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Power Mean Estimation in Stochastic Continuous Monte Carlo Tree Search

Tuan Dam

arXiv:2609.06489v1cs.LGcs.AI

TL;DR

MCTS lacks comparable convergence guarantees for continuous, stochastic MDPs, where non-stationary values and uncertain transitions complicate planning. Stochastic-Power-HOOT combines power-mean backups with polynomial exploration, achieving O(n^-ζ), ζ ∈ (0, 1/2), and empirical validation in stochastic control tasks.

  • Problem

    Continuous, stochastic MDPs lack convergence guarantees comparable to POLY-HOOT’s deterministic results, despite their relevance to uncertain real-world domains.

  • Method

    Stochastic-Power-HOOT combines a power mean backup operator with a polynomial exploration bonus for continuous-action, stochastic MDP planning.

  • Results

    O(n^-ζ), ζ ∈ (0, 1/2): Stochastic-Power-HOOT converges polynomially and empirical evaluations validate its robustness and versatility in stochastic tasks.

  • Takeaways & Limitations

    The approach extends continuous MCTS convergence analysis from deterministic to stochastic MDPs while supporting empirical performance across control challenges.

  • Takeaways & Limitations

    The analysis assumes infinite-horizon discounted MDPs with finite states, continuous actions, stochastic transitions, bounded rewards, and access to sampled transitions and rewards.

Abstract

from arXiv · show

Monte Carlo Tree Search (MCTS) has demonstrated success in online planning for deterministic environments, yet significant challenges remain in adapting it to stochastic Markov Decision Processes (MDPs), particularly in continuous state-action spaces. Existing methods, such as HOOT, which combines MCTS with the Hierarchical Optimistic Optimization (HOO) bandit strategy, address continuous spaces but rely on a logarithmic exploration bonus that lacks theoretical guarantees in non-stationary, stochastic settings. Recent advancements, such as POLY-HOOT, introduced a polynomial bonus term to achieve convergence in deterministic MDPs, though a similar theory for stochastic MDPs remains undeveloped. In this paper, we propose a novel MCTS algorithm, \Algname, designed for continuous, stochastic MDPs. \Algname integrates a power mean as a value backup operator, alongside a polynomial exploration bonus to address the non-stationarity inherent in continuous action spaces. Our theoretical analysis establishes that \Algname converges at a polynomial rate of $\mathcal{O}(n^{-ζ})$, $ζ\in (0,1/2)$, where \( n \) is the number of visited trajectories, thereby extending the non-asymptotic convergence guarantees of POLY-HOOT to stochastic environments. Experimental results on stochastic tasks validate our theoretical findings, demonstrating the effectiveness of \Algname in continuous, stochastic domains.

1. Introduction

The paper targets MCTS in continuous, stochastic environments, where non-stationarity complicates convergence and practical planning. It proposes Stochastic-Power-HOOT, combining power-mean backups with polynomial exploration and reporting polynomial convergence plus empirical validation.

  • Motivation: Continuous actions and stochastic dynamics complicate MCTS convergence and practical performance, limiting applications beyond deterministic planning domains.Traditional methods also assume discrete actions and near-stationary rewards.
  • Motivation: HOOT uses logarithmic exploration bonuses without convergence guarantees for non-stationary values, while POLY-HOOT addresses deterministic continuous MDPs.The stochastic case remains theoretically unresolved.
  • Approach: Stochastic-Power-HOOT combines a power mean backup operator with a polynomial exploration bonus for continuous, stochastic MDPs.The method adaptively searches the action space rather than naively discretizing it.
  • Contributions: O(n^-ζ), ζ ∈ (0, 1/2): Stochastic-Power-HOOT achieves polynomial convergence in stochastic domains, extending POLY-HOOT’s guarantees beyond deterministic environments.The contribution is presented as a theoretical extension to stochastic settings.
  • Contributions: Experiments on robotic tasks report better final performance and robustness than HOOT, discretized-UCT, and Voronoi MCTS.These experiments are presented as empirical validation of practical viability and theoretical soundness.

