pith. sign in

arxiv: 2512.06109 · v3 · pith:4O4D7MDDnew · submitted 2025-12-05 · 🧮 math.OC · cs.LG· cs.RO· cs.SY· eess.SY

Unifying Entropy Regularization in Optimal Control: From and Back to Classical Objectives via Iterated Soft Policies and Path Integral Solutions

Ajinkya Bhole , Mohammad Mahmoudi Filabadi , Guillaume Crevecoeur , Tom Lefebvre This is my paper

Pith reviewed 2026-05-17 00:12 UTC · model grok-4.3

classification 🧮 math.OC cs.LGcs.ROcs.SYeess.SY
keywords optimal controlKL regularizationentropy regularizationsoft policiespath integralstochastic optimal controlrisk-sensitive controlBellman operator
0
0 comments X

The pith

A KL-regularized umbrella problem unifies optimal control formulations and recovers the classical objectives through iteration of soft-policy solutions.

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

The paper introduces a central formulation that applies separate Kullback-Leibler penalties to policies and to transitions. This setup recovers the classical stochastic optimal control problem, its risk-sensitive variant, and their soft-policy surrogates. The soft-policy versions majorize the originals, so repeated solution of the surrogates converges back to the classical objectives. A synchronized weighting case produces a linear Bellman operator along with path-integral solutions and compositionality.

Core claim

By separating the policy KL weight from the transition KL weight, the formulation creates an umbrella that includes SOC, RSOC, soft-policy SOC, and soft-policy RSOC. The soft-policy problems majorize the classical ones, and iteration of their solutions therefore recovers the original objectives. When the two weights coincide in the soft-policy RSOC case, the Bellman operator becomes linear, admitting path-integral solutions and compositional properties that extend to a broad class of control problems.

What carries the argument

The central problem that separates the KL penalties on policies and transitions with independent weights, generalizing trajectory-level KL regularization.

If this is right

  • Soft-policy SOC and RSOC serve as tractable majorizing surrogates for the classical problems.
  • Iteration of the soft-policy solutions recovers the original SOC and RSOC objectives.
  • The synchronized soft-policy RSOC case yields a linear Bellman operator.
  • Path-integral solutions and compositionality extend to a broader class of control problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Algorithms developed for the soft-policy surrogates could be reused to solve classical problems simply by iterating them.
  • The compositionality in the synchronized case may allow breaking complex multi-stage tasks into modular subproblems whose solutions combine directly.
  • The majorization relation suggests similar surrogate-and-iterate schemes could be explored for other regularization choices in control.

Load-bearing premise

The soft-policy formulations majorize the original SOC and RSOC objectives so that iteration recovers them.

What would settle it

A numerical example on a linear-quadratic problem in which repeated solution of the soft-policy RSOC fails to converge to the risk-sensitive optimal value or policy.

read the original abstract

This paper develops a unified perspective on several optimal control formulations through the lens of Kullback-Leibler (KL) regularization. We propose a central problem that separates the KL penalties on policies and transitions with independent weights, thus generalizing the standard trajectory-level KL-regularization used in probabilistic optimal control. This umbrella formulation recovers various control problems: the classical Stochastic Optimal Control (SOC), Risk-Sensitive Stochastic Optimal Control (RSOC), and their policy-based KL-regularized counterparts, termed soft-policy SOC and RSOC, which yield tractable surrogates. Beyond being regularized variants, these soft-policy formulations majorize the original SOC and RSOC, thus, iterating their solutions recovers the original objectives. We further identify a synchronized case of soft-policy RSOC where the policy and transition KL weights coincide, yielding a linear Bellman operator, path-integral solution, and compositionality -- extending these computationally favourable properties to a broad class of control problems.

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

2 major / 2 minor

Summary. The manuscript develops a unified perspective on optimal control through Kullback-Leibler regularization. It introduces an umbrella formulation that applies independent weights to KL penalties on policies and on transitions, generalizing standard trajectory-level regularization. This central problem is claimed to recover classical Stochastic Optimal Control (SOC), Risk-Sensitive Stochastic Optimal Control (RSOC), and their soft-policy surrogates. The key technical claim is that the soft-policy formulations majorize the original SOC and RSOC objectives, so that iterating the soft solutions recovers the classical objectives. A synchronized case of soft-policy RSOC (where the two KL weights coincide) is further identified as yielding a linear Bellman operator, path-integral solution, and compositionality.

Significance. If the majorization relation and the iteration-recovery property can be established rigorously for the stated generality, the work would offer a coherent unification of several entropy-regularized control formulations and extend computationally attractive properties (linear Bellman operators and path-integral representations) to a wider class of problems. Such a bridge between surrogate and classical objectives could inform both theoretical analysis and the design of iterative algorithms in stochastic control.

