Pith. sign in

REVIEW 3 major objections 4 minor 57 references

Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions

T0 review · 3 major / 4 minor · reviewed 2026-07-11 · grok-4.5

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

Pith's one-line read Non-convex PMC regularization of LSTD yields better sparse RL feature selection by solving a non-monotone inclusion with FRBS under new convergence guarantees.

desk verdict Solid dual contribution: PMC-LSTD for sparse offline RL plus usable FRBS guarantees for monotone-Lipschitz + hypomonotone inclusions; existence of a solution is assumed, not proved, but the rest of the math and the empirical gains hold up. read the letter →

arxiv 2607.04990 v2 pith:AGF5JNUS submitted 2026-07-06 cs.LG

Kyohei Suzuki , Konstantinos Slavakis This is my paper · ORCID
classification cs.LG
keywords reinforcementlearningfeatureselectionLSTDprojectedminimaxconcavenon-monotoneinclusionsforward-reflected-backwardsplittingweakMintyvariationalinequalitysparsemodeling
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

Standard sparse RL methods rely on the ℓ1 penalty, which underestimates large coefficients and leaves residual bias in policy evaluation. This paper replaces that penalty inside the classical LSTD fixed-point equation with the non-convex projected minimax-concave (PMC) penalty, which debiases only on a tunable subspace while preserving enough convexity to keep the problem tractable. The resulting operator is no longer monotone; it is the sum of a monotone Lipschitz map and a hypomonotone map. The authors therefore extend the forward-reflected-backward splitting (FRBS) method to this broader class and prove Lyapunov stability plus existence of a limit point under mild step-size conditions, and exact weak convergence under a weak Minty variational inequality. On three classic control benchmarks the new iterates select more accurate sparse features and produce higher-success-rate policies than LSTD, LARS-TD or BPDN, especially when hundreds of irrelevant features are present. A free hyper-parameter that controls the dimension of the debiasing subspace further improves performance when the Gram matrix is nearly singular.

What carries the argument

The reformulation of the PMC-regularized LSTD fixed point as the zero of a monotone Lipschitz operator plus a hypomonotone operator, together with the closed-form resolvent of the hypomonotone part that lets unmodified FRBS be applied.

What would settle it

On the 50-state chain walk with 1000–2000 irrelevant features, replace PMC by plain ℓ1 (or set the subspace dimension q too large or too small) and check whether the reported NMSE gap of several dB and the success-rate advantage on mountain-car/acrobot disappear.

Watch

Extended reading notes

Core claim

Augmenting LSTD with the weakly convex PMC penalty produces a non-monotone inclusion that FRBS can solve, and the resulting sparse weights give substantially lower policy-evaluation error and higher success rates than existing convex sparse methods once many noisy features appear.

Load-bearing premise

A solution to the regularized fixed-point problem is simply assumed to exist, and the stronger exact-convergence claim further needs a weak Minty condition whose validity for the concrete operator is left open.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes sparse batch policy evaluation by embedding the non-convex projected minimax concave (PMC) penalty into the classical LSTD fixed-point problem, then recasts the resulting problem as a non-monotone inclusion (sum of a monotone Lipschitz operator and a hypomonotone operator). It supplies a free subspace-dimension hyperparameter q for PMC, derives a closed-form resolvent for the hypomonotone part, and applies the unmodified FRBS iteration. On the theory side it proves, under summable decreasing step sizes, quasi-Fejér monotonicity, Lyapunov stability and existence of a limit of the FRBS sequence; under an additional weak Minty variational inequality it proves weak convergence to a zero. An approximate policy-iteration wrapper is given together with a standard suboptimality bound. Numerical experiments on 50-state chain walk, mountain car and acrobot report clear gains over LSTD, LARS-TD and BPDN, especially with many irrelevant features.

