arxiv: 2605.07005 · v1 · submitted 2026-05-07 · 💻 cs.DS · cs.LG

Recognition: no theorem link

Equivalence of Coarse and Fine-Grained Models for Learning with Distribution Shift

Adam R. Klivans, Arsen Vasilyan, Konstantinos Stavropoulos, Shyamal Patel

Pith reviewed 2026-05-11 00:56 UTC · model grok-4.3

classification 💻 cs.DS cs.LG

keywords distribution shiftPQ learningTDS learningBoolean concept classeshalfspacesdistribution-free learningmembership queriesbranching programs

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The pith

An efficient black-box reduction shows PQ learning is equivalent to TDS learning for any Boolean concept class in the distribution-free setting.

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

The paper proves that two models for learning under distribution shift are interchangeable when no assumptions are placed on how the data is distributed. PQ learners reject individual points they flag as out of distribution, while TDS learners may reject an entire test set upon detecting shift. The authors supply a black-box reduction proving that, for Boolean classes, efficient learning in one model implies efficient learning in the other. This equivalence immediately yields the first hardness results for TDS learning of halfspaces. With membership queries added, the hardness disappears and halfspaces become efficiently PQ learnable through repeated application of Forster transforms.

Core claim

The central claim is that PQ learning and TDS learning are equivalent in the distribution-free setting: there exists an efficient black-box reduction from PQ to TDS learning for every Boolean concept class. The reduction is realized by a boosting procedure that converts weak distinguishing power of a TDS learner (after it has rejected the target domain) into strong distinguishing power via branching programs. As a direct corollary, basic classes such as halfspaces are hard to learn in the TDS model. Separately, membership queries allow an efficient PQ algorithm for halfspaces that iteratively recovers large-margin separators by successive Forster transforms on the training data.

What carries the argument

The black-box reduction from PQ learning to TDS learning, realized by boosting the distinguishing power of rejected TDS learners through branching programs.

If this is right

Distribution-free TDS learning of halfspaces is computationally hard.
The equivalence holds for every Boolean concept class.
Membership queries make halfspaces efficiently PQ learnable in the distribution-free setting via iterative Forster transforms.
Weak TDS distinguishers can be boosted to strong ones using branching programs after domain rejection.

Where Pith is reading between the lines

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

Pointwise rejection and whole-domain rejection have identical power for Boolean learning without distributional assumptions.
The membership-query algorithm indicates that query access can bypass hardness induced by distribution shift.
The branching-program boosting technique may simplify algorithm design for other shift models that allow partial rejection.

Load-bearing premise

The equivalence and reduction are stated only for the distribution-free setting and for Boolean concept classes.

What would settle it

An efficient distribution-free TDS learner for halfspaces whose output cannot be turned into a PQ learner by the reduction procedure would disprove the claimed equivalence.

read the original abstract

Recent work on provably efficient algorithms for learning with distribution shift has focused on two models: PQ learning (Goldwasser et al. (2020)) and TDS learning (Klivans et al. (2024)). Algorithms for TDS learning are allowed to reject a test set entirely if distribution shift is detected. In contrast, PQ learners may only reject points that are deemed out-of-distribution on an individual basis. Our main result is a surprising equivalence between these two models in the distribution-free setting. In particular, we give an efficient black-box reduction from PQ learning to TDS learning for any Boolean concept class. This equivalence implies the first hardness results for distribution-free TDS learning of basic classes such as halfspaces. The main technical contribution underlying our equivalence is a method for boosting, via branching programs, the weak distinguishing power of TDS learners that have rejected the target domain. We also show that giving a learner access to membership queries sidesteps these hardness results and allows for efficient, distribution-free PQ learnability of halfspaces. Our algorithm iteratively recovers large-margin separators obtained by applying successive Forster transforms on the training data.

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.

