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REVIEW 3 major objections 4 minor 162 references

Effective and secure federated online learning to rank

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This thesis claims that FPDGD, a federated adaptation of PDGD with differential privacy, is a reliable, stable, and secure federated online learning to rank method that significantly outperforms the only prior FOLTR baseline.

desk verdict Empirical thesis on federated learning to rank with solid experiments, but the claimed ε-DP guarantee does not follow from the L2 clipping + Laplace noise mechanism. read the letter →

arxiv 2412.19069 v1 pith:UGETOP4C submitted 2024-12-26 cs.LG cs.CRcs.IR

Shuyi Wang This is my paper · ORCID
classification cs.LGcs.CRcs.IR
keywords federatedlearningonlinetorankdifferentialprivacynon-IIDdatapoisoningattacksmachineunlearningrankingeffectivenessprivacy-preservingsearch
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The thesis tries to establish that online learning to rank can be moved into a federated, privacy-preserving setting without giving up ranking quality. Its central claim is that FPDGD, which trains rankers from implicit click feedback across distributed clients while sharing only weight updates, is effective, stable, and secure. Empirical evaluations are said to show FPDGD significantly outperforms the only prior federated OLTR method, FOLtR-ES, across datasets and click models. The thesis also claims to identify which non-IID data distributions actually hurt FOLTR, to show how poisoning attacks degrade it and which defenses help, and to supply an efficient unlearning method with a poison-based way to verify forgetting. If true, this would make privacy-preserving search ranking practical under real user behaviours, heterogeneous clients, and legal requirements to erase user contributions.

What carries the argument

The method that carries the argument is FPDGD: Federated Pairwise Differentiable Gradient Descent. It combines PDGD, which estimates a pairwise gradient from click preferences using a Plackett-Luce ranking model and inverse-propensity-style reweighting to reduce position bias, with the Federated Averaging algorithm, where each client runs local PDGD updates on a batch of queries and the server averages the returned weight vectors. Privacy is enforced by clipping each local update's norm to a bound $\Delta$/2 and having each client contribute Gamma noise whose sum behaves as a Laplace random variable, implementing an epsilon-differential privacy mechanism.

What would settle it

Measure the true global sensitivity of the FedAvg-aggregated PDGD update by computing the maximum L1 distance between aggregated updates obtained from two click datasets that differ in one client's interaction, and compare that value with the $\Delta$ used in the privacy analysis; if the measured sensitivity exceeds $\Delta$, the stated epsilon-differential privacy guarantee is invalid. A complementary test is to run a membership inference attack against the released local updates and check whether the empirical attack advantage matches what the claimed epsilon would permit.

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Extended reading notes

Core claim

The central discovery is a new method, FPDGD, that casts the state-of-the-art centralised OLTR algorithm PDGD into the Federated Averaging framework and overlays an epsilon-differential privacy mechanism on the communicated updates. Each client performs local PDGD updates from clicks, clips the resulting weights to a norm bound, and adds Gamma noise that sums to a Laplace-distributed perturbation, so that the server never sees raw queries, documents, or clicks. The thesis reports that FPDGD consistently and significantly outperforms FOLtR-ES, the only prior FOLTR method, on large-scale datasets and under noisy click models, and that it remains stable when privacy budgets change. It further claims that only certain types of non-IID data, chiefly document preference skew and extreme label-distribution skew, seriously degrade FPDGD, and that standard non-IID remedies from general federated learning do not transfer, while data sharing helps the label-skew case. It also reports that data and model poisoning attacks can reduce ranking effectiveness and that robust aggregation rules such as Krum, Multi-Krum, Trimmed Mean, and Median provide defense, and finally that unlearning a client via historical local updates yields a ranker comparable to retraining from scratch.

Load-bearing premise

The differential privacy guarantee holds only if clipping each local weight update to norm $\Delta$/2 actually bounds the global sensitivity of the aggregated update by $\Delta$; the paper selects $\Delta$ by grid search rather than computing the true sensitivity.

