---
title: HGEMM Optimization Problem
url: https://www.emergentmind.com/topics/hgemm-optimization-problem
type: topic
---

# HGEMM Optimization Problem

The HGEMM optimization problem refers to the determination of high-performance, energy-efficient scheduling and mapping strategies for half-precision General Matrix Multiply (HGEMM) kernels on advanced hardware architectures, notably including GPUs, FPGAs, and heterogenous SoCs. Recent research highlights multi-objective optimization techniques, combining analytical, machine-learned, and reinforcement-learning approaches to search vast configuration spaces, often targeting throughput (GFLOPS), energy efficiency (GFLOPS/W), and, in communication-bound domains, data movement cost. Several paradigms, including hypergraph partitioning for sparse-matrix kernels, compiler-level optimization tactics, and deep RL-guided kernel auto-tuning, represent the state-of-the-art.

## 1. Mathematical Formulation of the HGEMM Problem

HGEMM computes $C = \alpha AB + \beta C$, typically with $\alpha=1$, $\beta=0$, for $A \in \mathbb{R}^{M \times K}$, $B \in \mathbb{R}^{K \times N}$, $C \in \mathbb{R}^{M \times N}$ in half-precision (fp16). The optimization objective is commonly to maximize the achievable throughput:
$$
\text{TFLOPS} = \frac{2MNK}{10^{12}t_{\text{custom}}}
$$
where $t_{\text{custom}}$ is the elapsed execution time for a kernel evaluated on the target hardware. Other objective functions include energy efficiency $E = \text{TFLOPS}/\text{Power}$, and, in multi-objective settings, Pareto rankings over $(\text{TFLOPS}, E)$. The mapping $\theta$, specifying tiling, parallelism, and loop unrolling, must satisfy hardware constraints (register capacity, memory bandwidth, device-specific limits) [2511.06907][2512.02551][1909.10616].

## 2. Design Space: Parameters and Constraints

The parameter space for HGEMM mapping spans tile sizes $(\text{BM}, \text{BN}, \text{BK})$, parallelization degrees (e.g., $P_m, P_n, P_k$), loop orderings, vectorization widths, and buffering/prefetching modes:
- In a modern FPGA SoC (Versal ACAP), choices include:
  - Degree of parallelism $P_d$ along $M,N,K$ (AIE assignment)
  - Tile size $B_d$ buffered in PL for data reuse
  - Layout and assignment indicators $x_{t,r}$ ("tile $t$ assigned to resource $r$")
- On CUDA GPUs, action vectors specify thread/block configuration, WMMA abstraction controls, shared-memory pipelining parameters ($n_{\text{stage}}$), swizzle, zero-padding, register buffering, prefetch, and epilogue copy strategy [2512.02551].
Hardware constraints include:
- Register capacity for accumulators, broadcasted values, temporaries [1611.08035]
- On-chip memory (AIE scratchpad, PL buffer, GPU shared mem)
- Memory bandwidth (DDR, L2, transport channels)
- Physical resource usage (DSPs, LUTs, BRAM, URAM, etc.)

A plausible implication is that the combinatorial growth in configuration space (e.g., >6000 candidate tuples for ACAP, $10^3$ kernel configurations for CUDA-L2) necessitates automated exploration driven by model-based or RL strategies.

## 3. Optimization Algorithms and Search Strategies

Several distinct approaches have been proposed and evaluated:

**a. ML-Guided Design Space Exploration (ACAP)**  
Gradient-boosted trees (XGBoost) predict kernel latency, power, and resource usage using derived and raw features ($N_{\text{AIE}}, \rho=\text{FLOP}/N_{\text{AIE}}, R_{P_d}=d/P_d, R_{B_d}=d/B_d$). The online search iterates over tuples $(P_M, P_N, P_K, B_M, B_N, B_K)$, evaluating candidate mappings in milliseconds, and extracts the Pareto frontier for throughput/energy objectives [2511.06907].

**b. Reinforcement Learning with LLM Guidance (CUDA-L2)**  
The kernel tuning process is framed as a Markov Decision Process:
- State incorporates $(M,N,K)$, profiling metrics (NCU occupancy, memory throughput), and kernel history.
- Actions select kernel configuration vectors (tiling, pipelining, prefetch strategy, layout).
- Rewards quantify speedup over reference kernels, penalize numerical error and code length.
- LLMs (671B DeepSeek, SFT on CUDA code) generate kernel variants via contrastive prompting, guiding the RL agent towards non-obvious optimizations (zero-padding, swizzle, ping-pong buffering).
Policy optimization uses GRPO; the empirical search covers thousands of configurations [2512.02551].

