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Synthetic mixers for the published MQAR evidence.
Structure follows the described synthetic setup: a tiny transformer
(d_model = 64, 2 heads, < 100K matrix parameters of "intelligence") whose
sequence mixer is swapped between
* μLattice -- growing 2D lattice of gated-DeltaNet cells
* Attention -- causal softmax attention, no positional encodings
* GatedDeltaNet -- ONE fixed-size state per head (cell count capped at 1:
the recipe's own failure rule calls this Gated DeltaNet,
not μLattice)
Published for the synthetic mixer: a short conv1d over q,k,v before the mixer.
"""
from __future__ import annotations
import math
from dataclasses import dataclass
import torch
import torch.nn as nn
import torch.nn.functional as F
from .autoscan import AdaptiveScan
from .sparse_scan import SparseLatticeScan
class ShortConv(nn.Module):
"""Depthwise causal conv1d of width 4 (published: short conv1d on q,k,v)."""
def __init__(self, d: int, width: int = 4):
super().__init__()
self.width = width
self.conv = nn.Conv1d(d, d, width, groups=d)
# std = 1/sqrt(width) keeps the conv output at unit scale. A 0.02 init
# (the usual "small output layer" default) attenuates the signal ~50x, and
# because the address head reads the POST-conv activations that
# collapsed the address spread: 1.15 cells/token instead of ~9.
nn.init.normal_(self.conv.weight, std=1.0 / math.sqrt(width))
nn.init.zeros_(self.conv.bias)
def forward(self, x): # (B,T,d)
y = F.pad(x.transpose(1, 2), (self.width - 1, 0))
return self.conv(y).transpose(1, 2)
@dataclass
class MixerCfg:
d_model: int = 64
n_heads: int = 2
d_k: int = 32
d_v: int = 32
learn_addr_scale: bool = False # False = fixed scale (PROVEN to bootstrap; the learnable
# scale shrinks and collapses the lattice)
bound_address: bool = True # True = tanh-bounded address. Measured: the unbounded
# linear address (False) breaks routing -- key/query
# addresses drift apart and held-out stays 0.000.
addr_scale: float = 8.0 # fixed address scale (tanh radius)
read_tie: bool = False # addr_q = addr_k.detach() + Waq_delta(xc): the read is
# anchored to the cell the same token writes to (it cannot
# drift off the answer); the delta starts at ZERO so the
# read begins exactly on the write cell.
retention_init: float | None = None
# bias added to the alpha-gate so retention starts near 1.
# None = legacy (alpha~0.5), which KILLS credit beyond ~30
# steps (tests/test_kernel.py::test_credit_decay_...);
# 9.0 -> alpha0~0.999, credit survives 130+ steps.
conv: bool = True
class MuLatticeMixer(nn.Module):
def __init__(self, cfg: MixerCfg):
super().__init__()
d, H, dk, dv = cfg.d_model, cfg.n_heads, cfg.d_k, cfg.d_v
self.cfg = cfg
self.conv = ShortConv(d) if cfg.conv else nn.Identity()
# (q1, k1) SPLIT -- required by the blog and by the released C, and missing
# from every earlier version of this file:
# "Keys write to a small neighborhood around their addresses, and queries
# read from a neighborhood around theirs."
# reference: phi(p[2h],..) is the WRITE address, phi(p[2H+2h],..) the READ
# address -- separate projections. Passing one address for both forces a
# token's read and write to the same cell, so at the answer position the
# separator overwrites the very value it is about to retrieve.
# NOTE: the address is SHARED across heads -- each cell holds one d_k x d_v
# DeltaNet state PER HEAD, as the blog states, so the cell (not the head) is the
# routing unit.
self.Wa = nn.Linear(d, 2) # k1: write address
self.Waq_addr = nn.Linear(d, 2) # q1: read address (untied path)
if cfg.read_tie:
self.Waq_delta = nn.Linear(d, 2)
nn.init.zeros_(self.Waq_delta.weight)
nn.init.zeros_(self.Waq_delta.bias)
self.Wq = nn.Linear(d, H * dk)
self.Wk = nn.Linear(d, H * dk)
self.Wv = nn.Linear(d, H * dv)
self.Wg = nn.Linear(d, 2 * H) # alpha, beta logits
if cfg.retention_init is not None:
