Title: Stochastic Optimization in Semi-Discrete Optimal Transport: Convergence Analysis and Minimax Rate
Venue: NeurIPS 2025
Accepted: yes
Keywords: Stochastic Optimization

Gradient Descent

Optimal Transport

Statistical estimation
Abstract: We investigate the semi-discrete Optimal Transport (OT) problem, where a continuous source measure $\mu$ is transported to a discrete target measure $\nu$, with particular attention to the OT map approximation. In this setting, Stochastic Gradient Descent (SGD) based solvers have demonstrated strong empirical performance in recent machine learning applications, yet their theoretical guarantee to approximate the OT map is an open question. In this work, we answer it positively by providing both computational and statistical convergence guarantees of SGD. Specifically, we show that SGD methods can estimate the OT map with a minimax convergence rate of $\mathcal{O}(1/\sqrt{n})$, where $n$ is the number of samples drawn from $\mu$.  To establish this result, we study the averaged projected SGD algorithm, and identify a suitable projection set that contains a minimizer of the objective, even when the source measure is not compactly supported. Our analysis holds under mild assumptions on the source measure and applies to MTW cost functions,whic include $\|\cdot\|^p$ for $p \in (1, \infty)$. We finally provide numerical evidence for our theoretical results.


Peer reviews:
comments: SUMMARY:
Semi-discrete optimal transport refers to the scenario where we would like to learn the optimal transport map that transports a continuous source measure into a discrete target measure. This problem can be tackled using Stochastic Gradient Descent (SGD), which results in empirical successes but it is unclear how it is able to work so well. The paper considers the projected version of SGD (PSGD), and provide convergence guarantees of PSGD. There are convergence guarantee of PSGD in the general scenario, convergence guarantee of PSGD for MTW cost and compact case and for quadratic cost and non-compact case, among others. It was demonstrated that PSGD can learn the optimal transport map reasonably efficiently. Numerical experiments are provided, verifying the theoretical claims.

STRENGTHS AND WEAKNESSES:
Strengths:

1. The research questions are well-motivated.
2. The theoretical results are rigorous, demonstrating the reasonable convergence of PSGD for semi-discrete OT.

Weaknesses:

1. For the experiments, no error bars were provided: what are the standard deviations across the runs?

QUESTIONS:
Please see the weakness in Strengths and Weaknesses section.

Some minor issues: 

1. In TL;DR on OpenReview, "in the semi-discre setting" -> "in the semi-discrete setting"
2. MTW properties was mentioned on Line 82, while introduced only on Line 93
3. Line 146: "[XXXX]" bibtex error perhaps?
4. Theorem 5.3, the equation should end with "," instead of "."

