Stochastic quasi-Newton with line-search regularization

Adrian Wills; Thomas Sch\"on

REVIEW 5 major objections 5 minor 54 references

Stochastic quasi-Newton with line-search regularization

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Gaussian-process Hessian learning makes quasi-Newton work on noisy gradients.

desk verdict A genuinely new GP-Hessian stochastic quasi-Newton idea, but a sign error in Eq. (36b) inverts the Hessian estimate as printed, so the paper needs serious correction before it can be trusted. read the letter →

arxiv 1909.01238 v1 pith:AAEVN35K submitted 2019-09-03 eess.SY cs.SYstat.ML

Adrian Wills , Thomas Schön This is my paper · ORCID

classification eess.SYcs.SYstat.ML

keywords stochasticoptimizationquasi-NewtonmethodsGaussianprocesslinesearchsystemidentificationsequentialMonteCarloparticlefiltermaximumlikelihoodestimation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a quasi-Newton method for stochastic optimization when only noisy evaluations of the cost and its gradient are available. Instead of assuming the Hessian is constant between iterates, as classical secant equations do, it places a Gaussian-process prior on the Hessian and conditions it on noisy gradient differences, producing a closed-form posterior whose mean supplies curvature for the search direction. It also introduces a stochastic backtracking line search that enforces an Armijo condition in expectation and eventually settles into a $\alpha_k = \xi/k$ step-length schedule, so standard stochastic-gradient convergence conditions apply. The combination is demonstrated on maximum-likelihood identification of nonlinear state-space models using particle-filter estimates of likelihood and gradient, where the new algorithm matches a particle-smoother EM baseline while using only forward filtering. If the claims hold, this gives nonlinear system identification a parameter-estimation route that avoids the often difficult particle smoother.

What carries the argument

The load-bearing object is the stochastic quasi-Newton integral, Eq. (17), viewed as a line-integral observation model $y_k = \bar D_k \int_0^1 h(r_k(\tau))\,d\tau + w_k$ for the half-vectorized Hessian $h$. Because Gaussian processes are closed under linear operators, conditioning this GP on stacked differences $y_{\ell_k}$ and $s_{\ell_k}$ gives the closed-form posterior (36); its mean $\varphi_{\ell_k}(x)$ supplies the curvature matrix $H_k$ through the duplication matrix $D$. A second mechanism is the regularized scaling $B_k = (H_k + \lambda_k I)^{-1}$ with $\lambda_k = \epsilon - \min\{0, \eta_k\}$, where $\eta_k$ is the minimum eigenvalue of $H_k$, ensuring $B_k \succ 0$ and hence an expected descent direction. The third mechanism is the stochastic Armijo line search from Algorithm 1, which backtracks while the observed cost fails a sufficient-decrease test and, because backtracking is limited, eventually follows the deterministic schedule $\alpha_k = \xi/k$ needed for stochastic-gradient convergence.

What would settle it

Run Algorithm 2 on a nonlinear state-space benchmark while replacing the assumed gradient-noise covariance $R$ in the GP posterior and line search with a deliberately wrong value, say $10R$; if the final parameter estimates and accepted step lengths barely change, the known-$R$ assumption is not load-bearing, whereas a sharp degradation would confirm it is.

