From 77557a84f83d7a3977a0e50a7be2c5890fbe25b7 Mon Sep 17 00:00:00 2001 From: sergimichi Date: Sat, 4 Apr 2026 11:15:46 +0200 Subject: [PATCH 1/5] Add LegendreGPT non-record submission --- .../2026-03-31_LegendreGPT/README.md | 150 ++ .../final_model.mixed.ptz | Bin 0 -> 15630808 bytes .../2026-03-31_LegendreGPT/requirements.txt | 6 + .../2026-03-31_LegendreGPT/submission.json | 15 + .../2026-03-31_LegendreGPT/train.log | 1872 +++++++++++++++++ .../train_gpt_legendre.py | 1639 +++++++++++++++ 6 files changed, 3682 insertions(+) create mode 100644 records/track_non_record_16mb/2026-03-31_LegendreGPT/README.md create mode 100644 records/track_non_record_16mb/2026-03-31_LegendreGPT/final_model.mixed.ptz create mode 100644 records/track_non_record_16mb/2026-03-31_LegendreGPT/requirements.txt create mode 100644 records/track_non_record_16mb/2026-03-31_LegendreGPT/submission.json create mode 100644 records/track_non_record_16mb/2026-03-31_LegendreGPT/train.log create mode 100644 records/track_non_record_16mb/2026-03-31_LegendreGPT/train_gpt_legendre.py diff --git a/records/track_non_record_16mb/2026-03-31_LegendreGPT/README.md b/records/track_non_record_16mb/2026-03-31_LegendreGPT/README.md new file mode 100644 index 0000000000..3c99023ed9 --- /dev/null +++ b/records/track_non_record_16mb/2026-03-31_LegendreGPT/README.md @@ -0,0 +1,150 @@ +# LegendreGPT + +LegendreGPT generates all transformer layer weights from a small set of Legendre polynomial coefficients, compressing 22 middle layers into 6 coefficient matrices per weight type. As far as I know, this is the first time anyone has applied orthogonal polynomial weight parameterization to transformer language models. The greatest news is that it learns. + +## Result + +| Metric | Value | +|--------|-------| +| Pre-quantization val_bpb | 1.2079 | +| Post-quantization val_bpb (mixed INT8/INT7+LZMA) | 1.2266 | +| Compressed model size | 15.70 MB | +| Architecture | dim=512, 24L (2 groups), g5/2, GQA 8/4 | +| Training | 60k steps, 80 shards, 1x RTX 5090 (~27h) | + +## How It Works + +Each weight matrix in the transformer is a function of depth: + +``` +W(layer_l) = sum_{k=0}^{K-1} C_k * P_k(t_l) +``` + +`P_k` are Legendre polynomials. `t_l` maps layer index to [-1, 1]. `C_k` are learned matrices. With K=6 (degree 5), I specify 11 unique layers from 6 coefficient matrices per weight type — and I have two independent groups of 11. + +Think of it like an equalizer: the polynomials are fixed frequencies (constant, linear, quadratic...), and the coefficients are the sliders. Training only adjusts the sliders, never the frequencies. + +**Why Legendre and not monomials?** Orthogonality. Monomials (1, t, t^2...) become catastrophically ill-conditioned at higher degrees. Legendre polynomials stay well-behaved. NANODE (Massaroli et al., 2020) showed this matters for Neural ODEs. I confirmed it matters for transformers too. + +## Architecture + +``` +[Factorized Embedding] <- ALBERT-style, 1024 -> 128 -> 512 +[Independent Block 0] <- own weights +[Legendre Group A: 11 layers] <- coefficients A (degree 5 attn, 2 FFN) +[Legendre Group B: 11 layers] <- coefficients B (independent) +[Independent Block 23] <- own weights +[RMSNorm -> Tied Logit Head] +``` + +The 2-group split is important. With 1 group, each coefficient affects all 22 layers. If layer 5 wants the coefficient to go up but layer 15 wants it to go down, the gradients cancel and nothing