Pith. sign in

REVIEW 2 major objections 5 minor 32 references

On Convergence Analysis of Network-GIANT: An approximate Hessian-based fully distributed optimization algorithm

T0 review · 2 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves Network-GIANT, a distributed approximate-Newton algorithm, converges linearly with a computable rate, and locally at rate ≈1−η when the Hessian approximation error is small.

desk verdict The global linear rate for Network-GIANT is a real, usable result; the advertised fast local rate is conditional on an unverified Hessian-approximation bound and an informal asymptotic step. read the letter →

arxiv 2602.14830 v2 pith:H3YVUUWR submitted 2026-02-16 math.OC cs.SYeess.SPeess.SY

Souvik Das , Luca Schenato , Subhrakanti Dey This is my paper · ORCID
classification math.OCcs.SYeess.SPeess.SY MSC 90C2590C53
keywords Network-GIANTdistributedoptimizationapproximateNewtonmethodlinearconvergenceHessianapproximationgradienttrackingconsensusalgorithmstronglyconvex
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

Network-GIANT is a fully distributed optimization algorithm in which each node takes a damped Newton step using its own local Hessian and a gradient-tracking vector, with consensus averaging for both parameters and gradients. The paper's goal is to explain analytically why this approximate-Newton method converges faster than first-order gradient tracking. The authors prove a global linear convergence bound: the vector of consensus error, gradient-tracking error, and optimality gap is driven by a 3×3 matrix whose spectral radius gives an explicit rate. They also derive a mixed linear-quadratic inequality for the optimality gap, which yields an asymptotic local rate of 1−η(1−γ/μ), nearly 1−η when the Hessian approximation error γ is well below the strong-convexity constant μ. This local rate is network-independent and much faster than the 1−ημ typical of gradient-tracking methods.

What carries the argument

The main object is the 3×3 nonnegative matrix G(η) that couples the consensus error, gradient-tracking error, and optimality gap; its spectral radius provides the global linear rate, and Perron-Frobenius theory guarantees the largest eigenvalue is real. The second central object is the harmonic-mean Hessian H_app, the matrix inverse of the average of local Hessian inverses, which Network-GIANT implicitly uses in place of the true global Hessian H_tr. The approximation error γ=||H_tr−H_app|| enters the local rate through the coefficient 1−η(1−γ/μ).

What would settle it

Run Network-GIANT on a logistic regression problem engineered with strongly heterogeneous local datasets so that the harmonic-mean Hessian differs from the global Hessian by more than μ, and check whether the ratio r_k=||x_{k+1}−x*||/||x_k−x*|| still approaches 1−η or degrades; equivalently, directly compute γ for the real problem and test whether γ<μ holds on every iterate.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the convergence of Network-GIANT decomposes into three coupled error recursions that can be bounded by a single linear matrix inequality. Theorems 1 and 2 show the optimality gap decays at a rate given by the spectral radius of a 3×3 matrix G(η) built from L, μ, and the consensus spectral norm σ, provided the step size is below an explicit threshold. The sharper Theorem 3 shows that when the local Hessians are Lipschitz and the harmonic-mean Hessian approximates the global arithmetic-mean Hessian with uniform error γ<μ, the optimality gap satisfies a mixed linear-quadratic bound with linear coefficient 1−η(1−γ/μ). For small step sizes, the consensus and

Load-bearing premise

The fast local rate rests on the assumption that the difference between the true global Hessian and its harmonic-mean approximation is uniformly smaller than the strong-convexity constant μ; the paper gives no way to compute or verify this difference, and it is never measured in the experiments.

Editorial extensions

If this is right

  • Network-GIANT converges linearly on any connected undirected graph for step sizes below the explicit bound in Theorem 2, with a rate computable from L, μ, and σ.
  • When the Hessian approximation error γ is smaller than μ and the step size is small, the optimality gap eventually decays at rate about 1−η, independent of the graph spectrum.
  • The mixed linear-quadratic bound predicts a fast transient phase (quadratic in the gap) followed by a linear tail, matching the observed convergence plots.
  • The same proof technique applies to any distributed approximate-Newton algorithm whose Hessian approximation error is uniformly small, not just to Network-GIANT.
  • The explicit global rate allows quantifying the trade-off between network connectivity (σ) and step size, and shows that a larger η can yield a faster rate if it stays below the bound.

