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arxiv: 2605.17189 · v1 · pith:RJ2Y5242new · submitted 2026-05-16 · 📊 stat.ML · cs.IT· cs.LG· math.IT· math.ST· stat.TH

Sample efficient inductive matrix completion with noise and inexact side information

Yuepeng Yang , Cong Ma This is my paper

Pith reviewed 2026-05-20 14:00 UTC · model grok-4.3

classification 📊 stat.ML cs.ITcs.LGmath.ITmath.STstat.TH
keywords inductive matrix completionsample complexityside informationnonconvex optimizationprojected gradient descentlow-rank recoverynoisy observationsinexact side information
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The pith

Noisy inductive matrix completion achieves linear convergence with samples scaling only to side information dimension.

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

The paper seeks to resolve a split in prior work on inductive matrix completion: noiseless methods exploit side information for lower sample needs, while noisy methods revert to higher sample counts tied to the full matrix size. It introduces a nonconvex projected gradient descent method with spectral initialization and proves that a regularity condition on the loss function holds once samples reach a threshold set by the side information dimensions. This condition directly produces linear convergence of the iterates and estimation error governed solely by the effective dimension rather than the ambient matrix size. The same reduced sample complexity and order-optimal error scaling with inexactness are shown to persist when side information is only approximate.

Core claim

The authors establish a regularity condition for the inductive matrix completion loss that is satisfied at the reduced sample complexity determined by the side information dimension a. This condition yields linear convergence of the projected gradient descent algorithm and an estimation error that depends only on the effective problem size. The analysis extends to inexact side information while preserving the reduced sample complexity and delivering order-optimal error relative to the inexactness level.

What carries the argument

The regularity condition on the IMC loss function (analogous to restricted strong convexity) that holds at sample counts scaling with side information dimension a instead of ambient dimension n.

If this is right

  • The algorithm converges linearly to the underlying low-rank matrix at the reduced sample complexity.
  • Estimation error is controlled solely by the effective dimension induced by the side information.
  • Reduced sample complexity continues to hold when side information is inexact.
  • Estimation error scales in an order-optimal way with the degree of inexactness in the side information.

Where Pith is reading between the lines

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

  • The regularity condition may extend to other low-rank recovery tasks that incorporate auxiliary feature information.
  • Recommendation systems could adopt the method to recover preferences from sparse noisy ratings augmented by user or item attributes.
  • The scaling could be verified by running the algorithm on matrices with controlled growth in ambient versus effective dimension.

Load-bearing premise

The loss function for inductive matrix completion satisfies a regularity condition once the number of samples reaches the level set by the side information dimension.

What would settle it

A simulation or calculation in which the number of samples required for linear convergence or the final estimation error scales with the ambient dimension n rather than the side information dimension a would disprove the central claim.

Figures

Figures reproduced from arXiv: 2605.17189 by Cong Ma, Yuepeng Yang.

Figure 1
Figure 1. Figure 1: Exact recovery success rates for MC and IMC in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative error ∥Lb−L⋆∥F/∥L⋆∥F in log scale for MC and IMC with noisy observations and exact side information, with σ = 0.001. Results are averaged over 100 trials for p ∈ [0.01, 0.05]. 0.01 0.05 0.10 0.15 0.20 Sampling probability p 0.1 1 0.02 0.05 0.2 0.5 Relative error Noisy Obs ( = 0.001) + Inexact Side Info MC IMC (exact SI) IMC ( = 0.1) IMC ( = 0.2) [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative error ∥Lb − L⋆∥F/∥L⋆∥F for IMC with noisy observations (σ = 0.001) and inexact side information as δ varies from 0.02 to 0.20. Results are averaged over 100 trials. 10 2 10 1 10 0 10 1 Regularization parameter 0.1 0.02 0.05 0.2 Relative error Relative Error vs ( = 0.05, noiseless) p = 0.05 p = 0.1 p = 0.2 Min Error [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Test RMSE of IMC and non-inductive MC on MovieLens 100K as the training sample size [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Norms of IMC iterates XA and Y B across one trial of IMC objective optimization. The norm ∥XA∥2,∞ is in red, and the norm ∥Y B∥2,∞ is in purple. A reference line in blue at the value 2 pµ0r n ∥Z0∥ is shown for comparison. In this experiment, we take matrix parameters n1 = n2 = n = 1000, a1 = a2 = a = 50, r = 10 and sampling rate p = 0.01. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
read the original abstract