major comments (2)
  1. [Abstract and umbrella-formulation section] The central claim that the soft-policy SOC/RSOC formulations majorize the original objectives (and that iteration therefore recovers the classical problems) is load-bearing for the unification narrative. The manuscript asserts this majorization in the abstract and in the section introducing the umbrella problem, yet provides neither an explicit inequality derivation nor the regularity conditions (on dynamics, costs, or weight ratios) under which the inequality holds. Without these details it is not possible to verify whether the recovery property extends to the broad class claimed.
  2. [Synchronized-case subsection] The synchronized case (equal policy and transition KL weights) is presented as yielding a linear Bellman operator and path-integral solution. The manuscript should supply the precise algebraic steps showing how the Bellman operator becomes linear under this synchronization and confirm that the resulting path-integral representation remains valid for the general (non-quadratic) costs considered elsewhere in the paper.
minor comments (2)
  1. [Notation and definitions] Notation for the two independent KL weights should be introduced once and used consistently; occasional reuse of the same symbol for both weights creates ambiguity in the general (non-synchronized) case.
  2. [Recovery claims] The paper would benefit from a short table summarizing which classical and soft-policy problems are recovered for each choice of the two weight parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our unification framework for entropy-regularized optimal control. The comments highlight areas where additional rigor will strengthen the manuscript, and we address each point below with planned revisions.

read point-by-point responses

  1. Referee: [Abstract and umbrella-formulation section] The central claim that the soft-policy SOC/RSOC formulations majorize the original objectives (and that iteration therefore recovers the classical problems) is load-bearing for the unification narrative. The manuscript asserts this majorization in the abstract and in the section introducing the umbrella problem, yet provides neither an explicit inequality derivation nor the regularity conditions (on dynamics, costs, or weight ratios) under which the inequality holds. Without these details it is not possible to verify whether the recovery property extends to the broad class claimed.

    Authors: We agree that the majorization relation requires an explicit derivation and stated conditions to support the claimed recovery property. In the revised manuscript we will insert a dedicated subsection immediately following the umbrella formulation that derives the inequality between the soft-policy objectives and the classical SOC/RSOC objectives. The derivation will be accompanied by the necessary regularity assumptions (bounded continuous costs, Lipschitz dynamics, and strictly positive KL weights) under which the majorization holds and iterated soft policies converge to the original optima. revision: yes

  2. Referee: [Synchronized-case subsection] The synchronized case (equal policy and transition KL weights) is presented as yielding a linear Bellman operator and path-integral solution. The manuscript should supply the precise algebraic steps showing how the Bellman operator becomes linear under this synchronization and confirm that the resulting path-integral representation remains valid for the general (non-quadratic) costs considered elsewhere in the paper.

    Authors: We will expand the synchronized-case subsection with the full algebraic expansion of the Bellman operator. When the two KL weights are identical the policy and transition divergence terms combine to cancel the nonlinear dependence on the value function, producing a linear operator whose solution is expressed via a path integral. We will add an explicit remark confirming that this linearity and the associated path-integral representation hold for arbitrary (non-quadratic) running costs, as the cancellation depends solely on weight synchronization and not on the cost structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; majorization supplies independent recovery bridge

full rationale

The paper introduces an umbrella KL-regularized formulation that separates policy and transition penalties as a generalization of classical SOC and RSOC. It then asserts that the resulting soft-policy surrogates majorize the original objectives, from which iteration recovers the targets. This majorization step is presented as a derived inequality rather than a definitional identity or self-referential fit; the synchronized case yielding linear Bellman operators and path integrals is identified as a special parameter choice within the same framework. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the derivation chain. The argument therefore remains self-contained once the majorization inequality is established independently of the target recovery claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard Markov decision process assumptions and the existence of well-defined value functions under the chosen KL penalties; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The underlying process is a controlled Markov decision process with well-defined transition kernels and cost functions.
    Invoked implicitly when defining the central KL-regularized problem and its special cases.

pith-pipeline@v0.9.0 · 5496 in / 1298 out tokens · 31814 ms · 2026-05-17T00:12:22.323581+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    and Todorov, E

    Dvijotham, K. and Todorov, E. (2012). Linearly solvable optimal control.Reinforcement learning and approxi- mate dynamic programming for feedback control, 119–

    work page 2012

  2. [2]

    and Schied, A

    F¨ ollmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints.Finance and stochastics, 6(4), 429–447. Ito, K. and Kashima, K. (2024). Risk-sensitive control as inference with r´ enyi divergence.Advances in Neural Information Processing Systems, 37, 71381–71413. Kappen, H.J. (2005). Linear theory for control of nonlinear stochastic s...

    work page 2002

  3. [3]

    Lefebvre, T. (2024). Probabilistic control and majorisation of optimal control.Systems & Control Letters, 190, 105837. Levine, S. (2018). Reinforcement learning and control as probabilistic inference.arXiv:1805.00909. Levine, S. and Koltun, V. (2013a). Guided policy search. InInt. Conf. Mach. Learn., 1–9. Levine, S. and Koltun, V. (2013b). Variational pol...

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  4. [4]

    Neumann, G. (2011). Variational inference for policy search in changing situations. InInternational confer- ence on machine learning, 817–824. Nishimura, H., Mehr, N., Gaidon, A., and Schwager, M. (2021). Rat ilqr: A risk auto-tuning controller.IEEE Robotics and Automation Letters, 6(2), 763–770. Noorani, E. and Baras, J.S. (2022a). Embracing risk in re- ...

    work page 2011

  5. [5]

    Toussaint, M. (2009). Robot trajectory optimization via approximate inference. InInternational conference on machine learning, 1049–1056. Toussaint, M. and Storkey, A. (2006). Probabilistic infer- ence for mdps. Ininternational conference on machine learning, 945–952. Watson, J., Abdulsamad, H., Findeisen, R., and Peters, J. (2021). Efficient stochastic o...

    work page arXiv 2009