2. Related Works

Prior MCTS research established strong results in deterministic settings, but continuous and stochastic domains remain difficult. Existing exploration strategies introduce convergence concerns or lack guarantees under non-stationary rewards.

  • Deterministic MCTS: UCT demonstrated MCTS effectiveness for discrete actions in deterministic environments.Other methods explored entropy regularization and Boltzmann-based exploration.
  • Continuous and stochastic MCTS: HOOT and related continuous-domain approaches remain challenged by non-trivial convergence in continuous or stochastic settings.Temperature parameters in some alternative exploration strategies may also impede convergence.

3. Setup and Notations

The paper studies infinite-horizon discounted MDPs with finite states, continuous actions, stochastic transitions, bounded rewards, and discounting. Its objective is to identify an optimal policy using sampled transitions and rewards from a generative model.

  • MDP setting: The setting uses infinite-horizon discounted MDPs with finite state space, continuous action space, stochastic transitions, bounded rewards, and γ ∈ (0, 1).The action space is a subset of R^m and transitions map state-action pairs to distributions over states.
  • Objective: The objective is to find an optimal policy π* whose value equals the supremum over policy values.The algorithm assumes access to a generative model supplying sampled transitions and rewards.

4. Background material

The background introduces MDP and MCTS planning concepts, including trajectory collection, value estimation, convergence measurement, and continuous-action search through HOO. It motivates polynomial exploration by noting that logarithmic bonuses lack clear guarantees for non-stationary rewards.

  • MCTS overview: MCTS iteratively selects nodes, expands the tree, evaluates nodes through rollouts, and backpropagates rewards.Its effectiveness depends on the value update operator and node-selection strategy.
  • MCTS formalization: MCTS collects trajectories from an initial state to a fixed depth or leaf and outputs an estimated best action and root value.A playout policy estimates terminal values after each batch of trajectories.
  • Convergence: The background defines convergence rate as the performance measure for evaluating MCTS estimates.The supplied passage introduces the rate without giving its displayed formula.
  • Value estimation: Value functions are defined recursively from leaf values and expected rewards across tree depths.The resulting approximation goal is to estimate root action values and identify the optimal action.
  • HOO: HOO partitions continuous actions into a binary tree and selects actions by traversing toward leaves with larger upper confidence bounds.Each node represents an action-space subset that is recursively split.
  • Exploration guarantees: HOOT uses logarithmic exploration bonuses, creating unclear convergence guarantees because non-stationary rewards lack concentration.POLY-HOOT was introduced to address this issue, but its proof covered deterministic settings only.

5. Stochastic Power-HOOT

Stochastic-Power-HOOT combines continuous-action HOO search with power-mean value backups and polynomial exploration bonuses for stochastic MCTS. Its procedure alternates tree search, rollout evaluation, and backpropagation under bounded-horizon assumptions.

  • Algorithm overview: Stochastic-Power-HOOT targets continuous-action, stochastic environments with power-mean value estimation and polynomial exploration.The algorithm is organized into Selection, Expansion, Rollout, and Backpropagation phases.
  • Selection and expansion: Selection traverses HOO trees using upper confidence bounds and initializes a new HOO agent when a state-depth pair is unvisited.Expansion samples an action from the selected partition when the depth limit permits.
  • Rollout and backpropagation: Rollouts use a default policy to estimate terminal-node values, whose rewards are propagated back along the traversed tree path.Backpropagation updates visit counts, empirical means, and confidence bounds with a polynomial exploration bonus.
  • Value backup: The power-mean backup operator provides a tunable balance between mean- and max-based value updates for stochastic rewards.The formulation is presented as an alternative to simple averaging.
  • Assumptions: The theoretical framework assumes a bounded planning horizon and fixed-length trajectories with playouts applied only at terminal leaf nodes.The text notes that practical implementations could use infinite depth.

6. Theoretical analysis

The analysis models each MCTS node as a non-stationary bandit whose reward function changes as deeper value estimates evolve. It combines HOO structure, polynomial exploration, and power-mean concentration results to establish stochastic continuous-MDP convergence.

