✗ Failed on benchmark 2026

Critical stochastic min-plus tree layer

Implementation & benchmark of arXiv:2608.12295 — Finite-depth scaling and an exact Bernoulli-leaf identity for the min-plus process on the binary tree

Usefulness7/10

Difficulty5/10

Novelty8/10

Architecture Stability Generalization Sample efficiency Mlp Cnn Training dynamics Regularization Statistical mechanics Dynamical systems Probability Bifurcations

Source paper: Finite-depth scaling and an exact Bernoulli-leaf identity for the min-plus process on the binary tree arXiv:2608.12295 ⓘ · analyzed Sep 1, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace deterministic binary-tree pooling or hierarchical feature aggregation by a stochastic merge that chooses either elementwise addition or elementwise minimum. The mixing probability p controls whether zero or sparse states proliferate or disappear, with a predicted absorbing-state transition at p = 1/2.

Formulas

$M_p(x,y) = x+y$ with probability p, and $M_p(x,y) = min(x,y)$ with probability 1-p

$z_{n+1} = p z_n^2 + (1-p)(2z_n-z_n^2) = 2(1-p)z_n + (2p-1)z_n^2$

$z_{n+1}-z_n = (1-2p)z_n(1-z_n), \quad p_c = 1/2$

$p=1/2 \implies z_n=z_0=q \text{ for every } n$

Mathematical statement

For nonnegative scalar or coordinatewise feature values x and y, define a random merge M_p(x,y) equal to x + y with probability p and min(x,y) with probability 1 - p. If leaf activations are Bernoulli, with probability q of being exactly zero, and z_n = P(X_n = 0) is the zero probability at tree level n, independence of sibling subtrees gives z_(n+1) = p z_n^2 + (1-p)(2 z_n - z_n^2) = 2(1-p)z_n + (2p-1)z_n^2. The fixed points are z = 0 and z = 1. The critical point is p_c = 1/2; at criticality z_n = q for every depth n. For p > 1/2, zero mass decreases toward zero, whereas for p < 1/2, it increases toward one.

Implementation notes

Integration point: implement a binary-tree or hierarchical feature block in an MLP, CNN pyramid, or graph pooling module. For two feature vectors u and v in the nonnegative orthant, sample a Bernoulli gate B with parameter p and compute w = B times (u + v) plus (1-B) times min(u,v), coordinatewise. Follow each merge with normalization and a learned projection if the feature dimension changes. Pseudocode: initialize nonnegative leaf features; at every tree level pair siblings; sample B for each pair or channel; calculate the stochastic merge; record the fraction of exact zeros; continue to the root; optimize the task loss with straight-through sampling or ordinary autodiff through the selected branch. Compute from the paper the recursion and critical value p_c = 1/2. Estimate empirically the effect of finite width, sibling correlations, normalization, and gradient routing. First experiment: train a four- to eight-level tree MLP on MNIST or CIFAR-10, comparing sum pooling, min pooling, and fixed p values 0.25, 0.4, 0.5, 0.6, and 0.75. Measure zero fraction z_n at every level, gradient norm by depth, and accuracy. The quantitative prediction is that at p = 0.5, z_n remains approximately equal to the leaf zero rate q; for p above 0.5 it decreases, and for p below 0.5 it increases. Fit the observed recursion and require the estimated transition to be within 0.05 of p = 0.5.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a NumPy implementation of the stochastic min-plus binary-tree merge and verified the claimed mechanism. Monte Carlo matched the exact zero-mass recursion with maximum absolute error 0.00186; the fitted transition was p=0.500000, and depth/q sweeps confirmed critical invariance at p=0.5 plus convergence toward one below and zero above criticality. The toy classifier comparison did not show a general task win: sum/min/critical-stochastic accuracies were 0.498/0.604/0.542, so the positive result is specifically the predicted absorbing-state mechanism, not downstream accuracy.

Agent confidence: 9/10

Baseline: Deterministic sum pooling accuracy: 0.498; deterministic min pooling accuracy: 0.604.
Idea: Critical stochastic p=0.5 accuracy: 0.542. At q=0.37, p=0.25 observed zero fractions across depths [0,1,2,4,8] were [0.3700,0.4858,0.6082,0.8300,0.9853], versus predictions [0.3700,0.4866,0.6115,0.8287,0.9850]. For p=0.75, observed values were [0.3700,0.2540,0.1542,0.0488,0.0038], versus predictions [0.3700,0.2535,0.1588,0.0503,0.0035]. The fitted transition was p=0.500000.

Limitations: The downstream comparison is a small NumPy toy classifier rather than a full MNIST/CIFAR MLP or CNN. It does not measure gradient routing, learned projections, normalization effects, FLOPs, or finite-width feature correlations; the exact recursion check uses independent Bernoulli zero states, where its assumptions hold by construction.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.12295) Sep 2, 2026

Ahead of the baseline (−2.4%) but not statistically significant (p=0.2052, wins 5 / 8 paired seeds).

Benchmark: Vision (image classification)
Model: cnn_small
Paired seeds: 8
Baseline mean: 0.7975
Idea mean: 0.7787
Effect (Δ): -0.0188 (−2.4%; negative = idea better)
Wins: 5 / 8 paired seeds
p-value: 0.2052 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±3.8%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 6 configs

Limitations:

Only the registered vision track was tested, with 400 training examples, 200 test examples, and 8 epochs. Deeper CNNs, alternative pooling baselines, larger CIFAR subsets, and detailed gradient-routing diagnostics were not tested.

How to run: python3 bench_experiment.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 9/10

The full registered vision benchmark completed with a matched CNN system: deterministic max-pooling baseline versus stochastic min-plus spatial pooling. The idea achieved lower mean test error (0.77875 versus 0.79750; paired delta -0.01875), but the paired permutation p-value was 0.20525, so it was not a significant win. The trained-model mechanism signature at p=0.5 was also not confirmed, with mean absolute prediction error 0.15978.

Baseline: Tuned deterministic max-pooling CNN, lr=0.003 and weight_decay=0.0; 8-seed error mean 0.79750.
Idea: Best stochastic min-plus CNN, p=0.5, lr=0.003 and weight_decay=0.0; 8-seed error mean 0.77875; paired delta -0.01875; p=0.20525.

Benchmark result