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Extended reading notes

Core claim

The paper claims that a working quasi-Newton method can be built directly from noisy gradients by treating the Hessian as a Gaussian process and conditioning it on line integrals of gradient differences. Starting from the identity $\nabla f(x_{k+1}) - \nabla f(x_k) = [\int_0^1 \nabla^2 f(r_k(\tau))\,d\tau] s_k$, the authors derive a stochastic observation model $$y_k = \bar D_k \$int_0^{1}$ h(r_k(\tau))\,d\tau + w_k,$$ where $h$ is the half-vectorized Hessian and $\bar D_k = (s_k^T \otimes I)D$. With a GP prior on $h$, the closed-form posterior mean $\varphi_{\ell_k}(x)$ is obtained from the most recent $p+1$ gradient differences. The search direction is $p_k = -B_k g_k$ with $B_k = (H_k + \lambda_k I)^{-1}$ and $H_k = D\varphi_{\ell_{k-2}}(x_k)$, the regularization chosen so that $B_k \succ 0$ and therefore the direction is a descent direction in expectation. Step lengths come from a stochastic backtracking line search, Algorithm 1, which enforces an Armijo condition in expectation; Theorem 1 bounds the Armijo constant by $\bar c = \gamma/(\gamma + \beta)$ with $\gamma = \nabla f(x_k)^T B_k \nabla f(x_k)$ and $\beta = \operatorname{Tr}(B_k R)$. On maximum-likelihood identification of nonlinear state-space models with particle-filter likelihood and gradient estimates, the resulting Algorithm 2 matches a particle-smoother EM baseline while using only a forward filter.

Load-bearing premise

The load-bearing premise is that the noise in each gradient estimate is independent, Gaussian, and has a known covariance $R$; if the real particle-filter gradient noise is state-dependent or misspecified, both the learned Hessian and the stochastic Armijo test are miscalibrated.

Editorial extensions

If this is right

Algorithm 2 can be run with only a forward particle filter for the likelihood and its gradient, so maximum-likelihood parameter estimation becomes available in settings where particle smoothing is difficult or prohibitive.
Because backtracking is limited to $\max\{0, \tau-k\}$ steps, the line search eventually produces steps $\alpha_k = \xi/k$, placing the iterates under the standard stochastic-approximation convergence conditions.
The GP Hessian posterior uses only the most recent $p+1$ gradient and iterate differences, so per-iteration memory does not grow with the number of optimization steps.
On the reported benchmarks, the new method reaches parameter accuracy comparable to the PSEM baseline; in the nonlinear toy problem, 8 of 100 PSEM runs were trapped in poor local minima while none of the QNGP runs were.
The Hessian is learned continuously during optimization, which in principle lets the method adapt to non-quadratic cost surfaces rather than assuming constant curvature between iterates.

Reading between the lines

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

The paper keeps the gradient-noise covariance $R$ fixed and known, but particle-filter gradient noise is typically state-dependent; estimating $R$ online or modeling it with a second GP would be a direct test of how much the method's calibration depends on that assumption.
The GP construction over the half-vectorized Hessian involves a duplication matrix and squared-exponential kernel, so its cost grows with parameter dimension; extending it to large-scale problems would require sparse or inducing-point approximations.
The stochastic Armijo line search is developed for any search direction, not only the GP-scaled one; composing it with minibatch or adaptive diagonal direction methods would test whether the expectation-form Armijo condition alone improves step-length selection.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. The paper combines a GP model of the Hessian with a stochastic backtracking line search for stochastic optimization, aimed at SMC-based maximum likelihood system identification. That combination is new. Second, the printed mathematics has a load-bearing sign error: Eq. (36b) gives the GP conditional mean with a minus instead of a plus. Since H_k is formed as D times that mean, and the cross-covariance terms are nonnegative kernel integrals, the Hessian estimate is the negative of the intended regression estimate whenever the prior mean is not dominant. As written, Algorithm 2 is not the quasi-Newton algorithm the paper claims, and the Section 7 experiments cannot be reproduced from the text unless this is a typo.

What is genuinely good: the integral formulation of the quasi-Newton condition as a line integral of the Hessian, the use of a GP prior over the unique Hessian elements, and the stochastic Armijo line search that eventually decays like 1/k. The application to SMC identification is well motivated, and the comparison with PSEM on the toy and Hammerstein-Wiener problems is a reasonable demonstration. The paper is clearly organized and honestly cites the prior work, including the authors' own CDC 2017 paper.

