  
+Figure 1. An overview of the quantum machine learning protocols explored in our paper. In both protocols, the quantum algorithm  $A$  is tasked with processing training data  $T_{x}$  to output a classical description of a function  $h$  that correctly produces samples  $z^{*}$  for new input states  $\rho^{*}$  (i.e.,  $h$  should implement the quantum measurement). The difference between the two protocols is that, in the measure-first protocol, both the input quantum states and those in the training set  $T_{x}$  are subjected to a randomized measurement strategy  $M$ . Consequently, these quantum states  $\rho^{*}$  are transformed into a classical representation  $\hat{\rho}^{*}$ . Importantly, this strategy is allowed to depend only on the concept class  $\mathcal{C}$  (which captures the general learning problem), not on the specific target concept  $\pi_{x} \in \mathcal{C}$ . In other words, the same randomized measurement strategy  $M$  is used to preprocess the quantum data for all concepts  $\pi_{x} \in \mathcal{C}$ .
+
+# 1.1. High-level overview of learning setting and main result
+
+The machine learning problem concerns learning an unknown measurement acting on a set of input quantum states. Specifically, the data that the learning protocol gets consists of pairs of copies of  $n$ -qubit quantum states  $\rho$  drawn from some distribution  $\mathcal{D}$  together with a corresponding label  $z \in \{0,1\}^{2n + 1}$ . The first  $n$  bits of  $z$  encode a  $2^n$ -outcome POVM measurement  $\Lambda_{x} = \{E_{j}^{x} \mid j \in \{0,1\}^{n}\}$  from some set  $\Lambda = \{\Lambda_{x} \mid x \in \{0,1\}^{n}\}$ . The remaining  $n + 1$  bits are determined by the outcome of  $\Lambda_{x}$  on the quantum state  $\rho$ . The goal of the machine learning protocol is to "learn" how to reproduce the measurement  $\Lambda_{x}$ . More precisely, the trained learning protocol has to receive as input an unseen quantum state  $\rho'$  and output a sample  $z'$  in agreement with the probability distribution  $\pi_{x}(\rho)$ , where  $\pi_{x}(\rho)$  denotes the distribution of the measurement outcomes of  $\Lambda_{x}$ .
+
+This paper explores whether solving learning problems such as the above requires a quantum computer capable of adaptive measurements on the training data, or if a fixed measurement strategy that produces classical representations of the quantum states is sufficient. Specifically, we consider so-called "measure-first protocols" that are forced to measure the input states and use the obtained classical description to train the machine learning model to produce samples
+
+from the target distribution. Importantly, in a measure-first protocol the measurements are not allowed to depend on the training data, but are unconstrained otherwise. In particular, the measure-first protocol can depend on the set of measurements that is to be reproduced from the data, but not on the specific individual measurement hidden within some given training data. On the other hand, we consider so-called "fully-quantum protocols" that can coherently process the quantum states and adjust the measurements based on the training data. Our main result is the existence of a learning problem where the quantum data is efficiently generatable on a quantum computer for which a "measure-first protocol" requires an exponential amount of data to be able to reproduce measurement outcomes on new input states, whereas a fully-quantum protocol only requires a polynomial amount of data. Additionally, we require that the protocols must be efficient in the sense that they run in time polynomial in  $n$ . Using the notion of learning we presented above, we now give an informal description of the main result of this paper.
+
+Theorem 1.1. (Informal) There exist a concept class  $\mathcal{C}$ , where each concept is defined by a measurement and a distribution  $\mathcal{D}$  over efficiently preparable quantum states such that no "measure-first" protocol can learn to reproduce  $\mathcal{C}$  correctly on average with a polynomial amount of training data. On the other hand, there exists a "fully-quantum" protocol which can learn  $\mathcal{C}$  efficiently with respect to both sample and time complexity.
+
+The specific learning problem which exhibits a learning separation is carefully outlined in Section 3 and the precise proofs of Theorem 1.1 can be found in the Appendix.
+
+# 1.2. Related work
+
+In this section we discuss related works and highlight their relationships to our learning setting. Firstly, in (Huang et al., 2020), the authors introduced a randomized measurement technique tailored to extract a classical description of a quantum state  $\rho$ . This description enables the computation of the expectation values of any set of observables  $\{O_k\}_{k=1}^M$  - provided they have a low "shadow norm", such as when the observables are local - up to a precision of  $\epsilon$ . Notably, they showed that a number of copies of  $\rho$ , scaling logarithmically with the number of observables  $M$  and inverse-polynomially in the precision  $\epsilon$ , suffices for this task. As we want to learn the distribution associated of a  $2^n$ -outcome POVM with an  $\epsilon$  error in total variation, directly applying these techniques to estimate each of the  $2^n$  outcomes would require an exponential precision for each. In their upper bound estimates, this results in the requirement of an exponential number of samples.
+
+The concept of shadow tomography, introduced in (Aaronson, 2018), revolves around the problem of computing the expectation values of any set of  $M$  two-outcome measure
+
+ments on an  $n$ -qubit state  $\rho$  up to precision  $\epsilon$ . It has been shown that this can be done using a number of copies of  $\rho$  that scale polylogarithmically in  $M$ , linearly in  $n$ , and inverse-polynomially in  $\epsilon$  (Aaronson, 2018). In contrast to the methods in (Huang et al., 2020), the approach in (Aaronson, 2018) requires coherent measurements on multiple copies of  $\rho$ . Additionally, as demonstrated in (Huang et al., 2022; Chen et al., 2022a) for specific tasks, the capacity to coherently measure multiple copies of quantum states provides an exponential advantage in sample complexity over sequential measurements. Considering our framework, where measurements can act coherently on multiple copies of each input state, one might question whether such strategies enable a measure-first protocol to solve our learning task. However, it is important to note that coherent measurements do not improve the scaling with respect to the precision  $\epsilon$ , leaving room for the possibility of a separation since the learning task studied in this paper would require at least exponential precision to be solved.
+
+In (Gong & Aaronson, 2023), the authors extended the concept of shadow tomography to the scenario of learning a  $K$ -outcome POVMs (for  $K \geq 2$ ) selected from a set of  $M$  unknown quantum measurements. In contrast to the binary outcome case, the goal now is to approximate an unknown distribution up to precision  $\epsilon$  in total variation distance, rather than focusing on expectation values. Their procedure requires a number of copies of the quantum state scaling linearly with  $K$  and  $n$ , polylogarithmically with  $M$ , and inverse-polynomially with the precision  $\epsilon$ . Moreover, they establish the optimality of this scaling with respect to the dependence on the number of outcomes  $K$ . This result prompts the question of whether our separation between measure-first and fully-quantum protocols can be directly inferred from it. However, in establishing their lower bound, it is important to note that no assumptions were made regarding the complexity of the unknown quantum state. The crux of our study lies in demonstrating that measure-first protocols fall short of reproducing the unknown measurement, even on quantum states that are efficiently preparable. Moreover, it is important to highlight that while their shadow tomography procedures can be employed to construct a measure-first protocol by "shadowfying" input states to approximate the expected values of each measurement outcome, this is not strictly necessary for our task. Specifically, understanding the probability of each outcome allows for the creation of an "evaluator" that can compute the correct probability for every outcome. However, to resolve our learning problem, a "generator" (i.e., an algorithm generating samples with the correct probabilities) already suffices, and it does not necessarily require computing the output probabilities (Sweke et al., 2021).
+
+Next, in (Grier et al., 2024) the authors explore lower bounds on the number of copies of a quantum state required for solv-
+
+ing what they term the "classical shadow" task. This task involves estimating the expected values of observables using a measure-first approach. While they establish lower bounds on the number of copies of a quantum state required for this task, shedding light on the inherent limitations of measure-first protocols, it is crucial to note two significant bottlenecks of their approach when viewed from our more general machine learning perspective. Firstly, their analysis only focuses on worst-case scenarios, whereas in our machine learning scenarios we are concerned with average-case performance. Secondly, their investigation centers on protocols required to operate effectively even for unphysical quantum states that are not efficiently preparable.
+
+The authors of (Cheng et al., 2016) provide upper bounds for the dual problem of shadow tomography, or more specifically the problem of learning a measurement. In particular, they studied the task of learning an unknown two-outcome POVM denoted  $\{E, I - E\}$ , from data of the form  $\{(\rho_i, \mathrm{Tr}(E\rho_i))\}_{i=1}^m$ . They showed that to approximate the unknown  $E$  up to a precision of  $\epsilon$  on new  $n$ -qubit input states,  $\mathcal{O}(2^n / \epsilon^2)$  training samples are sufficient. Nonetheless, the number of required samples scales exponentially with the number of qubits.
+
+Finally, in (Jerbi et al., 2023) the authors study quantum process learning, where the task is to learn a unitary  $U$ , from data of the form  $\{|\psi_i\rangle ,U|\psi_i\rangle \}$ . In particular, they study the limitations of what they call incoherent learning, where the learner is constrained to first measure multiple copies of the data  $U|\psi_i\rangle$ . While they therefore also study the problem of extracting classical information from quantum data and utilizing it in the learning process, the setting in their work differs from ours. Namely, the quantum states in our scenario are labeled by a sample obtained from the unknown measurement process, whereas in (Jerbi et al., 2023) the labels are the input quantum states when evolved under some unknown target unitary.
+
+# 2. Background
+
+In this section, we introduce the notations and definitions needed to frame our main result, which involves a learning problem that demonstrate the separation between any measure-first protocol and a fully quantum protocol.
+
+The learning problem we study is the learning of a measurement. In particular, it involves generating samples from a distribution induced by measuring an (unknown) POVM measurement  $\Lambda_{x}$  on  $n$ -qubit quantum states. The (unknown) target measurement  $\Lambda_{x}$  is drawn from a set of POVMs  $\{\Lambda_{x} \mid x \in \{0,1\}^{n}\}$ , and each measurement  $\Lambda_{x}$  is a computational basis measurement preceded by an  $n$ -qubit unitary  $U_{x}$ , i.e.,
+
+$$
+\Lambda_ {x} = \left\{E _ {j} ^ {x} \mid j = 1, \dots , 2 ^ {n} \right\}, \text {w h e r e} E _ {j} ^ {x} = U _ {x} | j \rangle \langle j | U _ {x} ^ {\dagger}.
+$$
+
+During training the learner is given a set of examples  $T_{x}$ , where each example consists of a polynomial number of copies of a phase state  $|\psi_f\rangle$  together with a sample from the associated POVM-induced distribution. This is a special case of labeled quantum data, which was introduced in (Aimeur et al., 2006), where we are additionally allowed to have access to polynomially many copies of the input quantum state. We formalize our learning problem by generalizing the standard PAC learning framework (Valiant, 1984). In our generalization, a concept corresponds to a quantum randomized function, i.e., a function that on each quantum input state outputs a sample from a random variable (which in our case corresponds to the outcomes of a POVM on the input quantum state). Before we define the concept class studied throughout this paper, we first setup some auxiliary definitions.
+
+Definition 2.1 (Auxiliary definitions/notation).
+
+- Let  $N = 2^n$ , or equivalently  $n = \log_2 N$ .  
+- We identify a function  $f: \{0,1\}^n \to \{0,1\}$  with its truth table  $f \in \{0,1\}^N$ , and we denote its corresponding phase state with
+
+$$
+\left| \psi_ {f} \right\rangle = \frac {1}{\sqrt {N}} \sum_ {i = 1} ^ {N} (- 1) ^ {f _ {i}} | i \rangle . \tag {1}
+$$
+
+- Let  $S_{\mathrm{phase}} = \{|\psi_f\rangle^{\otimes \ell} \mid f \in \{0,1\}^N\}$  denote the set of  $\ell = \mathrm{poly}(n)$  copies of  $n$ -qubit phase states.  
+- We write  $x \sim \pi$  to denote that  $x$  was drawn according to a distribution  $\pi$ .  
+- We write  $\mathcal{U}(\mathcal{X})$  for the uniform distribution over a set  $\mathcal{X}$ .  
+- We write  $\Delta(\mathcal{X})$  for the set of all distributions over a set  $\mathcal{X}$ .  
+- For  $p, q \in \mathbb{R}_{\geq 0}^{2^n}$  probability distributions over  $\{0,1\}^n$  we define
+
+$$
+\left| \left| p - q \right| \right| _ {T V} = \frac {1}{2} \sum_ {x \in \{0, 1 \} ^ {n}} \left| p _ {x} - q _ {x} \right|
+$$
+
+to be the total variation distance.
+
+Definition 2.2 (Concept class). We define our concept class as  $\mathcal{C} = \{\pi_x\mid x\in \{0,1\} ^n\}$ , where
+
+$$
+\pi_ {x}: \mathcal {S} _ {\text {p h a s e}} \rightarrow \Delta (\{0, 1 \} ^ {2 n + 1}) \tag {2}
+$$
+
+$$
+\left| \psi_ {f} \right\rangle^ {\otimes \ell} \mapsto \pi_ {x} (f)
+$$
+
+where  $\pi_x(f)$  is a distribution over samples  $(x,y,b) \in \{0,1\}^{2n+1}$ , where  $(y,b) \in \{0,1\}^{n+1}$  are variables sampled from a problem-specific distribution characterizing the specific learning task.
+
+In particular,  $\pi_x$  is a randomized function which takes as input a polynomial number of copies of a phase state and outputs a sample  $z$  consisting of  $x \in \{0,1\}^n$  together with some  $(y,b) \in \{0,1\}^{n+1}$  sampled from some problem-specific probability distribution. As we will see in the next section, a learning separation for this kind of learning problem is achieved by considering  $(y,b)$  sampled uniformly from the set of variables which satisfy the Hidden Matching relation defined by the input functions  $f$  and the bitstring  $x \in \{0,1\}^n$ .
+
+A learner then is given several evaluations of the randomized function  $\pi_{x}$  in the form of training data and its objective is to implement a randomized function  $\widetilde{\pi}_{x}$  that closely approximates  $\pi_{x}$  on most input states.
+
+In this paper, we compare two categories of machine learning systems that can tackle problems of this type. First, we introduce what we call a "fully-quantum protocol".
+
+Definition 2.3 (Fully-quantum protocol). A fully-quantum protocol for the concept class  $\mathcal{C}$  in Definition 3.1 is a polynomial-time quantum algorithm  $A$  that takes as input training data of the form
+
+$$
+\begin{array}{l} T _ {x} = \left\{\left(\left| \psi_ {f ^ {(i)}} \right\rangle^ {\otimes \ell}, (x, y, b)\right) \mid (x, y, b) \sim \pi_ {x} \left(f ^ {(i)}\right), \right. \\ \text {a n d} f ^ {(i)} \sim \mathcal {U} \left(\{0, 1 \} ^ {N}\right) \bigg \} _ {i = 1} ^ {\operatorname {p o l y} (n)}, \\ \end{array}
+$$
+
+and outputs a classical description of a polynomial-time quantum algorithm that on input  $|\psi_f\rangle^{\otimes \ell}\in S_{\mathrm{phase}}$  generates a sample from a distribution  $\widetilde{\pi}_x(f)\in \Delta (\{0,1\}^{2n + 1})$
+
+We emphasize that for a "fully quantum" protocol, the learning algorithm must produce a classical description of the quantum algorithm generating samples from  $\widetilde{\pi}_x$ . Consequently, we do not store any quantum states from the training data in quantum memory, which would be more general but not studied in this paper. Ultimately, the goal of the protocol is to implement a randomized function  $\widetilde{\pi}_x$  that closely approximates the actual data-generating randomized function  $\pi_x$  for most of the input quantum states.
+
+Definition 2.4  $((\epsilon, \delta, p_{\mathrm{succ}})$ -fully-quantum learnable). We say that  $\mathcal{C}$  is  $(\epsilon, \delta, p_{\mathrm{succ}})$ -fully-quantum learnable if there exists a fully-quantum protocol  $A$  such that for every  $\pi_x \in \mathcal{C}$ , with probability at least  $p_{\mathrm{succ}}$  we have
+
+$$
+\Pr_ {f \sim \mathcal {U} (\{0, 1 \} ^ {N})} \left(\left| \left| \widetilde {\pi} _ {x} (f) - \pi_ {x} (f) \right| \right| _ {T V} \leq \epsilon\right) \geq 1 - \delta , \tag {4}
+$$
+
+where  $\widetilde{\pi}_x(f) \in \Delta(\{0,1\}^{2n+1})$  denotes the distribution that the polynomial-time quantum algorithm obtained from the learning algorithm  $A$  generates samples from on input  $|\psi_f\rangle^{\otimes \ell} \in S_{\mathrm{phase}}$ .
+
+Next, we introduce a "measure-first protocol" which consists of two components: (i) a randomized measurement
+
+strategy  $M$ , and (ii) a learning algorithm  $A$ . The main difference between a measure-first protocol and a fully-quantum protocol is that the former involves a randomized measurement procedure that first measures the quantum states before putting it into a learning algorithm. Importantly, the measure-first protocol is allowed to perform arbitrary coherent measurements on all input quantum states (i.e., the polynomially-many copies of the phase states). The only constraint is that the measurement strategy cannot depend on the specific target concept of the learning problem, although it is allowed to depend on the concept class (i.e., the set of all possible target concepts). In short, a randomized measurement strategy is a polynomial-time algorithms that maps a polynomial number of copies of a phase state  $|\psi_f\rangle$  to some classical description  $\widehat{\psi}_f \in \{0,1\}^m$  for some  $m = \mathrm{poly}(n)$ . These classical descriptions  $\widehat{\psi}_f$  are then used as input for the learning algorithm, that is tasked with implementing a randomized function close to  $\pi_x$ .
