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Stochastic Zeroth-Order Method for Computing Generalized Rayleigh Quotients

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Pith's one-line read A stochastic zeroth-order Riemannian algorithm that takes exact optimal steps along random tangent directions computes the maximum generalized Rayleigh quotient almost surely, at a sublinear rate, using only matrix-vector products with A an

desk verdict Solid one-sample convergence theory; the m-sample algorithm behind the numerics lacks the same guarantee — a real gap, not a fatal one. read the letter →

arxiv 2512.05520 v2 pith:VAZ2JBCM submitted 2025-12-05 math.OC cs.NAmath.NA

Jonas Bresch , Oleh Melnyk , Martin Schoen , Gabriele Steidl This is my paper · ORCID
classification math.OCcs.NAmath.NA MSC 65F3515A6068W20
keywords generalizedRayleighquotientstochasticzeroth-orderoptimizationRiemannianadjoint-freematrix-freealmostsureconvergencesublinearrateeigenvalueproblem
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

Maximizing the generalized Rayleigh quotient r(A,B,v)=⟨v,Av⟩/⟨v,Bv⟩ is a standard task in numerical linear algebra, but classical solvers need products with A^T and often the inverse of B. This paper proposes an iterative method that avoids both: at each step it samples a random direction from the B-tangent sphere of the current point, and moves to the exact maximizer of the quotient along that one-dimensional line, given by a closed-form formula. The central result is that when the eigenspace of the largest generalized eigenvalue has dimension < d−1, the quotient values converge almost surely to the global maximum R(A,B), the iterates converge to the set of maximizers, and the squared directional derivative b_k^2 — which measures the Riemannian gradient in expectation — vanishes at rate O(1/k). Since only products with A and B are used, the method is designed for problems where the transpose is unavailable or unreliable, such as imaging with approximate back-projectors.

What carries the argument

The central object is the one-sample exact-line-search update. For v∈S_B^{d−1} and x∈T_v, the restricted quotient g(τ)=r(A,B,v+τx) is a rational function of form (a+τ b+τ^2 c)/(1+τ^2 d), whose stationary equation g'(τ)=0 is a quadratic with closed-form solution (12). This reduces each stochastic iteration to an exact one-dimensional solve. The paper also relates b_k x_k to the zeroth-order gradient estimator (Theorem 7.1), showing E[b_k x_k | v_k] = (1/(d−1)) grad f(v_k), so the method is a zeroth-order Riemannian method with an optimal deterministic step along a random direction, and the monotone increase τ_k b_k/2 plays the role of the sufficient-decrease condition.

What would settle it

Run Algorithm 1 on a small diagonal instance, say d=3, B=I, A=diag(3,2,1), which has dim(Gmax)=1<d−1. The theorem says r(A,B,v_k)→3 a.s. and min_{k≤n} b_k^2 ≤ C/(n+1). If exact-arithmetic trials ever settle at a lower eigenvalue or the empirical minimal squared derivative decays slower than 1/n, the almost-sure convergence or the sublinear rate would be contradicted.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is: for real matrices A and symmetric positive definite B, maximizing r(A,B,v) can be achieved by repeatedly sampling a uniformly random unit tangent direction x_k at v_k, computing the step size τ_k via formula (12) that exactly maximizes the quotient on the line v_k+τ x_k, and setting v_{k+1} to the B-normalized v_k+τ_k x_k. Under the condition dim(Gmax)<d−1, this yields almost sure convergence r(A,B,v_k)→R(A,B) and dist(Gmax,v_k)→0, with b_k→0 and min_{k≤n} b_k^2 ≤ C/(n+1) almost surely (Theorem 5.6, Lemma 6.1). The formula's inputs are a=⟨v,Av⟩, b=⟨x,Av⟩+⟨Ax,v⟩, c=⟨x,Ax⟩ and d=⟨x,Bx⟩, all of which require only A and B products, not A^T or B^{-1

Load-bearing premise

The load-bearing premise for the practical claims is that the multi-sample Algorithm 2 inherits the one-sample convergence theorems: the paper states only an informal justification, noting that its search-direction distribution remains absolutely continuous, and numerical experiments rely on this unproved extension.

