PaperlayerView exact revision on arXiv

Source-linked AI summary

Blazing the trails before beating the path: Sample-efficient Monte-Carlo planning

Jean-Bastien Grill, Michal Valko, Rémi Munos

arXiv:2604.14974v1cs.LGstat.ML

TL;DR

TrailBlazer addresses sample-efficient planning in MDPs with a generative model, where actions and random transitions alternate. It adapts sampling to near-optimal structure and gives sample-complexity bounds for finite and infinite successor sets, including a 1/ε² case. One reported infinite-case result is guaranteed only in expectation, unlike Theorem 2’s high-probability guarantee.

  • Problem

    Sampling-based MDP planning with a generative model needs sample-complexity guarantees that can reflect near-optimal structure across finite or infinite successor sets.

    Source: §1 · pp. 1–2, §1.1 · p. 2

  • Method

    TrailBlazer adaptively samples an alternating MAX/AVG planning tree and bounds its oracle calls using measures of near-optimal nodes.

    Source: §1 · p. 2, §1.2 · p. 3, §2.1 · p. 4

  • Results

    For N<∞, its bound depends on κ; for N=∞, complexity is (1/ε)^(2+d), reaching 1/ε² when d=0 under stated conditions.

    Source: §1.2 · p. 3

  • Takeaways & Limitations

    TrailBlazer extends Monte-Carlo sampling to stochastic control problems in which action maximization and transition expectation alternate.

    Source: §1 · p. 2, §1.2 · p. 3

    One reported result holds only in expectation, unlike Theorem 2, which holds in high probability.

    Source: §4.3 · p. 9

Abstract

from arXiv · show

You are a robot and you live in a Markov decision process (MDP) with a finite or an infinite number of transitions from state-action to next states. You got brains and so you plan before you act. Luckily, your roboparents equipped you with a generative model to do some Monte-Carlo planning. The world is waiting for you and you have no time to waste. You want your planning to be efficient. Sample-efficient. Indeed, you want to exploit the possible structure of the MDP by exploring only a subset of states reachable by following near-optimal policies. You want guarantees on sample complexity that depend on a measure of the quantity of near-optimal states. You want something, that is an extension of Monte-Carlo sampling (for estimating an expectation) to problems that alternate maximization (over actions) and expectation (over next states). But you do not want to StOP with exponential running time, you want something simple to implement and computationally efficient. You want it all and you want it now. You want TrailBlazer.

1 Introduction

TrailBlazer plans with a generative model and bounds oracle-call complexity by measures of near-optimal nodes, including when the number of possible next states is infinite. Its results connect adaptive planning to Monte-Carlo sampling while comparing its guarantees with earlier planning methods.

1 Introduction

  • The setting uses costly generative-model calls to sample rewards and transitions; the goal is a high-probability ε-accurate root value in a tree alternating action maximization and transition expectation.Repeated states reached by different paths remain separate tree nodes rather than being merged.Source: §1 · pp. 1–2

1.1 Related work

  • UCT can expand apparently best branches deeply yet undersample a hidden narrow optimal path; uniform look-ahead sampling avoids this adaptivity risk but does not exploit favorable MDP structure.UCT’s sample complexity can be worse than doubly-exponential in 1/ε for some MDPs, while the uniform bound is non-polynomial in 1/ε.Source: §1.1 · p. 2
  • The paper seeks adaptive sampling that captures favorable MDP structure without doing worse than uniform sampling in the worst case.Earlier uniform sampling bounds can improve with finite branching, but uniform tree construction does not benefit from favorable structure.Source: §1.1 · p. 2
  • StOP has a difficult-to-interpret near-optimal-node measure, requires computationally costly optimistic-policy identification, and does not cover N=∞.Source: §1.1 · p. 3

1.2 Our contributions

  • For N<∞, TrailBlazer’s sample complexity is of order (1/ε)^max(2, log(Nκ)/log(1/γ)+o(1)), with κ related to near-optimal branching.Its exponent improves on the prior worst-case exponent 2 + log(NK)/log(1/γ), replacing it with max(2, log(NK)/log(1/γ)).Source: §1.2 · p. 3
  • For N=∞, complexity is (1/ε)^(2+d); when d=0 under stated action-gap or transition-probability conditions, it is 1/ε², with no previous polynomial bound reported for this case.The 1/ε² rate matches Monte-Carlo sampling, making TrailBlazer its extension to problems alternating MAX and AVG nodes.Source: §1.2 · p. 3
  • With deterministic transitions and stochastic rewards, TrailBlazer’s bound is (1/ε)^max(2, log κ/log(1/γ)), similar to Bubeck and Munos for open-loop policies.An open-loop policy is a state-independent sequence of actions.Source: §1.2 · p. 3
  • When K=1, TrailBlazer behaves exactly like Monte-Carlo sampling and achieves complexity 1/ε², even when N=∞.Source: §1.2 · p. 3

2 The algorithm

TrailBlazer works on an alternating MAX/AVG planning tree with bounded action branching and potentially unbounded transition branching. Its nodes make repeated, parameterized calls while retaining samples, using separate controls for variance and bias.

