MetalGrad

Differentiable Metal kernels for Apple Silicon training.

Custom Metal kernels written via mx.fast.metal_kernel don't auto-generate backward passes — calling mx.grad on one raises Primitive::vjp Not implemented for CustomKernel. That makes every hand-written fast kernel inference-only, which is fine for serving but blocks training.

MetalGrad fixes this by pairing a fast forward (Metal kernel, mx.fast.* call, or pure mx ops) with an explicit .vjp written in mx ops. The result behaves like a regular MLX op under mx.grad.

import mlx.core as mx
from metalgrad import differentiable

@differentiable
def my_op(x, w):
    return _my_metal_kernel(x, w)        # whatever fast forward you have

@my_op.vjp
def _(primals, cotangent, output):
    x, w = primals
    gy = cotangent
    gx = mx.matmul(gy, mx.swapaxes(w, -1, -2))     # backward in mx ops
    gw = mx.matmul(mx.swapaxes(x, -1, -2), gy)
    return gx, gw

def loss(x, w): return mx.sum(my_op(x, w) ** 2)
gx, gw = mx.grad(loss, argnums=(0, 1))(x, w)        # works

The forward stays fast (custom Metal). The backward stays correct (plain mx ops, can be slower than the forward — that's fine for training). You ship one op that works in both inference and training without losing autograd.

Pre-built ops

from metalgrad.ops import (
    # core layers (norms + linear + conv + attention)
    matmul, conv1d, conv2d, depthwise_conv2d,
    rms_norm, layer_norm, attention,
    # DiT / FCDM modulation
    adaln,
    # activations
    swiglu, geglu, squared_relu,
    # losses
    cross_entropy, mse, kl_div_logits,
    # FFN blocks (convenience APIs)
    swiglu_ffn, stack_gate_up,
    # RoPE variants (freqs builders + thin wrappers around mx.fast.rope)
    rope_standard, rope_linear_pi, rope_ntk_aware, rope_yarn, rope_llama3,
    # training-side fused kernels (not part of the autograd graph)
    adamw_step, ema_update, clip_grad_norm,
    # HuBERT / data2vec / sylber speech stack
    group_norm,
    l1_loss, smooth_l1_loss, cosine_loss, l2_normalize,
    sinusoidal_pe,
)

op	Forward	Backward	Technique
rms_norm	2.6–3.8×	1.00×	TG-cooperative simd_sum, register-tiled, mx.compile-fused backward
layer_norm	3.0–6.8×	1.15×	same forward pattern + canonical fused backward ((1/C)·inv·(C·gn − Σgn − ñ·Σgn·ñ))
swiglu	2.33×	—	mx.compile-fused silu(a) * b
geglu	6.51×	—	mx.compile-fused gelu(a) * b (large because GELU itself fuses many subexprs)
squared_relu	1.76×	—	mx.compile-fused max(x, 0)²
cross_entropy	0.65ׇ	1.76×	one-pass online softmax (Welford-style) forward + streaming fused-grad backward; fwd + bwd combined 1.88×
kl_div_logits	3.35×	(combined fwd+bwd 3.39×)	two parallel online-softmax passes (pred & target) in one kernel + streaming fused-grad backward writing (softmax(pred) − softmax(target))/N directly
mse	parity	parity	mx.compile-fused mean((p − t)²) with closed-form 2(p − t)/N VJP — useful as named API for regression / diffusion training
adaln	1.8–6.0×	(via mx.vjp)	affineless LayerNorm + (1 + scale[batch, c]) · norm + shift[batch, c] — DiT / FCDM block's per-batch modulation. Win grows with model size: DiT-XL (B=4, T=1024, C=1152) hits 5.95×
adamw_step	3.80× (per-tensor)	— (no VJP; optimizer kernel)	fused AdamW: (param, grad, m, v) → (new_param, new_m, new_v) in one pass. Every training step uses this
ema_update	parity	—	α · ema + (1 − α) · param. mx.compile already fuses; shipped as named API
clip_grad_norm	(small bench win)	—	global L2 norm over a list of grad tensors + scale. One pass per grad tensor
group_norm	1.3-4.0×	(via mx.vjp)	TG-cooperative reduction over C/num_groups channels per (B, T) row. Used by HuBERT / WavLM / data2vec feature extractors. Requires C/num_groups divisible by 32 (fallback to mx otherwise)
l1_loss	1.48×	0.99×	`mean(
smooth_l1_loss β=1	1.01×	0.98×	Huber loss with quadratic region — parity (mx.where chain is already optimal)
smooth_l1_loss β=0	1.23×	—	data2vec's loss_beta=0 default (= 0.5·MSE). mx.compile fuses
cosine_loss	1.04×	1.02×	1 − mean(cos_sim(a, b)). Parity — mx.compile boundary on the chain
l2_normalize	0.95-1.04×	—	unit L2 norm along last axis. Parity (noise)
sinusoidal_pe(T, D)	builder, no autograd	—	classical absolute positional encoding builder. Fills the mx.fast.rope gap for absolute PE
matmul, conv1d, conv2d, depthwise_conv2d	1.0×	1.0×	thin re-exports — mx is already MPSGraph-tuned with simdgroup_matrix MMA
attention	1.0×	1.0×	thin re-export — mx.fast.scaled_dot_product_attention is already FlashAttention-style (1.6–3.9× over manual SDPA)
rope_standard / rope_linear_pi / rope_ntk_aware / rope_yarn / rope_llama3	2.9× (via mx.fast.rope)	autograd via mx	thin wrappers around mx.fast.rope; each variant supplies a different frequency table to the same underlying fused rotation kernel