Editorial extensions

If this is right

  • FPDGD gives the first gradient-based federated online learning to rank method with reported consistent gains over the evolutionary-strategy baseline, making it a practical candidate for real federated search.
  • Adding differential privacy to FPDGD has little effect on ranking quality when enough clients participate, but it can badly hurt convergence when only a handful of clients are available.
  • Only some forms of non-IID data are dangerous for FPDGD: document-preference skew and single-label per-client skew degrade performance, while click-model variation and data-quantity skew do not.
  • Standard federated non-IID remedies such as FedProx and FedPer do not close the gap for Type 1 non-IID data; sharing a small global dataset helps for extreme label-distribution skew.
  • Untargeted data and model poisoning attacks can degrade FOLTR effectiveness, and robust aggregation rules mitigate them, with Krum behaving differently from Trimmed Mean and Median depending on attack strength and attack type.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The privacy guarantee is only as strong as the fitted clipping bound: because Delta is chosen by grid search rather than derived from PDGD's actual sensitivity, the reported epsilon values may understate the true privacy loss if the update's global sensitivity exceeds Delta.
  • The non-IID findings are tied to FPDGD's pairwise loss and FedAvg aggregation; other federated ranking methods, especially ones not using pairwise preferences, could respond differently to the same data distributions.
  • The poison-based unlearning verification is a transferable idea: a malicious-client probe can serve as a practical membership test for whether a federated model has erased a client's influence, beyond the ranking domain.
  • As federated rankers move toward pretrained-language-model architectures, the same attack, non-IID, and unlearning questions will need to be revisited because gradient statistics and update norms differ substantially from linear rankers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The thesis studies Federated Online Learning to Rank (FOLTR), motivated by privacy concerns in centralized OLTR. It makes four contributions: (1) an analysis of the existing FOLtR-ES method, showing instability on large datasets; (2) a new method FPDGD, which adapts PDGD to the FedAvg framework and adds a differential privacy mechanism via norm clipping and Gamma-split Laplace noise; (3) an empirical taxonomy and study of non-IID data in FOLTR; (4) a study of poisoning attacks and defenses, and an unlearning method with a poisoning-based verification. The effectiveness comparison of FPDGD against FOLtR-ES is supported by extensive experiments on MQ2007 and MSLR-WEB10K with significance tests, and the thesis honestly reports settings where FOLtR-ES wins (Table 3.1, MQ2007 perfect and navigational clicks). However, the central 'secured' claim rests on a differentially private mechanism whose stated privacy guarantee does not follow from the presented analysis, because the sensitivity is computed in the L2 norm while Laplace noise requires L1 sensitivity.

Significance. If the privacy guarantee were valid, FPDGD would be the first gradient-based FOLTR method with consistent gains over the evolutionary-strategy baseline, and the thesis would make a solid contribution to privacy-preserving ranking. The non-IID taxonomy (Type 1--4), the poisoning attack/defense analysis, and the unlearning verification approach are useful empirical benchmarks for a nascent subfield. The thesis is also commendable for releasing code and experimental scripts for most chapters, and for including statistical significance testing and honest reporting of negative results (e.g., MQ2007 cases where FOLtR-ES is better). However, the privacy analysis in Chapter 3 is load-bearing for the thesis's claim that FPDGD is 'secured' and 'privacy-preserving', and that analysis is incorrect as presented; this materially weakens the contribution.

