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REVIEW 3 major objections 6 minor 1 cited by

Correlated Chance Sampling for Monte Carlo Counterfactual Regret Minimization

T0 review · 3 major / 6 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read A one-line change to MCCFR’s chance sampler—persistent randomized Weyl streams at each chance node—cuts final exploitability by about 19–34% across Kuhn and Leduc poker with no new hyperparameters.

desk verdict A one-line persistent Weyl chance sampler that actually cuts tabular-poker exploitability 20–34% with honest local theory and clear regime boundaries. read the letter →

arxiv 2607.27035 v1 pith:3KJUNNO3 submitted 2026-07-29 cs.GT

Boning Li , Yu Chen , Longbo Huang This is my paper
classification cs.GT
keywords MCCFRchancesamplinglow-discrepancysequencesWeylsequenceexploitabilityimperfect-informationgamesExternaltabularpoker
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

Monte Carlo Counterfactual Regret Minimization repeatedly samples chance events such as card deals while strategies evolve, usually drawing independently on every visit. This paper argues that those repeated draws should instead be taken from a persistent low-discrepancy stream at each concrete chance node, so the node’s outcomes are forced to cover its distribution more evenly over time. The construction keeps every fixed-index draw correctly distributed and leaves the External Sampling estimator and regret updates unchanged, yet it guarantees much tighter local frequency error than independent sampling. In paired experiments the method lowers final exploitability by roughly one-fifth to one-third on Kuhn and several Leduc variants, with the gain holding at multi-million node-touch budgets and stacking with Linear CFR. A sympathetic reader cares because equilibrium computation in imperfect-information games is bottlenecked by sampling noise, and this is presented as a free, drop-in way to reduce that noise wherever chance nodes are revisited under private information.

What carries the argument

Correlated Chance Sampling via persistent randomized Weyl streams: each concrete chance node keeps a phase ϕ and visit count N, draws u = (ϕ + N·g) mod 1 with g the golden-ratio conjugate, and maps u through the node’s quantile function. That stream supplies the local discrepancy bound and the cross-visit temporal structure the authors credit for lower exploitability.

What would settle it

Run the same paired i.i.d.-versus-CCS protocol on a high-revisit private-information poker game at a large node-touch budget; if the paired-bootstrap interval on relative exploitability reduction includes zero or turns negative while local frequency error still tracks O(log N / N), the claimed link from local balance to lower exploitability fails.

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Extended reading notes

Core claim

CCS-MCCFR replaces independent chance draws with one persistent randomized Weyl stream per concrete chance node, mapped through that node’s chance law. Each fixed-index draw remains correctly distributed, the first N draws at a node have deterministic frequency error O(log(N+1)/N) rather than the usual i.i.d. scale, and along fixed strategy trajectories the regret estimates stay unbiased. Empirically this yields large, statistically detected exploitability reductions on tabular poker and a smaller but significant gain on Goofspiel-4, with no measurable time cost and no new hyperparameters.

Load-bearing premise

That evening out unweighted outcome frequencies at a chance node will, under fully adaptive regret updates, produce favorable cancellation with evolving downstream values often enough to lower exploitability.

Editorial extensions

If this is right

  • A one-line chance-sampler swap can deliver double-digit exploitability cuts on Kuhn and Leduc without changing the regret estimator or adding hyperparameters.
  • Local frequency error at a repeatedly visited chance node can be driven to O(log(N+1)/N) instead of the i.i.d. Θ(N^{-1/2}) scale.
  • CCS-MCCFR stacks with Linear CFR and Discounted CFR; LCFR plus CCS reaches the lowest measured Leduc cell in the composition grid.
  • Per-traversal phase reset restores the standard O(1/√T) External Sampling convergence guarantee when a global adaptive proof is not yet available.
  • Gains concentrate where chance nodes are revisited under private-information coupling; low-revisit or highly symmetric chance structures show little or no endpoint improvement.

Reading between the lines

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

  • Any other Monte Carlo tree procedure that revisits fixed chance locations—continual resolving, search-time sampling, or model-based rollouts—could try the same per-node persistent stream without touching its value estimator.
  • If δ_{c,t} (adaptive phase-selection distance from uniform) can be shown to shrink with visit count, the open global-convergence gap for non-reset CCS-MCCFR would close along the paper’s own bound.
  • The same placement idea extends naturally to higher-dimensional chance (multi-card deals) via randomized nets or lattices once per-node visit counts stay large enough for discrepancy to matter.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes CCS-MCCFR, a drop-in replacement for i.i.d. chance sampling in External Sampling MCCFR: each concrete chance node is assigned a persistent randomized Weyl stream whose phases are mapped through the node’s chance law. The authors prove fixed-index marginal correctness (Prop. 1), unbiasedness of External-Sampling regret estimates along fixed strategy trajectories (Prop. 2), deterministic O(log(N+1)/N) local frequency discrepancy for the first N draws at one concrete node (Thm. 1), a conditional total-variation bound on adaptive phase-selection bias (Prop. 3), and that a per-traversal phase-reset variant recovers the standard O(1/√T) External Sampling guarantee (Thm. 2). Empirically, paired experiments report 19–34% final exploitability reductions on Kuhn and a controlled Leduc deck family (all paired-bootstrap CIs above zero), a smaller significant gain on Goofspiel-4, persistence to 3M Leduc node touches, favorable composition with Linear CFR, and null or near-null endpoints on Liar’s Dice, reduced Flop, and four HUNL endgames, organized by revisit/symmetry/private-information diagnostics.