moves. With 2 groups, each coefficient only fights with ~11 layers instead of 22. + +Each layer also has cheap independent scalars: attention scale, MLP scale, residual mixing ratio, query gain, and RMSNorm params. These cost < 0.05 MB total and let each layer fine-tune its behavior without defeating the compression. + +Other details: GQA (8 heads, 4 KV heads), ReLU^2 MLP at 3x dim, RoPE, logit soft-capping at 30. + +## Parameter Budget + +| Component | Params | INT8 (MB) | +|-----------|--------|-----------| +| Factorized embedding | 196,608 | 0.20 | +| Sandwich block (first) | 2,361,352 | 2.36 | +| Legendre Group A coefficients | 9,437,250 | 9.44 | +| Legendre Group B coefficients | 9,437,250 | 9.44 | +| Per-layer lightweight params | 45,232 | 0.05 | +| Sandwich block (last) | 2,361,352 | 2.36 | +| **Total** | **23,839,044** | **15.70 MB compressed** | + +Compression: mixed precision quantization (INT8 for Legendre orders 0-1, INT7 for orders 2-5 and sandwich blocks) + LZMA. + +## Training + +- **Muon optimizer** for all 2D weight matrices, Adam for embeddings and scalars +- **Per-order learning rates:** 1.1x higher per polynomial order. Order 0 at 0.025, order 5 at 0.040. Higher orders capture finer detail and need more push. +- **Linear LR decay** from 0.2 to 0.0 over 60k steps +- **Momentum cooldown:** Muon momentum decays from 0.95 to 0.05 over steps 10k-60k. Discovered accidentally when a checkpoint resume zeroed the momentum buffers and the model learned 8x faster. High momentum dampens updates too much near convergence. +- **Batch size:** 393,216 tokens. Legendre coefficients serve 11+ layers and benefit from clean gradient estimates. +- **Data:** 80 FineWeb sp1024 shards (~8B tokens) +- **Hardware:** 1x RTX 5090, ~27 hours total + +## Training Curve + +| Step | val_bpb | lr_mul | +|------|---------|--------| +| 1,000 | 1.48 | 0.197 | +| 5,000 | 1.35 | 0.183 | +| 10,000 | 1.28 | 0.167 | +| 20,000 | 1.24 | 0.133 | +| 30,000 | 1.22 | 0.100 | +| 40,000 | 1.21 | 0.067 | +| 50,000 | 1.21 | 0.033 | +| 60,000 | 1.2054 | 0.000 | + +## What I Learned + +**Dimension matters most.** For a fixed byte budget, bumping dim from 512 to 640 gave 0.02 BPB. Raising polynomial degree from g5/2 to g8/4 gave ~0.01. Wider layers beat more variation between them. + +**2 groups beat 1 group.** Each Legendre coefficient affects all layers in its group. With 1 group of 22 layers, gradients from different layers partially cancel. Splitting into 2 groups of 11 halves the cancellation and improves convergence at the same parameter cost. + +**Wrap doesn't help.** Tested circular weight topology (W = W - round(W)) with standard init, 8x init, and wrap-aware gradient modifications. Smooth weights win consistently — the continuity prior between adjacent layers is correct. + +**LoRA is less efficient than higher degree.** Per-layer low-rank corrections (W = W_legendre + A*B) at rank 8 underperform simply raising Legendre degree. g6/3 without LoRA beats g5/2 + LoRA r8 — the bytes are better spent on polynomial expressivity. + +**Momentum cooldown helps late training.** High momentum (0.95) dampens updates too much near convergence. Decaying to 0.05 in the second half of training allows finer adjustments when the model is close to a good minimum. + +**Larger batches help disproportionately.** Going from 262k to 393k tokens/batch improved convergence visibly. + +**Mixed precision quantization is key.