2 The algorithm

  • The planning objective is to estimate the root value of an alternating MAX/AVG tree, with at most K children per MAX node and potentially unbounded children per AVG node.The paper also considers finite AVG branching, bounded by N.Source: §2 · p. 4

2.1 Description of the algorithm

  • TrailBlazer represents only the accessed finite subset of the potentially infinite tree, and persistent nodes can reuse stored samples across repeated calls.Source: §2.1 · p. 4
  • The call parameter m controls requested variance and ε controls maximum bias; the root call uses m=log(1/δ)/((1−γ)^2ε^2) and tolerance ε/2.Source: §2.1 · pp. 4–5
  • At MAX nodes, TrailBlazer maintains simultaneous high-probability child bounds, discards a child when its upper bound falls below the best lower bound, and stops when one child remains or precision is sufficient.Source: §2.1 · p. 4
  • At AVG nodes, it samples rewards and transitions, calls each of the first m sampled children with multiplicity k and tolerance ε/γ, then returns the reward estimate plus γ times the children’s average estimate.The active list may contain repeated children; k counts each child’s occurrences, and the return is r + γµ.Source: §2.1 · p. 5

3 Key ideas

TrailBlazer separates bias from variance, limits which child samples contribute to an AVG node, and stores planning information in a tree. Its non-adaptive bias avoids choosing a sample-count-dependent bias based on unknown subtree difficulty.

3 Key ideas

  • Each call’s (ε, m) parameters distinguish an absolute-bias bound from a variance term, a property used to prove Lemma 4 and Theorem 1.The output estimate satisfies this property conditioned on a high-probability event.Source: §3 · p. 6

3.1 Separate handling of bias and variance

  • A single confidence interval can mis-handle AVG nodes because biases add, but variances combine through their squares.TrailBlazer supports both high-variance, low-bias queries and low-variance, low-bias queries, and treats them differently.Source: §3.1 · p. 6

3.2 Picking the champion paths

  • An AVG node uses only the first m child samples; extra samples can aid a MAX decision but are discarded afterward and may be reused if an ancestor becomes unsure.The motivation is that extra child samples do not improve the parent’s variance bound when it needs only m independent samples.Source: §3.2 · p. 6

3.3 Tree-based algorithm

  • The planning tree stores information at nodes rather than explicitly representing policies, allowing information to be shared between policies without simultaneous policy updates.This addresses the challenges of policy overlap, where a reward can affect every policy reaching a state, and the infeasibility of keeping all policies in memory.Source: §3.3 · p. 7

3.4 Non-adaptive bias

  • A sample-count-dependent bias ε(k) could favor heavily sampled children, but its optimal form depends on unknown subtree difficulty; non-adaptive bias is optimal when infinite-case complexity is linear in k.The passage contrasts this setting with the proposed sample-count-dependent choice.Source: §3.4 · p. 7

4 TrailBlazer is good and cheap — consistency and sample complexity

TrailBlazer provides PAC-consistent value estimates and sample-complexity guarantees for planning with finite or infinite successor sets. Its bounds depend on measures of near-optimal nodes and, in the infinite case, the difficulty of identifying near-optimal paths.

4.1 Definition of the problem difficulty

  • The difficulty measure Δ→s(s′) compares optimal play from s′ with play that first moves toward descendant s and then acts optimally.Near-optimal nodes are defined using this difference along the path from their ancestors.Source: §4.1 · p. 7
  • With probability at least 1 − δ, TrailBlazer explores only near-optimal nodes, whose size reflects the sample complexity and contains the information needed to compute the root value.The section also establishes PAC consistency for finite and infinite state spaces.Source: §4.1 · p. 7, §4 · p. 7

4.2 Sample complexity in the finite case

  • For finite successor sets, κ is at most K and can be as low as 1 when the near-optimal set is small.The stated bound improves on the best-known worst-case bound [16], improves as κ decreases, and gives order (1/ε)^2 when κ = 1 and the MAX-node gap is uniformly lower-bounded.Source: §4.2 · p. 8

4.3 Sample complexity in the infinite case

  • For infinite successor sets, finite d yields polynomial sample-complexity guarantees, with d measuring both near-optimal-set size and the difficulty of identifying near-optimal paths.When every near-optimal MAX node has gap at least Δmin, the stated sample complexity is of order 1/ε^2; the infinite-case bound holds in expectation only.Source: §4.3 · pp. 8–9

5 Discussion

The discussion explains why TrailBlazer uses its near-optimality definition and contrasts the finite and infinite analyses. The chosen definition supports computing a strategy from descendant values, while the analyses use different bounds on reaching a successor.

