Register Allocation for Vector/Matrix Units

Managing the processor's registers is a main stage in generating high-performance code, occurring after target-specific instructions are selected. While register allocation is a primary compiler optimization, modern CPUs, GPUs, and accelerators feature wide vector units (SIMD) and specialized matrix multiplication units. These units introduce considerable complexity to traditional scalar register allocation. Effectively utilizing these large, often specialized, register files is essential for achieving the full throughput of these units.

The Unique Demands of Vector and Matrix Registers

Vector and matrix operations, common in ML workloads, operate on large amounts of data simultaneously. The hardware reflects this with correspondingly large register files:

• Size and Width: CPU SIMD extensions like AVX-512 offer 32 512-bit vector registers (ZMM), while Arm's SVE provides scalable vector lengths. GPUs often feature even larger vector register files per processing block (e.g., potentially hundreds of KBs per SM in NVIDIA GPUs, logically partitioned among threads). Matrix units (like NVIDIA Tensor Cores or AMD Matrix Cores) operate on tile-like structures that map to groups of registers or dedicated accumulators.
• Structure and Aliasing: Vector registers frequently allow access to sub-parts (e.g., AVX allows 128-bit (XMM) and 256-bit (YMM) views within 512-bit (ZMM) registers). The allocator must correctly model these aliasing relationships to ensure correctness and avoid unnecessary constraints.
• Specialization: Matrix units may have dedicated accumulator registers that behave differently from general-purpose vector registers. Some ISAs impose pairing constraints or require specific register types for certain operands of matrix instructions (e.g., mma instructions in PTX).
• High Pressure: Tensor operations, particularly dense matrix multiplications (GEMM) and convolutions implemented via im2col/GEMM, involve complex loop structures with many intermediate values. Optimizations like loop unrolling and software pipelining, designed to expose instruction-level parallelism, further increase the demand for registers (register pressure), especially within computationally intensive inner loops.
• Predicate/Mask Registers: Architectures like AVX-512 and Arm SVE utilize mask registers to control element-wise operations within vectors. These masks themselves represent live values that must be allocated to a separate, smaller predicate register file.

Limitations of Standard Allocation Approaches

Classical graph-coloring register allocators (based on Chaitin's or Briggs' algorithms) form the basis of many compilers. They build an interference graph where nodes represent live ranges and edges connect interfering ranges, then attempt to color the graph using a number of colors equal to the available physical registers. However, applying these directly to large vector/matrix register files encounters issues:

• Scalability: The sheer number of potential live ranges and the size of the registers can lead to enormous, dense interference graphs, making the coloring problem computationally expensive.
• Modeling Complexity: Representing sub-register aliasing, register pairing, or specialized accumulator constraints within a standard interference graph can be cumbersome or inefficient.
• Spill Cost: Spilling a 512-bit vector register is significantly more costly than spilling a 64-bit scalar register due to the increased memory bandwidth required. Naive spill choices (e.g., spilling the live range with the highest degree in the interference graph) might be suboptimal, ignoring the actual cost or frequency of the spill.

Advanced Allocation Strategies for Vector/Matrix Units

To address these challenges, compilers employ more sophisticated techniques tailored for vector and matrix registers:

1. Rematerialization: Instead of spilling and reloading a value, especially constants or values easily derived from others (e.g., generating a vector of zeros), the allocator can opt to recompute (rematerialize) it later. This avoids costly memory traffic for values that are cheap to regenerate. Compilers identify instructions whose results can be rematerialized and weigh the cost of recomputation against the cost of spilling/reloading.

2. Live Range Splitting and Register Packing: When a vector register holds multiple independent smaller values, or when a value is only live in a subset of the vector lanes, the allocator might split the live range. This allows different parts of the original live range to be allocated to different physical registers or spilled independently. Conversely, if multiple small, non-interfering values fit within a single vector register, they can be packed together, reducing overall register demand.

3. Optimized Spill Code: When spilling is unavoidable, the allocator must generate efficient spill code.

   • Partial Spills: If only certain lanes of a vector are needed after a potential spill point, the compiler might generate code to spill only those specific lanes, reducing memory bandwidth usage.
   • Placement: Spilling to the closest/fastest cache level (e.g., L1) is preferred over spilling to main memory. The allocator might use heuristics or profile data to guide spill slot placement.
   • Scheduling: Spill and fill instructions (vector loads/stores) are treated as memory operations that need careful scheduling to hide their latency, potentially overlapping them with computation.