Low-rank matrix completion is a widely studied problem with many variants. Inductive matrix completion (IMC) incorporates row and column side information to significantly narrow the search space. Prior work falls into two regimes: methods that exploit this structure to achieve reduced sample complexity but only in noiseless settings, and methods that handle noise but require sample complexity matching the ambient matrix dimension, forfeiting the sample efficiency that side information should provide. In this paper, we close this gap by studying noisy IMC with a nonconvex projected gradient descent algorithm with spectral initialization. Our main technical contribution is establishing a regularity condition for the IMC loss function that holds at the reduced sample complexity determined by the effective problem size, scaling with the side information dimension a rather than the ambient dimension n. This directly yields linear convergence and an estimation error that both depend only on the effective problem size rather than the ambient matrix dimension. We further extend our analysis to the inexact side information setting, demonstrating that the reduced sample complexity is maintained and the estimation error is order-optimal with respect to the inexactness of the side information. Extensive simulations and real-world experiments on the MovieLens dataset validate our theoretical findings.

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

2 major / 2 minor

Summary. The paper studies noisy inductive matrix completion (IMC) incorporating row/column side information, using nonconvex projected gradient descent with spectral initialization. The central claim is that a regularity condition (restricted strong convexity) on the IMC loss holds with high probability once the number of samples reaches a threshold governed by the side-information dimension a (rather than ambient n), directly implying linear convergence of the algorithm and an estimation error that depends only on the effective dimension. The analysis is extended to inexact side information while preserving the reduced sample complexity and achieving order-optimal error with respect to the inexactness level. Theoretical results are supported by simulations and experiments on the MovieLens dataset.

Significance. If the regularity condition and initialization analysis hold without reintroducing ambient-dimension factors, the result would meaningfully close the gap between noiseless sample-efficient IMC and noisy methods that previously forfeited the side-information benefit. The extension to inexact side information with order-optimal guarantees is a practical strength, and the nonconvex analysis with explicit linear convergence adds to the literature on scalable matrix completion.

major comments (2)
  1. [Abstract and spectral initialization analysis] Abstract and the main regularity theorem: the claim that the regularity condition holds at sample complexity scaling only with a is load-bearing for both the linear convergence and the reduced-sample error bound. The spectral initialization step must be shown to land inside the basin whose radius is controlled by the regularity parameters; standard matrix-Bernstein or covering arguments for the initializer can introduce n-dependent factors when the side-information matrices map from dimension a to ambient n, unless explicit boundedness or incoherence assumptions on the features are stated and used.
  2. [Main convergence theorem] Theorem establishing linear convergence: the proof sketch must verify that the regularity parameters (e.g., restricted strong convexity constant and smoothness) remain independent of n once the sample threshold set by a is met; any hidden dependence on the ambient dimension through the initialization or the projection step would invalidate the central sample-complexity reduction.
minor comments (2)
  1. [Inexact side-information extension] The abstract states that estimation error is order-optimal with respect to inexactness; the precise dependence (e.g., additive term linear in the inexactness level) should be stated explicitly in the corresponding theorem statement.
  2. [Experiments] Experimental section: report the number of independent trials and error bars (or standard deviations) for the synthetic and MovieLens results to allow readers to assess variability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points about the initialization analysis and the independence of regularity parameters from the ambient dimension. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and spectral initialization analysis] Abstract and the main regularity theorem: the claim that the regularity condition holds at sample complexity scaling only with a is load-bearing for both the linear convergence and the reduced-sample error bound. The spectral initialization step must be shown to land inside the basin whose radius is controlled by the regularity parameters; standard matrix-Bernstein or covering arguments for the initializer can introduce n-dependent factors when the side-information matrices map from dimension a to ambient n, unless explicit boundedness or incoherence assumptions on the features are stated and used.

    Authors: We agree that the spectral initialization requires explicit control to avoid reintroducing ambient-dimension factors. The manuscript assumes bounded row and column side-information features (a standard condition in the IMC literature), which ensures that the operator norm of the feature maps is controlled independently of n. Under this assumption, the matrix-Bernstein bound for the initializer yields an error that scales only with a and the sample size, placing the initializer inside the basin of attraction whose radius is set by the restricted strong convexity parameters. We will state the boundedness assumption explicitly in the problem formulation and expand the initialization analysis in the appendix with the full concentration argument. revision: yes

  2. Referee: [Main convergence theorem] Theorem establishing linear convergence: the proof sketch must verify that the regularity parameters (e.g., restricted strong convexity constant and smoothness) remain independent of n once the sample threshold set by a is met; any hidden dependence on the ambient dimension through the initialization or the projection step would invalidate the central sample-complexity reduction.

