Pith. sign in

REVIEW 3 major objections 5 minor 27 references

The Root Finding Problem Revisited: Beyond the Robbins-Monro procedure

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read SPRB root-finding reaches the parametric rate where Robbins–Monro stalls.

desk verdict SPRB is a genuinely new adaptive root-finding method with strong asymptotic results, but the nonasymptotic anytime-valid confidence sequence has a real proof gap at early stages. read the letter →

arxiv 2508.17591 v1 pith:APIEJJBH submitted 2025-08-25 math.ST stat.MEstat.TH

Yue Yu , Moulinath Banerjee , Ya'acov Ritov This is my paper · ORCID
classification math.STstat.MEstat.TH MSC 62L2062L1062F12
keywords SequentialProbabilityRatioBisectionstochasticapproximationrootfindinganytime-validconfidencesequenceRobbins-Monroprocedurerandomstoppingtimesasymptoticvariance
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

This paper introduces Sequential Probability Ratio Bisection (SPRB), a sequential algorithm for finding the root $\theta$ of a regression function $f$ observed with noise. The paper's central claim is that SPRB attains the parametric convergence rate and the minimal asymptotic variance $\sigma^2/\beta^2$ for stochastic approximation even when the derivative $f'(\theta)$ is positive but no larger than $1/(2\alpha)$, the regime in which the classical Robbins–Monro procedure slows down. It further claims exponential convergence when $f$ jumps at $\theta$, and rates near $n^{-1/(2\gamma)}$ when the first $\gamma-1$ derivatives vanish. Because the intervals produced at every stage are nonasymptotic anytime-valid confidence sequences, a user gets valid error control without estimating the convergence rate or asymptotic variance.

What carries the argument

The machinery is Sequential Probability Ratio Bisection. At each stage $t$, the procedure repeatedly samples the current location $X_t$ until the partial sum of responses crosses the moving boundary $T(j,\alpha_t)=\sigma\sqrt{-2j\log(j+1)\log\alpha_t}$, with $\alpha_t=\alpha 2^{-t}$; the stopped average $\hat f_t$ then carries both sign and magnitude information about $f(X_t)$. A bracketing interval $[X_{\ell t},X_{rt}]$ is updated with plain bisection until its width falls below $\delta$, then by the weighted-section update $X_{t+1}=(\hat f_{\ell t}X_{rt}-\hat f_{rt}X_{\ell t})/(\hat f_{\ell t}-\hat f_{rt})$, which recovers $\theta$ exactly in the noiseless linear case. Extra samples at the opposite endpoint reduce the variance of the endpoint estimates. The load-bearing analytic tools are the nonasymptotic false-sign bounds of Lemma 25, the expected-sample-size bounds of Proposition 7, and a generalized CLT for randomly stopped averages (Theorem 9), which together turn the sequential test nested inside SPRB into rate and coverage guarantees.

What would settle it

Run SPRB with $\alpha=\Delta/3$ on a linear function $f(x)=\beta(x-\theta)$, and record over many replications the first stage at which $\theta$ leaves $I_k$; if the empirical frequency of violations before the $K$ in Lemma 25 exceeds the per-stage budget $\alpha_k$ (or the overall $\Delta$), the nonasymptotic coverage claim fails as stated. Equivalently, exhibit any small $n,k$ with $-2\log\alpha_k\, n\log n < n(\log\log 2n + 0.72\log(5.2/\alpha_k))$, which contradicts the boundary inequality used in the proof of Lemma 25.