**c. Compiler-Level Search Algorithms**  
Within the TVM framework, two search algorithms are prominent:
- Greedy Best-First Search (G-BFS): Uses a heuristic cost model, prioritizing local improvement and limited exploration radius per step ($\rho=5$ neighbors).
- Neighborhood Actor-Advantage Critic (N-A2C): Restricts RL actions to locally neighboring configurations ($k$-nearest), updating actor/critic networks on observed runtime cost. Both methods demonstrate drastic reductions in search space explored (0.1%) while achieving superior runtime savings over XGBoost and RNN-based tuners [1909.10616].

**d. Analytical Outer Product Code Generation**  
Analytical micro-kernel synthesis (outer-product decomposition) selects SIMD-tile shapes $(M_r,N_r)$, optimizing register usage and pipeline saturation. The instruction mix is statically scheduled considering hardware port allocations and latency, yielding near-peak performance for DLA kernels [1611.08035].

## 4. Cost Models and Pareto Optimization

Cost assessment across methodologies integrates empirical measurement and analytic estimates:

| Model            | Evaluated Metric        | Features Used                 |
|------------------|------------------------|-------------------------------|
| ML (XGBoost)     | Latency, Power, Usage  | $M,N,K,P_*,B_*,N_{\text{AIE}},\rho,L/P_d,R/B_d$ |
| Compiler-level   | Execution Time         | Tile sizes, loop order, vector width, overhead factor |
| Analytical       | Cycles per micro-kernel| Outer-product shape, pipeline mix, register count |
| RL-Based         | Reward (Speedup > baseline, code quality) | state/action, historic metrics, LLM-generated proposals |

Pareto front extraction provides explicit trade-off curves. For Versal ACAP, throughput and energy-optimal solutions often diverge: maximal throughput uses maximal AIEs, while energy-optimized schedules reduce AIE count and increase PL tile sizes for data reuse [2511.06907]. CUDA-L2 demonstrates that RL-generated kernels outperform cuBLAS and cuBLASLt-AutoTuning across thousands of cases (up to +29% speedup in "server mode") [2512.02551].

## 5. Insights, Challenges, and Best Practices

Empirical observations across research publications suggest:
- Analytical models can misestimate latency (by 10–20%) and ignore power, causing suboptimal scheduling [2511.06907].
- Derived features (FLOP/AIE, size-to-tile ratios) are critical for generalizability of prediction models.
- RL agents, when informed by large-scale pretrained LLMs and accurate profiling, can explore and exploit non-obvious kernel optimizations not found in manual or heuristic methods [2512.02551].
- Greedy and neighborhood RL search are highly efficient, yielding >24% cost savings while exploring minimal search fractions [1909.10616].
- In outer-product codegen, explicit analysis of hardware pipeline constraints, register pressure, and memory hierarchy enables mapping that approaches the best human-tuned implementations [1611.08035].

Best-practice recommendations include restraining search-space complexity (limiting tiling depths, adhering to HW vector widths), precomputing valid divisors, batching measurement, exploiting asynchronous eval, and caching results to avoid redundant assessment [1909.10616].

## 6. Communication-Optimal HGEMM and Sparse Kernels

For sparse-matrix multiplication (SpGEMM), the optimization problem generalizes to the minimization of communication volume under computational and memory-load balancing. Hypergraph models formalize this mapping:
- Vertices correspond to products and nonzeros.
- Nets (hyperedges) encode dependencies and communication requirements.
Partitioning heuristics (e.g., PaToH, hMetis) minimize cut-net metrics subject to load-balance constraints, with fine-grained and coarsened models adapting to application needs. Empirical evidence shows choice of partition model impacts scalability and communication cost, with fine-grained approaches optimal for irregular sparsity [1603.05627]. This suggests algorithm design in HGEMM must adapt to data and architectural characteristics.

## 7. Experimental Benchmarks and Impact

Published frameworks have demonstrated strong performance relative to established libraries:

| Approach          | Throughput Gain    | Energy Efficiency Gain | Fraction of Search Space Used |
|-------------------|--------------------|-----------------------|-------------------------------|
| ACAP ML-DSE       | 1.23× geom. mean   | 1.25× geom. mean      | ~100% (6000 candidates, <2s eval) |
| CUDA-L2 RL+LLM    | +11–29% vs cuBLAS/cuBLASLt | —                 | 1000 configs, ~80% win rate     |
| TVM G-BFS/N-A2C   | -24% (G-BFS), -40% (N-A2C) runtime vs XGBoost/RNN | — | 0.1%                          |
| Analytical Codegen | 95% of peak vs. hand-tuned BLAS | —            | Small enumerative search        |

Adaptable search mechanisms, integration with hardware profiling, and tight coupling of model prediction with empirical measurement are essential for scalable, portable, and high-performance HGEMM optimization [2511.06907][2512.02551][1909.10616][1611.08035][1603.05627].

Source: https://www.emergentmind.com/topics/hgemm-optimization-problem