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Deep neural network approximations for Monte Carlo algorithms

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Pith's one-line read This paper proves a transfer theorem: if a discrete Monte Carlo scheme approximates a function without the curse of dimensionality and the scheme's components are DNN-approximable at polynomial cost, then the function is DNN-approximable…

desk verdict The abstract transfer theorem is not proven as written—(2.23) drops a nonnegative term—but the ANN calculus and the Kolmogorov application are real work and worth a revision. read the letter →

arxiv 1908.10828 v1 pith:XMBBTNXO submitted 2019-08-28 math.NA cs.LGcs.NAmath.APmath.PR

Philipp Grohs , Arnulf Jentzen , Diyora Salimova This is my paper · ORCID
classification math.NAcs.LGcs.NAmath.APmath.PR MSC 65C0565C3068T0735K1541A25
keywords deepneuralnetworksMonteCarloalgorithmscurseofdimensionalityKolmogorovPDEsEulerschemerectifierapproximationratestractability
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

Deep neural networks are known to approximate solutions of certain PDEs without the curse of dimensionality, and one recurring proof strategy is to show that a DNN can mimic a tractable Monte Carlo scheme. This paper extracts that strategy into a general transfer theorem. It proves that if a function can be approximated by a discrete Monte Carlo scheme with error bounded polynomially in the dimension and inverse accuracy, and if the scheme's update and averaging maps can themselves be realized by DNNs with polynomial parameter cost and a composition-closure property, then the function has DNN approximations with the same qualitative tractability. The theorem gives an explicit exponent for the number of parameters in terms of the scheme's exponents. Applied to Kolmogorov PDEs with constant diffusion and possibly nonlinear Lipschitz drift, it yields DNN approximations at $L^p$ error $\varepsilon$ with parameter count bounded polynomially in $d$ and $\varepsilon^{-1}$.

What carries the argument

The machinery is an explicit ANN calculus: formal operations of composition, parallelization, sum, and scalar multiplication on feedforward networks with rectifier activations, each with controlled growth of the parameter count (Definitions 3.1–3.28 and Lemmas 3.8–3.29). The load-bearing object is the class $\mathcal{N}_{d,\varepsilon}$ of admissible networks, together with a composition-closure hypothesis: for every admissible $\Phi$ and every sampled point $z$, there must be networks $\varphi_z$ realizing the composition $(R(f_{N,d}^{\varepsilon,z})) \circ (R(\Phi))$ with parameter overhead at most $C N^{n_1} d^{d_3} \varepsilon^{-e}$ and depth independent of $z$. A second construction, Lemma 3.29, builds a network whose output is the average of $M_N$ independent network evaluations, and Lemma 3.30 verifies the closure condition for the Kolmogorov application by composing with an exact identity network and a scaled drift network while controlling the width of the composed network. Proposition 4.4 supplies the underlying Monte Carlo Euler error estimates.

What would settle it

A concrete way to test the theorem: produce a discrete Monte Carlo scheme that satisfies the approximation, moment, and Lipschitz assumptions (2.9)–(2.13) and (2.14)–(2.15), but for which every DNN family approximating the update map forces the composition-closure parameter overhead to grow faster than any polynomial in $d$ or $\varepsilon^{-1}$, while the scheme itself still has polynomial-in-$d$, polynomial-in-$1/\varepsilon$ error. Such an example would show that the inheritance conclusion (2.16) fails in general, and it would isolate the composition-closure hypothesis as the obstruction.