The soft spots are real but uneven. The sign error in (36b) is fatal as printed. Eq. (22) also confuses vech and vec — it should be vec(A) = D vech(A), not vech(A) = D h. The Appendix proof of Theorem 1 flips a sign in the reduction; the final bound c < gamma/(gamma+beta) may be correct, but the proof as written does not establish it. The algorithm assumes the gradient-noise covariance R is known, but in the SMC examples it is neither known nor specified, which affects both the Hessian estimate and the line-search acceptance condition. Convergence of the full algorithm is not proven — only the Armijo condition in expectation. Experimental details such as kernel hyperparameters, R, and line-search parameters are missing, so the results are not reproducible from the text.

Bottom line: the paper deserves peer review because the core idea is substantive and the main flaw is a specific, checkable sign error. I would not accept this version, but I would engage with a revision that fixes the signs and supplies reproducible experimental settings. The right move is to send it to referees rather than desk-reject.

Referee Report

5 major / 5 minor

Summary. The paper proposes a stochastic quasi-Newton algorithm, called QN-GP, that models the Hessian as a Gaussian process and learns it from noisy gradient differences. A stochastic Armijo line search is introduced and the combined algorithm is applied to maximum likelihood identification of nonlinear state-space models using particle-filter estimates of the likelihood and gradient. The manuscript reports numerical experiments on a linear Gaussian model, a nonlinear benchmark, an interferometry problem, and a MIMO Hammerstein-Wiener system, comparing favorably to an EM-based method in the last example.

Significance. If the technical issues are resolved, the paper would contribute a genuinely novel way to bring second-order information into stochastic optimization via GP regression, with a plausible application to SMC-based system identification where it avoids particle smoothers. The paper also formulates a stochastic Armijo condition and a backtracking line-search that transitions to a classical step-length schedule. However, the current manuscript contains several load-bearing errors and gaps that undermine the claims as written.

major comments (5)

[Section 4, Eq. (36b)] The conditional mean of h(x) given y_ell_k is written as phi(x) = mu(x) - K_{x,ell} K_{ell,ell}^{-1} (y_ell - m_ell). The standard Gaussian conditioning formula has a plus sign, not a minus sign. Since K_{x,j} in Eq. (35) is a nonnegative kernel integral times Dbar_j^T, the sign error flips the learned Hessian estimate whenever the prior mean is not the dominant term; in the prior-mean-zero case H_k in Eq. (48b) is the negative of the intended regression estimate. This is not a cosmetic typo: it changes the search direction in Algorithm 2 and makes the algorithm described in the text different from the one that must have been run to produce the experimental results.
[Appendix A, Eq. (A.4)] The derivation of Theorem 1 contains an inequality sign error. The correct algebra gives alpha (gamma - c(gamma+beta)) >= -E[alpha^2/2 p^T H p], not <=. The paper's displayed inequality is reversed. The stated condition 0 < c < gamma/(gamma+beta) is consistent with the corrected inequality for small alpha, but as printed the proof and the theorem do not match. This makes the stochastic Armijo condition in Section 5 insufficiently supported.
[Sections 5--6] No convergence result is proved for the full Algorithm 2. The authors give a descent-direction lemma and a local Armijo condition, and then claim that the line search eventually reduces to alpha_k = xi/k, which 'affords standard convergence results.' But the search direction p_k = -B_k g_k with learned B_k is not the standard stochastic gradient direction, and the paper does not show that the required conditions for stochastic approximation theorems (e.g., on the direction and step sizes) hold for this B_k. A convergence theorem for the proposed algorithm is missing.
[Section 7, experiments] The algorithm requires the gradient noise covariance R as an input (Eq. (14)) and uses it both in the GP posterior (Eq. (31)) and in the stochastic Armijo bound (Theorem 1). In the SMC system identification setting the particle-filter gradient estimates have unknown, state-dependent covariance, and the manuscript never states how R was set in the experiments. If R is misspecified, both the Hessian estimate and the line-search acceptance condition are miscalibrated, so the reported results cannot be reproduced or interpreted from the text.
[Section 7, Tables 1 and 2] The numerical comparisons do not isolate the effect of the GP Hessian from that of the line search; there is no baseline that uses the same stochastic line search with a plain gradient direction. In Table 1, the EM results are reported after removing 8/100 runs that 'were trapped in minima far from the global minimum,' while no such removal is done for QNGP; this asymmetric protocol weakens the comparison.

