Trying-Early Adaptive Multilevel Splitting

• The paper introduces an innovative method that discovers intermediate levels dynamically, using empirical quantiles and adaptive resampling to efficiently estimate rare-event probabilities.
• Trying-Early AMS is defined by its on-the-fly selection of thresholds from current particle scores, maintaining unbiasedness and consistency through a rigorous particle system framework.
• The approach extends to various settings, including Fleming–Viot systems and surrogate-based extensions, with practical applications in high-cost simulations like catalytic reaction rate estimation.

Trying‑Early Adaptive Multilevel Splitting denotes a family of Adaptive Multilevel Splitting (AMS) strategies in which intermediate levels are discovered on the fly rather than fixed a priori, and in which trajectories may be pushed toward the rare set earlier or more aggressively through adaptive selection, cloning, or level updates. The designation is not standard in the AMS literature, but it is consistent with two established perspectives: AMS as an adaptive splitting method based on empirical order statistics, and AMS as a particle approximation of rare-event conditional laws that can be analyzed through level-indexed Markov processes, path-space sequential sampling, or idealized scalar models (Cérou et al., 2018, Bréhier et al., 2014).

1. Rare-event setting and terminological scope

Two canonical formulations recur. In the dynamic formulation, one considers a Markov process $(Y_s)_{s\ge 0}$ on a Polish space $E$, a continuous score or level function $\xi:E\to\mathbb R$, a rare set $A\subset E$, and stopping times

$$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$

The central rare-event probability is

$p_1=\mathbb P(S_1<S_A),$

that is, the probability of reaching level $1$ before hitting the unwanted set $A$. In the static idealized formulation, one considers a positive real random variable $X$, a high threshold $a>0$, and

$E$0

These two formulations are linked by the common role of a score or reaction coordinate that orders configurations or trajectories according to progress toward a rare event (Cérou et al., 2018, Bréhier et al., 2014).

Classical fixed-level multilevel splitting writes a rare probability as a product of less rare conditional probabilities over a deterministic sequence of intermediate thresholds. Adaptive Multilevel Splitting replaces those fixed thresholds by random, data-driven levels chosen from the particle system itself. In that sense, “trying‑early” refers not to a single formal algorithm but to a design principle: use currently observed scores to determine the next threshold, and allocate computational effort to trajectories that already show progress. In the idealized scalar setting this adaptive threshold is the empirical $E$1-quantile of the current particle scores; in the last-particle dynamic setting it is the minimum score achieved among the current particles (Bréhier et al., 2014, Cérou et al., 2018).

A basic misconception is that any adaptive thresholding rule is automatically an AMS algorithm in the strict probabilistic sense. The literature instead distinguishes carefully between standard AMS, generalized AMS, Fleming–Viot–type systems, and more aggressive modifications. This suggests that “trying‑early” is best understood as an umbrella label for adaptive splitting schemes whose levels are not predetermined and whose early selection pressure may vary, rather than as a single standardized method (Charles-Edouard et al., 2015, Cérou et al., 2018).

2. Canonical algorithmic forms and estimators

The dynamic last-particle version maintains $E$2 trajectories. At iteration $E$3, for each particle one computes the maximal level reached before stopping at $E$4 or before exceeding level $E$5; the worst particle is the one with minimal such score. If that minimal score is $E$6, the algorithm kills exactly that particle, chooses uniformly a donor among the remaining $E$7 particles, copies the donor path up to the first time it exceeds $E$8, and then resimulates independently from that point until reaching $E$9 or $\xi:E\to\mathbb R$0. Under the regularity assumptions used in the theory, the minimizer is almost surely unique and $\xi:E\to\mathbb R$1 is strictly increasing. The corresponding probability estimator at level $\xi:E\to\mathbb R$2 is

$\xi:E\to\mathbb R$3

where $\xi:E\to\mathbb R$4 is the number of branchings up to level $\xi:E\to\mathbb R$5. The empirical conditional-law estimator is

$\xi:E\to\mathbb R$6

and the unnormalized estimator is

$\xi:E\to\mathbb R$7

For bounded $\xi:E\to\mathbb R$8, $\xi:E\to\mathbb R$9 is unbiased: $A\subset E$0 This is the most direct continuous-time realization of AMS as progressive elimination of the least advanced trajectory (Cérou et al., 2018).

The idealized scalar $A\subset E$1-particle formulation instead keeps $A\subset E$2 replicas and kills the $A\subset E$3 least adapted ones at each iteration. Starting from level $A\subset E$4, the current threshold is

$A\subset E$5

the $A\subset E$6-th order statistic of the current particle scores. At each step the $A\subset E$7 worst particles are replaced by i.i.d. draws from the exact conditional law $A\subset E$8. The algorithm stops when the adaptive level crosses $A\subset E$9, and the estimator of $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$0 is

$$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$1

where

$$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$2

For $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$3, $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$4 estimates the original rare-event probability $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$5. In this formulation the “trying‑early” mechanism is explicit: the next level is always the empirical $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$6-quantile of the current sample, so each empirical conditional survival factor is approximately $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$7 (Bréhier et al., 2014, Bréhier et al., 2015).

