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I Dropped a Neural Net

Hyunwoo Park

TL;DR

Given unlabelled layers of a Residual Network and its training dataset, the exact ordering of the layers is recovered and stability conditions during training like dynamic isometry leave the product W_{\text{in}} W_{\text{in}} for correctly paired layers with a negative diagonal structure, allowing us to use diagonal dominance ratio as a signal for pairing.

Abstract

A recent Dwarkesh Patel podcast with John Collison and Elon Musk featured an interesting puzzle from Jane Street: they trained a neural net, shuffled all 96 layers, and asked to put them back in order. Given unlabelled layers of a Residual Network and its training dataset, we recover the exact ordering of the layers. The problem decomposes into pairing each block's input and output projections ($48!$ possibilities) and ordering the reassembled blocks ($48!$ possibilities), for a combined search space of $(48!)^2 \approx 10^{122}$, which is more than the atoms in the observable universe. We show that stability conditions during training like dynamic isometry leave the product $W_{\text{out}} W_{\text{in}}$ for correctly paired layers with a negative diagonal structure, allowing us to use diagonal dominance ratio as a signal for pairing. For ordering, we seed-initialize with a rough proxy such as delta-norm or $\|W_{\text{out}}\|_F$ then hill-climb to zero mean squared error.

I Dropped a Neural Net

TL;DR

Given unlabelled layers of a Residual Network and its training dataset, the exact ordering of the layers is recovered and stability conditions during training like dynamic isometry leave the product W_{\text{in}} W_{\text{in}} for correctly paired layers with a negative diagonal structure, allowing us to use diagonal dominance ratio as a signal for pairing.

Abstract

A recent Dwarkesh Patel podcast with John Collison and Elon Musk featured an interesting puzzle from Jane Street: they trained a neural net, shuffled all 96 layers, and asked to put them back in order. Given unlabelled layers of a Residual Network and its training dataset, we recover the exact ordering of the layers. The problem decomposes into pairing each block's input and output projections ( possibilities) and ordering the reassembled blocks ( possibilities), for a combined search space of , which is more than the atoms in the observable universe. We show that stability conditions during training like dynamic isometry leave the product for correctly paired layers with a negative diagonal structure, allowing us to use diagonal dominance ratio as a signal for pairing. For ordering, we seed-initialize with a rough proxy such as delta-norm or then hill-climb to zero mean squared error.
Paper Structure (32 sections, 1 theorem, 21 equations, 9 figures, 3 tables)

This paper contains 32 sections, 1 theorem, 21 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Puzzle
  3. Architecture
  4. Available Data
  5. Objective
  6. Pairing
  7. Diagonal Dominance Ratio
  8. Ordering
  9. Seed Ordering
  10. Delta-norm (data-dependent).
  11. $\|W_{\text{out}}\|_F$ (data-free).
  12. Near-Commutativity of Residual Blocks
  13. Bradley--Terry Ranking
  14. Transitivity violations.
  15. Bubble Repair
  16. ...and 17 more sections

Key Result

Proposition 1

If the block satisfies dynamic isometry pennington2017resurrecting in expectation, $\mathbb{E}_x[J_f(x)^\top J_f(x)] = I_d$, with non-trivial residual, and the ReLU activation probability is $\tfrac{1}{2}$ per unit (i.e. $\mathbb{E}_x[D(x)] = \tfrac{1}{2}I_h$), then $\mathop{\mathrm{tr}}\nolimits(W_

Figures (9)

  • Figure 1: Architecture. The 48-block ResNet. Each block applies input layer, ReLU, output layer, then a residual connection. The final layer produces the scalar output $\hat{y}$.
  • Figure 2: Dataset overview. Features and outputs are normalized.
  • Figure 3: Pairing via diagonal dominance. The diagonal dominance ratio achieves complete separation between correct pairs ($d \in [1.76, 3.28]$) and incorrect pairs ($d \in [0.00, 0.58]$), with a gap of $1.18$.
  • Figure 4: Correct vs. incorrect pairings. Top: Correctly paired matrices show negative diagonal structure. Bottom: Incorrect pairings with no structure.
  • Figure 5: All 48 product matrices from the original model, ordered by delta-norm $\delta_k$ (Section \ref{['sec:seed']}). Every block exhibits a negative diagonal structure. Earlier blocks (top-left) and later blocks (bottom-right). Traces range from $-13.5$ to $-7.4$. Compare with Figure \ref{['fig:toy_matrices']}.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Proposition 1: Negative diagonal structure
  • proof