Watch

Extended reading notes

Core claim

The core discovery is that a root-finding procedure can avoid the phase transition that slows Robbins–Monro down. Under Assumption 2, if $f$ has derivative $\beta=f'(\theta)>0$ at the root, the SPRB estimator $X_{k+1}$ after $k$ stages obeys $\sqrt{n}(X_{k+1}-\theta) \to N(0,\sigma^2/\beta^2)$ in distribution, where $n$ is the total sample size; this holds for every $0<\beta\le 1/(2\alpha)$, not only for $\beta>1/(2\alpha)$. Since $\sigma^2/\beta^2$ is the minimal asymptotic variance achievable by stochastic approximation, SPRB is efficient without knowing $\beta$ or tuning the step size. The same interval construction gives $P(\exists k\in\mathbb{N}^+: I_k\not\ni\theta)\le \Delta$ when $\alpha=\Delta/3$, a nonasymptotic anytime-valid confidence sequence requiring no rate calibration. In addition, for discontinuous $f$, SPRB converges at the rate $\exp(-\kappa\sigma^{-1}\sqrt{n}/(\log n)^\eta)$, against $O(n^{-1})$ for Robbins–Monro; for $f$ of order $\gamma$ it attains $n^{-1/(2\gamma)+\delta}$, against $(\log n)^{1/(1-\gamma)}$.

Load-bearing premise

The load-bearing premise is that the false-sign probability bounds of Lemma 25 hold from the very first stage; the proof as written establishes the key boundary inequality only for $k\ge K$ for a universal $K$, so the anytime-valid coverage of the earliest intervals is not established as stated.

Editorial extensions

If this is right

  • In the regime $0<\beta\le 1/(2\alpha)$, SPRB achieves $\sqrt{n}$ convergence and variance $\sigma^2/\beta^2$ where Robbins–Monro's rate degrades, and it does so without knowing $\beta$ or calibrating the step size.
  • The intervals $I_k=[X_{\ell k},X_{rk}]$ form a $\Delta$-time-uniform confidence sequence for every $k$, so inference may stop at any data-dependent time without losing coverage.
  • For discontinuous regression functions, SPRB converges exponentially rather than at the $O(n^{-1})$ rate of Robbins–Monro.
  • For $f(x)=\operatorname{sign}(x-\theta)\beta|x-\theta|^\gamma(1+o(1))$, $\gamma\ge 1$, SPRB converges at $n^{-1/(2\gamma)+\delta}$ for any $\delta>0$, beating the logarithmic rate of classical stochastic approximation.
  • The randomly stopped average CLT and the boundary-crossing expansions are stand-alone tools for sequential analysis beyond root finding.

Reading between the lines

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

  • Inference: Because the proof of Lemma 25 establishes its boundary inequality only for stages $k\ge K$, the anytime-valid coverage theorem as stated relies on unproved early-stage control; users of the confidence sequence from stage one should check whether the bound holds for $k<K$ or inflate $\alpha$ for early stages.
  • Inference: The boundary rule uses only sign and magnitude of $f(X_t)$, so SPRB should tolerate heavier-tailed noise if the boundary is built from a corresponding concentration inequality; the paper develops the sub-Gaussian case only.
  • Inference: The weighted-section update behaves like a one-dimensional bracketing root finder, and the paper itself names the multivariate problem as the open sequel; any such extension must decide how to aggregate sign information across orthants.
  • Inference: The cheapest test of Theorem 25 is empirical: run SPRB with $\alpha=\Delta/3$, record the first time $\theta$ leaves $I_k$, and check whether early-stage violations respect the per-stage $\alpha_k$ budget.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes SPRB (Sequential Probability Ratio Bisection), a multi-stage stochastic approximation algorithm for root finding under noisy evaluations. At each stage t the method repeatedly samples a current design point X_t until a moving-boundary stopping time N_t is crossed, forming a randomly stopped average whose sign is used to update a bracketing interval, first by bisection and later by a weighted-section update. The advertised contributions are: (i) under differentiability with positive derivative, √n(X_{k+1}-θ) → N(0, σ²/β²) with n the total sample size, so SPRB attains the parametric rate and the minimal asymptotic variance even when 0 < f'(θ) ≤ 1/(2α), a regime where Robbins-Monro slows down; (ii) exponential convergence for discontinuous regression functions; (iii) a rate n^{-1/(2γ)+δ} for vanishing derivatives of order γ; and (iv) nonasymptotic anytime-valid confidence intervals I_k = [X_{ℓk}, X_{rk}] satisfying P(∃k∈N⁺: I_k ∌ θ) ≤ Δ with the choice α = Δ/3. Supporting results include nonasymptotic bounds on the expected stopping time, a generalized CLT for randomly stopped sums, and sign-correctness lemmas.