Significance. If the claims hold, the work makes two concrete contributions: (i) a practical non-convex regularizer for batch sparse RL that measurably reduces the estimation bias of ℓ1 methods on standard benchmarks with noisy features, and (ii) an extension of FRBS convergence theory from monotone inclusions to the broader class of hypomonotone-plus-monotone-Lipschitz inclusions, with complete proofs of the quasi-Fejér / Lyapunov / weak-MVI statements supplied in the appendices. The free-q generalization of PMC and the closed-form resolvent are useful technical devices. The empirical tables (30-trial averages) are reproducible in principle and show large effect sizes, which is valuable for offline RL feature selection. The main caveats are that exact convergence to a solution of the regularized LSTD problem is not established for the concrete operators, and that solution existence is postulated rather than proved.

major comments (3)
  1. [§III-B, Prop. 9; §IV, Problem 11(iii)] Proposition 9 and Problem 11(iii) simply assume that the solution set K of the PMC-regularized LSTD fixed-point (17a) (equivalently of the inclusion (22)) is nonempty. No existence argument or sufficient condition is given for the concrete operators T and μ∂∥·∥1 that arise from rank-deficient Φ and many noisy columns. Without such a condition the Lyapunov-stability and limit-point statements remain conditional; if K is empty for typical RL feature matrices the theoretical claims do not apply to the reported experiments.
  2. [§IV-A–B, Thms 15, 17, 19; abstract] Under the mild step-size regime of Assumption 13 / Theorem 8 the paper establishes only that the FRBS sequence is Lyapunov stable and convergent (Theorems 15 and 17, Prop. 9(ii)). It does not show that the limit lies in zer(A+B). Exact identification of cluster points as solutions requires the weak Minty variational inequality (Assumption 18(b), Theorem 19), whose validity for the concrete operator (22) is explicitly left as future work (end of §IV-B). Consequently the abstract claim that the FRBS iterates solve the non-convexly regularized LSTD problem is not fully supported by the mild-conditions theory that is actually used in the experiments.
  3. [§V (all three tasks)] All reported gains rest on hyperparameters (μ, τ, q, α, (η_k)) that are described only as “carefully tuned for each method to achieve best performance” (§V). No search ranges, selection criterion, or validation protocol are supplied. Because the central empirical claim is that the proposed method “substantially outperform[s] state-of-the-art feature-selection methods,” the absence of a documented tuning procedure undermines reproducibility and makes it impossible to judge whether the gains are robust or the result of asymmetric tuning effort.
minor comments (4)
  1. [§IV-B, Remark after Thm 19; §V] The two incompatible step-size regimes (summable η_k → 0 for Lyapunov/limit-point results versus η_k bounded away from zero for weak-MVI exact convergence) are noted only briefly. A short practical recommendation on which regime is used in the numerical section would help readers.
  2. [§V-D, Fig. 3] Figure 3 caption and surrounding text discuss the influence of q but do not report the corresponding values of τ that satisfy the eigenvalue constraint (20); adding those values would make the “moderate range of q” claim more transparent.
  3. [§III-B, Alg. 1] Notation for the soft-shrinkage operator alternates between Soft_τ and soft_τ; a single consistent symbol would improve readability.
  4. [§I-E] The conference precursor [36] is cited; a one-sentence statement of what is new relative to that short version (already present in the introduction) could be repeated in the contributions list for clarity.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the FRBS extensions and PMC reformulation are self-contained once standard operator facts and nonempty solution set are granted.

full rationale

The derivation chain is independent. Proposition 4 rewrites the PMC-regularized LSTD fixed-point (17a) as the inclusion (18)/(22) by Fermat’s rule plus the known gradient of the Moreau envelope (Fact 1) under the eigenvalue condition (20) that restores convexity of the inner objective; this is a standard equivalence, not a definitional loop. The FRBS iteration itself is the unmodified scheme of Malitsky–Tam [26]; the novel material (Theorems 14–17, 19) supplies quasi-Fejér monotonicity, Lyapunov stability, existence of a limit point, and weak-MVI exact convergence for the broader hypomonotone+monotone-Lipschitz class. These proofs rely only on the stated step-size restrictions, maximal (–ρ)-monotonicity of A, Lipschitz monotonicity of B, and the standing assumption that the solution set is nonempty (Problem 11(iii), Prop. 9)—the usual hypothesis in monotone-operator theory, not a circular reduction. PMC is taken from Yukawa et al. [24] but is immediately generalized to a free subspace dimension q (Remark 5) with an independent closed-form resolvent (Prop. 12). Empirical claims are ordinary numerical comparisons after hyper-parameter tuning; no fitted constant is re-labeled a “prediction.” The sole minor self-reference is the conference precursor [36], which is explicitly extended rather than load-bearing. Existence of a solution is assumed rather than proved, but that is a correctness gap, not circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The central claims rest on standard convex-analysis facts, the existence of a fixed point, a free subspace dimension q that must be chosen by the user, and (for exact convergence) an unverified weak-MVI assumption. No new physical entities are postulated.

