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Complex SGD converges without analyticity, recovers superoscillations

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2026-07-04 16:14 UTC pith:UDGTCHTA

load-bearing objection Complex SGD convergence proofs are sound but incremental; the real issue is a missing 1/σ_k factor in the directional bias bound the 3 major comments →

arxiv 2604.23017 v2 pith:UDGTCHTA submitted 2026-04-24 cs.LG cs.NAmath.CVmath.NA

Complex Stochastic Gradient Descent and Directional Bias in Reproducing Kernel Hilbert Spaces

Natanael Alpay , Emeric Battaglia This is my paper
classification cs.LG cs.NAmath.CVmath.NA
keywords complexgradientkerneldescentproblemsresultsstochasticallows
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves that Stochastic Gradient Descent (SGD) can be extended to complex-valued parameters with full convergence guarantees, without requiring the objective function to be analytic. The authors use the Wirtinger gradient, a derivative notion from complex analysis that handles non-analytic, real-valued loss functions over complex domains. The key technical move is replacing the standard complex inner product with its real part and imposing an explicit assumption that minima are stationary points. With these adjustments, the authors show that the classical SGD convergence proofs (polynomial-time convergence, exponential convergence under strong convexity, and stationary convergence with adaptive step sizes) carry over essentially verbatim. The paper then extends a known directional bias result from real-valued SGD: for overdetermined linear systems with small step sizes, iterates converge primarily along the smallest singular-value direction. Finally, the authors demonstrate the method empirically by using complex SGD to solve kernel regression problems in complex reproducing kernel Hilbert spaces, recovering superoscillation functions in the Fock space and finite Blaschke products in the Hardy space to machine precision.

Core claim

The central object is the Wirtinger gradient, defined as twice the Wirtinger derivative, which serves as a replacement for the classical gradient when the objective function accepts complex inputs but need not be analytic. The authors establish that, under assumptions nearly identical to the real-valued SGD setting plus one additional assumption (stationarity of minima), this gradient yields the same convergence guarantees as classical SGD. The mechanism is straightforward: by using the real inner product (the real part of the standard complex inner product) in place of the complex inner product throughout the standard SGD convergence proofs, and by explicitly assuming that the Wirtinger梯度 (

What carries the argument

The Wirtinger gradient (2∇_z f for non-analytic f, 2∇_{z̄} f for analytic f), the real inner product ⟨z,w⟩_R := Re(⟨z,w⟩), Assumption 4 (stationarity of minima), and the connection between SGD and the Random Kaczmarz method for directional bias analysis.

Load-bearing premise

The Wirtinger gradient is not a true gradient in the classical sense: it does not automatically vanish at minima of real-valued functions over complex domains. The paper imposes this vanishing as an explicit assumption (Assumption 4) rather than deriving it, and verifies it only for the least-squares objective. The generality of the convergence results depends on whether this assumption holds broadly for other objectives, particularly non-convex ones.

What would settle it

Find a real-valued, convex objective function over a complex domain where the Wirtinger gradient does not vanish at a minimum, violating Assumption 4 and breaking the convergence guarantees.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Complex-valued neural networks can use SGD with the same convergence guarantees as real-valued networks, without splitting complex parameters into real and imaginary parts or requiring analyticity of the loss.
  • Directional bias in the small-step-size regime, previously known only for real-valued SGD, persists in the complex setting: iterates converge preferentially along the smallest singular-value direction of the data matrix.
  • Optimal functions in complex RKHS (Fock space, Hardy space, RBF spaces) that were previously obtainable only through closed-form analytic construction can be numerically recovered via stochastic iteration to machine precision.
  • The same proof strategy—replacing the inner product and imposing stationarity of minima—may extend other real-valued optimization results to the complex setting, including more refined SGD variants.