These formulations admit broader path-space generalizations. In the discrete-time Markov-chain setting, generalized AMS uses a path importance function $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$8, first entrance times $$S_t=\inf\{s\ge 0:\xi(Y_s)>t\},\qquad S_A=\inf\{s\ge 0:Y_s\in A\}.$$9, level-indexed resampling kernels $p_1=\mathbb P(S_1<S_A),$0, and weighted particle systems. The generic weighted estimator is

$p_1=\mathbb P(S_1<S_A),$1

with the rare-event probability estimator obtained by taking $p_1=\mathbb P(S_1<S_A),$2. This path-space viewpoint is the natural formal setting for more elaborate early-resampling or early-termination variants (Charles-Edouard et al., 2015).

3. Probabilistic frameworks underlying adaptive and early variants

A central theoretical development is the level-indexed or “stochastic wave” representation. Given the original process $p_1=\mathbb P(S_1<S_A),$3, one defines a new process $p_1=\mathbb P(S_1<S_A),$4 indexed by level rather than by physical time: $p_1=\mathbb P(S_1<S_A),$5 where $p_1=\mathbb P(S_1<S_A),$6 is a cemetery state. In this representation, time is the level $p_1=\mathbb P(S_1<S_A),$7, and the AMS particle system becomes exactly a Fleming–Viot particle system: particles evolve independently according to the level-indexed Markov process until one hits the cemetery, at which point it is reborn at the position of another surviving particle at the same level. This identification is what allows consistency and CLT results for AMS to be derived from general results on Fleming–Viot systems (Cérou et al., 2018).

In parallel, generalized AMS recasts splitting as sequential sampling in path space. For a path $p_1=\mathbb P(S_1<S_A),$8, the maximum level

$p_1=\mathbb P(S_1<S_A),$9

and the strict entrance time

$1$0

define the information retained when resampling at level $1$1. A resampling kernel $1$2 copies the path up to $1$3 and then evolves it with the original Markov dynamics. Under right-continuity and conditional-consistency assumptions on $1$4, and with weight updates based on conditional expectations of branching numbers, the resulting weighted estimator is unbiased whatever the importance function and whatever the number of replicas (Charles-Edouard et al., 2015).

The fluctuation theory of adaptive splitting further shows that the adaptive particle system asymptotically behaves like a fixed-level Feynman–Kac particle system with optimally placed levels. In that framework the adaptive levels are empirical quantiles $1$5 converging almost surely to deterministic quantiles $1$6, and the adaptive and fixed-level algorithms have the same asymptotic variance. A plausible implication is that early level adaptation is not, by itself, a statistical penalty; the critical issue is whether the adaptive scheme preserves the particle-system structure required by the limit theory (Cerou et al., 2014).

4. Unbiasedness, consistency, fluctuations, and large deviations

Several complementary asymptotic results define the current theoretical core of AMS. In the general continuous-time Markov-process setting, well-posedness holds under Feller regularity, strict entrance, and a uniformly positive success probability: the worst particle is almost surely unique at each step, and the algorithm terminates after finitely many iterations. For $1$7,

$1$8

and

$1$9

with

$A$0

For the probability estimator,

$A$1

and the variance satisfies

$A$2

The lower bound is attained when the score is the committor

$A$3

whereas the upper bound shows that a poor score can make AMS worse than crude Monte Carlo, though still with finite variance. The number of branchings satisfies

$A$4

so the total complexity is roughly

$A$5

in that implementation (Cérou et al., 2018).

In the idealized scalar setting, unbiasedness is exact for all $A$6, all $A$7, and all $A$8: $A$9 For fixed $X$0 and large $X$1,

$X$2

so the leading variance term is $X$3, independent of $X$4. The expected number of iterations satisfies

$X$5

and the asymptotic cost expansion implies that $X$6 is optimal for large $X$7 and fixed $X$8 (Bréhier et al., 2014).

The corresponding CLT states that, for fixed $X$9,

$a>0$0

so the asymptotic variance is again independent of $a>0$1. In logarithmic coordinates the proof proceeds through a functional equation for the characteristic function, then a linear ordinary differential equation of order $a>0$2, and finally an eigenvalue analysis of the associated characteristic polynomial (Bréhier et al., 2015).

At the large-deviation scale, the law of $a>0$3 satisfies an LDP with speed $a>0$4 and rate function

$a>0$5

again independent of $a>0$6. This rate function is minimized uniquely at $a>0$7, and it dominates the Cramér rate function of crude Monte Carlo, implying exponentially smaller large-error probabilities for AMS in the idealized regime (Bréhier, 2015).

