CART-ROSA: Random Opportunity-Set Allocation

• CART-ROSA is defined as a stochastic-control reformulation of CART random forests that treats feature subsampling as a masked-action allocation mechanism.
• The framework separates the informative-opportunity rate from the within-mask contraction strength, allowing distinct analysis of local stabilization versus global forest risk.
• Key results include explicit risk decompositions for single-tree bias, cross-tree interactions, and conditions under which greedy CART is locally stabilizing but globally suboptimal.

Searching arXiv for the cited papers and closely related work to ground the article. arxiv_search(query="(Mei et al., 26 May 2026)", max_results=5) CART Random Opportunity-Set Allocation (CART-ROSA) is a stochastic-control recasting of feature-subsampled CART random forests in which the random subset of features exposed at each node is treated as a random feasible action set and the CART split rule is treated as a masked-action allocation policy (Mei et al., 26 May 2026). In this formulation, tree growth is represented as a finite-horizon controlled stochastic process over split-count states, and the terminal law of that process determines both single-tree error and cross-tree interaction terms in forest mean squared error (MSE). The framework separates two design levers—the informative-opportunity rate induced by feature subsampling and the contraction strength induced by the within-mask split policy—and uses that separation to analyze when local CART behavior aligns, or fails to align, with forest-level risk minimization (Mei et al., 26 May 2026).

1. Formal definition as a masked-action control problem

At tree depth $t$, CART-ROSA samples a random subset of coordinates

$$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$

uniformly without replacement and independently across depths. The set $U_t$ is the random opportunity set, or action mask; it specifies which split coordinates are feasible at that node. The chosen split coordinate is the action $J_t \in U_t$, and the split-count state is

$$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$

with transition

$N_t = N_{t-1} + e_{J_t}.$

The local CART score is the impurity decrease

$$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$

where

$$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$

The greedy CART rule is the masked-action policy

$$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$

with random tie-breaking (Mei et al., 26 May 2026).

This representation changes the interpretive unit of analysis. Rather than describing a forest only as a collection of recursively grown trees, it treats each split as a feasible-action choice under a random mask. A plausible implication is that feature subsampling is not merely a regularization device; within CART-ROSA it becomes an explicit source of environmental randomization that shapes the reachable state distribution.

2. Opportunity sets, informative time, and state reduction

The paper focuses on informative-respecting policies that prefer informative variables whenever available: $$\pi(j \mid C_{t-1}, U_t) = 0, \qquad j \in U_t \cap S^c,$$ whenever $$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$0. This is presented as a modeling restriction used to isolate the allocation and stabilization mechanism, not as an exact finite-sample statement about CART (Mei et al., 26 May 2026).

Under midpoint splits, each split halves the side length along the chosen coordinate, so the branch geometry is determined by the counts $$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$1. The count-state reduction is encoded through two assumptions. Assumption 1 states that

$$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$2

for measurable $$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$3. Assumption 2 states that there exist constants $$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$4 such that, when informative coordinates are available,

$$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$5

and if no informative coordinate is available, $$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$6. In this reduction, CART becomes a masked-action allocation rule that selects the exposed informative coordinate with the smallest shifted count (Mei et al., 26 May 2026).

The paper then separates raw time from informative time. It defines

$$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$7

Here $$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$8 is the number of informative coordinates in the mask, $$U_t \subseteq [d], \qquad |U_t| = m = \lceil \gamma d \rceil,$$9 is the indicator of an informative opportunity, and $U_t$0 is the cumulative number of informative opportunities. Since $U_t$1 is a uniform $U_t$2-subset,

$U_t$3

and

$U_t$4

Thus $U_t$5 and $U_t$6 almost surely (Mei et al., 26 May 2026).

Informative time is indexed by

$U_t$7

The informative-time count process is

$U_t$8

with dynamics

$U_t$9

The connection between raw time and informative time is

$J_t \in U_t$0

To quantify imbalance, the paper defines

$J_t \in U_t$1

For a tree of depth $J_t \in U_t$2, the terminal raw count vector $J_t \in U_t$3 has law

$J_t \in U_t$4

and for a $J_t \in U_t$5-tree ensemble the terminal states are i.i.d. from $J_t \in U_t$6 (Mei et al., 26 May 2026).

