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arxiv: 2511.14482 · v2 · pith:VV6OED24new · submitted 2025-11-18 · 💻 cs.DB · cs.LG

Gradient-Based Join Ordering

Tim Schwabe , Maribel Acosta This is my paper

Pith reviewed 2026-05-21 18:55 UTC · model grok-4.3

classification 💻 cs.DB cs.LG
keywords join orderingquery optimizationcontinuous optimizationgraph neural networksdifferentiable constraints
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The pith

Join ordering can be reframed as gradient descent over a continuous relaxation of plans when the cost model is differentiable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that traditional discrete search for join orders can be replaced by continuous optimization. Query plans are relaxed into a soft adjacency matrix that encodes a superposition of possible trees. Differentiable constraints keep the search within the space of valid plans while a graph neural network supplies the cost signal. Gradients then drive the search toward low-cost configurations. On graph datasets this yields plans whose costs match or beat those of conventional methods and whose computation time grows more slowly.

Core claim

When the cost model is differentiable, query plans can be continuously relaxed into a soft adjacency matrix that represents a superposition of plans. This continuous relaxation, combined with differentiable constraints that enforce plan validity, enables a gradient-based search for low-cost plans within this relaxed space.

What carries the argument

Soft adjacency matrix that encodes a continuous superposition of binary join trees, optimized by gradient descent under differentiable validity constraints.

If this is right

  • Gradient search guided by a GNN cost model produces plans whose execution costs are comparable to or lower than those found by dynamic programming or other discrete methods on the tested graph datasets.
  • Runtime of the gradient procedure grows more slowly than discrete enumeration as query size increases.
  • The same continuous relaxation can be reused with any differentiable cost model, opening the door to joint optimization of ordering and other query parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be combined with learned cost models that are already being trained end-to-end for other database tasks.
  • It may extend naturally to join ordering under uncertainty, where the cost surface is stochastic rather than deterministic.
  • Similar continuous relaxations might apply to other combinatorial choices in query optimization such as predicate push-down or index selection.

Load-bearing premise

Local minima of the relaxed objective under the differentiable cost model correspond to valid, low-cost query plans.

What would settle it

On a benchmark query whose optimal plan cost is known exactly, run the gradient procedure from multiple random starts and check whether any recovered plan has cost equal to or lower than the known optimum.

Figures

Figures reproduced from arXiv: 2511.14482 by Maribel Acosta, Tim Schwabe.

Figure 1
Figure 1. Figure 1: Join Ordering using a learned Cost Model. Continuously relaxing the discrete Search Space of Plans allows [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A join plan p is represented as a (2n − 1) × (2n − 1) dimensional matrix f(p) = A. The first n rows denote the outgoing edges of triple patterns, while the last n − 1 rows are outgoing edges of join nodes. The root node always corresponds to the last row. Triple patterns and conjunctive queries. Such graphs are queried with triple patterns, i.e., triples in which any position may be a variable from an infi… view at source ↗
Figure 3
Figure 3. Figure 3: Examples of the used query shapes. Star queries have a single center node with multiple outgoing edges, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: True vs. predicted costs for different query shapes (2–14 triple patterns) on the LUBM dataset. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 1-D projections of the cost landscape between two random plans. Even though the space between plans does not represent valid plans, the local gradient still contains directional information towards the better plan. 1 2 3 4 5 6 7 8 9 10 k 400 600 800 1000 1200 Median Cost [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Increasing the number of gradient search fronts [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Predicted Costs of Plans found by Gradient- and Greedy-Search for LUBM and Wikidata. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Search runtime. DP exhibits exponential and Greedy Search quadratic increase of runtime, while Gradient [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Results of the hyperparameter search for different datasets ordered with respect to their correlation with the [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

Join ordering is the NP-hard problem of selecting the most efficient order in which to evaluate joins (conjunctive, binary operators) in a database query. Because query execution performance critically depends on this choice, join ordering lies at the core of query optimization. Traditional approaches cast this problem as a discrete combinatorial search over binary trees guided by a cost model, but they have trade-offs between effectiveness and efficiency. We show that when the cost model is differentiable, query plans can be continuously relaxed into a soft adjacency matrix that represents a superposition of plans. This continuous relaxation, combined with differentiable constraints that enforce plan validity, enables a gradient-based search for low-cost plans within this relaxed space. Using a Graph Neural Network as the cost model, we demonstrate that this gradient-based approach can find comparable and even lower-cost plans compared to traditional discrete search methods on two different graph datasets. Furthermore, we empirically show that the runtime of this approach scales better than discrete search algorithms. We believe this first step towards gradient-based join ordering can lead to more effective and efficient query optimizers in the future.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes relaxing the NP-hard join ordering problem into a continuous optimization task by representing query plans as a soft adjacency matrix over tables. This relaxation is optimized via gradient descent using a differentiable Graph Neural Network (GNN) cost model together with differentiable validity constraints that enforce plan validity. On two graph datasets the method is reported to produce plans whose costs are comparable to or lower than those found by traditional discrete search algorithms, while also exhibiting better runtime scaling.

