Pith. sign in

REVIEW 3 major objections 5 minor 47 references

Path Integral Value Matching for Linear Quadratic Stochastic Optimal Control

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The optimal value function for stochastic control satisfies a temporal recursion, so it can be learned by short-horizon TD matching instead of full-trajectory path integrals.

desk verdict A useful TD-style shortcut for path integral control, with real but overstated speed advantages and a convergence theorem that does not cover its own benchmarks. read the letter →

arxiv 2608.10777 v1 pith:KZ75KAL4 submitted 2026-08-11 cs.LG math.OCstat.ML

Bangyan Liao , Chenglei Yu , Yuchen Yang , Chuanrui Wang , Zhisheng Song , Peidong Liu , Tailin Wu This is my paper · ORCID
classification cs.LGmath.OCstat.ML MSC 93E2049L2560H30
keywords pathintegralcontrolstochasticoptimalvaluefunctiontemporaldifferencelearningGirsanovtheoremoff-policyhigh-dimensionallinearquadratic
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

The paper claims that the path integral representation of the optimal value function for linear-quadratic stochastic optimal control can be truncated and marginalized into a temporal recursion, so that the value at time t is an expectation of the value at a later time s discounted by the running cost over [t,s]. This recursion, its Proposition 3.1, turns the global Feynman-Kac expectation into a local bootstrapping target, which is the basis of a new algorithm, PI-VM, that learns the value function with temporal-difference updates and short M-step sample branches. If correct, it replaces the high-variance, expensive full-trajectory Monte Carlo used by policy-based solvers and classical path integral control with short-horizon, off-policy training. The experiments support this with an order-of-magnitude speedup in low dimensions and stable convergence in up to 200 dimensions.

What carries the argument

The central object is the recursive path integral identity exp(−V(X_t,t)) = $E^{{P0}}$[ exp(−V(X_s,s)) exp(−∫_t^s f dr) | F_t ], with V(x,1) = g(x). It turns the global path integral (Feynman-Kac representation) into a local transition operator that maps the value function at time s to the value at time t. The PI-VM algorithm fits a neural network V_θ(x,t) to satisfy this identity by minimizing a squared TD loss whose target is the Monte Carlo average of exp(−W − G) over N short M-step branches; the Girsanov theorem provides the density ratio that makes the same loss valid off-policy.

What would settle it

Run the iterative update of Definition 3.2 on a problem with f=0 and a linear terminal cost (as in the Linear OU benchmark) and measure ∥V_k − V∥; if the sequence converges to the optimal value despite f_inf=0, the stated contraction-based argument is not the reason, and if it diverges or oscillates, the theorem's assumption is violated. A more controlled test is to set f(x,t) = −c < 0 and check whether the fixed-point equation exp(−V) = E[exp(−V) exp(∫ c)] admits the claimed bounded solution, and to compare the empirical variance of the short-horizon estimator against the full-trajectory estimator on a fixed problem to verify Proposition 3.4's variance reduction term is positive.

Watch

Extended reading notes

Core claim

The discovery is that exp(−V(X_t,t)) = $E^{{P0}}$[ exp(−V(X_s,s)) exp(−∫_t^s f(X_r,r) dr) | F_t ] for any 0 < t < s, with V(x,1) = g(x), i.e., the path integral value function satisfies a multiplicative recursion under the uncontrolled measure. The paper proves this as Proposition 3.1 via the tower property of conditional expectation, and uses it to define an iterative bootstrapping rule (Definition 3.2). The central claim is that this recursion is the right object for practical computation: it converts a long-horizon, high-variance path integral into a short-horizon conditional expectation that can be learned by temporal-difference matching, with off-policy data re-weighted by the Girsanov density ratio so that the estimator variance vanishes as the sampling control approaches optimality (Theorem 4.3).