+
+Definition 2.5 (Measure-first protocol). A measure-first protocol is a tuple  $(M, A)$  where
+
+-  $M$  is a measurement strategy that in time  $\mathcal{O}(\mathrm{poly}(n))$  maps  $|\psi_f\rangle^{\otimes \ell}\in S_{\mathrm{phase}}^2$  to some  $\widehat{\psi}_f\in \{0,1\} ^m$  , where  $m = \mathrm{poly}(n)$  
+-  $A$  is a polynomial-time quantum algorithm that takes input of the form
+
+$$
+\begin{array}{l} T _ {x} ^ {M} = \left\{\left(\widehat {\psi} _ {f ^ {(i)}}, (x, y, b)\right) \mid (x, y, b) \sim \pi_ {x} (f ^ {(i)}) \right. \\ \left. \text {a n d} f ^ {(i)} \sim \mathcal {U} \left(\{0, 1 \} ^ {N}\right) \right\} _ {i = 1} ^ {\text {p o l y} (n)}, \tag {5} \\ \end{array}
+$$
+
+and outputs a description of a polynomial-time quantum algorithm that on input  $\widehat{\psi_f} = M(|\psi_f\rangle^{\otimes \ell})$  generates a sample from a distribution  $\widetilde{\pi}_x(f)\in \Delta (\{0,1\}^{2n + 1})$
+
+Note that the distinction between measure-first and fully-quantum protocols lies in Eq. 5, where the data is measured to give  $\hat{\psi}_f$  instead of remaining quantum states  $|\psi_f\rangle$ . Nonetheless, the measurements employed by the measurement strategy  $M$  are completely arbitrary and entirely unrestricted. We emphasize, however, that the sole constraint on the measurement strategy  $M$  is that it must not rely on the specific concepts it will be applied to later, although it can depend on the concept class it is intended for. Recall that the objective of the protocol is to implement a randomized function  $\widetilde{\pi}_x$  that closely approximates the actual data-generating randomized function  $\pi_x$  on most inputs.
+
+Definition 2.6  $((\epsilon, \delta, p_{\mathrm{succ}})$ -measure-first learnable). We say that  $\mathcal{C}$  is  $(\epsilon, \delta, p_{\mathrm{succ}})$ -measure-first learnable if there exists a measure-first protocol  $(M, A)$  such that for every  $\pi_x \in \mathcal{C}$ , with probability at least  $p_{\mathrm{succ}}$  we have
+
+$$
+\Pr_ {f \sim_ {U} \{0, 1 \} ^ {N}} \left(\left| \left| \widetilde {\pi} _ {x} (f) - \pi_ {x} (f) \right| \right| _ {T V} \leq \epsilon\right) \geq 1 - \delta , \tag {6}
+$$
+
+where  $\widetilde{\pi}_x(f) \in \Delta(\{0,1\}^{2n+1})$  denotes the distribution that the polynomial-time quantum algorithm obtained from the learning algorithm  $A$  generates samples from on input  $\widehat{\psi}_f = M(|\psi_f\rangle^{\otimes \ell})$ .
+
+Although we defined the two protocols in an idealized case, we will show in Appendix A.1 that our separation results are robust. In particular, they still hold even in the experimental setting where the input states  $|\psi_f\rangle$  are affected by preparation errors and they are just close in trace distance to the ones defined in Eq.(1). Furthermore, we also allow for measurement errors on the labels  $(y,b)$ .
+
+# 3. Main result
+
+In this section, we present the key findings of our paper. In Section 3.1, we define the specific learning problem considered and show how the fully-quantum machine learning model can efficiently solve it. In Section 3.2, we present our first main result showing that no measure-first quantum machine learning model can solve our learning problem efficiently. Finally, in Section 3.3, we show that this separation between the models still holds if the quantum states in the data are efficiently preparable.
+
+# 3.1. The learning problem
+
+In this section, we concretely define the learning problem for which we prove a learning separation. Then, we initiate the proof by showing that it is learnable using a fully-quantum protocol.
+
+The specific concept class we developed to prove our main result is a particular instance of the one defined in Def. 2.2, with the condition that the variables  $(y,b)$  are sampled from the set  $R_{x}(f)$  defined as follows:
+
+Definition 3.1 (Concept class).  $\mathcal{C} = \{\pi_x\mid x\in \{0,1\} ^n\}$  where
+
+$$
+\begin{array}{l} \pi_ {x}: \mathcal {S} _ {\text {p h a s e}} \rightarrow \Delta (\{0, 1 \} ^ {2 n + 1}) \tag {7} \\ \left| \psi_ {f} \right\rangle^ {\otimes \ell} \mapsto \pi_ {x} (f) \\ \end{array}
+$$
+
+where  $\pi_x(f)$  is a distribution over samples  $(x,y,b)$ , where  $(y,b) \sim \mathcal{U}(R_f(x))$  and
+
+$$
+\begin{array}{l} R _ {f} (x) = \{(y, b) \mid y \in \{0, 1 \} ^ {n}, b \in \{0, 1 \}, \tag {8} \\ f (y) \oplus f (y \oplus x) = b \}. \\ \end{array}
+$$
+
+Importantly, in (Aaronson et al., 2024) the authors showed that for each  $x \in \{0,1\}^n$  there exist a POVM measurement  $\Lambda_x$  which when applied to a phase state  $|\psi_f\rangle$  outputs a pair exactly satisfying the relation  $R_{f}(x)$ . With regards to our learning problem, the task of the learning protocols is to learn this measurement.
+
+We now describe how the concept class in Definition 3.1 is fully-quantum learnable.
+
+Proposition 3.2. The concept class in Definition 3.1 is  $(0,0,1)$ -fully-quantum learnable.
+
+The proof of Proposition 3.2 can be found in Appendix A, and we provide a high-level overview of the fully-quantum protocol here. Firstly, the fully-quantum protocol reads out  $x$  from one of the samples generated by  $\pi_x$  in the training data. Next, a quantum circuit denoted as  $U_x$  is constructed following the procedure outlined in (Aaronson et al., 2024). Measuring  $U_x | \psi_f \rangle$  in the computational basis results in a sample from the distribution  $\mathcal{U}(R_f(x))$ . Crucially, it is worth noting that these quantum circuits  $U_x$  are of size  $\mathcal{O}(\mathrm{poly}(n))$  and can be constructed in time  $\mathcal{O}(\mathrm{poly}(n))$ . We remark here that the concept class in Definition 3.1 remains fully quantum learnable even when errors are present in the training data, see Appendix A.1.
+
+It might seem that little genuine learning occurs when  $x$  can be readily read out from a single example in  $T_{x}$ . However, we can introduce various levels of learning by providing only partial information about  $x$  within the examples. This partial information should allow the recovery of  $x$  from a polynomial number of examples. Several examples illustrating this are discussed in more detail in Appendix A.
+
+# 3.2. Limitations of measure-first protocols with general quantum data
+
+In the last section, we discussed how the concept class in Definition 3.1 is fully-quantum learnable. Conversely, in this section we discuss the first part of our main result which states that this concept class is not measure-first learnable.
+
+Theorem 3.3. The concept class in Definition 3.1 is not  $(\epsilon, \delta, p_{\mathrm{succ}})$ -measure-first learnable for  $(1 - \epsilon) \cdot (1 - \delta) > 7/8$  and any  $p_{\mathrm{succ}} > 0$ .
+
+The full proof of Theorem 3.3 is provided in Appendix B, we present a concise overview of the proof here. At its core, the proof hinges on the notion that the existence of a measure-first protocol for the concept class described in Definition 3.1 implies the existence of an efficient classical one-way communication protocol for the Hidden Matching (HM) problem (Bar-Yossef et al., 2004). Notably, in (BarYossef et al., 2004), it has been shown that the HM problem cannot be solved with a communication cost of  $\mathcal{O}\left(\mathrm{poly}(n)\right)$  bits, even on a  $7/8$  fraction of possible inputs. In essence, if the concept class in Definition 3.1 was learnable, one of the
+
+two parties can employ the measurement protocol to encode their input for the HM problem, transmit it to the other party, who can then utilize the learning algorithm  $A$  to successfully solve the HM problem. Intuitively, the reason behind why measure-first learning fails is that due to (Bar-Yossef et al., 2004) it is not possible to compress a phase state  $|\psi_f\rangle$  into a polynomially-sized classical representation  $\widehat{\psi}_f$  that contains enough information to allow one to generate samples from the distributions  $\pi_x(f)$  for all possible  $x$ . Importantly, within the machine learning context, it is crucial to also consider the possibility of protocols making errors on a fraction of inputs and the requirement to gather the necessary data. To overcome this limitation, we use Yao's principle to show that the existence of a protocol which succeeds in the machine learning context would still violate the classical hardness of the HM problem. In particular, by applying Yao's principle, we can ensure the existence of a deterministic algorithm that solves the HM problem on a sufficiently large fraction of inputs, provided that there exists a machine learning measure-first protocol capable of solving the learning task defined in Definition 2.6.
+
+Since in our machine learning setting we are concerned with physical states appearing in the real-world, in the next section we show that our findings also apply to settings where states are efficiently preparable, which we achieve by using pseudorandom states (Brakerski & Shmueli, 2019) (see Section 3.3). Finally, it is crucial to note that even when the data in Eq.(5) is derived from a polynomial number  $\ell$  of copies of each input state  $|\psi_f\rangle$ , measure-first protocols remain incapable of solving the task. This holds true even when considering inefficient (i.e., superpolynomial-time) algorithms or an exponential number of training points, as the communication complexity bounds still apply regardless of the time resources utilized by the algorithms (since the communication protocol assumes both parties have unbounded resources).
+
+# 3.3. Learning separation with efficiently preparable quantum data
+
+From a pragmatic perspective, a crucial limitation of the learning problem outlined in the previous section is that preparing a general phase state is intractable (i.e., not realized by polynomial-time processes). In particular, this raises the question of whether separations could persist for states that are prepared by (natural or artificial) polynomial-time processes. To address this limitation, we show that the concept class in Definition 7 remains not measure-first learnable, even when we constrain the input of the random functions  $\pi_x$  to phase states of so-called pseudorandom functions. Notably, phase states corresponding to appropriately chosen pseudorandom functions can be efficiently prepared. Because of this additional consideration, our separations are notably more general than those in previous works (Aaron
+
+son et al., 2024; Bar-Yossef et al., 2004). Our definition of pseudorandom functions is as follows.
+
+Definition 3.4 (Quantum-secure pseudorandom function (QPRF) (Brakerski & Shmueli, 2019)). Let  $\mathcal{K} = \{\mathcal{K}_n\}_{n\geq 1}$  be an efficiently samplable key distribution, and let PRF =  $\{\mathrm{PRF}_n\}_{n\geq 1},\mathrm{PRF}_n:\mathcal{K}_n\times \{0,1\} ^n\to \{0,1\}$  be an efficiently computable function. We say PRF is a quantum-secure pseudorandom function if for every efficient nonuniform quantum algorithm  $A$  that can make quantum queries there exists a negligible function  $\mathrm{negl}(.)$  such that for every  $n\geq 1$
+
+$$
+\left| \Pr_ {k \sim \mathcal {U} \left(\mathcal {K} _ {n}\right)} \left[ A ^ {\operatorname {P R F} _ {n} (k)} () = 1 \right] - \right. \tag {9}
+$$
+
+$$
+\Pr_ {f \sim \mathcal {U} (\{0, 1 \} ^ {n})} [ A ^ {f} () = 1 ] \mid \leq \operatorname {n e g l} (n)
+$$
+
+Where the notation  $A^f()$  stands for the non-uniform quantum algorithm which can make queries to the function  $f$  and can take any quantum state as input. We remark that if every function  $\mathrm{PRF}_n$  admits a classical circuit of size  $s(n)$  and depth  $d(n)$ , then one can prepare the corresponding phase states using a quantum circuit of size  $\mathcal{O}(s(n))$  and depth  $d(n) + 1$  (Brakerski & Shmueli, 2019). Moreover, the existence of such  $\mathrm{PRF}_n$  is implied by the existence of quantum secure one-way functions (Zhandry, 2012).
+
+# 3.3.1. FULLY-QUANTUM LEARNABILITY WITH PSEUDORANDOM PHASE STATES
+
+Note that when we constrain the inputs of  $\pi_{x}$  to phase states of pseudorandom functions, we essentially modify the distribution over input states in Eq. 4 and Eq. 6. This new distribution now only has support on phase states that are efficiently preparable. While Proposition 3.2 examines general quantum phase states as input states (which are not typically efficiently preparable), we note that the fully-quantum learnability directly extends the setting where we limit ourselves to efficiently preparable phase states as well. We summarize this observation in the following proposition (whose proof is the same as that of Proposition 3.2).
+
+Proposition 3.5. Let  $S_{\mathrm{pr}} = \{\left|\psi_{f^{(k)}}\right\rangle^{\otimes \ell} \mid f^{(k)}(.)$  =  $\mathrm{PRF}_n(k,.)$ ,  $k\in \mathcal{K}\}),$  where PRF is a quantum-secure pseudorandom function with keys  $\mathcal{K}$ . The concept class in Definition 3.1 is  $(0,0,1)$ -fully-quantum learnable when the distribution over input states is uniform over  $S_{\mathrm{pr}}$ .
+
+# 3.3.2. LIMITATIONS OF MEASURE-FIRST PROTOCOLS WITH PSEUDORANDOM PHASE STATES
+
+In the last section, we discussed how the concept class in Definition 3.1 remains fully-quantum learnable when restricted to phase states of pseudorandom functions. Conversely, in this section we show that this concept class also
+
+remains not measure-first learnable when restricted to phase states of pseudorandom functions.
+
+Theorem 3.6. Let  $S_{\mathrm{pr}} = \{|\psi_{f^{(k)}}\rangle^{\otimes \ell} | f^{(k)}(\cdot) = \mathrm{PRF}_n(k,.)$ ,  $k \in \mathcal{K} \}$ , where PRF is a quantum-secure pseudorandom function with keys  $\mathcal{K}$ . The concept class in Definition 3.1 is not  $(\epsilon, \delta, p_{\mathrm{succ}})$ -measure-first learnable for  $(1 - \epsilon) \cdot (1 - \delta) \cdot p_{\mathrm{succ}} > c$  for any constant  $c > 7/8$  when the distribution over input states is uniform over  $S_{\mathrm{pr}}$ .
+
+The main results of this section directly follows from combining Theorem 3.6 and Proposition 3.5.
+
+Corollary 3.7 (informal). If there exist quantum-secure pseudorandom functions, then there exists a quantum supervised learning problem with efficiently generatable quantum data, which cannot be learned by any measure-first protocol according to Def. 2.6 while there exist a fully-quantum protocol which satisfies the learning condition of Def. 2.4
+
+The proof of Theorem 3.6 is provided in Appendix C, and we first present a concise overview of the proof here. The main idea behind the proof is to illustrate that if the concepts are measure-first learnable when restricted to pseudorandom phase states, then the corresponding measure-first learning protocol can be harnessed to create a non-uniform quantum algorithm that is able to distinguish between truly random functions and pseudorandom functions. More precisely, this "distinguisher" algorithm employs the measure-first learning protocol and evaluates its performance when applied to the phase state corresponding to the function it has been given oracular access to. In the proof of Theorem 3.3, we established an upper bound on the generalization performance of any measure-first protocol for truly random phase states. If, however, the measure-first protocol performs well on pseudorandom phase states, then the outcomes of the "distinguisher" algorithm would differ significantly based on whether it is given oracular access to a truly random or a pseudorandom function, thereby contradicting the pseudorandomness assumption. In other words, if the measure-first protocols were effective on pseudorandom states, there would be a clear difference in the performance of the distinguishing: poor accuracy when dealing with truly random functions and strong accuracy when dealing with pseudorandom functions. However, since this would violate Eq. 9, measure-first protocols have to fail when applied to pseudorandom functions.
+
+# 4. Conclusion
+
+In our study, we explored the constraints and capabilities of learning from quantum data. We established a formal machine learning framework that contrasts two protocols: "fully quantum", which adjusts measurements based on data, and "measure-first" restricted by fixed initial (though arbitrarily powerful) measurements. In particular, we pro
+
+vided an example of a learning problem efficiently solved by a fully-quantum protocol but beyond the capabilities of measure-first protocols. Moreover, we showed that this persists even when we limit from universal quantum states, which include also those intractable to prepare, to efficiently preparable quantum states. These findings underscore the crucial role of processing unmeasured quantum data in machine learning, presenting a setting where quantum advantages arise. In particular, they imply that certain learning tasks inherently require the "exponential capacity" of quantum states, distinct from classical data. In other words, the number of bits needed to store  $n$  qubits, in a way that allows a learner to successfully solve the learning problem, is exponential in  $n$ . Such a conclusion is analogous to what Montanaro refers as "anti-Holevo" theorems (Montanaro, 2019). While our proof relies on both separations in one-way communication complexity and pseudorandom states, we highlight the potential for more general constructions. Instead of considering states demonstrating a one-way communication complexity separation, any quantum advice state used in quantum advice complexity classes that cannot be classically simulated with polynomial overhead could suffice, leveraging the separation between the classes FBQP/qpoly and FBQP/poly showed in (Aaronson et al., 2024). In particular, by the result in (Aaronson & Drucker, 2010) any problem that can be efficiently solved with a polynomial-sized quantum advice state can also be solved with an advice state that is the ground state of a local Hamiltonian. This suggests that learning problems where the input quantum data consists of ground states of sufficiently complex local Hamiltonians are promising candidates for demonstrating a separation between measure-first and fully-quantum protocols. Finally, we note that beyond pseudorandom states one could use so-called "computationally indistinguishable" states, which are known to exist assuming the intractability of the graph isomorphism problem (Kawachi et al., 2012), or various other complexity theoretic assumptions (Brakerski et al., 2022).
+
+# Acknowledgements
+
+VD and CG acknowledge the support of the Dutch Research Council (NWO/OCW), as part of the Quantum Software Consortium programme (project number 024.003.037). This work was supported by the Dutch National Growth Fund (NGF), as part of the Quantum Delta NL programme. This publication is also part of the project Divide & Quantum (with project number 1389.20.241) of the research programme NWA-ORC which is (partly) financed by the Dutch Research Council (NWO).
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Quantum Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# References
+
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+
+# A. Proof of Proposition 3.2
+
+Proposition 3.2. The concept class in Definition 3.1 is  $(0,0,1)$ -fully-quantum learnable.
+
+Proof. To prove that the concept class  $\mathcal{C}$  in Definition 3.1 is fully-quantum learnable we will provide a fully-quantum protocol  $A$  that does so successfully. Suppose we are given training data  $T_{x}$  of the form provided in Eq. 3. Firstly, the fully-quantum protocol  $A$  reads out  $x$  from one of the examples in  $T_{x}$ . Next, it uses the construction of (Aaronson et al., 2024) to construct a circuit  $U_{x}$  of size  $\mathcal{O}(n)$  in time  $\mathcal{O}(n)$  such that when measuring the state  $|\phi_{f,x}\rangle = U_x|\psi_f\rangle$  in the computational basis it produces  $(y,b)\in \{0,1\}^{n + 1}$  such that  $b = f(y)\oplus f(x\oplus y)$ . The operator  $U_{x}$ , whose graphical representation is provided in Figure 3 of (Aaronson et al., 2024), is defined as follows. Consider an  $x\in \{0,1\}^n$  with a Hamming weight of  $k\geq 1$ . The operator  $U_{x}$  acts on  $n$  qubits based on the values of the entries of  $x$  as follows:
+
+1. A position  $i$  is selected such that  $x_{i} = 1$ . The corresponding qubit  $i$  is then chosen to output the value of  $b \in \{0,1\}$ .  
