What They Understand Well
1) Memory hierarchy intuition (HBM → VMEM → registers)
That mental model is good.
On modern TPUs (including v5e), the simplified flow is:
HBM (off-chip DRAM) → On-chip SRAM (VMEM) → Vector / scalar registers → Compute units (VPU/MXU)
The kitchen analogy works conceptually.
What’s especially good:
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They understand every operation must eventually happen in registers.
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They understand data movement dominates performance.
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They recognize tiling is required because VMEM and registers are limited.
That’s foundational systems thinking.
2) Parallelism intuition (8×128 tiles, 1024 elements)
Correct spirit.
The VPU processes large vector widths (implementation details vary slightly by generation), but the key idea is:
TPUs are throughput machines. They operate on wide tiles, not scalars.
That mental shift — from scalar programming to tile programming — is huge.
3) VPU vs MXU execution model
This is a strong observation:
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VPU → synchronous, elementwise ops
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MXU → asynchronous matrix multiply pipeline
The important concept they caught:
You can overlap scalar/vector work with MXU matmuls.
That’s essentially understanding latency hiding via pipeline parallelism.
That is exactly how high-performance kernels are written.
Very good.
4) BlockSpec + grid explanation
This explanation is clean and mostly accurate:
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block_shape→ tile size -
index_map→ mapping grid coords to array slice -
grid→ launch dimensions
And yes:
grid_index × block_shape = offset into tensor
That’s correct at a conceptual level.
5) Softmax tiling insight
This is 100% correct:
Never tile across the reduction dimension unless you know how to handle partial reductions.
Softmax reduces across columns (usually). So you:
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Tile rows
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Keep full reduction dimension intact
This is a core GPU/TPU rule too.
6) BF16 inputs, FP32 accumulators
Correct and important.
BF16:
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8 exponent bits (like FP32)
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7 mantissa bits
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Same dynamic range, lower precision
Accumulating in BF16 would:
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Introduce large rounding error
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Cause catastrophic loss in reductions
All high-performance kernels accumulate in FP32.
This shows strong numerical awareness.
🔧 What Could Be Clarified or Made More General
Now let’s refine some details.
1) “Every operation follows HBM → VMEM → VREG”
Conceptually true — but performance engineers think differently:
The real goal is:
Move data from HBM as few times as possible.
Good kernels:
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Load once from HBM
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Reuse heavily in VMEM / registers
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Write back once
So a more precise framing:
Instead of:
Every op follows that path
Better:
All data originates in HBM, but high-performance kernels maximize reuse in on-chip memory to avoid going back.
The difference matters.
2) “MXU runs asynchronously”
Yes — but nuance:
It’s not free parallelism.
It works because:
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The scalar unit issues MXU instructions.
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The MXU has its own deep pipeline.
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The compiler/runtime schedules around it.
The overlap works only if:
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Dependencies allow it.
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You structure the kernel to expose independent work.
So the deeper insight:
Asynchrony only helps if you create independent instruction streams.
Otherwise you just stall.
3) Softmax tiling rule — generalize it
Instead of:
Never tile along a dimension you're reducing over.
More general:
You can tile reduction dimensions — but then you must:
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Compute partial reductions
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Store partial sums
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Perform a second reduction pass
So the full rule is:
Tiling across reduction dims increases algorithmic complexity and synchronization cost.
They learned the simplified case. The advanced version is worth adding.
4) Missing concept: Bandwidth vs Compute Bound
They’re close to a big “aha” but didn’t state it:
There are two kernel regimes:
Compute-bound
MXU saturated
Performance limited by FLOPs
Memory-bound
HBM bandwidth limited
Performance limited by bytes/sec
Softmax is often memory-bound.
MatMul is compute-bound (if sized well).
This distinction is foundational in inference engineering.
🧠Extra “Ahas” They’re Ready For
These would level them up.
🔥 Aha 1: Tiling Is About Reuse, Not Fitting
Many beginners think:
We tile because memory is small.
Real reason:
We tile to maximize arithmetic intensity.
Arithmetic intensity = FLOPs / bytes moved.
MatMul works because:
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Each loaded value participates in many multiply-adds.
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So compute dominates memory cost.
Softmax has low reuse → often bandwidth bound.
That mental shift is huge.
🔥 Aha 2: MXU Is a Systolic Array
The MXU isn’t just a fast matmul unit.
It’s a systolic array:
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Data flows rhythmically across a 2D grid.
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Multiply-accumulate units arranged spatially.
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Partial sums propagate across the array.
That explains:
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Why tile shapes matter.
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Why alignment matters.
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Why padding matters.
🔥 Aha 3: Kernel performance is scheduling + memory choreography
The best TPU kernels:
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Double buffer VMEM
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Prefetch next tile while computing current tile
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Overlap MXU with VPU
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Avoid bank conflicts
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Align shapes to hardware tile sizes
The game is not just math.
It’s choreography.
🔥 Aha 4: Inference vs Training
If they’re reading inference engineering:
Key difference:
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Training → needs backward pass + activation storage
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Inference → cares about latency + throughput
Softmax in inference:
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Often fused
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Sometimes avoided entirely (logits used directly)
Kernel design differs.