Desk Editor's Note private letter to a colleague

The paper shows PQ and TDS are equivalent for Boolean classes in the distribution-free setting via a black-box reduction that transfers hardness to TDS halfspace learning, while membership queries give an efficient positive algorithm using Forster transforms.

read the letter

The core result is that PQ learning reduces to TDS learning in a black-box way for any Boolean concept class, which immediately imports hardness results and shows TDS halfspaces are hard in the distribution-free model. The authors also give a membership-query algorithm that learns halfspaces efficiently by iterating Forster transforms to find large-margin separators. This equivalence is the main new thing. It unifies two separate lines of work on distribution shift and lets hardness carry over cleanly. The branching-program boosting step that amplifies weak distinguishing power from a rejecting TDS learner is the technical engine behind the reduction, and it appears original. The positive membership-query result is also concrete and sidesteps the hardness without distributional assumptions. Both directions are useful to see in one paper. The main soft spot is whether the branching programs stay polynomial in size. The reduction is only efficient if the program size does not blow up with 1/ε or the weak advantage γ; if it does, the black-box claim fails to deliver polynomial time even when the base TDS learner is efficient. The abstract asserts efficiency, but that detail needs verification from the full proof. The result is also restricted to Boolean classes and the distribution-free setting, so extensions would require separate arguments. This is for researchers in computational learning theory who work on provable robustness and distribution shift. Anyone tracking hardness results or trying to connect PQ and TDS models will find the connections and the membership-query workaround worth reading. It has enough new technical content and both hardness and algorithmic results to merit a serious referee.

Referee Report

2 major / 1 minor

Summary. The paper claims an equivalence between PQ learning and TDS learning in the distribution-free setting for arbitrary Boolean concept classes, established via an efficient black-box reduction from PQ to TDS. The reduction boosts the weak distinguishing power of rejecting TDS learners using branching programs. This yields the first hardness results for distribution-free TDS learning of classes such as halfspaces. Separately, the paper shows that membership queries enable efficient distribution-free PQ learning of halfspaces via iterative application of Forster transforms to recover large-margin separators.

Significance. If the black-box reduction is polynomial-time and correct for all Boolean classes, the equivalence is a significant unification of two models of learning under distribution shift; it transfers existing PQ hardness to TDS and supplies the first such hardness results for basic classes. The membership-query algorithm for halfspaces is a positive algorithmic contribution that sidesteps the hardness. The branching-program boosting technique is a novel technical tool whose broader applicability could be explored.

major comments (2)

[technical contribution on branching-program boosting] The central reduction (described in the technical contribution section following the abstract): efficiency requires that the branching programs used to amplify weak per-point rejection signals into a reliable domain distinguisher have size polynomial in n, 1/ε, and 1/γ. The manuscript must supply an explicit size bound; absent this, the claimed black-box reduction from PQ to TDS is not guaranteed to be efficient, which would invalidate both the equivalence and the transferred hardness results for TDS learning of halfspaces.
[hardness implications] Hardness transfer paragraph (immediately after the reduction statement): the implication that TDS learning of halfspaces is hard in the distribution-free setting rests entirely on the reduction being efficient and black-box. If the branching-program construction incurs superpolynomial blow-up, this hardness claim does not follow.

minor comments (1)

[abstract] The abstract states the reduction is 'efficient' but does not list the precise polynomial dependence on the weak advantage γ; adding this would clarify the result for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the potential significance of the equivalence between PQ and TDS learning. We address the two major comments point by point below.

read point-by-point responses

Referee: The central reduction (described in the technical contribution section following the abstract): efficiency requires that the branching programs used to amplify weak per-point rejection signals into a reliable domain distinguisher have size polynomial in n, 1/ε, and 1/γ. The manuscript must supply an explicit size bound; absent this, the claimed black-box reduction from PQ to TDS is not guaranteed to be efficient, which would invalidate both the equivalence and the transferred hardness results for TDS learning of halfspaces.