+2. For each of the remaining positions  $j$  where  $x_{j} = 1$  and  $j \neq i$ , a CNOT gate is applied with qubit  $j$  as the control and qubit  $i$  as the target. This results in a total of  $k - 1$  CNOT gates.  
+3. Finally, a Hadamard gate is applied to qubit  $i$ .
+
+Measuring the resulting state  $U_{x}|\psi_{f}\rangle$  in the computational basis yields an  $n$ -bit string  $j_{1}j_{2},\ldots ,j_{n}$  such that the variables  $y = j_{1},\dots,j_{n - 1}$  and  $b = j_n^3$  are guaranteed to satisfy the relation  $b = f(y)\oplus f(x\oplus y)$ .
+
+The learning protocol outputs the description of the POVM measurement
+
+$$
+\Pi_ {x} = \left\{U _ {x} | j \rangle \langle j | U _ {x} ^ {\dagger} \mid j \in \{0, 1 \} ^ {n} \right\} \tag {10}
+$$
+
+as by the above measuring  $\Pi_x$  on an arbitrary phase state  $|\psi_f\rangle$  implements  $\pi_{x}$  with zero error.
+
+While it may appear that little learning is occurring when we can readily extract  $x$  from a single example in  $T_{x}$ , we can introduce varying degrees of learning by not appending the complete description of  $x$  to the examples. Instead, we include only partial information about  $x$  that still allow us to recover a full description of  $x$  using a polynomial number of examples. For instance, instead of appending  $x$  to the examples we can append certain functions  $g_{i}(x)$ , where  $g_{i}$  is drawn uniformly random from some set of  $\mathcal{G} = \{g_i\}_{i\in I}$ . For instance, for  $i\in \{0,1\} ^n$  we can consider functions like
+
+$$
+g _ {i} (x) = (i \cdot x, i) \in \{0, 1 \} ^ {n + 1}, \tag {11}
+$$
+
+where  $x\cdot i = \sum_{j = 1}^{n}x_{j}\cdot i_{j}$  mod 2. Another example of such a family of functions would be
+
+$$
+g _ {i} (x) = \left(\mathrm {D L P} (x) _ {i}, i\right) \in \{0, 1 \} ^ {n + 1}, \tag {12}
+$$
+
+where  $\mathrm{DLP}_i(x)$  denotes the  $i$ th bit of the discrete logarithm of  $x$  in a suitably chosen group. For these functions, one can show that  $x$  can be recovered with high probability from a polynomial number of evaluations of  $g_i(x)$  for randomly chosen  $g_i$  from  $\mathcal{G}$ . Moreover, functions similar to the  $g_i$  in Eq. 12 require a quantum computer to be able to efficiently recover  $x$  (Liu et al., 2021).
+
+# A.1. Learnability with noisy data
+
+In a realistic setting, quantum states and measurements will always be affected with experimental errors. In this section we study whether the fully-quantum protocol still manages to solve the learning task even in non ideal scenarios.
+
+We specifically address the scenario where the input states are not exactly the phase states  $|\psi_f\rangle$  from Eq.(1) but are instead approximate states  $|\psi_f\rangle_{\epsilon_{sp}}$  in trace distance, such that:
+
+$$
+d \left(\left| \psi_ {f} \right\rangle , \left| \psi_ {f} \right\rangle_ {\epsilon_ {s p}}\right) \leq \epsilon_ {s p} \tag {13}
+$$
+
+where  $d(|\psi_f\rangle ,|\psi_f\rangle_{\epsilon_{sp}}) = |||\psi_f\rangle \langle \psi_f| - |\psi_f\rangle \langle \psi_f|_{\epsilon_{sp}}||_1$  is the trace distance between the states  $|\psi_f\rangle$  and  $|\psi_f\rangle_{\epsilon_{sp}}$ .
+
+As mentioned before, we also allow for measurement errors on the training labels. Specifically we can consider the case that the labels  $(y,b)$  in the training data follow an approximate distribution  $\pi_x^{\epsilon M}(f)$  which is only close to the ideal one in total variation distance:
+
+$$
+\forall f \in \{0, 1 \} ^ {N} \quad | | \pi_ {x} ^ {\epsilon_ {M}} (f) - \pi_ {x} (f) | | _ {T V} \leq \epsilon_ {M} \tag {14}
+$$
+
+Note that only the two variables  $(y, b)$  are affected by measurement errors, while the variable  $x$ , which labels each concept, is not affected by any error.
+
+Even with both of the above sources of experimental error, the learning problem in Def. 3.1 remains fully quantum learnable in the following sense.
+
+Proposition A.1. Assuming a maximum error  $\epsilon_{sp}^{max}$  on the input states and a maximum error  $\epsilon_M^{max}$  on the corresponding labels, then there exists a fully-quantum protocol  $A$  such that for every  $\pi_x\in \mathcal{C}$  with probability 1 satisfies:
+
+$$
+P r _ {f \sim \mathcal {U} (\{0, 1 \} ^ {N})} \left(\left| \left| \tilde {\pi} _ {x} (f) - \pi_ {x} (f) \right| \right| _ {T V} \leq \epsilon_ {s p} ^ {m a x}\right) = 1 \tag {15}
+$$
+
+where  $\tilde{\pi}_x(f) \in \Delta(\{0,1\}^{2n+1})$  denotes the distribution that the polynomial-time quantum algorithm obtained from the learning algorithm  $A$  generates samples from on input  $|\psi_f\rangle_{\epsilon_{sp}}^{\otimes l}$ .
+
+We can adjust Def. 2.3 of fully-quantum protocols to take into account the presence of errors in the training data.
+
+Definition A.2 (Fully-quantum protocols with noisy data). A  $(\epsilon_{sp},\epsilon_M)$ -fully-quantum protocol for the concept class  $\mathcal{C}$  in Definition 3.1 is a polynomial-time algorithm  $A$  that takes as input training data of the form
+
+$$
+T _ {x, \epsilon_ {s p}, \epsilon_ {M}} = \left\{\left(\left| \psi_ {f ^ {(i)}} \right\rangle_ {\epsilon_ {s p}} ^ {\otimes \ell}, (x, y, b)\right) \mid (x, y, b) \sim \pi_ {x} ^ {\epsilon_ {M}} \left(f ^ {(i)}\right), \right. \tag {16}
+$$
+
+$$
+\left. \text {a n d} f ^ {(i)} \sim \mathcal {U} \left(\{0, 1 \} ^ {N}\right) \right\} _ {i = 1} ^ {\text {p o l y} (n)}, \tag {16}
+$$
+
+and outputs a classical description of a polynomial-time quantum algorithm that on input  $|\psi_f\rangle_{\epsilon_{sp}}^{\otimes \ell} \in S_{\mathrm{phase}}$  generates a sample from a distribution  $\widetilde{\pi}_x(f) \in \Delta(\{0,1\}^{2n+1})$ .
+
+Having noisy data, the natural modified learning condition for the fully-quantum protocol is the following.
+
+Definition A.3  $((\epsilon, \epsilon_{sp}, \epsilon_M, \delta, p_{\mathrm{succ}})$ -fully-quantum learnable with noisy data). We say that  $\mathcal{C}$  is  $(\epsilon, \epsilon_{sp}, \epsilon_M, \delta, p_{\mathrm{succ}})$ -fully-quantum learnable if there exists a  $(\epsilon_{sp}, \epsilon_M)$ -fully-quantum protocol  $A$  such that for every  $\pi_x \in \mathcal{C}$ , with probability at least  $p_{\mathrm{succ}}$  we have
+
+$$
+\Pr_ {f \sim \mathcal {U} (\{0, 1 \} ^ {N})} \left(\left| \left| \widetilde {\pi} _ {x} (f) - \pi_ {x} (f) \right| \right| _ {T V} \leq 1 - \epsilon_ {t o t}\right) \geq \delta , \tag {17}
+$$
+
+where  $\epsilon_{tot} = \epsilon +\epsilon_{sp} + \epsilon_{M}$  and  $\widetilde{\pi}_x(f)\in \Delta (\{0,1\}^{2n + 1})$  denotes the distribution that the polynomial-time quantum algorithm obtained from the learning algorithm  $A$  generates samples from on input  $|\psi_f\rangle^{\otimes \ell}\in S_{\mathrm{phase}}$
+
+We can now prove that the concept class  $\mathcal{C}$  is fully-quantum learnable even in the presence of noisy data.
+
+Proposition A.4. Assuming a maximum error of  $\epsilon_{sp}^{max}$  on the input states, the concept class in Definition 3.1 is  $(0,\epsilon_{sp}^{max},0,1)$ -fully-quantum learnable.
+
+Proof. The proof follows directly from the one for the ideal case in Appendix A. The fully-quantum protocol performs exactly the same steps as in the ideal case, in this way the error in the prediction is bounded by the error in state preparation. Specifically, assuming for every  $f \in \{0,1\}^N$  input states  $|\psi\rangle_{\epsilon_{sp}}$  are such that:
+
+$$
+d \left(\left| \psi_ {f} \right\rangle , \left| \psi_ {f} \right\rangle_ {\epsilon_ {s p}}\right) \leq \epsilon_ {s p} ^ {\max } \tag {18}
+$$
+
+Then by the definition of trace distance we have that given any set of POVM  $\{E_m\}$  (see Theorem 9.1 of (Nielsen & Chuang, 2010)) it holds that:
+
+$$
+\sum_ {m} \left| \operatorname {T r} \left[ E _ {m} \left(\left| \psi_ {f} \right\rangle \left\langle \psi_ {f} \right| - \left| \psi_ {f} \right\rangle \left\langle \psi_ {f} \right| _ {\epsilon_ {s p}}\right) \right] \right| \leq d \left(\left| \psi_ {f} \right\rangle , \left| \psi_ {f} \right\rangle_ {\epsilon_ {s p}}\right) \tag {19}
+$$
+
+Therefore, any probability distribution obtained from the measurement outcomes of the POVM  $E_{m}$  on each state  $|\psi_f\rangle_{\epsilon sp}$  will have a total variation distance of at most  $d(|\psi_f\rangle ,|\psi_f\rangle_{\epsilon sp})$  compared to the distribution induced by applying the same POVM  $E_{m}$  on the ideal states  $|\psi_f\rangle$ . As the inequality (19) holds for any set of POVM  $\{E_m\}$ , it will particularly be true for the set of POVM  $\Pi_x$  of Eq.(10) implemented by the fully-quantum protocol. This then concludes the proof as we previously showed that the fully-quantum protocol exactly reconstructs the target distribution  $\pi_x(f)$  for each  $f$  with zero error and we assumed  $\epsilon_{sp}^{max}$  to be the maximum difference in trace distance between the noisy input states and the ideal ones.
+
+Following the same reasoning as in the proof of Theorem 3.3 in Appendix B, it is straightforward to show that when  $\epsilon_{sp}^{max} \leq 7/8$ , the learning task becomes intractable for any measure-first protocol.
+
+# B. Proof of Theorem 3.3
+
+Theorem 3.3. The concept class in Definition 3.1 is not  $(\epsilon, \delta, p_{\mathrm{suc}})$ -measure-first learnable for  $(1 - \epsilon) \cdot (1 - \delta) > 7/8$  and any  $p_{\mathrm{suc}} > 0$ .
+
+Proof of Theorem 3.3. The main building block of our proof of Theorem 3.3 is a result in one-way communication complexity by Bar-Yossef, Jayram and Kerenidis (Bar-Yossef et al., 2004). They define a problem called Hidden Matching (HM). Here Alice is given a string  $f \in \{0,1\}^N$ , while Bob is given a perfect matching  $M$  on the set  $[N]$ , consisting of  $N/2$  edges. Bob's goal is to output some  $(i,j,f_i \oplus f_j)$  for some edge  $(i,j) \in M$ . Their main result is:
+
+Theorem B.1 (Classical hardness of HM (Bar-Yossef et al., 2004)). Let  $\mathcal{M}$  be any set of perfect matchings on  $[N]$  that is pairwise edge-disjoint and satisfies  $|\mathcal{M}| = \Omega(N)$ . Let  $\mu$  be the distribution over inputs to HM in which Alice's input is uniform in  $\{0,1\}^N$  and Bob's input is uniform in  $\mathcal{M}$ . Then, any deterministic one-way protocol for HM that errs with probability at most  $1/8$  with respect to  $\mu$  requires  $\Omega(\sqrt{N})$  bits of communication.
+
+Suppose the concept class in Definition 3.1 is  $(\epsilon, \delta, p_{\mathrm{suc}})$ -measure-first learnable using a measure-first protocol given by  $(M, A)$  with  $(1 - \epsilon) \cdot (1 - \delta) > 7/8$  and  $p_{\mathrm{suc}} > 0$ . Throughout the proof, we will show that the existence of such a measure-first learning protocol contradicts the classical hardness of HM outlined in Theorem B.1. To do so, consider the HM problem with  $\mathcal{M} = \{M_x \mid x \in \{0, 1\}^n\}$ , where
+
+$$
+M _ {x} = \{(y, y \oplus x) \mid y \in \{0, 1 \} ^ {n} \}. \tag {20}
+$$
+
+and note that  $|\mathcal{M}| = N$ . To solve this instance of the HM problem Bob first generates training data  $T_{x}^{M}$  as in Eq. 5. Note that Bob can do so because he has knowledge of the bitstring  $x$ . In particular, Bob can generate  $f^{(i)}$  from  $\{0,1\}^N$ , compute  $R_{f^{(i)}}(x)$  and pick an element  $(y,b)$  from it. Next, Alice applies the measure protocol  $M$  to  $|\psi_f\rangle$  for her input  $f\in \{0,1\} ^N$  and sends  $\widehat{\psi}_f = M(|\psi_f\rangle)$  to Bob. Finally, Bob applies  $A$  on the data  $T_{x}^{M}$  he generated and Alice's input  $\widehat{\psi}_f$  to obtain a sample  $(x,y,b)\sim \widetilde{\pi}_x(\widehat{\psi}_f)$ . Since we assumed that  $p_{\mathrm{suc}} > 0$ , we know that for any  $x\in \{0,1\} ^n$  there must exist training data  $\hat{T_x^M}$  and internal randomization of the learning algorithm  $A$  such that the polynomial-time quantum algorithm output by the protocol satisfies Eq. 6. Throughout the remainder of this proof, we assume Bob fixes this to be the training data and internal randomization he uses for his input  $x$  (note that Bob can do so because this does not depend on the input of Alice). Based on this fixed choice of training data and internal randomization we partition  $\{0,1\} ^N = \mathrm{F}_{\mathrm{good}}^x\sqcup \mathrm{F}_{\mathrm{bad}}^x$ , where  $\mathrm{F}_{\mathrm{good}}^x$  denotes the set of functions  $f$  for which
+
+$$
+\left| \left| \widetilde {\pi} _ {x} (f) - \pi_ {x} (f) \right| \right| _ {T V} \leq 1 - \epsilon , \tag {21}
+$$
+
+where  $\widetilde{\pi}_x$  is the random function implemented by the quantum algorithm output by the protocol when using the training data  $T_x^M$  and internal randomization as above. Moreover, we note  $|\mathrm{F}_{\mathrm{good}}^x| \geq (1 - \delta) \cdot 2^n$  by Eq. 6. Finally, due to Eq. 6 we find that the probability that  $(y, b) \in R_f(x)$  is at least
+
+$$
+\Pr \bigl ((y, b) \in R _ {f} (x) \bigr) \geq (1 - \epsilon) \tag {22}
+$$
+
+for all  $f \in \mathrm{F}_{\mathrm{good}}^x$ . In conclusion, we find that the above described protocol is a randomized one-way communication protocol for HM with success probability at least  $\epsilon$  for all inputs  $(x,f)$  in the subset
+
+$$
+\mathcal {X} := \bigcup_ {x \in \{0, 1 \} ^ {n}} \{x \} \times \mathrm {F} _ {\text {g o o d}} ^ {x}. \tag {23}
+$$
+
+In the remainder of our proof, we let  $A'(x, \widehat{\psi}_f)$  denote the protocol that Bob runs on his side (i.e., generating the training data  $T_x^M$ , running the algorithm  $A$  on it, and drawing a sample from  $\widetilde{\pi}_x(\widehat{\psi}_f)$ ). Also, we ensure Bob does so using only classical randomized computation by classically simulating the quantum algorithms. Next, we use Yao's principle to show that the above randomized one-way communication protocol implies the existence of a deterministic one-way communication protocol that errs with probability at most  $(1 - \epsilon) \cdot (1 - \delta)$  with respect to  $\mu$  (which would violate Theorem B.1 since  $(1 - \epsilon) \cdot (1 - \delta) > 7/8$ ). Let  $\mathcal{A}$  denote the family of deterministic protocols obtained by "hardwiring" all possible internal randomizations of the evaluation of  $\widetilde{\pi}_x$  by  $A'$ , i.e.,
+
+$$
+\mathcal {A} = \left\{A _ {r} ^ {\prime} (.,.) \mid r \in \{0, 1 \} ^ {\exp (n)} \right\}. \tag {24}
+$$
+
+Also, let  $\mathbb{X}$  be the random variable with values  $(x,f)$  distributed according to the uniform distribution over  $\mathcal{X}$ , and let  $\mathbb{A}$  be the random variable over  $\mathcal{A}$  where the  $r$  is uniformly random. Finally, we define the function  $s:\mathcal{X}\times \mathcal{A}\to \mathbb{R}$  as
+
+$$
+s ((x, f), A _ {r} ^ {\prime} (., .)) = \mathbb {1} \left[ A _ {r} ^ {\prime} (x, \widehat {\psi_ {f}}) \in R _ {f} (x) \right]. \tag {25}
+$$
+
+Theorem B.2 (Yao's principle). Let  $\mathbb{A}$  be a random variable with values in  $\mathbb{A}$  as defined in Eq. 24, and let  $\mathbb{X}$  be a random variable with values in  $\mathcal{X}$  as defined in Eq. 23. Then,
+
+$$
+\min  _ {(x, f) \in \mathcal {X}} \mathbb {E} [ s ((x, f), \mathbb {A}) ] \leq \max  _ {A _ {r} ^ {\prime} \in \mathcal {A}} \mathbb {E} [ s (\mathrm {X}, A _ {r} ^ {\prime}) ] \tag {26}
+$$
+
+where  $s$  is the function defined in Eq.25.