    Authors: The full proof in the appendix confirms that both the restricted strong convexity and smoothness constants depend only on the effective dimension a once the sample threshold is met. The projection step is performed in the feature space of dimension a, and the analysis uses the boundedness of the side information to bound the Lipschitz constant of the projected gradient without ambient factors. The linear convergence rate therefore inherits the same sample complexity. We will augment the main-text proof sketch to explicitly note the n-independence of these parameters and reference the corresponding appendix lemmas. revision: partial

Circularity Check

0 steps flagged

No circularity: regularity condition derived from first principles at reduced sample size

full rationale

The paper's central contribution is establishing a regularity condition (restricted strong convexity) for the IMC loss that holds once samples reach the threshold governed by side-information dimension a. This is presented as a new technical result that directly implies linear convergence and error bounds depending only on effective dimension. No quoted equations or self-citations reduce this claim to a fitted parameter, renamed input, or prior result by the same authors. The derivation chain remains self-contained against external benchmarks such as standard matrix concentration and nonconvex optimization analyses; the skeptic concern about spectral initialization is not supported by any exhibited reduction in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard low-rank matrix and side-information model assumptions plus the newly asserted regularity condition at reduced sample size; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption The underlying matrix is approximately low-rank and the side information matrices have effective dimension a.
    This is the standard modeling premise of inductive matrix completion invoked throughout the abstract.

pith-pipeline@v0.9.0 · 5741 in / 1326 out tokens · 51267 ms · 2026-05-20T14:00:36.085365+00:00 · methodology

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Reference graph

Works this paper leans on

196 extracted references · 196 canonical work pages · 8 internal anchors

  1. [1]

    2006 , publisher=

    Prediction, learning, and games , author=. 2006 , publisher=

    work page 2006

  2. [2]

    2014 , publisher=

    Markov decision processes: discrete stochastic dynamic programming , author=. 2014 , publisher=

    work page 2014

  3. [3]

    2002 , publisher=

    Learning with kernels: support vector machines, regularization, optimization, and beyond , author=. 2002 , publisher=

    work page 2002

  4. [4]

    1999 , publisher=

    The nature of statistical learning theory , author=. 1999 , publisher=

    work page 1999

  5. [5]

    2002 , publisher=

    A distribution-free theory of nonparametric regression , author=. 2002 , publisher=

    work page 2002

  6. [6]

    2012 , publisher=

    Convergence of stochastic processes , author=. 2012 , publisher=

    work page 2012

  7. [7]

    2012 , publisher=

    Density ratio estimation in machine learning , author=. 2012 , publisher=

    work page 2012

  8. [8]

    2000 , publisher=

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

    work page 2000

  9. [9]

    2012 , publisher=

    Machine learning in non-stationary environments: Introduction to covariate shift adaptation , author=. 2012 , publisher=

    work page 2012

  10. [10]

    2019 , publisher=

    High-dimensional statistics: A non-asymptotic viewpoint , author=. 2019 , publisher=

    work page 2019

  11. [11]

    2020 , publisher =

    Bandit algorithms , author =. 2020 , publisher =

    work page 2020

  12. [12]

    2018 , publisher =

    Reinforcement learning: An introduction , author =. 2018 , publisher =

    work page 2018

  13. [13]

    2011 , publisher =

    Mathematical theory of statistics: Statistical experiments and asymptotic decision theory , author =. 2011 , publisher =

    work page 2011

  14. [14]

    1998 , publisher =

    Online algorithms: The state of the art , author =. 1998 , publisher =

    work page 1998

  15. [15]

    2014 , publisher =

    Theory of approximation of functions of a real variable , author =. 2014 , publisher =

    work page 2014

  16. [16]

    Journal of machine learning research , volume=

    Nash Q-learning for general-sum stochastic games , author=. Journal of machine learning research , volume=

    work page

  17. [17]

