Approximate full-conformal multi-task regression with reproducing kernels

Alain Celisse (SAMM); Davidson Lova Razafindrakoto (SAMM); J\'er\^ome Lacaille

arxiv: 2607.00645 · v1 · pith:NNETWEP5new · submitted 2026-07-01 · 🧮 math.ST · stat.ML· stat.TH

Approximate full-conformal multi-task regression with reproducing kernels

Davidson Lova Razafindrakoto (SAMM) , Alain Celisse (SAMM) , J\'er\^ome Lacaille This is my paper

Pith reviewed 2026-07-02 04:47 UTC · model grok-4.3

classification 🧮 math.ST stat.MLstat.TH

keywords multi-task regressionconformal predictionreproducing kernel Hilbert spaceprediction regionsvector-valued kernelscovariance estimationcoverage guarantees

0 comments

The pith

An approximating prediction region for multi-task regression contains the full-conformal one and admits a volume upper bound when the inter-task covariance matrix is known.

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

The paper addresses the intractability of full-conformal prediction in multi-task regression by constructing an approximating prediction region that contains the exact full-conformal region. This is done for vector-valued functions in a reproducing kernel Hilbert space, first when the inter-task covariance matrix is known and second when it is estimated from data. A theoretical upper bound on the volume of the approximation is provided in the known-covariance case. The construction is shown empirically to produce smaller regions than split-conformal prediction on synthetic data in both scenarios.

Core claim

The authors design an approximating prediction region that contains the full-conformal prediction region for multi-task regression in an RKHS setting. This is done for both known and estimated inter-task covariance matrices. When the covariance is known, the volume of this approximation is bounded above by a derived quantity. Empirically, it improves upon split-conformal prediction on synthetic data.

What carries the argument

The approximating prediction region constructed to contain the full-conformal one, based on the reproducing kernel Hilbert space of vector-valued functions and the inter-task covariance matrix.

If this is right

The construction ensures the same finite-sample coverage guarantee as the full-conformal method since the region contains it.
When the inter-task covariance matrix is known, the volume of the approximating region is bounded above by a specific quantity derived from the method.
In both known and estimated covariance cases, the approximating region improves upon split-conformal prediction on synthetic data.
The approach allows practical computation of a prediction region while maintaining the theoretical guarantees of full-conformal prediction.

Where Pith is reading between the lines

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

The volume bound could be used to compare the efficiency of this method against other conformal approaches in settings with many tasks.
If covariance estimation error remains moderate, the same approximation might apply directly to real multi-output problems with correlated responses.
The containment property suggests the method could serve as a conservative starting point for developing faster conformal procedures in kernel spaces.

Load-bearing premise

The vector-valued functions belong to a reproducing kernel Hilbert space and the data satisfy the exchangeability condition required for conformal coverage guarantees.

What would settle it

A dataset or calculation showing that the constructed approximating region fails to contain the full-conformal prediction region, or that its volume exceeds the derived upper bound when the covariance matrix is known.

Figures

Figures reproduced from arXiv: 2607.00645 by Alain Celisse (SAMM), Davidson Lova Razafindrakoto (SAMM), J\'er\^ome Lacaille.

read the original abstract

Multi-task regression aims at jointly solving multiple regression problems, called tasks. Compared to solving each task separately, better performances can be achieved as long as the tasks are sufficiently related. Full-conformal prediction is a framework that formulates a data-dependent prediction-region containing the unknown output-vector at any prescribed confidence level. However, explicit computation of this prediction-region is intractable in general since it requires training infinitely many predictors. The present work focuses on multi-task regression in a Reproducing Kernel Hilbert Space (RKHS) of vector-valued functions. This computational issue is addressed by designing an approximating predictionregion containing the full-conformal one. This construction is carried out in two scenarios: piq when the inter-task covariance-matrix is known, and piiq when this matrix is estimated. In terms of volume, the tightness of this approximation is assessed theoretically by means of an upper-bound in the first scenario. It is also empirically proved to improve upon the split-conformal prediction on synthetic data in both scenarios.

