Optimizing INT4 Quantized Matrix Multiplication Kernels on Next-Generation GPUs

On next-generation data center GPUs, low-bit matrix multiplication is not just a matter of “changing the input type to 4-bit.” Once the main compute is completed at high speed by hardware matrix-multiply units, what actually determines throughput tends to be data movement, shared memory capacity, pipeline scheduling, CTA cooperation, and how the residual branches are organized.

Using a typical 4-bit quantized compute operator as a case study, this article discusses how to refactor a single-CTA template implementation into a dual-CTA persistent kernel. The operator includes three types of computation: a 4-bit floating-point scaled matrix multiply, a relatively small low-rank residual branch, and a per-column affine transform. Its mathematical expression can be written as:

[
Y = \operatorname{GEMM}(X_q,W_q,S_x,S_w) + X A B + \beta
]

where (X_q) and (W_q) are the low-bit inputs and weights, (S_x) and (S_w) are the corresponding scale factors, (A) and (B) are the two matrices of the low-rank decomposition, and (\beta) is a per-column offset or affine parameter. The formula is compact, but executing it efficiently on a GPU requires simultaneously handling quantized decoding, scale factor loading, matrix-multiply instruction issue, residual prefetching, and result fusion.

1. Why Single-CTA Implementations Hit a Performance Ceiling

An initial implementation is usually built directly on a generic GEMM template. Its advantages are clear structure and low validation cost — you can quickly confirm that:

• The quantized format decoding logic is correct;
• Scale factor broadcasting works as expected;
• The GEMM results match a reference implementation;
• The low-rank residual and per-column affine are fused correctly;
• No out-of-bounds access occurs at various edge-case sizes.

However, this kind of implementation typically has three structural limitations.

First, a single CTA takes on too wide a scope of work. One CTA must move inputs and weights, execute the main matrix multiply, and handle the residual branch and final write-back. The different phases easily block each other, leaving the matrix-multiply unit starved of data.

Second, a generic pipeline usually only describes “block-level producer–consumer relationships” and fails to exploit the warp-level role specialization that next-generation GPUs provide. The warps responsible for loading, decoding, matrix multiplication, and write-back lack clear division of labor, resulting in either excessive synchronization or local idling.

Third, the single-CTA model cannot adequately amortize fixed overhead. When matrix dimensions are large and the main loop is long, a single CTA can complete all the work, but its internal waiting time gradually becomes exposed. At that point, adding more threads does not necessarily help; what is actually needed is for multiple CTAs to cooperatively process the same output tile, with persistent scheduling to reduce the cost of repeatedly launching and retiring CTAs.

In actual testing, the initial single-CTA kernel achieved an effective compute rate of about 27%. Converting it directly to dual-CTA — changing only the block cooperation parameters while keeping the data layout, pipeline, and shared memory budget untouched — yielded only around 28%, essentially no real gain. This shows that dual-CTA is not a switch, but a complete execution model.

2. Basic Design of the Dual-CTA Persistent Kernel

The core idea of the dual-CTA mode is to have two CTAs jointly process a larger output work unit. They can share part of the work, or each handle different slices along the K dimension, then coordinate through cluster-level synchronization or a predefined shared memory layout.

A reasonable division of execution looks like this:

1. Load warps read quantized inputs, weights, and scale factors from global memory;
2. Decode/convert warps organize the low-bit data into the layout required by matrix-multiply instructions;
3. Compute warps continuously issue scaled matrix-multiply instructions;
4. Residual warps prefetch the data needed by the low-rank branch;
5. Write-back warps handle result fusion, the affine transform, and writing to global memory.

The key here is not simply adding more warps, but letting different warps overlap in time. For example, while the current compute warps are working on the (k)-th K block, the load warps should be preparing the (k+1)-th K block, and the residual warps can prefetch upcoming low-rank weights. Only when these tasks genuinely run in parallel can the dual-CTA mode translate into throughput gains.