# alpha for head h is Wg output feature h (view(B,T,2,H): gate*H + head);
# biasing only the alpha features leaves beta at its default init.
with torch.no_grad():
self.Wg.bias[0:H] += cfg.retention_init
self.Wo = nn.Linear(H * dv, d)
# NOT zero-init. With empty cells the read is zero, so a zero gate makes
# d(loss)/d(gate) = 0 and every mixer parameter is multiplied by a zero
# gate -- a deadlock: measured gates stuck at 0.002 after 1,200 steps and
# accuracy pinned at chance forever. A 1.0 init lets the mixer learn.
self.out_gate = nn.Parameter(torch.ones(1))
self.scan = AdaptiveScan(dk, dv, H)
nn.init.normal_(self.Wa.weight, std=1.0 / (cfg.d_model ** 0.5))
# CHOICE, and load-bearing: the address map. A tanh-bounded address
# (addr = scale*tanh(Wh)) confines coordinates to [-(scale+1), scale+1],
# i.e. a hard ceiling of (2*scale+3)^2 cells per sequence, and at init the
# addresses barely spread (+-1.3), so the partition is chaotic: CPU and MPS
# landed on 258 vs 19 cells for identical inputs, and the collapsed case
# overloads cells until gradients go inf. A linear address with a learned
# scale is unbounded and spreads from step 0.
self.addr_log_scale = nn.Parameter(
torch.full((2,), math.log(math.expm1(cfg.addr_scale))))
self.q_scale = nn.Parameter(torch.zeros(H)) # per-head softplus scale on q2
self.write_mask = None # optional (B,T,1,1) gate on writes, for diagnostics
# Address source: POST-conv (context-dependent). A pre-conv (token-identity)
# address was tested and REJECTED: the value token writes at its own cell, not
# the key's, breaking the value->key-cell induction (measured 0.000 held-out).
def forward(self, x, read_mode="all"):
B, T, _ = x.shape
H, dk, dv = self.cfg.n_heads, self.cfg.d_k, self.cfg.d_v
xc = self.conv(x)
if self.cfg.learn_addr_scale:
sc = F.softplus(self.addr_log_scale)
else:
sc = self.cfg.addr_scale
# (q1, k1) SPLIT: the write address k1 and the read address q1 are separate
# projections, as in the released C (phi(p[2h]) writes, phi(p[2H+2h]) reads).
# UNBOUNDED linear address (released C and the blog's "infinitely growing
# memory"): a tanh cap confines every key to (2*scale+1)^2 cells and makes
# the capacity claim impossible. The scale is learned (softplus), so it can
# grow when more distinct keys need more distinct cells.
if self.cfg.bound_address:
addr_k = sc * torch.tanh(self.Wa(xc))
addr_q = sc * torch.tanh(self.Waq_addr(xc)) if not self.cfg.read_tie \
else addr_k.detach() + self.Waq_delta(xc)
else:
addr_k = self.Wa(xc) * sc
addr_q = self.Waq_addr(xc) * sc if not self.cfg.read_tie \
else addr_k.detach() + self.Waq_delta(xc)
# CHOICE: L2-normalise the content key/query before the delta rule
# (standard in gated-delta implementations; keeps the recurrence bounded).
q2 = F.normalize(self.Wq(xc).view(B, T, H, dk), dim=-1) * F.softplus(self.q_scale).view(1, 1, H, 1)
kap = F.normalize(self.Wk(xc).view(B, T, H, dk), dim=-1)
v = self.Wv(xc).view(B, T, H, dv)
gates = self.Wg(xc).view(B, T, 2, H).permute(0, 1, 3, 2) # (B,T,H,2)
alpha = torch.sigmoid(gates[..., 0:1]) * 0.999 + 1e-4
beta = torch.sigmoid(gates[..., 1:2])
if self.write_mask is not None:
beta = beta * self.write_mask
out, n_cells = self.scan(addr_q, addr_k, q2, kap, v, alpha, beta, read_mode=read_mode)
self.n_cells = n_cells
y = self.Wo(out.reshape(B, T, H * dv))
return x + self.out_gate * y
class AttentionMixer(nn.Module):
def __init__(self, cfg: MixerCfg):
super().__init__()
d, H = cfg.d_model, cfg.n_heads
self.cfg = cfg
self.conv = ShortConv(d) if cfg.conv else nn.Identity()
self.Wqkv = nn.Linear(d, 3 * d)
self.Wo = nn.Linear(d, d)
self.out_gate = nn.Parameter(torch.ones(1))
self.head_dim = d // H
def forward(self, x, read_mode="all"):
B, T, _ = x.shape
q, k, v = self.Wqkv(self.conv(x)).chunk(3, dim=-1)
q = q.view(B, T, self.cfg.n_heads, self.head_dim).transpose(1, 2)
k = k.view(B, T, self.cfg.n_heads, self.head_dim).transpose(1, 2)
v = v.view(B, T, self.cfg.n_heads, self.head_dim).transpose(1, 2)
y = F.scaled_dot_product_attention(q, k, v, is_causal=True)
y = y.transpose(1, 2).reshape(B, T, -1)
return x + self.out_gate * self.Wo(y)
class GatedDeltaNetMixer(nn.Module):
"""One state per head, fixed size -- no lattice, cell count capped at 1."""