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-06-11

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the semi-discrete optimal transport problem, whereby one seeks to compute the optimal transport cost and map between a continuous distribution and a discrete one. The authors attack the problem using stochastic gradient descent and study their computational and statistical properties, establishing the convergence of optimal transport costs, Kantorovich potentials, and optimal transport maps with a rate of $1/\\sqrt N$ even when the continuous measure does not have compact support. In a specific setting, the authors also show that the rate for optimal transport maps is minimax optimal. Numerical experiments support the theoretical findings.\n\nSTRENGTHS AND WEAKNESSES:\n**Strenghts:**\n- The paper studies an important problem, namely the design of computational algorithms for the semi-discrete optimal transport, which is relevant for a variety of machine learning applications. \n- The algorithm proposed by the authors is intuitive and simple to implement, and enjoys rigorous theoretical guarantees of convergence (rate of $1/N$ for the squared error of the potentials and $1/\\sqrt N$ for OT maps, under mild assumptions).  \n- The paper includes several technical results, in particular boundedness of Kantorovich potentials in the non-compact case (Lemma 3.1) and Lipschitzness of the “error” in the optimal transport map in the non-compact case (Theorem 5.1) which are of independent interest.\n\n**Weaknesses and comments:**\n-  The paper should significantly improve its positioning w.r.t. the existing literature, especially in the following two ways: \n  - What are the novel contributions of the paper compared to [5, 20, 29]? The paper definitely contains novel contributions (e.g., the non-compact case), but the existing work should be better clarified (not only with a citation but with references to the statements). Also, it seems that [29] considers the entropic optimal transport problem.\n  - The authors should review existing results on the regularity of Kantorovich potential and their boundedness.\n  - There is existing work by Taşkesen and co-authors (see reference below and references therein) on the computational aspects of semi-discrete optimal transport. The authors should position their work w.r.t. this literature. \n- Lemma 3.1 and its proof:\n  - I suspect Lemma 3.1 cannot be true without assumptions on transportation cost; indeed, if the transportation cost fails to be lower semi-continuous, then Kantorovich duality might fail (indeed, reference [31] assumes lower semi-continuity of the transportation cost to derive duality). Thus, I suppose some regularity of the transportation cost is necessary.  \n  - The authors say “suppose that (4) admits a minimum”, but already write max in (3). In fact, Kantorovich potentials are well-defined under minor assumptions on the transportation cost, so I do not see the benefit of being more general here. \n  - The equation after line 954 in the proof of Lemma 3.1 needs to be better explained, saying that it follows from the definition of $\\mathbb{L}$.\n- The results in the non-compact case are predicated under Assumption B, which is however never discussed in the paper (in particular, (B3), which does not have an analog in the compact case). Given that the focus that the authors put the non-compact case, the assumption must be explained and justified.  \n- Theorem 5.3: \n  - I fail to understand why Theorem 5.3 is proven in the one-dimensional case when $\\nu$ is only supported on two points. \n  - I fail to understand why the proof implies the presented result. For instance, the set $\\mathcal{P}_\\mathrm{lip}$ appears in the statement of the theorem, but never in the proof.\n  - In the proof, the authors focus on Gaussian distributions (in E1) and specific distributions (in E2). I do not see why this continues to imply the general result in Theorem 5.3. \n- Minor comments: \n  - The structure of the appendix of the paper is confusing, with some of the results hard to find (e.g., the proof of Theorem 5.3 is not clearly identified, same for the one of proposition 4.1 and Lemma 4.2).\n  - Defining measures as on line 73 is not rigorous. A measure “eats” sets of reals and not reals. \n  - The acronym MTW is used in the abstract and section 2 before being defined. \n  - There is a [XXXX] on line 146.\n  - In Section 6, $\\mathbf{g}$ should be $\\mathbf{g}^\\ast$, in lines 317-318. \n  - The norms and inner product defined in the equation after line 219 are in fact seminorms. I suggest making this point clear in the text and in the proofs (e.g., Proposition C1). \n  - In Proposition B1, the term $\\mathcal{H}$ appears undefined. \n  - In Proposition C1, there is a 2 missing in the exponent in the left-hand side of the equation (but the proof appears correct). \n  - In the appendix, Theorem 4.5 is in fact 4.4 and Theorem 4.4 is 4.5.  \n  - In the proof of Theorem 4.5., $p_k, \\xi_{k+1}, \\delta_k$ are vectors. I suggest to use bold to unify the notation. \n  - $\\bar g_n$ should be bold in the equation after line 934. \n  - In the equation after line 955, I suspect that the last sum should run from $k=1$ to $n+1$ (and not $n$), implying that the last term is     $8\\gamma_1\\sqrt{n+2}$.\n  - There should no max over j in the second line of equation 955. \n  - Many equations in the appendix go out of margin, and one equation (after line 939) does not remain visible. \n- Typos: \n  - Period missing on line 47. \n  - There should be no period at end of line 607 and a period at the end of line 613.