Load-bearing premise

The convergence proof (Theorem 3.3) assumes the running cost f(x,t) is bounded below by a positive constant, giving a contraction factor exp(−f_inf ε) < 1; every benchmark in the paper uses f=0 (Linear OU, GMM, Many Well) or f=x^T P x that vanishes at the origin (Quadratic OU), so the contraction condition is not satisfied in the experimental regime.

Editorial extensions

If this is right

  • Full-trajectory Monte Carlo is replaced by short-horizon TD targets, so the per-iteration cost becomes independent of the time horizon.
  • Because the value function is learned directly, control can be recovered as u = −σ^T ∇V, and the paper reports that this indirect route matches or beats direct policy optimization in accuracy on quadratic benchmarks.
  • Off-policy training via experience replay and Girsanov reweighting is legitimate, with the estimator variance provably vanishing as the sampling policy approaches the optimal one (Theorem 4.3).
  • The recursive estimator's variance reduction grows with the distance from the terminal time, making long-horizon problems relatively easier than naive PIC.
  • The method scales to at least 200 state dimensions without the out-of-memory failures of full-trajectory policy-search baselines.

Reading between the lines

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

  • The recursion is not limited to LQ-SOC: the same log-sum-exp identity appears in any linearly solvable MDP with the Cole-Hopf structure, so a path integral value matching target could be applied to risk-sensitive RL where the log-sum-exp Bellman operates on the same object.
  • The paper's own convergence theorem excludes its experimental f=0 cases; a natural extension is to prove convergence with nonnegative (rather than strictly positive) running cost, e.g., via a weaker contraction in a weighted norm, or to show that in practice the TD update acts as a damped iteration even when the contraction factor equals one.
  • A testable extension is to apply PI-VM to Schrödinger bridge / diffusion-model sampling problems where the terminal cost g is a log-density; the off-policy Girsanov reweighting should allow training on data from a suboptimal sampler and still recover the optimal value.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes PI-VM, a value-based solver for linear-quadratic stochastic optimal control. Starting from the classical path-integral representation of the optimal value function, it derives a temporal recursion (Proposition 3.1) and turns it into a temporal-difference training objective with short M-step Monte Carlo branches, Girsanov reweighting, and experience replay. The paper states a convergence theorem for the value-iteration analog (Theorem 3.3), a variance-decomposition claim (Proposition 3.4), and an off-policy variance bound (Theorem 4.3). It reports experiments on three unimodal SOC tasks and two multimodal sampling tasks, with ablations and a high-dimensional scalability study.

Significance. The central idea is attractive and potentially important: replacing full-trajectory path-integral Monte Carlo by a recursive, short-horizon value-matching objective could remove the main computational bottleneck of path-integral control. The tower-property derivation of Eq. (7) is clean, the connection to Feynman-Kac is standard, and the experimental results are consistent with the claimed behavior on the reported benchmarks. The paper also includes useful ablations and analytic ground-truth benchmarks. However, the theoretical support is currently narrower than the claims: the convergence theorem excludes all five experimental settings, and the off-policy variance bound has a gap in its proof. The efficiency claim also needs a clearer definition of the reported times.

