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arxiv: 2604.20369 · v1 · submitted 2026-04-22 · 💻 cs.IT · cs.SY· eess.SY· math.IT· math.OC

Rate-Cost Tradeoffs in Nonlinear Control

Eray Unsal Atay , Venkat Chandrasekaran , Victoria Kostina This is my paper

Pith reviewed 2026-05-09 23:12 UTC · model grok-4.3

classification 💻 cs.IT cs.SYeess.SYmath.ITmath.OC
keywords rate-cost tradeoffdirected informationnonlinear controlstochastic controlrate-limited controlfunctional representation lemmacausal source codingLQG control
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The pith

For general nonlinear stochastic control, the minimal rate to meet a control cost D lies within a logarithmic additive gap of the directed-information minimum.

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

The paper examines rate-limited control over finite horizons for arbitrary stochastic systems, where an encoder sends a limited-rate description of the observed state to a remote controller that must choose actions to keep average costs below a threshold D. It proves that the lowest achievable rate R_n(D) is bounded from below by the minimum directed information F_n(D) between states and controls under the cost constraint, and from above by that same value plus log(F_n(D) + 3.4) plus small constants and a 1/n term. The upper bound is obtained by explicitly constructing an encoding-and-control policy at every step. This result shows that directed information, previously known to govern causal source coding and linear-quadratic-Gaussian control, remains the key quantity even for fully nonlinear dynamics.

Core claim

We show that F_n(D) ≤ R_n(D) ≤ F_n(D) + log(F_n(D) + 3.4) + 2 + 1/n bits, where F_n(D) is the minimum directed information between the state and control sequences subject to an average cost constraint D. The lower bound follows from standard information inequalities, while the upper bound is achieved constructively by applying the strong functional representation lemma to the conditional distributions at each time step. This establishes directed information as the operationally relevant quantity for rate-limited control of general nonlinear systems and recovers known nonasymptotic bounds for causal rate-distortion and LQG control as special cases.

What carries the argument

The strong functional representation lemma applied sequentially to the conditional distributions of the stochastic control system, which constructs an encoding-and-control policy achieving the stated upper bound on R_n(D).

Load-bearing premise

An encoding-and-control policy can be constructed at each time step using the strong functional representation lemma applied to the conditional distributions of the general stochastic control system.

What would settle it

A concrete finite-horizon nonlinear system and cost threshold D for which the true minimal rate exceeds F_n(D) + log(F_n(D) + 3.4) + 2 + 1/n bits.

Figures

Figures reproduced from arXiv: 2604.20369 by Eray Unsal Atay, Venkat Chandrasekaran, Victoria Kostina.

Figure 1
Figure 1. Figure 1: Control system with a rate-limited link to the controller. The shaded [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Separation principle for LQG control: the communication side only needs to produce the MMSE state estimate ˆ [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Control system over a rate-limited link and per-stage control cost. [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: , the source {Xt}, characterized by kernels PXt|X[t−1] , is unaffected by the encoder/decoder; one designs causal map￾pings to produce Xˆ [n] under a distortion constraint between X[n] and Xˆ [n] . The key quantity is the normalized directed information 1 n I  X[n] → Xˆ [n]  [16], whose minimization over causal reproduction kernels defines the sequential (causal) rate-distortion function Fn(D) = inf PXˆ … view at source ↗
Figure 5
Figure 5. Figure 5: Action-dependent source-coding setting with side information at the [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
read the original abstract

We study the rate-cost tradeoff in rate-limited control of general stochastic control systems, including nonlinear systems, over a finite horizon. At each time step, an encoder observes the state and transmits a description to a controller, which then selects the control action. For an average control-cost threshold $D$, we characterize the minimum achievable communication rate $R_n(D)$ via a nonasymptotic bound: $R_n(D)$ lies within an additive logarithmic gap of the optimal value of a directed-information minimization $F_n(D)$, namely, we show that $F_n(D) \le R_n(D) \le F_n(D)+\log \bigl(F_n(D)+3.4\bigr)+2+\frac{1}{n}$, in bits. This establishes directed information as the operationally relevant quantity governing rate-limited control, thereby broadening its utility beyond its previously established roles in causal source coding and linear quadratic Gaussian (LQG) control to general nonlinear control systems. We prove the upper bound constructively by building an encoding-and-control policy using the strong functional representation lemma at each time step. As special cases of our setting, our framework yields nonasymptotic bounds for sequential (causal) rate-distortion and LQG control.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper considers finite-horizon rate-cost tradeoffs for general (including nonlinear) stochastic control systems in which an encoder observes the state and sends a description to a controller that chooses the action. For a given average cost threshold D it defines R_n(D) as the minimum achievable rate and F_n(D) as the minimum directed information. It claims the non-asymptotic sandwich F_n(D) ≤ R_n(D) ≤ F_n(D) + log(F_n(D) + 3.4) + 2 + 1/n (in bits) and proves the upper bound constructively by applying the strong functional representation lemma at each time step to the conditional distributions P(U_t | X^t, U^{t-1}). Special cases recover non-asymptotic bounds for causal rate-distortion and LQG control.

