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Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem

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Pith's one-line read This paper proves that a full-history multilevel Picard estimator, which beats the curse of dimensionality for max-norm Lipschitz semilinear heat equations, suffers that curse for the Hamilton-Jacobi-Bellman equation of a Gaussian control…

desk verdict A real lower bound proving that the m=n MLP schedule blows up in dimension, but not the general curse-of-dimensionality claim the title announces. read the letter →

arxiv 2506.23969 v1 pith:EGDES6MI submitted 2025-06-30 math.NA cs.NA

Martin Hutzenthaler , Tuan Anh Nguyen This is my paper · ORCID
classification math.NAcs.NA MSC 65C9960H99
keywords fullhistorymultilevelPicardapproximationscurseofdimensionalityHamilton-Jacobi-Bellmanequationstochasticcontrolsemilinearheathigh-dimensionalPDEapproximationEuclidean-normLipschitznonlinearityseminorm-type
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

Full-history recursive multilevel Picard (MLP) approximations are proved to suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a classical Gaussian stochastic control problem. The counterexample is minimal: the value function depends only on the first spatial coordinate, so the underlying approximation problem is not inherently high-dimensional, and fixed-point Picard iterations converge with error growing only polynomially in the dimension. Yet the L2 error of the MLP estimator $U_{n,n}^{d,0}(0,0)$ at the origin grows faster in the dimension $d$ than any polynomial, for every fixed number of levels $n$. The paper isolates the mechanism: the nonlinearity $f(v)=(\sum_{j=2}^d |v_j|^2)^{1/2}$ is Lipschitz with respect to the Euclidean norm but its Lipschitz constant with respect to the max-norm grows like $\sqrt{d-1}$, so the standard MLP theory does not apply. If the theorem is right, the max-norm Lipschitz condition in earlier MLP results is not a technical convenience but a genuine complexity boundary.

What carries the argument

The central object is the full-history MLP recursion (8)/(6): at level $n$, the estimator $V_{n,m}$ is a telescoping sum of $mn$ terminal increments and $n$ levels of difference terms $(f\circ V_{\ell,m} - 1_{N(\ell)} f\circ V_{\ell-1,m})$ evaluated at random times $R_t = t+(1-t)r$ with density $\varrho(t,s)=1/(2\sqrt{1-t}\sqrt{s-t})$, scaled by the weight $m^{n-\ell}\varrho(t,R_t)$. The nonlinearity $f(v)=(\sum_{j=2}^d |v_j|^2)^{1/2}$ is of seminorm-type: it satisfies $|f(v_1+v_2)|\le |f(v_1)|+|f(v_2)|$ but has max-norm Lipschitz constant $\sqrt{d-1}$, unbounded in $d$. The lower-bound proof uses evenness of $g$ and $f$ to center the MLP sums, Lemma 3.1 to lower-bound the variance of a sum of independent random fields evaluated at one random point by the sum of their variances, and an iterated-integral recursion that produces the growth $\varepsilon_{0,m}(n)\ge d^{n/2}/((34m)^{n/2}\sqrt{n!})$ once $d\ge (1224m)^n n!$.

What would settle it

Compute the empirical second moment $E[\|f(V_{n,m}^{d}(0,0))\|^2]$ for fixed $n,m$ at dimensions $d$ beyond the stated threshold $d\ge(1224m)^n n!$; Theorem 3.2 predicts it grows at least like $d^n/((34m)^n n!)$, so observing it bounded or growing polynomially there would contradict the proof. Alternatively, if a concrete definition of $N(\ell)$ in the recursion can be exhibited under which the symmetry and variance steps fail, the lower-bound chain breaks at that point.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for the HJB equation $\partial_t u + \tfrac12\Delta u + (\sum_{j=2}^d |\partial_{x_j}u|^2)^{1/2}=0$ with terminal data $u(1,x)=|x_1|$, for every $p\in[0,\infty)$ and every integer $n>2p$, $$\liminf_{d\to\infty} $d^{{-p}}$\Big(E\big[\|U_{n,n}^{d,0}(0,0)-(u_d,\nabla u_d)(0,0)\|_{\mathbb{R}^{d+1}}^2\big]\Big)^{1/2}=\infty,$$ where $U_{n,n}^{d,0}$ is the $n$-level MLP approximation with $m=n$ samples per level, zero initialization, and random times with density $\varrho(t,s)=1/(2\sqrt{1-t}\sqrt{s-t})$. Equivalently, the error is not bounded by any polynomial in $d$, no matter how many levels are fixed. The paper stresses the contrast with the facts that the solution depends only on $x_1$ and that Picard fixed-point iterations for the same equation converge with errors growing at most polynomially; the failure is a property of the MLP estimator itself.

Load-bearing premise

The result concerns the specific full-history MLP estimator with zero initialization $V_{-1}=V_0=0$, the stated random-time density, and the convention that the undefined set $N(\ell)$ acts only through the bound $0\le 1_{N(\ell)}\le1$; a different estimator, initialization, or a clarified definition of $N(\ell)$ could in principle escape the lower bound.