Reading between the lines

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

  • If the 1−η local rate holds whenever γ<μ, then the practical bottleneck for highly connected graphs is the consensus and gradient-tracking dynamics, not the Newton direction; acceleration should therefore target those error channels.
  • Since the paper offers no computable bound for γ or a way to measure it, a natural next step is to estimate γ empirically on the actual data and verify that γ<μ, turning the asymptotic rate into a testable prediction.
  • The result suggests a design principle: cheap harmonic-mean Hessian consensus can capture most of Newton's benefit even without exact Hessian averaging, as long as data heterogeneity keeps γ small.
  • The observation that sparse graphs (σ close to 1) do not exhibit the 1−η asymptote within the iteration budget suggests that the number of iterations needed to reach the local regime scales with 1/(1−σ), so the local rate is only relevant after consensus has nearly settled.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper analyzes Network-GIANT, a fully distributed approximate-Newton method that combines consensus updates with gradient tracking and local Hessian inverses. It provides two main theoretical contributions: (i) a global linear convergence rate for the optimality gap, given explicitly as the spectral radius of a 3x3 matrix G(η) depending on L, μ, σ, and η, together with an explicit sufficient step-size bound η̄ ensuring ρ(G(η))<1 (Theorems 1–2); and (ii) a mixed linear-quadratic inequality for the optimality gap under a uniform Hessian-approximation error bound γ (Theorem 3, Eq. (14)), from which the paper claims an asymptotic local rate of 1−η(1−γ/μ) ≈ 1−η. Numerical experiments on logistic regression over expander and Erdős–Rényi graphs illustrate the convergence behavior and compare with gradient-tracking baselines.

Significance. If the local-rate claim were fully established, the paper would provide a formal explanation for the empirically observed speedup of Network-GIANT over first-order distributed methods and would give the first explicit rate formulas for this algorithm. The global linear convergence result in Theorems 1–2 is explicit and parameter-free, and its proof appears structurally sound; this is a useful contribution. Theorem 3's mixed linear-quadratic inequality is also a valuable conditional bound. However, the advertised asymptotic local rate is not rigorously derived from (14): the paper does not prove that the consensus and gradient-tracking error terms are asymptotically negligible relative to the optimality gap, and the key assumption γ<μ is never verified numerically or analytically. Thus the central explanatory claim remains conditional in a way the abstract and conclusions do not fully convey.

major comments (2)
  1. [Section IV-B, Eq. (14) and following paragraph] The claimed local asymptotic rate 1−η(1−γ/μ) does not follow from inequality (14) alone. The right-hand side contains additive consensus/tracking terms and a bilinear term in ||x^k−1\bar x^k||·||\bar x^k−x*||. To conclude that the optimality gap contracts at the stated rate, one must prove these terms are o(||\bar x^k−x*||). The paper's justification—that for small η the matrix in (12) is diagonally dominant and hence consensus/tracking errors decay faster—is not sufficient: for η>0, G(η) is a positive matrix, so the coupled upper-bound recursion gives the same Perron-root asymptotic rate to all components; diagonal dominance does not imply permanent rate separation. A rigorous small-gain or comparison argument for the actual trajectories is needed, or the 'rigorous proof' wording must be downgraded.
  2. [Assumption 3, Remark 7, and Section V (Eq. (15))] The advertised local rate is conditional on γ<μ, but no analytical bound on γ is provided and the experiments never measure γ. Consequently the numerical observation r_k≈1−η cannot be positively linked to Theorem 3; it is only consistent with the theorem under an unverified hypothesis. Please either estimate γ for the logistic-regression problems or explicitly state that the numerical section does not verify the key assumption of the local-rate result.
minor comments (5)
  1. [Eq. (4)] The sums in the algorithm definition should be over j=1,...,N and should involve x^k_j and s^k_j, not x^k_i and s^k_i. The compact matrix form (5) is correct, but (4) as written does not match the analyzed consensus update.
  2. [Assumption 1] The identity matrix in µI ≤ ∇²f_i(x) ≤ LI should be n×n, not N×N.
  3. [Appendix II] The displayed formula \barη = (1−σ)µ/L / [2(2−σ)(1+(L/µ)^2)] is inconsistent with Theorem 2's \barη = (1−σ)^2/(2(2−σ)(κ+κ^3)). The derivation in the appendix supports the main-text formula; the appendix appears to have dropped a square on (1−σ).
  4. [Eq. (15)] The rate r_k is defined with x^{k+1} and x^k, but from context it should be the ratio for the average iterate \bar x^k. Please use \bar x^k throughout to avoid ambiguity.
  5. [Section V] The acronym for the accelerated algorithm is ACC-NGD-SC in the text but Acc-DNGD-SC in the figure captions; please harmonize. Also, L and μ are not reported for the experiments, so the reader cannot check whether the chosen step sizes satisfy the sufficient conditions of Theorem 2.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the advertised local rate is a conditional consequence of Assumption 3, not a disguised input; the only self-citation ([21]) is non-load-bearing.