    SIAM Journal on Optimization , volume=

    Excessive gap technique in nonsmooth convex minimization , author=. SIAM Journal on Optimization , volume=. 2005 , publisher=

    work page 2005

  18. [18]

    Science , volume=

    Superhuman AI for multiplayer poker , author=. Science , volume=. 2019 , publisher=

    work page 2019

  19. [19]

    Proceedings of the national academy of sciences , volume=

    Stochastic games , author=. Proceedings of the national academy of sciences , volume=. 1953 , publisher=

    work page 1953

  20. [20]

    Nature , volume=

    Grandmaster level in StarCraft II using multi-agent reinforcement learning , author=. Nature , volume=. 2019 , publisher=

    work page 2019

  21. [21]

    IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews) , volume=

    A comprehensive survey of multiagent reinforcement learning , author=. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews) , volume=. 2008 , publisher=

    work page 2008

  22. [22]

    Handbook of Reinforcement Learning and Control , pages=

    Multi-agent reinforcement learning: A selective overview of theories and algorithms , author=. Handbook of Reinforcement Learning and Control , pages=. 2021 , publisher=

    work page 2021

  23. [23]

    Journal of the ACM (JACM) , volume=

    Robust principal component analysis? , author=. Journal of the ACM (JACM) , volume=. 2011 , publisher=

    work page 2011

  24. [24]

    Bernoulli , volume=

    Noisy low-rank matrix completion with general sampling distribution , author=. Bernoulli , volume=. 2014 , publisher=

    work page 2014

  25. [25]

    Journal of the American Statistical Association , pages=

    Convex and nonconvex optimization are both minimax-optimal for noisy blind deconvolution under random designs , author=. Journal of the American Statistical Association , pages=. 2021 , publisher=

    work page 2021

  26. [26]

    Noisy matrix completion: Understanding statistical guarantees for convex relaxation via nonconvex optimization , year =

    Chen, Yuxin and Chi, Yuejie and Fan, Jianqing and Ma, Cong and Yan, Yuling , journal =. Noisy matrix completion: Understanding statistical guarantees for convex relaxation via nonconvex optimization , year =

    work page

  27. [27]

    Hanson--

    Chen, Xiaohui and Yang, Yun , journal=. Hanson--. 2021 , publisher=

    work page 2021

  28. [28]

    Annals of the Institute of Statistical Mathematics , volume=

    Direct importance estimation for covariate shift adaptation , author=. Annals of the Institute of Statistical Mathematics , volume=. 2008 , publisher=

    work page 2008

  29. [29]

    The Journal of Machine Learning Research , volume=

    A least-squares approach to direct importance estimation , author=. The Journal of Machine Learning Research , volume=. 2009 , publisher=

    work page 2009

  30. [30]

    Domain Adaptation for Medical Image Analysis: A Survey , year=

    Guan, Hao and Liu, Mingxia , journal=. Domain Adaptation for Medical Image Analysis: A Survey , year=

    work page

  31. [31]

    IEEE Transactions on knowledge and data engineering , volume=

    A survey on transfer learning , author=. IEEE Transactions on knowledge and data engineering , volume=. 2009 , publisher=

    work page 2009

  32. [32]

    Bartlett, Peter L and Bousquet, Olivier and Mendelson, Shahar , journal=. Local. 2005 , publisher=

    work page 2005

  33. [33]

    Proceedings of the National Academy of Sciences , volume=

    Inference and uncertainty quantification for noisy matrix completion , author=. Proceedings of the National Academy of Sciences , volume=. 2019 , publisher=

    work page 2019

  34. [34]

    Journal of statistical planning and inference , volume =

    Improving predictive inference under covariate shift by weighting the log-likelihood function , author =. Journal of statistical planning and inference , volume =. 2000 , publisher =

    work page 2000

  35. [35]

    Foundations and Trends

    An Introduction to Matrix Concentration Inequalities , doi =. Foundations and Trends. 2015 , volume =

    work page 2015

  36. [36]

    On some extensions of

    Minsker, Stanislav , journal=. On some extensions of. 2017 , publisher=

    work page 2017

  37. [37]

    The Annals of Statistics , volume=

    Randomized sketches for kernels: Fast and optimal nonparametric regression , author=. The Annals of Statistics , volume=. 2017 , publisher=

    work page 2017

  38. [38]

    The Annals of Statistics , volume=

    A new perspective on least squares under convex constraint , author=. The Annals of Statistics , volume=. 2014 , publisher=

    work page 2014

  39. [39]