\n\n\nTaşkesen, Bahar, Soroosh Shafieezadeh-Abadeh, and Daniel Kuhn. \"Semi-discrete optimal transport: Hardness, regularization and numerical solution.\" Mathematical Programming 199.1 (2023): 1033-1106.\n\nQUESTIONS:\n1. Can the authors please clarify the position of their paper w.r.t. the literature (see comment above)? \n2. Can the authors please clarify my concerns for Lemma 3.1 and Theorem 5.3 (see comment above)?\n3. Can the authors please clarify assumption B (see comment above)? \n4. The authors repeatedly used “linear convergence”. What do they mean by that? The algorithm does not enjoy “linear convergence” as commonly meant in the optimization literature.\n5. In Section 6  and Fig. 3c, the empirical results do not seem to precisely match the theoretical bound, at least not as accurately as in the other experiments. Can the authors explain?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nSummary: This paper considers the semi-discrete optimal transport problem where we are given a discrete distribution \\mu and a continuous distribution \\nu and one wishes to compute the optimal transport from \\mu to \\nu. The semi-discrete optimal transport is more expressive than the discrete OT and more tractable than the continuous OT. Moreover, unlike the continuous OT, estimating the semi-discrete OT via samples does not suffer from the curse of dimensionality. In fact, with n samples, we can get an estimate of the semi-discrete OT cost and map with a convergence rate of 1/\\sqrt{n} (I believe this convergence rate hides the support size of the discrete distribution). \n\nFrom an algorithmic stand-point, even with the sampling result, one has to compute the exact discrete OT to estimate the cost. The authors suggest SGD based algorithm that uses these samples and estimates the cost and plan efficiently. The authors also complement their theoretical results with an implementation and analyze the performance. \nAnother major contribution of this paper is in reducing the assumptions (for instance, compactness assumption) from the convergence rate.\n \nThe authors, however, miss an important reference which provides an alternate approach for low-dimensional settings (the result presented at the end of [1]). In this alternate approach, one can pre-process the sample in a data structure in time that is linear in the number of samples, and obtain the exact OT cost (between discrete distributions) using only sub-linearly many calls to this data structure. So, by combining this with the result of Pooladian et al., one should be able to get a linear-time estimator in low dimensions. \n\nOverall, I think this is a solid paper that can be accepted.\n\n[1] \tPankaj K. Agarwal, Sharath Raghvendra, Pouyan Shirzadian, Keegan Yao:\nA Combinatorial Algorithm for the Semi-Discrete Optimal Transport Problem. NeurIPS 2024\n\nSTRENGTHS AND WEAKNESSES:\nPositives: This paper establishes near optimal convergence rate for SGD based approaches to estimate both the semi-discrete OT cost and map. The paper generalizes the convergence rate to various cost functions as well as for distributions that are not necessarily compact.\n\nOne the flip side, the paper does not discuss the dependence of the convergence rate on the size of the discrete support. Could the authors provide the dependence of the convergence rate as a function of the size of the support of discrete distribution?\n\nThere are also a few typos across the paper. For instance, L13 (whic -> which) and L146 ([XXXX] must be removed)\n\nQUESTIONS:\nPlease answer the questions I raised in weakness.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the convergence of an online SGD-type method for the semi-discrete optimal transport problem, where the source measure $\\mu$ is discrete and the target measure $\\nu$ is continuous. The analysis is applicable to a broad range of settings, including: 1) cases where costs satisfy the Ma-Trudinger-Wang properties (MTW), and $\\operatorname{supp}\\mu$ is compact; 2) cases where costs are quadratic (in which case compactness of $\\operatorname{supp}\\mu$ is not required). The authors consider a semi-dual formulation and introduce a compact set $\\mathcal{C}$, which they prove contains a solution. This set is then used as a projection set in the iterations of an SGD-type method. This leads to an $\\mathcal{O}(1/\\sqrt{n})$ rate for OT map estimation and $\\mathcal{O}(1/n)$ rates in terms of the objective and potential. The authors prove that these rates are minimax optimal. Numerical experiments illustrate the performance of the algorithm.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths are as follows:\n- The theoretical analysis of the method is applicable to a broad range of settings,\n- Lower bounds are established,\n- Numerical experiments illustrate the performance of the method.\n\nAs for the weaknesses, the clarity and quality of the presentation should be improved, and typos should be corrected. In my opinion, more discussion about the compact set $K$ used in the definition of $\\mathcal{C}$ is needed (see questions).\n\nQUESTIONS:\n1. In which cases is the compact set $K$ (used in defining $\\mathcal{C}$) difficult or impossible to identify?\n2. The objective $H$ is referenced in lines 95-96 prior to its formal definition, exemplifying clarity issues in the presentation.\n3. Please correct typos (e.g., replace \"whic\" with \"which\" in line 13).\n\nThe paper's final score will depend on satisfactory resolution of these issues.\n\nLIMITATIONS:\nThe limitations related to finding the set $K$ may be discussed in more detail. Apart from that, limitations are mostly addressed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}