minor comments (5)

[Section 3.2, Eq. (22)] The relation vech(nabla^2 f) = D h(r_k(tau)) is incorrect; the standard identity is vec(nabla^2 f) = D vech(nabla^2 f). The subsequent equations appear to use the correct identity, but the displayed equation is misleading.
[Proof of Lemma 1] In the proof of Lemma 1, the term '- E[-v_k]^T B nabla f' contains an erroneous sign; it should be '+ E[v_k]^T B nabla f' or simply omitted since E[v_k]=0.
[Section 7.1] The comparison methodology in the first experiment adds artificial noise only to the QNGP evaluations; this is a profiling choice, but it should be explicitly framed as such and not as a head-to-head stochastic optimization comparison.
[Section 4, after Eq. (37)] The computation of the integrals in Eqs. (31) and (35) is deferred to the authors' technical report [19]. The manuscript would benefit from a brief self-contained description or at least a statement of the main formulas used.
[Throughout] There are multiple typos and grammatical slips, e.g., 'indecies', 'conevient', 'psuedo-code', and an incomplete reference entry [54]. These should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Hessian surrogate is learned from noisy gradient differences by a standard GP posterior, and the stochastic Armijo line search follows from Taylor's theorem; the authors' self-citations are modular implementation references, not load-bearing premises.

full rationale

The paper's central derivation is self-contained and does not reduce to its own inputs. Section 3.2 builds the stochastic quasi-Newton integral (17)-(23) directly from the fundamental theorem of calculus and the noisy gradient model (13)-(14). Section 4 then treats the Hessian as a Gaussian process and computes its posterior from the joint Gaussian distribution (27), yielding the mean update (36b) and covariance (36c) by the standard conditional-Gaussian formula. The search direction (46)-(48) and the stochastic Armijo bound (57)-(58) are derived from the positive-definite regularized Hessian estimate and a Taylor expansion in Appendix A. No parameter is fitted to a target quantity and then renamed as a prediction; the GP Hessian is learned from the same first-order gradient information the algorithm is designed to consume, which is the intended mechanism rather than a circularity. The numerical comparisons in Section 7 are against independent external benchmarks: the EM method of Schon et al. (2011) and the PSEM method of Wills et al. (2013). The self-citations are not load-bearing in a circular sense: [19] supplies computational details for evaluating the squared-exponential covariance integrals, [52] is acknowledged as earlier preliminary work that the paper explicitly says lacked key ingredients, and [50] is mentioned only as future large-scale work. None of these citations is invoked to establish a uniqueness theorem or to forbid alternative constructions. The skeptical observation that Eq. (36b) carries a minus sign where the standard conditional mean has a plus sign concerns correctness and reproducibility of the algorithm as written, not circularity; even if true, it would mean the implemented algorithm differs from the claimed one, not that the derivation is equivalent to its inputs by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central algorithm depends on a set of user-specified hyperparameters (GP prior, noise covariance, line-search constants) that are not reported for the benchmarks, and on domain assumptions (known gradient noise covariance, GP adequacy for the Hessian) that are not validated in the SMC setting.

free parameters (6)

GP prior mean function mu = constant, e.g., 100 in the 1D example
Defines the baseline Hessian element values; no adaptation or selection procedure is given.
GP covariance hyperparameters M, V = Not reported for the experiments
The squared-exponential kernel (37) requires M and V; the paper does not state how these are chosen for the benchmark problems.
Gradient noise covariance R = Not reported for the SMC experiments
Assumed known in Eq. (14) and used in the GP posterior; no estimator is proposed for the particle-filter setting.
Line-search parameters rho, xi, tau, c = Not reported
Algorithm 1 requires a shrink factor, reduction limit, backtracking limit, and Armijo constant; none are specified for the simulations.
Memory length p = Not reported
The GP posterior uses the most recent p+1 gradient differences; p is not specified in the experiments.
Regularization tolerance epsilon = Not reported
Used in lambda_k = epsilon - min(0, eta_k) to ensure B_k is positive definite; value not given.