+
+Observe that the quantity  $\mathbb{E}\left[s(\mathbb{X},A_r^{\prime})\right]$  is precisely the success probability of the deterministic algorithm  $A_{r}^{\prime}\in \mathcal{A}$  with respect to the uniform distribution over  $\mathcal{X}$ . Thus, Eq. 26 implies the existence of a deterministic algorithm  $A_{r}^{\prime}$  such that
+
+$$
+\begin{array}{l} \mathbb {E} \left[ s \left(\mathbb {X}, A _ {r} ^ {\prime}\right) \right] = \Pr_ {(x, f) \sim \mathcal {U} (\chi)} \left(A _ {r} ^ {\prime} (x, \widehat {\psi} _ {f}) \in R _ {f} (x)\right) \\ \geq \min  _ {(x, f) \in \mathcal {X}} \mathbb {E} [ s ((x, f), \mathbb {A}) ]. \tag {27} \\ \end{array}
+$$
+
+Moreover, observe that the quantity  $\min_{(x,f)\in \mathcal{X}}\mathbb{E}\left[s((x,f),\mathbb{A})\right]$  is precisely the success probability of the randomized algorithm  $A^{\prime}$ , which we have previously shown to be at least  $1 - \epsilon$ . By combining this with Eq. 27 we find that
+
+$$
+\Pr_ {(x, f) \sim \mathcal {U} (\mathcal {X})} \left(A _ {r} ^ {\prime} (x, \widehat {\psi} _ {f}) \in R _ {f} (x)\right) \geq 1 - \epsilon . \tag {28}
+$$
+
+Moreover, since  $|\mathrm{F}_{\mathrm{good}}^x |\geq (1 - \delta)\cdot 2^n$  we find that
+
+$$
+\Pr_ {(x, f) \sim \mu} \left(A _ {r} ^ {\prime} (x, \widehat {\psi} _ {f}) \in R _ {f} (x)\right) \geq (1 - \epsilon) \cdot (1 - \delta). \tag {29}
+$$
+
+Finally, since  $(1 - \epsilon)\cdot (1 - \delta) > 7 / 8$ , this violates the classical hardness of HM outlined in Theorem B.1.
+
+# C. Proof of Theorem 3.6
+
+Theorem 3.6. Let  $S_{\mathrm{pr}} = \{\left|\psi_{f^{(k)}}\right\rangle^{\otimes \ell} \mid f^{(k)}(.) = \mathsf{PRF}_n(k,.)$ ,  $k \in \mathcal{K}\}$ , where PRF is a quantum-secure pseudorandom function with keys  $\mathcal{K}$ . The concept class in Definition 3.1 is not  $(\epsilon, \delta, p_{\mathrm{suc}})$ -measure-first learnable for  $(1 - \epsilon) \cdot (1 - \delta) \cdot p_{\mathrm{suc}} > c$  for any constant  $c > 7/8$  when the distribution over input states is uniform over  $S_{\mathrm{pr}}$ .
+
+Proof of Theorem 3.6. Suppose the concept class in Definition 3.1 is  $(\epsilon, \delta, p_{\mathrm{succ}})$ -measure-first learnable with  $p_{\mathrm{succ}} \cdot (1 - \epsilon) \cdot (1 - \delta) > c$  for a constant  $c > 7/8$  when the distribution over input states is uniform over  $\mathcal{S}_{\mathrm{pr}}$  using a measure-first protocol given by  $(M, A)$ . That is, for every  $\pi_x \in \mathcal{C}$ , with probability at least  $p_{\mathrm{succ}}$  we have
+
+$$
+\Pr_ {k \sim \mathcal {U} \left(\mathcal {K} _ {n}\right)} \left(\left| \left| \widetilde {\pi} _ {x} \left(f ^ {(k)}\right) - \pi_ {x} \left(f ^ {(k)}\right) \right| \right| _ {T V} \leq \epsilon\right) \geq 1 - \delta , \tag {30}
+$$
+
+where  $f^{(k)}(.) = \mathrm{PRF}(k,.)$  and  $\widetilde{\pi}_x$  is the randomized quantum function obtained from  $A$  on input of the form
+
+$$
+\begin{array}{l} T _ {x} ^ {M} = \left\{\left(\widehat {\psi} _ {f ^ {(k)}}, (x, y, b)\right) \mid (x, y, b) \sim \pi_ {x} (f ^ {(k)}) \right. \tag {31} \\ \left. \text {a n d} k \sim \mathcal {U} \left(\mathcal {K} _ {n}\right) \right\} _ {i = 1} ^ {\operatorname {p o l y} (n)}. \\ \end{array}
+$$
+
+The main goal of the remainder of the proof is to show that the above assumptions violate the assumption that PRF is a quantum-secure pseudorandom function. To achieve this, we devise a quantum algorithm, denoted as  $A^f$ , which has query access to a function  $f$ , and which will exhibit a significant difference in the probability of outputting 1 when provided with either a truly random function  $f$  or a pseudorandom function  $f^{(k)}$ . In essence,  $A^f$  will train a measure-first protocol on phase states of pseudorandom functions and evaluate its performance on the provided function  $f$ , outputting 1 if it produces a correct sample  $(x,y,b)$  with  $(y,b)\in R_f(x)$ . Assuming our measure-first protocol can successfully learn the concepts for phase states of pseudorandom functions,  $A^f$  will most likely output 1 when  $f$  is pseudorandom. Conversely, if  $f$  is truly random, then based on arguments similar to those used in the proof of Theorem 3.3, the measure-first learning protocol is likely to be incorrect, leading  $A^f$  to most of the time output 0. In particular, we consider the polynomial-time quantum algorithm  $A^f$  that does the following:
+
+1. Sample  $x \sim \mathcal{U}(\{0, 1\}^n)$ .  
+2. Generate a set of examples  $T_x^M$  as in Eq. 31<sup>4</sup>.  
+3. Use the learning algorithm  $A$  with set of examples  $T_x^M$  to obtain a quantum algorithm  $A'$  for  $\widetilde{\pi}_x$ .  
+4. Using quantum query access to  $f$  prepare  $|\psi_f\rangle^{\otimes \ell}.^5$  
+5. Apply  $M$  to  $\left|\psi_f\right\rangle^{\otimes \ell}$  to obtain  $\widehat{\psi}_f = M\left(\left|\psi_f\right\rangle^{\otimes \ell}\right)$ .  
+6. Apply  $A'$  to  $\widehat{\psi}_f$  to obtain a sample  $(x, y, b)$  and output 1 if  $y \in R_f(x)$ , and 0 otherwise.
+
+By the Eq. 30 and the paragraph leading up to it, we know that
+
+$$
+\Pr_ {k \sim \mathcal {U} \left(\mathcal {K} _ {n}\right)} \left[ A ^ {f ^ {(k)}} = 1 \right] \geq p _ {\text {s u c c}} \cdot (1 - \epsilon) \cdot (1 - \delta) > c. \tag {32}
+$$
+
+On the other hand, from the classical lower bound for the HM problem in Theorem B.1, we know that
+
+$$
+\Pr_ {f \sim \mathcal {U} (\{0, 1 \} ^ {N})} [ A ^ {f} (.) = 1 ] \leq 7 / 8. \tag {33}
+$$
+
+In particular, if Eq. 33 does not hold, then one can construct a one-way communication protocol for HM that succeeds with probability at least  $7/8$  with respect to  $\mu$  by having Bob perform steps  $(2) - (3)$ , having Alice perform steps  $(4) - (5)$ , and sending  $\widehat{\psi}_f$  to Bob to perform the first part of step (6) where they obtain the sample  $(x, y, b)$ . In summary, we conclude that the measure-first protocol, when trained on phase states of pseudorandom functions, cannot generalize well to truly random functions based on the lower-bound established for the HM problem in Theorem B.1. Moreover, given our assumption that the concept class  $\mathcal{C}$  in Definition 3.1 is  $(\epsilon, \delta, p_{\mathrm{suc}})$ -measure-first learnable on phase states of pseudorandom states, it has to generalize well to other pseudorandom states. This implies a distinctive behavior of the "benchmarking algorithm"  $A^f$  when provided with access to either a pseudorandom function  $f^{(k)}$  or a truly random function  $f$ . In other words, we thus conclude that Eq. 32 and Eq. 33 are in contradiction with the assumption that PRF is a quantum-secure pseudorandom function.
+
+# D. Two examples of measure-first protocols
+
+To elaborate on the definition of measure-first (MF) protocols (i.e., Definition 2.5) and their potential capabilities, we present two concrete MF protocols that, at first glance, appear capable of solving the learning task. However, in each case, we identify fundamental reasons for their failure, which highlight the limitations of MF protocols.
+
+# D.1. Measure-first protocol based on classical shadows
+
+One plausible MF protocol leverages the classical shadow framework (Huang et al., 2020). Here, the measurement strategy  $M$  takes as input multiple copies of each state  $|\psi_f\rangle$  and generates a classical representation  $\widehat{\psi}_f$  by performing randomized measurements as prescribed by the classical shadow protocol (Huang et al., 2020). The learning algorithm  $A$  can then use the training data  $\hat{T}_x$  to determine  $x$  and output a hypothesis  $h_x$ . This hypothesis would use the classical representation  $\widehat{\psi}_f$  to predict the correct labels associated with  $f$  by approximating the outcomes of the POVMs  $E_j^x$ .
+
+However, for the hypothesis  $h_x$  to generate samples  $(y, b)$  that are close in total variation distance to the target distribution  $\Lambda_x$ , the precision required on each POVM  $E_j^x$  must scale exponentially. As a result, the number of copies  $N$  of  $|\psi_f\rangle$  required by the classical shadow protocol to achieve such precision also grows exponentially. Therefore, this MF protocol fails due to fundamental limitations in efficiently compressing pseudorandom phase states into classical representations capable of achieving the required exponential precision. In this sense, our result can also be interpreted as a lower bound on the number of copies required for any shadow-based procedure to recover observables with exponential precision.
+
+# D.2. Measure-first protocol with circuit learning
+
+Another potential MF protocol could attempt to directly learn a polynomially-sized description of the circuit that prepares the pseudorandom states  $|\psi_f\rangle$ . By definition, pseudorandom states are efficiently preparable and always admit such a circuit. In this approach, the measurement scheme  $M$  would take as input multiple copies of  $|\psi_f\rangle$  and attempt to infer this circuit description. The learning algorithm  $A$  would then use the training data  $\hat{T}_x$  to determine  $x$  and output a hypothesis  $h_x$ . The hypothesis  $h_x$  could use the learned circuit to recreate the state  $|\psi_f\rangle$  and implement the measurement  $\Lambda_x$ .
+
+However, despite the existence of a polynomial-depth circuit that prepares  $|\psi_f\rangle$ , our results show that no efficient measurement strategy  $M$  can extract a succinct classical description of this circuit. This failure arises from the inherent pseudorandomness of the states, which ensures that no efficient measurement strategy can compress the information contained in  $|\psi_f\rangle$  into a usable classical description. Consequently, this MF protocol also fails to achieve the learning goal.
+
+These examples highlight not only the challenges faced by MF protocols but also the fundamental separations between MF and fully quantum (FQ) protocols. They illustrate the difficulty of efficiently compressing pseudorandom phase states into classical descriptions that retain enough information for the machine learning task.

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2025/LineFlow_ A Framework to Learn Active Control of Production Lines/full.md

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+# Kai Müller<sup>1</sup> Martin Wenzel<sup>1</sup> Tobias Windisch<sup>1</sup>
+
+# Abstract
+
+Many production lines require active control mechanisms, such as adaptive routing, worker reallocation, and rescheduling, to maintain optimal performance. However, designing these control systems is challenging for various reasons, and while reinforcement learning (RL) has shown promise in addressing these challenges, a standardized and general framework is still lacking. In this work, we introduce LineFlow, an extensible, open-source Python framework for simulating production lines of arbitrary complexity and training RL agents to control them. To demonstrate the capabilities and to validate the underlying theoretical assumptions of LineFlow, we formulate core subproblems of active line control in ways that facilitate mathematical analysis. For each problem, we provide optimal solutions for comparison. We benchmark state-of-the-art RL algorithms and show that the learned policies approach optimal performance in well-understood scenarios. However, for more complex, industrial-scale production lines, RL still faces significant challenges, highlighting the need for further research in areas such as reward shaping, curriculum learning, and hierarchical control.
+
+# 1. Introduction
+
+At its core, manufacturing is about transforming raw materials into finished goods, often on a large scale. In most production systems, the necessary process steps are carried out by work stations which each component has to traverse sequentially. Typically, the work steps necessary have to be applied in a fixed order, creating an interdependence between stations in the sense that performance issues of one station directly affect stations down- and upstream.
+
+In theory, finding optimal layouts for production lines that
+
+<sup>1</sup>University of Applied Sciences Kempten, Kempten, Germany. Correspondence to: Tobias Windisch <tobias.windisch@hs-kempten.de>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+take the interdependence of the individual work steps into account is a computationally hard but well understood problem called assembly line balancing problem (Padron et al., 2009; Scholl & Becker, 2006). Solution strategies are powered by good stochastic models for interlinked production lines, like in (Bierbooms, 2012; Liberopoulos et al., 2006), that help computing the equilibrium state of the throughput for a given layout. However, in practice, optimal layouts and theoretic knowledge on the equilibrium of the throughput does not guarantee that the production line unfolds its full theoretical potential. This may be due to situations like changing processing conditions, adverse coincidences in the stochasticity of the processes, or machine failures. These circumstances can suddenly lead to full buffers, jams, and shifting bottlenecks (Roser et al., 2003; Mahmoodi et al., 2022). Thus, reacting in a dynamic and optimal way at runtime is key to keep the performance of the production line at its best. At present, this is either a completely manual task accomplished by line workers on-site or by line control systems, that actively adjust line parameters in real-time (see Figure 1). Possible interventions are rerouting tasks to alternative stations (Huang et al., 2019), reallocating workers (Pérez-Wheelock et al., 2022; Dolgui et al., 2018), or scheduling maintenance (Geurtsen et al., 2023). Classical
+
+![](images/884250c2be873ae9970381c96d242d62ab2744bb8e11c66ebf15907e934aa45a.jpg)  
+Figure 1. Active line control based on real-time data.
+
+active line control systems typically combine rule-based strategies and mathematical models to optimize scheduling and resource allocation, focusing mostly on bottleneck identification, see for instance (Roser et al., 2014; 2017; Li et al., 2009; Lai et al., 2021) and references therein. Finding optimal control strategies is a complex task that not only requires a detailed understanding of the production line dynamics but also involves time-intensive trial-and-error session on the real system and simulations. While effective
+
+in predictable scenarios, classical approaches often struggle in dynamic and complex environments with high uncertainties, which has motivated the use of machine learning for tasks like predicting bottlenecks (Subramaniyan et al., 2021) and forecasting demand (Gonçalves et al., 2021).
+
+Since active production line control is, at its core, a sequence of observing a state and taking action, it seems like a natural next step to learn line control in an end-to-end fashion through interactions with a simulation using reinforcement learning (RL). RL algorithms have demonstrated the ability to learn complex relationships between states and actions (Vinyals et al., 2019; Silver et al., 2016; Wu et al., 2023; Jumper et al., 2021). Significant research has been conducted on learning control policies for production lines (see Section 1.1); however, progress is hindered because existing studies each rely on their own ad hoc or domain-specific simulations, making benchmarking, reproducibility, and the assessment of RL progress in active line control difficult. One reason for this is the absence of a theoretical or practical framework to train and evaluate RL agents in various production settings in a standardized way.
+
+In this work, we address exactly this research gap by introducing LineFlow<sup>1</sup>, a free and extensible Python package specifically tailored to simulate production lines in order to train RL agents for active line control at scale (Section 2). LineFlow not only is a fine-graded and fast discrete-event simulation of production lines, it also comes with an implementation of conventional action- and state-spaces as well as a flexible score (Section 2.3) inspired by classical performance indicators that allows to train agents for various production settings (Section 3). To evaluate its theoretic underpinnings, we study well understood and realistic production scenarios by computing their mathematical optimum (Section 4) and then comparing it with the performance of control policies interacting with LineFlow (Section 5). To showcase the current limits, we also analyze a very complex production setting as mentioned above with unclear optimum and train RL agents via curriculum learning (Section 4.4). As there exist strong privacy concerns among manufacturers when it comes to releasing production line layouts, publicly available datasets holding performance information and non-trivial layouts are rare. Here, we hope that LineFlow will close this gap by allowing manufacturers to provide synthetic non-confidential digital twins preserving their key runtime challenges to the public. In Section D, we model a real-world production line based on publicly available data in LineFlow. Our research is purely application-driven by the needs of end-users in manufacturing and opens up a wide range of challenges for machine learning researchers to investigate. Since implementing active control systems is currently a highly manual task guided
+
+by domain expertise, we aim for LineFlow to accelerate research on learning active line control in the future.
+
+# 1.1. Related Work
+
+There has been a lot of research where RL methods are studied to control a given manufacturing technology, like for welding (Jin et al., 2019; Masinelli et al., 2020; Zou & Lan, 2020) or molding (Szarski & Chauhan, 2023; Guo et al., 2019) just to name a few. We refer to (Nian et al., 2020) and references therein for an overview. In contrast, our work treats the industrial technology happening inside the processes as a black box and solely considers the processing time of the stations and their statistical interplay. This means, we neither model physical relations nor monitor physical parameters of the processes, which is a fruitful application field for RL on its own (Viharos & Jakab, 2021).
+
+A large body of research directly related to our work has been done for production scenarios where tasks of varying length have to be scheduled on heterogeneous machines with different processing capabilities or resources. Such scheduling problems are tackled with RL-based approaches in many works, like in (Shi et al., 2020; Shiue et al., 2018; Kim & Lee, 1998; Tortorelli et al., 2022; Ali & Tirel, 2023; Overbeck et al., 2021). A comprehensive overview is provided in (Kuhnle et al., 2021). Other research focuses on machine interactions, as in (Wang et al., 2016), where RL is used to control a layout consisting of two stations and one buffer to maximize throughput or in (Loffredo et al., 2024; 2023), where energy costs are minimized by switching stations dynamically to standby mode. In (Geurtsen et al., 2023), the scheduling of predictive maintenance of a production line using RL has been studied.
+
+# 2. Simulating Production Lines
+
+Production lines can be characterized in many ways, like by the types of their processes or parts, their production volume, or how their stations are arranged (Kang et al., 2020; Liberopoulos et al., 2006). Here, we focus on production lines that produce discrete items. This section introduces general principles and mechanisms of such production systems and explains how these are reflected in LineFlow.