Authors: We agree that an explicit size bound is required to rigorously establish polynomial-time efficiency. The proof of the main reduction (Theorem 3.1) constructs the branching program by iteratively amplifying the weak per-point rejection signal using a standard majority-vote construction over O(1/γ² log(1/δ)) independent paths, each of length O(log(1/ε)), yielding an overall size of at most (n/(εγ))^{O(1)}. To address the concern directly, we will insert a new lemma (Lemma 3.4) in the revised manuscript that states this bound and derives it from first principles. revision: yes
Referee: Hardness transfer paragraph (immediately after the reduction statement): the implication that TDS learning of halfspaces is hard in the distribution-free setting rests entirely on the reduction being efficient and black-box. If the branching-program construction incurs superpolynomial blow-up, this hardness claim does not follow.

Authors: The hardness transfer for classes such as halfspaces follows immediately once the reduction is shown to be polynomial-time, which rests on the branching-program size bound discussed above. We will revise the hardness-implications paragraph to cite the new lemma explicitly, thereby making the dependence on polynomial efficiency transparent. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for model definitions; central reduction remains independent

full rationale

The paper cites Goldwasser et al. (2020) for PQ learning and Klivans et al. (2024) for TDS learning (with author overlap on the latter). These supply the model definitions used in the claimed equivalence. However, the core technical contribution—an efficient black-box reduction from PQ to TDS via branching-program boosting of weak distinguishing power—is presented as a new construction that does not reduce to the cited definitions by construction or tautology. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the derivation. The equivalence and implied hardness results for classes such as halfspaces follow from the new reduction rather than being presupposed by the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions and assumptions from prior work on PQ and TDS learning (Goldwasser et al. 2020, Klivans et al. 2024). No free parameters, new axioms beyond domain assumptions in learning theory, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard definitions of PQ and TDS learning in the distribution-free setting for Boolean concept classes
The equivalence and reduction are stated to hold under these prior definitions.

pith-pipeline@v0.9.0 · 5507 in / 1375 out tokens · 32487 ms · 2026-05-11T00:56:43.107581+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

259 extracted references · 23 canonical work pages

[1]

International Conference on Computational Learning Theory , pages=

Martingale boosting , author=. International Conference on Computational Learning Theory , pages=. 2005 , organization=

2005
[2]

The Thirty Seventh Annual Conference on Learning Theory , pages=

Testable learning with distribution shift , author=. The Thirty Seventh Annual Conference on Learning Theory , pages=. 2024 , organization=

2024
[3]

Advances in Neural Information Processing Systems , volume=

Beyond perturbations: Learning guarantees with arbitrary adversarial test examples , author=. Advances in Neural Information Processing Systems , volume=
[4]

refutation , author=

On learning vs. refutation , author=. Conference on Learning Theory , pages=. 2017 , organization=

2017
[5]

9th Innovations in Theoretical Computer Science Conference (ITCS 2018) , pages=

Improper learning by refuting , author=. 9th Innovations in Theoretical Computer Science Conference (ITCS 2018) , pages=. 2018 , organization=

2018
[6]

Algorithmic Learning Theory , pages=

Efficient learning with arbitrary covariate shift , author=. Algorithmic Learning Theory , pages=. 2021 , organization=

2021
[7]

Journal of Computer and System Sciences , volume=

Cryptographic hardness for learning intersections of halfspaces , author=. Journal of Computer and System Sciences , volume=. 2009 , publisher=

2009
[8]

The Thirty Seventh Annual Conference on Learning Theory , pages=

Improved hardness results for learning intersections of halfspaces , author=. The Thirty Seventh Annual Conference on Learning Theory , pages=. 2024 , organization=

2024
[10]

1994 , publisher=

An introduction to computational learning theory , author=. 1994 , publisher=

1994
[11]

Advances in neural information processing systems , volume=

Analysis of representations for domain adaptation , author=. Advances in neural information processing systems , volume=
[12]

Advances in neural information processing systems , volume=

Learning bounds for domain adaptation , author=. Advances in neural information processing systems , volume=
[13]

Machine learning , volume=

A theory of learning from different domains , author=. Machine learning , volume=. 2010 , publisher=

2010
[14]

Proceedings of The 22nd Annual Conference on Learning Theory (COLT 2009) , address =