full rationale

The derivation chain in Sections IV and Appendices I–III is self-contained. Theorem 1 is proved from a damped-Newton lemma proved in the paper (Lemma 2) and from standard gradient-tracking inequalities quoted from [6]; Theorem 2 is a spectral-radius/continuity argument. Theorem 3 starts from the exact algebraic identity x̃^{k+1}−x̃* = (1−η)(x̃^k−x̃*) + η(H_app)^{-1}(H_app−δ)(x̃^k−x̃*) + S1 and bounds ∥H_app−δ∥ using the Hessian Lipschitz Assumption 2 and the approximation Assumption 3. The coefficient 1−η(1−γ/μ) in (14) is therefore the base linear coefficient of an upper bound, not the target rate assumed in advance. The paper explicitly records in Remark 7 that "it is not generally possible to obtain useful explicit analytical bounds on ∥H_tr(x)−H_app(x)∥," and Section V does not measure γ; hence the condition γ<μ is unverified for the experiments. That is a significant conditionality/validation gap, but it is not circular reasoning: the theorem is honestly conditional. The transition from (14) to the clean asymptotic rate also relies on the stated qualitative premise that consensus and gradient-tracking errors decay faster than the optimality gap; this is argued from diagonal dominance of G(η) for small η and is not imported from the result being claimed. The only self-citation, [21], supplies the algorithm and earlier empirical comparisons; no theorem from [21] is used as a premise in the new proofs. Consequently, no step reduces to its own input, and the paper should not receive a circularity score above 2 (reflecting only the non-load-bearing self-citation, not circular derivation).

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The theoretical results introduce no fitted parameters or new physical/mathematical entities. The main unverified input is Assumption 3 (the uniform γ bound), which is a domain assumption rather than a fitted constant. The experimental comparisons do involve hand-tuned step sizes and regularization, but these do not determine the stated convergence theorems.

free parameters (3)
  • step-size η for Network-GIANT experiments = 0.05–0.08 depending on graph (Tables II and III)
    Hand-picked per graph for the numerical illustrations; the theory treats η as a free variable with a sufficient upper bound, but the experimental validation relies on chosen values.
  • step-size η for baseline algorithms = GradTrack 0.08–0.18; ACC-NGD-SC 0.09–0.3
    Baseline step sizes were tuned on a grid [0.035, 0.3] and the best values reported; this is standard practice but introduces hand-tuned constants in the comparison.
  • regularizer λ = 0.01 (reduced CovType), 0.001 (full CovType)
    Regularization strength chosen by hand; it indirectly sets μ for the loss and affects the Hessian-approximation error γ, but is not part of the theoretical derivation.
assumptions (5)
  • domain assumption Assumption 1: each local f_i is μ-strongly convex and L-smooth
    Standard in second-order optimization; the entire analysis assumes this for all local functions.
  • domain assumption Assumption 2: local Hessians are L̄-Lipschitz
    Used in Theorem 3 to bound the quadratic term involving ||x_k − x*||².
  • domain assumption Assumption 3: ||H_tr(x) − H_app(x)|| ≤ γ with γ < μ for all x
    Central to the advertised local rate 1−η(1−γ/μ). The paper itself states that no useful explicit analytical bound for γ is generally available (Remark 7), and it is not verified experimentally.
  • domain assumption Lemma 1 from [6]: gradient difference and tracking error bounds
    Imported from prior gradient-tracking literature and used as a black box in the proof of Theorem 1 (Appendix I).
  • standard math Perron-Frobenius theory and continuity of eigenvalues
    Used in Theorem 2 to relate ρ(G(η)) to the largest positive eigenvalue and to analyze crossing of 1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Convergence Analysis of Network-GIANT: An approximate Hessian-based fully distributed optimization algorithm." pith.science (2026). https://pith.science/paper/H3YVUUWR

