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Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality

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Pith's one-line read This paper proves that Q-functions of discounted Markov decision processes can be approximated by leaky ReLU neural networks with polynomial parameter growth in the state-space dimension and in the reciprocal error, provided the reward…

desk verdict The central theorem survives the stress test: the 'every probability measure' claim is justified because w≡1 satisfies Corollary 5.4's Lyapunov condition, and the apparent counterexample violates the explicit first-moment hypothesis in Theorem 1.1. read the letter →

arxiv 2506.22851 v1 pith:6WGDUODO submitted 2025-06-28 math.OC cs.LGcs.NAmath.NAmath.PRstat.ML

Arnulf Jentzen , Konrad Kleinberg , Thomas Kruse This is my paper · ORCID
classification math.OCcs.LGcs.NAmath.NAmath.PRstat.ML MSC 68T0790C4049L20
keywords BellmanequationQ-functionMarkovdecisionprocessleakyReLUdeepneuralnetworkscurseofdimensionalitymultilevelfixed-pointapproximationreinforcementlearning
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 proves that the Q-function of an infinite-horizon discounted Markov decision process with a finite action set can be approximated, in $L^2$ with respect to any probability measure on the state space, by a leaky ReLU deep neural network using at most $c d^c \varepsilon^{-c}$ parameters. The dimension $d$ enters only polynomially, so the exponential cost that usually plagues high-dimensional dynamic programming is absent. The result is conditional: the reward functions and the random transition maps must themselves be approximable by leaky ReLU networks with pointwise error at most $\varepsilon c d^c$ and with polynomially many parameters. Under that hypothesis, the Bellman fixed point inherits the approximation, and the proof gives an explicit network construction built from the full-history recursive multilevel fixed-point (MLFP) scheme.

What carries the argument

The load-bearing object is the full-history recursive multilevel fixed-point (MLFP) approximation scheme of reference [5], coupled with an explicit ANN calculus. The scheme replaces the Bellman fixed point by a multilevel Monte Carlo recursion in which each level averages over $M^{n-l}$ independent nested transition samples and subtracts lower-level corrections, and the paper's stability estimates show that the error of the $n$-th level decays geometrically in $n$. The ANN side supplies the concrete representations: a two-layer leaky ReLU block realizes the finite maximum function, composition and parallelization combine networks for the reward and transition maps, and linear combinations sum the multilevel terms. The key meta-identity is that every MLFP iterate is itself a function of the same form—composed, summed, and maxed primitives—so the whole iteration compiles into one leaky ReLU network whose depth and width stay polynomial.

What would settle it

Take an MDP with an analytically known Q-function, such as a linear-quadratic regulator or a threshold policy, run the MLFP network construction at dimensions $d=10,100,1000$ and accuracies $\varepsilon=10^{-2},10^{-4},10^{-6}$, and record the smallest parameter count $P$ achieving the $L^2$ error; the central claim predicts $P \le c d^c \varepsilon^{-c}$ for a fixed $c$, so a growth rate visibly faster than any polynomial in $d$ and $1/\varepsilon$ would falsify it.

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Extended reading notes

Core claim

The paper's central claim, stated as Theorem 1.1 and generalized in Theorem 5.1, is that the unique bounded solution $\mathscr{q}$ of the Bellman equation $(\mathscr{q}(x))(a) = (g_d(x))(a) + \delta \mathbb{E}[\max_{b \in A} (\mathscr{q}(t_d^{(a)}(x, \xi_d)))(b)]$ admits, for every $d \in \mathbb{N}$, every $\varepsilon \in (0,1)$, and every probability measure $\mu$, a leaky ReLU network $Q$ with $\left(\int_{\mathbb{R}^d} \|\mathscr{q}(x) - (\mathcal{R}(Q))(x)\|^2 \mu(dx)\right)^{1/2} \le \varepsilon$ and $P(Q) \le c d^c \varepsilon^{-c}$. The mechanism is inheritance: the Bellman equation is a fixed-point equation, and the multilevel fixed-point scheme approximates its solution using only compositions, sums, and finite maxima of the primitive functions $g_d$ and $t_d^{(a)}$; each of those operations is realized by a leaky ReLU network, so the network representation of the fixed point has the same activation type and polynomial complexity as the primitives. Uniqueness and existence of $\mathscr{q}$ are also established under the stated contraction and Lipschitz assumptions.

Load-bearing premise

The load-bearing premise is that the primitive data—the reward function $g_d$ and the random transition maps $t_d^{(a)}$—are themselves approximable by leaky ReLU networks with error at most $\varepsilon c d^c$ and at most $c d^c \varepsilon^{-c}$ parameters; the proof also takes the MLFP error estimate (reference [5, Proposition 3.8]) as an external input rather than proving it.