Domain Adaptation: Learning Bounds and Algorithms , author =. Proceedings of The 22nd Annual Conference on Learning Theory (COLT 2009) , address =. 2009 , URL =

2009
[15]

Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics , pages=

Impossibility theorems for domain adaptation , author=. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics , pages=. 2010 , organization=

2010
[16]

Proceedings of the 55th Annual ACM Symposium on Theory of Computing , pages=

A strongly polynomial algorithm for approximate forster transforms and its application to halfspace learning , author=. Proceedings of the 55th Annual ACM Symposium on Theory of Computing , pages=
[17]

Journal of Computer and System Sciences , volume=

Reliable agnostic learning , author=. Journal of Computer and System Sciences , volume=. 2012 , publisher=

2012
[18]

The Thirty Seventh Annual Conference on Learning Theory , year=

Learning Intersections of Halfspaces with Distribution Shift: Improved Algorithms and SQ Lower Bounds , author=. The Thirty Seventh Annual Conference on Learning Theory , year=
[20]

The Thirteenth International Conference on Learning Representations , year=

Learning Neural Networks with Distribution Shift: Efficiently Certifiable Guarantees , author=. The Thirteenth International Conference on Learning Representations , year=
[21]

Tolerant Algorithms for Learning with Arbitrary Covariate Shift , year =

Goel, Surbhi and Shetty, Abhishek and Stavropoulos, Konstantinos and Vasilyan, Arsen , journal =. Tolerant Algorithms for Learning with Arbitrary Covariate Shift , year =
[23]

Testing distributional assumptions of learning algorithms , year =

Rubinfeld, Ronitt and Vasilyan, Arsen , journal =. Testing distributional assumptions of learning algorithms , year =
[24]

Philosophical Transactions of the Royal Society A , year=

Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations , author=. Philosophical Transactions of the Royal Society A , year=
[25]

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) , pages=

Classical lower bounds from quantum upper bounds , author=. 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) , pages=. 2018 , organization=

2018
[26]

Journal of the ACM (JACM) , volume=

Polylogarithmic independence fools AC 0 circuits , author=. Journal of the ACM (JACM) , volume=. 2008 , publisher=

2008
[27]

Advances in Neural Information Processing Systems , volume=

Tester-learners for halfspaces: Universal algorithms , author=. Advances in Neural Information Processing Systems , volume=
[28]

Bakry, D. and \'. Diffusions hypercontractives , BOOKTITLE =. 1985 , MRCLASS =. doi:10.1007/BFb0075847 , URL =

work page doi:10.1007/bfb0075847 1985
[29]

Polylogarithmic independence fools

Braverman, Mark , journal =. Polylogarithmic independence fools
[30]

Bounded independence fools halfspaces , volume =

Diakonikolas, Ilias and Gopalan, Parikshit and Jaiswal, Ragesh and Servedio, Rocco A and Viola, Emanuele , journal =. Bounded independence fools halfspaces , volume =
[31]

Polylogarithmic independence can fool DNF formulas , volume =

Bazzi, Louay MJ , journal =. Polylogarithmic independence can fool DNF formulas , volume =
[32]

Agnostically learning halfspaces , volume =

Kalai, Adam Tauman and Klivans, Adam R and Mansour, Yishay and Servedio, Rocco A , journal =. Agnostically learning halfspaces , volume =
[33]

Moment-matching polynomials , year =

Klivans, Adam and Meka, Raghu , journal =. Moment-matching polynomials , year =
[34]

Conference on Learning Theory , pages=

Learning halfspaces under log-concave densities: Polynomial approximations and moment matching , author=. Conference on Learning Theory , pages=. 2013 , organization=

2013
[35]

Almost k-wise independence versus k-wise independence , volume =

Alon, Noga and Goldreich, Oded and Mansour, Yishay , journal =. Almost k-wise independence versus k-wise independence , volume =
[36]

Proximity of probability measures with common marginals in a finite number of directions , year =

Klebanov, LB and Rachev, ST , journal =. Proximity of probability measures with common marginals in a finite number of directions , year =
[37]

