tools/factory/specs/comba_backward.py

"""Spec for `comba-backward` — the training-side counterpart of comba-forward."""
import pathlib
import sys

sys.path.insert(0, str(pathlib.Path(__file__).resolve().parents[1]))
from spec import TaskSpec

# the chunked Comba forward, reused verbatim as the differentiable spec
_FWD = '''
CHUNK_SIZE = 64


def _comba_forward(q, k, v, p, beta, g, scale=None):
    """Chunked Comba forward in fp32 — differentiable; autograd through this defines the gradients."""
    B, T, H, K = q.shape
    BT = CHUNK_SIZE
    if scale is None:
        scale = K ** -0.5
    q, k, v, p = [rearrange(x.to(torch.float32), 'b (n c) h d -> b h n c d', c=BT) for x in (q, k, v, p)]
    beta, g = [rearrange(x.to(torch.float32), 'b (n c) h -> b h n c', c=BT) for x in (beta, g)]
    gc = g.cumsum(-1)                                    # cumulative log-decay INSIDE each chunk
    L = (gc[..., :, None] - gc[..., None, :]).exp()      # L[i, j] = exp(gc_i - gc_j) <= 1
    pb = p * beta[..., None]

    # linearise the chunk's rank-1 writes: u = (I + X)^-1 (beta*v - beta*exp(gc)*p @ S)
    tri0 = torch.triu(torch.ones(BT, BT, dtype=torch.bool, device=q.device), diagonal=0)
    M = (-((pb @ k.transpose(-1, -2)) * L)).masked_fill(tri0, 0)
    eye = torch.eye(BT, dtype=torch.float32, device=q.device)
    A = torch.linalg.solve_triangular(eye - M, eye.expand(*M.shape[:-2], BT, BT), upper=False)
    u = A @ (v * beta[..., None])
    w = A @ (pb * gc[..., None].exp())

    tri1 = torch.triu(torch.ones(BT, BT, dtype=torch.bool, device=q.device), diagonal=1)
    S = q.new_zeros(B, H, K, v.shape[-1])                # the (K, V) state carried across chunks
    outs = []
    for i in range(T // BT):
        q_i, k_i, gc_i = q[:, :, i] * scale, k[:, :, i], gc[:, :, i]
        attn = ((q_i @ k_i.transpose(-1, -2)) * L[:, :, i]).masked_fill(tri1, 0)
        u_i = u[:, :, i] - w[:, :, i] @ S
        outs.append((q_i * gc_i[..., None].exp()) @ S + attn @ u_i)
        last = gc_i[..., -1]
        S = S * last[..., None, None].exp() + \\
            (k_i * (last[..., None] - gc_i)[..., None].exp()).transpose(-1, -2) @ u_i
    return rearrange(torch.stack(outs, dim=2), 'b h n c d -> b (n c) h d')


def comba_backward(q, k, v, p, beta, g, do, scale=None):
    """Comba backward — the baseline runs the chunked forward under autograd."""
    ins = [x.detach().clone().requires_grad_(True) for x in (q, k, v, p, beta, g)]
    o = _comba_forward(*ins, scale=scale)
    return torch.autograd.grad(o, ins, do.float())
'''

SPEC = TaskSpec(
    name="comba-backward",
    title="Write a fast Comba BACKWARD kernel",
    blurb=("The training-side counterpart of Comba (2025): the gated delta rule with a DECOUPLED read key — "
           "the state correction is read with `p` while the write still uses `k`, so the within-chunk "
           "transform is not the symmetric one DeltaNet's WY trick relies on, and neither is its transpose in "
           "the backward. The reference obtains the six gradients by running the chunked fp32 forward under "
           "autograd, including the triangular solve; a fused backward recomputes the chunk transform instead "
           "and carries the reverse state scan in registers."),
    keywords=["mle", "kernel-generation", "comba", "linear-attention", "delta-rule", "backward", "gpu"],
    module="comba_bwd.py",
    func="comba_backward",
    signature="comba_backward(q, k, v, p, beta, g, do, scale=None)",
    returns_doc="""Comba backward.

Args:
    q, k, p: (B, T, H, K) bfloat16 — queries, write-keys, read-keys.
    v:       (B, T, H, V) bfloat16 — values.
    beta:    (B, T, H)    bfloat16 — delta-rule step size in (0, 1).
    g:       (B, T, H)    float32  — per-step log-decay (<= 0); exp(g) is the gate.
    do:      (B, T, H, V) bfloat16 — incoming gradient w.r.t. the forward output.
    scale: float or None — query scale; None means K ** -0.5.