free parameters (4)
  • µ (regularization weight)
    Chosen by hand for each method 'to achieve best performance'; controls sparsity-bias trade-off.
  • τ (PMC index)
    Must satisfy µ/τ ≤ λ_q; larger τ collapses PMC toward L1. Tuned jointly with µ and q.
  • q (subspace dimension for PMC)
    Free integer in 1…rank(Φ⊤Φ); moderate values [30,200] work best on the chain-walk task. Directly controls the support of the debiasing effect.
  • α and (η_k) step-size sequence
    Must obey the explicit upper bounds involving the Lipschitz constant of Ω and ho; practical choice η_k = c/(k+2)^ζ with ζ>1.
assumptions (4)
  • domain assumption Solution set of the regularized fixed-point problem (17a) is nonempty
    Invoked without proof in Prop. 9 and Problem 11(iii); all convergence statements are conditional on it.
  • ad hoc to paper Weak Minty variational inequality holds for A+B (Assumption 18(b))
    Required for exact weak convergence (Thm 19); authors explicitly leave verification for the RL operator (22) as future work.
  • standard math µ au^{-1} ≤ λ_q guarantees convexity of the inner u-problem
    Direct consequence of Fact 2 / Prop. 4; used to obtain the monotone+hypomonotone splitting.
  • standard math A(S) is bounded for every bounded S (Assumption 16)
    Used to upgrade quasi-Fejér monotonicity to actual convergence of the whole sequence (Prop. 17); holds for the soft-thresholding resolvent.
invented entities (1)
  • PMC penalty with free subspace dimension q
    purpose: Generalizes the original PMC (fixed to null⊥ Φ) so that the debiasing subspace can be chosen smaller than the full range of Φ⊤ when the smallest positive eigenvalue is tiny.
    Defined in Prop. 4 / Remark 5; independent evidence is only the empirical NMSE-vs-q curve in Fig. 3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions." pith.science (2026). https://pith.science/paper/AGF5JNUS

@misc{pith2026260704990,
  author       = {Pith},
  title        = {Pith review of: Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGF5JNUS}},
  note         = {Machine review of arXiv:2607.04990}
}
read the original abstract

This work delivers two key contributions: one to efficient feature selection in reinforcement learning (RL), the other to the theory of non-monotone inclusions. On the RL side, the estimation bias inherent in conventional regularization schemes is addressed by augmenting classical least-squares temporal-difference (LSTD) policy evaluation with the sparsity-inducing, non-convex projected minimax concave (PMC) penalty. Because the PMC penalty is weakly convex, the resulting fixed-point problem is no longer monotone; instead, it falls under a broader class of non-monotone inclusions involving the sum of a monotone Lipschitz operator and a hypomonotone operator. On the theory side, novel convergence conditions are developed for the forward-reflected-backward splitting (FRBS) method applied to this broader class of non-monotone inclusion problems. Under mild conditions, Lyapunov stability and the existence of a limit point of the sequence of FRBS iterates are established; alternatively, under the weak Minty variational inequality assumption, exact convergence is guaranteed. Numerical tests on benchmark datasets show that the proposed FRBS iterates, applied to the non-convexly regularized LSTD problem, substantially outperform state-of-the-art feature-selection methods, especially when many noisy features are present.

Figures

Figures reproduced from arXiv: 2607.04990 by the authors.