Semi-infinite programming , pages=

On duality theory of conic linear problems , author=. Semi-infinite programming , pages=. 2001 , publisher=

2001
[38]

Summer school on machine learning , pages=

Introduction to statistical learning theory , author=. Summer school on machine learning , pages=. 2003 , organization=

2003
[39]

Acta numerica , volume=

Deep learning: a statistical viewpoint , author=. Acta numerica , volume=. 2021 , publisher=

2021
[40]

2008 49th Annual IEEE Symposium on Foundations of Computer Science , pages=

Learning geometric concepts via Gaussian surface area , author=. 2008 49th Annual IEEE Symposium on Foundations of Computer Science , pages=. 2008 , organization=

2008
[41]

Journal of the ACM (JACM) , volume=

On the Fourier spectrum of monotone functions , author=. Journal of the ACM (JACM) , volume=. 1996 , publisher=

1996
[42]

Theoretical Computer Science , volume=

Learnability with respect to fixed distributions , author=. Theoretical Computer Science , volume=. 1991 , publisher=

1991
[43]

Theory of Probability & Its Applications , volume=

Probability metrics , author=. Theory of Probability & Its Applications , volume=. 1984 , publisher=

1984
[44]

Journal of Soviet Mathematics , volume=

Estimation of the closeness of distributions in terms of identical moments , author=. Journal of Soviet Mathematics , volume=. 1986 , publisher=

1986
[45]

2014 , publisher=

Analysis of boolean functions , author=. 2014 , publisher=

2014
[46]

The American mathematical monthly , volume=

A remark on Stirling's formula , author=. The American mathematical monthly , volume=. 1955 , publisher=

1955
[47]

UC Berkeley CS281B/Stat241B: Statistical Learning Theory, Lecture 7

Peter L Bartlett. UC Berkeley CS281B/Stat241B: Statistical Learning Theory, Lecture 7
[48]

2011 , eprint=

k -Independent Gaussians Fool Polynomial Threshold Functions , author=. 2011 , eprint=

2011
[49]

Proceedings of the forty-fourth annual ACM symposium on Theory of computing , pages=

Making polynomials robust to noise , author=. Proceedings of the forty-fourth annual ACM symposium on Theory of computing , pages=
[50]

2014 , publisher=

Understanding machine learning: From theory to algorithms , author=. 2014 , publisher=

2014
[51]

The Journal of Machine Learning Research , volume=

Learnability, stability and uniform convergence , author=. The Journal of Machine Learning Research , volume=. 2010 , publisher=

2010
[52]

The Annals of Statistics , volume=

Local rademacher complexities , author=. The Annals of Statistics , volume=. 2005 , publisher=

2005
[53]

The Annals of Statistics , volume=

Local Rademacher complexities and oracle inequalities in risk minimization , author=. The Annals of Statistics , volume=. 2006 , publisher=

2006
[54]

Machine Learning , volume=

Model selection and error estimation , author=. Machine Learning , volume=. 2002 , publisher=

2002
[55]

Innovations in Theoretical Computer Science , year=

Convex Influences , author=. Innovations in Theoretical Computer Science , year=
[56]

Proceedings of the fifth annual workshop on Computational learning theory , pages=

Toward efficient agnostic learning , author=. Proceedings of the fifth annual workshop on Computational learning theory , pages=
[57]

1998 , publisher=

Statistical Learning Theory , author=. 1998 , publisher=

1998
[58]

Conference on Learning Theory , pages=

On the Minimal Error of Empirical Risk Minimization , author=. Conference on Learning Theory , pages=. 2021 , organization=

2021
[59]

High dimensional probability II , pages=

Rademacher processes and bounding the risk of function learning , author=. High dimensional probability II , pages=. 2000 , publisher=

2000
[60]

Journal of Machine Learning Research , volume=

Rademacher and Gaussian complexities: Risk bounds and structural results , author=. Journal of Machine Learning Research , volume=
[61]

IEEE Transactions on Information Theory , volume=

Rademacher penalties and structural risk minimization , author=. IEEE Transactions on Information Theory , volume=. 2001 , publisher=