Significance. If the results hold, this is a substantial contribution to stochastic approximation. The method is genuinely adaptive across regimes that are usually treated separately: positive slope, small slope, vanishing higher-order derivatives, and jump discontinuities. The claimed confidence-sequence property is notable because it avoids estimating the convergence rate or the asymptotic variance. Strengths of the paper include the detailed main-text and Appendix D development of the differentiable case, including the expansion of X_{k+1}-θ and the proof that n/N_k → 1; the nonasymptotic expected-stopping-time bounds of Proposition 7; and simulation comparisons that support the stated qualitative conclusions. The proofs are not circular: the asymptotic rates and the confidence-sequence error are derived from the algorithm's own stopping rule using external concentration inequalities and CLTs, and α, δ, Δ are design hyperparameters rather than fitted constants. However, the anytime-valid theorem currently rests on a proof gap in Lemma 25, and the discontinuous and higher-order cases are only sketched in the appendices as submitted.

major comments (3)
  1. [§4.5 (Theorem 25) and Appendix A (Lemma 25)] Lemma 25 states P(f̂_k f_k < 0) < α_k and P(f^♯_{·,k} f_{·,k} < 0) < 2α_k for every k ∈ N⁺, but its proof establishes the controlling inequality −2 log α_k · n log n ≥ n(log log(2n) + 0.72 log(5.2/α_k)) only for k ≥ K for a universal K, and the LIL-based concentration bound is applied only there. Thus the all-k statement is not proved as written. Since Theorem 25 sums these bounds over all k ≥ 2 through the union bound, the claimed Δ-time-uniform coverage in Eq. (4.24) is not established: early-stage false-sign probabilities above the α_k levels would contribute an unaccounted additive constant. The asymptotic theorems can tolerate finitely many early failures by Borel-Cantelli, but the nonasymptotic anytime-valid claim cannot. The authors should either prove the early-stage bounds, restate the theorem with an explicit finite-stage correction term, or restrict the confidence sequence to k ≥ K with the error level adjusted accordingly. Additionally, the derivation of inequality (A.10) for the extra endpoint samples is sketched and appears to mix the stopping time at the original update stage with the mean f_{·,k} at the endpoint; this part of the lemma needs a complete proof as well.
  2. [§4.4 / Theorem 18 and Appendix F] The rate n^{-1/(2γ)+δ} for regression functions with vanishing first-order derivatives is one of the paper's central claims, but the proof in Appendix F is a sketch: Lemma 40 is stated with 'the argument parallels that of Lemma 34 and is therefore omitted,' and the final display relies on unstated high-probability events and on Lemma 39 in a compressed way. If the supplemental material contains the complete proof, the manuscript should point to the specific supplement locations; as submitted, the main text and Appendix F do not contain enough detail to verify this advertised result.
  3. [§4.3 / Theorem 16 and Appendix E] The exponential convergence claim for discontinuous regression functions is likewise only sketched. The proof relies on Corollary 32, Lemma 37, and a Cesàro summation argument, but the key inequalities (E.4)-(E.5) are asserted with 'with probability 1-o(1)' without the full event decomposition needed to control the remainder in the product over stages. Since exponential convergence in this regime is a distinctive advertised improvement over Robbins-Monro's n^{-1} rate, the proof should be completed in the text or the supplement should be made available with full details.
minor comments (5)
  1. [Abstract and §1] The abstract's phrase 'derivative at the root is small (at most half the step size)' is imprecise; the condition is 2α f'(θ) ≤ 1, i.e. f'(θ) ≤ 1/(2α). Please rephrase to avoid confusion between the step-size α and the inverse step size.
  2. [Throughout] The manuscript contains numerous typos and grammatical slips, e.g. 'varinace', 'substracting', 'neglible', 'constistutes', 'the the', and 'terminate SPRB either when...'. A careful proofreading pass is needed.
  3. [Section 3 and Algorithm descriptions] The algorithm boxes are not explicitly labeled in the text (the main SPRB procedure and the simplified variant are referenced informally). Adding numbered algorithm environments would improve readability and make the comparison between the simplified and full versions easier to follow.
  4. [Assumption 2, Eq. (4.9)] Equation (4.9) writes o(|X−θ|^ν) with an uppercase X; this should be o(|x−θ|^ν) to match the function argument. The same uppercase/lowercase slip appears in a few other Taylor-expansion displays.
  5. [arXiv rendering, Sections 1-2] In the submitted arXiv rendering, large portions of Section 1 and parts of Section 2 appear as mojibake/replacement characters, making the introduction's stated question and several remarks unreadable. The authors should regenerate the source so that the manuscript is fully readable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SPRB results are derived from its own stopping rule using external concentration inequalities and CLTs; the k≥K proof gap in Lemma 25 is a rigor concern, not circularity.