@misc{pith2026260214830,
  author       = {Pith},
  title        = {Pith review of: On Convergence Analysis of Network-GIANT: An approximate Hessian-based fully distributed optimization algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H3YVUUWR}},
  note         = {Machine review of arXiv:2602.14830}
}
abstract

This paper presents a detailed convergence and performance analysis of a recently developed approximate Newton-type fully distributed optimization method for \(L\)-smooth, \(\mu\)-strongly convex local loss functions, called Network-GIANT (inspired by the Federated learning algorithm GIANT possessing mixed linear-quadratic convergence properties). Network-GIANT has been empirically seen to achieve faster linear convergence properties compared to its gradient-based counterparts, and several other existing second order distributed algorithms, while having the same communication complexity (per iteration) as its first order distributed counterparts. We first explicitly characterize a \emph{global linear convergence rate} for Network-GIANT, which can be computed as the spectral radius of a $3 \times 3$ matrix dependent on $L$, $\mu$, and the spectral norm ($\sigma$) of the consensus matrix of the underlying undirected graph. We provide an explicit bound on the step size parameter $\eta$, below which this spectral radius is guaranteed to be less than $1$. Furthermore, we derive a mixed linear-quadratic inequality based upper bound for the optimality gap norm, and provide a rigorous proof of a local asymptotic convergence rate of \(1 - \eta \big(1 - \frac{\gamma}{\mu}\big)\) given the Hessian approximation error $\gamma < \mu$, which formally explains the faster convergence rate of Network-GIANT. Numerical experiments are carried out with a reduced CovType dataset for binary logistic regression over a variety of graphs, including heterogeneous data distributions, to illustrate the above theoretical results.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 1 linked inside Pith

  1. [21]

    Network-GIANT: Fully distributed Newton-type optimization via harmonic hessian con- sensus,

    A. Maritan, G. Sharma, L. Schenato, and S. Dey, “Network-GIANT: Fully distributed Newton-type optimization via harmonic hessian con- sensus,” in2023 IEEE Globecom Workshops (GC Wkshps), 2023, pp. 902–907

    2023

  2. [6]

    Harnessing smoothness to accelerate distributed optimization,

    G. Qu and N. Li, “Harnessing smoothness to accelerate distributed optimization,”IEEE Transactions on Control of Network Systems, vol. 5, no. 3, pp. 1245–1260, 2018

    2018

  3. [24]

    Second-order stochastic optimization for machine learning in linear time,

    N. Agarwal, B. Bullins, and E. Hazan, “Second-order stochastic optimization for machine learning in linear time,”Journal of Machine Learning Research, vol. 18, no. 116, pp. 1–40, 2017. [Online]. Available: http://jmlr.org/papers/v18/16-491.html

    2017

  4. [1]

    Decentralized federated learning: A survey and perspective,

    L. Yuan, Z. Wang, L. Sun, P. S. Yu, and C. G. Brinton, “Decentralized federated learning: A survey and perspective,”IEEE Internet of Things Journal, vol. 11, no. 21, pp. 34 617–34 638, 2024

    2024

  5. [2]

    A survey on secure decentralized optimization and learning,

    C. Liu, N. Bastianello, W. Huo, Y . Shi, and K. H. Johansson, “A survey on secure decentralized optimization and learning,”arXiv preprint arXiv:2408.08628, 2024

    arXiv 2024

  6. [3]

