Adversarial Reinforcement Learning: A Duality-Based Approach To Solving Optimal Control Problems

Mengzhou Liu; Nan Chen; Nanyi Zhang; Xiaoyan Wang

REVIEW 4 major objections 5 minor 26 references

Adversarial Reinforcement Learning: A Duality-Based Approach To Solving Optimal Control Problems

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a min-max duel between an adversary and a controller can yield both a near-optimal policy and a tight bracket around its true value.

desk verdict A plausible neural extension of information-relaxation duality with an uncertified inner optimization that may make the advertised gaps too small. read the letter →

arxiv 2506.00801 v2 pith:PGR3FZTG submitted 2025-06-01 math.OC

Nan Chen , Mengzhou Liu , Xiaoyan Wang , Nanyi Zhang This is my paper · ORCID

classification math.OC MSC 93E2068T07

keywords adversarialreinforcementlearningstochasticcontrolinformationrelaxationdualitydualgapdeepneuralnetworksoptimaltradeexecutionmin-maxoptimizationempiricalriskminimization

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

This paper is trying to establish that the information-relaxation dual of a high-dimensional stochastic control problem can be turned into a learnable min-max game, and that solving that game with neural-network penalties yields both a near-optimal control policy and a tight bracket around the true optimal value. If the claim is right, a simulation-based control workflow gets something it usually lacks: a certificate of how far a proposed policy is from optimal, with the certificate tightening as training proceeds. The authors demonstrate the bracket on a ten-asset, twenty-period optimal trade-execution problem, where the gap between the dual lower bound and the policy upper bound falls to roughly two to three percent. They also show that the in-sample overfitting of deep empirical risk minimization can be detected against these bounds.

What carries the argument

The machinery is the information-relaxation dual value with a neural-network-generated penalty. For a generating function $W=(W_0,\ldots,W_T)$, the penalty at time $t$ is $z^W_t(s,a,\xi)=W_{t+1}(s_{t+1})-\mathbb{E}[W_{t+1}(f_t(s_t,a_t,\eta_{t+1}))\mid s_t]$, and summing over $t$ gives a feasible penalty whose expected value is non-positive for every non-anticipative policy. The dual problem moves the maximization inside the expectation, allowing the agent to see the future, and subtracts the penalty to remove that informational advantage. Strong duality says that minimizing the dual value over $W$ recovers the true value, with the value functions $V_t^*$ attaining the minimum. ADRL replaces $W$ by neural networks $\rho_t(\cdot,\phi_t)$, solves the resulting min-max by alternating an action stage and an adversarial stage, and uses either Robbins-Monro with an envelope-theorem gradient estimator or Kiefer-Wolfowitz with SPSA. The same learned functions then act as continuation values in the one-step Bellman equation, defining the greedy non-anticipative policy whose simulated value supplies the other side of the bracket.

What would settle it

For the unconstrained trade-execution case, take the trained ADRL penalty network, draw an independent large sample of noise paths, and solve the inner maximization with a much more exhaustive solver; if the resulting dual value is not below the known closed-form optimal cost, the claimed lower-bound property is violated.

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Extended reading notes

Core claim

The central claim is that the dual side of information-relaxation duality is trainable end to end. Parameterize the penalty-generating functions by feedforward neural networks, alternate between a deterministic inner maximization over action sequences and a gradient step on the penalty parameters, and the dual value converges to the true value from below while a greedy policy built from the same penalty functions converges from above. Under boundedness, Lipschitz, and uniqueness assumptions, a pathwise envelope-theorem argument makes the gradient of the expected maximum computable as the gradient of the penalty alone at the optimal action, which supports an unbiased Robbins-Monro update. In the paper's numerical experiments the bracket narrows to a 2.82 percent gap after 5,000 iterations in the unconstrained execution problem and to average gaps of 1.85 and 2.86 percent for wider networks in the no-shorting problem, bracketing the true optimal cost between 607.47 and 621.08.

Load-bearing premise

The bracketing property holds only if the learned penalty is truly feasible in expectation and the inner maximization is solved exactly; the implementation approximates the expectation with a 2,000-sample average and the inner problem with SLSQP, so the reported bounds are not certified.