Editorial extensions

If this is right

  • For dim(Gmax)<d−1, Algorithm 1 (one sample) converges to a global maximizer almost surely and the objective increases monotonically at every step; as a corollary, it never terminates prematurely.
  • The stopping statistic b_k^2 is certified: it goes to zero almost surely, has the sublinear bound min_{k≤n} b_k^2 ≤ 8∥A∥^2∥B^{-1}∥(1+3κ(B))/(n+1), and in expectation controls the squared Riemannian gradient up to a factor d−1.
  • For dim(Gmax)=d−1 the algorithm terminates after one step almost surely; for dim(Gmax)=d it terminates immediately.
  • The method extends to complex matrices A,B∈C^{d×d} by realification, preserving the convergence guarantees for the maximal real generalized Rayleigh quotient.

Reading between the lines

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

  • If the informal m-sample extension can be made rigorous, the multi-sample algorithm would provide a practical variance-reduced eigensolver for streaming or large-scale generalized eigenproblems, with each iteration still costing only products with A and B and with no step-size parameter to tune.
  • Because the step is an exact one-dimensional maximizer, the method is closer to a randomized coordinate descent with optimal steps than to standard zeroth-order gradient ascent; this suggests that local linear convergence under an eigengap may hold, a stronger rate the paper does not claim.
  • The exact line-search step could be replaced by deterministic or low-discrepancy tangent directions, yielding a quasi-Monte Carlo variant with deterministic error control while remaining adjoint-free and inverse-free.
  • A clean empirical check of the variance-reduction claim: for a fixed v, the conditional variance of the estimator b_k x_k should scale as 1/m; the paper's Remark 7.1 predicts this, and it is directly measurable from samples.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a stochastic zeroth-order Riemannian algorithm for maximizing the generalized Rayleigh quotient R(A,B)=max_v ⟨v,Av⟩/⟨v,Bv⟩ without using A^T or B^{-1}. Algorithm 1 samples a uniformly random direction in the tangent space of the B-unit sphere and performs an exact line search along that direction; the step size has a closed form (Eq. 12). For dim(Gmax)<d−1, the paper proves monotone increase of the Rayleigh quotient, a.s. convergence of the objective to R(A,B), a.s. convergence of the iterates to the maximal eigenspace, and sublinear best-iterate rates for the quantity b_k (and a probabilistic bound for the eigen-residual). Algorithm 2 is a multi-sample variant used in the numerical experiments, but the paper only informally asserts that the convergence proofs extend to it.

Significance. If the claims are established, Algorithm 1 gives a conceptually simple, adjoint-free, inverse-free stochastic method with global a.s. convergence to the generalized Rayleigh quotient maximum, and the closed-form line search is a nice feature. The proofs are largely self-contained and include measure-theoretic details, a Borel–Cantelli argument, and explicit constants. The paper also makes code available and compares against Gen-Oja. The main limitation is that the practically recommended multi-sample Algorithm 2—used in all numerical demonstrations and for the advertised m>1 speedup—is not covered by the proofs; this weakens the significance of the multi-sample claims, though the one-sample method remains a meaningful contribution.