Editorial extensions

If this is right

  • The full-history MLP estimator as defined here cannot be the basis of a general polynomial-in-dimension solver for HJB equations whose nonlinearities are Lipschitz only in the Euclidean norm.
  • The earlier curse-of-dimensionality-free guarantees for MLP require the max-norm Lipschitz constant to stay bounded; this paper shows that condition is essential, not just an artifact of the proof.
  • For the counterexample, fixed-point Picard iterations remain polynomial, so the difference in complexity is caused by the randomized multilevel estimator itself, giving a clean separation between the two approximation strategies.
  • The quantitative lower bound grows like $d^{n/2}$ for fixed $n$, so increasing the number of levels only raises the polynomial degree, while the sample count required for the bound scales like $m^n n!$; the trade-off does not remove the curse.

Reading between the lines

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

  • An immediate testable extension is to vary the estimator: nonzero initialization, a different density for the random times, or a fully defined $N(\ell)$; if any variant keeps the error polynomial, the curse is an artifact of this recursion rather than of MLP-type randomized estimators in general.
  • The same lower-bound strategy should apply to other semilinear PDEs whose nonlinearity is a seminorm of the gradient, such as $\|\nabla u\|$, suggesting that the Euclidean-norm class is broadly the wrong setting for MLP guarantees.
  • Because the target has bounded derivatives and depends on one coordinate, the example is a natural stress test: any proposed high-dimensional PDE method can be benchmarked on this equation to expose hidden dimension dependence in its variance.
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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 / 3 minor

Summary. The manuscript studies the full-history recursive multilevel Picard (MLP) approximation U^θ_{n,m} for the Hamilton-Jacobi-Bellman equation (5) with terminal data g(x)=|x_1| and nonlinearity f(z)=(Σ_{j=2}^d z_j^2)^{1/2}, a Euclidean-norm Lipschitz function whose Lipschitz constant in the maximum norm is unbounded in d. Section 2 proves moment upper bounds under a seminorm-type subadditivity assumption. Section 3 proves a lower bound, Theorem 3.2, of the form (E|f(V^0_{n,m}(0,0))|^2)^{1/2} ≥ d^{n/2}/((34m)^{n/2}√(n!)) for d ≥ (1224m)^n n!, and Corollary 3.3 converts this into a lower bound for the L^2 approximation error of both the value and the gradient. Taking m=n yields Theorem 1.1: for every p ≥ 0 and every integer n > 2p, liminf_{d→∞} d^{-p}(E|U^0_{n,n}(0,0)-(u_d,∇u_d)(0,0)|^2)^{1/2} = ∞. The paper concludes that MLP approximations suffer from the curse of dimensionality for this equation.

Significance. If read as a statement about the specific parameter schedule m=n, the proof is original and technically substantial: it uses symmetry to center the MLP increments, a variance lower bound for sums of i.i.d. random fields evaluated at a common random argument, and a recursive lower bound iterated back to the base level. The upper and lower bounds are consistent, and no parameters are fitted; the proof is a genuine mathematical contribution. The result also shows that the maximum-norm Lipschitz condition underlying earlier MLP convergence results is not merely technical. However, the advertised method-level conclusion is not established: m is a free parameter in Setting 2.1, and for n=1 the explicit variance (d+2)/m shows that choosing m = ⌈d ε^{-2}⌉ gives L^2 error O(ε) at cost O(d^2 ε^{-2}). Thus the abstract's sweeping claim that MLP approximations suffer from the curse of dimensionality is too strong; it should be restricted to the diagonal schedule or to a precisely stated class of schedules.