Significance. If the empirical pattern holds, this is a high-value practical contribution: a one-line, hyperparameter-free change to the chance sampler that yields double-digit exploitability reductions on standard tabular poker benchmarks with no measurable runtime cost, and that composes with Linear CFR to the best measured cell. The local theory is carefully scoped rather than overclaimed—classical discrepancy and External Sampling facts, no fitted free parameters, and an explicit conditional bias object δ_{c,t} for the adaptive case. Strengths include paired multi-seed designs with 10k-resample bootstrap CIs, an antithetic control, a persistence ablation, composition tests with update rules and a restricted control variate, and honest boundary diagnostics (HUNL, Liar’s Dice, low-revisit Flop). The main scientific limitation is that global rates for fully adaptive non-reset CCS-MCCFR remain open; that gap is disclosed and does not erase the local guarantees or the empirical result, but it does bound how far the paper can claim a complete theoretical replacement for i.i.d. chance sampling.

major comments (3)
  1. [§1, §4–5, §7; App. A–B] The central empirical claim rests on the temporal-cancellation hypothesis (§4–5): Thm. 1 controls only unweighted local frequencies, while exploitability reductions require favorable cumulative covariance between the persistent stream and adaptive downstream values g_t. Global convergence of non-reset CCS-MCCFR is left open (App. A–B); only fixed-trajectory unbiasedness, |B_c(t)| ≤ 2G δ_{c,t} (Prop. 3), and the reset variant (Thm. 2) are proved. This gap is disclosed, but the main text (esp. Intro/Conclusion) still frames CCS-MCCFR primarily as delivering “explicit local guarantees and large exploitability reductions” without a equally prominent main-body statement that the algorithm used in all positive experiments lacks a proven O(1/√T) guarantee. Please elevate a short, explicit caveat next to the main claims (abstract is already careful; §1 and §7 should match) so readers cannot mist
  2. [Prop. 3; §4–5; App. H; Table 4] Prop. 3 isolates adaptive bias in δ_{c,t}, but the manuscript never estimates or upper-bounds δ_{c,t} on the runs that produce the gains. Appendix H’s adaptive frequency diagnostic (visit-weighted max |p̂−p| after 30k iterations, seed 0) is a useful descriptive check on outcome frequencies, yet it is not a measurement of conditional phase law given g_t and reach, and ordinary count discrepancy need not control δ_{c,t} (as App. A notes). Because the load-bearing step from local balance to lower exploitability is exactly this adaptive coupling, please either (i) add a direct diagnostic of phase non-uniformity conditional on reach/downstream scalars on Kuhn/Leduc, or (ii) clearly state in §4–5 that no empirical bound on δ_{c,t} is provided and that favorable covariance is supported only indirectly (persistence ablation, Table 4; revisit table; null boundaries). Without one of these, the mec
  3. [§6; Table 5; Figure 4; App. A, E, G] Table 5 and Figure 4 present revisit statistics as the main organizer of when CCS helps, but the diagnostic is run for 1000 vanilla iterations and then compared to final gains at very different budgets and solvers (OpenSpiel vs. standalone C++ HUNL). The reduced Flop row is explicitly a structural low-exposure probe, not full Flop Hold’em (App. A), and HUNL exploitability is within retained abstract endgames under a protocol-mixed archive (App. G). These caveats belong in the main experimental narrative when the paper claims the “operating regime” and “transfer boundary,” not only in appendices. Please qualify Table 5/Fig. 4 in §6 as descriptive correlates on this benchmark set, and avoid language that reads as a validated transfer predictor for full-scale Hold’em.
minor comments (6)
  1. [§4; Tables 1–3] Notation: N (per-node consumed draws), T (iterations), and node-touch budgets are distinguished in §4 but easy to conflate in figure captions and Table 1–3 headers. A single notation paragraph or consistent subscripting (N_c vs. T) in all tables would help.
  2. [Figure 1–2; §6] Figure 1 bands are mean±1.96 SEM over seeds, while significance is decided by paired-bootstrap intervals on differences (Fig. 2). State this dual convention once in the Fig. 1 caption to avoid readers treating SEM ribbons as paired tests.
  3. [§6; Table 1; App. C] The antithetic baseline is a useful simple control; briefly define the exact pairing (u vs 1−u per node vs global) in §6 or App. C so the comparison is reproducible from the main text alone.
  4. [§2] Related work correctly separates VR-MCCFR (estimator baselines) from CCS (temporal chance allocation). A single sentence on whether public chance sampling [Johanson et al., 2012] already correlates chance across infosets—versus CCS’s across-visit correlation at one concrete node—would sharpen the novelty claim.
  5. [§4–5] Typos/style: “V ariance” and “T emporal” appear with stray spaces in headings (§5, §4); “W eyl” similarly. Normalize heading capitalization.
  6. [§6 HUNL paragraph; App. G] Appendix G’s mixed legacy/v2 HUNL protocol is handled carefully; still flag in the main HUNL paragraph that Subgame 1 mixes solver revisions so the four-endgame set is not a homogeneous scaling study.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: local discrepancy and marginals are classical QMC facts; exploitability is an external benchmark, not defined from the sampler.