Editorial extensions

If this is right

For any problem satisfying the paper's assumptions, ADRL returns two numbers that sandwich the true optimal value, so the dual gap is an in-situ certificate of policy quality rather than an afterthought.
The learned generating functions double as continuation-value approximations, so the same network that produces the dual bound also produces the greedy policy, eliminating a separate policy-training phase.
ADRL supplies the "good" value approximation required by the Balseiro-Brown construction, making tight dual gaps a systematic objective rather than a matter of luck.
The dual bounds give a principled early-stopping rule for deep empirical risk minimization: stop when the in-sample training loss falls inside the ADRL interval, because strong duality then implies near-optimality.
The unbiased gradient estimator from the envelope theorem lets the adversarial update use one inner optimization per sample path, while the SPSA variant removes the differentiability requirement at the cost of two solves per iteration.

Reading between the lines

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

An extension the paper does not pursue: if the dual-primal gap is monotone decreasing on a validation set, the gap would double as a convergence diagnostic for the entire training run.
The paper leaves implicit that the same bracket could certify any deep-Monte-Carlo policy; testing ADRL on optimal stopping or hedging would show how far the construction generalizes.
One natural model-free variant, not discussed by the authors, would estimate the conditional expectation inside the penalty from a learned transition model instead of the known dynamics, extending ADRL beyond simulation-based settings.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Take a look at this paper if you care about turning the information-relaxation dual into a practical learning algorithm. The authors parameterize the penalty with a neural net, train it via a min-max game, and get both a dual bound and a greedy policy from the same value approximation. That specific combination is genuinely new, and the one experiment (a 10-asset execution problem with no-shorting constraints) is sensible. The comparison to DERM and the overfitting behavior is a nice practical touch.

The problems start with the machinery. Proposition 3.2, the envelope-theorem gradient estimator that the Robbins-Monro step depends on, is stated and then the proof is deferred to the first author's website. For a paper that asks the reader to trust the method, that's a real gap. More serious is the inner maximization in Eq. (12). The authors hand it to SLSQP, which is a local method. The penalty network is nonconvex, so SLSQP generally returns a local solution, and if that solution is below the true maximum, the computed dual value is biased toward the primal side. That makes the advertised gap (2.82% after 5,000 iterations) potentially too small. The paper gives no optimality certificate, no multi-start, nothing to show the inner problem was solved globally. This is the load-bearing issue: without a global inner max, the 'confidence intervals' are not valid bounds.

The Monte Carlo approximation of the conditional expectation in Eq. (5) is less of a concern—with independent inner samples, the randomized penalty stays feasible in expectation, so the dual bound remains valid. The circularity worry is also modest: the dual gap is indeed the training objective, but the greedy policy is evaluated out-of-sample, so the reported primal values are not a tautology.

There is also some sign confusion in the text: the theory is set up as reward maximization, the experiment is cost minimization, and the statement that 'dual values should provide lower bounds' only makes sense in the minimization view. That needs cleaning up. No code or data is released, which makes it hard to verify the plots.

Overall, this is a plausible algorithmic construction with one good experiment and several technical loose ends. The idea deserves serious refereeing, but the current version needs a full proof of the gradient estimator, a careful treatment of the inner optimization, and a reproducibility package before the bounds can be taken at face value.

I would send it to peer review if it crossed my desk. For a reading group, it's a good starting point for a discussion on when information-relaxation dual bounds are actually computable.

Referee Report

4 major / 5 minor

Summary. The paper proposes ADRL, an adversarial deep reinforcement learning algorithm for finite-horizon stochastic control problems. The method is based on the information relaxation dual formulation: the dual problem is recast as a min-max game in which the adversary learns a neural-network generating function for the penalty and the agent solves an inner deterministic optimization problem for each simulated path. The learned generating functions are also used to construct a greedy policy via a one-step Bellman equation, yielding a primal upper bound; the dual value gives a lower bound, and the gap between the two is reported as a performance guarantee. Numerical experiments on an optimal trade execution problem (one unconstrained case with a known analytical optimum and one constrained case) are used to claim that ADRL produces tight confidence intervals and improves on deep empirical risk minimization by avoiding overfitting.