Returns:
    (dq, dk, dv, dp, dbeta, dg) with shapes
    (B,T,H,K), (B,T,H,K), (B,T,H,V), (B,T,H,K), (B,T,H), (B,T,H).""",

    reference_imports="import torch\nfrom einops import rearrange",
    reference_src=_FWD,
    make_inputs_src='''
def _mk(B, T, H, K, V, seed):
    import torch.nn.functional as F
    gen = torch.Generator(device="cuda").manual_seed(seed)
    q = torch.randn(B, T, H, K, device="cuda", dtype=torch.bfloat16, generator=gen)
    k = F.normalize(torch.randn(B, T, H, K, device="cuda", generator=gen), dim=-1).to(torch.bfloat16)
    v = torch.randn(B, T, H, V, device="cuda", dtype=torch.bfloat16, generator=gen)
    p = F.normalize(torch.randn(B, T, H, K, device="cuda", generator=gen), dim=-1).to(torch.bfloat16)
    beta = torch.rand(B, T, H, device="cuda", generator=gen).sigmoid().to(torch.bfloat16)
    g = (F.logsigmoid(torch.randn(B, T, H, device="cuda", generator=gen)) / 4.0).to(torch.float32)
    do = torch.randn(B, T, H, V, device="cuda", dtype=torch.bfloat16, generator=gen)
    return q, k, v, p, beta, g, do
''',
    flops_src='''
def canonical_work(B, T, H, K, V, C=64):
    """FLOPs attributed to one Comba BACKWARD, from the SHAPE ALONE (chunked form, chunk length C=64).

    Per (b, h) and per token the forward costs the chunk transform's two K-contractions and the intra-chunk
    score matrix (2*C*2*K), the application to the corrected values (2*C*V), and three (K, V)-state products
    — the correction read, the readout, and the state update (6*K*V). The backward is credited the standard
    2x the forward. The triangular solve and the elementwise gate work are not counted.
    """
    return 2 * (B * H * T * (2 * C * (2 * K + V) + 6 * K * V))
''',
    flops_formula="2 * ( B*H*T * (2*C*(2*K + V) + 6*K*V) )        with C = 64   # 2x the forward",

    metric="TFLOP/s",
    compare="tuple",
    tuple_names=("dq", "dk", "dv", "dp", "dbeta", "dg"),
    tol=2e-2,
    shape_names=("B", "T", "H", "K", "V"),
    grader_shapes=[(8, 4096, 32, 128, 128), (16, 2048, 32, 128, 128), (12, 4096, 32, 64, 128),
                   (16, 4096, 16, 128, 128), (12, 4096, 16, 128, 128)],
    measure_shapes=[(8, 2048, 32, 128, 128), (12, 2048, 32, 128, 128), (8, 4096, 16, 128, 128),
                    (16, 2048, 32, 64, 128), (10, 4096, 32, 128, 128)],
    measure_quick_shapes=[(2, 1024, 16, 128, 128), (4, 1024, 8, 128, 128), (2, 2048, 8, 64, 128)],
    correct_shapes=[(1, 512, 4, 128, 128), (2, 1024, 4, 64, 64), (1, 1024, 8, 128, 64),
                    (2, 256, 4, 64, 128)],

    spec_md="""The forward, per batch `b` and head `h`, with a recurrent state `S` of shape `(K, V)`
initialised to zero, for `t = 0 … T-1`:

```
S   = exp(g_t) * S                       # scalar forget gate on the whole state
u_t = beta_t * ( v_t - S^T p_t )         # correction, read with p (NOT with k), AFTER the decay
S   = S + k_t u_t^T                      # rank-1 write, using k
o_t = S^T (scale * q_t)                  # readout
```

You must return the gradients of that forward with respect to `q, k, v, p, beta, g`, given the incoming
gradient `do` of the loss with respect to `o`. `scale` defaults to `K ** -0.5`.

The one thing that distinguishes Comba from a gated delta rule is that the state is **read with `p` and
written with `k`**. In DeltaNet those are the same vector, which makes the within-chunk transform
`(I - tril(beta k kᵀ))⁻¹` symmetric-ish and lets the WY/UT trick apply directly; here the corresponding
matrix is built from `p` against `k`, is **not** symmetric, and its transpose — which is what the backward
needs — is a *different* triangular system.

`/app/reference.py` gives you `_comba_forward` — the same forward in its equivalent chunked form (chunk
length 64: the chunk transform obtained by a triangular solve, then a sequential state scan) — and obtains
the gradients by running it under **autograd**. That is the specification, and it is what torch gives you
for free, but it replays the entire chunked graph, differentiates through the triangular solve, and
materialises every chunk intermediate in HBM.