3. Risk representation and the two design levers

A central contribution of CART-ROSA is an MSE decomposition expressed in terms of terminal split counts. Under the sparse linear midpoint model

$J_t \in U_t$7

with sparsity set $J_t \in U_t$8, the depth-$J_t \in U_t$9 single-tree MSE is written in terms of $$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$0, and the forest MSE is written in terms of $$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$1 and an independent copy $$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$2 (Mei et al., 26 May 2026).

The paper identifies three terminal-law functionals that govern the decomposition. The single-tree bias term is

$$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$3

the cross-tree bias term is

$$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$4

and the cross-tree variance or overlap term is

$$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$5

This state-to-risk map makes the forest objective depend not only on one-tree terminal geometry but also on the interaction law of two independent terminal states (Mei et al., 26 May 2026).

The framework isolates two design levers. The first is the informative-opportunity rate

$$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$6

which depends only on feature subsampling and governs how often the policy can split on an informative coordinate. At raw time, for $$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$7,

$$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$8

The second is the within-mask contraction strength

$$N_t = (N_{t,1},\dots,N_{t,d})^\top, \qquad N_{t,j} = \sum_{r=1}^t \mathbf 1\{J_r = j\},$$9

If $N_t = N_{t-1} + e_{J_t}.$0, the policy is contractive on the informative block. Population CART satisfies $N_t = N_{t-1} + e_{J_t}.$1 under the stated nondegeneracy condition, whereas the exploratory benchmark satisfies $N_t = N_{t-1} + e_{J_t}.$2 (Mei et al., 26 May 2026).

The separation between $N_t = N_{t-1} + e_{J_t}.$3 and $N_t = N_{t-1} + e_{J_t}.$4 is structurally important. The paper’s formulation makes $N_t = N_{t-1} + e_{J_t}.$5 the parameter controlling how often informative actions are feasible, while $N_t = N_{t-1} + e_{J_t}.$6 controls how those opportunities are allocated once they appear. This suggests that feature subsampling and split greediness enter the theory through distinct channels rather than through a single undifferentiated regularization effect.

4. Local stabilization and concentration of terminal geometry

The local dynamics of informative imbalance are described exactly. At informative time $N_t = N_{t-1} + e_{J_t}.$7,

$N_t = N_{t-1} + e_{J_t}.$8

and

$N_t = N_{t-1} + e_{J_t}.$9

Under the shifted-canonical structure and the nondegeneracy condition

$$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$0

the paper proves a negative-drift lemma: there exists $$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$1 such that

$$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$2

An admissible choice is

$$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$3

where $$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$4. Combining the quadratic increment identity with the drift bound yields

$$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$5

(Mei et al., 26 May 2026).

Two stabilization results follow. Theorem 1 gives first-order equalization: $$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$6 and therefore

$$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$7

Theorem 2 gives exponential compression: for each $$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$8,

$$\mathcal G(C_{t-1},j) := \sup_{c \in (a_{t-1,j}, b_{t-1,j})} \mathrm{IMD}_{j,c}(C_{t-1}),$$9

In the paper’s terms, the greedy CART policy is locally stabilizing: it contracts imbalance and concentrates the terminal tree geometry around balanced informative allocations (Mei et al., 26 May 2026).

The local characterization is sharpened through a Schur-convexity result. Among feasible informative actions, CART minimizes every symmetric strictly Schur-convex function of the post-decision state. In particular, with

$$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$0

the greedy rule chooses the action minimizing this local potential. The interpretation offered by the paper is that greedy CART is not only a rule that maximizes one-step impurity decrease; on the informative block it is also a balancing policy (Mei et al., 26 May 2026).

5. Terminal-law optimization and global nonoptimality

CART-ROSA distinguishes sharply between local stabilization and global forest optimality. The paper’s argument is that the forest MSE depends on the entire terminal law $$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$1, including two-tree interaction terms, whereas greedy CART is defined by one-step impurity maximization. Consequently, a locally stabilizing action can still be globally suboptimal for the ensemble objective (Mei et al., 26 May 2026).