full rationale

The load-bearing derivation chain does not collapse into its inputs. Theorem 1 is the standard Kronecker/Weyl star-discrepancy bound (Niederreiter) applied to per-node quantile bins; g = (√5−1)/2 is the classical badly approximable constant, not fitted to poker. Proposition 1 is measure-preservation of a random circle shift; Proposition 2 is fixed-trajectory independence of visit index from phase; Proposition 3 is a generic TV-to-expectation bound |B_c(t)| ≤ 2G δ_{c,t}; Theorem 2 recovers Lanctot et al.'s External Sampling guarantee by per-traversal re-randomization. None of these equalities is self-definitional or a fitted parameter renamed as prediction. Exploitability reductions are measured against OpenSpiel (and a standalone C++ HUNL solver) under paired seeds—external quantities, not quantities defined from the Weyl stream. Author self-citations (Li & Huang abstraction/parallel CFR, PokerSkill) appear only as complementary axes in Related Work and do not justify the sampler, the discrepancy constants, or the empirical endpoints. The open adaptive-convergence gap (Appendix A–B) is a correctness/scope limitation, not circularity. Score 0 is the honest finding.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central empirical claim rests on standard EFG/MCCFR definitions, classical Weyl discrepancy, and the modeling choice that persistent per-node phases are the right intervention. No numeric parameters are fitted to exploitability. The load-bearing extra structure is the temporal-cancellation hypothesis under adaptive weights, which is not proved globally.

assumptions (5)
  • domain assumption External Sampling MCCFR regret estimates are unbiased under independent chance draws with the standard O(1/√T) average-regret guarantee (Lanctot et al., 2009).
    Used as the baseline estimator algebra left unchanged by CCS and as the theorem recovered by the per-traversal reset variant (Thm. 2).
  • standard math Kronecker/Weyl sequences with golden-ratio step have star discrepancy N D_N^* = O(log(N+1)), yielding the deterministic frequency bound after quantile mapping (Niederreiter, 1992).
    Theorem 1 invokes this classical one-dimensional discrepancy fact; g = (√5−1)/2 is chosen for bounded partial quotients, not fitted.
  • domain assumption In an acyclic extensive-form tree, a concrete chance node is reached at most once per traversal, so the current draw’s phase index is not determined by the outcome at that same node on that traversal.
    Essential to Prop. 2 trajectory unbiasedness and to the reset argument in Thm. 2.
  • ad hoc to paper Under fully adaptive regret updates, improved local unweighted frequency balance often induces favorable cumulative covariance between chance outcomes and evolving downstream values (temporal-cancellation hypothesis).
    Stated as mechanism hypothesis in §4–5; not proved. Global adaptive convergence left open in Appendix A–B. Empirical moderators (revisit, symmetry, private-info coupling) are correlates of this hypothesis.
  • domain assumption Two-player zero-sum vanishing average external regret implies average-strategy convergence to Nash at O(T^{-1/2}) (Zinkevich et al., 2007).
    Background CFR convergence fact used to interpret exploitability reductions and the reset-variant guarantee.
invented entities (2)
  • CCS-MCCFR persistent per-concrete-chance-node randomized Weyl stream independent evidence
    purpose: Replace i.i.d. chance draws with a correlated low-discrepancy schedule while preserving fixed-index marginals and the External Sampling estimator.
    The algorithmic object the paper introduces. Classical Weyl math plus standard MCCFR; novelty is the persistent binding and adaptive analysis.
  • Conditional scalar phase-selection bias B_c(t) and expected conditional TV δ_{c,t}
    purpose: Isolate the only adaptive bias channel at one node and time when visit index can covary with the persistent phase.
    Definitional bookkeeping for Prop. 3; not a physical entity. Useful but does not by itself yield a global rate.

