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The Risk-Sensitive Schr\"odinger Bridge: Is Not a KL Projection

Hamidreza Behjoo

arXiv:2609.27250v1cs.ITcs.AImath.IT

TL;DR

The standard Schrödinger bridge is a KL projection onto a fixed reference, but risk sensitivity with hard endpoint constraints breaks that structure. The paper formulates a terminal-multiplier fixed point and shows that its value has no fixed-reference KL representation with a regular endpoint law, with an explicit θ/2 heat-equation defect.

  • Problem

    Replacing expected path cost with entropic risk while retaining hard endpoint constraints raises whether the bridge remains a KL projection onto a fixed reference.

    Source: §I · p. 1

  • Method

    The paper enforces the terminal constraint through a terminal-multiplier fixed point and uses an asymmetric factorisation that penalises the backward factor’s score energy.

    Source: §I · p. 1, A. Gauge and the bridge value · p. 2, D. Asymmetric factorisation · p. 3

  • Results

    The Gaussian bridge value has a θ/2 heat-equation defect, and no endpoint-regular fixed path-space reference represents the fixed-point value; no single reference serves two distinct θ values.

    Source: §I · p. 1, §IV · p. 3, §VII · p. 6

  • Takeaways & Limitations

    The fixed-point construction preserves exact terminal matching, but risk-sensitive bridge values do not admit the fixed-reference KL dual used by the standard bridge.

    Source: §IV · p. 3, §VII · p. 6

    A general existence theory and contraction of the fixed-point iteration for θ > 0 remain open problems.

    Source: §VII · p. 6

Abstract

from arXiv · show

The Schr\"odinger bridge owes its computational power to a single structural fact: by Girsanov's theorem the controlled problem is a Kullback--Leibler (KL) projection onto a fixed reference measure, solvable by alternating projections. This letter shows that the fact does not survive risk sensitivity. When the expected path cost is replaced by the entropic risk measure and both endpoint marginals are kept as hard constraints, the resulting fixed-point bridge value $J_\theta$ (the soft-problem value at the multiplier that enforces the terminal constraint) admits no representation as a constrained KL minimum against any fixed path-space reference with a regular endpoint law (a class strictly larger than the uniformly elliptic diffusion references: no Markov property is required), even allowing an additive normalisation depending on the initial marginal. Moreover, no single reference generates the one-parameter family in the risk parameter. The obstruction is computed in closed form: the Gaussian bridge value violates, by exactly $\theta/2$, a heat equation that any Gaussian smoothing of a fixed endpoint density must obey. In place of the projection, the theory rests on a terminal-multiplier fixed point and an asymmetric factorisation penalising the score energy of the backward factor.

I. INTRODUCTION

Risk sensitivity with hard endpoint constraints breaks the fixed-reference KL projection structure underlying the standard Schrödinger bridge. The paper introduces a terminal-multiplier fixed point and identifies both the obstruction and the structure that replaces projection.

I. INTRODUCTION

  • The Gaussian bridge value violates a heat equation by exactly θ/2, obstructing a fixed-reference KL representation under risk sensitivity and hard endpoint constraints.The paper distinguishes the fixed-point bridge value from the hard-constrained risk-sensitive value; they separate at first order by a variance–covariance gap.Source: §I · p. 1
  • The proposed bridge enforces the hard terminal constraint through a terminal-multiplier fixed point and has an asymmetric factorisation penalising the backward factor’s score energy.The paper also gives a Gaussian Riccati reduction to decoupled scalar shooting equations.Source: §I · p. 1
  • Prior risk-sensitive control work considered unconstrained problems with a free terminal law, while the stated contribution addresses hard endpoint constraints.The paper describes this as the first Schrödinger bridge formulation under the entropic risk measure with hard endpoint constraints.Source: §I · p. 1

II. PRELIMINARIES: THE STANDARD BRIDGE

  • By Girsanov’s theorem, the standard bridge is a KL projection onto a fixed reference, with a symmetric factorisation of the optimally controlled density.The reference incorporates the running potential through an exponential change of measure.Source: §II · p. 2

III. THE RISK-SENSITIVE BRIDGE

The risk-sensitive bridge replaces the linear expectation with a cumulant-generating objective for a prescribed terminal cost. Its terminal potential becomes a multiplier chosen to enforce the hard terminal constraint.

III. THE RISK-SENSITIVE BRIDGE

  • The soft risk-sensitive value replaces the linear expectation in the standard objective with a cumulant-generating function for a prescribed terminal cost.The soft problem imposes only the initial constraint.Source: §III · p. 2
  • With the terminal constraint restored, the bridge value recovers the classical problem at θ = 0.Source: §III · p. 2
  • To enforce the hard terminal law, λ becomes a terminal multiplier—a shooting parameter—rather than a prescribed datum.Source: §III · p. 2

A. Gauge and the bridge value

The bridge value is defined through a gauge-invariant fixed-point combination of the soft value and terminal multiplier. For θ > 0, it differs from the hard-constrained risk-sensitive value, and the fixed point is not a Lagrangian saddle point.