Benched on M3 Pro, FP32. Norm ops at (4, 512, 1024); activations at (4, 512, 2048); losses at (N=1024, V=32000). Every op passes gradcheck with rel_err < 1e-5 (or 1e-4 for compound losses) vs the mx reference VJP.

‡ The forward for cross_entropy is one pass over logits, which is exactly what mx.logsumexp does. There's no bandwidth left to take. The win is the backward, where mx materialises a full (N, V) softmax tensor while we stream the gradient directly.

Design principle. An op only ships wrapped with @differentiable when there is a real forward speedup over mx.{op}. For ops where mx is already optimal — matmul, conv, attention, RoPE — metalgrad.ops is a thin re-export. Wrapping a no-win op costs ~2× on the backward without any forward gain, so we don't.

How the speedups work

Each fast op uses one or more of the following techniques, all chosen so the result still participates in mx.grad.

1. Threadgroup-cooperative SIMD reductions. A norm op (RMSNorm, LayerNorm) reads every element of a row, computes a sum, and writes a result. The naive mx implementation has the sum become a materialised tensor (mx.mean(x, ...)), then a separate kernel reads it back. We collapse the whole row into one threadgroup of 32 threads, accumulate locally, and reduce across the SIMD in a single shuffle pass (simd_sum). One read of x, one write of y — and no intermediate buffers in DRAM.

2. Register-tiled streaming. Each thread loads its slice of x into a local register array on first touch. LayerNorm needs two reductions (mean, variance) and a final write — all three operate on the register-held values. A previous version re-read x for each pass and wasted 2× the input bandwidth.

3. Online softmax (Welford-style log-sum-exp). Cross-entropy reads 50k+ vocabulary logits per row. The textbook softmax is two passes (max, then sum-exp); the online version maintains running (m, s) in a numerically-stable update and finishes in one pass. We use it for the forward; for cross-SIMD combine we use simd_shuffle_xor to tree-reduce (m, s) pairs.

4. Backward = recompute, don't materialise. mx implements the cross-entropy gradient by materialising a (N, V) softmax tensor and subtracting a one-hot. Our backward kernel streams over logits, recomputes (max, sum) per row, and writes (softmax − 1{j == label}) / N directly into the gradient buffer. One read, one write — no intermediate softmax tensor at all. This is the kernel that gives us the 1.88× combined forward+backward win.

5. mx.compile fusion for pure-elementwise ops. MLX's own compiler will fuse a chain of elementwise ops into a single dispatched kernel. Empirically mx.compile(lambda a, b: silu(a) * b) beats a hand-rolled mx.fast.metal_kernel doing the same thing — the MLX runtime has lower wrapper overhead than the FFI path. For ops like SwiGLU / GeGLU / SquaredReLU we let MLX do the heavy lifting and just attach an explicit VJP.

6. Closed-form mx.compile-fused backward. Where the forward uses a custom Metal kernel (RMSNorm, LayerNorm), the corresponding VJP is a hand-derived closed form (e.g. the canonical LayerNorm gradient (1/C) · inv · (C·gn − Σgn − ñ·Σgn·ñ)) wrapped in mx.compile. The backward is a single fused kernel chain, not a generic chain-rule unrolling through mx.vjp.