Significance. If the theoretical and numerical gaps are closed, the paper would provide a practically useful simulation-based workflow that outputs both a policy and a bracketing interval for the value of a high-dimensional stochastic control problem. The connection between information relaxation duality and adversarial training is a natural and potentially valuable extension of the Balseiro-Brown framework, and the paper explicitly aims to address the overfitting problem identified by Reppen and Soner. The paper does not, however, currently deliver a proof of its key gradient estimator, a convergence guarantee for the min-max training, or certified numerical bounds; the strength of the contribution therefore rests on the correctness of the unproved Proposition 3.2 and on the exactness of the inner optimization in the implementation. The paper is honest about some of its limitations (it acknowledges the computational cost of ADRL and the lack of a convergence analysis), but the main claims require substantially more support before they can be accepted.

major comments (4)

[Section 3.1.1, Proposition 3.2 and Eq. (18)] Proposition 3.2 is the theoretical foundation of the Robbins-Monro gradient estimator, but no proof is given in the manuscript; the text states 'Due to the page limit, we omit the complete proof' and refers the reader to the first author's website. The interchange of expectation and differentiation in Eq. (15), the envelope theorem for the constrained inner program, and the representation in Eq. (17) require conditions that are not stated or verified. In particular, the experiments use ReLU activations, which are not differentiable, while Assumptions 2.1 and 3.1 do not ensure differentiability of the value function or of the argmax with respect to φ. The proof must be included in the manuscript or in a supplement, and the regularity assumptions must cover the actual network class used in the experiments.
[Section 4, implementation of Eq. (12)] The bracketing claim that the dual value is a lower bound on V* requires that, for each outer sample path, the inner maximization in Eq. (12) be solved exactly. The paper uses SLSQP, a local optimizer, on an objective that is nonconcave in the action because the learned penalty zφ is a neural-network function of the state, and the state depends on the action through the dynamics in Eq. (1). A local solution below the true maximum biases the estimated dual value toward the primal side, so the reported gaps of 2.82%, 1.85%, and 2.86% are not certified upper bounds on the optimality gap. The agreement with the Bertsimas-Lo optimum in the unconstrained case is a useful sanity check, but it is shown only visually and only for one configuration; the constrained case has no external benchmark. The paper should either provide a global-optimization guarantee, report multi-start and optimality-condition checks, or present the results as heuristic approximations rather than certified intervals.
[Section 3.1, Algorithm 1] No convergence theorem is provided for the stochastic approximation scheme that solves the min-max problem in Eq. (10). The outer problem is nonconvex in φ, the gradient estimator in Eq. (19) relies on the unproved Proposition 3.2, and the inner maximization is solved only approximately. The conditions cited for the SPSA variant in Section 3.1.2 are standard textbook conditions, but no analogous convergence result is given for the RM/Adam update actually used in the experiments. Consequently, the central claim that ADRL produces tight confidence intervals is an empirical assertion supported by a single experiment. A convergence result for a simplified setting, or a clear statement that the empirical claims are heuristic, is needed.
[Section 4, numerical experiments] The numerical evidence consists of one problem instance (order execution with n=10, m=3, T=20). The paper reports no multiple seeds, no out-of-sample evaluation for the dual bounds, and no numerical table of the values in Figures 1 and 2; the reported gap is computed on the training trajectory. The abstract's claim that ADRL yields tight dual gaps 'even for high-dimensional control problems' is broader than what the evidence supports. Please add tables with means and standard errors over independent runs, out-of-sample dual evaluations, and at least one additional problem class or benchmark.

minor comments (5)

[Section 3.1, Algorithm 1] The text says 'In each iteration, a single sample trajectory of ξ is generated as a mini-batch,' but Algorithm 1 and the preceding batch construction describe minibatches K_b of l paths; please reconcile the two descriptions.
[Figures 1 and 2] The shaded bands are described as 95% confidence intervals, but it is not clear whether the shading applies to the blue dual line, the red primal line, or both; the captions should specify this and report the number of replications used to form the intervals.
[Section 4] The sentence 'The results for more complex network configurations (k=50 and k=100) exhibit trends similar to those shown in Figure 1. Therefore, we omit the corresponding figures for brevity' is confusing because Figures 2a and 2b are for the constrained problem, not for the unconstrained case with k=50 and k=100; please clarify what was computed and what was omitted.
[Section 3.1.2, after Eq. (20)] The sentence 'It significantly reducing computational cost' is a grammatical error, and the displayed regularity conditions for SPSA are incomplete; for example, the condition ∑ γ_b^2 c_b^{-2} < ∞ appears in the text but should be stated consistently with the other summability conditions.
[Appendix A.1 and Section 3.1.1] The use of ReLU activations conflicts with the differentiability requirements of the envelope theorem invoked in Proposition 3.2; the paper should either use smooth activations in the experiments or explain how nondifferentiability is handled in the gradient estimator.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the reported dual bounds are the very objective minimized in Eq. (10), so tight dual gaps partly reflect training convergence; independent anchoring comes only from the unconstrained analytical benchmark and out-of-sample policy evaluation.

fitted input called prediction [Section 3.1, Eq. (10); Section 4, Figure 1 discussion]
"Invoking (7), we train optimal φt by solving the following minimization problem φ* = arg min φ E[max aaa ... ] (10) ... At 5,000 iterations, the gap reaches 2.82%. Since the true value lies between the dual and primal values (by strong duality), the relative error of the greedy policy at 5,000 iterations must be less than 2.82%."