Finally, adaptive fluctuation analysis shows that the precision of the adaptive version is the same as that of the fixed-level version where the levels would have been placed in an optimal manner. The additional computational burden of adaptivity is the sorting step needed to compute empirical quantiles, which contributes a factor $a>0$8 in time complexity per level (Cerou et al., 2014).

5. Design principles, structural constraints, and recurrent misconceptions

The primary design variable is the score or reaction coordinate. The general-process variance formula shows that the quantity driving asymptotic variance is the propagated potential

$a>0$9

For the probability estimator, the committor $E$00 is optimal in the precise sense that it cancels the intermediate variance term $E$01. This is the main reason why “trying‑early” design is inseparable from reaction-coordinate design: an aggressive early policy with a poor coordinate can amplify variance rather than reduce it (Cérou et al., 2018).

A second structural principle is that unbiasedness and limit theorems are not consequences of adaptation alone but of specific particle-system properties. In the generalized AMS framework, early resampling must be expressible through level-indexed filtrations, resampling kernels consistent with conditional expectations, and weight updates of the form

$E$02

for each child of a parent $E$03. Early stopping rules must be stopping times for the natural filtration, and conditional continuation must follow the original Markov law. These conditions allow broad generalizations—randomized level computation, variable population size, or state-dependent branching—while preserving unbiasedness (Charles-Edouard et al., 2015).

The literature also documents failure modes. In the generalized path-space analysis, enforcing exactly $E$04 resampled replicas when ties occur, or branching from $E$05 instead of from the strict level rule $E$06, produces significant underestimation in the benchmark examples. This directly counters the common misconception that small implementation changes are harmless as long as the broad splitting logic is retained (Charles-Edouard et al., 2015).

A third issue concerns how far the present theory extends to more aggressive “trying‑early” variants. If one changes cloning probabilities, kills more than one particle at a time, or uses threshold rules based on higher quantiles or other statistics, then the Fleming–Viot structure may be lost and the CLT may change or require new proofs. Likewise, adaptive schemes in which the number of resampling times grows with $E$07 fall outside the fluctuation results derived for a fixed finite number of adaptive levels. The established theory therefore serves as a baseline rather than a blanket guarantee for all early-adaptive modifications (Cérou et al., 2018, Cerou et al., 2014).

6. Related formulations, surrogate-based extensions, and applications

One limiting “trying‑early” formulation is the moving-particles viewpoint. Instead of explicitly defining subsets or levels, one repeatedly moves the worst particle upward by conditional simulation. In the ideal setting, the number of samples required to get one realization in the rare set follows a Poisson law with parameter $E$08, and for $E$09 particles the number of moves is Poisson with mean $E$10. This yields a parallel optimal multilevel splitting method in which there is no subset to define any more, and it can also be used for extreme-quantile estimation and for the construction of initial designs of experiments in meta-model–based algorithms (Walter, 2014).

A different extension embeds static rare-event problems into continuous-time Markov processes. The universal splitting estimator does this by constructing $E$11 so that $E$12, typically via Gamma-process or Poisson-process embeddings, and then applying multilevel splitting in the time variable. The same splitting algorithm then applies to sums of random variables, partial sums of ordered random variables, ratios of random variables, and weighted sums of Poisson random variables. The levels are selected so that conditional survival probabilities are approximately constant, either by a pilot inverse-complementary-CDF procedure or by analytical lower bounds when available (Rached et al., 2019).

When score evaluation is itself extremely expensive, Adaptive Reduced Multilevel Splitting extends AMS to the case of surrogate scores $E$13 equipped with certified error bounds $E$14. At each surrogate iteration, the algorithm defines pessimistic and optimistic rare-event distributions using $E$15 and $E$16, chooses a critical level $E$17 through a Kullback–Leibler cost threshold, runs AMS on the surrogate to approximate $E$18, and then evaluates the expensive true score only on selected snapshots. The outer estimator is an importance-sampling estimator based on those snapshots. In the reported examples, relative cost-versus-RMSE curves show roughly an order-of-magnitude reduction in cost compared with standard AMS, especially for rare events associated with parametric PDEs approximated by reduced bases (Cérou et al., 2023).

A concrete high-cost application is the computation of catalytic surface reaction rates by coupling AMS with ab initio molecular dynamics. There the transition probability $E$19 is estimated by AMS and combined with a flux estimate through the Hill relation. The implementation uses reaction coordinates derived from linear SVM decision functions and from interpolated SOAP path collective variables, and the study of chemisorbed water on $E$20-alumina reports AMS-based rate constants that can be smaller than those from a harmonic Eyring–Polanyi approach by up to two orders of magnitude because of entropic effects involved in the chemisorbed water (Pigeon et al., 2023).

Across these formulations, the unifying feature of “trying‑early” AMS is not a single canonical algorithm but a strategy: push the particle system toward rarer regions as soon as current information makes that statistically and computationally defensible, while preserving the probabilistic structure needed for unbiasedness, variance control, and asymptotic analysis.