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Separable Approximations of Optimal Value Functions and Their Representation by Neural Networks

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Pith's one-line read A value function with decaying sensitivity has a separable approximation whose error does not grow with the number of agents or dimension, and a neural network with polynomially many parameters can represent it.

desk verdict Worth a serious referee: genuine new results and a clear conditional framework, but the directed-graph claim of Theorem 3.4 is not proved as stated. read the letter →

arxiv 2502.08559 v2 pith:2GB4ETVI submitted 2025-02-12 math.OC

Mario Sperl , Luca Saluzzi , Dante Kalise , Lars Grüne This is my paper · ORCID
classification math.OC MSC 49N1049L2065D15
keywords separableapproximationsdecayingsensitivityvaluefunctionsneuralnetworksoptimalcontrolcurseofdimensionalityRiccatiequations
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

Decaying sensitivity means the influence of one subsystem's state on another fades as their graph distance grows. The paper proves that any function with this property, and in particular optimal-control value functions from LQR problems, can be approximated by the sum of local terms involving only nearby subsystems. Its main estimate, Theorem 3.4, holds when sensitivity decays exponentially and graph neighborhoods grow at most exponentially with a compatible base; then the approximation error is bounded by a constant times $\delta^{l+1}\|x\|^2$, with constants independent of the number of agents $s$ and dimension $n$. That result is used to show that a feedforward neural network with a number of neurons and parameters polynomial in $s$ can approximate the value function to any desired accuracy on a bounded box.

What carries the argument

The central object is the $\gamma$-decaying sensitivity property (Definition 2.1): for the maps $g_j(x)=V(x)-V(x\text{ with }z_j=0)$, the Lipschitz constant with respect to $z_i$ is at most $\gamma(\operatorname{dist}_G(i,j))\|z_j\|$. This condition links the value function to the graph geometry. The second ingredient is the separable ansatz $\Psi_j^l(z_{j,l})=V(x_{j,l})-V(\Lambda_j x_{j,l})$, where $x_{j,l}$ keeps only $z_j$ and its graph ball of radius $l$. Lemma 3.3, the norm inequality $\|A\|_2\le\sqrt{\|A\|_1\|A\|_\infty}$, converts row- and column-sum estimates of the truncated decay matrix $D_l$ into the final error. The graph growth bound $r(k)$ controls these sums uniformly as $s$ grows. In Theorem 5.1, the same local terms are the targets of independent shallow subnetworks, which are then concatenated into one feedforward network.

What would settle it

For a fixed $\gamma(l)=C\rho^l$ and a graph whose neighborhood growth $r(k)$ exceeds $b\mu^k$ with $\mu\ge\rho^{-1}$, compute $\|D_l\|_2$ and the actual separable error $|V-\Psi_l-V(0)|$ as $s$ grows; Theorem 3.4 predicts the bound loses its $s$-independence in this regime, so observing the error remain uniformly bounded would falsify the role of the growth condition.

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Extended reading notes

Core claim

Under the $\gamma$-decaying sensitivity property, the separable approximation $V(x)\approx V(0)+\sum_{j=1}^s \Psi_j^l(z_{j,l})$ has error bounded by $\|x\|_2^2 \|D_l\|_2$, where $D_l$ collects the decay constants for pairs at graph distance greater than $l$. Theorem 3.4 sharpens this to $|V(x)-\Psi_l(x)-V(0)|\le \tilde C \delta^{l+1}\|x\|_2^2$ when $\gamma(l)=C\rho^l$ and the graph growth satisfies $r(k)\le b\mu^k$ with $\mu<\rho^{-1}$, with $\delta\in(\rho\mu,1)$ and constants independent of $s$ and $n$. For polynomial decay $\gamma(l)=C(l+1)^{-\alpha}$ and $r\in O(l^\beta)$ with $\beta<\alpha-1$, Theorem 3.5 gives an analogous bound with a tail sum. Theorem 5.1 then constructs a feedforward network with $s^{d+1}C^d+n+1$ neurons and $s^{d+1}C^d(d+s+1)+1$ parameters that approximates $V$ to error $\varepsilon(1+\|x\|_2^2)$ on $[-R,R]^n$. The paper also proves an explicit off-diagonal decay rate for the solution of the continuous-time Riccati equation when $A$ is symmetric and banded, and provides numerical evidence for exponential sensitivity decay in a nonlinear control problem.