Figure 1
Figure 1. 50-state chain walk: NMSE vs. number of irrelevant features for 2000 samples. 1000 2000 3000 4000 Number of samples −20 −10 NMSE [dB] LSTD LARS-TD BPDN proposed [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. 50-state chain walk: NMSE vs. number of samples for 1000 irrelevant features. A. 50-state chain walk In the 50-state chain walk [13], the MDP consists of 50 states, two actions, “left” and “right”, and the success probability of either action is 0.9. When an action fails, the state changes to the opposite direction. A reward is 1 (one￾step loss is −1) only at states 10 and 41 and zero everywhere else. The discount f… view at source ↗
Figure 3
Figure 3. 50-state chain walk: NMSE vs. q for 500 irrelevant features, 2000 samples, and µ = 0.5. trials. The NMSE attains an almost minimum (about −18 dB) for a moderate range [30, 200] of q, and degrades outside of this range. Since q controls the dimension of the subspace M onto which the debiasing effect of the PMC penalty is confined, the influence of estimation bias is large when q is too small. When q is too large, on … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 4 linked inside Pith

  1. [1]

    R. S. Sutton, A. G. Barto,et al.,Reinforcement Learning: An Introduc- tion, 1st ed. Cambridge, MA: MIT press, 1998

    1998

  2. [2]

    Bertsekas,Reinforcement Learning and Optimal Control

    D. Bertsekas,Reinforcement Learning and Optimal Control. Belmant, MA, USA: Athena Sci., 2019

    2019

  3. [3]

    Deep reinforcement learning: A brief survey,

    K. Arulkumaran, M. P. Deisenroth, M. Brundage, and A. A. Bharath, “Deep reinforcement learning: A brief survey,”IEEE Signal Pro- cess. Mag., vol. 34, no. 6, pp. 26–38, 2017

    2017

  4. [4]

    Offline reinforcement learning: Tutorial, review, and perspectives on open problems,

    S. Levine, A. Kumar, G. Tucker, and J. Fu, “Offline reinforcement learning: Tutorial, review, and perspectives on open problems,”arXiv preprint arXiv:2005.01643, 2020

    arXiv 2005

  5. [5]

    Regularization and feature selection in least- squares temporal difference learning,

    J. Z. Kolter and A. Y . Ng, “Regularization and feature selection in least- squares temporal difference learning,” inProc. Int. Conf. Mach. Learn., 2009, pp. 521–528. 13

    2009

  6. [6]

    ℓ 1-penalized projected Bellman residual,

    M. Geist and B. Scherrer, “ℓ 1-penalized projected Bellman residual,” in Proc. Eur. Worksh. Reinforce. Learn., 2011, pp. 1–12

    2011

  7. [7]

    Sparse reinforcement learning via convex optimization,

    Z. Qin, W. Li, and F. Janoos, “Sparse reinforcement learning via convex optimization,” inProc. Int. Conf. Mach. Learn., vol. 32, no. 2, Beijing, China, 2014, pp. 424–432

    2014

  8. [8]

    Feature selection and feature learning for high-dimensional batch reinforcement learning: A survey,

    D.-R. Liu, H.-L. Li, and D. Wang, “Feature selection and feature learning for high-dimensional batch reinforcement learning: A survey,” Int. J. Autom. Comput., vol. 12, no. 3, pp. 229–242, 2015

    2015

Show all 57 references
  1. [9]

    Sparse Q-learning with mirror descent,

    S. Mahadevan and B. Liu, “Sparse Q-learning with mirror descent,” in Proc. Uncertainty Artif. Intell. Conf., 2012

    2012

  2. [10]

    Sparse proximal reinforcement learning via nested optimization,

    T. Song, D. Li, Q. Jin, and K. Hirasawa, “Sparse proximal reinforcement learning via nested optimization,”IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 50, no. 11, pp. 4020–4032, 2018

    2018

  3. [11]

    Online sparse temporal difference learning based on nested optimization and regularized dual averaging,

    T. Song, D. Li, and X. Xu, “Online sparse temporal difference learning based on nested optimization and regularized dual averaging,”IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 52, no. 4, pp. 2042–2052, 2021

  4. [12]

    L1 regularized linear temporal difference learning,

    C. Painter-Wakefield, R. Parr, and N. Durham, “L1 regularized linear temporal difference learning,”Technical report: Department of Com- puter Science, Duke University, Durham, NC, TR-2012–01, 2012