+
+# 2.1. Objects of Production Lines
+
+The main objective of production lines studied in this work is to produce discrete items called parts. Typically, the production of a single part requires a specific set of work steps that need to be applied in a certain order involving the production and assembly of a series of sub-components. Each work step is carried out in a specific station. One of the objectives of LineFlow is to model the statistical dynamics of a production line by considering the individual processing
+
+![](images/5f4a9a518ad94f0fc64cc2141a8a5e21eb203e59e7270de2802dc4a9e98992f7.jpg)  
+Figure 2. A production line visualized with LineFlow.
+
+times of the stations and their interplay. To be more precise, the processing times are assumed to be exponentially distributed like in (Bierbooms, 2012) as  $\mathcal{T} = T + \mathrm{Exp}_S$ , where  $T \geq 0$  is the minimal processing time possible and where  $\mathrm{Exp}_S$  denotes the exponential distribution with mean  $S$ . We distinguish different types of stations, among them are sources that set up components, processes that apply a piece of work on a single component, assemblies that join two or more components into one, and sinks that remove a fully build part from the line (see Figure 2). Components are transported by carriers from station to station via buffers, which work on a first-in-first-out principle. Buffers can only hold a predefined number of carriers, called their capacity. Time-intensive work steps may be distributed over identical parallel stations, while switches handle the routing of carriers. The arrangement of stations and buffers is typically called the layout of the production line.
+
+# 2.2.Active Line Control
+
+The overall performance of a production line depends sensitively on the individual performances of its stations as performance issues of any station are propagated down- and upstream by their linking buffers. For instance, consider the production line shown in Figure 2: If the processing time of Process is higher than the processing time of Assembly, the buffer in between fills up and once its maximal capacity is reached, the previous station is blocked as it cannot push finished components to the buffer. Such a jam can be propagated backwards through the line and may even block the source. Similarly, subsequent stations like the two sinks with faster processing times are not served with components and have to wait till new components arrive. In the equilibrium, the performance of the full system depends essentially on the bottleneck station, which is in simple cases just the station having the highest processing time. Buffers constitute an important parameter of production lines as they can dampen fluctuations in processing times at the stations. However, buffers and their capacity are fixed parameters of the layout, whereas stochasticity in the processing times of the processes can result in a shifting runtime bottleneck (Roser et al., 2003). Thus, the bottle
+
+neck can vary, whereas the buffer capacities cannot. We refer to (Bierbooms, 2012) for a detailed analysis of the interplay of stations and buffers. Bottlenecks in production lines do not only reduce throughput by slowing down the overall part flow, but they can also lead to quality issues and even scrap. One prominent example arises in processes with strict timing constraints between consecutive steps. In adhesive bonding, for instance, components must be joined within a certain time window after the adhesive is applied. If a downstream bottleneck causes delays and this threshold is exceeded, the bond can no longer be guaranteed to meet quality standards, and the affected part must be discarded (Chen et al., 2005). Such timing-sensitive constraints make bottleneck prevention critical not only for efficiency, but also for product quality and material waste reduction.
+
+To circumvent or mitigate such harmful runtime effects, we essentially study three corrective actions and their interplay: Changing worker assignments to speed up bottleneck stations, changing the distribution of components, and including waiting times to prevent scrap. On the one hand, increasing the waiting time is an important counter activity to prevent scrap, but there is a thin line between preventing scrap and slowing down the bottleneck and, in turn, the whole line (see Section 4.1). Switches, on the other hand, allow to change the routing and distribution of components to keep the bottleneck loaded constantly. For instance, if the processing time of Sink1 starts to increase, the switch could change the distribution ratio between the sinks (see Section 4.2). Finally, reassigning workers from one station to another directly impacts their processing time and can help to handle peak loads but can create a bottleneck elsewhere (see Section 4.3). A more involved scenario where all these actions have to be combined effectively is studied in Section 4.4. Even if a bottleneck is correctly detected, choosing an optimal action sequence to mitigate is challenging since actions take time to show results. For example, reassigning a worker requires task completion and relocation first. While this reduces the bottleneck's mean processing time, high variance makes changes hard to detect. In general, even with the right actions, high data variability delays recognizing improvements. Compounding this issue, production lines often operate near equilibrium states, where minor disruptions can cascade into buffer overflows, bottlenecks, or system-wide jams. Moreover, the combination of different actions may have unintended negative consequences on the performance. For instance, if Sink1 is the current bottleneck, then assigning more workers to Sink1 and sending more components to Sink2 additionally, may result in a worse overall performance. To apply optimal actions, line control systems rely on real-time data from various production line objects, which is often noisy, incomplete, and lagged (see Section 3.2 and Section A.7).
+
+# 2.3. Performance Measurements
+
+There are several ways to measure the performance of production lines. A commonly used metric is the Overall Equipment Effectiveness, short OEE, which quantifies the effectiveness of equipment and machinery used (Nakajima, 1988). Assume we have given a control policy  $\pi$  for a production line decomposed into stations  $P_{1},\ldots ,P_{k}$ . Denote by  $n_{\mathrm{ok}}^{\pi}(t,i)$  and  $n_{\mathrm{noK}}^{\pi}(t,i)$  the cumulated number of OK and not OK (NOK) parts produced at station  $P_{i}$  at time  $t$ . Then, following (Nakajima, 1988), the OEE for  $P_{i}$  is essentially defined as  $\mathrm{OEE}_i(t) = \frac{T_i}{t}\cdot n_{\mathrm{ok}}^{\pi}(t,i)$  where  $T_{i}$  is the minimum of the processing time of  $P_{i}$ . By construction, we have  $n_{\mathrm{ok}}^{\pi}(t,i)\cdot T_{i}\leq t$  for all  $t$  and thus the maximal possible OEE is 1. Analogously, the OEE of a production line can be defined. The OEE definition is mainly focusing on productivity and quality, but a more economic view requires to also take costs of the individual stations and components into account, for instance as some stations require more energy for their processing, like in (Loffredo et al., 2024), or because only some components are scraped, not the full part. Thus, we assume that applying work at station  $P_{i}$  comes with a cost  $c_{i}$ , which may be material costs for joining a component or costs for energy as in (Loffredo et al., 2024). The value of the final part is considered to be the sum of all costs  $c = \sum_{i = 1}^{k}c_{i}$ . At simulation time  $t$ , the production has produced an aggregated value of
+
+$$
+C _ {\pi} (t) = \frac {T _ {C}}{t} \left(c \cdot n _ {\mathrm {o k}} ^ {\pi} (t) - \sum_ {i = 1} ^ {k} c _ {i} \cdot n _ {\mathrm {n o k}} ^ {\pi} (t, i)\right) \tag {1}
+$$
+
+where  $n_{\mathrm{ok}}^{\pi}(t) \in \mathbb{N}$  denotes the number of parts produced at all sinks combined and  $T_{C}$  denotes the minimal possible time the line needs to produce a single part. One can think of Equation 1 as the value the production line generates: If  $C_{\pi}(t) > 0$ , the line generates revenue, if  $C_{\pi}(t) < 0$ , it generates costs. If no scrap can occur, maximizing  $C_{\pi}(t)$  is equivalent to maximize the number of produced parts.
+
+Generally, one wants to find a control policy  $\pi$  that maximizes  $C_{\pi}(t)$  and simultaneously minimizes the time  $t$  to reach the maximum. It then depends on the exact usecase to weight a smaller  $t$  needed with a higher  $C(t)$  reached by. To circumvent this distinction in our work and to create a general benchmarking environment, one either has to fix the available time  $T_{\mathrm{sim}}$  and to maximize the output  $C_{\pi}(T_{\mathrm{sim}})$ , or to fix the desired output  $\tilde{C}$  and to minimize the time needed by a policy  $\pi$  to reach it. In our benchmark studies, we decided to fix  $T_{\mathrm{sim}}$  as this typically originates from high-volume production where the output needs to be maximized in a given work shift of length  $T_{\mathrm{sim}}$ .
+
+# 3. RL for Active Line Control
+
+In this section, we describe how RL can be used to learn policies for active line control. Specifically, we consider active line control as an episodic and partially-observed Markov decision process. Let  $S$  be a set of states and  $\mathcal{A}$  a set of actions. Transition probabilities  $P$  dictate how likely transition from a state  $s \in S$  to a state  $s' \in S$  is when choosing action  $a \in \mathcal{A}$  at  $s$ , namely  $P(s'|s, a)$ . When selecting  $a$  at  $s$ , a reward  $R(s, a)$  is observed. Additionally given a desired length  $T \in \mathbb{N}$ , this gives a Markov decision process  $(S, \mathcal{A}, P, R, T)$  where the goal is to find a policy  $\pi: S \to \mathcal{A}$  to maximize  $\mathbb{E}_{P,\pi}\left[\sum_{t=0}^{T} R(s_t, \pi(s_{t-1})\right]$  where  $(s_t)_{t \in \mathbb{N}}$  is chosen under the interplay of  $P$  and  $\pi$ . If the states  $s \in S$  cannot be observed directly by the policy but only features  $f \in \mathcal{F}$  instead controlled by another conditional probability density function  $O$ , i.e., the likelihood of observing  $f$  at state  $s$  is  $O(f|s)$ , then the tuple  $(S, \mathcal{A}, P, R, \mathcal{F}, O)$  defines a partially-observed Markov decision process where a policy  $\pi: \mathcal{F} \to \mathcal{A}$  needs to be found maximizing  $\mathbb{E}_{P,O,\pi}\left[\sum_{t=0}^{T} R(s_t, \pi(f_{t-1})\right]$ . Note that the definitions given here are streamlined versions of the common way of defining Markov decision processes, where the process typically ends in a set of defined terminal states and additionally, a discount factor  $\gamma \in [0,1]$  is given to trade-off rewards in early and late states.
+
+While the formulation above assumes a discrete-time decision process, we emphasize that the underlying dynamics of the production line still can be continuous time and only the agent interaction—i.e., observation and action execution—is restricted to fixed intervals. This modeling choice reflects real-world production control, where decisions like worker reallocation or routing changes are typically made at regular intervals, such as every few seconds or at shift-level granularity. This approach offers a practical compromise between physical realism and learning efficiency, and is consistent with prior work in industrial reinforcement learning. Section 3.3 provides details how this is realized in LineFlow.
+
+# 3.1. Episodes and Rewards
+
+As described in Section 2.3, we consider constant operation times  $T_{\mathrm{sim}}$  of the manufacturing setting to be optimized. In principle, a control policy can interact with a production line at any time. We, however, assume that the policy can only interact with the production line at fixed and equidistant time points each  $T_{\mathrm{step}}$  apart, thus we get  $T = \frac{T_{\mathrm{sim}}}{T_{\mathrm{step}}}$  many interactions in an episode. This approach introduces the possibility that the model may miss certain state changes occurring between observation intervals, such as a very brief buffer overflow. However, from a RL point of view, this has multiple advantages: First, the trajectories all become of fixed length and second, the policy does not need the time scale.
+
+Allowing arbitrary interactions of a policy, particularly at the early stage of the training, can lead to many thousands of interactions within only a few time units. Although in LineFlow,  $T_{\mathrm{step}}$  and  $T_{\mathrm{sim}}$  are considered to be constants coming from the situation to be optimized, both can be varied sequentially in a curriculum learning fashion (Narvekar et al., 2020). The goal of an episode is to find a policy  $\pi$  that maximizes  $C_{\pi}(T_{\mathrm{sim}})$ . To allow temporal difference learning (see (Sutton & Barto, 2020, Chapter 6)), we decompose  $C_{\pi}$  into discrete quantities for  $t \in \{0, \dots, T_{\mathrm{sim}}\}$  by defining
+
+$$
+R (s _ {t}, \pi (s _ {t - 1})) = C _ {\pi} (T _ {\mathrm {s t e p}} \cdot (t + 1)) - C _ {\pi} (T _ {\mathrm {s t e p}} \cdot t).
+$$
+
+That way, we get  $\sum_{i=0}^{T} R(t) = C_{\pi}(T_{\mathrm{sim}})$  as desired.
+
+# 3.2. States, Observations, and Actions
+
+Due to the stochastic nature of production processes and sensor inaccuracies, the true system state of a production line is not directly observable. Instead, it must be inferred from incomplete, noisy, and delayed observations. Key features relevant for performance estimation include buffer fill levels, station processing times, production rates, station modes, and routing information from switches (Roser et al., 2003; 2014). Whenever available, features in LineFlow are constrained by known upper and lower bounds. For example, fill levels are normalized to [0, 1], while production rates or processing times are positive but unbounded.
+
+The action space in LineFlow primarily consists of discrete control decisions, such as assigning workers or routing components across stations. Additionally, stations can be switched on or off, similar to (Loffredo et al., 2023). Workers are drawn from predefined pools associated with specific station groups. Each worker is modeled as an individual and independent dimension in the action space, where the possible values correspond to the stations the worker can be assigned to. While this introduces symmetries in the action space when workers are indistinguishable, it enables fine-grained modeling of worker attributes—such as skill levels or expertise—when they are not. Furthermore, representing workers as explicit objects within the production line ensures that side constraints, such as a fixed number of available workers, are respected at all times. Switches typically feature two discrete actions: one controlling incoming buffers and another for outgoing buffers. Beyond the core actions, LineFlow also supports modifications to the physical production environment, such as actively adding or removing carriers from the line. While not included in our case studies, these actions provide additional flexibility for modeling dynamic settings. A list of all observable and actionable dimensions is in Section A.7.
+
+![](images/90f0e5367bdba2f81e80a94cdd54341723ef9551d237758591f25faceef74f90.jpg)  
+(a) WT and WTJ
+
+![](images/72841c1ab1fcc4ac114423a7d07089538ec47326f83b87017e00b2e52c5c3247.jpg)  
+(b)  $\mathrm{WA}_{3,10}$  
+Figure 3. Representatives of the three atomic production line challenges analyzed in this case study.
+
+![](images/22c57805ac2ca7aee41b112c9d7d467ba3558b2d77a8a4345312f20a90dc18fb.jpg)  
+(c)  $\mathrm{PD}_5$
+
+# 3.3. Implementation Details
+
+LineFlow is designed to address key challenges in optimizing active line controls while meeting the needs of RL researchers. The discrete event simulation, handling station interactions and stochasticity, is built on SimPy (SimPy, 2025). Its object-oriented structure enables easy customization of stations and rapid setup of complex production scenarios. A visualization module based on pygame provides insights into layout dynamics and agent interactions. Fully implemented in Python, LineFlow integrates seamlessly with data science tools like pandas (Wes McKinney, 2010) and numpy (Harris et al., 2020). RL interaction follows the gymnasium API (Towers et al., 2024), allowing training with stable-baselines3 (Raffin et al., 2021) or skrl (Serrano-Muñoz et al., 2023). Environments support vectorization and parallelization to accelerate training. Further details are provided in Section A.1, with a complete example in Section A, illustrating the layout in Figure 2. Although agent interaction follows the discrete-time process described in Section 3, the underlying production dynamics in LineFlow is simulated in continuous time using a discrete-event engine. The interaction frequency of the agent can be adjusted via a parameter, allowing the discrete-time control to approximate continuous-time behavior with high fidelity.
+
+# 4. Case Studies
+
+In this section, we introduce and study realistic production scenarios where active control is required. The scenarios have been selected for two reasons: First, their theoretical optimum can be computed statistically allowing us to quantify whether RL algorithms learn optimal policies. Second, they appear as subproblems in many production scenarios.
+
+# 4.1. Optimal Waiting Time WT and WTJ
+
+In this scenario, the optimal waiting time between parts produced by a source station  $S_{C}$  has to be found (see also Figure 3a). The source  $S_{C}$  serves together with another source  $S_{M}$  an assembly station  $A$ , which joins the components from both sources to produce the final product and sends it to sink  $S$ . Components from source  $S_{C}$  have a fixed
+
+![](images/4e45372c6df2e6e8d5a365e6468198a83e26892f6860ad9234e615531a053ae6.jpg)  
+Figure 4. The jumps in the processing time of the assembly  $A$  in WTJ for different simulations of length 4000.
+
+expiration time  $T_{\mathrm{AC}}$  called assembly condition: If the time from their setup at  $S_{C}$  to the time the process starts at  $A$  is larger than  $T_{\mathrm{AC}}$ , then  $A$  has to dispose this component and has to wait for the next one. The removal of a scrap part not only generates costs, it also blocks the assembly for a certain amount of time as the defective component has to be removed first. Such a situation is typical in many joining processes, like gluing applications, where adhesives can dry out if not processed in time (Chen et al., 2005). If the waiting time is chosen too small,  $S_{C}$  produces to many components which cannot be handled by  $A$  in time and thus drives scrap costs. On the other hand, if the waiting time is chosen to high,  $A$  has to wait for components which delays the production of final parts. The goal is to balance the waiting time of  $S_{C}$  to maintain a continuous component supply at  $A$ . Assuming costs  $c_{M}$  and  $c_{S}$  for components produced at  $S_{M}$  and  $S_{C}$  respectively and that only components of  $S_{C}$  can potentially be scraped, the performance of the line as described in Section 2.3 within a given time frame  $t$  is
+
+$$
+C _ {\pi} (t) = \frac {T _ {C}}{T _ {\text {s i m}}} \left(\left(c _ {M} + c _ {S}\right) \cdot n _ {\mathrm {o k}} ^ {\pi} (t) - c _ {S} \cdot n _ {\mathrm {n o k}} ^ {\pi} (t, A)\right).
+$$
+
+We generally assume that the processing times without potential waiting times are such that  $A$  is the bottleneck of the line. Thus, the maximum number of parts produced depends on the time  $A$  needs to get one carriers and one component, to assemble them, and to push the final product to the buffer. The optimal waiting time essentially fills the between the times  $A$  and  $S_{C}$  need to handle and process their parts. We give an explicit equation for the optimal waiting time and the maximum number of parts in Section B.1.
+
+To force a dynamic adaption we now introduce the related scenario Waiting time jump (WTJ). Here, we uniformly sample a length  $T_{\mathrm{jump}}$  and a trigger time  $T_{\mathrm{trigger}}$  such that  $[T_{\mathrm{trigger}}, T_{\mathrm{trigger}} + T_{\mathrm{jump}}] \subset [0, T_{\mathrm{sim}}]$ . Then, the processing time of  $A$  at simulation time  $t$  from
+
+$$
+\left\{ \begin{array}{l l} T + \operatorname {E x p} _ {S}, & \quad \text {i f} t \notin [ T _ {\text {t r i g g e r}}, T _ {\text {t r i g g e r}} + T _ {\text {j u m p}} ] \\ f \cdot T + \operatorname {E x p} _ {S}, & \quad \text {i f} t \in [ T _ {\text {t r i g g e r}}, T _ {\text {t r i g g e r}} + T _ {\text {j u m p}} ] \end{array} \right..