full rationale

The paper's derivation chain is self-contained with respect to circularity. SPRB's stopping rule (Eq. (3.1)-(3.2)), the false-sign bounds (Lemma 25), the expected stopping-time bounds (Proposition 7), the randomly-stopped CLT (Theorem 9), and the asymptotic expansions for the bisection/weighted-section updates are all derived from the algorithm's own construction using external concentration inequalities (Schreuder et al. 2020; Howard et al. 2021), Anscombe-type random-time CLTs, and Taylor expansions under the stated local assumptions on f. No parameter is fitted to data and then renamed a prediction: alpha = Delta/3 is fixed before the coverage proof, and delta, beta, and sigma appear as assumptions or design thresholds, not as fitted outputs. The asymptotic variance sigma^2/beta^2 in Theorem 13 emerges from the algebra of the weighted-section update, not from an assumption of that variance. The confidence-sequence proof in Theorem 25 calibrates alpha = Delta/3 so that 3 sum_k alpha_k <= Delta; this is calibration, not circularity. There are no load-bearing self-citations: the cited results are external and do not presuppose SPRB's conclusions. One non-circular proof-gap caveat: Lemma 25's proof invokes the boundary inequality only for k >= K (a universal K), while Theorem 25 claims all k in N+; if early-stage false-sign probabilities are not alpha_k-controlled, the nonasymptotic anytime-valid coverage would not be established as stated. That is a correctness and rigor concern, not a circularity, because the claimed bound is still derived from the stopping rule rather than assumed as an input.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central rate results rely on the standard model classes (Assumptions 1-4) and external concentration and CLT results; the algorithm introduces no invented physical entities. The free parameters are the noise level σ, the error-control constant α, the bisection/weighted-section threshold δ, and the initial bracket, none of which are fitted to data.