major comments (3)
  1. [Section 3.2.4 and Table 3.1] The differential privacy mechanism clips each local weight vector to L2 norm Delta/2 and adds Gamma-split Laplace noise with scale Delta/epsilon. The Laplace mechanism (Definition 3.2.2) requires the global sensitivity in the L1 norm, not the L2 norm. Two d-dimensional vectors each with L2 norm at most Delta/2 can differ in L1 norm by up to Delta*sqrt(d). For the linear ranker on MSLR-WEB10K, d = 136, so a claimed epsilon = 1.2 corresponds to an effective privacy budget of at least 1.2*sqrt(136) ≈ 14.0; for the neural ranker the gap is larger. The statement in Section 3.2.1 that clipping to Delta/2 'can meet the global sensitivity Delta' is therefore unsupported. The claimed epsilon-DP guarantee does not follow from the presented analysis, undermining the central 'secured' and 'privacy-preserving' claims of the thesis. The mechanism should either clip in L1 norm, calibrate the noise to the true L1 sensitivity Delta*sqrt(d), or use a different privacy accounting such as the moments accountant with L2 sensitivity and Gaussian noise.
  2. [Section 3.2.4 and Table 3.1] The summary states that 'Empirical evaluation shows FPDGD significantly outperforms the only other federated OLTR method (i.e., FOLtR-ES)'. Table 3.1 shows the opposite for MQ2007 under the perfect and navigational click models, where FOLtR-ES is significantly better than FPDGD (marked with ▼, p < 0.01) for all privacy levels epsilon. The claim should be qualified to the large-scale dataset (MSLR-WEB10K) or to the informational click model on MQ2007. As written, the claim is contradicted by the thesis's own results and overstates the reliability of FPDGD.
  3. [Section 6.1.4 and Section 6.3.1] The unlearning verification injects a poisoning attack designed by the authors (a reversed CCM click model and amplified updates) and measures whether its impact diminishes after unlearning. This is a plausible approach, but it only verifies removal of a contrived signal, not the removal of a client's actual data contributions. The thesis should explicitly state that the verification is conditional on the attack model and that it does not provide a formal or general guarantee of unlearning. This limitation is important because Chapter 6 claims, based on this verification, that the proposed method effectively forgets client contributions.
minor comments (4)
  1. [Definition 3.2.1] Typo: 'datesets' should be 'datasets'.
  2. [Section 3.2.1] Typo: 'outputed' should be 'output'.
  3. [Section 3.2.2] The mapping from FOLtR-ES privacy parameter p in {0.25, 0.5, 0.9, 1.0} to epsilon values in {1.2, 2.3, 4.5, 10} is asserted but not derived. Please show the calculation using Eq. (3.6) and state the number of discrete metric values n used in that formula.
  4. [Section 4.5-4.6] The conclusions that Type 3 (click preferences) and Type 4 (data quantity) non-IID data do not impact FPDGD effectiveness are based on experiments with a linear ranker and the MSLR-WEB10K dataset only. Please qualify the generality of these findings, as the thesis itself notes for Type 4 that the result 'may be specific to FPDGD'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FPDGD's effectiveness is externally benchmarked; the DP guarantee has a correctness gap but is not circular.

full rationale

The thesis's central effectiveness claim (FPDGD outperforms FOLtR-ES) is an empirical comparison against an external baseline and against the non-federated PDGD method, using standard LTR datasets (MQ2007, MSLR-WEB10K), standard OLTR click simulation, and nDCG metrics. FPDGD is an adaptation of the externally published PDGD algorithm to the FedAvg framework; its gains are measured, not derived from its own definitions. The non-IID taxonomy (Chapter 4) and the poisoning attack/defense study (Chapter 5) define new evaluation conditions but do not use the target conclusion as an input. The unlearning verification (Chapter 6) uses a self-designed poisoning signal as an experimental proxy, with retraining-from-scratch as the baseline, so it is not a circular derivation. Self-citations are present but not load-bearing: the FOLtR-ES reproducibility study is conducted with the original author's code and external datasets, and the non-IID and attack experiments are new. The one load-bearing theoretical claim, the ε-DP guarantee in Section 3.2.1, is under-supported: Eq. 3.19 clips the L2 norm of updates while the Laplace mechanism requires an L1 global sensitivity, so the stated privacy budget is not established from the presented analysis. This is a correctness / omitted-proof gap rather than a circularity: the privacy parameter Δ is tuned by grid search, but the reported ranking effectiveness is an empirical outcome, not a quantity forced by that choice. No step reduces by construction to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The central effectiveness claims rest on simulated click experiments and on the unproven convergence of PDGD under FedAvg. The privacy claim rests on an assumed sensitivity bound that is selected by grid search, and the unlearning verification uses an attack designed by the authors. These are the main items the reader pays for beyond standard OLTR background.