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Extended reading notes

Core claim

The central claim is Theorem 2.3: under the moment, approximation, Lipschitz, and composition-closure assumptions (2.9)–(2.15), for every $d \in \mathbb{N}$ and $\varepsilon \in (0,1]$ there are rectified DNNs $\Psi_{d,\varepsilon}$ with $\left(\int_{\mathbb{R}^d} |u_d(x) - (R(\Psi_{d,\varepsilon}))(x)|^p \, \nu_d(dx)\right)^{1/p} \leq \varepsilon$ and parameter count bounded by $c \, d^{d_0(n_1+n_2+1)/n_0 + d_3 + e \delta} \varepsilon^{-(n_1+n_2+1)/n_0 - e}$, where $\delta = \max\{d_5 + \theta(d_1+d_2),\, d_4 + d_6 + 2\theta(d_1+d_2)\}$. The proof replaces the Monte Carlo update $f_{N,d}$ by its DNN approximant, runs the recursion through composed networks, evaluates a DNN realization of the averaging function $g$, and chooses the number of steps $N$ and an internal accuracy $E_{d,\varepsilon}$ so that the Monte Carlo error, the $g$-approximation error, and the perturbation error from replacing $f$ by a network all stay below $\varepsilon$. The conclusion is a direct inheritance: the discrete scheme's polynomial-in-$d$, polynomial-in-$1/\varepsilon$ error becomes a polynomial parameter bound for the DNN.

Load-bearing premise

The load-bearing premise is the composition-closure condition: every admissible network must remain admissible after composition with the DNN version of the Monte Carlo update, with parameter overhead at most $C N^{n_1} d^{d_3} \varepsilon^{-e}$ and depth independent of the sampled point; if this width and depth control fails, the transfer theorem does not apply.

Editorial extensions

If this is right

  • Any function tractably approximated by a discrete Monte Carlo scheme whose components admit polynomial-cost DNN representations inherits DNN tractability, with explicit parameter exponents (Theorem 2.3).
  • Solutions of Kolmogorov PDEs with constant diffusion matrix, possibly nonlinear Lipschitz drift, and terminal data that are Lipschitz and polynomially growing are DNN-approximable without the curse of dimensionality (Theorem 4.5).
  • On the unit cube with uniform measure, the parameter count for these PDE solutions is at most $c \, \varepsilon^{-(e+6)}$ times a polynomial in $d$ whose exponent is given in closed form by the data-approximation constants (Corollary 4.6).
  • The result makes explicit the dimension and accuracy exponents for classes previously known only through existence results, including Black-Scholes PDEs, semilinear heat equations, and nonsmooth value functions in zero-sum games.

Reading between the lines

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

  • The transfer statement suggests a general recipe: any future Monte Carlo sampler that is tractable and whose components are network-friendly in the composition sense automatically yields DNN tractability; samplers such as multilevel or quasi-Monte Carlo variants could be compared through the induced parameter exponents.
  • The composition-closure hypothesis is likely the practical bottleneck: for nonlinear update maps it requires controlling the width of composed networks, not just their depth, so schemes with high-dimensional or ill-conditioned updates may fail the hypothesis even when they are otherwise tractable.
  • The explicit exponents imply that improving the scheme's accuracy in $\varepsilon$, the dimension growth of the moments, or the Lipschitz constants directly improves the DNN parameter bound, giving quantitative targets for designing network-friendly samplers.
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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

1 major / 4 minor

Summary. The paper develops an abstract transfer theorem (Theorem 2.3) asserting that if a function can be approximated by a suitable discrete Monte Carlo scheme without the curse of dimensionality, and if the steps of that scheme can be realized by DNNs without the curse of dimensionality, then the function itself can be approximated by DNNs without the curse of dimensionality. The theorem also provides explicit polynomial exponents in the dimension d and in the reciprocal accuracy ε. The main application is to Kolmogorov PDEs with constant diffusion and possibly nonlinear drift, for which the authors prove in Theorem 4.5 that viscosity solutions can be approximated by rectified DNNs without the curse of dimensionality. The proof combines discrete Gronwall-type moment bounds with a Monte Carlo Euler error analysis and an ANN calculus developed in Section 3.