assumptions (5)

domain assumption The gradient noise v_k is i.i.d. N(0,R) with known R
Assumed in Eq. (14) and used throughout Sections 4-5; in the SMC application R is unknown and state-dependent.
domain assumption f is twice continuously differentiable on an open set
Assumption A1 in Theorem 1, used for Taylor's theorem in the stochastic Armijo proof.
domain assumption The Hessian can be modeled as a Gaussian process with a mean function and squared-exponential kernel
Section 4 postulates h ~ GP(mu, kappa); this is the modeling choice that the whole algorithm rests on.
ad hoc to paper The GP posterior mean provides a useful surrogate Hessian for the search direction
No convergence analysis shows that H_k from Eq. (48b) yields a direction satisfying SG convergence conditions.
ad hoc to paper Standard results for stochastic approximation apply to the scaled direction p_k = -B_k g_k
Section 5 claims the deterministic step decay 'affords standard convergence results,' but the angle condition for p_k is not verified.

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Cite this review

Pith. "Pith review of Stochastic quasi-Newton with line-search regularization." pith.science (2026). https://pith.science/paper/AAEVN35K

@misc{pith2026190901238,
  author       = {Pith},
  title        = {Pith review of: Stochastic quasi-Newton with line-search regularization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AAEVN35K}},
  note         = {Machine review of arXiv:1909.01238}
}

read the original abstract

In this paper we present a novel quasi-Newton algorithm for use in stochastic optimisation. Quasi-Newton methods have had an enormous impact on deterministic optimisation problems because they afford rapid convergence and computationally attractive algorithms. In essence, this is achieved by learning the second-order (Hessian) information based on observing first-order gradients. We extend these ideas to the stochastic setting by employing a highly flexible model for the Hessian and infer its value based on observing noisy gradients. In addition, we propose a stochastic counterpart to standard line-search procedures and demonstrate the utility of this combination on maximum likelihood identification for general nonlinear state space models.

Figures

Figures reproduced from arXiv: 1909.01238 by the authors.

**Figure 3.** Parameter iterations for nonlinear benchmark prob [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 2.** Bode plots of estimated mean (light grey) and true [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Parameter iterations for nonlinear interferometry [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 6.** Figure 6: Bode magnitude response using the QNGP method [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Bode magnitude response using the PSEM method [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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Works this paper leans on

54 extracted references · 50 canonical work pages

[1]

Observation of gravitational waves from a binary black hole merger

Benjamin P Abbott, Richard Abbott, TD Abbott, MR Abernathy, Fausto Acernese, Kendall Ackley, Carl Adams, Thomas Adams, Paolo Addesso, RX Adhikari, et al. Observation of gravitational waves from a binary black hole merger. Physical review letters, 116(6):061102, 2016

2016
[2]

L. Armijo. Minimization of functions having Lipschitz continuous ﬁrst partial derivatives. Paciﬁc Journal of Mathematics, 16(1):1–3, 1966

work page 1966
[3]

Asi and J

H. Asi and J. C. Duchi. Stochastic (approximate) proximal point methods: convergence, optimality, and adaptivity. SIAM Journal on Optimization , To appear, 2019

work page 2019
[4]

Neuro-dynamic programming, volume 5

Dimitri P Bertsekas and John N Tsitsiklis. Neuro-dynamic programming, volume 5. Athena Scientiﬁc Belmont, MA, 1996

work page 1996
[5]

Bollapragada, D

R. Bollapragada, D. Mudigere, J. Nocedal, H.-J. M. Shi, and P. T. P. Tang. A progressive batching L-BFGS method for machine learning. In Proceedings of the 35th International Conference on Machine Learning (ICML) , Stockholm, Sweden, 2018

work page 2018
[6]