Load-bearing premise

The entire argument rests on Definition 2.1, the assumption that the value function's cross-sensitivity has a Lipschitz constant decaying as $\gamma(\operatorname{dist}_G(i,j))\|z_j\|$; for nonlinear problems this property is only verified numerically from an SDRE surrogate, not proved.

Editorial extensions

If this is right

  • For a family of problems with uniformly decaying sensitivity and uniformly bounded graph-neighborhood dimension, value functions can be represented by neural networks with polynomial parameter growth, bypassing the exponential curse of dimensionality.
  • The local radius $l$ sets an explicit accuracy-versus-sparsity trade-off: larger $l$ reduces the error as $\delta^{l+1}\|x\|^2$ but increases the dimension of the local terms.
  • Discrete-time LQR problems that satisfy the uniform stabilizability and detectability conditions inherit the separable error bounds through the spatially exponential decay of the Riccati solution.
  • For continuous-time LQR with symmetric banded $A$, $B=I$, $Q=cI$, $R=\gamma I$, the paper proves an explicit off-diagonal decay rate for the Riccati solution, which feeds directly into the separable approximation.
  • Standard low-rank approximation cannot exploit this structure, so the separable network architecture offers a complementary route for high-dimensional value functions.

Reading between the lines

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

  • The same separable construction might be used to design control laws for PDE-constrained multi-agent systems if the decaying-sensitivity property is verified for their value functions; the nonlinear example in the paper uses an SDRE surrogate rather than a direct proof.
  • Replacing the fixed anchor $0$ by data-dependent anchors could improve the constants, since the construction is a graph-localized anchored decomposition.
  • The numerical finding that two neurons per sublayer suffice across dimensions suggests the worst-case polynomial exponents in Theorem 5.1 may be pessimistic, and a sharper bound might depend on the range of local sensitivity rather than the maximal neighborhood dimension.
  • An independent test would be to train the $s$ local terms $\Psi_j^l$ separately and sum them, then compare accuracy and training cost against the end-to-end separable network.
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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

2 major / 5 minor

Summary. The manuscript develops a framework for approximating high-dimensional value functions by separable sums of local contributions. Setting 2.1 formalizes a gamma-decaying sensitivity property of a function V with respect to a directed graph of interacting subsystems. The central construction (4)-(5) approximates V(x) by V(0)+Psi_l(x)=V(0)+sum_j Psi^j_l(z_{j,l}), where each local term depends on a bounded graph neighborhood of subsystem j. Lemma 3.2 bounds the approximation error by x^T D_l x, and Theorems 3.4 and 3.5 derive explicit error bounds under exponential and polynomial sensitivity decay together with graph growth bounds, with constants independent of the number of agents s and the dimension n. Section 4 verifies the sensitivity property for discrete-time LQR (citing prior work) and for a continuous-time LQR class with symmetric A, B=I, Q=cI, R=gamma I (Proposition 4.3). Section 5 proves a neural-network representation result: under the exponential-decay assumptions, a feedforward network with polynomially many neurons and parameters achieves accuracy epsilon(1+||x||^2) on [-R,R]^n. Section 6 reports numerical experiments for LQR problems, non-quadratic functions with prescribed decay, and an Allen-Cahn SDRE surrogate, with code available online.