4. Register Tiling: This technique closely ties register allocation to loop tiling optimizations (discussed in Chapter 4). Inner loops are structured such that the working set for a tile of computation (e.g., a sub-block of a matrix multiplication) fits within the available vector/matrix registers. For GEMM ($C += A * B$), this often means keeping a tile of the C matrix ($C_{sub}$) in registers (often accumulators) and streaming blocks of A and B through other registers. The allocator's goal is to minimize reloading of the $C_{sub}$ tile between iterations.

5. Handling Matrix Accumulators: Allocators targeting matrix units need specific strategies. Partial sums accumulated within these units are extremely valuable and costly to spill. The allocator must prioritize keeping these partial sums resident, often by carefully scheduling the outer loops that iterate over matrix tiles. The specific instructions (e.g., PTX mma, HLSL wave matrix intrinsics) often dictate how operands and accumulators map to the register file.

6. Phase Ordering Considerations: The classic dilemma of whether to perform register allocation before or after instruction scheduling is exacerbated with vector/matrix units. Early allocation constrains the scheduler, while late allocation might force more spills if the schedule creates high register pressure. Modern compilers often use iterative approaches or integrated scheduling and allocation phases, especially for performance-critical loops.

GPU Register Allocation: The Occupancy Trade-off

On GPUs, register allocation has a direct, significant impact on occupancy. Occupancy refers to the number of active warps (groups of threads) that can reside concurrently on a Streaming Multiprocessor (SM). Each SM has a large physical register file, but it's shared among all threads running on that SM.

• Per-Thread Limit: The GPU hardware and runtime impose a limit on the number of registers each thread can use.
• Occupancy Calculation: The total physical registers available on the SM divided by the registers allocated per thread determines (along with other factors like shared memory usage) how many threads, and thus warps, can run concurrently. $\text{Max Warps per SM} \approx \frac{\text{Total Registers per SM}}{\text{Registers per Thread} \times \text{Threads per Warp}}$
• Performance Impact: Higher occupancy allows the SM to hide memory latency more effectively by switching between ready warps. However, allocating fewer registers per thread to increase occupancy might require more spill code within the thread, potentially slowing down individual thread execution.

Compilers must navigate this trade-off. Aggressively allocating registers might enable better instruction-level parallelism within a thread but reduce thread-level parallelism (occupancy). Conversely, minimizing register usage increases occupancy but might lead to performance loss from spills or reduced unrolling. GPU compilers often use heuristics, profile data, or allow programmer hints (like __launch_bounds__ in CUDA) to guide this balance.

Relationship between the number of registers allocated per thread and the maximum number of warps that can run concurrently on an SM, assuming registers are the limiting factor.

Example: Register Pressure in Tiled GEMM

Consider a simplified inner loop for matrix multiplication ($C_{ij} += A_{ik} \times B_{kj}$), where we aim to keep a $4 \times 4$ tile of C in registers. This requires 16 accumulator registers (scalar or vector, depending on the target). To compute this tile, we might need to load, say, 4 vector registers for a panel of A and 4 vector registers for a panel of B in each iteration of the innermost ($k$) loop.

• Minimum Register Need: 16 (for C tile) + 4 (for A panel) + 4 (for B panel) = 24 vector registers, plus temporary registers for intermediate results and address calculations.
• Challenge: If the available vector register file is small (e.g., only 16 vector registers), the compiler cannot keep the entire C tile, A panel, and B panel resident simultaneously.
• Strategy: The allocator, guided by the loop transformations, might prioritize keeping the C accumulators resident and spill/reload parts of A or B, or use a smaller C tile size. Alternatively, it might employ register rotation or other sophisticated techniques if supported by the ISA, potentially overlapping loads for the next iteration's A/B panels with the current computation.

Effectively managing vector and matrix registers demands more than just applying standard allocation algorithms. It requires deep knowledge of the target architecture's capabilities and constraints, careful interaction with instruction scheduling and loop optimization phases, and sophisticated strategies for minimizing the high cost associated with spilling wide vector or specialized matrix data. The choices made here are critical for bridging the gap between optimized IR and high-performance machine code on modern heterogeneous hardware.