Bordes, L

A. Bordes, L. Bottou, and P. Gallinari. SGD-QN: Careful quasi-Newton stochastic gradient descent. Journal of Machine Learning Research (JMLR), 10:1737–1754, 2009

work page 2009
[7]

Bottou, F

L. Bottou, F. E. Curtis, and J. Nocedal. Optimization methods for large-scale machine learning. SIAM Review , 60(2):223–311, 2018

work page 2018
[8]

C. G. Broyden. A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation , 19(92):577–593, 1965

work page 1965

Show all 54 references

[9]

C. G. Broyden. Quasi-Newton methods and their application to function minimization. Mathematics of Computation , 21:368–381, 1967

1967
[10]

C. G. Broyden. The convergence of a class of double- rank minimization algorithms. Journal of the Institute of Mathematics and Its Applications , 6(1):76–90, 1970

1970
[11]

Del Moral

P. Del Moral. Feynman-Kac formulae: Genealogical and Interacting Particle Systems with Applications . Springer, New York, USA, 2004

2004
[12]

Duchi, E

J. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research (JMLR) , 12:2121– 2159, 2011

2011
[13]

Fletcher

R. Fletcher. A new approach to variable metric algorithms. The computer journal , 13(3):317–322, 1970

1970
[14]

Fletcher

R. Fletcher. Practical methods of optimization . John Wiley & Sons, Chichester, UK, second edition, 1987

1987
[15]

Fletcher and M

R. Fletcher and M. J. D. Powell. A rapidly convergent descent method for minimization. The computer journal , 6(2):163– 168, 1963. 13

1963
[16]

Goldfarb

D. Goldfarb. A family of variable metric updates derived by variational means. Mathematics of Computation, 24(109):23– 26, 1970

1970
[17]

Discrete time stochastic adaptive control

Graham C Goodwin, Peter J Ramadge, and Peter E Caines. Discrete time stochastic adaptive control. SIAM Journal on Control and Optimization , 19(6):829–853, 1981

1981
[18]

N. J. Gordon, D. J. Salmond, and A. F. M. Smith. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. In IEE Proceedings on Radar and Signal Processing, volume 140, pages 107–113, 1993

1993
[19]

J. N. Hendriks, C. Jidling, A. Wills, and T. B. Sch¨ on. Evaluating the squared-exponential covariance function in Gaussian processes with integral observations. Technical report, arXiv:1812.07319, 2018

2018 arXiv
[20]

P. Hennig. Probabilistic interpretation of linear solvers. SIAM Journal on Optimization , 25(1):234–260, 2015

2015
[21]

Hennig and M

P. Hennig and M. Kiefel. Quasi-Newton methods: a new direction. Journal of Machine Learning Research (JMLR) , 14:843–865, 2013

2013
[22]

Kantas, A

N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski, and N. Chopin. On particle methods for parameter estimation in state-space models. Statistical Science, 30(3):328–351, 2015

2015
[23]

Stochastic estimation of the maximum of a regression function

Jack Kiefer and Jacob Wolfowitz. Stochastic estimation of the maximum of a regression function. The Annals of Mathematical Statistics, 23(3):462–466, 1952

1952
[24]

D. P. Kingma and J. Ba. Adam: a method for stochastic optimization. In Proceedings of the 3rd international conference on learning representations (ICLR) , San Diego, CA, USA, 2015

2015
[25]

Kitagawa

G. Kitagawa. A Monte Carlo ﬁltering and smoothing method for non-Gaussian nonlinear state space models. In Proceedings of the 2nd US-Japan joint Seminar on Statistical Time Series Analysis , pages 110–131, 1993

1993
[26]

Lindsten and T

F. Lindsten and T. B. Sch¨ on. Backward simulation methods for Monte Carlo statistical inference.Foundations and Trends in Machine Learning , 6(1):1–143, 2013

2013
[27]