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

This paper builds an approximating region that contains the full-conformal prediction set for vector-valued kernel multi-task regression, with a volume bound when covariance is known and synthetic gains over split conformal.

read the letter

The core contribution is a construction that produces a prediction region guaranteed to contain the full-conformal one in the vector-valued RKHS multi-task setting. They handle two cases: known inter-task covariance, where they also prove a volume upper bound, and estimated covariance. Both versions show smaller average volume than split conformal on the synthetic examples they report.

The work is clearest on the containment property, which follows directly from how the approximation is defined rather than from extra assumptions. The volume bound for the known-covariance case is a concrete theoretical statement that can be checked. The empirical section is limited but consistent with the claim that the approximation improves on the simpler split method.

The main limitations are straightforward. The volume bound is stated only for the known-covariance scenario, so the estimated case rests entirely on the synthetic runs. Those runs use synthetic data only, with no real multi-task datasets or scaling tests shown in the abstract. The approach inherits the usual exchangeability and RKHS modeling requirements, which are explicit but restrict the range of problems where the guarantees apply.

This is a narrow but self-contained piece aimed at researchers already working on conformal prediction for kernel or multi-output problems. The construction is explicit enough and the claims are narrow enough that the paper deserves referee time rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper constructs an approximating prediction region for full-conformal multi-task regression in a vector-valued RKHS that contains the exact full-conformal region by design. It derives a theoretical upper bound on the volume of this approximation when the inter-task covariance matrix is known, and reports empirical improvements over split-conformal prediction on synthetic data for both known and estimated covariance cases, under exchangeability.

Significance. If the containment and volume bound hold, the work makes full-conformal prediction computationally tractable for multi-task vector-valued regression while preserving coverage guarantees. The RKHS setting and explicit construction when covariance is known or estimated are standard in the field; the synthetic experiments provide supporting evidence of practical benefit over split-conformal baselines. The design choice to enforce containment is a clear strength.

minor comments (2)

Notation for the vector-valued RKHS and the inter-task covariance matrix should be introduced with explicit definitions in §2 before use in the approximation construction.
The synthetic data generation protocol (including how tasks are correlated and sample sizes) needs to be stated more precisely to allow reproduction of the reported volume and coverage results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description accurately captures the paper's contributions on the approximating prediction region for full-conformal multi-task kernel regression, the volume bound when covariance is known, and the empirical comparisons.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper explicitly designs an approximating prediction region to contain the full-conformal region (by construction of the approximation), then derives a volume upper bound for the known-covariance case and demonstrates empirical improvement over split-conformal on synthetic data. No step reduces a claimed independent derivation or prediction to a fitted parameter, self-citation chain, or input by definition; the containment, bound, and comparisons are presented as direct consequences of the RKHS modeling choice and exchangeability assumption without hidden reductions. The derivation chain is self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The method rests on standard conformal prediction assumptions and the vector-valued RKHS modeling framework; no additional free parameters or invented entities are visible from the abstract.

free parameters (1)

inter-task covariance matrix
Estimated from data in the second scenario, introducing a data-dependent quantity.

axioms (2)

domain assumption Data points are exchangeable
Required for the coverage guarantee of conformal prediction.
domain assumption Regression functions lie in a vector-valued RKHS
Core modeling assumption enabling the kernel approach.

pith-pipeline@v0.9.1-grok · 5717 in / 1277 out tokens · 38188 ms · 2026-07-02T04:47:52.334041+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 5 canonical work pages

[1]

URL https://scikit-learn.org/stable/modules/generated/sklearn.metrics.pairwise.laplacian\_kernel.html

Laplacian\_kernel. URL https://scikit-learn.org/stable/modules/generated/sklearn.metrics.pairwise.laplacian\_kernel.html
[2]

URL https://docs.scipy.org/doc/scipy/reference/optimize.minimize-newtoncg.html

Minimize(method=' Newton-CG ') --- SciPy v1.18.0 Manual . URL https://docs.scipy.org/doc/scipy/reference/optimize.minimize-newtoncg.html
[3]

Optimization in infinite-dimensional Hilbert spaces

Alen Alexanderian. Optimization in infinite-dimensional Hilbert spaces. North Carolina State University, Raleigh, NC, USA, 2019

2019
[4]