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Cite this review

Pith. "Pith review of Correlated Chance Sampling for Monte Carlo Counterfactual Regret Minimization." pith.science (2026). https://pith.science/paper/3KJUNNO3

@misc{pith2026260727035,
  author       = {Pith},
  title        = {Pith review of: Correlated Chance Sampling for Monte Carlo Counterfactual Regret Minimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3KJUNNO3}},
  note         = {Machine review of arXiv:2607.27035}
}
abstract

Monte Carlo Counterfactual Regret Minimization (MCCFR) repeatedly allocates chance outcomes while its strategy evolves, yet standard sampling draws those outcomes independently on every visit. We introduce Correlated Chance Sampling MCCFR (CCS-MCCFR), a drop-in replacement that assigns each concrete chance node a persistent randomized Weyl stream and maps its phases through the node's chance distribution. Each fixed-index draw has the correct marginal law, while the first $N$ draws consumed during $N$ visits to one concrete node achieve deterministic local frequency error $O(\!\log(N+1)/N)$, compared with the $O(N^{-1/2})$ expected scale of i.i.d. frequencies. We further establish unbiasedness along fixed strategy trajectories, isolate adaptive phase selection through a conditional scalar bound, and show that a per-traversal reset variant retains the standard $O(1/\sqrt{T})$ External Sampling guarantee. In paired experiments, CCS-MCCFR reduces final exploitability by 19.05\% to 34.01\% across Kuhn poker and four Leduc poker configurations, with every paired-bootstrap confidence interval above zero, and by a significant 4.27\% on Goofspiel-4. The gain survives to 3M Leduc node touches and combines with Linear CFR to reach the lowest measured exploitability. The sampler introduces no new hyperparameters and no measurable time overhead, so CCS-MCCFR turns a one-line change to the chance sampler into explicit local guarantees and large exploitability reductions across tabular poker.

Figures

Figures reproduced from arXiv: 2607.27035 by the authors.

Figure 1
Figure 1. Exploitability vs. node touches for four games. Each panel shows vanilla, antithetic, and [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Paired reduction (vanilla − CCS-MCCFR)/vanilla (%) versus node touches for the four main games, with the 95% paired-bootstrap CI band and a zero reference line (n=200 paired seeds). This uses the same statistic and convention as the HUNL reduction figures ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Exploitability vs. node touches for the Leduc family poker games (6/10/12 card decks). [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: CCS-MCCFR gain vs. median visits per observed concrete chance node. The high-revisit tabular poker configurations show the largest gains; the low-exposure re￾duced Flop and four HUNL endgames local￾ize regimes with no statistically detectable difference. HUNL triangles…
Figure 5
Figure 5. Figure 5: Pointwise paired reduction on the two Turn endgames, with 95% paired-bootstrap inter [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Pointwise paired reduction on the two River endgames ( [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: MCCFR regret update accelerators vs. node touch budget on Leduc (i.i.d. chance, 10 [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Leduc at 3 × 105 node touches (100 seeds). Within each tested update rule, CCS￾MCCFR (blue) lowers exploitability relative to i.i.d. chance (grey); across rules, LCFR has the lowest i.i.d. exploitability. LCFR plus CCS-MCCFR is the lowest measured cell. The differing r…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Agents That Certify Their Own Exploits: Confidence-Scheduled Restricted Responses for Safe Opponent Exploitation

    cs.GT 2026-07 accept novelty 6.5 of 10

    CS-RNR certifies each candidate exploit by full-tree best response before atomic deployment, so model error can cost gain but not the reference-relative safety budget.

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Works this paper leans on

94 extracted references · 8 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Science , volume=

    Morav. Science , volume=

  2. [2]

    Science , volume=

    Superhuman AI for heads-up no-limit poker: Libratus beats top professionals , author=. Science , volume=

  3. [3]

    Science , volume=

    Superhuman AI for multiplayer poker , author=. Science , volume=

  4. [4]

    Advances in neural information processing systems , volume=

    Regret minimization in games with incomplete information , author=. Advances in neural information processing systems , volume=

  5. [5]

    Advances in neural information processing systems , volume=

    Monte Carlo sampling for regret minimization in extensive games , author=. Advances in neural information processing systems , volume=

  6. [6]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Generalized sampling and variance in counterfactual regret minimization , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  7. [7]

    Advances in neural information processing systems , volume=

    Efficient Monte Carlo counterfactual regret minimization in games with many player actions , author=. Advances in neural information processing systems , volume=

  8. [8]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Variance reduction in monte carlo counterfactual regret minimization (VR-MCCFR) for extensive form games using baselines , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

Show all 94 references
  1. [9]

    International Conference on Machine Learning , pages=

    Low-variance and zero-variance baselines for extensive-form games , author=. International Conference on Machine Learning , pages=. 2020 , organization=

    2020

  2. [10]

    The Eleventh International Conference on Learning Representations , year=

    ESCHER: Eschewing Importance Sampling in Games by Computing a History Value Function to Estimate Regret , author=. The Eleventh International Conference on Learning Representations , year=

  3. [11]

    International conference on machine learning , pages=

    Deep counterfactual regret minimization , author=. International conference on machine learning , pages=. 2019 , organization=

    2019

  4. [12]

    International Conference on Machine Learning , pages=

    Stochastic regret minimization in extensive-form games , author=. International Conference on Machine Learning , pages=. 2020 , organization=

    2020

  5. [13]