A. Gauge and the bridge value

  • The risk-sensitive bridge value is the gauge-invariant fixed-point value, computable in the gauge ⟨λ∗, µ1⟩ = 0 as Sθ(λ∗).The definition is independent of the gauge representative.Source: A. Gauge and the bridge value · p. 2

B. The hard value and the first-order gap

  • For θ > 0, the fixed-point bridge value differs from the hard-constrained risk-sensitive value, although they coincide at θ = 0.Source: B. The hard value and the first-order… · p. 2
  • The two values separate at first order by a generically nonzero variance–covariance gap that may have either sign.Source: B. The hard value and the first-order… · p. 2
  • The hard constraint’s KKT multiplier remains outside the entropic exponential because kinetic cost depends on the controlled law, so the hard problem is not a linear program over path measures.The fixed-point construction instead keeps the terminal potential inside the exponential, making the generalised Cole–Hopf transform available.Source: B. The hard value and the first-order… · p. 2

C. HJB equation and generalised Cole–Hopf

The section introduces a dynamic-programming HJB equation for the state-conditioned soft value and a generalised Cole–Hopf transform.

C. HJB equation and generalised Cole–Hopf

D. Asymmetric factorisation

Risk sensitivity breaks the symmetric factorisation behind the standard bridge’s KL projection: the backward factor contributes a score-energy term to the forward potential. The paper’s no-KL result concerns the fixed-point value, not merely the optimizer.

D. Asymmetric factorisation

  • The forward factor’s effective potential adds a term proportional to the squared score of the backward factor, ∥∇log ψ∥^2.The added coefficient is κ_t θ/[2(1 − θ)^2].Source: D. Asymmetric factorisation · p. 3
  • For θ ≥ 0, the effective potential is at least V pointwise and is strictly larger wherever the backward score is nonzero.Risk sensitivity concentrates the added potential where the backward score is steep.Source: D. Asymmetric factorisation · p. 3
  • The hard terminal constraint is enforced through a terminal multiplier determined implicitly as a shooting parameter.In factorised form, the constraint is φ(1, ·)e^−λ*(·) = μ1.Source: D. Asymmetric factorisation · p. 3

IV. THE FIXED-POINT VALUE HAS NO KL DUAL

  • For θ > 0, the bridge’s asymmetric factorisation prevents the equal-weight Gibbs form underlying the standard static KL dual.The no-KL result concerns the fixed-point value; a solution-dependent reference can still reproduce the optimizer.Source: §IV · p. 3

A. The per-θ representation is circular

A fixed-θ KL representation can match the risk-sensitive bridge only by constructing its reference from the unknown solution. The accompanying static reduction and Gaussian point-mass result support the impossibility arguments developed in the following sections.

A. The per-θ representation is circular

  • A fixed-θ reference matches the bridge value only when its drift and potential satisfy a matching condition built from the risk-sensitive solution itself.With h = f and U = Veff, the reference I-projection reproduces the risk-sensitive bridge optimizer when λ* enforces the terminal constraint.Source: A. The per-θ representation is circular · p. 3
  • Because the matching reference presupposes ∇Sθ, the representation is circular rather than an independent duality.The matching condition can instead be read constructively as a fixed-point iteration on the multiplier coset.Source: A. The per-θ representation is circular · p. 3

B. Two ingredients

  • The point-mass Gaussian value is parameterized by a shooting map whose variance relation is a strictly decreasing bijection from a1 ∈ (−1, ∞) onto s2 ∈ (0, ∞).At θ = 0, the value reduces to KL(N(m, s2) ∥ N(0, 1)).Source: B. Two ingredients · p. 4
  • Endpoint regularity requires a positive continuous endpoint density with two-sided Gaussian bounds.This class includes uniformly elliptic diffusions with bounded measurable drift and bounded Feynman–Kac potential, as well as Ornstein–Uhlenbeck references.Source: B. Two ingredients · p. 4
  • For an endpoint-regular reference, the path-space constrained KL minimum reduces to a static Schrödinger problem on the endpoint pair.The reduction follows from disintegrating the KL divergence over endpoints.Source: B. Two ingredients · p. 4
  • Restricting the additive correction to depend only on μ0 makes the representation question meaningful, unlike allowing dependence on both marginals.The standard bridge’s Feynman–Kac normalisation is independent of both marginals.Source: B. Two ingredients · p. 4

C. No reference works uniformly in θ

No single Markov reference with marginal-independent additive constants represents the risk-sensitive bridge value at two distinct risk parameters. In the Gaussian point-mass limit, the values’ difference depends on the target mean, contradicting what a shared reference would require.