\'A lvarez, Lorenzo Rosasco, and Neil D

Mauricio A. \'A lvarez, Lorenzo Rosasco, and Neil D. Lawrence. Kernels for Vector-Valued Functions : A Review . Foundations and Trends in Machine Learning , 4 0 (3): 0 195--266, 2012. ISSN 1935-8237, 1935-8245. doi:10.1561/2200000036

work page doi:10.1561/2200000036 2012
[5]

Theory of reproducing kernels

Nachman Aronszajn. Theory of reproducing kernels. Transactions of the American mathematical society, 68 0 (3): 0 337--404, 1950

1950
[6]

Stability of multi-task kernel regression algorithms

Julien Audiffren and Hachem Kadri. Stability of multi-task kernel regression algorithms. In Asian Conference on Machine Learning , pages 1--16. PMLR, 2013

2013
[7]

Learning Theory from First Principles

Francis Bach. Learning Theory from First Principles . 2024

2024
[8]

Bartlett, Philip M

Peter L. Bartlett, Philip M. Long, G \'a bor Lugosi, and Alexander Tsigler. Benign overfitting in linear regression. Proceedings of the National Academy of Sciences, 117 0 (48): 0 30063--30070, December 2020. ISSN 0027-8424, 1091-6490. doi:10.1073/pnas.1907378117

work page doi:10.1073/pnas.1907378117 2020
[9]

Stability and generalization

Olivier Bousquet and Andr \'e Elisseeff. Stability and generalization. Journal of machine learning research, 2 0 (Mar): 0 499--526, 2002

2002
[10]

Jordan, and Francis Bach

Sacha Braun, Liviu Aolaritei, Michael I. Jordan, and Francis Bach. Minimum volume conformal sets for multivariate regression. arXiv preprint arXiv:2503.19068, 2025

work page arXiv 2025
[11]

Jordan, and Francis Bach

Sacha Braun, Eug \`e ne Berta, Michael I. Jordan, and Francis Bach. Multivariate Standardized Residuals for Conformal Prediction , May 2026

2026
[12]

Micchelli, Massimiliano Pontil, and Yiming Ying

Andrea Caponnetto, Charles A. Micchelli, Massimiliano Pontil, and Yiming Ying. Universal multi-task kernels. The Journal of Machine Learning Research, 9: 0 1615--1646, 2008

2008
[13]

Two deterministic half-quadratic regularization algorithms for computed imaging

Pierre Charbonnier, Laure Blanc-Feraud, Gilles Aubert, and Michel Barlaud. Two deterministic half-quadratic regularization algorithms for computed imaging. In Proceedings of 1st international conference on image processing, volume 2, pages 168--172. IEEE, 1994

1994
[14]

A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression , February 2025

Victor Dheur, Matteo Fontana, Yorick Estievenart, Naomi Desobry, and Souhaib Ben Taieb. A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression , February 2025

2025
[15]

Micchelli, Massimiliano Pontil, and John Shawe-Taylor

Theodoros Evgeniou, Charles A. Micchelli, Massimiliano Pontil, and John Shawe-Taylor . Learning multiple tasks with kernel methods. Journal of machine learning research, 6 0 (4), 2005

2005
[16]

Exact and Approximate Conformal Inference for Multi-Output Regression , June 2024

Chancellor Johnstone and Eugene Ndiaye. Exact and Approximate Conformal Inference for Multi-Output Regression , June 2024

2024
[17]

Leave- One-Out Stable Conformal Prediction , April 2025

Kiljae Lee and Yuan Zhang. Leave- One-Out Stable Conformal Prediction , April 2025

2025
[18]

Optimal rates for regularized conditional mean embedding learning

Zhu Li, Dimitri Meunier, Mattes Mollenhauer, and Arthur Gretton. Optimal rates for regularized conditional mean embedding learning. Advances in Neural Information Processing Systems, 35: 0 4433--4445, 2022

2022
[19]

Towards optimal sobolev norm rates for the vector-valued regularized least-squares algorithm

Zhu Li, Dimitri Meunier, Mattes Mollenhauer, and Arthur Gretton. Towards optimal sobolev norm rates for the vector-valued regularized least-squares algorithm. Journal of Machine Learning Research, 25 0 (181): 0 1--51, 2024