    Sankhya A , volume=

    On concentration for (regularized) empirical risk minimization , author=. Sankhya A , volume=. 2017 , publisher=

    work page 2017

  40. [40]

    Random Structures & Algorithms , volume=

    A sharp concentration inequality with applications , author=. Random Structures & Algorithms , volume=. 2000 , publisher=

    work page 2000

  41. [41]

    The Annals of Probability , volume=

    Concentration around the mean for maxima of empirical processes , author=. The Annals of Probability , volume=. 2005 , publisher=

    work page 2005

  42. [42]

    IEEE Transactions on Information Theory , volume =

    Minimax rates of entropy estimation on large alphabets via best polynomial approximation , author =. IEEE Transactions on Information Theory , volume =. 2016 , publisher =

    work page 2016

  43. [43]

    Journal of the ACM , volume =

    Estimating the unseen: Improved estimators for entropy and other properties , author =. Journal of the ACM , volume =. 2017 , publisher =

    work page 2017

  44. [44]

    The Annals of Statistics , volume =

    Minimax estimation of linear functionals over nonconvex parameter spaces , author =. The Annals of Statistics , volume =. 2004 , publisher =

    work page 2004

  45. [45]

    On a problem of adaptive estimation in

    Lepskii, OV , journal =. On a problem of adaptive estimation in. 1991 , publisher =

    work page 1991

  46. [46]

    The Annals of Statistics , volume =

    A constrained risk inequality with applications to nonparametric functional estimation , author =. The Annals of Statistics , volume =. 1996 , publisher =

    work page 1996

  47. [47]

    Geometrizing rates of convergence,

    Donoho, David L and Liu, Richard C , journal =. Geometrizing rates of convergence,. 1991 , publisher =

    work page 1991

  48. [48]

    Statistical Science , volume =

    Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data , author =. Statistical Science , volume =. 2007 , publisher =

    work page 2007

  49. [49]

    On estimation of the

    Lepski, Oleg and Nemirovski, Arkady and Spokoiny, Vladimir , journal =. On estimation of the. 1999 , publisher =

    work page 1999

  50. [50]

    SIAM Journal on Computing , volume =

    The nonstochastic multiarmed bandit problem , author=. SIAM Journal on Computing , volume =. 2002 , publisher =

    work page 2002

  51. [51]

    Econometrica , pages =

    On the role of the propensity score in efficient semiparametric estimation of average treatment effects , author =. Econometrica , pages =. 1998 , publisher =

    work page 1998

  52. [52]

    Testing composite hypotheses,

    Cai, T Tony and Low, Mark G , journal =. Testing composite hypotheses,. 2011 , publisher =

    work page 2011

  53. [53]

    The Journal of Machine Learning Research , volume =

    Counterfactual reasoning and learning systems: The example of computational advertising , author =. The Journal of Machine Learning Research , volume =. 2013 , publisher =

    work page 2013

  54. [54]

    Journal of Computational and Graphical Statistics , volume =

    Truncated importance sampling , author =. Journal of Computational and Graphical Statistics , volume =. 2008 , publisher =

    work page 2008

  55. [55]

    Dynamic Games and Applications , pages=

    Provably efficient reinforcement learning in decentralized general-sum markov games , author=. Dynamic Games and Applications , pages=. 2022 , publisher=

    work page 2022

  56. [56]

    International Conference on Machine Learning , pages=

    A sharp analysis of model-based reinforcement learning with self-play , author=. International Conference on Machine Learning , pages=. 2021 , organization=

    work page 2021

  57. [57]

    Advances in Neural Information Processing Systems , volume=

    Model-based multi-agent rl in zero-sum markov games with near-optimal sample complexity , author=. Advances in Neural Information Processing Systems , volume=

    work page

  58. [58]

    Dualizing

    Polyanskiy, Yury and Wu, Yihong , journal =. Dualizing

    work page

  59. [59]

    arXiv preprint arXiv:2007.09055 , year =

    Hyperparameter selection for offline reinforcement learning , author =. arXiv preprint arXiv:2007.09055 , year =

    work page arXiv 2007

  60. [60]

    IEEE Transactions on Information Theory , volume =

    Minimax estimation of functionals of discrete distributions , author =. IEEE Transactions on Information Theory , volume =. 2015 , publisher =

    work page 2015

  61. [61]