major comments (3)
  1. [Section 3 / Lemma 5.5 / Theorem 5.6] Algorithm 2 is not covered by the convergence theory. The only support is the informal sentence in Section 3: "We give only an informal justification that all these results can be extended to Algorithm 2." The convergence proof for Algorithm 1 hinges on Lemma 5.5, which provides a uniform lower bound p>0 on the surface measure of D_v under the uniform distribution on T_v∩S^{d−1}. For Algorithm 2 the search direction x_k is no longer uniform, and absolute continuity alone does not imply a uniform lower bound on P_{X_k|V_k=v}(D_v). Without such a bound, the Borel–Cantelli argument in Theorem 5.6 cannot be applied. Because Figures 2–7 and the claimed multi-sample improvement use Algorithm 2, this gap is load-bearing. The paper should either prove the required uniform lower bound for Algorithm 2's conditional distribution or explicitly restrict the theoretical claims to Algorithm 1 and prese
  2. [Section 7 / Theorem 7.1] As stated, the equality b_k x_k = 2 dgrad1 f(v_k) does not follow from the definition (11), where x_i∼N(0,I_d), the update uses R_v(μP_v x_i), and the denominator is μ. Algorithm 1 uses the normalized tangent vector x_k and the line-search parameter τ_k. The claimed equality holds only if dgrad1 is redefined with normalized tangent directions and the finite-difference scaling is adjusted accordingly. Since Remark 7.1 uses this equality to motivate the variance reduction in Algorithm 2, the statement should be corrected or the estimator redefined.
  3. [Section 6 / Abstract] The abstract states that the norm of the Riemannian gradient vanishes at a sublinear rate with probability one. The body proves a.s. convergence to Gmax (hence a.s. gradf→0 by continuity), but the quantitative statements are: Lemma 6.1 gives min_{0≤k≤n} b_k^2 ≤ O(1/n) a.s.; Theorem 6.3 is a Markov bound on the squared eigen-residual in probability. No theorem states an a.s. sublinear rate for ∥gradf(v_k)∥ itself. The final paragraph of Section 6 also switches between 'error' and 'squared error' when interpreting ε. Please state the precise best-iterate/probabilistic rates and adjust the abstract accordingly.
minor comments (4)
  1. [Appendix A, Lemma 5.5 proof] The spherical-cap area formula appears to have a typo: the integral upper limit and exponent are inconsistent with the usual beta-integral expression. Since only positivity of p is needed, this does not affect the proof, but the formula should be corrected.
  2. [Section 5, Theorem 5.4 proof] In the displayed chain after Eq. (19), the expression tends to λ_j, not to 0. The surrounding text uses the intended limit λ_j, so this is a typo.
  3. [Section 6, final paragraph] The phrase "the error is bounded by 1/√(n+1)" and "the error is O(1/n)" should be stated consistently for the squared eigen-residual; otherwise the reader may infer a rate for the norm that is not proved.
  4. [Algorithm 2] If \bar{x}_k = 0, the normalization in Step 4 makes x_k undefined. This event has probability zero for continuous sampling, but adding an explicit guard or stopping condition would make the algorithm statement rigorous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Algorithm 1's convergence is derived self-containedly; the informal extension to Algorithm 2 is an unproved gap, not a circular assumption.

full rationale

The paper's derivation chain is self-contained for Algorithm 1 and does not reduce any prediction to its inputs. The one-sample algorithm samples x_k uniformly on the tangent sphere (Lemma 4.1), computes the step size by the closed-form maximizer (Theorem 3.1) using only products with A and B, establishes monotone increase via exact algebra (Theorem 5.1), proves a positive-probability geometric lemma (Lemma 5.5) with a full proof in Appendix A, and then uses Borel-Cantelli to conclude convergence to R(A,B) and dist(Gmax,v_k)->0 (Theorem 5.6). The rate bounds in Lemma 6.1 and Theorem 6.3 are obtained by telescoping sums and Markov's inequality, not by assuming the conclusion. The only self-citation, [41], is used for Lemma 5.5, but the appendix reproduces the proof, so the citation is not load-bearing. The paper explicitly flags the extension to the m-sample Algorithm 2 as informal ('We give only an informal justification that all these results can be extended to Algorithm 2'), which is a genuine proof gap rather than circularity: Algorithm 2's convergence is not assumed in the proofs for Algorithm 1, it is merely asserted as a plausible inheritance. I find no self-definitional step, no fitted parameter renamed as a prediction, and no uniqueness or ansatz imported solely from the authors' prior work.

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

No fitted constants or invented objects. The only design choice is the random projection/line-search direction, which is not a new physical or mathematical entity. Free-parameter count is zero; all constants in the rates are explicit in terms of ||A||, ||B||, κ(B), and d.

assumptions (5)
  • domain assumption B is symmetric positive definite (B ∈ Sym_{≻0}^d).
    Defines the B-inner product, tangent space, B-sphere, and generalized eigenvalues; stated in Section 2.1.
  • domain assumption The main convergence theorem assumes dim(Gmax) < d−1; the cases dim(Gmax)=d and d−1 are handled separately in Remark 4.2 and Theorem 4.5.
    This is a condition on the problem, not an ad hoc fit; it excludes degenerate eigenspaces where the random directions have zero probability of escaping.
  • domain assumption Sampled search directions are uniformly distributed on the tangent unit sphere, obtained by projecting Gaussian vectors (Lemma 4.1).
    Algorithmic construction; the uniformity is what makes the measure-zero arguments and probability lower bounds work.
  • standard math Affine subspaces of dimension < d−1 have surface measure zero on the sphere (Lemma 4.2).
    Used in termination and convergence proofs; standard result proved in Appendix A.
  • standard math Borel-Cantelli lemma and dominated convergence theorem are available.
    Used in Theorem 5.6 and Theorem 5.4 to convert probability bounds into almost-sure statements.