2024
[20]

A vector-contraction inequality for Rademacher complexities, May 2016

Andreas Maurer. A vector-contraction inequality for Rademacher complexities, May 2016

2016
[21]

Copula-based conformal prediction for multi-target regression

Soundouss Messoudi, S \'e bastien Destercke, and Sylvain Rousseau. Copula-based conformal prediction for multi-target regression. Pattern Recognition, 120: 0 108101, 2021

2021
[22]

Ellipsoidal conformal inference for multi-target regression

Soundouss Messoudi, S \'e bastien Destercke, and Sylvain Rousseau. Ellipsoidal conformal inference for multi-target regression. In Conformal and Probabilistic Prediction with Applications , pages 294--306. PMLR, 2022

2022
[23]

Kernels for Multi --task Learning

Charles Micchelli and Massimiliano Pontil. Kernels for Multi --task Learning . Advances in neural information processing systems, 17, 2004

2004
[24]

Micchelli and Massimiliano Pontil

Charles A. Micchelli and Massimiliano Pontil. On learning vector-valued functions. Neural computation, 17 0 (1): 0 177--204, 2005

2005
[25]

Stable conformal prediction sets

Eugene Ndiaye. Stable conformal prediction sets. In International Conference on Machine Learning , pages 16462--16479. PMLR, 2022

2022
[26]

Inductive Conformal Prediction: Theory and Application to Neural Networks

Harris Papadopoulos. Inductive Conformal Prediction: Theory and Application to Neural Networks . INTECH Open Access Publisher Rijeka, 2008

2008
[27]

Approximate full conformal prediction in an RKHS , January 2026

Davidson Lova Razafindrakoto, Alain Celisse, and J \'e r \^o me Lacaille. Approximate full conformal prediction in an RKHS , January 2026

2026
[28]

Saleh and A

Resve A. Saleh and A. K. Saleh. Statistical properties of the log-cosh loss function used in machine learning. arXiv preprint arXiv:2208.04564, 2022

work page arXiv 2022
[29]

Learning theory estimates via integral operators and their approximations

Steve Smale and Ding-Xuan Zhou. Learning theory estimates via integral operators and their approximations. Constructive approximation, 26 0 (2): 0 153--172, 2007

2007
[30]

Multi-task regression using minimal penalties

Matthieu Solnon, Sylvain Arlot, and Francis Bach. Multi-task regression using minimal penalties. The Journal of Machine Learning Research, 13 0 (1): 0 2773--2812, 2012

2012
[31]

Hush, and Clint Scovel

Ingo Steinwart, Don R. Hush, and Clint Scovel. Optimal Rates for Regularized Least Squares Regression . In COLT , pages 79--93, 2009

2009
[32]

Algorithmic Learning in a Random World

Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic Learning in a Random World . Springer Science & Business Media, 2005

2005
[33]

Algorithmic Learning in a Random World

Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic Learning in a Random World. 2 edition, December 2022. ISBN 978-3-031-06648-1. doi:10.10007/978-3-031-06649-8

work page doi:10.10007/978-3-031-06649-8 2022
[34]

A survey on multi-task learning

Yu Zhang and Qiang Yang. A survey on multi-task learning. IEEE transactions on knowledge and data engineering, 34 0 (12): 0 5586--5609, 2021

2021
[35]

Gradient descent algorithms for quantile regression with smooth approximation

Songfeng Zheng. Gradient descent algorithms for quantile regression with smooth approximation. International Journal of Machine Learning and Cybernetics, 2: 0 191--207, 2011

2011

[1] [1]

URL https://scikit-learn.org/stable/modules/generated/sklearn.metrics.pairwise.laplacian\_kernel.html

Laplacian\_kernel. URL https://scikit-learn.org/stable/modules/generated/sklearn.metrics.pairwise.laplacian\_kernel.html

[2] [2]

URL https://docs.scipy.org/doc/scipy/reference/optimize.minimize-newtoncg.html

Minimize(method=' Newton-CG ') --- SciPy v1.18.0 Manual . URL https://docs.scipy.org/doc/scipy/reference/optimize.minimize-newtoncg.html