    IEEE Journal on Selected Areas in Information Theory , volume =

    Minimax rate-optimal estimation of divergences between discrete distributions , author =. IEEE Journal on Selected Areas in Information Theory , volume =. 2020 , publisher =

    work page 2020

  62. [62]

    Some problems on nonparametric estimation in

    Ibragimov, Ildar A and Nemirovskii, Arkadi S and. Some problems on nonparametric estimation in. Theory of Probability & Its Applications , volume =. 1987 , publisher =

    work page 1987

  63. [63]

    Journal of the American Statistical Association , volume =

    A generalization of sampling without replacement from a finite universe , author =. Journal of the American Statistical Association , volume =. 1952 , publisher =

    work page 1952

  64. [64]

    ACM Transactions on Interactive Intelligent Systems , volume =

    The movielens datasets: History and context , author =. ACM Transactions on Interactive Intelligent Systems , volume =. 2015 , publisher =

    work page 2015

  65. [65]

    Econometrica , volume =

    Efficient estimation of average treatment effects using the estimated propensity score , author =. Econometrica , volume =. 2003 , publisher =

    work page 2003

  66. [66]

    Synthesis Lectures on Artificial Intelligence and Machine Learning , volume =

    Algorithms for reinforcement learning , author =. Synthesis Lectures on Artificial Intelligence and Machine Learning , volume =. 2010 , publisher =

    work page 2010

  67. [67]

    Statistical Science , volume =

    Doubly robust policy evaluation and optimization , author =. Statistical Science , volume =. 2014 , publisher =

    work page 2014

  68. [68]

    Statistical methods in medical research , volume=

    Review of inverse probability weighting for dealing with missing data , author=. Statistical methods in medical research , volume=. 2013 , publisher=

    work page 2013

  69. [69]

    Journal of Machine Learning Research , year =

    Alberto Maria Metelli and Matteo Papini and Nico Montali and Marcello Restelli , title =. Journal of Machine Learning Research , year =

    work page

  70. [70]

    Minimax Estimation of Discrete Distributions Under

    Han, Yanjun and Jiao, Jiantao and Weissman, Tsachy , journal =. Minimax Estimation of Discrete Distributions Under. 2015 , publisher =

    work page 2015

  71. [71]

    The Annals of Statistics , volume =

    Chebyshev polynomials, moment matching, and optimal estimation of the unseen , author =. The Annals of Statistics , volume =. 2019 , publisher =

    work page 2019

  72. [72]

    The Annals of Statistics , volume =

    Marginal singularity and the benefits of labels in covariate-shift , author =. The Annals of Statistics , volume =. 2021 , publisher =

    work page 2021

  73. [73]

    The Annals of Statistics , volume =

    Transfer learning for nonparametric classification: Minimax rate and adaptive classifier , author =. The Annals of Statistics , volume =. 2021 , publisher =

    work page 2021

  74. [74]

    IEEE transactions on medical imaging , volume=

    Convolutional neural networks for medical image analysis: Full training or fine tuning? , author=. IEEE transactions on medical imaging , volume=. 2016 , publisher=

    work page 2016

  75. [75]

    Journal of the American Statistical Association , volume=

    Matrix completion methods for causal panel data models , author=. Journal of the American Statistical Association , volume=. 2021 , publisher=

    work page 2021

  76. [76]

    ACM Transactions on Sensor Networks (TOSN) , volume=

    Semidefinite programming based algorithms for sensor network localization , author=. ACM Transactions on Sensor Networks (TOSN) , volume=. 2006 , publisher=

    work page 2006

  77. [77]

    Annals of mathematics , pages=

    Non-cooperative games , author=. Annals of mathematics , pages=. 1951 , publisher=

    work page 1951

  78. [78]

    Advances in Neural Information Processing Systems , volume=

    Optimization, learning, and games with predictable sequences , author=. Advances in Neural Information Processing Systems , volume=

    work page

  79. [79]

    Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms , pages=

    Near-optimal no-regret algorithms for zero-sum games , author=. Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms , pages=. 2011 , organization=

    work page 2011

  80. [80]

    Machine learning proceedings 1994 , pages=

    Markov games as a framework for multi-agent reinforcement learning , author=. Machine learning proceedings 1994 , pages=. 1994 , publisher=

    work page 1994

Showing first 80 references.