Significance. If the central theorem were established, the paper would be a valuable contribution: it systematizes a widely used two-step argument (Monte Carlo scheme first, DNN realization second), gives explicit exponents in the parameter-count bounds, and extends prior Kolmogorov PDE results by making the exponents explicit. The ANN calculus in Lemmas 3.29–3.30 is coherent and useful, and the Kolmogorov PDE application is well structured. However, the main proof contains a gap in the derivation of the growth bound (2.23) from (2.22), and that gap is load-bearing for the moment estimates and the final conclusion. The abstract theorem is therefore not established as written, although the argument appears repairable by adding an explicit linear-growth hypothesis on the discrete scheme, which is satisfied by the Euler scheme used in Section 4.

major comments (1)
  1. [§2.2, Eqs. (2.22)–(2.23)] The deduction of (2.23) from (2.22) is invalid: the term ε C d^{d4}(d^{θ(d1+d2)} + ‖x‖^θ) in (2.22) is nonnegative and is simply discarded. Since θ ≥ 1 and ‖x‖ is unbounded, this term is not bounded by the right-hand side of (2.23), which is affine in ‖x‖. The subsequent moment bounds (2.26)–(2.28), the perturbation estimate (2.30), the error bound (2.38), and the final parameter bound (2.16) all depend on (2.23). The hypotheses (2.11)–(2.13) do not imply (2.23); for instance, take d1 = d2 = 0, C = 1, N = 1, ε = 1, θ = 2, d4 > 0, f_{N,d}(z,x) = x + d^{d4}, R(f^{ε,z}_{N,d})(x) = x. Then (2.11)–(2.13) hold, but (2.23) fails at x = 0 for large d. The proof can be repaired by adding an explicit growth assumption such as ‖f_{N,d}(z,x)‖ ≤ (1 + C/N)‖x‖ + C d^{d2}(d^{d1} + ‖z‖), which is satisfied by the Euler scheme in Section 4; without such an assumption, Theorem 2.3 is not established as stated.
minor comments (4)
  1. [Theorem 2.3, composition-closure assumption] The assumption on the composed networks is stated as "D(φz) = D(φz)" with two bound variables named z; this is tautological as written. It should presumably read D(φz) = D(φ_z̃) for all z, z̃ ∈ R^d, expressing that the depth does not depend on the point z.
  2. [Proof of Theorem 2.3, Eq. (2.18)] The notation N_{d,ε} is used both for the set of networks in the theorem statement and for the integer step-size parameter in (2.18). This overloading makes the proof harder to follow; a different symbol, such as N̂_{d,ε}, would help.
  3. [Lemma 3.30] The width condition l_{2,L2−1} ≤ l_{1,L1−1} + i is essential for the composition to remain in the admissible class, but it is not explained. A short remark on why this condition is natural for the ANN calculus would improve readability.
  4. [General typography] There are several typographical artifacts, such as "www.univie.ac.at" split as "uni vie.ac.at" and the notation "(4eC+1C3)" which should read "(4e^{C+1} C^3)". These should be corrected in a revised version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the transfer theorem and Kolmogorov application are conditional on external Monte Carlo and Feynman-Kac inputs, not on their own conclusions.

full rationale

The main result Theorem 2.3 is a conditional transfer theorem: assumptions (2.9)-(2.15) postulate that a discrete Monte Carlo scheme approximates u and that DNNs represent the scheme's transition f and output functional g with controlled complexity. The conclusion (2.16) is a parameter bound for a DNN approximating u, obtained by propagating errors through Lemma 2.2 and composing networks. The exponents in (2.16) are sums and products of the input exponents, not renamings of them, and no hypothesis states directly that u itself is DNN-approximable with the claimed rate. Thus the theorem is not self-definitional. In the Kolmogorov application, u_d(T,.) is identified with E[phi0,d(X_T^.)] via the standard Feynman-Kac representation cited from [32], and the Euler/Monte Carlo error is bounded in Proposition 4.4 from Lemmas 4.1-4.3; these are external analytic estimates, not the paper's own target. The assumed DNN approximability of phi0,d and phi1,d is a genuine input; the PDE solution is obtained by composing and averaging those input approximations, not by assuming the conclusion. The repeated citations to the authors' earlier ANN calculus [22] and Monte Carlo results [32] supply elementary matrix-composition rules and standard stochastic estimates; none of those cited results asserts Theorem 2.3 or Theorem 4.5 and none is a uniqueness or completeness theorem that forces the answer. The proof gap noted by the skeptic at (2.23), where the nonnegative epsilon term is dropped when passing from (2.22) to (2.23), is a correctness concern about an unproven moment bound, not a circularity: it does not make the conclusion equal to an input by construction. Therefore no circular step is identified.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The theorem is built on assumptions about the Monte Carlo scheme and network expressibility, which are explicit hypotheses rather than hidden free parameters. The only 'free parameters' are the abstract exponents d0...d6, n0...n2, e, theta, C, p, which are quantified inputs of the theorem, not fitted values.