Persistence means the CTAs are assigned to a work loop with a long lifetime, rather than exiting after processing each small tile. The pseudocode can be abstracted as:

while (has_next_tile()) {
    load_next_quantized_tile();
    decode_and_scale();
    issue_mma_for_current_tile();
    prefetch_low_rank_tile();
    fuse_residual_and_affine();
    store_output();
}

In practice, at least three pipeline stages are usually needed:

• A data pipeline for the main matrix multiply;
• A data pipeline for the low-rank residual;
• A pipeline for output fusion and write-back.

If all data shares one pipeline, a slow access from one branch can stall the main computation. Splitting them apart allows more precise control over the prefetch depth and synchronization boundaries for each data type.

3. Shared Memory Budget Is the Key Tuning Constraint

Shared memory is not “the more the better.” Before the kernel launches, the runtime needs to estimate the shared memory required per CTA based on:

• The input and weight tiles each CTA is responsible for;
• The number of tiles in flight during the main K loop;
• The quantized scale factor cache;
• The prefetch buffers for the low-rank residual;
• The extra space needed for double or triple buffering;
• The coordination region in dual-CTA mode;
• Alignment, padding, and synchronization metadata.

The budget can be abstracted as:

[
M{\text{total}} =
M{\text{main}} +
M{\text{scale}} +
M{\text{residual}} +
M{\text{sync}} +
M{\text{padding}}
]

Based on the total budget, the runtime decides how many concurrent stages the main loop can open and how many prefetch buffers the low-rank branch can retain. If the budget is too large, the following can happen:

• The number of concurrent stages drops;
• The main computation can no longer hide global memory latency;
• Some configurations fail to launch;
• The kernel falls back to a more conservative pipeline;
• Occupancy drops and the matrix-multiply unit idles intermittently.

A more insidious problem: the budget formula itself may be wrong, yet the kernel still runs.

4. The Most Representative Shared Memory Accounting Error

In dual-CTA mode, the low-rank up-projection weight tile is shared between two CTAs. The hardware and cluster layout already split that tile across the two CTAs, but the budget function in the code still computes it as if “each CTA owns the full tile”:

smem_bytes_for_rank_up =
    tile_elements * sizeof(Element);

If the actual layout gives each of the two CTAs half of the tile, the correct estimate should be close to:

smem_bytes_for_rank_up =
    tile_elements * sizeof(Element) / 2;

The key issue here is not the code form, but that “logical tile size” and “the actual per-CTA allocation size” were conflated.

This kind of error usually does not cause a compile failure, nor necessarily an out-of-bounds access. What it affects is resource decision-making during the trace phase: the runtime mistakenly believes shared memory consumption is higher, so it reduces the number of concurrent stages in the main K loop or shrinks the prefetch depth of the low-rank branch. The symptoms ultimately appear as:

• Smoke tests pass;
• Numerical validation is completely correct;
• Small inputs show no anomalies;
• But throughput on production shapes is noticeably lower.

After fixing that one byte calculation, the kernel’s available pipeline depth can be restored, and performance can improve by nearly 2x. In one similar case, this single-line fix yielded about +198% effective throughput. This does not mean the hardware’s compute capability suddenly increased — rather, the parallelism previously suppressed by the erroneous resource budget was finally restored.

Therefore, performance debugging should not focus solely on matrix-multiply instruction counts. Shared memory budgets, stage selection, and occupancy decisions are equally part of the kernel’s algorithm.

5. How to Verify the Shared Memory Accounting

Validation is recommended at four levels.

1. Verify the physical layout first

Answer these questions explicitly:

• How many CTAs jointly process one logical tile?
• How many elements does each CTA actually own?
• Is any data replicated across CTAs?
• Are there regions held by only one CTA and remotely accessed by the other?
• Does alignment and padding happen before or after the tiling split?

Only after clarifying these questions can you build a correct byte-count formula.

2. Separate logical capacity from physical capacity

The code should keep two distinct concepts:

logical_tile_bytes
per_cta_tile_bytes

The former describes the algorithmic data volume; the latter is used to compute the actual shared memory allocation. Do not feed the logical tile size directly into CTA resource budgeting, otherwise it is easy to double-count in cooperative modes.