major comments (3)
  1. [§3.2 / Appendix B (Theorem 3.3)] The stated convergence guarantee is not applicable to any of the five benchmarks. Theorem 3.3 relies on Assumption A1, f(x,t) >= f_inf > 0, which gives the contraction factor exp(-f_inf epsilon) < 1 in the Appendix B proof. But Linear OU, GMM, and Many Well all use f ≡ 0 (Appendix G.2.1, H.2.1, H.3.1), and Quadratic OU uses f = x^T P x, which vanishes at x = 0 (Appendix G.1.1). The paper never discloses that its own experimental regime violates the theorem's main assumption. In addition, the main-text statement of Theorem 3.3 assumes A2: 'V bounded below', while Appendix B's restatement assumes g continuous and bounded and works in C_b; these are not equivalent, and neither holds for the unbounded quadratic value functions. This is load-bearing because Theorem 3.3 is presented as the theoretical foundation for PI-VM. The authors should either prove convergence under a weaker norm or a weighted contraction that covers the quadratic setting, or explicitly restrict the theoretical claims and describe the experimental section as outside the theorem.
  2. [Appendix C (Theorem 4.3)] The proof of the off-policy variance bound contains an unjustified step. After the Girsanov change to the measure Q^u with drift b - sigma u + 2 sigma u*, the proof bounds E^{Q^u}[ exp(-2V(X_r,r) + ∫_t^r ||u-u*||^2 dτ) | F_t ] by exp(-2V(X_t,t)) exp(kappa(r-t)). This treats exp(-2V(X_r,r)) as if it were exp(-2V(X_t,t)), but V(X_r,r) is random under Q^u and is not shown to be monotone along the shifted process. No pointwise comparison V(X_r,r) >= V(X_t,t) is available from the assumptions. The claimed variance-vanishing behavior of the off-policy estimator is therefore not established. The theorem should either be proved with a valid bound (for example, using a pathwise bound or a different localization argument) or removed from the theoretical claims.
  3. [§5.1 / Table 1] The manuscript reports a 'TIME' column in Tables 1 and 2 but never defines what it measures. If it is per-update or per-iteration wall-clock time, the statement that PI-VM is '10–20x faster' is not justified unless PI-VM also reaches the reported control error in the same number of updates as the baselines. The text says that training hyperparameters are aligned with SOCM for the first three tasks, but it does not state that all methods are run for the same number of updates or that the reported error is the final error after a fixed budget. Please report total training time to a fixed control-error threshold, or explicitly report per-update time together with learning curves that allow a total-time comparison.
minor comments (5)
  1. [§3.1] The phrase 'truncating and marginalizing the original path integral formulation' is not quite accurate; Eq. (7) follows directly from the tower property of conditional expectation. The wording could be adjusted to avoid implying a new approximation.
  2. [Definition 4.2] The text says the off-policy loss has 'the same setup with Theorem 4.1'; this should be Definition 4.1, not Theorem 4.1.
  3. [Algorithm 1 / Algorithm 2] The pseudocode does not state whether the N short-term branches sampled at line 7/8 follow the uncontrolled measure P^0 or the controlled measure P^u. This matters because the Girsanov correction S in Eq. (17) is only valid if the branches are generated under the control appearing in S. Please specify the sampling measure explicitly, including whether the stored S corresponds to the control used at collection time or to the current control during replay.
  4. [Table 3] The ablation table reports only point estimates without standard deviations, unlike Tables 1, 2, and 4. Adding error bars would make the 'no reduction from M=8 to M=16' conclusion more robust.
  5. [Eq. (10)] Using the same symbol V for both the value function and variance in Proposition 3.4 is confusing; consider using Var(·) for variance.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the recursive path-integral identity is a self-contained Feynman-Kac semigroup decomposition.

full rationale

Proposition 3.1 / Eq. (7) is obtained directly from the Feynman-Kac representation in Eq. (20) by the tower property of conditional expectation: the expectation from t to 1 is split at s, and the inner expectation from s to 1 is exactly exp(-V(X_s, s)) by the very same representation. This is a semigroup identity, not a fitted relation; no parameter is adjusted to make Eq. (7) hold, and the TD loss in Eqs. (12)-(14) estimates this identity with Monte Carlo branches and a target network, which is standard bootstrapping rather than a prediction forced by construction. The paper cites external prior work for benchmarks and baselines, and no load-bearing argument reduces to a self-citation. The genuinely weak spots are correctness risks rather than circularity: Theorem 3.3's contraction factor exp(-f_inf * epsilon) < 1 requires Assumption A1 (f >= f_inf > 0), whereas the Linear OU, GMM, and Many Well tasks set f = 0 and Quadratic OU uses f = x^T P x, which vanishes at x = 0; and Appendix B's proof asserts 'V_k -> V' after establishing contraction without explicitly verifying that exp(-V) is the unique fixed point of the operator (though this follows from Eq. (7)). The conclusion's admission that computing the control still requires automatic differentiation is a stated limitation, not a circular step. None of these issues makes the central derivation equivalent to its own inputs.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The central derivation uses standard stochastic calculus tools. The only hand-chosen quantities are algorithm hyperparameters (M, N, B, eta, tau). The main theoretical gap is that the strict positivity assumption on the running cost, needed for the contraction proof, is not satisfied by any of the test problems.