L. Ljung. Asymptotic behavior of the extended Kalman ﬁlter as a parameter estimator for linear systems. IEEE Transactions on Automatic Control , AC–24(1):36–50, February 1979

1979
[28]

Ljung and T

L. Ljung and T. S¨ oderstr¨ om. Theory and Practice of Recursive Identiﬁcation. The MIT Press series in Signal Processing, Optimization, and Control. The MIT Press, Cambridge, Massachusetts, 1983

1983
[29]

Analysis of recursive stochastic algorithms

Lennart Ljung. Analysis of recursive stochastic algorithms. IEEE transactions on automatic control , 22(4):551–575, 1977

1977
[30]

Strong convergence of a stochastic approximation algorithm

Lennart Ljung. Strong convergence of a stochastic approximation algorithm. The Annals of Statistics , pages 680–696, 1978

1978
[31]

Stochastic approximation and optimization of random systems , volume 17

Lennart Ljung, Georg Pﬂug, and Harro Walk. Stochastic approximation and optimization of random systems , volume 17. Birkh¨ auser, 2012

2012
[32]

Adaptive gradient methods with dynamic bound of learning rate

Liangchen Luo, Yuanhao Xiong, and Yan Liu. Adaptive gradient methods with dynamic bound of learning rate. In International Conference on Learning Representations (ICLR), New Orleans, LA, USA, 2019

2019
[33]

J. R. Magnus and H. Neudecker. The elimination matrix: some lemmas and applications. SIAM Journal on Algebraic Discrete Methods, 1(4):422–449, 1980

1980
[34]

Mahsereci and P

M. Mahsereci and P. Hennig. Probabilistic line searches for stochastic optimization. Journal of Machine Learning Research (JMLR), 18(119):1–59, 2017

2017
[35]

Malik and M

S. Malik and M. K. Pitt. Particle ﬁlters for continuous likelihood evaluation and maximisation. Journal of Econometrics, 165(2):190–209, 2011

2011
[36]

Mokhtari and A

A. Mokhtari and A. Ribeiro. RES: regularized stochastic BFGS algorithm. IEEE Transactions on Signal Processing , 62(23):6089–6104, 2014

2014
[37]

Moulines and F

E. Moulines and F. Bach. Non-asymptotic analysis of stochastic approximation algorithms for machine learning. In Advances in Neural Information Processing Systems (NIPS), Granada, Spain, 2011

2011
[38]

Nocedal and S

J. Nocedal and S. J. Wright. Numerical Optimization . Springer Series in Operations Research. Springer, New York, USA, second edition, 2006

2006
[39]

M. K. Pitt, R. dos Santos Silva, R. Giordani, and R. Kohn. On some properties of Markov chain Monte Carlo simulation methods based on the particle ﬁlter. Journal of Econometrics, 171(2):134–151, 2012

2012
[40]

Poyiadjis, A

G. Poyiadjis, A. Doucet, and S.S. Singh. Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika, 98(1):65–80, 2011

2011
[41]

C. E. Rasmussen and C. K. I. Williams. Gaussian processes for machine learning . MIT Press, 2006

2006
[42]

Robbins and S

H. Robbins and S. Monro. A stochastic approximation method. Annals of Mathematical Statistics , 22(3):400–407, 1951

1951
[43]

T. B. Sch¨ on, F. Lindsten, J. Dahlin, J. W˚ agberg, A. C. Naesseth, A. Svensson, and L. Dai. Sequential Monte Carlo methods for system identiﬁcation. In Proceedings of the 17th IFAC Symposium on System Identiﬁcation (SYSID), Beijing, China, October 2015

2015
[44]

T. B. Sch¨ on, A. Wills, and B. Ninness. System identiﬁcation of nonlinear state-space models. Automatica, 47(1):39–49, January 2011

2011
[45]

N. N. Schraudolph, J. Yu, and S. G¨ unter. A stochastic quasi-Newton method for online convex optimization. In Proceedings of the 11th international conference on Artiﬁcial Intelligence and Statistics (AISTATS) , 2007