major comments (3)
  1. [Abstract, Theorem 1.1, Setting 2.1] The title and abstract claim that MLP approximations suffer from the curse of dimensionality, but the proved statement is only about the diagonal schedule m=n. In Setting 2.1, m is a free algorithmic parameter, and the lower bound in Corollary 3.3 is d^{n/2}/((34m)^{n/2}√(n!)) - 1, which becomes vacuous once m grows linearly in d. For n=1, equation (8) gives V_{1,m}(0,0) = m^{-1} Σ_{i=1}^m |Z_{1,i}| Z_i with Z_i ∼ N(0,I_d), so E‖V_{1,m}(0,0)‖² = (d+2)/m; choosing m = ⌈d ε^{-2}⌉ yields error O(ε) with cost O(d² ε^{-2}). Therefore the MLP family with dimension-dependent sample sizes does not suffer the curse for this example. The manuscript should explicitly restrict its claim to the m=n schedule, adjust the title and abstract accordingly, or prove a lower bound for all polynomial-cost schedules (which the n=1 calculation shows is impossible).
  2. [Setting 2.1, Eq. (8)] The recursion (8) contains the factor 1_{N(ℓ)}, but the random set N(ℓ) is never defined in Setting 2.1 or anywhere else in the paper. The subsequent estimates use only the fact that 0 ≤ 1_{N(ℓ)} ≤ 1, so the bounds may in fact hold for any such indicator, but as written the algorithm, and hence Theorem 1.1, is not fully specified. Please define N(ℓ), or state explicitly that the theorem and corollaries hold for every choice of N(ℓ) with 0 ≤ 1_{N(ℓ)} ≤ 1.
  3. [Section 3, Theorem 3.2 and Corollary 3.3] The term 'curse of dimensionality' is used without a precise definition. If it means the standard complexity-theoretic notion (no algorithm in the considered class can achieve accuracy ε with cost polynomial in d and 1/ε), then Theorem 1.1 does not establish it, because the theorem fixes n and lets d tend to infinity, and because m is a free parameter. The paper should define the notion of curse it proves and state the resulting complexity of the m=n schedule; otherwise the reader cannot tell whether the negative result is about a method or about a particular parameter choice.
minor comments (3)
  1. [Section 1, paragraph 2] The word 'counded' in 'the Lipschitz constant is counded in the dimension' should be 'bounded'.
  2. [Before Corollary 3.3] The word 'approixmation' in 'immediately implies a lower bound for the approixmation errors' should be 'approximation'.
  3. [Theorem 3.2 proof] The notation cW^θ for the reflected Brownian motion is typographically heavy and easy to confuse with W^θ; consider using a simpler symbol such as W̃^θ or B^θ.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lower bound is a direct variance estimate, and the self-cited MLP lemmas are independent technical facts rather than restatements of the theorem.

full rationale

The central claim (Theorem 1.1) is proved by direct estimates. Theorem 3.2 lower-bounds E[|(f∘V^0_{n,m})(0,0)|^2] using Lemma 3.1, symmetry identities (48) and (52), and explicit Brownian-moment computations (53)-(56); Corollary 3.3 converts this into an approximation-error lower bound via the reverse triangle inequality. Nothing in this chain is fitted to data, and no conclusion quantity is inserted by definition: the bound dn/2/(34m)n/2 sqrt(n!) - 1 is derived, not assumed. The proof relies on [9, Lemma 3.2/3.3] for independence and distributional properties of the MLP hierarchy and on [12, Lemma 2.2] for disintegration; these are prior published results about the MLP construction itself, not about the HJB counterexample or the lower bound, so they are independent support even though the author sets overlap. The 'uniqueness of the solution of PDE (5)' invoked after Corollary 3.3 is standard linear heat-equation uniqueness, not an imported self-citation. The theorem is narrow in scope: it lower-bounds the diagonal schedule m=n for the specific density rho(t,s)=1/(2 sqrt(1-t) sqrt(s-t)), with the never-defined set N(l) appearing in recursions (6) and (8); changing the sampling schedule or the density is not addressed. That is a scope/rigor limitation, not circularity. In particular, allowing m to grow with d (e.g., m approximately d/epsilon^2 for n=1) would evade the fixed-schedule lower bound, but Theorem 1.1 states only the m=n diagonal result, and the abstract's 'curse' language is an interpretation of that theorem. No circularity pattern 1-6 is present.

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

The paper is a pure proof paper: no fitted constants, no invented entities. The proofs depend on standard probabilistic tools and on prior MLP distributional lemmas from the same research group; these are unproved in the present manuscript but are not restatements of the target claim.

assumptions (4)
  • domain assumption Dynamic programming principle identifying the value function as the solution of the HJB equation (3).
    Invoked in the introduction and in the proof of Theorem 1.1; cited to Yong-Zhou [22, Prop. 4.3.5]. Standard in stochastic control.
  • domain assumption Distributional and independence properties of MLP approximations, cited to [9, Lemma 3.2, 3.3].
    Used throughout Sections 2 and 3 to decouple the random fields in the MLP recursion; not proved in this paper, so the reader must rely on the cited prior work.
  • standard math Disintegration theorem (cited to [12, Lemma 2.2]).
    Used in Lemma 3.1 and throughout the moment computations.
  • standard math Uniqueness of the classical solution of the semilinear heat equation (5).
    Invoked after Corollary 3.3 to conclude Theorem 1.1; standard for this Lipschitz semilinear setting.

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Cite this review

Pith. "Pith review of Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem." pith.science (2026). https://pith.science/paper/EGDES6MI

@misc{pith2026250623969,
  author       = {Pith},
  title        = {Pith review of: Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EGDES6MI}},
  note         = {Machine review of arXiv:2506.23969}
}
read the original abstract

Full history recursive multilevel Picard (MLP) approximations have been proved to overcome the curse of dimensionality in the numerical approximation of semilinear heat equations with nonlinearities which are globally Lipschitz continuous with respect to the maximum-norm. Nonlinearities in Hamilton-Jacobi-Bellman equations in stochastic control theory, however, are often (locally) Lipschitz continuous with respect to the standard Euclidean norm. In this paper we prove the surprising fact that MLP approximations for one such example equation suffer from the curse of dimensionality.

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