    2012

  5. [13]

    Least-squares policy iteration,

    M. G. Lagoudakis and R. Parr, “Least-squares policy iteration,” J. Mach. Learn. Res., vol. 4, no. Dec, pp. 1107–1149, 2003

    2003

  6. [14]

    Regression shrinkage and selection via the lasso,

    R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. Roy. Stat. Soc. B, vol. 58, no. 1, pp. 267–288, 1996

    1996

  7. [15]

    Hastie, R

    T. Hastie, R. Tibshirani, and M. Wainwright,Statistical Learning with Sparsity: the Lasso and Generalizations. CRC Press, 2015

    2015

  8. [16]

    M. J. Wainwright,High-dimensional statistics: A non-asymptotic view- point. Cambridge university press, 2019, vol. 48

    2019

  9. [17]

    Variable selection via nonconcave penalized likelihood and its oracle properties,

    J. Fan and R. Li, “Variable selection via nonconcave penalized likelihood and its oracle properties,”J. Amer. Stat. Assoc., vol. 96, no. 456, pp. 1348–1360, 2001

    2001

  10. [18]

    Sparse recovery via differential inclusions,

    S. Osher, F. Ruan, J. Xiong, Y . Yao, and W. Yin, “Sparse recovery via differential inclusions,”Appl. Comput. Harmon. Anal., vol. 41, no. 2, pp. 436–469, 2016

    2016

  11. [19]

    Nearly unbiased variable selection under minimax con- cave penalty,

    C. H. Zhang, “Nearly unbiased variable selection under minimax con- cave penalty,”Ann. Statist., vol. 38, no. 2, pp. 894–942, Apr. 2010

    2010

  12. [20]

    Exact reconstruction of sparse signals via nonconvex minimization,

    R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,”IEEE Signal Process. Lett., vol. 14, no. 10, pp. 707– 710, Oct. 2007

    2007

  13. [21]

    Sparse regularization via convex analysis,

    I. Selesnick, “Sparse regularization via convex analysis,”IEEE Trans. Signal Process., vol. 65, no. 17, pp. 4481–4494, 2017

    2017

  14. [22]

    On the convergence of the iterative shrinkage/thresholding algorithm with a weakly convex penalty,

    I. Bayram, “On the convergence of the iterative shrinkage/thresholding algorithm with a weakly convex penalty,”IEEE Trans. Signal Process., vol. 64, pp. 1597–1608, 2016

    2016

  15. [23]

    Linearly involved generalized Moreau enhanced models and their proximal splitting algorithm under overall convexity condition,

    J. Abe, M. Yamagishi, and I. Yamada, “Linearly involved generalized Moreau enhanced models and their proximal splitting algorithm under overall convexity condition,”Inverse Problems, vol. 36, no. 3, pp. 1–36, Feb. 2020

    2020

  16. [24]

    Linearly-involved Moreau-enhanced-over-subspace model: Debiased sparse modeling and stable outlier-robust regression,

    M. Yukawa, H. Kaneko, K. Suzuki, and I. Yamada, “Linearly-involved Moreau-enhanced-over-subspace model: Debiased sparse modeling and stable outlier-robust regression,”IEEE Trans. Signal Process., vol. 71, pp. 1232–1247, 2023

    2023

  17. [25]

    Linear least-squares algorithms for temporal difference learning,

    S. J. Bradtke and A. G. Barto, “Linear least-squares algorithms for temporal difference learning,”Machine learning, vol. 22, no. 1, pp. 33– 57, 1996

    1996

  18. [26]

    A forward-backward splitting method for monotone inclusions without cocoercivity,

    Y . Malitsky and M. K. Tam, “A forward-backward splitting method for monotone inclusions without cocoercivity,”SIAM J. Optim., vol. 30, no. 2, pp. 1451–1472, 2020

    2020

  19. [27]

    Efficient methods for structured nonconvex-nonconcave min-max optimization,

    J. Diakonikolas, C. Daskalakis, and M. I. Jordan, “Efficient methods for structured nonconvex-nonconcave min-max optimization,” inInter- national Conference on Artificial Intelligence and Statistics. PMLR, 2021, pp. 2746–2754