Where Pith is reading between the lines

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

  • If Assumption 4 fails for some non-convex objective encountered in complex-valued neural networks, the convergence guarantees would not hold, potentially limiting applicability to well-behaved losses such as least-squares-type objectives.
  • The directional bias result suggests that complex-valued models trained with small-step SGD may inherit an implicit bias toward low-variance directions, which could affect generalization behavior in ways analogous to the real-valued case.
  • The Mittag-Leffler–Fock space, mentioned by the authors as a natural next setting, would provide a test of whether the method extends to RKHS with more intricate kernel structure.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper proposes a complex variant of Stochastic Gradient Descent (SGD) based on the Wirtinger gradient, providing convergence guarantees (Theorems 4–6) under assumptions paralleling the real-valued setting, without requiring analyticity of the objective. The proofs are direct adaptations of classical SGD proofs from [21], with the real inner product ⟨·,·⟩_R replacing the standard inner product. The paper also extends directional bias results for SGD in kernel regression (Corollary 1) from the real to the complex setting, borrowing analysis from the Random Kaczmarz method [28]. Finally, the paper provides numerical experiments demonstrating recovery of superoscillation functions in the Fock space and Blaschke products in the Hardy space, using analytically known optima from [13] as targets.

Significance. The paper provides a clean and self-contained treatment of complex SGD with convergence guarantees that avoid analyticity assumptions, which is a useful contribution for practitioners working with complex-valued neural networks and complex RKHS problems. The verification that the least-squares objective satisfies all assumptions (Section D.2) is thorough. The numerical experiments are well-designed, using independently established analytical targets (Theorems 8 and 9 from [13]) as ground truth, and the recovery to near machine precision in both Fock and Hardy space settings is convincing. The connection to the Random Kaczmarz method for directional bias analysis is a natural and interesting bridge. However, the convergence proofs are straightforward adaptations of existing results, and the directional bias result contains a quantitative error (see Major Comment 1) that weakens one of the paper's claimed contributions.

major comments (3)
  1. Corollary 1 (and its proof in Appendix C): The error bound for the inconsistent-system case is incorrect. The recurrence yields E[δ_{t+1}] = α·δ_t + β, where α = (1 − ηnσ_k²/m) and β = (ηnσ_k/m)⟨ε, u_k⟩. Iterating gives δ_{t+1} = α^{t+1}δ_0 + β(1−α^{t+1})/(1−α). Since |β| ≤ (ηnσ_k/m)||ε|| and |1−α| = ηnσ_k²/m, the steady-state error term is bounded by ||ε||/σ_k, not ||ε|| as stated in Corollary 1. This factor of 1/σ_k is significant: the directional bias result is most interesting for the smallest singular value σ_min, where 1/σ_min could be large, potentially inflating the error floor well beyond what the paper claims. The consistent-system case (where ε = 0) is unaffected and remains correct. The bound in Corollary 1 should be corrected to ||ε||/σ_k (times (1−|α|^{t+1})), and the qualitative discussion should be updated accordingly.
  2. Section 5.1, Assumptions 4–7: The paper presents Assumption 4 (stationarity of minima) as a load-bearing assumption needed because 'the gradient is not truly a gradient in the usual sense.' However, for any real-valued function F: C^n → R that is differentiable in the real sense, the real gradient on R^{2n} vanishes at local minima, which implies both Wirtinger derivatives ∂F/∂z and ∂F/∂z̄ vanish. Since the paper defines ∇F := 2∇_{z̄}F for non-analytic F, Assumption 4 holds automatically—it is a theorem, not an assumption. The same reasoning applies to Assumptions 5–7: for real-valued objectives, the Wirtinger gradient is a representation of the real gradient on R^{2n}, so the descent lemma, convexity, and strong convexity all follow from their real-valued counterparts. The paper should acknowledge this explicitly, as it affects how the contribution is framed (the results are more direct
  3. Section 5.1, Assumptions 4–7 (continued): The paper should acknowledge that these assumptions are automatically satisfied for real-valued differentiable objectives, rather than presenting them as additional conditions that must be imposed. This does not invalidate the proofs, but it strengthens the paper's positioning and clarifies that the convergence results are essentially immediate consequences of the real-valued theory via the Wirtinger calculus correspondence. The verification in Section D.2 for least-squares is still valuable as a concrete example, but the general principle should be stated.
minor comments (7)
  1. Section 2: The definition of the Wirtinger derivative for the analytic case appears to have a sign error. The paper writes ∂_z f = (1/2)(∂f/∂x − i∂f/∂y) when f is analytic, but the standard convention for ∂f/∂z̄ (which is what should vanish for analytic functions) is (1/2)(∂f/∂x + i∂f/∂y). Please verify the conventions are consistent throughout.
  2. Section 6, Corollary 1 statement: The bound uses ||ε|| without defining the norm explicitly (presumably the Euclidean norm on C^m). This should be stated.
  3. Appendix C, proof of Corollary 1: The line '⟨a_i^*, v_k⟩ is the conjugate of the i-th entry of Av_k' should be clarified—Av_k = σ_k u_k, so the i-th entry of Av_k is σ_k times the i-th entry of u_k, and ⟨a_i^*, v_k⟩ = conj(a_i · v_k) = conj((Av_k)_i).
  4. Section 7.1.1, Equation (13): The normalized update rule is introduced without full justification of why it preserves convergence guarantees. The paper states it is 'purely to improve numerical stability' and that results are 'virtually identical,' but a brief remark on why the normalization does not affect the theoretical analysis would be helpful.
  5. Figure 1a caption: The text mentions α^{(100000)} but the figure label says 'Iteration' up to 10^5. Please ensure consistency in iteration counts between text and figures.
  6. Section D.2: The statement 'Assumption 6 follows similarly, except for our choice of f_j, we have σ_min = 0' is somewhat terse. A brief expansion would help readers verify that convexity (not strong convexity) of each f_j is indeed satisfied.
  7. References: [13] is cited as a 2026 preprint. Please ensure the citation is complete and accessible to readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and for identifying a genuine quantitative error in Corollary 1, as well as a framing issue regarding Assumptions 4–7. Both points are well-taken and will be addressed in the revision.