2001
[62]

Festschrift for lucien le cam , pages=

From model selection to adaptive estimation , author=. Festschrift for lucien le cam , pages=. 1997 , publisher=

1997
[63]

2008 , publisher=

Introduction to Nonparametric Estimation , author=. 2008 , publisher=

2008
[64]

2000 , publisher=

Empirical Processes in M-estimation , author=. 2000 , publisher=

2000
[65]

Acta Numerica , volume=

Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation , author=. Acta Numerica , volume=. 2021 , publisher=

2021
[66]

The Annals of Statistics , volume=

Surprises in high-dimensional ridgeless least squares interpolation , author=. The Annals of Statistics , volume=. 2022 , publisher=

2022
[67]

Proceedings of the National Academy of Sciences , volume=

Benign overfitting in linear regression , author=. Proceedings of the National Academy of Sciences , volume=. 2020 , publisher=

2020
[68]

Communications of the ACM , volume=

Understanding deep learning (still) requires rethinking generalization , author=. Communications of the ACM , volume=. 2021 , publisher=

2021
[69]

Advances in Neural Information Processing Systems , volume=

Uniform convergence may be unable to explain generalization in deep learning , author=. Advances in Neural Information Processing Systems , volume=
[70]

and Jordan, Michael I

Zhang, Yuchen and Lee, Jason D. and Jordan, Michael I. , booktitle =. L1-regularized Neural Networks are Improperly Learnable in Polynomial Time , volume =
[71]

2010 IEEE 51st Annual Symposium on Foundations of Computer Science , pages=

Bounded independence fools degree-2 threshold functions , author=. 2010 IEEE 51st Annual Symposium on Foundations of Computer Science , pages=. 2010 , organization=

2010
[72]

Journal of Soviet Mathematics , volume=

Stability of the characterization of the multivariate normal distribution in the Skitovich-Darmois theorem , author=. Journal of Soviet Mathematics , volume=. 1981 , publisher=

1981
[73]

SIAM Journal on Computing , volume=

Hardness of learning halfspaces with noise , author=. SIAM Journal on Computing , volume=. 2009 , publisher=

2009
[74]

Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 1 , pages =

Livni, Roi and Shalev-Shwartz, Shai and Shamir, Ohad , title =. Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 1 , pages =. 2014 , publisher =

2014
[75]

SIAM Journal on Computing , volume=

Agnostic learning of monomials by halfspaces is hard , author=. SIAM Journal on Computing , volume=. 2012 , publisher=

2012
[76]

Proceedings of the forty-eighth annual ACM symposium on Theory of Computing , pages=

Complexity theoretic limitations on learning halfspaces , author=. Proceedings of the forty-eighth annual ACM symposium on Theory of Computing , pages=
[77]

Conference on Learning Theory , pages=

Complexity theoretic limitations on learning DNF’s , author=. Conference on Learning Theory , pages=. 2016 , organization=

2016
[78]

Journal of the ACM (JACM) , volume=

Cryptographic limitations on learning boolean formulae and finite automata , author=. Journal of the ACM (JACM) , volume=. 1994 , publisher=

1994
[79]

Journal of the ACM (JACM) , volume=

Constant depth circuits, Fourier transform, and learnability , author=. Journal of the ACM (JACM) , volume=. 1993 , publisher=

1993
[80]

computational complexity , volume=

The Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions , author=. computational complexity , volume=. 2011 , publisher=

2011
[81]

2018 , publisher=

High-dimensional probability: An introduction with applications in data science , author=. 2018 , publisher=

2018
[82]

Advances in Neural Information Processing Systems , volume=

Near-optimal sq lower bounds for agnostically learning halfspaces and relus under gaussian marginals , author=. Advances in Neural Information Processing Systems , volume=
[83]

Conference on Learning Theory , pages=

The optimality of polynomial regression for agnostic learning under gaussian marginals in the SQ model , author=. Conference on Learning Theory , pages=. 2021 , organization=

2021

Showing first 80 references.