The dual value plotted as one side of the reported confidence interval is exactly the expectation E[max_a Y(φ,a,ξ)] that Eq. (10) minimizes during training. Thus the 'tight dual gap' at iteration 5,000 is, on the dual side, the residual of the training objective: the optimizer was explicitly run to make this number small, so reporting it as evidence of near-optimality is partly reporting the convergence of the loss. The interval is not an independent test of optimality; it is the algorithm's own objective paired with a separately evaluated greedy policy. The unconstrained Bertsimas-Lo analytical optimum is the only external anchor; for the constrained case no independent value is available.

full rationale

The derivation chain is not fundamentally circular: the strong duality invoked from Rogers (2007) and Brown et al. (2010) is an external theorem; the greedy policy in Eq. (14) is evaluated out-of-sample; and the unconstrained experiment is checked against the Bertsimas-Lo analytical optimum, which is independent of the fitted network. The same-author citation (Chen et al. 2024) is used only as motivating prior work, not as the load-bearing justification for ADRL. The main circular content is the dual side of the reported confidence interval: the dual value plotted in Figures 1-2 and used in the gap is the very objective minimized in Eq. (10), so 'tight dual gaps' is in part a restatement of successful loss minimization rather than an independent certificate. Feasibility of the penalty and exact inner maximization are not guaranteed in the implementation (2,000-sample Monte Carlo and SLSQP), which breaks the bracketing property; these are correctness risks, not circularity. The omitted proof of Proposition 3.2 (deferred to the first author's website) is likewise a verifiability gap. Overall, partial circularity of the reported gap as evidence, with some independent anchoring, justifies a score of 4 rather than a higher score.

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

The algorithm introduces no new physical or mathematical entity; all ingredients, information relaxation, penalties, and DNNs, exist in prior work. The central claim rests on the background duality theorem plus a chain of unverified regularity, approximation, and numerical-solver assumptions, as well as several hand-set hyperparameters.

free parameters (5)

Neural network size (hidden layers, neurons k) = 2 hidden layers, k=10,50,100
The approximation capacity of the dual penalty network is chosen by hand; the reported gap depends on it (for example 1.85% for k=50 versus 2.86% for k=100).
Monte Carlo sample size for penalty expectation = 2,000
Used to estimate the conditional expectation in Eq. (5); the finite-sample bias is not accounted for in the dual bound.
Quadratic feature map = R263: P, X, R, X X^T, R R^T, P R^T, X R^T
Hand-designed nonlinear features fed into the network; essential for the DNN approximation but not derived from the control problem.
Reporting iteration = 5,000 or 6,000 iterations
Gaps are quoted at these iterations with no pre-registered stopping rule, so the reported tightness could reflect convenient selection.
Learning rate and optimizer schedules = Adam, rates not reported
The stochastic approximation step sizes and perturbation sizes are not specified, though convergence of RM and KW methods depends on them.

assumptions (5)

domain assumption Assumption 2.1: rewards, terminal reward, and dynamics are bounded and Lipschitz.
Used throughout the theoretical development; no verification is provided for the price-impact model or trading cost functions.
domain assumption Assumption 3.1: compact convex action sets satisfying Slater, with a unique inner maximizer for almost every random draw.
The trade execution no-shorting set is a simplex, which is compact and convex, but uniqueness of the SLSQP solution is asserted rather than proved.
standard math Strong duality of information relaxation (Rogers 2007 Theorem 1; Brown et al. 2010 Theorem 2.3).
Eq. (7), the theoretical foundation of ADRL, is taken from prior literature; the paper relies on it without re-proving it.
ad hoc to paper Interchange of expectation and differentiation and validity of the envelope theorem for the constrained inner problem (Eqs. 15-17).
Proposition 3.2 is not proved in the text; the required regularity at nondifferentiable max points is not stated in full, and the inner problem is solved numerically with SLSQP.
ad hoc to paper The neural networks rho_t are expressive enough to approximate V*_t for the problem class.
The greedy policy in Eq. (14) is near-optimal only if the learned generating function is close to the true value function; no approximation or generalization bounds are provided.