free parameters (5)
  • TD horizon M = 8 (default)
    Number of forward steps for the short-horizon path branches; hand-chosen and ablated in Table 3, with (N=8, M=8) selected as the default trade-off.
  • Monte Carlo branches N = 8 (default)
    Number of short branches per state for the TD target; ablated in Table 3 and Appendix D.3.
  • Training batch size B = 6400 or 12800 per task
    Batch size set per task in Appendices G and H; ablation in D.2 shows larger batch helps.
  • Learning rate eta = 1e-4 or 5e-4 per task
    Adam learning rate chosen per task; no sensitivity analysis reported.
  • Target network EMA tau = not reported
    Exponential moving average coefficient for the target network (Algorithm 1); the value is not given in the paper.
assumptions (8)
  • standard math Feynman-Kac theorem represents the HJB solution as a conditional expectation under the uncontrolled path measure (used in Eq (5) and Prop 3.1).
    Standard stochastic calculus result, cited as [14].
  • standard math Cole-Hopf transform linearizes the HJB equation (used to obtain Eq (5)).
    Classical transform, cited as [17].
  • standard math Girsanov theorem gives the density ratio between controlled and uncontrolled path measures (Eq (15) and Appendix A).
    Standard result, cited as [14].
  • standard math Tower property of conditional expectation (used in the proof of Prop 3.1 and Appendix A).
    Basic probability identity.
  • domain assumption Running cost bounded below by a positive constant f >= f_inf > 0 (Theorem 3.3 A1, Appendix B).
    Needed for the contraction factor exp(-f_inf epsilon) < 1. All benchmark tasks violate it (f=0 or f=x^T P x vanishing at x=0), so the convergence theorem does not apply to the experiments.
  • domain assumption Lipschitz and growth conditions giving the Feller property for the transition semigroup (Theorem 3.3 A3).
    Standard regularity condition for existence and continuity of the semigroup; plausible for the linear/OU tasks but not verified for all settings.
  • domain assumption Boundedness of the terminal cost and of the iterates in C_b (Appendix B).
    The contraction is taken in C_b; Appendix B states g is continuous and bounded, while the main text A2 instead bounds the value function from below, an inconsistency.
  • domain assumption The optimal control relation u* = -sigma^T nabla V (Eq (6)).
    Standard first-order condition for the quadratic control cost; this is what turns the learned value into a control.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Path Integral Value Matching for Linear Quadratic Stochastic Optimal Control." pith.science (2026). https://pith.science/paper/KZ75KAL4

@misc{pith2026260810777,
  author       = {Pith},
  title        = {Pith review of: Path Integral Value Matching for Linear Quadratic Stochastic Optimal Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZ75KAL4}},
  note         = {Machine review of arXiv:2608.10777}
}
read the original abstract

Linear Quadratic Stochastic Optimal Control (LQ-SOC) establishes a fundamental framework for steering noisy dynamical systems and has recently gained renewed interest in the machine learning community. However, current state-of-the-art policy-based methods suffer from prohibitive computational costs and instability due to their heavy reliance on full-trajectory simulation. To overcome these limitations, we propose a paradigm shift toward a value-based approach by revisiting Path Integral Control (PIC). Although standard PIC suffers from the same high-variance bottleneck as policy-based methods, we discover that by truncating and marginalizing the original path integral formulation, we can derive a temporal recursive form of the value function. Building upon this theoretical foundation, we propose the Path Integral Value Matching (PI-VM) algorithm. Specifically, we employ temporal-difference learning to approximate the recursive value dynamics, and further integrate the Girsanov theorem with experience replay to enable off-policy training. We benchmark PI-VM against SOTA policy-based methods across various SOC benchmarks and sampling tasks. Empirical results demonstrate that PI-VM matches SOTA precision with an order-of-magnitude efficiency gain in low-dimensional settings, while effectively mitigating mode collapse in high-dimensional scenarios. Consequently, PI-VM offers a scalable solution for solving complex SOC problems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 26 canonical work pages