    2021

  20. [28]

    Extending the reach of first- order algorithms for nonconvex min-max problems with cohypomono- tonicity,

    A. Alacaoglu, D. Kim, and S. J. Wright, “Extending the reach of first- order algorithms for nonconvex min-max problems with cohypomono- tonicity,”arXiv preprint arXiv:2402.05071, 2024

    2024 arXiv

  21. [29]

    Beyond the golden ratio for variational inequality algorithms,

    A. Alacaoglu, A. B ¨ohm, and Y . Malitsky, “Beyond the golden ratio for variational inequality algorithms,”J. Mach. Learn. Res., vol. 24, no. 172, pp. 1–33, 2023

    2023

  22. [30]

    Solving nonconvex-nonconcave min-max problems exhibit- ing weak minty solutions,

    A. B ¨ohm, “Solving nonconvex-nonconcave min-max problems exhibit- ing weak minty solutions,”arXiv preprint arXiv:2201.12247, 2022

    2022 arXiv

  23. [31]

    On the convergence properties of non- euclidean extragradient methods for variational inequalities with gen- eralized monotone operators,

    C. D. Dang and G. Lan, “On the convergence properties of non- euclidean extragradient methods for variational inequalities with gen- eralized monotone operators,”Computational Optimization and appli- cations, vol. 60, no. 2, pp. 277–310, 2015

    2015

  24. [32]

    Douglas-rachford algorithm for nonmonotone multioperator inclusion problems,

    J. H. Alcantara and A. Takeda, “Douglas-rachford algorithm for nonmonotone multioperator inclusion problems,”arXiv preprint arXiv:2501.02752, 2025

    2025

  25. [33]

    Convergence of the preconditioned proximal point method and douglas–rachford splitting in the absence of monotonicity,

    B. Evens, P. Pas, P. Latafat, and P. Patrinos, “Convergence of the preconditioned proximal point method and douglas–rachford splitting in the absence of monotonicity,”Math. Prog., pp. 1–55, 2025

    2025

  26. [34]

    Variance-reduced forward-reflected-backward splitting methods for nonmonotone generalized equations,

    Q. Tran-Dinh, “Variance-reduced forward-reflected-backward splitting methods for nonmonotone generalized equations,” in Proc. Int. Conf. Mach. Learn., 2025, pp. 1–68

    2025

  27. [35]

    A forward-reflected-anchored-backward splitting algo- rithm with double inertial effects for solving non-monotone inclusion problems,

    N. Van Tran, “A forward-reflected-anchored-backward splitting algo- rithm with double inertial effects for solving non-monotone inclusion problems,”arXiv preprint arXiv:2503.08432, 2025

    2025 arXiv

  28. [36]

    Nonconvex regularization for feature selection in reinforcement learning,

    K. Suzuki and K. Slavakis, “Nonconvex regularization for feature selection in reinforcement learning,” inProc. Int. Conf. Acoust., Speech, Signal Process.IEEE, 2026, pp. 5886–5890

    2026

  29. [37]

    D. P. Bertsekaset al.,Dynamic programming and optimal control, volume II, 4th ed. Belmont, MA: Athena Scientific, 2012, vol. 2

    2012

  30. [38]

    Convergence results for some temporal difference methods based on least squares,

    H. Yu and D. P. Bertsekas, “Convergence results for some temporal difference methods based on least squares,”IEEE Trans. Autom. Control, vol. 54, no. 7, pp. 1515–1531, 2009

    2009

  31. [39]

    Monotone Lipschitz-gradient denoiser: Explainability of operator regularization approaches free from Lipschitz constant control,

    M. Yukawa and I. Yamada, “Monotone Lipschitz-gradient denoiser: Explainability of operator regularization approaches free from Lipschitz constant control,”IEEE Trans. Signal Process., pp. 1–16, 2025

    2025

  32. [40]

    D ´ecomposition orthogonale d’un espace Hilbertien selon deux c ˆones mutuellement polaires,

    J. J. Moreau, “D ´ecomposition orthogonale d’un espace Hilbertien selon deux c ˆones mutuellement polaires,”Comptes rendus hebdomadaires des s´eances de l’Acad´emie des sciences, vol. 255, pp. 238–240, 1962