Significance. If the main theorems hold as stated, the paper gives explicit, dimension-independent approximation guarantees for a well-defined class of value functions and a constructive neural-network complexity bound, thereby offering a principled route around the curse of dimensionality for separable structures. The strongest assets are the explicit constants in Theorems 3.4 and 3.5, the closed-form continuous-time LQR decay result in Proposition 4.3, the explicit connection to anchored decompositions, and a reproducible numerical study that compares the separable architecture against fully connected networks and tests non-quadratic and polynomial-decay examples. The paper is also honest in labeling the nonlinear verification in Section 6.3 as numerical evidence rather than proof. The main caveat is the directed-graph proof gap identified below, which is localized and does not affect the undirected and sequential examples actually treated in Section 6.

major comments (2)
  1. [Section 3.2, Eq. (8), Lemma 3.2, Theorem 3.4] The dimension-independent bound for genuinely directed graphs is not proved as stated. Lemma 3.2's telescoping sum runs over nodes i with dist_G(i,j)=k+1, i.e. over the in-neighborhood of j, and the column sums of D_l that give ||D_l||_1 in Theorem 3.4 require a bound on |{i : dist_G(i,j)=l}|. The growth bound (8), however, is stated as |{i : dist_G(j,i)=l}| <= r(l), i.e. an out-neighborhood bound. These two cardinalities agree only when G is undirected. Therefore the assertion in Setting 2.1 that the main results of Section 3 hold for directed graphs is unsupported, and Theorem 3.4 -- and hence the neural-network bound in Theorem 5.1 -- needs either to be restricted to undirected graphs or supplemented by an analogous in-neighborhood growth assumption. The fix is local, and the practical conclusions are unaffected because all LQR and numerical examples use undirected or sequential graphs.
  2. [Section 6.3 and Section 7] The applicability of the theory to nonlinear control is presented as empirically supported rather than proven, and this should be stated more explicitly as a limitation. The Lipschitz estimates in Figure 13 are first-order difference quotients computed at two reference points with one perturbation per coordinate; they do not certify the uniform, state-dependent inequality in Definition 2.1 on the whole domain. I do not regard this as an error, because the wording 'strong numerical evidence' is honest, but the abstract and Section 7 should state that for nonlinear problems the gamma-decaying sensitivity assumption is a conjecture supported by numerical evidence rather than a theorem, so that readers do not infer that Theorem 5.1 has been verified for the Allen-Cahn problem.
minor comments (5)
  1. [Proof of Lemma 3.2] The set of newly added nodes is written once as {i : dist_G(j,i)=l~} and later as {i : dist_G(i,j)=k+1}. Please use a single orientation, preferably dist_G(i,j), throughout the proof, since the definition of z_{j,l} in (4)-(5) uses the distance from i to j.
  2. [Proof of Theorem 3.4] As typeset, the displayed chain of inequalities contains an extra factor delta^{l+1}: it reads '<= C b_C delta^{l+1}/(1-delta) delta^{l+1}', which would make the constant ilde C depend on l. The intended bound should be C b_C delta^{l+1}/(1-delta), with ilde C = C b_C/(1-delta). Please correct this display.
  3. [Table 1 and surrounding text] There is an inconsistency in the reported maximum number of epochs for the 50-dimensional case with nu_g=nu_z=0: Table 1 says training stopped after max epochs 1000, while the text says the MSE was 1.3e-2 after 10^4 epochs. Please align the numbers.
  4. [Section 6.3] The sentence beginning 'or a detailed discussion of the theoretical underpinnings' should read 'For a detailed discussion...'.
  5. [Proof of Theorem 5.1] The definition of l uses nonstandard bracket notation and should specify the floor/ceiling operation and require epsilon small enough that the chosen l is nonnegative; currently the formula is hard to parse and appears as 'l := l ... m'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are conditional on the explicitly stated decaying-sensitivity assumption, and the one self-citation (discrete LQR decay) is not load-bearing for the separable-approximation results.