Significance. If the central empirical claims hold after addressing validity enforcement and experimental rigor, the work would open a new direction for query optimization that integrates continuous relaxation, gradient-based search, and learned differentiable cost models. The approach could improve scalability for complex queries where discrete enumeration becomes intractable, and the use of GNNs as cost models is a concrete strength that enables end-to-end differentiability.

major comments (3)
  1. [§3] §3 (Method, validity constraints): The paper describes differentiable validity terms but does not specify whether these are implemented as hard differentiable projections onto the space of binary trees or as soft Lagrangian penalties. Without an explicit operator or post-hoc validity rate after rounding/argmax, it is unclear whether converged points decode to executable plans with exactly n-1 binary joins, no cycles, and a single root; this directly affects whether the reported lower-cost plans are genuine or artifacts of incomplete enforcement.
  2. [Experimental evaluation] Experimental evaluation (results on two graph datasets): The abstract and results section report comparable or lower costs and better scaling but supply no error bars, exact baseline implementations, dataset cardinalities, number of queries, or ablation studies on the validity mechanism. This leaves the headline claim that the gradient-based method outperforms or matches discrete search on a preliminary footing.
  3. [§4] §4 (Empirical results): The claim that local minima in the relaxed space correspond to valid low-cost plans relies on the unstated assumption that the continuous relaxation plus validity terms produce a search landscape whose optima are decodable to tree-structured plans; no supporting analysis or counter-example checks are provided.
minor comments (2)
  1. [Abstract] The abstract would benefit from naming the two graph datasets and briefly indicating their sizes or characteristics.
  2. [§3] Notation for the soft adjacency matrix and the decoding step (rounding or argmax) should be introduced with a small illustrative example for a 3- or 4-table query.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments on our manuscript. We address each of the major comments below and will make revisions to improve clarity and rigor as suggested.

read point-by-point responses

  1. Referee: [§3] §3 (Method, validity constraints): The paper describes differentiable validity terms but does not specify whether these are implemented as hard differentiable projections onto the space of binary trees or as soft Lagrangian penalties. Without an explicit operator or post-hoc validity rate after rounding/argmax, it is unclear whether converged points decode to executable plans with exactly n-1 binary joins, no cycles, and a single root; this directly affects whether the reported lower-cost plans are genuine or artifacts of incomplete enforcement.

    Authors: We implemented the validity constraints as soft Lagrangian penalties to preserve end-to-end differentiability throughout the optimization process. In the revised manuscript, we will explicitly describe the penalty formulation, the operator used for decoding, and include a post-hoc validity rate measured after applying argmax to the soft adjacency matrix. This will confirm that the optimized plans decode to valid tree-structured join orders with the required properties. revision: yes

  2. Referee: [Experimental evaluation] Experimental evaluation (results on two graph datasets): The abstract and results section report comparable or lower costs and better scaling but supply no error bars, exact baseline implementations, dataset cardinalities, number of queries, or ablation studies on the validity mechanism. This leaves the headline claim that the gradient-based method outperforms or matches discrete search on a preliminary footing.

    Authors: We agree that additional experimental details are necessary for reproducibility and to strengthen the claims. In the revision, we will add error bars to the reported costs, provide exact descriptions of the baseline implementations, specify dataset cardinalities and the number of queries evaluated, and include ablation studies isolating the effect of the validity mechanism. These additions will place the empirical results on a firmer footing. revision: yes

  3. Referee: [§4] §4 (Empirical results): The claim that local minima in the relaxed space correspond to valid low-cost plans relies on the unstated assumption that the continuous relaxation plus validity terms produce a search landscape whose optima are decodable to tree-structured plans; no supporting analysis or counter-example checks are provided.

    Authors: To address this, we will include in the revised manuscript an analysis of the optimization landscape, such as the frequency with which local minima decode to valid plans, along with checks for counter-examples where invalid structures arise. This will support the correspondence between continuous optima and valid low-cost plans. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external GNN cost model and empirical comparison to discrete baselines

full rationale

The paper relaxes join ordering to a soft adjacency matrix optimized via gradients from a GNN cost model plus differentiable validity terms. No equation or step in the provided abstract or description reduces a reported performance number to a quantity defined by parameters fitted inside the same paper. The GNN is treated as an external learned component, results are benchmarked against traditional discrete search methods on separate graph datasets, and no self-citation chain or self-definitional construction is present. The central claim therefore remains independent of its own fitted inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the differentiability of the cost model and on the assumption that the soft-adjacency relaxation plus differentiable constraints yields a usable optimization landscape; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The cost model is differentiable.
    Explicitly required in the abstract for gradient-based search to be applicable.
  • domain assumption Differentiable constraints can enforce plan validity inside the continuous relaxation.
    Stated as the mechanism that keeps the relaxed space meaningful.

pith-pipeline@v0.9.0 · 5704 in / 1393 out tokens · 49538 ms · 2026-05-21T18:55:12.450173+00:00 · methodology

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