  1. [1]

    Dynamic programming.science, 153(3731):34–37, 1966

    Richard Bellman. Dynamic programming.science, 153(3731):34–37, 1966

    1966

  2. [2]

    Springer Science & Business Media, 2012

    Wendell H Fleming and Raymond W Rishel.Deterministic and stochastic optimal control, volume 1. Springer Science & Business Media, 2012

    2012

  3. [3]

    Stochastic optimal control matching.Advances in Neural Information Processing Systems, 37:112459–112504, 2024

    Carles Domingo i Enrich, Jiequn Han, Brandon Amos, Joan Bruna, and Ricky TQ Chen. Stochastic optimal control matching.Advances in Neural Information Processing Systems, 37:112459–112504, 2024

    work page 2024

  4. [4]

    Carles Domingo-Enrich, Michal Drozdzal, Brian Karrer, and Ricky T. Q. Chen. Adjoint matching: Fine- tuning flow and diffusion generative models with memoryless stochastic optimal control. InThe Thirteenth International Conference on Learning Representations, 2025

    work page 2025

  5. [5]

    Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schrodinger bridge.Siam Review, 63(2):249–313, 2021

    Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schrodinger bridge.Siam Review, 63(2):249–313, 2021

    2021

  6. [6]

    Adjoint schr\" odinger bridge sampler.arXiv preprint arXiv:2506.22565, 2025

    Guan-Horng Liu, Jaemoo Choi, Yongxin Chen, Benjamin Kurt Miller, and Ricky TQ Chen. Adjoint schr\" odinger bridge sampler.arXiv preprint arXiv:2506.22565, 2025

    arXiv 2025

  7. [7]

    Trust region constrained measure transport in path space for stochastic optimal control and inference.arXiv preprint arXiv:2508.12511, 2025

    Denis Blessing, Julius Berner, Lorenz Richter, Carles Domingo-Enrich, Yuanqi Du, Arash Vahdat, and Gerhard Neumann. Trust region constrained measure transport in path space for stochastic optimal control and inference.arXiv preprint arXiv:2508.12511, 2025

    arXiv 2025

  8. [8]

    A taxonomy of loss functions for stochastic optimal control.arXiv preprint arXiv:2410.00345, 2024

    Carles Domingo-Enrich. A taxonomy of loss functions for stochastic optimal control.arXiv preprint arXiv:2410.00345, 2024

    arXiv 2024

Show all 47 references
  1. [9]

    Linearly-solvable markov decision problems.Advances in neural information processing systems, 19, 2006

    Emanuel Todorov. Linearly-solvable markov decision problems.Advances in neural information processing systems, 19, 2006

    2006

  2. [10]

    An introduction to stochastic control theory, path integrals and reinforcement learning

    Hilbert J Kappen. An introduction to stochastic control theory, path integrals and reinforcement learning. InAIP conference proceedings, volume 887, pages 149–181. American Institute of Physics, 2007

    2007

  3. [11]

    Linear theory for control of nonlinear stochastic systems.Physical review letters, 95(20):200201, 2005

    Hilbert J Kappen. Linear theory for control of nonlinear stochastic systems.Physical review letters, 95(20):200201, 2005

    2005

  4. [12]

    Path integral control and state-dependent feedback.Physical Review E, 91(3):032104, 2015

    Sep Thijssen and HJ Kappen. Path integral control and state-dependent feedback.Physical Review E, 91(3):032104, 2015

    2015

  5. [13]