You may reach the same gradients any way you like: derive and fuse the analytic backward, recompute
intermediates instead of storing them, use a different chunk length, or restructure the reverse scan. Only
the returned numbers are specified.""",

    contract_md="""| arg | shape | dtype | meaning |
|-----|-------|-------|---------|
| `q` | `(B, T, H, K)` | `bfloat16` | queries |
| `k` | `(B, T, H, K)` | `bfloat16` | **write** keys (L2-normalised along `K`) |
| `v` | `(B, T, H, V)` | `bfloat16` | values |
| `p` | `(B, T, H, K)` | `bfloat16` | **read** keys (L2-normalised along `K`) |
| `beta` | `(B, T, H)` | `bfloat16` | delta-rule step size, in `(0, 1)` |
| `g` | `(B, T, H)` | `float32` | per-step log-decay, `<= 0` (`exp(g)` is the gate) |
| `do` | `(B, T, H, V)` | `bfloat16` | incoming gradient w.r.t. the forward output `o` |
| `scale` | scalar | `float` or `None` | query scale; `None` means `K ** -0.5` |

Note the argument order is `(q, k, v, p, beta, g, do)` — `p` comes after `v`, and `do` is last.

**Return** a 6-tuple `(dq, dk, dv, dp, dbeta, dg)` **in that order**, with shapes
`(B,T,H,K)`, `(B,T,H,K)`, `(B,T,H,V)`, `(B,T,H,K)`, `(B,T,H)`, `(B,T,H)`.
Each may be `bfloat16` or `float32`.

**All six are graded.** Getting five of six right scores **0**. `dbeta` and `dg` are per-`(b, t, h)`
scalars — they are reductions over the whole `(K, V)` state, not per-channel vectors.

All tensors are CUDA and contiguous. `T` is a multiple of 64. No initial state, no state gradient, no
variable-length packing, no GQA. Treat all inputs as read-only.""",

    regime_md="""**Shape regime you are graded in** (the exact grader sizes are *not* disclosed): `B` in 8–16,
`T` in 2048–4096, `H` in 16–32, `K` in {64, 128}, `V` = 128, with `B*H >= 128`. The batch/head product is
large and `T` moderate on purpose: the sequential dimension is short enough that the scan does not starve the
GPU, and wide enough that every SM has a `(b, h)` pair to work on. Write a **general** kernel.""",

    correctness_md="""**All six** gradients must match the reference (evaluated in fp32) within **relative
Frobenius error `2e-2`** at every graded shape, including the timed ones.""",

    perf_md="""The backward of a chunked delta-rule scan is memory-bound when written through autograd: the
chunk transform `A`, the linearised values `u`, the correction operand `w`, the per-chunk `u_i` and the state
all stay alive for the reverse pass, and torch also differentiates the triangular solve as a *second*
triangular solve against the same matrix.

A fused backward instead **recomputes** the chunk-local quantities from `q, k, v, p, beta, g` during the
reverse scan, keeps the reverse state `dS` in registers/shared memory across chunks, and uses bf16 tensor
cores for the chunk matmuls with fp32 accumulation. The triangular solve should be replaced by the forward
substitution it stands for and fused into the same kernel; its adjoint is a *backward* substitution against
the transposed system, which is the same primitive run in the other direction — a matrix inverse never has
to be materialised.

Because the read key `p` and the write key `k` are different, the two sides of that system are different, so
`dp` and `dk` come out of separate contractions and cannot be merged. `dbeta` and `dg` are full reductions
over the `(K, V)` state; fold the scalar decay into the matmul operands (`exp(gc_i - gc_j)` factorises into
per-row and per-column scalings) rather than materialising `exp(g)` tensors.""",

    precision_md="""All inputs and outputs are **bfloat16** (the gate `g` is `float32` because it is a log) —
this is an LLM-training kernel and that is the precision it runs at in production. Your kernel is expected to
do its matmuls on **bf16 tensor cores with fp32 accumulation**. **fp8** is acceptable anywhere you can still
hold the tolerance.

For calibration, a faithful bf16 implementation of this backward lands around **4e-3 – 5e-3** relative error
on every one of the six gradients — roughly 4x inside the `2e-2` gate — and that number is **flat in `T`**. A
wrong algorithm misses by 0.2 or more.

The **gradient reductions drift first**. `dbeta` and `dg` are contracted over the entire `(K, V)` state at
every step, and the reverse state `dS` is accumulated along the whole sequence — all of them need **fp32
accumulators**. If `dbeta`/`dg` sit at 1e-2 while `dq`/`dk`/`dv`/`dp` are at 4e-3, that is a bf16
accumulator, not noise. The chunk-local triangular solve is also worth keeping in fp32: it is a sequential
substitution, so rounding there compounds across the chunk.

Do **not** infer from the reference that fp32 compute is wanted. It runs in fp32 purely to be a stable
numerical *specification*, and its speed has no bearing on your score.""",
).validate()