Significance. If the claimed gap holds, the result would make directed information the operationally relevant quantity for rate-limited nonlinear control, extending its known roles in causal source coding and LQG. The non-asymptotic character and explicit constructive policy are positive features.

major comments (1)
  1. [Abstract] Abstract (upper-bound construction): the claimed additive gap is only logarithmic in F_n(D). The stated proof applies the strong functional representation lemma separately at each of the n time steps, each application contributing an overhead of log(I_t + 3.4) + O(1) where I_t = I(X_t; U_t | past). The total overhead is therefore ∑ log(I_t + 3.4) + O(n). Jensen’s inequality then yields at most n log(F_n/n + 3.4) + O(n), which is linear in n rather than logarithmic in F_n. No global block-coding, shared randomness, or non-causal compression step that would reduce the overhead to a single log(F_n) term while preserving causality and the state-dependent dynamics is described. This discrepancy is load-bearing for the central sandwich claim.
minor comments (1)
  1. The abstract states that the framework yields non-asymptotic bounds for sequential rate-distortion and LQG as special cases, but the precise specialization steps (e.g., how the general directed-information functional reduces to the known LQG expression) are not visible in the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and for highlighting this critical point regarding our upper bound. We provide a point-by-point response below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (upper-bound construction): the claimed additive gap is only logarithmic in F_n(D). The stated proof applies the strong functional representation lemma separately at each of the n time steps, each application contributing an overhead of log(I_t + 3.4) + O(1) where I_t = I(X_t; U_t | past). The total overhead is therefore ∑ log(I_t + 3.4) + O(n). Jensen’s inequality then yields at most n log(F_n/n + 3.4) + O(n), which is linear in n rather than logarithmic in F_n. No global block-coding, shared randomness, or non-causal compression step that would reduce the overhead to a single log(F_n) term while preserving causality and the state-dependent dynamics is described. This discrepancy is load-bearing for the central sandwich claim.

    Authors: We appreciate the referee's careful analysis of the upper bound construction. The manuscript indeed proposes applying the strong functional representation lemma at each time step to construct the encoding policy. As the referee correctly notes, this leads to a cumulative overhead of ∑_t log(I_t + 3.4) + O(n). Using the concavity of the logarithm and Jensen's inequality, the overhead is at most n log(F_n(D)/n + 3.4) + O(n), which is linear in the horizon n. The paper does not provide a mechanism such as shared randomness across time steps or a block-coding approach that preserves causality to achieve a single logarithmic term. We acknowledge that the claimed bound F_n(D) ≤ R_n(D) ≤ F_n(D) + log(F_n(D) + 3.4) + 2 + 1/n does not follow from the described construction. In the revised version, we will update the abstract, the statement of the main result, and the proof to reflect the correct overhead, changing the upper bound to R_n(D) ≤ F_n(D) + n log(F_n(D)/n + 3.4) + C for some constant C. We will also discuss whether a logarithmic gap is achievable under additional assumptions or with a modified construction. This revision will be made. revision: yes

Circularity Check

0 steps flagged

No significant circularity; F_n(D) and R_n(D) defined independently with explicit constructive upper bound

full rationale

F_n(D) is defined as the minimum directed information I(X^n → U^n) subject to the average cost constraint, while R_n(D) is the infimum operational rate over all causal encoding-control policies achieving the same cost. The lower bound follows from standard directed-information data-processing inequalities that hold for any joint distribution induced by a policy. The upper bound is established by an explicit per-step construction that invokes the strong functional representation lemma on the conditional distributions P(U_t | X^t, U^{t-1}), producing a concrete policy whose rate is bounded by F_n(D) plus the stated logarithmic term. This construction is independent of the definition of F_n(D) and does not rely on fitting parameters to data, self-referential definitions, or load-bearing self-citations. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the strong functional representation lemma to the conditional distributions arising in the stochastic control problem and on standard properties of directed information.

axioms (1)
  • standard math The strong functional representation lemma applies to the relevant conditional distributions at each time step of the control system.
    Invoked constructively to build the encoding-and-control policy that achieves the upper bound.

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