  • Analysis framework: Each tree node is analyzed as a non-stationary bandit because action values shift as deeper MCTS estimates are refined.The analysis first studies concentration for non-stationary bandits and then applies it to MCTS.
  • HOO structure: HOO adaptively partitions the continuous action space into a binary tree whose root-to-leaf paths act as refined pseudo-actions.The framework relies on covering and smoothness assumptions for the action space and limiting payoff function.
  • Exploration: Polynomial upper-confidence bonuses replace logarithmic bonuses to support exploration when reward distributions change over time.The stated bonus has the form C n^b/β / T_a(n)^α/β.
  • Concentration: Power-mean aggregation preserves polynomial concentration rates for the bandit estimators.This result supplies the concentration component used in the broader MCTS analysis.
  • Convergence and scope: The convergence theorem extends guarantees to stochastic continuous MDPs, unlike POLY-HOOT's deterministic setting, while supporting power means with p ≥ 1.The paper characterizes the extension as maintaining theoretical rigor across stochastic dynamics and continuous actions.
  • Practical implications: The paper reports reliable performance under environmental noise, tunable power parameters without losing convergence guarantees, and scalability across tree depths and action-space dimensions.These are presented as practical implications of the theoretical guarantees.

7. Experiments

Experiments evaluate Stochastic-Power-HOOT on noisy continuous-action control and high-dimensional robotic tasks. The results indicate strong performance, scalability, stochastic robustness, and sensitivity to power-parameter selection.

  • Experimental setup: Stochastic-Power-HOOT is evaluated on classic control and high-dimensional robotic environments with continuous actions and stochastic perturbations.The evaluation includes action, dynamics, and observation noise in the robotic settings.
  • Classic control tasks: Stochastic-Power-HOOT optimally handles increased-gravity CartPole, sparse-reward MountainCar, and high-gravity Acrobot challenges.All methods solve basic CartPole, but only Stochastic-Power-HOOT solves CartPole-IG optimally.
  • High-dimensional robotic environments: 3.1× improvement over UCT is achieved in the 17-dimensional Humanoid environment.This result demonstrates performance under substantially increased action-space complexity.
  • Power-parameter sensitivity: The optimal power parameter varies by environment: Humanoid benefits from p = 2, whereas Hopper requires p = 8.The reported settings reflect different environmental stochasticity and search-concentration requirements.
  • Stochastic robustness: 12.7× worse than UCT on Hopper contrasts with 2.8× better performance on Humanoid for Voronoi MCTS.The contrast is presented as evidence that explicit stochasticity handling matters in these environments.
  • Power-parameter sensitivity: Moderate power settings p = 2 and p = 4 achieve optimal CartPole-IG performance.The sensitivity analysis describes these values as balancing exploration and exploitation.

8. Conclusion

The paper introduces Stochastic-Power-HOOT for continuous-action stochastic environments and combines power-mean backups with polynomial exploration. It reports polynomial convergence guarantees and empirical robustness, while the supplied conclusion materials also describe implementation and supporting theoretical components.

  • Conclusion: Stochastic-Power-HOOT is a continuous-action MCTS algorithm designed for stochastic environments.The method is presented as the paper’s central algorithmic contribution.
  • Conclusion: Polynomial exploration and power-mean backups are combined to balance exploration and exploitation in stochastic MDPs.The conclusion links these two components to the algorithm’s handling of stochastic planning.
  • Conclusion: O(n^-ζ), ζ ∈ (0, 1/2) is the reported convergence rate for stochastic continuous MDPs.The rate extends prior continuous-MCTS analyses from deterministic to stochastic MDPs.
  • Conclusion: The paper reports empirical robustness and versatility, including the impact of tuning the power mean for complex reward structures.The supplied conclusion characterizes the evaluations as validation on custom tasks.
  • Supporting material: The supporting material includes a non-stationary Power Mean HOO algorithm with query and update procedures.The listed procedures initialize paths, select actions, observe rewards, and update statistics and confidence bounds.

E. Technical Lemmas

The technical lemmas develop concentration and node-selection arguments for the continuous stochastic MCTS analysis. They extend bounds to small suboptimality gaps and use inductive, union-bound reasoning to control node visits.