full rationale

The paper's central derivation chain is: (i) assume gamma-decaying sensitivity (Definition 2.1); (ii) construct Psi_l and prove Lemma 3.2 by telescoping, with the error depending on D_l whose entries are exactly gamma(dist) outside the l-neighborhood; (iii) bound ||D_l|| using the growth bound (8) and the assumed decay rate, yielding Theorems 3.4 and 3.5; (iv) approximate each Psi_l^j by a standard universal-approximation network. None of these steps fits a parameter to data and then renames the fit as a prediction; the error constants are explicit functions of C, rho, mu, and b_C. The continuous-time LQR sensitivity (Proposition 4.3) is proved in the paper from the closed-form ARE solution and an external matrix-function decay result (Benzi-Golub), so it is independent support. The discrete-time LQR sensitivity (Proposition 4.1) is recalled from the authors' prior CDC paper, but it is a previous peer-reviewed theorem building on Shin et al. and is not used to prove the separable-approximation theorems; therefore the self-citation is not load-bearing. The nonlinear validation in Section 6.3 estimates Lipschitz constants from an SDRE surrogate; this is empirical evidence for the assumption, not an input to the theorem, and no circular prediction is claimed. The directed-graph mismatch between Lemma 3.2's in-neighborhood sums and the growth bound (8)'s out-neighborhood cardinality is a genuine proof gap for directed graphs, but it is a correctness issue, not circularity: for undirected graphs the argument is valid, and the error bound never assumes its own conclusion.

Assumptions & free parameters 3 free parameters · 9 assumptions · 0 invented entities

The central theorems are conditional on explicitly stated decay and graph-growth assumptions; the paper's contribution is the derivation of the error bounds and neural network complexity, not the values of any constants. Free parameters are experiment hyperparameters only. Axioms include standard background results (Riccati theory, matrix function bounds, network approximation theory) and domain assumptions that define the class of problems to which the results apply.

free parameters (3)
  • Loss weights nu_g, nu_z = nu_g = nu_z = 0.5 in LQR experiments; nu_g = 1e4, nu_z = 10 in Allen-Cahn experiment
    Chosen by hand to improve training; affect reported MSE and epoch counts but not the theoretical results.
  • Graph-neighborhood radius l = l = 10 (200-dim LQR), l = 5 (non-quadratic), l = 3 (Allen-Cahn and dimension scaling)
    Hand-picked architecture parameter trading approximation accuracy against separability; in Theorem 5.1 it is chosen depending on epsilon, eC, and delta.
  • Subnetwork hidden size M = M = 16, 2, or 32 depending on experiment
    Hand-picked; in the dimension scaling experiment M was minimized to achieve test MSE below 1e-4.
assumptions (9)
  • domain assumption V satisfies the gamma-decaying sensitivity property (Definition 2.1): L_{i,j} = gamma(dist_G(i,j)) * ||z_j||.
    Invoked in Lemma 3.2, Theorems 3.4, 3.5, and Theorem 5.1; the core premise of the framework. Verified in closed form only for special LQR cases, empirically via SDRE surrogate for nonlinear problems.
  • domain assumption For discrete LQR, the Riccati solution P and feedback matrix K are spatially exponentially decaying with constants independent of s (Prop 4.1; Theorem 3.3 of [30]).
    Imported from own conference paper [33] and Shin et al. [30]; the paper does not reprove this, so the dimension-uniform claim rests on it.
  • domain assumption Uniform stabilizability, uniform detectability, and boundedness conditions i)-iv) hold for the family of LQR problems (Section 4.1).
    Standard LQR well-posedness assumptions used to ensure existence of stabilizing Riccati solutions.
  • domain assumption For Proposition 4.3, A is symmetric, p-banded with spectrum in [a,b], and B = I, Q = cI, R = gamma*I.
    Restrictive structural assumption enabling the closed-form Riccati solution P = gamma( sqrt(A^2 + c/gamma I) + A ).
  • domain assumption Graph growth bound r(k) <= b_C * mu^k with mu < rho^{-1} (Theorem 3.4), or r in O(l^beta) with beta < alpha-1 (Theorem 3.5).
    Needed for dimension-independent constants in the exponential and polynomial decay error bounds.
  • standard math Benzi-Golub bound for analytic functions of banded symmetric matrices (Theorem 4.2, cited from [5]).
    Used to prove exponential off-diagonal decay of the continuous-time Riccati solution in Proposition 4.3.
  • standard math Existence and uniqueness of the stabilizing symmetric positive semidefinite Riccati solution (Theorem 4.1, cited from [21]) and the definition of functions of diagonalizable matrices (Definition 4.5, [16]).
    Background results used in the continuous-time LQR analysis.
  • standard math Approximation property of one-hidden-layer networks with non-polynomial C-infinity activation (Theorem 2.1 in [23], extended to cubes in [17]).
    Used in the proof of Theorem 5.1 to approximate each local component Psi_j^l.
  • ad hoc to paper The construction (4) uses one-sided neighborhoods defined by an arbitrary ordering of agents j = 1..s; the approximation depends on this numbering.
    Stated in Section 3.2 after (5): 'in general the resulting approximation will depend on the numbering of the subsystems z_j'.