[3] [3]

Optimization in infinite-dimensional Hilbert spaces

Alen Alexanderian. Optimization in infinite-dimensional Hilbert spaces. North Carolina State University, Raleigh, NC, USA, 2019

2019

[4] [4]

\'A lvarez, Lorenzo Rosasco, and Neil D

Mauricio A. \'A lvarez, Lorenzo Rosasco, and Neil D. Lawrence. Kernels for Vector-Valued Functions : A Review . Foundations and Trends in Machine Learning , 4 0 (3): 0 195--266, 2012. ISSN 1935-8237, 1935-8245. doi:10.1561/2200000036

work page doi:10.1561/2200000036 2012

[5] [5]

Theory of reproducing kernels

Nachman Aronszajn. Theory of reproducing kernels. Transactions of the American mathematical society, 68 0 (3): 0 337--404, 1950

1950

[6] [6]

Stability of multi-task kernel regression algorithms

Julien Audiffren and Hachem Kadri. Stability of multi-task kernel regression algorithms. In Asian Conference on Machine Learning , pages 1--16. PMLR, 2013

2013

[7] [7]

Learning Theory from First Principles

Francis Bach. Learning Theory from First Principles . 2024

2024

[8] [8]

Bartlett, Philip M

Peter L. Bartlett, Philip M. Long, G \'a bor Lugosi, and Alexander Tsigler. Benign overfitting in linear regression. Proceedings of the National Academy of Sciences, 117 0 (48): 0 30063--30070, December 2020. ISSN 0027-8424, 1091-6490. doi:10.1073/pnas.1907378117

work page doi:10.1073/pnas.1907378117 2020

[9] [9]

Stability and generalization

Olivier Bousquet and Andr \'e Elisseeff. Stability and generalization. Journal of machine learning research, 2 0 (Mar): 0 499--526, 2002

2002

[10] [10]

Jordan, and Francis Bach

Sacha Braun, Liviu Aolaritei, Michael I. Jordan, and Francis Bach. Minimum volume conformal sets for multivariate regression. arXiv preprint arXiv:2503.19068, 2025

work page arXiv 2025

[11] [11]

Jordan, and Francis Bach

Sacha Braun, Eug \`e ne Berta, Michael I. Jordan, and Francis Bach. Multivariate Standardized Residuals for Conformal Prediction , May 2026

2026

[12] [12]

Micchelli, Massimiliano Pontil, and Yiming Ying

Andrea Caponnetto, Charles A. Micchelli, Massimiliano Pontil, and Yiming Ying. Universal multi-task kernels. The Journal of Machine Learning Research, 9: 0 1615--1646, 2008

2008

[13] [13]

Two deterministic half-quadratic regularization algorithms for computed imaging

Pierre Charbonnier, Laure Blanc-Feraud, Gilles Aubert, and Michel Barlaud. Two deterministic half-quadratic regularization algorithms for computed imaging. In Proceedings of 1st international conference on image processing, volume 2, pages 168--172. IEEE, 1994

1994

[14] [14]

A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression , February 2025

Victor Dheur, Matteo Fontana, Yorick Estievenart, Naomi Desobry, and Souhaib Ben Taieb. A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression , February 2025

2025

[15] [15]

Micchelli, Massimiliano Pontil, and John Shawe-Taylor

Theodoros Evgeniou, Charles A. Micchelli, Massimiliano Pontil, and John Shawe-Taylor . Learning multiple tasks with kernel methods. Journal of machine learning research, 6 0 (4), 2005

2005

[16] [16]

Exact and Approximate Conformal Inference for Multi-Output Regression , June 2024

Chancellor Johnstone and Eugene Ndiaye. Exact and Approximate Conformal Inference for Multi-Output Regression , June 2024

2024

[17] [17]

Leave- One-Out Stable Conformal Prediction , April 2025

Kiljae Lee and Yuan Zhang. Leave- One-Out Stable Conformal Prediction , April 2025

2025

[18] [18]

Optimal rates for regularized conditional mean embedding learning

Zhu Li, Dimitri Meunier, Mattes Mollenhauer, and Arthur Gretton. Optimal rates for regularized conditional mean embedding learning. Advances in Neural Information Processing Systems, 35: 0 4433--4445, 2022