    A survey of distributed optimization,

    T. Yang, X. Yi, J. Wu, Y . Yuan, D. Wu, Z. Meng, Y . Hong, H. Wang, Z. Lin, and K. H. Johansson, “A survey of distributed optimization,”Annual Reviews in Control, vol. 47, pp. 278–305, 2019. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1367578819300082

    2019

  7. [4]

    Distributed optimization for control,

    A. Nedi ´c and J. Liu, “Distributed optimization for control,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 1, no. 1, pp. 77–103, 2018

    2018

  8. [5]

    Distributed gradient methods for convex machine learning problems in networks: Distributed Optimization,

    A. Nedic, “Distributed gradient methods for convex machine learning problems in networks: Distributed Optimization,”IEEE Signal Process- ing Magazine, vol. 37, no. 3, pp. 92–101, 2020

    2020

Show all 32 references
  1. [7]

    Linear convergence in optimization over directed graphs with row-stochastic matrices,

    C. Xi, V . S. Mai, R. Xin, E. H. Abed, and U. A. Khan, “Linear convergence in optimization over directed graphs with row-stochastic matrices,”IEEE Transactions on Automatic Control, vol. 63, no. 10, pp. 3558–3565, 2018

    2018

  2. [8]

    DONE: distributed approximate Newton-type method for federated edge learning,

    C. T. Dinh, N. H. Tran, T. D. Nguyen, W. Bao, A. R. Balef, B. B. Zhou, and A. Y . Zomaya, “DONE: distributed approximate Newton-type method for federated edge learning,”IEEE Transactions on Parallel and Distributed Systems, vol. 33, no. 11, pp. 2648–2660, 2022

    2022

  3. [9]

    On analog distributed approximate Newton with determinantal averaging,

    G. Sharma and S. Dey, “On analog distributed approximate Newton with determinantal averaging,” in2022 IEEE 33rd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC). IEEE, 2022, pp. 1–7

    2022

  4. [10]

    FedNL: Making Newton-type methods applicable to federated learning,

    M. Safaryan, R. Islamov, X. Qian, and P. Richt ´arik, “FedNL: Making Newton-type methods applicable to federated learning,” in39th Inter- national Conference on Machine Learning, ICML 2022, 2022

    2022

  5. [11]

    FAGH: Accelerating federated learning with approximated global hessian,

    M. Sen, A. Qin, and K. M. C, “FAGH: Accelerating federated learning with approximated global hessian,”arXiv e-prints, pp. arXiv–2403, 2024

    2024

  6. [12]

    Network Newton distributed optimization methods,

    A. Mokhtari, Q. Ling, and A. Ribeiro, “Network Newton distributed optimization methods,”IEEE Transactions on Signal Processing, vol. 65, no. 1, pp. 146–161, 2016

    2016

  7. [13]

    Newton-like method with diagonal correction for distributed optimization,

    D. Bajovic, D. Jakovetic, N. Krejic, and N. K. Jerinkic, “Newton-like method with diagonal correction for distributed optimization,”SIAM Journal on Optimization, vol. 27, no. 2, pp. 1171–1203, 2017

    2017

  8. [14]

    Multiagent Newton–Raphson optimization over lossy networks,

    N. Bof, R. Carli, G. Notarstefano, L. Schenato, and D. Varagnolo, “Multiagent Newton–Raphson optimization over lossy networks,”IEEE Transactions on Automatic Control, vol. 64, no. 7, pp. 2983–2990, 2018

    2018

  9. [15]

    A Newton tracking algorithm with exact linear convergence for decentralized consensus optimization,

    J. Zhang, Q. Ling, and A. M.-C. So, “A Newton tracking algorithm with exact linear convergence for decentralized consensus optimization,” IEEE Transactions on Signal and Information Processing over Networks, vol. 7, pp. 346–358, 2021

    2021

  10. [16]

    Accelerated distributed approximate Newton method,

    H. Ye, C. He, and X. Chang, “Accelerated distributed approximate Newton method,”IEEE Transactions on Neural Networks and Learning Systems, vol. 34, no. 11, pp. 8642–8653, 2022