  • Node-selection bounds: The technical development includes HOO optimality and suboptimal-node lemmas governing confidence bounds and node selection.The lemmas distinguish nodes with gaps above and below the HOO resolution scale.
  • Relation to prior analysis: The technical proofs follow the high-level strategy of Mao et al. (2020), while adding a concentration bound for the small-gap case.The attribution note identifies this missing small-gap bound as an important difference.
  • Proof strategy: The proofs use induction, concentration properties, recursive path arguments, and union bounds across times and nodes.These arguments bound events involving upper confidence values and excessive visits to suboptimal nodes.
  • Visit-count control: For a suboptimal node, simultaneous control of two events implies its visit count remains at most the chosen threshold.The events concern the local upper bound or visit count and the confidence values along the optimal path.
  • Small-gap case: The analysis extends the node-visit bound to gaps satisfying ∆h,i ≤ ν1ρh by inflating the gap and applying the corresponding concentration result.This addresses the regime excluded from the original larger-gap lemma.

F. Convergence of Stochastic-Power-HOOT n Non-stationary multi-armed bandits

The bandit-level analysis establishes polynomial concentration for power-mean estimates in non-stationary continuous-armed settings. Under parameter conditions, suboptimal-arm contributions become negligible and the estimator converges at a polynomial rate.

  • Theoretical foundation: The analysis builds on polynomial concentration and convergence guarantees for enhanced HOO in non-stationary continuous-armed bandits.These guarantees provide the foundation for analyzing power-mean aggregation.
  • Suboptimal-arm control: Suboptimal arms are controlled through visit-count bounds and concentration arguments across three regimes determined by p, α, and β.The regimes cover 1 ≤ p ≤ 2 and two cases with p > 2.
  • Power-mean concentration: Power-mean estimation preserves polynomial concentration around the optimal value while allowing p to tune the exploration–exploitation balance.The theorem synthesizes the optimal-arm and suboptimal-arm bounds.
  • Rate conditions: The derived convergence exponent satisfies 0 < ζ ≤ 1−bα/(1+d′+βα), under conditions ensuring positive polynomial exponents.The parameter restrictions prevent the concentration bounds from degenerating.

G. Convergence of Stochastic-Power-HOOT in Monte-Carlo Tree Search

The full MCTS analysis propagates polynomial concentration from leaves and stochastic transitions through every tree depth by induction. The resulting root payoff converges at rate O(n^-ζ), with ζ ∈ (0, 1/2).

  • Theoretical convergence: Theorem 3 establishes polynomial concentration for value and Q-value estimates at every node in trees up to depth D.The proof uses induction over tree depth.
  • Base case: Leaf estimates concentrate through i.i.d. rollout averaging, while stochastic-transition Q-values are handled by the bandit concentration lemma.These results form the base case for the tree-depth induction.
  • Inductive step: The induction step propagates concentration from depth D−1 subtrees to the root using the power-mean value estimates.The result holds for any node in any tree of depth up to D.
  • Final convergence rate: O(n^-ζ), ζ ∈ (0, 1/2): the expected root payoff converges at a polynomial rate under best parameter tuning.The rate follows by integrating the polynomial concentration bound.
  • Empirical evaluation: Experiments evaluate the method on stochastic classic-control and high-dimensional robotic tasks with action, dynamics, and observation noise.Comparisons include discretized-UCT, Progressive Widening, HOOT, POLY-HOOT, and Voronoi MCTS.

H.2. Practical Branching Factor Analysis

Practical branching remains compact despite a theoretical maximum of 2D, with branching concentrated at key decision points. This indicates focused search in promising regions.

  • Observed branching: 1.86–2.17: average branching factors across different power values remain low in practice.The measured branching factors are far below the theoretical maximum of 2D.
  • Branching distribution: More than 95% of nodes have only one child, while branching concentrates at key decision points.This pattern indicates that expansion is localized rather than uniformly distributed.
  • Search behavior: Power-mean backpropagation focuses search on promising regions while limiting resources on less promising areas.The reported pattern is interpreted as an exploration–exploitation balance.
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