2022

[19] [19]

Towards optimal sobolev norm rates for the vector-valued regularized least-squares algorithm

Zhu Li, Dimitri Meunier, Mattes Mollenhauer, and Arthur Gretton. Towards optimal sobolev norm rates for the vector-valued regularized least-squares algorithm. Journal of Machine Learning Research, 25 0 (181): 0 1--51, 2024

2024

[20] [20]

A vector-contraction inequality for Rademacher complexities, May 2016

Andreas Maurer. A vector-contraction inequality for Rademacher complexities, May 2016

2016

[21] [21]

Copula-based conformal prediction for multi-target regression

Soundouss Messoudi, S \'e bastien Destercke, and Sylvain Rousseau. Copula-based conformal prediction for multi-target regression. Pattern Recognition, 120: 0 108101, 2021

2021

[22] [22]

Ellipsoidal conformal inference for multi-target regression

Soundouss Messoudi, S \'e bastien Destercke, and Sylvain Rousseau. Ellipsoidal conformal inference for multi-target regression. In Conformal and Probabilistic Prediction with Applications , pages 294--306. PMLR, 2022

2022

[23] [23]

Kernels for Multi --task Learning

Charles Micchelli and Massimiliano Pontil. Kernels for Multi --task Learning . Advances in neural information processing systems, 17, 2004

2004

[24] [24]

Micchelli and Massimiliano Pontil

Charles A. Micchelli and Massimiliano Pontil. On learning vector-valued functions. Neural computation, 17 0 (1): 0 177--204, 2005

2005

[25] [25]

Stable conformal prediction sets

Eugene Ndiaye. Stable conformal prediction sets. In International Conference on Machine Learning , pages 16462--16479. PMLR, 2022

2022

[26] [26]

Inductive Conformal Prediction: Theory and Application to Neural Networks

Harris Papadopoulos. Inductive Conformal Prediction: Theory and Application to Neural Networks . INTECH Open Access Publisher Rijeka, 2008

2008

[27] [27]

Approximate full conformal prediction in an RKHS , January 2026

Davidson Lova Razafindrakoto, Alain Celisse, and J \'e r \^o me Lacaille. Approximate full conformal prediction in an RKHS , January 2026

2026

[28] [28]

Saleh and A

Resve A. Saleh and A. K. Saleh. Statistical properties of the log-cosh loss function used in machine learning. arXiv preprint arXiv:2208.04564, 2022

work page arXiv 2022

[29] [29]

Learning theory estimates via integral operators and their approximations

Steve Smale and Ding-Xuan Zhou. Learning theory estimates via integral operators and their approximations. Constructive approximation, 26 0 (2): 0 153--172, 2007

2007

[30] [30]

Multi-task regression using minimal penalties

Matthieu Solnon, Sylvain Arlot, and Francis Bach. Multi-task regression using minimal penalties. The Journal of Machine Learning Research, 13 0 (1): 0 2773--2812, 2012

2012

[31] [31]

Hush, and Clint Scovel

Ingo Steinwart, Don R. Hush, and Clint Scovel. Optimal Rates for Regularized Least Squares Regression . In COLT , pages 79--93, 2009

2009

[32] [32]

Algorithmic Learning in a Random World

Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic Learning in a Random World . Springer Science & Business Media, 2005

2005

[33] [33]

Algorithmic Learning in a Random World

Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic Learning in a Random World. 2 edition, December 2022. ISBN 978-3-031-06648-1. doi:10.10007/978-3-031-06649-8

work page doi:10.10007/978-3-031-06649-8 2022

[34] [34]

A survey on multi-task learning

Yu Zhang and Qiang Yang. A survey on multi-task learning. IEEE transactions on knowledge and data engineering, 34 0 (12): 0 5586--5609, 2021

2021

[35] [35]

Gradient descent algorithms for quantile regression with smooth approximation

Songfeng Zheng. Gradient descent algorithms for quantile regression with smooth approximation. International Journal of Machine Learning and Cybernetics, 2: 0 191--207, 2011

2011