free parameters (2)
  • DP sensitivity Delta = Delta in {3,3,5,5} for epsilon in {1.2,2.3,4.5,10}
    Chosen by grid search over {1,3,5,7,9} in Section 3.2.2. The claimed differential privacy guarantee depends on this value being a valid sensitivity bound, but no derivation of PDGD sensitivity is provided.
  • Unlearning hyperparameters n'i and Delta t = n'i in {1,2,3,4}, Delta t in {5,10,20}
    Tuned empirically in Section 6.3.3; they control how many historical updates are replayed and over what window, and the reported unlearning effectiveness depends on them.
assumptions (4)
  • domain assumption CCM click models (perfect, navigational, informational, poison) faithfully simulate real user and attacker click behavior.
    All effectiveness, attack, and unlearning results are measured on clicks simulated with these click models in Sections 2.2.1 and 5.2.2. If real users or attackers behave differently, the reported gains may not transfer.
  • domain assumption Weighted averaging of local PDGD-updated parameters in FedAvg produces a valid global ranker for online learning to rank.
    FPDGD relies on Algorithm 2 without a convergence or unbiasedness proof in the federated setting.
  • ad hoc to paper Clipping local weights to norm Delta/2 yields a global sensitivity of Delta for the PDGD update, so Laplace/Gamma noise provides epsilon-DP.
    Equation 3.19 and the text in Section 3.2.1 assert this; Delta is grid-searched rather than derived. This is the load-bearing privacy assumption.
  • standard math Laplace noise can be split into per-client Gamma noise as in Equations 3.21 and 3.22.
    Taken from prior differential privacy work and used without proof. Standard, but still an unproved background result.
invented entities (2)
  • Poison click model instantiation (reverse CCM)
    purpose: Simulate a malicious client that clicks irrelevant documents to poison FOLTR training in Section 5.2.2 and Table 5.1.
    This is a simulation artifact introduced by the authors; no external data shows that real attackers behave exactly this way.
  • Type 1-4 non-IID taxonomy for FOLTR
    purpose: Organize and benchmark non-IID data scenarios specific to FOLTR in Section 4.2.
    A conceptual taxonomy introduced in the thesis; it is an internal benchmark with no independent validation.

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Cite this review

Pith. "Pith review of Effective and secure federated online learning to rank." pith.science (2026). https://pith.science/paper/UGETOP4C

@misc{pith2026241219069,
  author       = {Pith},
  title        = {Pith review of: Effective and secure federated online learning to rank},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UGETOP4C}},
  note         = {Machine review of arXiv:2412.19069}
}
read the original abstract

Online Learning to Rank (OLTR) optimises ranking models using implicit user feedback, such as clicks. Unlike traditional Learning to Rank (LTR) methods that rely on a static set of training data with relevance judgements to learn a ranking model, OLTR methods update the model continually as new data arrives. Thus, it addresses several drawbacks such as the high cost of human annotations, potential misalignment between user preferences and human judgments, and the rapid changes in user query intents. However, OLTR methods typically require the collection of searchable data, user queries, and clicks, which poses privacy concerns for users. Federated Online Learning to Rank (FOLTR) integrates OLTR within a Federated Learning (FL) framework to enhance privacy by not sharing raw data. While promising, FOLTR methods currently lag behind traditional centralised OLTR due to challenges in ranking effectiveness, robustness with respect to data distribution across clients, susceptibility to attacks, and the ability to unlearn client interactions and data. This thesis presents a comprehensive study on Federated Online Learning to Rank, addressing its effectiveness, robustness, security, and unlearning capabilities, thereby expanding the landscape of FOLTR.