    1962

  33. [41]

    Fonctions convexes duales et points proximaux dans un espace Hilbertien,

    ——, “Fonctions convexes duales et points proximaux dans un espace Hilbertien,”C. R. Acad. Sci. Paris Ser. A Math., vol. 255, pp. 2897–2899, 1962

    1962

  34. [42]

    H. H. Bauschke and P. L. Combettes,Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed. New York: Springer, 2017

    2017

  35. [43]

    Generalized monotone operators and their averaged resolvents,

    H. H. Bauschke, W. M. Moursi, and X. Wang, “Generalized monotone operators and their averaged resolvents,”Math. Prog., vol. 189, pp. 55– 74, 2021

    2021

  36. [44]

    Learning near-optimal policies with Bellman-residual minimization based fitted policy iteration and a single sample path,

    A. Antos, C. Szepesv ´ari, and R. Munos, “Learning near-optimal policies with Bellman-residual minimization based fitted policy iteration and a single sample path,”Machine Learning, vol. 71, no. 1, pp. 89–129, 2008

    2008

  37. [45]

    Mercier,Lectures on Topics in Finite Element Solution of Elliptic Problems

    B. Mercier,Lectures on Topics in Finite Element Solution of Elliptic Problems. Bombay: Tata Inst. Fundam. Res., 1979

    1979

  38. [46]

    Solving monotone inclusions via compositions of nonexpansive averaged operators,

    P. L. Combettes, “Solving monotone inclusions via compositions of nonexpansive averaged operators,”Optimization, vol. 53, no. 5-6, pp. 475–504, 2004

    2004

  39. [47]

    Splitting algorithms for the sum of two nonlinear operators,

    P.-L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,”SIAM J. Numer. Anal., vol. 16, no. 6, pp. 964– 979, 1979

    1979

  40. [48]

    On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone oper- ators,

    J. Eckstein and D. P. Bertsekas, “On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone oper- ators,”Math. Prog., vol. 55, no. 1-3, pp. 293–318, 1992

    1992

  41. [49]

    J. P. LaSalle,The Stability and Control of Discrete Processes. Springer Science & Business Media, 2012, vol. 62

    2012

  42. [50]

    Rockafellar and R

    R. Rockafellar and R. J.-B. Wets,Variational Analysis. Heidelberg, Berlin, New York: Springer Verlag, 1998

    1998

  43. [51]

    Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problems,

    T. Pethick, P. Latafat, P. Patrinos, O. Fercoq, and V . Cevher, “Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problems,” inInternational Conference on Learning Represen- tations (ICLR), 2022

    2022

  44. [52]

    A method for finding the zeros of polynomi- als using a companion matrix,

    X.-M. Niu and T. Sakurai, “A method for finding the zeros of polynomi- als using a companion matrix,”Japan journal of industrial and applied mathematics, vol. 20, no. 2, p. 239, 2003

    2003

  45. [53]

    Nonparametric Bellman mappings for reinforcement learning: Application to robust adaptive filtering,

    Y . Akiyama, M. Vu, and K. Slavakis, “Nonparametric Bellman mappings for reinforcement learning: Application to robust adaptive filtering,” IEEE Trans. Signal Process., vol. 72, pp. 5644–5658, 2024

    2024

  46. [54]

    Finite- sample analysis of Lasso-TD,

    M. Ghavamzadeh, A. Lazaric, R. Munos, and M. Hoffman, “Finite- sample analysis of Lasso-TD,” inInternational Conference on Machine Learning, 2011

    2011

  47. [55]

    F. H. Clarke,Optimization and Nonsmooth Analysis. New York: John Wiley & Sons Inc., 1983

    1983

  48. [56]

    B. S. Mordukhovich,Variational Analysis and Applications. Germany: Springer, 2018

    2018

  49. [57]

    Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions

    D. P. Bertsekas, A. Nedi ´c, and A. E. Ozdaglar,Convex Analysis and Optimization. Cambridge, MA: Athena Sci., 2003. 1 Supplemental Material to “Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions” In this supplemental material, we refer to the equation numbers...

    2003

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.