    A deeper look at experience replay.arXiv preprint arXiv:1712.01275, 2017

    Shangtong Zhang and Richard S Sutton. A deeper look at experience replay.arXiv preprint arXiv:1712.01275, 2017

    2017 arXiv

  6. [14]

    Springer Science & Business Media, 2013

    Bernt Oksendal.Stochastic differential equations: an introduction with applications. Springer Science & Business Media, 2013

    2013

  7. [15]

    Recent developments in machine learning methods for stochastic control and games.arXiv preprint arXiv:2303.10257, 2023

    Ruimeng Hu and Mathieu Lauriere. Recent developments in machine learning methods for stochastic control and games.arXiv preprint arXiv:2303.10257, 2023

    2023 arXiv

  8. [16]

    q-learning in continuous time.Journal of Machine Learning Research, 24(161):1–61, 2023

    Yanwei Jia and Xun Yu Zhou. q-learning in continuous time.Journal of Machine Learning Research, 24(161):1–61, 2023

    2023

  9. [17]

    American mathematical society, 2022

    Lawrence C Evans.Partial differential equations, volume 19. American mathematical society, 2022

    2022

  10. [18]

    Stochastic optimal control for collective variable free sampling of molecular transition paths.Advances in Neural Information Processing Systems, 36:79540–79556, 2023

    Lars Holdijk, Yuanqi Du, Ferry Hooft, Priyank Jaini, Berend Ensing, and Max Welling. Stochastic optimal control for collective variable free sampling of molecular transition paths.Advances in Neural Information Processing Systems, 36:79540–79556, 2023

    2023

  11. [19]

    Nikolas Nüsken and Lorenz Richter. Solving high-dimensional hamilton–jacobi–bellman pdes using neural networks: perspectives from the theory of controlled diffusions and measures on path space.Partial differential equations and applications, 2(4):48, 2021

    2021

  12. [20]

    Learning boltzmann generators via constrained mass transport.arXiv preprint arXiv:2510.18460, 2025

    Christopher von Klitzing, Denis Blessing, Henrik Schopmans, Pascal Friederich, and Gerhard Neumann. Learning boltzmann generators via constrained mass transport.arXiv preprint arXiv:2510.18460, 2025

    2025

  13. [21]

    Levine, Brandon M Wood, Bin Hu, Brandon Amos, Brian Karrer, Xiang Fu, Guan-Horng Liu, and Ricky T

    Aaron J Havens, Benjamin Kurt Miller, Bing Yan, Carles Domingo-Enrich, Anuroop Sriram, Daniel S. Levine, Brandon M Wood, Bin Hu, Brandon Amos, Brian Karrer, Xiang Fu, Guan-Horng Liu, and Ricky T. Q. Chen. Adjoint sampling: Highly scalable diffusion samplers via adjoint matchin...

    2025

  14. [22]

    A fast algorithm for the two dimen- sional hjb equation of stochastic control.ESAIM: M2AN, 38(4):723–735, 2004

    Frédéric Bonnans, J., Ottenwaelter, Élisabeth, and Zidani, Housnaa. A fast algorithm for the two dimen- sional hjb equation of stochastic control.ESAIM: M2AN, 38(4):723–735, 2004

    2004

  15. [23]

    On the convergence of finite element methods for hamilton–jacobi–bellman equations.SIAM Journal on Numerical Analysis, 51(1):137–162, 2013

    Max Jensen and Iain Smears. On the convergence of finite element methods for hamilton–jacobi–bellman equations.SIAM Journal on Numerical Analysis, 51(1):137–162, 2013

    2013

  16. [24]

    Model predictive path integral control: From theory to parallel computation.Journal of Guidance, Control, and Dynamics, 40(2):344–357, 2017

    Grady Williams, Andrew Aldrich, and Evangelos A Theodorou. Model predictive path integral control: From theory to parallel computation.Journal of Guidance, Control, and Dynamics, 40(2):344–357, 2017

    2017

  17. [25]