Editorial extensions

If this is right

  • For every MDP in the assumed class, the Q-function has a leaky ReLU representation with at most $c d^c \varepsilon^{-c}$ parameters, so achieving error $\varepsilon$ remains feasible when the state-space dimension grows.
  • Because the maximum over the finite action set is built into the network, the approximation applies directly to the optimal Q-function and not to a softened or regularized surrogate.
  • The same $L^2$ guarantee holds for any probability measure $\mu$ on the state space, so the result is compatible with the measure-based error notions used in reinforcement learning.
  • The construction is fully explicit: given networks for $g_d$ and $t_d^{(a)}$, one can write down the Q-network by composing the MLFP iteration, which means the theorem is a proof of existence rather than an abstract counting argument.

Reading between the lines

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

  • Editorial extension: the same MLFP-plus-ANN argument should carry over to finite-horizon problems and to related fixed-point equations, because the paper's stability machinery only uses a contraction in Wasserstein distance and does not appear to need the discount factor in a special way.
  • Editorial extension: a practical reading is that deep Q-learning architectures could be initialized or structured with the MLFP network when the transition model is known; the paper does not address training dynamics or sample complexity.
  • Editorial extension: the theorem locates the actual bottleneck of high-dimensional reinforcement learning in the primitive functions—problems whose rewards or transitions oscillate too fast to be leaky-ReLU approximable at polynomial cost will remain hard regardless of the Bellman solver.
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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

0 major / 6 minor

Summary. This paper proves that, for a class of discrete-time discounted Markov decision processes, the Q-function can be approximated in L^2 by leaky ReLU neural networks with parameter counts growing at most polynomially in the state dimension d and in 1/ε. The main theorem, Theorem 1.1, assumes that the reward and transition functions are themselves approximable pointwise by such networks with polynomial complexity, together with Lipschitz regularity and a global first-moment condition on the transitions. The proof builds on the full-history recursive MLFP scheme imported from Beck, Jentzen, Kleinberg, and Kruse [5], and combines it with a detailed ANN calculus that represents MLFP iterations as explicit networks. The paper states Theorem 1.1 as an immediate consequence of Corollary 5.4, which is formulated with a general weight function w_d.

Significance. If the result holds, it is a meaningful contribution: it transfers the curse-of-dimensionality-free approximation results previously obtained for MLP/DNN methods for PDEs to Bellman equations and Q-functions, a central object in reinforcement learning. The proof is detailed, with explicit parameter bounds, and the ANN constructions are concrete. The main theorem is conditional on strong structural assumptions—pointwise DNN approximability of the primitives and a uniform-in-state first moment of the transition—but these are stated explicitly and are not hidden. The paper relies on [5, Proposition 3.8] as an external input; since [5] is a published parameter-free MLFP error bound, this is an acceptable, though visible, dependence. The central derivation is coherent, and the stress-test concern about the passage from Corollary 5.4 to Theorem 1.1 is resolved by the uniform first-moment assumption already present in Theorem 1.1, with the choice w_d≡1.

minor comments (6)
  1. [§1, Theorem 1.1 and §5.3, Corollary 5.4] The claimed 'immediate consequence' is valid but not transparent: taking w_d ≡ 1 and λ=1 reduces hypotheses (5.59)-(5.64) to assumptions (1.3)-(1.4) of Theorem 1.1, with the uniform first-moment condition in Theorem 1.1 supplying (5.62). Because the stress-test concern turns on this reduction, I recommend adding one explicit sentence in §1 to spell out this verification.
  2. [§5.3, proof of Corollary 5.4] The definition 'q_d(x))(a) = (g_d(x))(a) + (u_d(x))(x)' is not well formed, since x is a vector in R^d and u_d(x) is indexed by actions; it should read '(u_d(x))(a)'.
  3. [Corollary 5.4, proof] The line 'let K = max{2|A|, c, c}' repeats c; one of the entries is presumably a different constant, and the notation should be cleaned up.
  4. [§1, Theorem 1.1 and §2.2, Lemma 2.6] There are typographical slips, including '∪d,∈N' in the introduction and the stray 'y' in the integral inequality (2.6); these should be corrected.
  5. [§5.2, Corollary 5.3 proof] Equation (5.71) writes 'Rϕ(Sd,ε)' where the ANN is S_{|A|,2}; the subscript is inconsistent with the construction in (5.69).
  6. [References] Reference [25] duplicates [24], and the name 'Tistsiklis' in [11] is misspelled; the reference list should be deduplicated and corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Q-function approximation is derived compositionally from assumed DNN approximability of the primitive data and from an external MLFP error bound.