Figures

Figures reproduced from arXiv: 2412.19069 by the authors.

Figure 5
Figure 5. b- 5.8c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 1.1
Figure 1.1. Illustration of Online Learning to Rank (OLTR) system. [PITH_FULL_IMAGE:figures/full_fig_p021_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Schematic representation of Federated Online Learning to Rank (FOLTR) setting. [PITH_FULL_IMAGE:figures/full_fig_p022_1_2.png] view at source ↗
Figures from the paper (40 more)
Figure 3
Figure 3. Figure 3: a reports the results obtained by FOLtR-ES on the MQ2007 dataset with respect to [PITH_FULL_IMAGE:figures/full_fig_p046_3.png]
Figure 3.1
Figure 3.1. Figure 3.1: Results for RQ1.1.1: performance of FOLtR-ES across datasets under three different click [PITH_FULL_IMAGE:figures/full_fig_p047_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Results for RQ1.1.2: performance of FOLtR-ES with respect to number of clients (averaged [PITH_FULL_IMAGE:figures/full_fig_p048_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Results for RQ1.1.3: performance of FOLtR-ES and PDGD across datasets with privatiza [PITH_FULL_IMAGE:figures/full_fig_p050_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Results for RQ1.1.4: performance of FOLtR-ES in terms of online nDCG@10 computed [PITH_FULL_IMAGE:figures/full_fig_p051_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: Results for RQ1.1.4: performance of FOLtR-ES and PDGD in terms of offline nDCG@10 [PITH_FULL_IMAGE:figures/full_fig_p053_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: Offline performance (nDCG@10) across datasets, under different click models, averaged [PITH_FULL_IMAGE:figures/full_fig_p061_3_6.png]
Figure 3
Figure 3. Figure 3: displays the offline performance (nDCG@10) of the proposed FPDGD method across [PITH_FULL_IMAGE:figures/full_fig_p061_3.png]
Figure 3.7
Figure 3.7. Figure 3.7: Offline performance in terms of MaxRR across datasets, under three different click models, [PITH_FULL_IMAGE:figures/full_fig_p063_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: Offline performance on MSLR-WEB10K under three different click models, averaged [PITH_FULL_IMAGE:figures/full_fig_p063_3_8.png]
Figure 3.9
Figure 3.9. Figure 3.9: Investigation of the influence of batch size [PITH_FULL_IMAGE:figures/full_fig_p064_3_9.png]
Figure 3.10
Figure 3.10. Figure 3.10: Investigation of the influence of batch size [PITH_FULL_IMAGE:figures/full_fig_p065_3_10.png]
Figure 4.1
Figure 4.1. Figure 4.1: Illustration of the model divergence problem in FL, adapted from Zhu et al. [6]. [PITH_FULL_IMAGE:figures/full_fig_p068_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Schematic representation of the FOLTR setting. [PITH_FULL_IMAGE:figures/full_fig_p069_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: Offline performance (nDCG@10) on Type 1 data; results averaged across dataset splits [PITH_FULL_IMAGE:figures/full_fig_p073_4_3.png]
Figure 4.4
Figure 4.4. Figure 4.4: Offline performance on Type 1 data for FedProx and FedPer; results averaged across [PITH_FULL_IMAGE:figures/full_fig_p074_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Offline performance (nDCG@10) on MSLR-WEB10K for Type 2 ( [PITH_FULL_IMAGE:figures/full_fig_p077_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: Offline performance (nDCG@10) on MSLR-WEB10K for Type 2 ( [PITH_FULL_IMAGE:figures/full_fig_p077_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: Offline performance on MSLR-WEB10K when using Data-sharing, FedProx and FedPer [PITH_FULL_IMAGE:figures/full_fig_p079_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Offline performance (nDCG@10) on MSLR-WEB10K for Type 3, separately under CCM [PITH_FULL_IMAGE:figures/full_fig_p080_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Offline performance (nDCG@10) on MSLR-WEB10K and intent-change for Type 4, under [PITH_FULL_IMAGE:figures/full_fig_p081_4_9.png]