    Adaptive importance sampling for control and inference

    Hilbert Johan Kappen and Hans Christian Ruiz. Adaptive importance sampling for control and inference. Journal of Statistical Physics, 162(5):1244–1266, 2016

    2016

  18. [26]

    Recent advances in path integral control for trajectory optimization: An overview in theoretical and algorithmic perspectives

    Muhammad Kazim, Jungee Hong, Min-Gyeom Kim, and Kwang-Ki K Kim. Recent advances in path integral control for trajectory optimization: An overview in theoretical and algorithmic perspectives. Annual Reviews in Control, 57:100931, 2024

    2024

  19. [27]

    Reinforcement learning in continuous time and space: A stochastic control approach.Journal of Machine Learning Research, 21(198):1–34, 2020

    Haoran Wang, Thaleia Zariphopoulou, and Xun Yu Zhou. Reinforcement learning in continuous time and space: A stochastic control approach.Journal of Machine Learning Research, 21(198):1–34, 2020

    2020

  20. [28]

    Policy evaluation and temporal-difference learning in continuous time and space: A martingale approach.Journal of Machine Learning Research, 23(154):1–55, 2022

    Yanwei Jia and Xun Yu Zhou. Policy evaluation and temporal-difference learning in continuous time and space: A martingale approach.Journal of Machine Learning Research, 23(154):1–55, 2022

    2022

  21. [29]

    A temporal difference method for stochastic continuous dynamics.arXiv preprint arXiv:2505.15544, 2025

    Haruki Settai, Naoya Takeishi, and Takehisa Yairi. A temporal difference method for stochastic continuous dynamics.arXiv preprint arXiv:2505.15544, 2025

    2025

  22. [30]

    Continuous q-score matching: Diffusion guided reinforce- ment learning for continuous-time control.arXiv preprint arXiv:2510.17122, 2025

    Chengxiu Hua, Jiawen Gu, and Yushun Tang. Continuous q-score matching: Diffusion guided reinforce- ment learning for continuous-time control.arXiv preprint arXiv:2510.17122, 2025

    2025

  23. [31]

    Risk-sensitive reinforcement learning.Machine learning, 49(2):267– 290, 2002

    Oliver Mihatsch and Ralph Neuneier. Risk-sensitive reinforcement learning.Machine learning, 49(2):267– 290, 2002

    2002

  24. [32]

    Exponential td learning: A risk-sensitive actor-critic reinforcement learning algorithm

    Erfaun Noorani, Christos N Mavridis, and John S Baras. Exponential td learning: A risk-sensitive actor-critic reinforcement learning algorithm. InProceedings of the American Control Conference. IEEE, 2023

    2023

  25. [33]

    Gaussian error linear units (gelus).arXiv preprint arXiv:1606.08415, 2016

    D Hendrycks. Gaussian error linear units (gelus).arXiv preprint arXiv:1606.08415, 2016

    2016 arXiv

  26. [34]

    Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014

    Diederik P Kingma. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014

    2014 arXiv

  27. [35]

    Nu- merical approximation of a system of hamilton–jacobi–bellman equations arising in innovation dynamics

    L’ubomír Baˇnas, Herbert Dawid, Tsiry Avisoa Randrianasolo, Johannes Storn, and Xingang Wen. Nu- merical approximation of a system of hamilton–jacobi–bellman equations arising in innovation dynamics. Journal of Scientific Computing, 92(2):54, 2022

    2022

  28. [36]

    Semi-lagrangian schemes for linear and fully non-linear diffusion equations.Mathematics of Computation, pages 1433–1462, 2013

    Kristian Debrabant and Espen R Jakobsen. Semi-lagrangian schemes for linear and fully non-linear diffusion equations.Mathematics of Computation, pages 1433–1462, 2013

    2013

  29. [37]

    A semi-lagrangian scheme for hamilton–jacobi– bellman equations on networks.SIAM Journal on Numerical Analysis, 58(6):3165–3196, 2020