3. Print runtime decisions

In debug builds, log:

• The total shared memory per CTA;
• The number of stages for the main GEMM;
• The number of prefetch stages for the residual branch;
• Whether dual-CTA or single-CTA mode is used;
• The selected tile shape;
• The actual occupancy and number of active CTAs.

When performance drops, first compare whether these decisions have changed abnormally, rather than immediately tweaking matrix-multiply instructions or thread counts.

4. Use controlled experiments

Prepare at least three configurations:

• Single-CTA, original budget;
• Dual-CTA, original budget;
• Dual-CTA, corrected budget.

If the third configuration clearly outperforms the second while producing numerically identical results, the problem can basically be confirmed to sit at the resource-decision layer, not the computation-logic layer.

6. How to Interpret Performance Metrics

To compare implementations, relative peak utilization is commonly used:

[
\text{MFU} =
\frac{\text{measured throughput}}
{\text{theoretical peak of the target chip for the given precision}}
]

It should be emphasized that MFU is only a reference metric valid for the same hardware, same precision, and same problem size. If two sets of numbers come from different chip generations, or use different matrix-multiply instruction sets and compiler toolchains, you cannot directly conclude that one implementation is “better.”

For example, the fact that a mature handwritten kernel achieves a high MFU on its target hardware does not mean it will maintain the same performance when ported to another chip generation. Different architectures may differ in:

• The input layout of matrix-multiply instructions;
• How scale factors are loaded;
• Warp cooperation rules;
• Shared memory bank conflict characteristics;
• How the compiler allocates registers and pipelines.

A more meaningful comparison, on the same chip with the same data type, same matrix sizes, and same fusion logic, is to measure separately:

• The single-CTA baseline;
• The dual-CTA non-persistent version;
• The dual-CTA persistent version;
• The final version with the corrected shared memory budget;
• The pure main-GEMM version with the residual and affine removed.

The last item helps estimate the remaining performance headroom. If the pure main GEMM is still clearly faster than the fused kernel, the optimization focus should be on the residual, affine, and write-back paths. If the two are close, the bottleneck has shifted to the main matrix multiply or the memory system.

7. Practical Engineering Advice

First, establish a correctness baseline before introducing dual-CTA. Compare quantized decoding, scale factors, the low-rank residual, and the affine transform against high-precision reference results individually, so that performance optimization does not mask numerical errors.

Second, isolate the resource estimation code. Do not scatter shared memory byte calculations across multiple template branches. Provide a clear budget function for each execution mode, and print each component during debugging.

Third, document each cooperation mode clearly. In particular, state “how many CTAs jointly own a logical tile” and “whether the returned value is the total capacity or the per-CTA capacity” — this is where maintenance most often goes wrong.

Fourth, when profiling, observe instruction throughput, memory throughput, shared memory usage, register usage, active CTA count, and pipeline stage counts together. No single metric can explain the degradation of a complex fused kernel.

Fifth, do not treat “it runs” as “it’s optimized.” A smoke test only proves the path is executable — not that the resource budget is correct, and certainly not that the pipeline reaches the expected depth on production shapes.

Conclusion

Optimizing a 4-bit quantized GEMM kernel is, at its core, the co-design of the hardware execution model, data layout, and resource budget. A single-CTA template is suitable for fast validation, but achieving stable throughput usually requires dual-CTA persistent scheduling, warp specialization, and multi-stage pipelines.

The shared memory accounting deserves particular attention. An seemingly ordinary byte-count formula can determine whether the main K loop maintains enough concurrent stages, and whether low-rank residual prefetching crowds out the main computation. Many performance problems come not from complex matrix-multiply instructions, but from a resource estimate that fails to reflect the real CTA tiling relationship.

When optimizing operators of this kind, therefore, stick to three principles: confirm the physical data ownership before computing per-CTA resources; verify the runtime scheduling decisions before analyzing instruction-level throughput; and use a baseline on the same hardware at the same scale before interpreting peak utilization. Only then can a kernel that “appears to run” be pushed to a high-performance implementation that is sustainable, explainable, and scalable.