+$$
+
+As  $f$  and  $T_{\mathrm{jump}}$  can be different in each episode, the maximal possible reward can vary, too. To ease the comparison of agents over multiple runs, we construct  $f$  for given  $T_{\mathrm{jump}}$
+
+in a way that the maximal possible reward remains a constant (see Section B.1 for details). More precisely, we fix a constant  $0.5 < R < 1.0$  and construct  $f$  such that expected maximal parts produced is  $R \cdot N$ , where  $N$  is the expected number of parts of WT without jump. A visualization of the processing time of  $A$  for multiple simulations is in Figure 4. Although, by design, the expected maximal number of parts produced by the line is  $R \cdot N$ , this value cannot be reached by a control agent. The reason is that at time  $T_{\mathrm{trigger}}$  if the processing time of the assembly jumps to a higher level, the new processing time of  $A$  can first be observed once the first part has been produced. Thus, from  $T_{\mathrm{trigger}}$  to  $T_{\mathrm{trigger}} + f \cdot T + \mathrm{Exp}_S$ , the source  $S_C$  runs with a too low waiting time and possibly sends components to  $A$  which are going to expire. The same happens once the processing time of  $A$  jumps to the lower level. Here, the source may run at a higher waiting time causing  $A$  to wait. To quantify the optimum reachable by agents learning from observations, we evaluate the performance of an agent learning the means of the processing times online (see Section B.1).
+
+# 4.2. Optimal Part Distributions  $\mathrm{PD}_k$
+
+Our next case study is a subproblem of scheduling problems in manufacturing and is already extensively researched in literature (see Section 1.1). Here, components have to be distributed onto  $k$  many parallel processes  $P_{1},\ldots ,P_{k}$  with processing time distributions  $\mathcal{T}_1,\dots ,\mathcal{T}_k$  by a single switch in an optimal way. Moreover, another switch needs to fetch the outputs of the processes and sends them to a sink (see Figure 3c). We assume that  $\mathcal{T}_i = T_i + \mathrm{Exp}_{S_i,T_i}$  where  $T_{i}$  and  $S_{i}$  are constants for all  $i\in [k]$ . To compute the optimum, we assume for simplicity that get and put times from the connecting buffers are already contained in  $\mathcal{T}_i$ . Within a given time frame  $T_{\mathrm{sim}}$ , the expected maximal number of parts  $\mathcal{N}_i$  produced by process  $i$  is obtained when  $P_{i}$  is served with parts constantly without time gaps:  $\mathbb{E}[\mathcal{N}_i] = \frac{T_{\mathrm{sim}}}{\mathbb{E}[\mathcal{T}_i]} = \frac{T_{\mathrm{sim}}}{(1 + S_i)\cdot T_i}$ . When we assume that the processing times of the source and the sink are negligible compared to the processing times of the processes  $P_{i}$ . The maximal number of parts  $\mathcal{N}$  produced by the production line within  $T_{\mathrm{sim}}$  is the sum of the maximal parts produced by all processes, i.e.,  $\mathcal{N} = \sum_{i = 1}^{k}\mathcal{N}_{i}$  as all processes can work in parallel. This allows us to compute the maximum number of parts the layout can produce theoretically, see Section B.2. Clearly, the greedy policy deployed at both switches which pushes components on the buffer with lowest fill and fetches components from the buffer with highest fill loads the processes in an optimal way (see also Section B.2).
+
+# 4.3. Optimal Worker Assignments  $\mathrm{WA}_{k,N}$
+
+In many production scenarios, processes involve manual work which can be completed faster by multiple workers
+
+collaborating. In this case study, the goal is to find the right distribution of a limited number of workers over multiple stations each having a varying processing time. We assume that the required processing time of a station, where  $n$  workers are assigned to, is sampled from the distribution  $\mathcal{T}_{T,S,n} \coloneqq T \cdot p_c(n) + \mathrm{Exp}_{S,T}$  where  $T, S \in \mathbb{R}_{>0}$  are constants and where  $p_c(n) = \exp(-c \cdot n)$  is a performance coefficient (see also Section B.3). In our benchmarks, we set  $c = 0.3$ , which means that an additional worker reduces the processing time by approximately  $74\%$ . Now, consider a layout of  $k$  sequential stations  $A_1, \ldots, A_k$  with constants  $T_1, \ldots, T_k$  and  $S_1, \ldots, S_k$ . Moreover, assume the line has a pool of  $N$  workers to distribute. As the stations are sequentially arranged and connected via buffers (see Figure 3b), the slowest process determines the speed of the overall production line. This means, a partition of  $N$  into  $k$  integer summands, i.e.,  $n_1 + n_2 + \dots + n_k = N$  with  $n_i \in \mathbb{N}_0$  needs to be found such that the maximum of all processing times is minimal. Clearly, the assignment process is bound to certain constraints, like delayed start of processes due to the waiting for assigned workers because of traversal times. We refer to Section 3.2 for more details about how the worker assignment is modeled in LineFlow. Let  $\mathcal{N}_{N,k} \subseteq [N]^k$  be the set of partitions of  $N$  into many  $k$  non-negative integer numbers. More formally, the following max-min integer optimization problem needs to be solved:
+
+$$
+T _ {C} ^ {*} = \min  _ {(n _ {1}, \dots , n _ {k}) \in \mathcal {N} _ {N, k}} \max  _ {i \in [ k ]} \mathbb {E} \left[ \mathcal {T} _ {T _ {i}, S _ {i}, n _ {i}} \right] \tag {2}
+$$
+
+where in our case, the expected value of the processing time is  $\mathbb{E}[\mathcal{T}_{T_i,S_i,n_i}] = T_i\cdot p_c(n_i) + S_i\cdot T_i$ . The maximum number of parts that can be produced is then essentially given by  $\frac{T_{\mathrm{sim}} - T_{\mathrm{off}}}{T_C^*}$  after removing some initial offsets  $T_{\mathrm{off}}$  due to the ramp-up of the production line. Mathematically, solving Equation (2) is computationally hard not alone because it is a min-max problem but also because it involves integer partitions. Learning an optimal distribution in an interactive environment is even more challenging, as first good estimates for  $T_{i}$  and  $S_{i}$  need to be learned as well as for  $p_c$ . In Section B.3, we formulate Equation 2 as mixed-integer optimization problem and compare the empirical results obtained with LineFlow with the theoretical optimum.
+
+# 4.4. Complex Line CL
+
+Finally, we combined all studied problems into a problem we call complex line, short CL (see Figure 5). Here the agent has to distribute workers and components over  $k$  sequential assembly stations such that an assembly condition  $T_{\mathrm{AC}}$  as in WT for the components is kept. Interestingly, simply combining the individual and optimal solutions of the individual subproblems does not lead to an optimal solution for CL. A straightforward distribution approach with a fixed distribution of workers, parts and a fixed waiting time at the source can cause excessive scrap or a slow buffer
+
+![](images/889aeb61d144a12ec48f182003649e198774ea1cf61f6ddb1299bfce7a3c96c5.jpg)  
+Figure 5. Scenario CL with 8 assemblies.
+
+filling process between the source, switch, and assembly stations. In Section B.4, we present a profitable heuristic found by extensive testing meaning the number of produced parts outweighs the cost of scrap.
+
+# 5. Benchmarks
+
+# 5.1. Layout of Experiments
+
+First, we want to emphasize that our benchmark is intended to serve as a proof of principle how evaluation can be done with LineFlow and to show that optimal control can be learned. For that, we only select a handful of algorithms and hyperparameters to consider, although many more would be feasible as well. Concretely, we use the following algorithms in our experiments: PPO (Schulman et al., 2017) and its recurrent version (Pleines et al., 2022), A2C (Mnih et al., 2016), and TRPO (Schulman et al., 2015), all of which can deal with multi-dimensional observations of mixed types and multi-dimensional categorical action spaces. We used the implementations provided by the stable-baselines (Raffin et al., 2021) package and evaluated multiple hyperparameters with three different random seeds for each case study. More details on the hyperparameters used can be found in Section C. The performance of the agents was measured in online evaluations, reporting the mean reward of 5 episodes on a separate evaluation environment 5 times (see also Figure 6 for the evolution of the reward over global steps). When not stated differently, the deterministic version of a trained policy is evaluated, that is, the action with the highest probability is selected. In all experiments, the agents are trained and evaluated on five environments stacked into a vectorized environment.
+
+# 5.2. Results for WT, WTJ, WA, and PD
+
+All benchmarks we set  $T_{\mathrm{sim}} = 4000$ . For WT and WTJ, we set  $\mathcal{T}_A = 20 + \mathrm{Exp}_2$ ,  $\mathcal{T}_{S_C} = 5 + \mathrm{Exp}_{0.5}$ , and use a constant get time  $\mathcal{T}_g = 1$ . The trigger time  $T_{\mathrm{trigger}}$  is uniformly sampled from the interval [500, 1500] whereas  $T_{\mathrm{jump}}$  is uniformly sampled from [1600, 2000]. The assembly condition is set to  $T_{\mathrm{AC}} = 35$ . In  $\mathrm{PD}_k$ , the time distributions are  $\mathcal{T}_i = 10 \cdot (i + 1) + \mathrm{Exp}_{0.1}$  for  $i \in [k]$ . In  $\mathrm{WA}_{k,N}$ , we set  $N = 3 \cdot k$  and use  $c = 0.3$  for the performance coefficient. Moreover, we set  $T_i = (16 + i \cdot 4)$ .
+
+![](images/fb83f9f5de5953f658cfb43d564c2098b9018f73af9273c4c1377dfc79f46040.jpg)  
+Figure 6. Evaluation reward over global steps for  $\mathrm{PD}_3$ .
+
+<table><tr><td></td><td>PPO</td><td>TRPO</td><td>rPPO</td><td>A2C</td><td>Optimal</td></tr><tr><td>WT</td><td>155.0 ± 3.0 (158.4)</td><td>157.1 ± 1.9 (158.6)</td><td>157.7 ± 1.3 (159.0)</td><td>154.4 ± 5.9 (158.6)</td><td>156.2 ± 1.5</td></tr><tr><td>WTJ</td><td>93.7 ± 6.9 (101.4)</td><td>93.9 ± 7.6 (102.4)</td><td>105.9 ± 11.6 (113.0)</td><td>100.5 ± 11.1 (110.2)</td><td>114.8 ± 0.9</td></tr><tr><td>WA3,9</td><td>274.7 ± 13.3 (288.6)</td><td>244.6 ± 18.0 (262.8)</td><td>273.7 ± 21.9 (288.4)</td><td>278.9 ± 17.4 (289.4)</td><td>287.1 ± 2.7</td></tr><tr><td>WA4,12</td><td>243.1 ± 9.0 (252.0)</td><td>194.6 ± 7.7 (202.6)</td><td>231.9 ± 21.3 (246.4)</td><td>244.7 ± 15.4 (255.0)</td><td>252.8 ± 2.1</td></tr><tr><td>WA5,15</td><td>210.5 ± 22.0 (235.0)</td><td>151.2 ± 6.9 (155.6)</td><td>211.9 ± 27.8 (230.0)</td><td>207.7 ± 38.4 (234.2)</td><td>236.3 ± 2.1</td></tr><tr><td>PD3</td><td>578.7 ± 0.6 (579.2)</td><td>539.1 ± 61.3 (577.8)</td><td>359.6 ± 95.1 (469.4)</td><td>304.3 ± 0.5 (304.8)</td><td>582.7 ± 2.6</td></tr><tr><td>PD4</td><td>651.1 ± 9.3 (661.2)</td><td>582.1 ± 101.5 (657.2)</td><td>585.8 ± 31.7 (620.2)</td><td>305.1 ± 0.1 (305.2)</td><td>670.7 ± 2.2</td></tr><tr><td>PD5</td><td>651.3 ± 107.0 (715.6)</td><td>664 ± 77.0 (712.6)</td><td>305.1 ± 0.2 (305.4)</td><td>358.3 ± 92.5 (465.2)</td><td>738.3 ± 3.5</td></tr></table>
+
+Table 1. Overview of rewards obtained by the best hyperparameter combination. Number in brackets denotes maximal reward obtained (see Section B for details). Note that optimal policies are shaped by stochasticity as well and trained RL agents may outperform the expected mean performance in individual episodes.
+
+The performance of the best hyperparameter combination for each algorithm, averaged over multiple seeds, is shown in Table 1. More details of the rewards obtained during online evaluation of all hyperparameter sets is in Section C. Scenario WT is feasible for all algorithms, but only recurrent PPO approaches the optimal value for WTJ. The actor-critic method A2C outperforms policy-based methods in WA by achieving a better performance with fewer steps. However, in part distribution, actor-critic methods fail to learn good controls while policy-based approaches succeed.
+
+# 5.3. Results for CL
+
+Informed decisions for CL require to keep past actions in memory. For instance, sending a component to station  $i$  at a give step requires to send a part to station  $i + 1$  in the one of the subsequent step in order to keep the part flow on the main track. We found that, no matter which hyper
+
+![](images/eb1e58bd896a605afea090e135fcceaf5db10a8fc98df386a31669043095adeb.jpg)  
+Figure 7. An agent trained with the PPO algorithm causing a blocking state of the line: All components send to  $A_0$  causing  $A_1$  to wait. As  $A_1$  is blocked, the buffer between  $A_0$  and  $A_1$  gets full, which in turn blocks  $A_0$  and consequently the full line.
+
+![](images/c644ef0f2880708429cfc87408fe6421d8c8eb147db2f3588da586d57b43ba09.jpg)  
+Figure 8. Upper figure shows a comparison of recurrent PPO trained with (blue) and without (red) curriculum using three different seeds. Lower figure shows a comparison of the stacked PPO trained using curriculum with a stack of 40 (red) and 100 (blue).
+
+parameters have been used, hardly an algorithm reached a strictly positive reward. Since, at the start of the training the number of scrap parts heavily outweighs the number of produced parts the reward starts significantly negative. Interestingly the agents all converged to a reward of zero. An investigation revealed that agents learn to dead-lock the line, leading to no produced parts but also, no scrap parts (see Figure 7). From there, the same train data is generated during all subsequent rollouts leading to poor update steps of the agent. In our experiments, no algorithm tested managed to escape from these dead-lock situation when trained from scratch. Thus, we applied a curriculum which slightly increases the scrap costs of a component scraped at one of the assemblies. The intention behind is that first, the agents learn the general mechanics of the production line and which actions are necessary to bring parts to the final station. Then, they have to get more efficient by reducing the number of scrap parts. We increase the scrap weight by a constant factor until  $\frac{1}{k}$  is reached (in our experiments 0.006 for the scenario having  $k = 3$ ) once the reward is above 100 in five subsequent evaluations. As optimizing CL requires to memorize past actions, we consider only two algorithms here: A stacked PPO (Schulman et al., 2017) receiving the past  $k$  observations (with  $k = 40$  and  $k = 100$ ) and a recurrent PPO (Heess et al., 2015) as in (Pleines et al., 2023). First, we can see that using an curriculum is effective: Agents obtained through a curriculum obtain a large positive reward right from the beginning and they manage to keep it also when the scrap costs are increased. Figure 8 shows a comparison of the recurrent PPO trained with and without the curriculum for different random seeds and learning rates. In addition, we see that the stacked version of PPO does not manage to reach the performance of the recurrent version, even when trained 10 times longer.
+
+When implementing a baseline for CL, it is essential to consider the waiting time of the source as well as the distribution of components and workers. Our heuristics prioritized the buffer with the lowest fill level fills, while buffers feeding into later assembly stages receive higher priority. We then conducted a grid search over various waiting times and worker distributions. The best heuristic we identified achieved a reward of  $254.6 \pm 1.1$ . Overall, even the best-performing RL algorithms in our CL benchmark still fall short of the manually implemented heuristic.
+
+# 6. Discussion
+
+In this work, we explore a rich problem class from manufacturing and make it accessible for RL research by introducing a novel framework LineFlow. We demonstrated that RL algorithms can learn effective control policies for various production line challenges, achieving optimal performance in well-understood scenarios. Moreover, we showed that for more complex, dynamic production settings, traditional RL approaches struggle without additional techniques such as curriculum learning and memory-based policies. We found that learning effective control strategies from interactions alone is difficult as the ramp-up phase of production plays a crucial role in training stability and poor early decisions can lead to deadlocks and stalled learning. Thus, more complex settings require structured learning approaches, including curriculum learning and hierarchical control.
+
+Researchers and engineers can now use LineFlow to implement their specific production settings and systematically generate simulation data. It enables both the testing of custom policies and the training of RL agents. The behavior of trained agents can be used for analysis, but these policies can also be deployed in the real-world. LineFlow enables research in diverse directions. It supports the study of non-stationary problems, such as machine breakdowns or processing time drifts, which are common in real-world manufacturing. Additionally, improving reward structures for curriculum learning and advancing algorithms could significantly reduce training times and enhance performance in complex production settings. Another promising avenue is transfer learning, which could facilitate knowledge transfer across related tasks for more efficient and effective solutions. Beyond RL, LineFlow serves as a data generator, enabling analysis of production line dynamics, bottleneck prediction, and maintenance forecasting through supervised learning. To advance RL research in production control, a unified training and evaluation framework is essential for tracking and improving the state of the art. LineFlow fills this gap and drives further research into optimizing agent performance in complex, dynamic manufacturing environments.
+
+# Acknowledgments
+
+The project LineFlow is funded by the Bavarian state ministry of research. TW is funded by the Hightech Agenda Bavaria. We thank our research partners from Robert Bosch GmbH and DMG Mori AG for helpful discussions, particularly Martin Roth, Dominik Bohnlein, Markus Guggemoos, and Thomas Stark. Many thanks to Matthias Burkhardt, Cindy Buhl, Kilian Führer, Andreas Fritz, Fabian Hueber, Lea Müller, and Edgar Wolf for their helpful suggestions.
+
+# Impact Statement
+
+This work introduces a reinforcement learning framework for active control of production lines, enabling optimized operations with improved efficiency. By addressing the challenges of real-time decision-making in dynamic manufacturing environments, our method supports sustainable industrial practices by using existing resources of manufacturing lines in an optimal way. This research aligns with global efforts toward smarter and greener production systems, benefiting industries, workers, and society.