    1992 , publisher=

    Random number generation and quasi-Monte Carlo methods , author=. 1992 , publisher=

    1992

  6. [14]

    SIAM Journal on Numerical Analysis , volume=

    Randomization of number theoretic methods for multiple integration , author=. SIAM Journal on Numerical Analysis , volume=. 1976 , publisher=

    1976

  7. [15]

    SIAM Journal on Numerical Analysis , volume=

    Monte Carlo variance of scrambled net quadrature , author=. SIAM Journal on Numerical Analysis , volume=. 1997 , publisher=

    1997

  8. [16]

    USSR Computational mathematics and mathematical physics , volume=

    Distribution of points in a cube and approximate evaluation of integrals , author=. USSR Computational mathematics and mathematical physics , volume=

  9. [17]

    Numerische Mathematik , volume=

    On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , author=. Numerische Mathematik , volume=. 1960 , publisher=

    1960

  10. [18]

    The Annals of Mathematical Statistics , volume=

    On the theory of systematic sampling, I , author=. The Annals of Mathematical Statistics , volume=. 1944 , publisher=

    1944

  11. [19]

    AAMAS , pages=

    Online Monte Carlo Counterfactual Regret Minimization for Search in Imperfect Information Games , author=. AAMAS , pages=

  12. [20]

    Proceedings of the Nineteenth International Conference on Machine Learning , pages=

    Learning from Scarce Experience , author=. Proceedings of the Nineteenth International Conference on Machine Learning , pages=

  13. [21]

    Advances in neural information processing systems , volume=

    Simple random search of static linear policies is competitive for reinforcement learning , author=. Advances in neural information processing systems , volume=

  14. [23]

    AAMAS , pages=

    Efficient Nash equilibrium approximation through Monte Carlo counterfactual regret minimization , author=. AAMAS , pages=

  15. [25]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Solving imperfect-information games via discounted regret minimization , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  16. [26]

    Proceedings of the 41st International Conference on Machine Learning , pages=

    RL-CFR: improving action abstraction for imperfect information extensive-form games with reinforcement learning , author=. Proceedings of the 41st International Conference on Machine Learning , pages=

  17. [27]

    The Thirteenth International Conference on Learning Representations , year=

    Efficient online pruning and abstraction for imperfect information extensive-form games , author=. The Thirteenth International Conference on Learning Representations , year=

  18. [31]

    The Twelfth International Conference on Learning Representations , year=

    Dynamic discounted counterfactual regret minimization , author=. The Twelfth International Conference on Learning Representations , year=

  19. [32]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Deep (Predictive) Discounted Counterfactual Regret Minimization , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  20. [33]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Faster game solving via hyperparameter schedules , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  21. [34]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Faster Game Solving via Asymmetry of Step Sizes , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  22. [35]

    The Fourteenth International Conference on Learning Representations , year=

    A Faster Parameter-Free Regret Matching Algorithm , author=. The Fourteenth International Conference on Learning Representations , year=

  23. [36]

    Forty-second International Conference on Machine Learning , year=

    Reducing variance of stochastic optimization for approximating Nash equilibria in normal-form games , author=. Forty-second International Conference on Machine Learning , year=

  24. [37]

    German Conference on Artificial Intelligence (K

    Efficiently Training Neural Networks for Imperfect Information Games by Sampling Information Sets , author=. German Conference on Artificial Intelligence (K. 2024 , organization=

    2024

  25. [38]

    Advances in Neural Information Processing Systems , volume=

    Accelerating Nash equilibrium convergence in Monte Carlo settings through counterfactual value based fictitious play , author=. Advances in Neural Information Processing Systems , volume=

  26. [41]

    Science Advances , volume=

    Student of Games: A unified learning algorithm for both perfect and imperfect information games , author=. Science Advances , volume=

  27. [42]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Faster game solving via predictive blackwell approachability: Connecting regret matching and mirror descent , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  28. [43]

    Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems , pages=

    Monte Carlo Continual Resolving for Online Strategy Computation in Imperfect Information Games , author=. Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems , pages=

  29. [44]

    Advances in neural information processing systems , volume=

    Combining deep reinforcement learning and search for imperfect-information games , author=. Advances in neural information processing systems , volume=

  30. [45]

    Advances in Neural Information Processing Systems 31 (NeurIPS 2018) , pages=

    Actor-Critic Policy Optimization in Partially Observable Multiagent Environments , author=. Advances in Neural Information Processing Systems 31 (NeurIPS 2018) , pages=

    2018

  31. [46]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Solving games with functional regret estimation , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  32. [47]

    Proceedings of the 35th International Conference on Machine Learning , series=

    Quasi-Monte Carlo Variational Inference , author=. Proceedings of the 35th International Conference on Machine Learning , series=

  33. [48]

    Journal of Machine Learning Research , volume=

    Monte Carlo Gradient Estimation in Machine Learning , author=. Journal of Machine Learning Research , volume=

  34. [49]