2007
[46]

A. Shah, A. G. Wilson, and Z. Ghahramani. Student-t processs as alternatives to Gaussian processes. InProceedings of the 17th international conference on artiﬁcial intelligence and statistics (AISTATS), Reykjavik, Iceland, May 2014

2014
[47]

D. F. Shanno. Conditioning of quasi-Newton methods for function minimization. Mathematics of Computation , 24(111):647–656, 1970

1970
[48]

Introduction to stochastic search and optimization: estimation, simulation, and control, volume 65

James C Spall. Introduction to stochastic search and optimization: estimation, simulation, and control, volume 65. John Wiley & Sons, 2005

2005
[49]

Stewart and P

L. Stewart and P. McCarty. The use of Bayesian belief networks to fuse continuous and discrete information for target recognition and discrete information for target recognition, tracking, and situation assessment. In Proceedings of SPIE Signal Processing, Sensor Fusion and Ta...

1992
[50]

Wills and T

A. Wills and T. B. Sch¨ on. Stochastic quasi-Newton with adaptive step lengths for large-scale problems. arXiv:1802.04310, 2018

2018 arXiv
[51]

Wills, T

A. Wills, T. B. Sch¨ on, L. Ljung, and B. Ninness. Identiﬁcation of Hammerstein-Wiener models. Automatica, 49(1):70–81, 2013

2013
[52]

A. G. Wills and T. B. Sch¨ on. On the construction of probabilistic Newton-type algorithms. In Proceedings of the 14 56th IEEE Conference on Decision and Control (CDC) , Melbourne, Australia, December 2017

2017
[53]

P. Wolfe. Convergence conditions for ascent methods. SIAM Review, 11(2):226–235, 1969

1969
[54]

P. Wolfe. Convergence conditions for ascent methods II: some corrections. SIAM Review, 13(2):185–188, 1971. A The stochastic Armijo condition All expectation below are conditioned on the variables {xk,s𝓁k−2,ˆy𝓁k−2}. This conditioning is dropped from the notation in order to im...

1971

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Observation of gravitational waves from a binary black hole merger

Benjamin P Abbott, Richard Abbott, TD Abbott, MR Abernathy, Fausto Acernese, Kendall Ackley, Carl Adams, Thomas Adams, Paolo Addesso, RX Adhikari, et al. Observation of gravitational waves from a binary black hole merger. Physical review letters, 116(6):061102, 2016

2016

[2] [2]

L. Armijo. Minimization of functions having Lipschitz continuous ﬁrst partial derivatives. Paciﬁc Journal of Mathematics, 16(1):1–3, 1966

work page 1966

[3] [3]

Asi and J

H. Asi and J. C. Duchi. Stochastic (approximate) proximal point methods: convergence, optimality, and adaptivity. SIAM Journal on Optimization , To appear, 2019

work page 2019

[4] [4]

Neuro-dynamic programming, volume 5

Dimitri P Bertsekas and John N Tsitsiklis. Neuro-dynamic programming, volume 5. Athena Scientiﬁc Belmont, MA, 1996

work page 1996

[5] [5]

Bollapragada, D

R. Bollapragada, D. Mudigere, J. Nocedal, H.-J. M. Shi, and P. T. P. Tang. A progressive batching L-BFGS method for machine learning. In Proceedings of the 35th International Conference on Machine Learning (ICML) , Stockholm, Sweden, 2018

work page 2018

[6] [6]

Bordes, L

A. Bordes, L. Bottou, and P. Gallinari. SGD-QN: Careful quasi-Newton stochastic gradient descent. Journal of Machine Learning Research (JMLR), 10:1737–1754, 2009

work page 2009

[7] [7]

Bottou, F

L. Bottou, F. E. Curtis, and J. Nocedal. Optimization methods for large-scale machine learning. SIAM Review , 60(2):223–311, 2018

work page 2018

[8] [8]

C. G. Broyden. A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation , 19(92):577–593, 1965

work page 1965