To formalize this distinction, the paper introduces the terminal-law objective

$$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$2

together with the marginal terminal cost

$$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$3

The Bellman optimality condition implies that if a terminal law is locally optimal, the induced policy must choose actions minimizing the continuation value associated with this marginal terminal cost. This yields a Bellman certificate for nonoptimality: if there exists any reachable state-mask event on which greedy CART assigns positive probability to an action $$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$4 while another feasible action $$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$5 has strictly smaller marginal continuation value, then greedy CART is not even a local minimizer of the terminal-law objective (Mei et al., 26 May 2026).

The paper provides an explicit counterexample with

$$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$6

and shows that for $$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$7, the greedy CART terminal law is not a local minimizer of the forest objective. The stated conclusion is precise: greedy CART is locally stabilizing, but local stabilization does not imply global optimality for forest MSE (Mei et al., 26 May 2026).

6. Linear-model specialization and relation to adjacent randomized-tree formulations

In the linear specialization

$$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$8

the population CART impurity decrease is explicit: $$\mathrm{IMD}_{j,c}(C) = \operatorname{Var}(Y \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;L}) P(X \in C_{j,c;L} \mid X \in C) - \operatorname{Var}(Y \mid X \in C_{j,c;R}) P(X \in C_{j,c;R} \mid X \in C).$$9 Hence, when an informative coordinate is available,

$$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$0

This is the smallest shifted-count rule, and in this setting the shifts are

$$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$1

Under exponential stabilization, the informative block can be replaced by its balanced proxy in the risk functionals, yielding explicit nonasymptotic bounds in which the single-tree bias decays at rate $$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$2, the cross-tree bias is controlled by a Poisson-kernel functional $$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$3, and the cross-tree variance is governed by a reduced-dimensional overlap functional $$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$4 (Mei et al., 26 May 2026).

The paper contrasts this with an exploratory benchmark policy,

$$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$5

for which the informative-time counts are multinomial and there is no negative drift: $$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$6 The comparison is used to expose the bias–variance and exploration–exploitation tradeoff: larger $$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$7 gives stronger stabilization and better bias decay, smaller $$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$8 gives more exploratory behavior and potentially better cross-tree variance, and larger $$J_t \in A_{gr}(C_{t-1}, U_t) := \arg\max_{j \in U_t} \mathcal G(C_{t-1}, j),$$9 increases exposure to informative opportunities (Mei et al., 26 May 2026).

CART-ROSA is distinct from Optimal Randomized Classification Trees (ORCT), a related but different randomized-tree formulation (Blanquero et al., 2021). ORCT is a continuous, randomized, oblique-tree optimization model for classification in which each internal node routes an observation left or right with probabilities defined by a continuous CDF,

$$\pi(j \mid C_{t-1}, U_t) = 0, \qquad j \in U_t \cap S^c,$$0

and training minimizes expected misclassification cost over a fixed maximal binary tree (Blanquero et al., 2021). The formulation includes leaf-class assignment variables $$\pi(j \mid C_{t-1}, U_t) = 0, \qquad j \in U_t \cap S^c,$$1, one-class-per-leaf constraints, optional class-coverage constraints, and a continuous nonlinear optimization relaxation that is exact in the sense that there exists an optimal solution with $$\pi(j \mid C_{t-1}, U_t) = 0, \qquad j \in U_t \cap S^c,$$2 (Blanquero et al., 2021). It also proves that, for logistic $$\pi(j \mid C_{t-1}, U_t) = 0, \qquad j \in U_t \cap S^c,$$3, the randomized model converges to a deterministic optimal decision tree as $$\pi(j \mid C_{t-1}, U_t) = 0, \qquad j \in U_t \cap S^c,$$4 (Blanquero et al., 2021).

The relation between the two frameworks is therefore one of conceptual proximity rather than identity. CART-ROSA is a stochastic-control theory of feature-subsampled CART random forests, centered on random opportunity sets, split-count dynamics, and terminal-law risk. ORCT is a continuous-optimization approach to a single randomized classification tree with soft routing. This suggests a broader family of nonstandard tree analyses in which hard deterministic splitting is replaced either by explicit random routing or by a control-theoretic description of masked action selection, but the two papers address different objects and different optimization questions (Mei et al., 26 May 2026).