    Journal of Machine Learning Research , volume=

    Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels , author=. Journal of Machine Learning Research , volume=

  35. [50]

    Artificial Intelligence , volume=

    Rethinking formal models of partially observable multiagent decision making , author=. Artificial Intelligence , volume=

  36. [52]

    Quasi-monte carlo feature maps for shift-invariant kernels

    Haim Avron, Vikas Sindhwani, Jiyan Yang, and Michael W Mahoney. Quasi-monte carlo feature maps for shift-invariant kernels. Journal of Machine Learning Research, 17 0 (120): 0 1--38, 2016

    2016

  37. [53]

    Solving pasur using gpu-accelerated counterfactual regret minimization

    Sina Baghal. Solving pasur using gpu-accelerated counterfactual regret minimization. arXiv preprint arXiv:2508.06559, 2025

    2025 arXiv

  38. [54]

    u rnkranz, and Martin M \

    Timo Bertram, Johannes F \"u rnkranz, and Martin M \"u ller. Efficiently training neural networks for imperfect information games by sampling information sets. In German Conference on Artificial Intelligence (K \"u nstliche Intelligenz) , pages 17--29. Springer, 2024

    2024

  39. [55]

    Superhuman ai for heads-up no-limit poker: Libratus beats top professionals

    Noam Brown and Tuomas Sandholm. Superhuman ai for heads-up no-limit poker: Libratus beats top professionals. Science, 359 0 (6374): 0 418--424, 2018

    2018

  40. [56]

    Solving imperfect-information games via discounted regret minimization

    Noam Brown and Tuomas Sandholm. Solving imperfect-information games via discounted regret minimization. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 1829--1836, 2019 a

    2019

  41. [57]

    Superhuman ai for multiplayer poker

    Noam Brown and Tuomas Sandholm. Superhuman ai for multiplayer poker. Science, 365 0 (6456): 0 885--890, 2019 b

    2019

  42. [58]

    Deep counterfactual regret minimization

    Noam Brown, Adam Lerer, Sam Gross, and Tuomas Sandholm. Deep counterfactual regret minimization. In International conference on machine learning, pages 793--802. PMLR, 2019

    2019

  43. [59]

    Combining deep reinforcement learning and search for imperfect-information games

    Noam Brown, Anton Bakhtin, Adam Lerer, and Qucheng Gong. Combining deep reinforcement learning and search for imperfect-information games. Advances in neural information processing systems, 33: 0 17057--17069, 2020

    2020

  44. [60]

    Quasi-monte carlo variational inference

    Alexander Buchholz, Florian Wenzel, and Stephan Mandt. Quasi-monte carlo variational inference. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 667--676. PMLR, 2018

    2018

  45. [61]

    Efficient monte carlo counterfactual regret minimization in games with many player actions

    Neil Burch, Marc Lanctot, Duane Szafron, and Richard Gibson. Efficient monte carlo counterfactual regret minimization in games with many player actions. Advances in neural information processing systems, 25, 2012

    2012

  46. [62]

    Randomization of number theoretic methods for multiple integration

    Roy Cranley and Thomas NL Patterson. Randomization of number theoretic methods for multiple integration. SIAM Journal on Numerical Analysis, 13 0 (6): 0 904--914, 1976

    1976

  47. [63]

    Low-variance and zero-variance baselines for extensive-form games

    Trevor Davis, Martin Schmid, and Michael Bowling. Low-variance and zero-variance baselines for extensive-form games. In International Conference on Machine Learning, pages 2392--2401. PMLR, 2020

    2020

  48. [64]

    Stochastic regret minimization in extensive-form games

    Gabriele Farina, Christian Kroer, and Tuomas Sandholm. Stochastic regret minimization in extensive-form games. In International Conference on Machine Learning, pages 3018--3028. PMLR, 2020

    2020

  49. [65]

    Faster game solving via predictive blackwell approachability: Connecting regret matching and mirror descent

    Gabriele Farina, Christian Kroer, and Tuomas Sandholm. Faster game solving via predictive blackwell approachability: Connecting regret matching and mirror descent. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 5363--5371, 2021

    2021

  50. [66]

    Generalized sampling and variance in counterfactual regret minimization

    Richard Gibson, Marc Lanctot, Neil Burch, Duane Szafron, and Michael Bowling. Generalized sampling and variance in counterfactual regret minimization. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 26, pages 1355--1361, 2012

    2012

  51. [67]

    On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals

    John H Halton. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numerische Mathematik, 2 0 (1): 0 84--90, 1960

    1960

  52. [68]

    Efficient nash equilibrium approximation through monte carlo counterfactual regret minimization

    Michael Johanson, Nolan Bard, Marc Lanctot, Richard G Gibson, and Michael Bowling. Efficient nash equilibrium approximation through monte carlo counterfactual regret minimization. In AAMAS, pages 837--846, 2012

    2012

  53. [69]