read point-by-point responses

  1. Referee: Major Comment 1: The error bound in Corollary 1 for the inconsistent-system case is incorrect. The steady-state error should be ||ε||/σ_k, not ||ε||.

    Authors: The referee is correct. Tracing through the proof in Appendix C: the recurrence is E[δ_{t+1}] = α·δ_t + β with α = (1 − ηnσ_k²/m) and β = (ηnσ_k/m)⟨ε, u_k⟩. Iterating gives δ_{t+1} = α^{t+1}δ_0 + β(1−α^{t+1})/(1−α). The steady-state term satisfies |β/(1−α)| = |⟨ε, u_k⟩|/σ_k ≤ ||ε||/σ_k. In the proof, we incorrectly bounded the geometric sum Σ_{i=0}^{t} (ηnσ_k/m)|α|^{t-i}·||ε|| by ||ε||, when in fact it equals (||ε||/σ_k)(1−|α|^{t+1}). The consistent-system case (ε = 0) is unaffected. We will correct Corollary 1 to state the bound as (||ε||/σ_k)(1−|α|^{t+1}) and update the qualitative discussion to note that the error floor depends on 1/σ_k, which is most significant for the smallest singular value. We thank the referee for catching this. revision: yes

  2. Referee: Major Comment 2: Assumptions 4–7 are automatically satisfied for real-valued differentiable objectives, since the Wirtinger gradient is a representation of the real gradient on R^{2n}. The paper should acknowledge this rather than presenting them as additional conditions.

    Authors: The referee is right that for a real-valued differentiable function F: C^n → R, the Wirtinger gradient ∇F := 2∇_{z̄}F is a faithful representation of the real gradient on R^{2n} under the identification C^n ≅ R^{2n}. Consequently, Assumption 4 (stationarity of minima) is indeed a theorem rather than an additional hypothesis: if F has a local minimum at z*, the real gradient vanishes, which implies both Wirtinger derivatives vanish. The same logic applies to Assumptions 5–7: the descent lemma, convexity, and strong convexity in the Wirtinger formulation follow directly from their real-valued counterparts because ⟨∇F(z), w−z⟩_R coincides with the real inner product of the corresponding real gradient with w−z in R^{2n}. We will revise Section 5.1 to state this principle explicitly and reframe Assumptions 4–7 as properties that hold automatically for real-valued differentiable objectives, rather than as additional conditions. The verification in Section D.2 for the least-squares objective remains valuable as a concrete instantiation. This reframing does not affect any of the proofs but does clarify that the convergence results are immediate consequences of the real-valued theory via the Wirtinger calculus correspondence, which is a point we should have made from the outset. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain rests on external proofs and parameter-free analytical targets