free parameters (4)
  • sigma (noise std dev) = assumed known (Eq. (3.2))
    The stopping boundary T(j,α_t) = σ sqrt(-2j log(j+1) log α_t) requires σ; the theory assumes it is known, and the remark that σ can be sequentially estimated is not formalized.
  • alpha (error-control hyperparameter) = α = Δ/3 for confidence sequences; otherwise arbitrary
    Controls false-sign error probabilities in the stopping rule; asymptotic rates hold for any α>0, but finite-sample coverage depends on α.
  • delta (bisection-to-weighted-section threshold) = not specified numerically
    Determines when the algorithm switches from midpoint updates to weighted-section updates; asymptotics are insensitive to it, but finite-sample behavior may depend on the choice.
  • initial bracketing interval I and sign seeds = [0,1] and ±1 in simulations
    The algorithm requires an interval [X_ℓ1,X_r1] and initial sign estimates satisfying f_ℓ1<0 and f_r1>0; the theory assumes such a bracket is given.
assumptions (7)
  • domain assumption Sub-Gaussian noise with known variance proxy σ (Assumption 1)
    Used for LIL concentration bounds, stopping-time expectations, and the random-stopping CLT; the known-σ case is the paper's formal scope.
  • domain assumption Zero-crossing and boundedness: inf over η≤|x-θ|≤1/η of f(x)(x-θ)>0, and |f(x)|≤C1|x|+C2 (Assumption 1)
    Defines the root-finding problem and ensures a.s. finiteness of stopping times.
  • domain assumption Differentiable case: f(x)=β(x-θ)+o(|x-θ|^ν), ν>1 (Assumption 2)
    Used in Theorem 13 to justify the Taylor expansion in the weighted-section update.
  • domain assumption Discontinuous case: one-sided limits μ_+, μ_->0 (Assumption 3)
    Used in Theorem 16 to guarantee a constant contraction factor ω_k∨(1-ω_k)≤1-κ.
  • domain assumption Higher-order case: f(x)=β sign(x-θ)|x-θ|^γ(1+o(1)), γ>1 (Assumption 4)
    Used in Theorem 18 for the near-minimax rate n^{-1/(2γ)+δ}.
  • standard math Nonasymptotic law-of-the-iterated-logarithm inequalities from Schreuder et al. (2020) and Howard et al. (2021)
    Used as black boxes in Lemma 25 and in the finiteness-of-stopping-time argument.
  • standard math Anscombe-style random-stopping CLT and Wald's identity
    Used in Theorem 9 and the proof of Proposition 7 to handle randomly stopped averages.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Root Finding Problem Revisited: Beyond the Robbins-Monro procedure." pith.science (2026). https://pith.science/paper/APIEJJBH

@misc{pith2026250817591,
  author       = {Pith},
  title        = {Pith review of: The Root Finding Problem Revisited: Beyond the Robbins-Monro procedure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/APIEJJBH}},
  note         = {Machine review of arXiv:2508.17591}
}
abstract

We introduce Sequential Probability Ratio Bisection (SPRB), a novel stochastic approximation algorithm that adapts to the local behavior of the (regression) function of interest around its root. We establish theoretical guarantees for SPRB's asymptotic performance, showing that it achieves the optimal convergence rate and minimal asymptotic variance even when the target function's derivative at the root is small (at most half the step size), a regime where the classical Robbins-Monro procedure typically suffers reduced convergence rates. Further, we show that if the regression function is discontinuous at the root, Robbins-Monro converges at a rate of $1/n$ whilst SPRB attains exponential convergence. If the regression function has vanishing first-order derivative, SPRB attains a faster rate of convergence compared to stochastic approximation. As part of our analysis, we derive a nonasymptotic bound on the expected sample size and establish a generalized Central Limit Theorem under random stopping times. Remarkably, SPRB automatically provides nonasymptotic time-uniform confidence sequences that do not explicitly require knowledge of the convergence rate. We demonstrate the practical effectiveness of SPRB through simulation results.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 10 canonical work pages

  1. [2]

    doi: 10.2307/1905525

    ISSN 0012-9682. doi: 10.2307/1905525. URL ������ ������������������������������ . Publisher: [Wiley, Econometric Society]. Julius R. Blum. Approximation Methods which Converge with Probability one. ��� ������ �� ��������� ���� ����������, 25(2):382–386,

    work page doi:10.2307/1905525

  2. [13]

    doi: 10.1214/20-STS784

    ISSN 0883-4237. doi: 10.1214/20-STS784. URL ������������������������������������������������������������������������� ����������������������������������������������������������������������������������� ������������������� . Sokbae Lee, Yuan Liao, Myung Hwan Seo, and Youngki Shin. Fast and Robust Online Inference with Stochastic Gradient Descent via Random...

    work page doi:10.1214/20-sts784

  3. [17]

    arXiv:2002.06673 [cs, stat]