** Legendre order 0-1 (constant and linear components) carry most of the weight information and need INT8 precision. Higher orders (finer detail) tolerate INT7. This gives near-INT8 quality at near-INT7 size. + +## Experiments + +| Config | Steps | val_bpb | Takeaway | +|--------|-------|---------|----------| +| dim=256, 8L, g5/2, 1 shard | 3,000 | 1.67 | Architecture works | +| dim=512, 24L, g5/2, 1 shard | 2,000 | 1.39 | Full model, 8.6 MB | +| dim=640, 24L, g6/3, 80 shards | 30,000 | 1.214 | Best BPB but 18.4 MB | +| dim=576, 24L, g6/3, 80 shards | 30,000 | 1.22 | Fits budget, tight | +| dim=512, 2-group g5/2, wrap | 3,000 | 1.70 | Wrap hurts | +| dim=640, g5/2 + LoRA r8 | 5,000 | 1.30 | LoRA < higher degree | +| **dim=512, 2-group g5/2, 80 shards** | **60,000** | **1.2054** | **Final submission** | + +## Related Work + +**NANODE** (Massaroli et al., NeurIPS 2020) used Legendre polynomials to parameterize Neural ODE weights for PDE surrogate modeling. LegendreGPT extends this idea to transformer language models. + +**ALBERT** (Lan et al., 2020) shares identical weights across all layers. LegendreGPT generalizes this: degree 0 (single constant coefficient) is exactly ALBERT. Higher degrees let layers diverge smoothly. + +**Subformer** (Reid et al., 2021) showed that sandwich-style sharing (independent first/last layers, shared middle) works better than uniform sharing. I use the same structure. + +**PR #831 in this repo** ("Why Novel Architectures Fail at 16MB") found that 6 tested innovations failed because they couldn't co-optimize throughput and quantization. LegendreGPT avoids this: weight generation happens once per layer, and the resulting weights are standard dense matrices. torch.compile, tensor cores, and INT8 quantization all work unchanged. The novelty is in how weights are stored, not in the forward pass. + +## What I'd Try Next + +- **2D compression:** Legendre polynomials for the depth axis, DCT for the width axis. Could push dim to 1024+ in 16 MB. +- **Learned basis:** PCA from a pretrained model's weights instead of fixed Legendre. The optimal basis probably isn't polynomial. +- **Low-rank high orders:** Full-rank for orders 0-2, low-rank for orders 3+. More expressivity per byte. +- **Learnable layer positions:** Let the model learn optimal spacing in [-1, 1] instead of uniform. +- **Proper 8xH100 run:** All my runs were on a single RTX 5090. The competition target is 8xH100 in 10 minutes. Larger batch, fewer steps, different schedule. + +## Reproducibility + +```bash +git clone https://github.com/openai/parameter-golf.git +cd parameter-golf +pip install sentencepiece huggingface-hub datasets +python3 data/cached_challenge_fineweb.py --variant sp1024 --train-shards 80 + +# Copy train_gpt_legendre.py to parameter-golf/ + +MODE=full RUN_ID=legendregpt \ + LEGENDRE_GROUPS=2 \ + NUM_VIRTUAL_LAYERS=24 MODEL_DIM=512 \ + LEGENDRE_DEGREE_ATTN=5 LEGENDRE_DEGREE_FFN=2 \ + ITERATIONS=60000 TRAIN_BATCH_TOKENS=393216 \ + MAX_WALLCLOCK_SECONDS=0 LR_SCHEDULE=linear \ + python3 train_gpt_legendre.py +``` + +## Author + +**Sergio Cernuda Cueto** -- AI Engineer at Samsung Zhilabs, CS student at Universidad Carlos III de Madrid (Spain). diff --git a/records/track_non_record_16mb/2026-03-31_LegendreGPT/final_model.mixed.ptz b/records/track_non_record_16mb/2026-03-31_LegendreGPT/final_model.mixed.ptz new file mode 100644 index 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