5.1 On the choice of the near-optimality definition

  • The chosen near-optimal set retains the reward information needed to compute an optimal policy, while rewards at non-near-optimal nodes do not change that policy.TrailBlazer is intended to explore a set not significantly larger than the near-optimal set.Source: §5.1 · p. 9
  • TrailBlazer’s definition is larger than Alternative definition 2, but lets the algorithm compute a node’s strategy from descendant values and has computational complexity linear in generative-model calls.The definitions become close when transition probabilities are low and coincide in the stated infinite case with continuous transition probabilities.Source: §5.1 · p. 9

5.2 Comparison of the finite and the infinite analysis

  • The analyses differ in bounding the probability of reaching a successor after m samples: the infinite case uses a linear upper bound, while the finite case bounds it by 1.The probability is 1 − (1 − p(s|s′))^m; the linear bound is tight in the stated infinite-case condition.Source: §5.2 · p. 10

A Additional notation for the analysis

This section introduces quantities for use in the appendix analysis, including notation for MAX nodes and time.

A Additional notation for the analysis

B Maximum depth of the search

The section establishes that TrailBlazer terminates and bounds search depth, using the parameter reductions at alternating MAX and AVG nodes.

B Maximum depth of the search

C Consistency

The consistency proof proceeds bottom-up, treating leaf, AVG, and MAX nodes separately and using events to control estimates.

C Consistency

  • Lemma 4 states that for each node, time, ε, and m, there is an event As(t) associated with the output of a call to that node.Source: C Consistency · p. 14
  • The proof uses bottom-up induction on the explored tree, starting with leaves and then establishing the properties at each node from its children.Source: C Consistency · p. 14
  • An AVG leaf can return deterministic zero when called with ε ≥ 1/(2(1−γ)); the text bounds its error by 1/(2(1−γ)).Source: C Consistency · p. 15
  • For MAX nodes, the proof tracks potentially good children and assumes confidence intervals hold when analyzing the candidates retained by the algorithm.Source: C Consistency · p. 15
  • The proof concludes that the stated consistency properties hold in both cases, |Gs| = 1 and |Gs| > 1.Source: C Consistency · pp. 15–16
  • An AVG node averages independent estimates, with child transition counts and estimates used in its analysis.Source: C Consistency · p. 17

C.1 Proof of Theorem 1

The proof of Theorem 1 applies Lemmas 3 and 4 to the root node.

C.1 Proof of Theorem 1

  • Theorem 1 follows directly by applying Lemma 3 together with Lemma 4 to the root node.Source: §C.1 · p. 17

D Sample complexity

The section bounds calls to the generative model by analyzing how often near-optimal nodes are reached. Its proof tracks undecidable nodes along paths and treats AVG and MAX nodes separately.

D Sample complexity

  • On the event that all model outputs stay within their confidence intervals, clearly sub-optimal nodes are not explored, and the proof bounds calls to near-optimal nodes.The proof seeks a maximum parameter m for calls to a near-optimal node at depth h.Source: D Sample complexity · p. 17
  • The proof counts undecidable nodes along a path and uses induction that handles AVG and MAX ancestors separately.For MAX nodes, it distinguishes whether the target node is decidable; the induction is then generalized over H.Source: D Sample complexity · pp. 17–18
  • The proof concludes by bounding the total number of calls to the generative model.It states that the bound follows after a choice of η.Source: D Sample complexity · p. 19

D.1 Proof of Theorem 2

The proof bounds samples generated by near-optimal nodes at depth h, then states a choice of η for the resulting bound.

D.1 Proof of Theorem 2

  • At depth h, each near-optimal node generates at most 18 log(t/δ) (1−η)^2(1−γ)^2ε^2 samples.The proof uses this per-node cap to bound the maximum number of samples.Source: §D.1 · p. 19
  • The proof states that the per-node sample cap is used to bound the maximum number of samples.The passage introduces this overall bound after giving the cap for a near-optimal node.Source: §D.1 · p. 19
  • The stated bound uses the parameter choice η = γmax(1, 1 log(1/ε)).Source: §D.1 · p. 19

Related papers

Loading 2604.14974v1…