full rationale

The derivation chain is Theorem 1.1 -> Corollary 5.4 -> Corollary 5.3 -> Theorem 5.1, with the MLFP error estimate imported from [5, Proposition 3.8]. The assumptions in (1.3) concern DNN approximations of the reward g_d and the transition maps t_d^{(a)}; the conclusion in (1.6) concerns DNN approximations of the Q-function q_d. These are distinct objects. No parameter is fitted to q_d: the approximating ANN Q is constructed explicitly, as a leaky-ReLU network realizing the MLFP iteration, by composing the given ANNs G_{d,epsilon}, t_{d,epsilon}^{(a)}, and the max-network built in Lemmas 4.1 and 4.2. Lemma 4.3 realizes the MLFP approximation as an ANN, and the error estimate is then obtained from the stability result in Proposition 2.9 and from [5, Proposition 3.8]. The cited MLFP result is a published, parameter-free theorem about Monte Carlo estimators for fixed-point equations, not about DNN approximation, and its assumptions do not include the target claim. The overlapping authorship is therefore self-citation, but it is not circular in the prohibited sense. The same is true for the ANN-calculus lemmas reused from [1], [18], and [19], which are standard composition and parallelization facts. The statement that Theorem 1.1 is an immediate consequence of Corollary 5.4 is also structurally coherent: Theorem 1.1 explicitly assumes sup_x E[||t_d^{(a)}(x, xi_d)||] < infinity, so taking w_d identically equal to 1, lambda = 1, and absorbing constants into c makes the weight hypotheses (5.62)-(5.64) satisfiable, and the measure restriction in Corollary 5.4, namely (int |w_d(x)|^2 mu(dx))^{1/2} <= c d^c, holds for every probability measure mu. Whether all details of that application are fully spelled out is a presentation or correctness matter, not circularity. No equation in the paper is defined in terms of the conclusion, and no fitted value is relabeled as a prediction.

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

No fitted parameters appear: the constants M, α_{d,ε}, γ_{d,ε}, and δ_{d,ε} are explicitly constructed in the proof, not tuned to data. The central claim rests on three domain assumptions: DNN approximability of primitives, contraction and moment conditions, and the imported MLFP error estimate from [5]. No new physical or mathematical entities are introduced.

assumptions (5)
  • domain assumption For each d and ε, the reward g_d and transition t_d are approximable by leaky ReLU ANNs with pointwise error at most ε c d^c and parameter count at most c d^c ε^{-c} (Theorem 1.1, Eq. (1.3)).
    This is the main hypothesis of the theorem; without DNN-approximable primitives the conclusion is not asserted.
  • domain assumption The stochastic kernels and transition kernels satisfy the moment and contraction conditions in (5.4)-(5.6), including sup_d λ_d L_d < 1 and sup_{d,ε} λ_d ℓ_{d,ε} < 1.
    These conditions ensure uniqueness of the Bellman fixed point and geometric contraction of the MLFP error; they restrict the class of MDPs covered.
  • domain assumption The full-history recursive multilevel fixed-point error bound from [5, Proposition 3.8] is valid as stated.
    This imported theorem supplies the central O(α^n) error decay for the MLFP estimator; the present paper does not reprove it. It is published and parameter-free, but it is a load-bearing external result.
  • standard math The ANN calculus in Section 3, including composition, parallelization, linear combinations, and the leaky ReLU maximum representation in Lemma 4.2, holds as stated.
    The proof relies on these lemmas, which are either proven in the paper or adapted from the cited literature.
  • standard math Measurability and integrability of the random fields are assumed so that conditional expectations, Fubini's theorem, and coupling arguments apply.
    Stated in Setting 2.4 and used throughout Sections 2, 4, and 5.

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Cite this review

Pith. "Pith review of Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality." pith.science (2026). https://pith.science/paper/6WGDUODO

@misc{pith2026250622851,
  author       = {Pith},
  title        = {Pith review of: Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WGDUODO}},
  note         = {Machine review of arXiv:2506.22851}
}
abstract

Discrete time stochastic optimal control problems and Markov decision processes (MDPs) are fundamental models for sequential decision-making under uncertainty and as such provide the mathematical framework underlying reinforcement learning theory. A central tool for solving MDPs is the Bellman equation and its solution, the so-called $Q$-function. In this article, we construct deep neural network (DNN) approximations for $Q$-functions associated to MDPs with infinite time horizon and finite control set $A$. More specifically, we show that if the the payoff function and the random transition dynamics of the MDP can be suitably approximated by DNNs with leaky rectified linear unit (ReLU) activation, then the solutions $Q_d\colon \mathbb R^d\to \mathbb R^{|A|}$, $d\in \mathbb{N}$, of the associated Bellman equations can also be approximated in the $L^2$-sense by DNNs with leaky ReLU activation whose numbers of parameters grow at most polynomially in both the dimension $d\in \mathbb{N}$ of the state space and the reciprocal $1/\varepsilon$ of the prescribed error $\varepsilon\in (0,1)$. Our proof relies on the recently introduced full-history recursive multilevel fixed-point (MLFP) approximation scheme.

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