+
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+
+# A. An Example Use Case in LineFlow
+
+To demonstrate how LineFlow can be used to model and simulate production lines, and let an agent interact with it, we provide a step-by-step implementation of a simple production line.
+
+# A.1. Implementing the Layout
+
+In LineFlow, a production line is represented by a class that extends the Line class. Inside the Line.build method, we define and connect the various stations, setting their processing times and capacities. Below is an example of a small production line, which consists of two source stations, one supplying the main component and the other providing an additional component required for assembly. These components are fed into an assembly station, which combines them into a single unit. The assembled product then passes through a process station. After processing, a switch directs the product to one of two possible output paths, each leading to a sink station that collects the finished items.
+
+from lineflowsimulation import Line, Source, Sink, Assembly, Process, Switch
+
+```txt
+class ShowCase(Line):   
+def build(self):   
+source_main  $=$  Source( 'Source1' processing_time  $= 5$  carrier_capacity  $= 2$  actionable Waiting_time  $\equiv$  True,   
+） source_comp  $=$  Source( 'Source2' processing_time  $= 5$  partspecs  $\coloneqq$  ["assembly_condition":100]}, actionable Waiting_time  $\equiv$  True,   
+） assembly  $=$  Assembly('Assembly',processing_time  $= 40$  NOK_part_error_time  $= 5$    
+process  $=$  Process('Process'，processing_time  $= 15$  switch  $=$  Switch('Switch'，processing_time  $= 1$  ) sink_1  $=$  Sink('Sink1'，processing_time  $= 70$  ) sink_2  $=$  Sink('Sink2'，processing_time  $= 70$  ）   
+assembly.connect_to_component_input(station  $\equiv$  source_comp，capacity  $= 2$  transition_time  $= 5$  assembly.connect_to_input.source_main，capacity  $= 3$  transition_time  $= 5$  process.connect_to_input(assembly，capacity  $= 2$  ，transition_time  $= 2$  ) switch.connect_to_input(process，capacity  $= 2$  ，transition_time  $= 2$  ) sink_1.connect_to_input(switch，capacity  $= 3$  ，transition_time  $= 2$  ) sink_2.connect_to_input(switch，capacity  $= 3$  ，transition_time  $= 2$
+```
+
+# A.2. Interaction with Agents
+
+In LineFlow, agents interact with the production line by receiving observations from the environment and taking actions to optimize the system's performance. The state space includes features such as buffer fill levels, processing times, and the number of workers assigned to a station. Based on these observations, the agent can adjust parameters like switch routing decisions, worker assignments, or waiting times at sources. Agents can of course be trained, but also defined as a function. The following agent dynamically adjusts the output buffer selection at the switch based on the minimum buffer fill level.
+
+```python
+def agent(state,env): fills  $=$  np.array([state[f'Buffer_Switch_to_Sink{i}] ['fill'].value for i in [1,2]) return { "Switch": {"index_buffer_out": fills.argmin()} "Source1": {"waiting_time": 5}, "Source2": {"waiting_time": 2}, }
+```
+
+To observe the simulation in action, we enable visualization by setting realtime=True. This ensures that the system updates in real-time, allowing for a clear understanding of how products move through the production line. The simulation speed can be adjusted using the factor parameter, which controls the scaling of time units. For instance, setting factor=0.1 speeds up the process by a factor of ten, making it easier to analyze system behavior over extended periods
+
+line = ShowCase(realtime=True, factor=0.1)
+
+line.run(simulation_end=150, agent=agent, visualize=True)
+
+# A.3. Analysing the Data
+
+The user can get all the simulation data in form of a pandas dataframe, consisting of the states of all lineobjects using line.get Observations(). LineFlow also provides preinstall analysis for basic performance indicators like line.get_n_parts_produced() and line.get_n_scrap_parts(). A sample visualization is shown in Figure 9.
+
+![](images/c66379bc99171bfff3f9b12fbcadaa73706ca4b7fec9ac58734995ecbe6ea41b.jpg)
+
+![](images/5887c9c99384323af7279a820e650106d438f60cabcd45e135fa2155ad4c9165.jpg)
+
+![](images/a56b833ff8c50a85d1166d554bd5ad8c1856606e99e9cb8eff8d37fc47a1c2df.jpg)
+
+![](images/0ed15a00a0eb76060f2424abd7a1a918eb1b2ed35e2a21a4cc0c975659099a1b.jpg)
+
+![](images/cc0c74f5185b31174a99985ddb1cd304f0ace6de75846431e893e22a2e303c60.jpg)
+
+![](images/25597920b3b69006763dfc0334e3d70741e25e692353c0699ff8784c944420bb.jpg)
+
+![](images/09b263626523c5f20590729e102757da7e2adc653b42d0c4d033873ebe065b62.jpg)
+
+![](images/183fe8cc84a0b255bbbfd47673c9ad8b46d411474a7f4a23ad580722fc10f3e8.jpg)
+
+![](images/42d3fb7ca87ed0163edd61803e38c5e37e6d89012f5a029507f9845ffdd4e05e.jpg)
+
+![](images/0216cd3a4e810d9f6a28e9c5342b8eac924f62a4dd44d1cdc25326d583449161.jpg)
+
+![](images/464df858c599e24947973b4fbb687e6b4798b3d4e05253c5b1b5d8ee1db88520.jpg)
+
+![](images/262884dd1d4f9e8bafd3ca653ca999c455cb6a7024298fc65af05b9865c149c4.jpg)
+
+![](images/190f0b56403780ec7146cb4b316e3c88b9bb03174cffdc49efc65d7fc11657b3.jpg)
+
+![](images/c521757ce71210a209686e1846ced042269eb1f55c9fc7751716030f099cf62e.jpg)  
+Figure 9. Selection of features extracted from the line displayed in Figure 2 visualized over time.
+
+![](images/1f18ae72f86db0dfd79f07197516af0e844f28e76bafbf27f8c99d2bdf33aa32.jpg)
+
+![](images/e7f3eefc865299460eda1414b0cc258cdfb71a4caea427ce3a1f9b05b7190680.jpg)
+
+![](images/5ad4c83d52107977b0fa5aa4b5a19f684932e805edc82dcc4a2aee9db6d27754.jpg)
+
+![](images/5bf93da1a92c43974564bfffa72e7241727da67c851fc0f450a4d7ff931c3adf.jpg)
+
+![](images/609f4c589192e506acc5e9181ceb083d20efb5ac1667c4412f131b50920fa278.jpg)
+
+![](images/0f1ecb11e222029441ecbf771252da794aed30e522ca57f299e218a2e5aa1dfa.jpg)
+
+# A.4. Training RL Agents
+
+To train reinforcement learning (RL) agents, we implemented the LineSimulation (gym.Env) class, which takes a Line object and a simulation duration as initialization parameters. This environment integrates seamlessly with stable baselines3 (Raffin et al., 2021), allowing RL models to interact with the simulation by specifying it via the env parameter. The following example demonstrates how to train an RL agent using PPO:
+
+from stable的基础上3 import PPO
+
+```julia
+line = ShowCase()
+env = LineSimulation(line, simulation_time=4000)
+model = PPO("MlpPolicy", env, verbose=1)
+modellearn(total_timesteps=100_000)
+```
+
+# A.5. General Overview
+
+LineFlow is implemented in an object-oriented fashion. The central object is the Line object, which is decomposed of multiple LineObjects (Section A.6). A LineObject can either be stationary, like an assembly, or movable,
+
+like a carrier. Each LineObject has a ObjectState representing parameters of the system a parameter. Each ObjectState consists of atomic states where each can be actionable or non-actionable and observable or non-observable (see Section A.7). The states of all objects in a line are accumulated in the LineState which provides a centralized interface for Agents interacting with the Line during simulation.
+
+# A.6. Different Types of line objects available in LineFlow.
+
+Typical properties when instantiating stationary objects are  $T$  and  $S$  that define their processing time  $\mathcal{T} = T + \mathrm{Exp}_S$  as well as a rework probability, which models rework by letting a process run multiple times in a row. Any stationary object must be connected to other objects. This is typically done via one or multiple Buffer objects. For instance, a Switch can have arbitrarily many incoming and outgoing buffers where an Assembly must have exactly one incoming and one outgoing for the main carrier and arbitrarily many incoming for components that need to be joined with the carrier from the main track.
+
+<table><tr><td>Name</td><td></td><td>Type</td><td>Description</td></tr><tr><td>Process</td><td>Process</td><td>stationary</td><td>A station that simulates a processing step on the part. The processing step can be set to repeat due to a simulated human error, which doubles the processing time. This leads to a doubled processing time.</td></tr><tr><td>Sink</td><td>Sink</td><td>stationary</td><td>Removes components from a carrier. Carriers arrived here are marked as OK. Empty carrier either removed or returned to a Magazine or Source using a separate out-buffer depending on the layout. This way, the station can be used in both linear and circular lines.</td></tr><tr><td>Source</td><td>Source</td><td>stationary</td><td>Places parts onto carriers. Can set individual properties, called PartSpec, to every part set up, like the assembly condition \(T_{\mathrm{AC}}\). Carriers are either created, taken from a Magazine, or fetched separate incoming buffer.</td></tr><tr><td>Assembly</td><td>Setup.or.7AIW=4SATor.7</td><td>stationary</td><td>A Station for simulating assembly activities on the line. Individual parts and components are delivered with individual carriers, assembled with a simulated processing time and forwarded to the downstream station. Can be connected to a WorkerPool that can assign Worker objects to it modifying its processing time.</td></tr><tr><td>WorkerPool</td><td>Setup.or.7AIW=4SATor.7</td><td>stationary</td><td>Holds a predefined set of Worker objects and is attached to a fixed number of stations. Multiple pools can coexist for a production line allowing to modelling different skills or experience of workers.</td></tr><tr><td>Magazine</td><td>Magaz InC=1</td><td>stationary</td><td>Magazine station is used to manage the carriers. The total number of carriers available to the line can be controlled via this station. The capacity of the carriers, i.e. the possible number of components that can be added at the assembly station, is also determined by this station. If the number of carriers is not of interest, the source can place the parts directly on carriers and no magazine is required.</td></tr><tr><td>Switch</td><td>Switch</td><td>stationary</td><td>The Switch distributes carriers to different stations, enabling parallel structures within the line.</td></tr><tr><td>Buffer</td><td></td><td>stationary</td><td>The Buffer transports carriers from one station to another. Time needed to push and get carriers to and from can be specified as well as its capacity and time a carrier needs to traverse the buffer.</td></tr><tr><td>Carrier</td><td>Magaz in 26Magazin 26</td><td>movable</td><td>Is set up at a Source station or a Magazine and holds a predefined number of Part objects.</td></tr><tr><td>Part</td><td>Magazin_or_3</td><td>movable</td><td>Single unit which is initially created at a Source. Holds a PartSpec each station handling it can access and individually adapt to.</td></tr><tr><td>Worker</td><td>Setup_or_7 A1 W=4 SA1_or_7</td><td>movable</td><td>Belongs to a WorkerPool and can be assigned to a station. Traversal time can be configured.</td></tr></table>
+
+# A.7. Different Types of States
+
+This section provides an overview over the elementary states implemented in LineFlow. Every state is associated to at least one LineObject and the mapping of selected objects to states is given in Table 5. In general, a state can be actionable or non-actionable meaning that agents can set them to other values within their value range via runtime. Per default, every state is observable, meaning that they are part of the observational space agents have access to. However, to facilitate ablation studies or simulate sensor failures, any state can be selectively masked as non-observable. Many states are consequences of events taking place at the line, like how many carriers are on a buffer, and cannot be influenced directly. All states kept updated constantly and represent at any time the current situation. Some states, however, are lagged in the sense that their update happens once an certain event is triggered. For instance, the state holding the processing time of a station is updated once the process is over with the value of the last process.
+
+The states can be categorized into two main categories: discrete and numeric. Discrete States on the one side, handle categorical items such as station modes. Internally, the values of a discrete state are labeled encoded to integer numbers. A subclass are count states that represent integer numbers, like the number of scrap parts at a station. Numeric states, on the other side, hold continuous numerical values, like processing times or buffer fill levels. Their value range may be restricted by upper and lower tolerances. Table 3 provides a list of states implemented in LineFlow. All states are mapped to the respective states of gymnasium (Towers et al., 2024). At any simulation step  $t$ , the states of all LineObjects are fetched and put into an agent, which returns new values for the list of actionable features to be updated accordingly.
+
+Table 2: Different types of LineObjects.  
+
+<table><tr><td>Name</td><td>Type</td><td>Actionable</td><td>Description</td></tr><tr><td>Mode</td><td>discrete</td><td>non-actionable</td><td>The mode a station is currently in. Is either working, failing, or waiting.</td></tr><tr><td>Processing time</td><td>numeric</td><td>non-actionable</td><td>The processing time of the last process.</td></tr><tr><td>Waiting time</td><td>numeric</td><td>actionable</td><td>Value used to wait till two parts produced.</td></tr><tr><td>Station-Assignment</td><td>discrete</td><td>actionable</td><td>Discrete value denoting to which station a worker is assigned to. Must be one of the predefined stations attached to the worker pool the worker belongs to.</td></tr><tr><td>#Workers</td><td>count</td><td>non-actionable</td><td>Number of workers assigned to a station. Aggregated feature computed from the station-assignment of all workers</td></tr><tr><td>#OKs and #NOKs</td><td>count</td><td>non-actionable</td><td>Number of OK and NOK parts the station has produced so far.</td></tr><tr><td>Buffer-Indices</td><td>discrete</td><td>actionable</td><td>The incoming and outgoing index of a stationary object, like a Switch, it should get and push components to.</td></tr><tr><td>Buffer-Fill</td><td>numeric</td><td>non-actionable</td><td>Relative number of carriers on a buffer.</td></tr></table>
+
+Table 3: Examples of states.
+
+<table><tr><td></td><td>Processing time</td><td>Current mode</td><td>Number of workers</td><td>\(n_{\text{ok}}\)</td><td>Waiting time</td><td>In and out buffer</td></tr><tr><td>Assembly</td><td>O, N-A</td><td>O, N-A</td><td>O, A</td><td>O, N-A</td><td>-</td><td>-</td></tr><tr><td>Process</td><td>O, N-A</td><td>O, N-A</td><td>O, A</td><td>O, N-A</td><td>-</td><td>-</td></tr><tr><td>Source</td><td>O, N-A</td><td>O, N-A</td><td>-</td><td>O, N-A</td><td>O, A</td><td>-</td></tr><tr><td>Switch</td><td>O, N-A</td><td>O, N-A</td><td>-</td><td>-</td><td>-</td><td>O, A</td></tr></table>
+
+Table 5. Default mapping of selected stationary objects to states. Letters  $O$  and  $A$  stand for observable and actionable respectively,where  $N$  marks non-observable and non-actionable.
+
+# A.8. Visualization
+
+Once a layout is implemented in LineFlow, an object of the respective Line object allows to set visualize=True during a simulation run, which renders the current state in pygame. Figure 10. As layouts are implemented in custom line classes, users have control over how these lines are instantiated and can use the power of the programming language of python, i.e., parametrizing the number of sinks or assemblies via constructor arguments or using for-loops to set them up.
+
+![](images/cc2a3df6d463ab0c3e123d3e36f22f5d883fbe303a5d17807186c66c1aa5c190.jpg)  
+Figure 10. Some layouts implemented in LineFlow.
+
+# B. Optimality Proofs for Case Studies
+
+In this section, we give proofs for optimal control policies for the scenarios WT and WTJ in Section B.1, PD in Section B.2, and WA in Section B.3. In Section B.4, we state a heuristic for CL yielding near-optimal reward.
+
+# B.1. Waiting Time
+
+Let  $\mathcal{T}_{S_C}$  and  $\mathcal{T}_A$  be the process time distributions of  $S_{C}$  and  $A$  respectively and denote by  $\mathcal{T}_p$  and  $\mathcal{T}_g$  the time distributions to push and get a component to and from a buffer respectively. First, we state a formula for the optimal waiting time at  $S_{C}$ . Essentially, the waiting time has to fill the gap between the times  $A$  and  $S_{C}$  need to handle and process their parts, which are  $\mathcal{T}_A + 2\cdot \mathcal{T}_g + \mathcal{T}_p$  and  $\mathcal{T}_{S_C} + \mathcal{T}_p$ . Thus, the optimal waiting time can be globally determined as the expected value of their difference
+
+$$
+T _ {W} ^ {*} = \mathbb {E} \left[ \mathcal {T} _ {A} + 2 \cdot \mathcal {T} _ {g} - \mathcal {T} _ {S _ {C}} \right]. \tag {3}
+$$
+
+Particularly, it suffices in WT to learn a static waiting time. See also Figure 11 for how the waiting time at  $S_{C}$  affects the reward. To calculate the maximum number of parts that can be produced, we have to consider the time that it takes for the assembly to start processing as well as the time the last part produced in  $T_{\mathrm{sim}}$  needs to be finished by the sink. Let  $\mathcal{T}_{S_C}, \mathcal{T}_{S_M}, \mathcal{T}_A$ , and  $\mathcal{T}_S$  be the process time distributions of  $S_{C}, S_{M}, A$ , and the sink  $S$  respectively. Moreover, let  $T_{S_C \to A}, T_{S_M \to A}$ , and  $T_{A \to S}$  be the times a part needs to traverse on the buffers between  $S_{C}$  and  $A$ ,  $S_{M}$  and  $A$ , and  $A$  and the sink  $S$ . The assembly can start its processing once a part from  $S_{M}$  and  $S_{C}$  has arrived, that is
+
+$$
+\mathcal {T} _ {\rightarrow A} := \max  \left\{\mathcal {T} _ {S _ {C}} + \mathcal {T} _ {p} + T _ {S _ {C} \rightarrow A}, \mathcal {T} _ {S _ {M}} + \mathcal {T} _ {p} + T _ {S _ {M} \rightarrow A} \right\}
+$$
+
+where  $\mathcal{T}_p$  and  $\mathcal{T}_g$  denote the times to push and get a component to and from a buffer respectively. As the first part also needs to transfer to the sink  $S$  and has to be processed it, the expected value of the maximal parts produced from 0 to  $T_{\mathrm{sim}}$  is:
+
+$$
+\mathbb {E} \left[ \frac {T _ {\text {s i m}} - \mathcal {T} _ {\rightarrow A} - T _ {A \rightarrow S} - \mathcal {T} _ {g} - \mathcal {T} _ {S}}{\mathcal {T} _ {A} + 2 \cdot \mathcal {T} _ {g} + \mathcal {T} _ {p}} \right]. \tag {4}
+$$
+
+![](images/4074ad3543caf6a7b05bdd03db5afd21f570185b16fe24d7c03c06d2da9da5ec.jpg)  
+Figure 11. Visualisation of the setting in WT with optimal waiting time  $T_W^* = 18.5$  as in Equation (3) and the maximal number of expected parts as computed in Equation (4).