full rationale

The paper's three main components are each grounded in external sources rather than self-referential loops. (1) Theorems 4–6 are explicit adaptations of classical SGD convergence proofs from [21] (Garrigos and Gower, external authors), with the only modification being the use of ⟨·,·⟩_R and Assumptions 4–7. The proofs in Appendix B follow the standard structure and do not reduce to their own inputs. (2) Corollary 1 (directional bias) is attributed to Steinerberger [28] (external) with minor modifications for the complex setting. The skeptic correctly identifies a potential error in the bound (should be ||ε||/σ_k rather than ||ε||), but this is a correctness issue, not circularity. (3) The numerical experiments use analytical targets from [13] (self-citation by first author Alpay), but Theorems 8 and 9 of [13] provide explicit, parameter-free formulas for the optimal coefficient vectors (e.g., α* = (C_0(n,a),...,C_n(n,a))^T with C_j given by closed-form binomial expressions). The experiments verify that complex SGD recovers these known analytical solutions to machine precision—this is standard validation against independently derived targets, not a fitted input renamed as a prediction. The self-citation [13] is load-bearing for the experiments but provides parameter-free analytical results, not fitted values, so it does not raise the circularity score. Score 1 reflects the presence of a load-bearing self-citation that is nonetheless independent of the present paper's algorithmic claims.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, forces, or dimensions. The 'complex gradient' defined via Wirtinger calculus is a known mathematical object. The assumptions are reformulations of standard SGD assumptions adapted to the complex setting, with Assumption 4 being the most paper-specific addition. No free parameters are fitted to make derivations work; L, μ, σ² are standard optimization constants determined by the problem structure.

free parameters (5)
  • Step size η_t
    Standard SGD hyperparameter; bounded by 1/(4L) or 1/(2L) in theorems but chosen in experiments.
  • L (Lipschitz constant)
    Assumed to exist (Assumption 5); for least squares, L = σ²_max = ∥A∥² as shown in Section D.2.
  • μ (strong convexity constant)
    Assumed to exist (Assumption 7); for least squares, related to σ_min.
  • λ (regularization parameter) = 1
    Set to 1 in both numerical experiments; standard RKHS regularization parameter.
  • n (problem dimension) = 40 (Fock), 50 (Hardy)
    Chosen for numerical experiments; not a fitted parameter in the theoretical results.
axioms (5)
  • ad hoc to paper Assumption 4: Minima of F are stationary points of the complex gradient
    Section 5.1. This is imposed because the Wirtinger gradient does not automatically vanish at minima of real-valued functions over complex domains, unlike the real gradient. Verified for least squares in D.2 but imposed generally.
  • domain assumption Assumption 5: L-Lipschitz alternative for complex gradient
    Section 5.1. The standard L-Lipschitz condition on the gradient is reformulated as an inequality involving the complex gradient and the real inner product. In the real setting this follows from gradient Lipschitzness; here it is imposed separately.
  • domain assumption Assumption 6: Convexity alternative for complex gradient
    Section 5.1. Convexity reformulated using the complex gradient and real inner product.
  • domain assumption Assumption 7: μ-strong convexity of F
    Section 5.1. Strong convexity imposed on F directly rather than on component functions.
  • domain assumption Complex representer theorem (Theorem 10)
    Appendix A. Attributed to [13]; extends Kimeldorf-Wahba [12] to complex RKHS. Used to reduce infinite-dimensional optimization to finite-dimensional coefficient recovery.

pith-pipeline@v1.1.0-glm · 26204 in / 2880 out tokens · 216105 ms · 2026-07-04T16:14:51.285205+00:00 · methodology

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Stochastic Gradient Descent (SGD) is a known stochastic iterative method popular for large-scale convex optimization problems due to its simple implementation and scalability. Some objectives, such as those found in complex-valued neural networks, benefit from updates like in SGD and Gradient Descent (GD) with a newly defined ``gradient'' that allows for complex parameters. This complex variant of the SGD/GD methods has already been proposed, but convergence guarantees without analyticity constraints have not yet been provided. We propose a variant of SGD (complex SGD) that allows for complex parameters, and we provide convergence guarantees under assumptions that parallel those from the real setting. Notably, these results extend to GD as well, and with the same set of assumptions, we confirm that some directional bias results extend from the real to the complex setting for kernel regression problems. We provide empirical results demonstrating the efficacy of the complex SGD in kernel regression problems utilizing complex reproducing kernel Hilbert spaces. In particular, we demonstrate we may recover superoscillation functions and Blaschke products from the Fock Space and Hardy Space, respectively, as the optimal functions for a particular choice of a loss function.

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