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Cite this review

Pith. "Pith review of Stochastic Zeroth-Order Method for Computing Generalized Rayleigh Quotients." pith.science (2026). https://pith.science/paper/VAZ2JBCM

@misc{pith2026251205520,
  author       = {Pith},
  title        = {Pith review of: Stochastic Zeroth-Order Method for Computing Generalized Rayleigh Quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VAZ2JBCM}},
  note         = {Machine review of arXiv:2512.05520}
}
read the original abstract

The maximization of the (generalized) Rayleigh quotient is a central problem in numerical linear algebra. Conventional algorithms for its computation typically rely on matrix-adjoint products, making them sensitive to errors arrissing from adjoint mismatches. To address this issue, we introduce a stochastic zeroth-order Riemannian algorithm that maximizes the generalized Rayleigh quotient without requiring adjoint or matrix inverse computations. We provide theoretical convergence guarantees showing that the iterates converge to the set of global maximizers of the (generalized) Rayleigh quotient and the norm of the Riemannian gradient vanishes at a sublinear rate with probability one. Our theoretical results are supported by numerical experiments, which demonstrate the excellent performance of the proposed method compared to state-of-the-art algorithms.

Figures

Figures reproduced from arXiv: 2512.05520 by the authors.

Figure 1
Figure 1. Visualization of the generalized Rayleigh quotient r(A, B, v) for v = (v1, v2) T ∈ R 2 \ {0} and A = ( 3 1 1 2 ). Its values on S 1 B are highlighted by the solid line. Left: B = I2, Right: B = ( 2 0.5 0.5 1 ). A value λ ∈ R is called a generalized eigenvalue of (AH, B), it there exists v ∈ R d \ {0} fulfilling A Hv = λBv. (4) Indeed, such λ is also an eigenvalue of B−1AH, resp. of B − 1 2 AHB − 1 2 and we denote th… view at source ↗
Figure 3
Figure 3. Error estimation towards Riemannian gradient for d = 100 and m ∈ {1, 10, 100}. Left: Comparison of |bk| 2 (solid lines) with 1 d−1 ∥grad f(v k )∥ 2 (dashed lines). Right: Error ∥(d − 1)x k − grad f(v k )∥. Gaussian matrices B˜ ∈ R d×d and performing their QR-decomposition. Finally, we set B = Q diag(λ)QT. In contrast to the setup of [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 5
Figure 5. Runtime of our algorithm for m ∈ {1, 10, 100} for the matrices from [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figures from the paper (3 more)
Figure 6
Figure 6. Figure 6: Comparison of our algorithm with the two variants of ZO-RGA for m = 100 and d ∈ {10, 50, 100, 500}. Comparison of m = 10 (dashed lines) and m = 100 (solid lines). 8.5. Karhunen-Lo`eve Problem In the last experiment, we compare our approach with Gen-Oja methods for the …
Figure 7
Figure 7. Figure 7: Solution of a Karhunen–Lo`eve problem. Top: The sin2 B-error for the Gen-Oja methods and our algorithm. Bottom: The approximations of the generalized eigenfunction (black dotted) after 500 iterations. Studies in the Mathematical Sciences, JHU Press, Baltimore, MD, 2013…
Figure 8
Figure 8. Figure 8: Visualization in 3d of the situation in the proof of Lemma 5.5. where S ⊆ Bε,B(±v ∗ ). Then, inf v∈S d−1 B σBv(DB,v) ≥ inf v∈S d−1 B σBv(D˜B,v) ≥ pε,d,B. In the following, without loss of generality, we construct S for ⟨v, v∗ ⟩B ≥ 0 and otherwise, it can be done analog…

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    In this case, we takeS=B r,B(˜u) wherer :=ε 2/8,c :=ε/4 √ 2 and ˜u:= u ∥u∥B withu :=v ∗ −c(v ∗ −v) = (1−c)v ∗ +cv.(22) Next, we step-by-step show that this choice ofSsatisfies all desired properties. Since B∈Sym d ≻0, the norm∥·∥ B is strongly. By definition,uis a convex combi...

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