    Rethinking formal models of partially observable multiagent decision making

    Vojt e ch Kova r \' k, Martin Schmid, Neil Burch, Michael Bowling, and Viliam Lis \'y . Rethinking formal models of partially observable multiagent decision making. Artificial Intelligence, 303: 0 103645, 2022

    2022

  54. [70]

    Monte carlo sampling for regret minimization in extensive games

    Marc Lanctot, Kevin Waugh, Martin Zinkevich, and Michael Bowling. Monte carlo sampling for regret minimization in extensive games. Advances in neural information processing systems, 22, 2009

    2009

  55. [71]

    Efficient online pruning and abstraction for imperfect information extensive-form games

    Boning Li and Longbo Huang. Efficient online pruning and abstraction for imperfect information extensive-form games. In The Thirteenth International Conference on Learning Representations, 2025

    2025

  56. [72]

    Effective, efficient, and general information abstraction for imperfect-information extensive-form games

    Boning Li and Longbo Huang. Effective, efficient, and general information abstraction for imperfect-information extensive-form games. arXiv preprint arXiv:2605.10900, 2026 a

    2026 arXiv

  57. [73]

    Real-time parallel counterfactual regret minimization

    Boning Li and Longbo Huang. Real-time parallel counterfactual regret minimization. arXiv preprint arXiv:2605.19928, 2026 b

    2026 arXiv

  58. [74]

    Rl-cfr: improving action abstraction for imperfect information extensive-form games with reinforcement learning

    Boning Li, Zhixuan Fang, and Longbo Huang. Rl-cfr: improving action abstraction for imperfect information extensive-form games with reinforcement learning. In Proceedings of the 41st International Conference on Machine Learning, pages 27752--27770, 2024

    2024

  59. [75]

    Pokerskill: Llms can play expert-level poker without training or solvers

    Boning Li, Baoxiang Wang, and Longbo Huang. Pokerskill: Llms can play expert-level poker without training or solvers. arXiv preprint arXiv:2605.30094, 2026

    2026 arXiv

  60. [76]

    Online monte carlo counterfactual regret minimization for search in imperfect information games

    Viliam Lis \'y , Marc Lanctot, and Michael H Bowling. Online monte carlo counterfactual regret minimization for search in imperfect information games. In AAMAS, pages 27--36, 2015

    2015

  61. [77]

    On the theory of systematic sampling, i

    William G Madow and Lillian H Madow. On the theory of systematic sampling, i. The Annals of Mathematical Statistics, 15 0 (1): 0 1--24, 1944

    1944

  62. [78]

    Simple random search of static linear policies is competitive for reinforcement learning

    Horia Mania, Aurelia Guy, and Benjamin Recht. Simple random search of static linear policies is competitive for reinforcement learning. Advances in neural information processing systems, 31, 2018

    2018

  63. [79]

    Escher: Eschewing importance sampling in games by computing a history value function to estimate regret

    Stephen Marcus McAleer, Gabriele Farina, Marc Lanctot, and Tuomas Sandholm. Escher: Eschewing importance sampling in games by computing a history value function to estimate regret. In The Eleventh International Conference on Learning Representations, 2023

    2023

  64. [80]

    Reducing variance of stochastic optimization for approximating nash equilibria in normal-form games

    Linjian Meng, Wubing Chen, Wenbin Li, Tianpei Yang, Youzhi Zhang, and Yang Gao. Reducing variance of stochastic optimization for approximating nash equilibria in normal-form games. In Forty-second International Conference on Machine Learning, 2025

    2025

  65. [81]

    Faster game solving via asymmetry of step sizes

    Linjian Meng, Tianpei Yang, Youzhi Zhang, Zhenxing Ge, and Yang Gao. Faster game solving via asymmetry of step sizes. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 40, pages 17161--17169, 2026 a

    2026

  66. [82]

    A faster parameter-free regret matching algorithm

    Linjian Meng, Youzhi Zhang, Shangdong Yang, Wenbin Li, Tianyu Ding, and Yang Gao. A faster parameter-free regret matching algorithm. In The Fourteenth International Conference on Learning Representations, 2026 b

    2026

  67. [83]

    Monte carlo gradient estimation in machine learning

    Shakir Mohamed, Mihaela Rosca, Michael Figurnov, and Andriy Mnih. Monte carlo gradient estimation in machine learning. Journal of Machine Learning Research, 21 0 (132): 0 1--62, 2020

    2020

  68. [84]

    DeepStack : Expert-level artificial intelligence in heads-up no-limit poker

    Matej Morav c \' k, Martin Schmid, Neil Burch, Viliam Lis \'y , Dustin Morrill, Nolan Bard, Trevor Davis, Kevin Waugh, Michael Johanson, and Michael Bowling. DeepStack : Expert-level artificial intelligence in heads-up no-limit poker. Science, 356 0 (6337): 0 508--513, 2017

    2017

  69. [85]

    Random number generation and quasi-Monte Carlo methods

    Harald Niederreiter. Random number generation and quasi-Monte Carlo methods. SIAM, 1992