C. No reference works uniformly in θ

  • No θ-independent Markov reference represents Jθ as a constrained KL minimum for two distinct θ values across all absolutely continuous marginals.The proposition allows additive constants Cθ independent of the marginals.Source: C. No reference works uniformly in θ · p. 4
  • A shared reference would make the difference between the two point-mass Gaussian values constant in the target mean, but the difference is 1/2(θ1 − θ2)m2 at s2 = 1.The contradiction follows by taking initial Gaussians to the point-mass limit and comparing Gaussian targets.Source: C. No reference works uniformly in θ · p. 4
  • The risk-sensitive bridge is therefore a one-parameter family of different control problems, not a reweighting of one fixed reference measure.Source: C. No reference works uniformly in θ · p. 4

D. No fixed reference works at all

No endpoint-regular path-space reference, Markovian or otherwise, represents the fixed-point value as a constrained KL minimum with an initial-marginal-only correction. The scalar Brownian Gaussian value violates the heat equation required of Gaussian smoothing of a fixed endpoint density by exactly θ/2.

D. No fixed reference works at all

  • No endpoint-regular reference represents Jθ as a constrained KL minimum with an additive correction depending only on μ0, even for full-support absolutely continuous marginals.The impossibility allows non-Markov references and applies for θ ∈ (0, 1) in the scalar Brownian setting.Source: D. No fixed reference works at all · p. 4
  • A fixed endpoint density would make the Gaussian-smoothed value obey a heat equation in target mean m and variance s2.The point-mass limit reduces the proposed representation to Gaussian smoothing of log ρ0, up to a constant.Source: D. No fixed reference works at all · pp. 4–5
  • The heat-equation defect is exactly θ/2 at (m, s2) = (0, 1), ruling out every θ > 0.At θ = 0, the condition holds with the Brownian endpoint law N(0, 1).Source: D. No fixed reference works at all · p. 5
  • The structural contrast is an inf–sup of KL-type objects for entropic risk versus a single inf for the standard bridge.The inner supremum collapses only at θ = 0.Source: D. No fixed reference works at all · p. 5

V. THE GAUSSIAN CASE: CLOSED-FORM RICCATI

For Gaussian targets, a Riccati reduction turns the risk-sensitive bridge into decoupled scalar shooting problems. The fixed point is well-posed in this setting, while the bridge value can diverge at finite risk sensitivity for an unbounded initial law.

V. THE GAUSSIAN CASE: CLOSED-FORM RICCATI

  • The Gaussian ansatz yields a closed system for the precision and covariance, with covariance evolution matching the controlled Ornstein–Uhlenbeck SDE.In the eigenbasis of the terminal covariance, the system decouples into scalar equations; each terminal condition becomes one shooting equation per coordinate.Source: §V · p. 5
  • A unique terminal-multiplier coset exists for every positive terminal variance in the Gaussian setting.The shooting map is strictly decreasing and surjective onto positive values; the Brownian limit recovers the point-mass map.Source: §V · p. 5
  • For θ > 0, terminal precision from a point mass varies with target variance near s^2 = 1, whereas at θ = 0 it is target-independent.This factor-level discrepancy is identified as the counterpart of the value-level defect.Source: §V · p. 5
  • For the illustrated unbounded Gaussian initial law, the bridge value grows superlinearly and diverges at finite θ, while the point-mass soft value remains finite for every θ < 1.The terminal constraint is met exactly at every θ, and the intermediate variance contracts as θ grows.Source: §V · p. 5

VI. WHAT REPLACES THE PROJECTION

The risk-sensitive bridge retains a linearising Cole–Hopf structure, but replaces projection geometry with asymmetric factorisation. Its first-order correction localizes forward-mass depletion to regions where the standard bridge’s backward-factor score energy is large.

VI. WHAT REPLACES THE PROJECTION

  • For θ > 0, the bridge retains the generalised Cole–Hopf linearisation and factors as p* = φ ψ^(1/(1−θ)) under an effective potential.Its value remains an inf–sup of KL-type objects, but there is no single fixed reference or projection geometry.Source: §VI · pp. 5–6
  • The first-order density correction is governed by a linearised Fokker–Planck equation and has zero total mass at every time.It follows by matching first-order terms in the exact Fokker–Planck equation.Source: §VI · p. 6
  • Forward mass is depleted to first order where the standard bridge’s backward-factor score energy is large, such as near saddles, ridges, or ambiguous zones between mixture modes.The paper describes this as a design handle on which trajectories a generative sampler avoids.Source: §VI · p. 6

VII. CONCLUSION

The paper constructs a risk-sensitive Schrödinger bridge that preserves exact terminal matching and Cole–Hopf linearisation, while proving that its value has no fixed-reference KL dual. The conclusion identifies the θ/2 heat-equation defect and lists open problems in existence, iteration, the θ → 1 limit, and duality.

VII. CONCLUSION

  • The bridge preserves exact terminal matching and Cole–Hopf linearisation, but its value admits no fixed-reference KL dual.The obstruction is an explicit θ/2 heat-equation defect; per-θ representations are circular, and no single reference serves two distinct θ.Source: §VII · p. 6
  • The paper leaves general existence theory, contraction of the fixed-point iteration for θ > 0, the θ → 1 limit, and a duality theory for Jθ as open problems.Source: §VII · p. 6
  • The fixed-reference impossibility result extends to every endpoint-regular reference, and the paper rules out a single reference for two distinct risk parameters.Source: §VII · p. 6
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