    URL ������������������������������� . arXiv:2002.06673 [cs, stat]. M. Pollak and D. Siegmund. Approximations to the Expected Sample Size of Certain Sequential Tests. ��� ������ �� ����������, 3(6):1267–1282,

    arXiv 2002

  4. [21]

    doi: 10.1214/aos/1051027870

    ISSN 0090-5364, 2168-8966. doi: 10.1214/aos/1051027870. URL �������������������������������������������������������������������������� ��������������������������������������������������������������������������������� . Pub- lisher: Institute of Mathematical Statistics. Weijie J. Su and Yuancheng Zhu. HiGrad: Uncertainty Quantification for Online Learning ...

    arXiv

  5. [25]

    doi: 10.1093/jrsssb/qkad009

    ISSN 1369-7412, 1467-9868. doi: 10.1093/jrsssb/qkad009. URL ����������������������������������������������� ������� . C. Z. Wei. Multivariate Adaptive Stochastic Approximation. ��� ������ �� ����������, 15(3):1115–1130,

    doi:10.1093/jrsssb/qkad009

  6. [27]

    arXiv:2410.15057 [stat] version:

    URL �������������������������� �����. arXiv:2410.15057 [stat] version:

    arXiv

  7. [1948]

    doi: 10.1214/aoms/ 1177730197

    ISSN 0003-4851, 2168-8990. doi: 10.1214/aoms/ 1177730197. URL ��������������������������������������������������������������������� ������������������������������������������������������������������������������������� �������������������� . Publisher: Institute of Mathematical Statistics. Ian Waudby-Smith and Aaditya Ramdas. Estimating means of bounded ra...

    doi:10.1214/aoms/

  8. [1949]

    doi: 10.2307/2332681

    ISSN 0006-3444. doi: 10.2307/2332681. URL ������������������������������������ . Publisher: [Oxford University Press, Biometrika Trust]. F. J. Anscombe. Sequential Estimation. ������� �� ��� ����� ����������� �������� ������ � ����������������, 15(1):1–29,

    work page doi:10.2307/2332681

Show all 27 references
  1. [1967]

    doi: 10.1214/aoms/1177699069

    ISSN 0003-4851, 2168-8990. doi: 10.1214/aoms/1177699069. URL ��������������������������������������������������������������������������������������� ������������������������������������������������������������������������ . Publisher: Institute of Mathematical Statistics. A. W...

  2. [1968]

    URL ���������������������������������������������������

    doi: 10.1073/pnas.60.4.1175. URL ��������������������������������������������������� . Publisher: Proceedings of the National Academy of Sciences. Yixin Fang, Jinfeng Xu, and Lei Yang. Online Bootstrap Confidence Intervals for the Stochastic Gradient Descent Estimator. �������...

    doi

  3. [1973]

    doi: 10.1007/BF00736010

    ISSN 1432-2064. doi: 10.1007/BF00736010. URL ���������������������������������� . Eric Moulines and Francis Bach. Non-Asymptotic Analysis of Stochastic Approximation Algo- rithms for Machine Learning. In �������� �� ������ ����������� ���������� ������� , vol- ume

    doi

  4. [1974]

    doi: 10.1214/aos/1176342704

    ISSN 0090-5364, 2168-8966. doi: 10.1214/aos/1176342704. URL ������������������������������������������������������������������������� ������������������������������������������������������������������������������� . Pub- lisher: Institute of Mathematical Statistics. Herbert Ro...

  5. [1977]

    doi: 10.1214/aos/1176343943

    ISSN 0090-5364, 2168-8966. doi: 10.1214/aos/1176343943. URL ������������������������������������������������������������������������� ���������������������������������������������������������������� . Publisher: Institute of Mathematical Statistics. 39 Tze Leung Lai. Stochasti...

  6. [1979]

    doi: 10.1214/ aos/1176344840

    ISSN 0090-5364, 2168-8966. doi: 10.1214/ aos/1176344840. URL ����������������������������������������������������������������� �������������������������������������������������������������������������������� . Pub- lisher: Institute of Mathematical Statistics. T. L. Lai and D....