    2022

  11. [17]

    Communication-efficient dis- tributed optimization using an approximate Newton-type method,

    O. Shamir, N. Srebro, and T. Zhang, “Communication-efficient dis- tributed optimization using an approximate Newton-type method,” in International conference on machine learning. PMLR, 2014, pp. 1000– 1008

    2014

  12. [18]

    Distributed second-order method with diffusion strategy,

    Z. Qu, X. Li, L. Li, and Y . Hong, “Distributed second-order method with diffusion strategy,” in2023 IEEE International Conference on Systems, Man, and Cybernetics (SMC). IEEE, 2023, pp. 2022–2027

    2023

  13. [19]

    Fully-distributed optimization with Network Exact Consensus-GIANT,

    A. Maritan, G. Sharma, S. Dey, and L. Schenato, “Fully-distributed optimization with Network Exact Consensus-GIANT,” in2024 IEEE 25th International Workshop on Signal Processing Advances in Wireless Communications (SPA WC). IEEE, 2024, pp. 436–440

    2024

  14. [20]

    Distributed adaptive Newton methods with global superlinear convergence,

    J. Zhang, K. You, and T. Bas ¸ar, “Distributed adaptive Newton methods with global superlinear convergence,”Au- tomatica, vol. 138, p. 110156, 2022. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0005109821006865 11

    2022

  15. [22]

    GIANT: Globally improved approximate Newton method for distributed optimization,

    S. Wang, F. Roosta, P. Xu, and M. W. Mahoney, “GIANT: Globally improved approximate Newton method for distributed optimization,” Advances in Neural Information Processing Systems, vol. 31, 2018

    2018

  16. [25]

    Boyd and L

    S. Boyd and L. Vandenberghe,Convex Optimization. Cambridge University Press, March 2004

    2004

  17. [26]

    Introduction to Optimization,

    B. Polyak, “Introduction to Optimization,”Translations Series in Math- ematics and Engineering. New York: Optimization Software Inc. Publi- cations Division, 1987

    1987

  18. [27]

    Communication-efficient distributed optimization in networks with gradient tracking and variance reduction,

    B. Li, S. Cen, Y . Chen, and Y . Chi, “Communication-efficient distributed optimization in networks with gradient tracking and variance reduction,” The Journal of Machine Learning Research, vol. 21, no. 1, 2020

    2020

  19. [28]

    Network Newton distributed optimization methods,

    A. Mokhtari, Q. Ling, and A. Ribeiro, “Network Newton distributed optimization methods,”IEEE Transactions on Signal Processing, 2016

    2016

  20. [29]

    Convergence rates of sub-sampled Newton methods,

    M. A. Erdogdu and A. Montanari, “Convergence rates of sub-sampled Newton methods,” inAdvances in Neural Information Processing Sys- tems, C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, Eds., vol. 28. Curran Associates, Inc., 2015

    2015

  21. [30]

    Accelerated distributed Nesterov gradient descent,

    G. Qu and N. Li, “Accelerated distributed Nesterov gradient descent,” IEEE Transactions on Automatic Control, vol. 65, no. 6, pp. 2566–2581, 2019

    2019

  22. [31]

    UCI Machine Learning Repository,

    D. Dua and C. Graff, “UCI Machine Learning Repository,” 2019, DOI: http://archive.ics.uci.edu/ml

    2019

  23. [32]

    Distributed average consensus with least-mean-square deviation,

    L. Xiao, S. Boyd, and S.-J. Kim, “Distributed average consensus with least-mean-square deviation,”Journal of parallel and distributed computing, vol. 67, no. 1, pp. 33–46, 2007

    2007

  24. [33]

    Hastie, R

    T. Hastie, R. Tibshirani, and J. Friedman,The Elements of Statistical Learning, 2nd ed., ser. Springer Series in Statistics. Springer, New York, 2009, Data Mining, Inference, and Prediction. [Online]. Available: https://doi.org/10.1007/978-0-387-84858-7 Souvik Dasis a postdoct...

    2009 doi

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.