    1992

  70. [86]

    Monte carlo variance of scrambled net quadrature

    Art B Owen. Monte carlo variance of scrambled net quadrature. SIAM Journal on Numerical Analysis, 34 0 (5): 0 1884--1910, 1997

    1910

  71. [87]

    Learning from scarce experience

    Leonid Peshkin and Christian R Shelton. Learning from scarce experience. In Proceedings of the Nineteenth International Conference on Machine Learning, pages 498--505, 2002

    2002

  72. [88]

    Accelerating nash equilibrium convergence in monte carlo settings through counterfactual value based fictitious play

    Ju Qi, Hei Falin, Feng Ting, Yi Dengbing, Fang Zhemei, and Luo Yunfeng. Accelerating nash equilibrium convergence in monte carlo settings through counterfactual value based fictitious play. Advances in Neural Information Processing Systems, 37: 0 108088--108115, 2024

    2024

  73. [89]

    Variance reduction in monte carlo counterfactual regret minimization (vr-mccfr) for extensive form games using baselines

    Martin Schmid, Neil Burch, Marc Lanctot, Matej Moravcik, Rudolf Kadlec, and Michael Bowling. Variance reduction in monte carlo counterfactual regret minimization (vr-mccfr) for extensive form games using baselines. In Proceedings of the AAAI Conference on Artificial Intelligen...

    2019

  74. [90]

    Student of games: A unified learning algorithm for both perfect and imperfect information games

    Martin Schmid, Matej Morav c \' k, Neil Burch, Rudolf Kadlec, Josh Davidson, Kevin Waugh, Nolan Bard, Finbarr Timbers, Marc Lanctot, G Zacharias Holland, et al. Student of games: A unified learning algorithm for both perfect and imperfect information games. Science Advances, 9...

    2023

  75. [91]

    Distribution of points in a cube and approximate evaluation of integrals

    Ilya M Sobol. Distribution of points in a cube and approximate evaluation of integrals. USSR Computational mathematics and mathematical physics, 7: 0 86--112, 1967

    1967

  76. [92]

    Actor-critic policy optimization in partially observable multiagent environments

    Sriram Srinivasan, Marc Lanctot, Vin \'i cius Flores Zambaldi, Julien P \'e rolat, Karl Tuyls, R \'e mi Munos, and Michael Bowling. Actor-critic policy optimization in partially observable multiagent environments. In Advances in Neural Information Processing Systems 31 (NeurIP...

    2018

  77. [93]

    Dream: Deep regret minimization with advantage baselines and model-free learning

    Eric Steinberger, Adam Lerer, and Noam Brown. Dream: Deep regret minimization with advantage baselines and model-free learning. arXiv preprint arXiv:2006.10410, 2020

    2006 arXiv

  78. [94]

    Monte carlo continual resolving for online strategy computation in imperfect information games

    Michal S ustr, Vojt e ch Kova r \' k, and Viliam Lis \'y . Monte carlo continual resolving for online strategy computation in imperfect information games. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, pages 224--232, 2019

    2019

  79. [95]

    Sound search in imperfect information games

    Michal S ustr, Martin Schmid, Matej Morav c \' k, Neil Burch, Marc Lanctot, and Michael Bowling. Sound search in imperfect information games. arXiv preprint arXiv:2006.08740, 2020

    2006 arXiv

  80. [96]

    Meta-learning in self-play regret minimization

    David Sychrovsk \'y , Martin Schmid, Michal Sustr, and Michael Bowling. Meta-learning in self-play regret minimization. arXiv preprint arXiv:2504.18917, 2025

    2025 arXiv

  81. [97]

    Solving large imperfect information games using cfr+

    Oskari Tammelin. Solving large imperfect information games using cfr+. arXiv preprint arXiv:1407.5042, 2014

    2014 arXiv

  82. [98]

    Solving games with functional regret estimation

    Kevin Waugh, Dustin Morrill, James Bagnell, and Michael Bowling. Solving games with functional regret estimation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, 2015

    2015

  83. [99]

    Dynamic discounted counterfactual regret minimization

    Hang Xu, Kai Li, Haobo Fu, Qiang Fu, Junliang Xing, and Jian Cheng. Dynamic discounted counterfactual regret minimization. In The Twelfth International Conference on Learning Representations, 2024

    2024

  84. [100]

    Deep (predictive) discounted counterfactual regret minimization

    Hang Xu, Kai Li, Haobo Fu, Qiang Fu, Junliang Xing, and Jian Cheng. Deep (predictive) discounted counterfactual regret minimization. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 40, pages 17284--17292, 2026

    2026

  85. [101]

    Faster game solving via hyperparameter schedules

    Naifeng Zhang, Stephen Marcus McAleer, and Tuomas Sandholm. Faster game solving via hyperparameter schedules. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 40, pages 17319--17326, 2026

    2026

  86. [102]

    Regret minimization in games with incomplete information

    Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. Advances in neural information processing systems, 20, 2007

    2007

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.