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Cite this review

Pith. "Pith review of Adversarial Reinforcement Learning: A Duality-Based Approach To Solving Optimal Control Problems." pith.science (2026). https://pith.science/paper/PGR3FZTG

@misc{pith2026250600801,
  author       = {Pith},
  title        = {Pith review of: Adversarial Reinforcement Learning: A Duality-Based Approach To Solving Optimal Control Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PGR3FZTG}},
  note         = {Machine review of arXiv:2506.00801}
}

read the original abstract

We propose an adversarial deep reinforcement learning (ADRL) algorithm for high-dimensional stochastic control problems. Inspired by the information relaxation duality, ADRL reformulates the control problem as a min-max optimization between policies and adversarial penalties, enforcing non-anticipativity while preserving optimality. Numerical experiments demonstrate ADRL's superior performance to yield tight dual gaps. Our results highlight the potential of ADRL as a robust computational framework for high-dimensional stochastic control in simulation-based optimization contexts.

Figures

Figures reproduced from arXiv: 2506.00801 by the authors.

**Figure 2.** ADRL bounds (k = 50,100) showing learning curve and policy-based upper bound with 95% confidence intervals. The dotted line indicates the unconstrained theoretical optimal value. Finally, we apply a popular reinforcement learning (RL) algorithm from the literature—the deep empirical risk minimization (DERM) algorithm Han and E [2016]—to the problem with the no-shorting constraint to demonstrate the advantages of wor… view at source ↗

**Figure 3.** Training and evaluation loss of the ERM-based policy. The blue line shows in-sample training loss; The red [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 19 canonical work pages

[1]

Deep learning approximation for stochastic control problems

Jiequn Han and Weinan E. Deep learning approximation for stochastic control problems. arXiv preprint arXiv:1611.07422, 2016

arXiv 2016
[2]

Solving high-dimensional partial differential equations using deep learning

Jiequn Han, Arnulf Jentzen, and Weinan E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115 0 (34): 0 8505--8510, 2018

2018
[3]

Deep optimal stopping

Sebastian Becker, Patrick Cheridito, and Arnulf Jentzen. Deep optimal stopping. Journal of Machine Learning Research, 20 0 (1): 0 2712--2736, 2019

work page 2019
[4]

Deep hedging

Hans Buehler, Lukas Gonon, Josef Teichmann, and Ben Wood. Deep hedging. Quantitative Finance, 19 0 (8): 0 1271--1291, 2019

2019
[5]

Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis

C \^o me Hur \'e , Huy \^e n Pham, Achref Bachouch, and Nicolas Langren \'e . Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis. SIAM Journal on Numerical Analysis, 59 0 (1): 0 525--557, 2021

work page 2021
[6]

Deep empirical risk minimization in finance: Looking into the future

Anders Max Reppen and Halil Mete Soner. Deep empirical risk minimization in finance: Looking into the future. Mathematical Finance, 33 0 (1): 0 116--145, 2023

work page 2023
[7]

L. C. G. Rogers. Pathwise stochastic optimal control. SIAM Journal on Control and Optimization, 46 0 (3): 0 1116--1132, 2007

work page 2007
[8]

Brown, James E

David B. Brown, James E. Smith, and Peng Sun. Information relaxations and duality in stochastic dynamic programs. Operations Research, 58 0 (4-part-1): 0 785--801, 2010

work page 2010

Show all 26 references

[9]

Brown and James E

David B. Brown and James E. Smith. Information relaxations and duality in stochastic dynamic programs: A review and tutorial. Foundations and Trends in Optimization, 5 0 (3): 0 246--339, 2022

2022
[10]

Information relaxation and a duality-driven algorithm for stochastic dynamic programs

Nan Chen, Xiang Ma, Yanchu Liu, and Wei Yu. Information relaxation and a duality-driven algorithm for stochastic dynamic programs. Operations Research, 72 0 (6): 0 2302--2320, 2024

2024
[11]

Explaining and harnessing adversarial examples

Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. 2015. URL https://arxiv.org/abs/1412.6572

2015 arXiv
[12]