free parameters (1)
  • Assumption constants d0,...,d6, n0,n1,n2, e, theta, C, p = Not numeric; abstract constants in Theorem 2.3
    These are not fitted to data; they are abstract exponents and constants quantifying the assumed polynomial growth of the Monte Carlo scheme, network realizations, and moment bounds. They serve as inputs to the theorem rather than free parameters chosen to make the derivation work, so I do not count them as fitted parameters.
assumptions (5)
  • domain assumption The discrete approximation scheme satisfies the error bound (2.9) with rate N^{-n0} and dimension factor d^{d0}.
    Invoked as assumption (2.9) in Theorem 2.3; this is the 'without the curse of dimensionality' property of the Monte Carlo scheme that the theorem transfers.
  • domain assumption The random variables Z_{N,d,m,n} have uniformly bounded moments (2.10), and the initial measure nu_d has moment bounds (2.10).
    Invoked in the proof of Theorem 2.3 to control the growth of the Euler scheme trajectories via Lemma 2.2.
  • domain assumption For every admissible network Phi there exist networks phi_z realizing the composition of the approximated drift with Phi with parameter cost bounded by P(Phi) + C N^{n1} d^{d3} epsilon^{-e} (paragraph before (2.14)).
    This composition-closure hypothesis is the key structural assumption in Theorem 2.3. In the PDE application it is verified via Lemma 3.30 using the width bound DL-1(Psi) <= l_{1,L1-1} + i.
  • domain assumption The network realization map R preserves the required operations: composition, parallelization, and summation of networks (Section 3, Lemmas 3.29, 3.30).
    The ANN calculus in Section 3 is developed from definitions and proved in the paper, but its correctness relies on the standard interpretation of feedforward networks as compositions of affine maps and activation functions.
  • domain assumption In the Kolmogorov PDE application, the PDE solution u_d has the Feynman-Kac representation u_d(T,x) = E[phi_{0,d}(X^d_x_T)] for the SDE (4.46).
    Invoked in the proof of Theorem 4.5 via [32, Theorem 3.1]. This is a classical result requiring appropriate growth and regularity conditions on phi_{0,d} and phi_{1,d}.

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Pith. "Pith review of Deep neural network approximations for Monte Carlo algorithms." pith.science (2026). https://pith.science/paper/XMBBTNXO

@misc{pith2026190810828,
  author       = {Pith},
  title        = {Pith review of: Deep neural network approximations for Monte Carlo algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XMBBTNXO}},
  note         = {Machine review of arXiv:1908.10828}
}
read the original abstract

Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs can approximate solutions of certain PDEs without the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially both in the PDE dimension and the reciprocal of the prescribed approximation accuracy. One key argument in most of these results is, first, to use a Monte Carlo approximation scheme which can approximate the solution of the PDE under consideration at a fixed space-time point without the curse of dimensionality and, thereafter, to prove that DNNs are flexible enough to mimic the behaviour of the used approximation scheme. Having this in mind, one could aim for a general abstract result which shows under suitable assumptions that if a certain function can be approximated by any kind of (Monte Carlo) approximation scheme without the curse of dimensionality, then this function can also be approximated with DNNs without the curse of dimensionality. It is a key contribution of this article to make a first step towards this direction. In particular, the main result of this paper, essentially, shows that if a function can be approximated by means of some suitable discrete approximation scheme without the curse of dimensionality and if there exist DNNs which satisfy certain regularity properties and which approximate this discrete approximation scheme without the curse of dimensionality, then the function itself can also be approximated with DNNs without the curse of dimensionality. As an application of this result we establish that solutions of suitable Kolmogorov PDEs can be approximated with DNNs without the curse of dimensionality.

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Pith tools

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