Figure 5.1
Figure 5.1. Figure 5.1: Overview of a FOLTR system with attack and defense modules (the arrows point to where [PITH_FULL_IMAGE:figures/full_fig_p086_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Offline performance (nDCG@10) for MSLR-WEB10K under data poisoning attack and [PITH_FULL_IMAGE:figures/full_fig_p094_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Offline performance (nDCG@10) under model poisoning attacks, simulated using three [PITH_FULL_IMAGE:figures/full_fig_p095_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Offline performance (nDCG@10) of FOLTR system when no attack is present but defense [PITH_FULL_IMAGE:figures/full_fig_p097_5_4.png]
Figure 5.6
Figure 5.6. Figure 5.6: Results on Istella-S is shown in Figure 5.7. Further analysis on results is detailed in [PITH_FULL_IMAGE:figures/full_fig_p098_5_6.png]
Figure 5.5
Figure 5.5. Figure 5.5: Offline performance (nDCG@10) for MQ2007 under data poisoning attack and defense [PITH_FULL_IMAGE:figures/full_fig_p098_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: Offline performance (nDCG@10) for Yahoo under data poisoning attack and defense strate [PITH_FULL_IMAGE:figures/full_fig_p099_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Offline performance (nDCG@10) for Istella-S under data poisoning attack and defense [PITH_FULL_IMAGE:figures/full_fig_p100_5_7.png]
Figure 5.8
Figure 5.8. Figure 5.8: Offline performance (nDCG@10) for MQ2007 under model poisoning attacks, sim [PITH_FULL_IMAGE:figures/full_fig_p101_5_8.png]
Figure 5.9
Figure 5.9. Figure 5.9: Offline performance (nDCG@10) for Yahoo under model poisoning attacks, simu [PITH_FULL_IMAGE:figures/full_fig_p102_5_9.png]
Figure 5.10
Figure 5.10. Figure 5.10: Offline performance (nDCG@10) for Istella-S under model poisoning attacks, sim [PITH_FULL_IMAGE:figures/full_fig_p103_5_10.png]
Figure 5.11
Figure 5.11. Figure 5.11: Offline performance (nDCG@10) of FOLTR system when no attack is present but defense [PITH_FULL_IMAGE:figures/full_fig_p104_5_11.png]
Figure 6
Figure 6. Figure 6: ), we set [PITH_FULL_IMAGE:figures/full_fig_p111_6.png]
Figure 6
Figure 6. Figure 6: reports the results obtained by these [PITH_FULL_IMAGE:figures/full_fig_p112_6.png]
Figure 6.1
Figure 6.1. Figure 6.1: Relationships between FOLTR con￾figurations: 9H-1M (green line), 10H-0M (black), 9H-0M (pink). Circles are clients [PITH_FULL_IMAGE:figures/full_fig_p112_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: Offline effectiveness (nDCG@10) obtained under the 9H-1M (green line), 10H-0M (black [PITH_FULL_IMAGE:figures/full_fig_p113_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: Comparison between the offline effectiveness (nDCG@10) after the unlearning method is [PITH_FULL_IMAGE:figures/full_fig_p114_6_3.png]
Figure 6
Figure 6. Figure 6: , which instead contains the effectiveness of 9H-1M. [PITH_FULL_IMAGE:figures/full_fig_p115_6.png]
Figure 7.1
Figure 7.1. Figure 7.1: A high-level architecture and key components in FOLTR system. [PITH_FULL_IMAGE:figures/full_fig_p121_7_1.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.