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Cite this review

Pith. "Pith review of Separable Approximations of Optimal Value Functions and Their Representation by Neural Networks." pith.science (2026). https://pith.science/paper/2GB4ETVI

@misc{pith2026250208559,
  author       = {Pith},
  title        = {Pith review of: Separable Approximations of Optimal Value Functions and Their Representation by Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2GB4ETVI}},
  note         = {Machine review of arXiv:2502.08559}
}
read the original abstract

The use of separable approximations is proposed to mitigate the curse of dimensionality related to the approximation of high-dimensional value functions in optimal control. The separable approximation exploits intrinsic decaying sensitivity properties of the system, where the influence of a state variable on another diminishes as their spatial, temporal, or graph-based distance grows. This property allows the efficient representation of global functions as a sum of localized contributions. A theoretical framework for constructing separable approximations in the context of optimal control is proposed by leveraging decaying sensitivity in both discrete and continuous time. Results extend prior work on decay properties of solutions to Lyapunov and Riccati equations, offering new insights into polynomial and exponential decay regimes. Connections to neural networks are explored, demonstrating how separable structures enable scalable representations of high-dimensional value functions while preserving computational efficiency.

Figures

Figures reproduced from arXiv: 2502.08559 by the authors.

Figure 1
Figure 1. Graph neighborhoods z1,2 (left el￾lipse) and z2,2 (right ellipse) in a sequential graph. z3 z1 z2 Ψ1 l Ψ2 l Ψ3 l [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Decay of the first column’s absolute value of the solution of the CARE (17) and the corre [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Illustration of the separable structured neural network (S-NN) described in the proof of The [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: Plot of the optimal value function V projected onto the (x1, x2)-axis (upper left) and onto the (x100, x200)-axis (lower left), the S-NN approximation onto the same hyperplanes (middle column), and the decay in the first row of P (upper right) and the row 100 of P (low…
Figure 6
Figure 6. Figure 6: On the other hand, for larger amounts of training data, where both architectures meet the [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 6
Figure 6. Figure 6: Comparison of test error (log-scale) against the number of training points. 17 16 15 14 Log2(Number of Training Data Points) 10 2 10 3 Number of Epochs Fully Connected S-NN [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 8
Figure 8. Figure 8: Number of neurons and parameters for bandwidths 1 and 3 in dependence of the dimension. z1 z2 z3 z4 z5 z6 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 10
Figure 10. Figure 10: Comparison of the decaying sensitivity with the singular values of [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Test error (log-scale) for approxi￾mating V (exponential decay) in dependence of ρ. 0 1 2 3 10 4 10 3 10 2 10 1 10 0 10 1 10 2 10 3 Mean Squared Test Error [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 13
Figure 13. Figure 13: Plot of V projected onto the (x1, x2)-axis (upper left) and onto the (x25, x50)-axis (lower left), the S-NN approximation onto the same hyperplanes (middle column), and the decay of L˜ i,j for j = 1 (upper right) and j = 25 (lower right) varying the index i. 7 Conclus…

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