7. Strip wrappers when they don't pay rent. For ops where the mx baseline is already at hardware peak — matmul (MPSGraph + simdgroup MMA), attention (FlashAttention-style), RoPE (already fused) — wrapping with @differentiable adds backward overhead and zero forward upside. We re-export these as plain Python functions calling mx, so callers pay nothing.

8. What we tried that didn't pay off (documented so future work doesn't repeat the experiment): a naive K=7 depthwise Conv2d kernel loses to MPSGraph by 2–3× because MPSGraph uses TG-shared input tiling. A FlashAttention reimplementation has no upside because mx.fast.scaled_dot_product_attention already does it. Float4 vectorised loads on the norm kernels are within noise — Apple GPU is bandwidth-bound on these shapes, not instruction-bound, so packing loads doesn't help.

Scaling

Speedups widen with workload size, because mx's intermediate-tensor multi-pass cost grows linearly while our single-read single-write fused kernel stays constant. From scripts/bench_scaling.py:

shape	data	rms_norm	layer_norm
(4, 512, 1024)	8 MB	2.59×	3.75×
(8, 512, 2048)	34 MB	2.76×	5.82×
(4, 2048, 2048)	67 MB	3.08×	6.33×
(8, 1024, 4096)	134 MB	3.76×	6.81×

At the largest shape our kernels reach 60–67% of the 150 GB/s theoretical bandwidth peak; mx is at 18–27% peak.

Higher-level convenience APIs

• swiglu_ffn(x, W_gate_up, W_down, ...) — Llama-style SwiGLU FFN block with the standard fused gate+up matmul (W_gate and W_up stacked along the output axis). Use stack_gate_up(W_gate, W_up) once at model init. The fusion saves one matmul dispatch but speed is neutral at training-scale shapes (matmul compute dominates), so this is shipped as ergonomic API, not as a speedup.

• attention — re-export of mx.fast.scaled_dot_product_attention.

RoPE variants

The rotation kernel is the same across every "variant" in the literature — only the per-dimension angular frequencies differ. mx.fast.rope already accepts a freqs argument; this module supplies the frequency-table builders and matched wrappers for each variant seen in modern LLMs.

variant	wrapper	what it does
vanilla (Su 2021)	rope_standard	θ_i = base^(−2i/d), no scaling
Position Interpolation (Chen 2023)	rope_linear_pi	scales positions: θ → θ / scale
NTK-aware (bloc97 2023)	rope_ntk_aware	scales the base: base' = base · scale^(d/(d−2))
YaRN (Peng 2023)	rope_yarn	smooth piecewise interpolation between PI and standard, ramp based on rotation count
Llama 3.1 (Meta 2024)	rope_llama3	piecewise scaling with (low_freq_factor, high_freq_factor, original_max_pos)

You can also call the frequency builders directly (rope_freqs_yarn(...) etc.) and pass the result to mx.fast.rope with freqs=... for full control.

Testing

Every op has a gradcheck test:

from metalgrad.testing import gradcheck
gradcheck(lambda x, w: mx.sum(my_op(x, w) ** 2), [x, w], rtol=5e-2)

The 34 tests in tests/test_gradcheck.py cover both finite-difference checks (against autograd) and direct VJP-vs-reference checks (against mx.grad of an mx-only reference forward, requiring FP32 bit-precise agreement at rel_err < 1e-5).

Scope

See docs/scope.md for the full statement.

• In: the @differentiable wrapper + a curated set of common training ops + gradient testing infrastructure. MLX backend, Apple Silicon only.
• Out: CUDA, optimizers, dataloaders, higher-order gradients, forward-mode AD, inference-only paths, source sharing with sister repos.

Install

git clone git@github.com:feiyuehchen/MetalGrad.git
cd MetalGrad
uv sync
uv run python tests/test_gradcheck.py     # all 24 should PASS

Relationship to other repos

• conv1d_for_apple_silicon — sister project, inference-only Conv1d dispatcher. Source-independent from MetalGrad. If conv1d_for_apple_silicon's training path benefits from our training-safe ops in the future, it can import them; MetalGrad itself takes no dependency on the conv1d repo.

License

MIT.

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