Huang, Nicolas Papernot, Ian J

Sandy H. Huang, Nicolas Papernot, Ian J. Goodfellow, Yan Duan, and Pieter Abbeel. Adversarial attacks on neural network policies. In Proceedings of the 5th International Conference on Learning Representations (ICLR), pages 1--9, 2017. URL https://arxiv.org/abs/1702.02284

2017 arXiv
[13]

Generative adversarial networks

Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. Communications of the ACM, 63 0 (11): 0 139--144, 2020

2020
[14]

Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, New York, 2006

2006
[15]

Bertsekas and John N

Dimitri P. Bertsekas and John N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, Belmont, Massachusetts, 1997

1997
[16]

Balseiro and David B

Santiago R. Balseiro and David B. Brown. Approximations to stochastic dynamic programs via information relaxation duality. Operations Research, 67 0 (2): 0 577--597, 2019

2019
[17]

Monte Carlo Methods in Financial Engineering, volume 53

Paul Glasserman. Monte Carlo Methods in Financial Engineering, volume 53. Springer, New York, 2004

2004
[18]

S ren Asmussen and Peter W. Glynn. Stochastic Simulation: Algorithms and Analysis, volume 57. Springer, New York, 2007. ISBN 978-0-387-30679-7

2007
[19]

Marie Chau and Michael C. Fu. An overview of stochastic approximation. In Handbook of Simulation Optimization, pages 149--178. 2015

2015
[20]

Envelope theorems for arbitrary choice sets

Paul Milgrom and Ilya Segal. Envelope theorems for arbitrary choice sets. Econometrica, 70 0 (2): 0 583--601, 2002

2002
[21]

James C. Spall. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on Automatic Control, 37 0 (3): 0 332--341, 1992

1992
[22]

Stochastic estimation of the maximum of a regression function

Jack Kiefer and Jacob Wolfowitz. Stochastic estimation of the maximum of a regression function. The Annals of Mathematical Statistics, pages 462--466, 1952

1952
[23]

James C. Spall. Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. John Wiley & Sons, 2005

2005
[24]

Dimitris Bertsimas and Andrew W. Lo. Optimal control of execution costs. Journal of Financial Markets, 1 0 (1): 0 1--50, 1998

1998
[25]

Optimal execution of portfolio transactions

Robert Almgren and Neil Chriss. Optimal execution of portfolio transactions. Journal of Risk, 3: 0 5--40, 2001

2001
[26]

Dynamic portfolio execution and information relaxations

Martin Haugh and Chun Wang. Dynamic portfolio execution and information relaxations. SIAM Journal on Financial Mathematics, 5 0 (1): 0 316--359, 2014

2014

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Deep learning approximation for stochastic control problems

Jiequn Han and Weinan E. Deep learning approximation for stochastic control problems. arXiv preprint arXiv:1611.07422, 2016

arXiv 2016

[2] [2]

Solving high-dimensional partial differential equations using deep learning

Jiequn Han, Arnulf Jentzen, and Weinan E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115 0 (34): 0 8505--8510, 2018

2018

[3] [3]

Deep optimal stopping

Sebastian Becker, Patrick Cheridito, and Arnulf Jentzen. Deep optimal stopping. Journal of Machine Learning Research, 20 0 (1): 0 2712--2736, 2019

work page 2019

[4] [4]

Deep hedging

Hans Buehler, Lukas Gonon, Josef Teichmann, and Ben Wood. Deep hedging. Quantitative Finance, 19 0 (8): 0 1271--1291, 2019

2019

[5] [5]

Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis

C \^o me Hur \'e , Huy \^e n Pham, Achref Bachouch, and Nicolas Langren \'e . Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis. SIAM Journal on Numerical Analysis, 59 0 (1): 0 525--557, 2021

work page 2021

[6] [6]

Deep empirical risk minimization in finance: Looking into the future

Anders Max Reppen and Halil Mete Soner. Deep empirical risk minimization in finance: Looking into the future. Mathematical Finance, 33 0 (1): 0 116--145, 2023

work page 2023

[7] [7]

L. C. G. Rogers. Pathwise stochastic optimal control. SIAM Journal on Control and Optimization, 46 0 (3): 0 1116--1132, 2007

work page 2007

[8] [8]

Brown, James E

David B. Brown, James E. Smith, and Peng Sun. Information relaxations and duality in stochastic dynamic programs. Operations Research, 58 0 (4-part-1): 0 785--801, 2010

work page 2010