def __init__(self, cfg: MixerCfg):
super().__init__()
d, H, dk, dv = cfg.d_model, cfg.n_heads, cfg.d_k, cfg.d_v
self.cfg = cfg
self.conv = ShortConv(d) if cfg.conv else nn.Identity()
self.Wq = nn.Linear(d, H * dk)
self.Wk = nn.Linear(d, H * dk)
self.Wv = nn.Linear(d, H * dv)
self.Wg = nn.Linear(d, 2 * H)
self.Wo = nn.Linear(H * dv, d)
self.out_gate = nn.Parameter(torch.ones(1))
def forward(self, x, read_mode="all"):
B, T, _ = x.shape
H, dk, dv = self.cfg.n_heads, self.cfg.d_k, self.cfg.d_v
xc = self.conv(x)
# unit-norm content keys: without this the delta rule's erase term is
# ||k||^2 ~ d_k times too strong and the state never accumulates
q = F.normalize(self.Wq(xc).view(B, T, H, dk), dim=-1)
k = F.normalize(self.Wk(xc).view(B, T, H, dk), dim=-1)
v = self.Wv(xc).view(B, T, H, dv)
g = self.Wg(xc).view(B, T, 2, H).permute(0, 1, 3, 2)
alpha = torch.sigmoid(g[..., 0:1]) * 0.999 + 1e-4
beta = torch.sigmoid(g[..., 1:2])
S = torch.zeros(B, H, dv, dk, device=x.device, dtype=x.dtype)
outs = []
for t in range(T):
v_bar = alpha[:, t] * (S @ k[:, t].unsqueeze(-1)).squeeze(-1)
r = v[:, t] - v_bar
S = alpha[:, t].unsqueeze(-1) * S + beta[:, t].unsqueeze(-1) * (
r.unsqueeze(-1) * k[:, t].unsqueeze(-2))
outs.append((S @ q[:, t].unsqueeze(-1)).squeeze(-1))
out = torch.stack(outs, 1) # (B,T,H,dv)
self.n_cells = 1
return x + self.out_gate * self.Wo(out.reshape(B, T, H * dv))
class SwiGLU(nn.Module):
def __init__(self, d: int, mult: int = 2):
super().__init__()
self.w1 = nn.Linear(d, mult * d, bias=False)
self.w2 = nn.Linear(d, mult * d, bias=False)
self.w3 = nn.Linear(mult * d, d, bias=False)
def forward(self, x):
return self.w3(F.silu(self.w1(x)) * self.w2(x))
class Block(nn.Module):
def __init__(self, d: int, mixer: nn.Module, mlp_mult: int = 2):
super().__init__()
self.n1 = nn.RMSNorm(d)
self.mixer = mixer
self.n2 = nn.RMSNorm(d)
self.mlp = SwiGLU(d, mlp_mult)
def forward(self, x, read_mode="all"):
x = self.mixer(self.n1(x), read_mode=read_mode)
return x + self.mlp(self.n2(x))
class MQARModel(nn.Module):
def __init__(self, vocab: int, kind: str = "mulattice", n_layers: int = 2,
cfg: MixerCfg | None = None, mlp_mult: int = 2):
super().__init__()
cfg = cfg or MixerCfg()
self.cfg = cfg
self.kind = kind
self.emb = nn.Embedding(vocab, cfg.d_model)
mixers = {
"mulattice": MuLatticeMixer,
"attention": AttentionMixer,
"gdn": GatedDeltaNetMixer,
}
self.blocks = nn.ModuleList(
[Block(cfg.d_model, mixers[kind](cfg), mlp_mult) for _ in range(n_layers)]
)
self.norm = nn.RMSNorm(cfg.d_model)
self.head = nn.Linear(cfg.d_model, vocab, bias=False)
self.n_cells = 0
def forward(self, tokens, read_mode="all"):
x = self.emb(tokens)
for b in self.blocks:
x = b(x, read_mode=read_mode)
return self.head(self.norm(x))
def mixer_params(self) -> int:
return sum(p.numel() for n, p in self.named_parameters() if ".mixer." in n)
def memory_cells(self) -> int:
return max(getattr(b.mixer, "n_cells", 0) for b in self.blocks)
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