  7. [1981]

    doi: 10.1007/BF00536178

    ISSN 1432-2064. doi: 10.1007/BF00536178. URL ������������������������ ���������� . Tze Leung Lai and Hongsong Yuan. Stochastic Approximation: From Statistical Origin to Big-Data, Multi- disciplinary Applications. ����������� �������, 36(2), May

    doi

  8. [1987]

    doi: 10.1214/aop/1176992258

    ISSN 0091-1798, 2168-894X. doi: 10.1214/aop/1176992258. URL ��������������������������������������������������������������������������� ������������������������������������������������������������������������������� . Pub- lisher: Institute of Mathematical Statistics. Chuhan X...

  9. [1992]

    doi: 10.1007/978-3-0348-8609-3

    ISBN 978-3-7643-2733-0 978-3-0348-8609-3. doi: 10.1007/978-3-0348-8609-3. URL ��������������������������������������������������� . P. Major and P. R´ ev´ esz. A limit theorem for the Robbins-Monro approximation.����������� �� �� ����������� ���������������� ��� ��������� ����...

    doi

  10. [1995]

    URL �������������������������������������������������������������

    doi: 10.1080/15326349508807332. URL ������������������������������������������������������������� . Publisher: Marcel Dekker, Inc. G¨ otz D. Kersting. A Weak Convergence Theorem with Application to the Robbins-Monro Process.��� ������ �� �����������, 6(6):1015–1025,

    doi

  11. [2003]

    doi: 10.1007/b97441

    ISBN 978-0-387-00894-3. doi: 10.1007/b97441. URL ��������������������������������������� . T. L. Lai and Herbert Robbins. Adaptive Design and Stochastic Approximation. ��� ��� ���� �� ����������, 7(6):1196–1221, November

    doi

  12. [2004]

    ISBN 978-1-58113-838-2

    Association for Computing Machinery. ISBN 978-1-58113-838-2. doi: 10.1145/1015330.1015332. URL ���������������������������������������������� . 42

  13. [2009]

    doi: 10.1007/978-0-387-87835-5

    ISBN 978-0-387-87834-8 978-0-387-87835-5. doi: 10.1007/978-0-387-87835-5. URL ��������������������������������������������������� . Godfrey H. Hardy. ��������� ������. American Mathematical Society, June

    doi

  14. [2011]

    doi: 10.1214/11-AOS874

    ISSN 0090-5364, 2168-8966. doi: 10.1214/11-AOS874. URL �������������������������������������������������������������������������� ������������������������������������������������������������������������������������� . Publisher: Institute of Mathematical Statistics. David Sieg...

    doi

  15. [2017]

    arXiv:1609.04747 [cs]

    URL ������������� ������������������ . arXiv:1609.04747 [cs]. Jerome Sacks. Asymptotic Distribution of Stochastic Approximation Procedures. ��� ������ �� ������ ������� ����������, 29(2):373–405,

    arXiv

  16. [2021]

    doi: 10.1214/20-AOS1991

    ISSN 0090-5364. doi: 10.1214/20-AOS1991. URL ������������������������������� . arXiv:1810.08240 [math, stat]. Yu Kaniovski and G. Pflug. Non-standard limit theorems for urn models and stochastic approximation proce- dures. �������������� �� ����������� ���������� ������, January

    arXiv

  17. [2022]

    doi: 10.1609/aaai.v36i7.20701

    ISSN 2374-3468. doi: 10.1609/aaai.v36i7.20701. URL ������ ������������������������������������������������ . Number:

    doi

  18. [2023]

    doi: 10.1145/3597428

    ISSN 0360-0300. doi: 10.1145/3597428. URL ������������������� ������������������� . J. H. Venter. An Extension of the Robbins-Monro Procedure. ��� ������ �� ������������ ����������, 38(1):181–190, February

    doi

  19. [2024]

    arXiv:2207.04173 [math]

    URL �������������������������� �����. arXiv:2207.04173 [math]. D. A. Darling and Herbert Robbins. Confidence Sequences for Mean, Variance, and Median. ����������� �� ��� �������� ������� �� �������� �� ��� ������ ������ �� �������, 58(1):66–68, 1967a. ISSN 0027-8424. URL �����...

    arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.