+
+Next, we explain the construction of  $f$  and  $T_{\mathrm{jump}}$  of  $\mathrm{WT}_J$  with a jumping processing time of the assembly  $A$ . Following Equation (4) and ignoring the ramp-up and jumps of the assembly station, the expected number of parts  $N$  produced in  $T_{\mathrm{sim}}$  is
+
+$$
+N = \frac {T _ {\mathrm {s i m}}}{\mathbb {E} [ \mathcal {T} _ {A} + 2 \mathcal {T} _ {g} + \mathcal {T} _ {p} ]} = \frac {T _ {\mathrm {s i m}}}{T + S + E}
+$$
+
+with  $E = \mathbb{E}[2\mathcal{T}_g + \mathcal{T}_p]$ . When the assembly processing with  $f \cdot T + \mathrm{Exp}_S$  instead of  $T + \mathrm{Exp}_S$  for a period of  $T_{\mathrm{jump}}$ , the number of expected parts is:
+
+$$
+\frac {T _ {\mathrm {s i m}} - T _ {\mathrm {j u m p}}}{T + S + E} + \frac {T _ {\mathrm {j u m p}}}{f T + S + E}
+$$
+
+Clearly, if  $f > 1$ , the expected number of produced parts must be smaller than  $N$ . Thus, we want to construct  $f$  from a sampled  $T_{\mathrm{jump}}$  such that the expected number of produced parts is  $R \cdot N$  for a fixed constant  $0 < R < 1$ . Then, for a sampled  $T_{\mathrm{jump}}$ , we look for  $f$  such that the expected parts satisfies:
+
+$$
+\frac {T _ {\mathrm {s i m}} - T _ {\mathrm {j u m p}}}{T + S + E} + \frac {T _ {\mathrm {j u m p}}}{f T + S + E} = R \cdot N
+$$
+
+It is not hard to see that this is the case for
+
+$$
+f = \frac {1}{T} \left(\frac {T _ {\mathrm {j u m p}} \cdot (T + S + E)}{(R - 1) \cdot T _ {\mathrm {s i m}} + T _ {\mathrm {j u m p}}} - S - E\right).
+$$
+
+To verify that the maximal number of produced parts is in fact  $R \cdot N$ , we run simulations in LineFlow using varying values of  $R$  and a fixed waiting time of 0 at the source to keep the assembly constantly loaded. As this would lead to scrap, which in turn blocks the assembly time for a certain amount of time, we additionally set a very large assembly condition  $T_{AC}$ . Figure 12 shows the empirical frequency of the number of parts produced for WTJ for different  $R$  values as well as  $R \cdot N$  with  $N$  the expected value of WT showing that  $f$  and  $T_{\mathrm{jump}}$  are constructed as desired.
+
+![](images/3a8f2b53d31bb07e79b0ee684ec79ea0c953008240d95d00962b93d4104e323b.jpg)  
+Figure 12. Number of parts produced for WTJ and WT with large assembly condition  $T_{\mathrm{AC}}$  and no waiting at  $S_{C}$  for varying  $R$ .
+
+To get an optimal policy for WTJ, we estimate  $\mathbb{E}[\mathcal{T}_A + 2\cdot \mathcal{T}_g - \mathcal{T}_{S_C}]$  from Equation (3) by regressing on  $\mathbb{E}[\mathcal{T}_A]$  with the processing times reported from  $A$ . Essentially, we take a rolling mean of the last  $l$  processing times observations from  $A$ . Clearly, the larger  $l$ , the better the estimate for  $\mathbb{E}[\mathcal{T}_A]$  in a time period where the mean does not jump. However, the larger  $l$ , the worse the new estimate of the waiting time adjusts to a new level. By varying  $l$  and testing the agent for WTJ (see Figure 13), we found that  $l = 1$  gives the best reward. This reward is used as optimal value in Section 5.
+
+![](images/fdedf225f8621c8e174f38ff89005bfb5c2d12f95cdac4c5dc783620fe9ec947.jpg)  
+Figure 13. Reward for agent for varying look-back  $l$ .
+
+# B.2. Part Distribution
+
+In Section 4.2, we showed that the maximal number of parts that can be produced in  $\mathrm{PD}_k$  is
+
+$$
+\mathbb {E} [ \mathcal {N} ] := \sum_ {i = 1} ^ {k} \frac {T _ {\text {s i m}}}{\left(1 + S _ {i}\right) \cdot T _ {i}} \tag {5}
+$$
+
+Particularly, the fraction of all parts produced by  $P_{i}$  is
+
+$$
+\rho_ {i} = \mathbb {E} \left[ \frac {\mathcal {N} _ {i}}{\mathcal {N}} \right] = \frac {1}{\sum_ {j = 1} ^ {k} \frac {T _ {i}}{T _ {j}}} \tag {6}
+$$
+
+and consequently, the optimal distribution policy for the switch also needs to deliver that number to  $P_{i}$  to reach the maximal number of parts. Put differently, the optimal policy sends a part with probability  $\rho_{i}$  to  $P_{i}$ . We have implemented in LineFlow an agent that greedily controls the switches: A carrier is pushed on a buffer having lowest fill and fetched from a buffer with highest fill. This greedy policy have been evaluated for  $k \in \{3, 4, 5\}$  and its part distribution is compared to the optimal distribution  $(\rho_{1}, \ldots, \rho_{k})$  from Equation (6). The result is shown in Figure 14.
+
+![](images/3f9318affbe7d0cfc3a51cf3c5ffecdab3ec325a7e31b8a822c2e041d46cb4c1.jpg)
+
+![](images/f98a6a0282e1875060509cb4cd8a8f2d1a565fa5169f49c62fff71f5bf8ec9d4.jpg)  
+(a)  $\mathrm{WA}_{3,9}$
+
+![](images/3889ba4445087de78cfe16155ad3aa25c838beca940e9ead899a8bd263a668fa.jpg)
+
+![](images/4848b3c1cd78a5b749a27037d8cd5bf717b5c49e4bc981a88031298bf713a326.jpg)  
+(b)  $\mathrm{WA}_{4,12}$
+
+![](images/1bdf5bf84775a3217f5cf99071add36c704f0ae9b18f6aa8bd44528a5cf0fe32.jpg)
+
+![](images/ed12b207736efc9091b3fa1a672d6264b12b09e2e3f1689c6a155cd5282583ab.jpg)  
+(c)  $\mathrm{WA}_{5,15}$  
+Figure 15. Performance of all possible monotone partitions for  $T_{i} = (16 + i \cdot 4)$ , and  $c = 0.3$  for a simulation length of 2000 in the worker distribution example.
+
+![](images/a0879ec65e5d7225b48234a6c01ea8e80d1ba9578eea1baa8e28db6964c6e6a4.jpg)  
+Figure 14. Part distributions of the greedy policy compared with optimal distribution for  $k \in \{3, 4, 5\}$ .
+
+![](images/f33dccabcb8ba65df756dc8b0c3f5a991197b65377e2153b1a97007e435191f5.jpg)
+
+![](images/7f094e6950aa58cbeae0bb79f00971e524d00809202f32d681b56077d00f2615.jpg)
+
+# B.3. Worker Assignments
+
+In this section, we formulate
+
+$$
+T _ {C} ^ {*} = \min  _ {(n _ {1}, \dots , n _ {k}) \in \mathcal {N} _ {N, k}} \max  _ {i \in [ k ]} \mathbb {E} \left[ \mathcal {T} _ {T _ {i}, S _ {i}, n _ {i}} \right]
+$$
+
+as given in Equation (2) as integer optimization problem. Recall that  $\mathbb{E}[\mathcal{T}_{T_i,S_i,n_i}] = T_i\cdot (p_c(n_i) + S_i)$ . We equivalently reformulate the minimization problem by introducing a real-valued auxiliary variable  $m$  as follows:
+
+$$
+\min  _ {\dots m, m} m
+$$
+
+subject to  $m\geq T_{i}\cdot (p_{c}(n_{i}) + S_{i})\quad \forall i\in [k]$
+
+$$
+\sum_ {i = 1} ^ {k} n _ {i} = N \tag {7}
+$$
+
+$$
+n _ {i} \in \mathbb {N} _ {\geq 0} \quad \forall i \in [ k ]
+$$
+
+$$
+m \in \mathbb {R} _ {\geq 0}
+$$
+
+In our benchmarks, we used  $T_{i} = (16 + i \cdot 4)$  and  $c = 0.3$  for varying  $k$  and  $N = 3 \cdot k$ . We implemented this optimization problem in gekko (Beal et al., 2018). Solving Equation (7) yields (2, 3, 4) for  $k = 3$ , (2, 3, 3, 4) for  $k = 4$ , and (2, 2, 3, 4, 4) for  $k = 5$ . To evaluate the correctness of LineFlow, we enumerated and evaluated all monotonic worker assignment, that is  $n_{i} \leq n_{j}$  for  $i \leq j$ , for  $k \in \{3, 4, 5\}$  for multiple runs with simulation length 2000 (see Figure 15). We found the optimal assignment obtained empirically matches the exact optimum obtained by using Equation 7.
+
+<table><tr><td>Name</td><td>Value</td></tr><tr><td>Simulation time Tsim</td><td>4,000</td></tr><tr><td>Step size Tstep</td><td>1</td></tr><tr><td>Rollout steps</td><td>1,000</td></tr><tr><td>Discount factor γ</td><td>0.99</td></tr><tr><td>Number of environments (vectorized)</td><td>5</td></tr><tr><td>Number of stacked observations</td><td>1</td></tr><tr><td>Advantage normalization (only PPO)</td><td>False</td></tr><tr><td>Batch size in update</td><td>1,000</td></tr><tr><td>Number of epochs in update</td><td>5</td></tr><tr><td>Clip range (only PPO and A2C)</td><td>0.2</td></tr><tr><td>Maximal gradient norm (only PPO)</td><td>0.5</td></tr><tr><td>Policy (stable-baselines3)</td><td>MlpPolicy (MlpLstmPolicy for recurrent PPO)</td></tr></table>
+
+Table 7. General hyperparameters used for all algorithms.
+
+# B.4. Complex Line
+
+We encountered several challenges when designing an effective control policy for the complex line. Due to the small buffer capacities between assembly stages, the line is prone to blockages, leading to scrap parts and a significant drop in reward. Maintaining a steady production flow proved to be crucial. To resolve potential jams as efficiently as possible, we implemented a switch distribution policy that prioritizes later assembly stations. While this approach helped mitigate blockages, it also introduced new challenges, such as imbalanced utilization of early-stage buffers, which could lead to delays in part availability. Regarding worker distribution, our analysis showed that keeping a fixed distribution throughout the episode consistently outperformed any of our redistribution strategies. We attribute this effect to the time required for worker redistribution, which likely disrupts the system's stability. Additionally, frequent reassignments may introduce inefficiencies due to the time needed for workers to relocate and adapt to new tasks. We also explored adaptive worker allocation strategies that dynamically reassign workers based on buffer fill levels. However, these approaches often led to oscillatory behavior, where workers continuously moved between stations without improving throughput. This suggests that the overhead of frequent redistribution outweighs its potential benefits in our setup.
+
+# C. Benchmarking Details
+
+In our experiments, we used stable-baselines3 (Version 2.3.2). In total, 22 models have been trained and evaluated for every scenario, each model using three different random seeds resulting in 66 trained models for each scenario. The runtimes vary: Roughly 14 hours for  $\mathrm{PD}_5$  are required to train a single model, whereas only 7 hours for WTJ and 2 hours for WT. To train a stacked and recurrent version of PPO for scenario CL NVIDIA H100 GPUs were used. A single stacked PPO model trains roughly 20 hours, where the recurrent models train for almost 6 days.
+
+# C.1. Waiting Time
+
+![](images/b0918ddd9929eeaf5789b4af487ba8db3c7023cc0cf40be2f60e0b0655ba9e86.jpg)  
+(a) WT
+
+![](images/fdbcfae2a5a8dc314ad5257208bbb0d3714db83baa66bde02af4e683f986dfb0.jpg)  
+(b) WTJ  
+Figure 16. Best performance of algorithms on evaluation environments for WT and WTJ.
+
+![](images/28df5a5b87a47baf073c0e6f3841206f7f454824947fb77fa14346d4fec96ecb.jpg)  
+(a) WT
+
+![](images/3c32e51f5069be0b0ca1d64850af41c534fdec568e23021fc7ed07971b575822.jpg)  
+(b) WTJ  
+Figure 17. Reward over steps for WT and WTJ.
+
+# C.2. Part Distribution
+
+![](images/e16b2cabe72e97fcda2ac8fba7f462d68f1aaf4a2483558d7e94f740c519268b.jpg)  
+(a)  $k = 3$  
+Figure 18. Best performance of algorithms on evaluation environments for  $\mathrm{PD}_k$ .
+
+![](images/6735670487b07146214f3f8239640acf7adb2c41988208f884d8557c8ba569da.jpg)  
+(b)  $k = 4$
+
+![](images/2bd8ee82f93f44316d73873bbabf0cffd67dfd4dc318f590891940fe9ef0af03.jpg)  
+(c)  $k = 5$
+
+![](images/299a1a91436cacea4cec858d7ab34c723ed6f006161ae3c93f2ecef94c4f1a13.jpg)  
+(a)  $k = 3$
+
+![](images/e7ecd2c5818531b3a36cf07a564f219d3808f7c6c37e0e5ad975cd76b1448d1c.jpg)  
+(b)  $k = 4$  
+Figure 19. Reward over steps for  $\mathrm{PD}_k$ .
+
+![](images/4567d1f307afe95bec9aa5a31a7ef7dd5a6ce705b290033b55e1185375309eae.jpg)  
+(c)  $k = 5$
+
+# C.3. Worker Assignment
+
+![](images/fba29c952d6a43db5c2b74ffb6d5b868f0e0e792c3d7eb6ba9b63c6353755f11.jpg)  
+(a)  $k = 3$
+
+![](images/30dd0503342c645d4a6fccc0e2b71512e7194b2fa99442c32f115554cf1e4423.jpg)  
+(b)  $k = 4$  
+Figure 20. Best performance of algorithms on evaluation environments for  $\mathrm{WA}_{k,3k}$ .
+
+![](images/82323bfbe1a4fbff17c3a320ea12e69c28126fa6f33bbb096359619f815fe92c.jpg)  
+(c)  $k = 5$
+
+![](images/a854eab05f4eeea09c56c73b37a061d292b5f26382753518083864b192c30a19.jpg)  
+(a)  $k = 3$
+
+![](images/6dac3d5d4bc7cba26f298282894e321399de038806daacaff838938b14ba6f9c.jpg)  
+(b)  $k = 4$
+
+![](images/f7342ace6246cb08d71d611ba4d73cf36a65d3c0bf0318f39b917b20988831e7.jpg)  
+(c)  $k = 5$  
+Figure 21. Reward over steps for  $\mathrm{WA}_{k,3k}$ .
+
+# D. Validation on Real Production Data
+
+In this section, we evaluate the sim-to-real gap of LineFlow using the publicly available production line dataset from (Risdal et al., 2016). In this dataset, each row corresponds to a produced part, and the columns record timestamped feature activations at various stations. Based on this information, we reverse-engineered the production layout, which consists of two parallel pre-assembly lines with 12 stations each (see Figure 23), both feeding into a shared subsequent line. For our analysis, we focused on the pre-assembly line responsible for the majority of component production.
+
+We first analyzed the distribution of processing times at each station and found that they closely follow exponential distributions, which supports the modeling assumptions used in LineFlow (see Figure 22). The limited resolution of the provided timestamps required a pooling of the processing times of successive parts. Specifically, because individual job durations could not be precisely resolved, we applied a rolling, non-overlapping window over the production sequence and computed the average processing time across 100 consecutive parts. This smoothing technique allowed us to approximate the underlying processing time distribution while mitigating the effects of timestamp granularity.
+
+![](images/e907235bc7d81bede66d16eb1fc3fe9d982ced4ba52506b879c0adefc54301c9.jpg)
+
+![](images/7b473fbbe7d5aa1e216de3c5aced6a5808097dbc0a9d77d0ad50484d36676ab8.jpg)
+
+![](images/602f542399ba2187c8f878818d8f0032ec244301a22461cba855ad3cf3236fa6.jpg)
+
+![](images/b9e7b97237a95d807ccbed2528704e595c0b800884ed96b27602cd2b6bf48898.jpg)
+
+![](images/5a10e809c59400213a02c5c23860f5a006b312025c8902071949cf4ec6958268.jpg)
+
+![](images/0b24ccc531ded0a2e3bc13c2cd7f99ae513eea63a0be2b32b6e474286a203b6a.jpg)
+
+![](images/7925fb69886078cdb53f144ae2ef9d0f537ff4b57339b1951f2430602acea1fb.jpg)
+
+![](images/b119a2df2ed49802e251ed38757023dae13dfad9b8fa8942ecf2ef32c45b3fb6.jpg)
+
+![](images/1f2ec3d54860082cdb2f9f8a3d508c1073bfe596da6c812a2e58b87e1574a68d.jpg)  
+Figure 22. Histograms of the averaged processing times in a log scale together with a fitted exponential distribution in red.
+
+![](images/414d6b16704e3c0c6eca5bbfdef5158023e7ea425f72f3eb5f3edeea11751d27.jpg)
+
+![](images/c8da44ccbd7fa6db423d62812ac043311c175699cba90de4b0f8084992d35425.jpg)
+
+![](images/74abc59db2fff8237fc967343cb17c756de888e744c24820534e4e1d7aa5f8d6.jpg)
+
+Next, we implemented the reconstructed layout in LineFlow and with the aim to compare the number of parts produced in the simulation with those observed in the real system. Since the dataset does not provide information about traversal
+
+times or the number of slots in the connecting buffer, we made the simplifying assumption that traversal times are negligible compared to station processing times. Additionally, we assumed that the switches before and after the stations distribute parts in a round-robin fashion. The results showed a close match (see Figure 23): The number of parts produced by our simulation matches the number of parts produced by the real system, providing empirical evidence that it accurately models real-world production dynamics.
+
+![](images/98ed3a9efb7b3a7c7166e2b5d4018c0053eb3569b26092d29b4a3d1b2a52c2dc.jpg)  
+Figure 23. The implemented layout in LineFlow (left) and its simulated output compared with the output of the real production line (right).

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