    Elisabetta Carlini, Adriano Festa, and Nicolas Forcadel. A semi-lagrangian scheme for hamilton–jacobi– bellman equations on networks.SIAM Journal on Numerical Analysis, 58(6):3165–3196, 2020

    2020

  30. [38]

    A generalized path integral control approach to reinforcement learning.The Journal of Machine Learning Research, 11:3137–3181, 2010

    Evangelos Theodorou, Jonas Buchli, and Stefan Schaal. A generalized path integral control approach to reinforcement learning.The Journal of Machine Learning Research, 11:3137–3181, 2010

    2010

  31. [39]

    Aggressive driving with model predictive path integral control

    Grady Williams, Paul Drews, Brian Goldfain, James M Rehg, and Evangelos A Theodorou. Aggressive driving with model predictive path integral control. In2016 IEEE international conference on robotics and automation (ICRA), pages 1433–1440. IEEE, 2016

    2016

  32. [40]

    Policy search for path integral control

    Vicenç Gómez, Hilbert J Kappen, Jan Peters, and Gerhard Neumann. Policy search for path integral control. InJoint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 482–497. Springer, 2014

    2014

  33. [41]

    Applications of the cross-entropy method to importance sampling and optimal control of diffusions.SIAM Journal on Scientific Computing, 36(6):A2654–A2672, 2014

    Wei Zhang, Han Wang, Carsten Hartmann, Marcus Weber, and Christof Schütte. Applications of the cross-entropy method to importance sampling and optimal control of diffusions.SIAM Journal on Scientific Computing, 36(6):A2654–A2672, 2014

    2014

  34. [42]

    Deep learning for continuous-time stochastic control with jumps.arXiv preprint arXiv:2505.15602, 2025

    Patrick Cheridito, Jean-Loup Dupret, and Donatien Hainaut. Deep learning for continuous-time stochastic control with jumps.arXiv preprint arXiv:2505.15602, 2025

    2025

  35. [43]

    exp −V(X s, s)− Z s t f(X r, r)dr Ft # =E Pu

    Jannes Quer and Enric Ribera Borrell. Connecting stochastic optimal control and reinforcement learning. Journal of Mathematical Physics, 65(8), 2024. 11 A Recursive Structure of the Optimal Value Function In this section, we demonstrate detailed proofs for the conclusion about...

    2024

  36. [44]

    exp −2V− Z r t ∥u−u ∗∥2dτ−2 Z r t (u−u ∗)·dB τ Ft # =E Pu

    under the controlled measureP u satisfies: VPu (M(X[t,s])|Ft)≤exp (−2V(X t, t)) [exp (κ(s−t))−1]. Proof. Consider the log-estimator defined by Yt,s = logM(X [t,s]). By applying Itô’s formula to the definition ofY t,s, we obtain: dYt,s =− ∂V ∂s +∇V·b+∇V·(σu) + 1 2tr(σσ ⊤∇2V) +f...

  37. [45]

    Forward Score ( ∇logP t):Since the prior dynamics preserve the stationary distribution N(0, η2I): ∇x logP t(x) =− x η2 (54)

  38. [46]

    Backward Path Measure (Qt):The target distribution evolves as a time-varying GMM: Qt(x) = KX k=1 πkN(x|µ k(t),Σ k(t))(55) The parameters evolve according to the scale factor E(t) = exp − R T t ζ(s)ds . The explicit integral of the schedule is: Z s t ζ(τ)dτ= Cmax +C min 2 (s−t)...

  39. [47]

    Final Expression:Substituting these into the control formula: u∗(x, t) =η p 2ζ(t) " KX k=1 γk(x, t)Σk(t)−1(µk(t)−x) # + x η2 ! (59) whereγ k(x, t)is the time-dependent responsibility: γk(x, t) = πkN(x|µ k(t),Σ k(t))P j πjN(x|µ j(t),Σ j(t)) (60) H.2.3 Analytical Optimal Value F...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.