Vectorized CUDA Kernel

• Vectorized CUDA kernels are GPU programs that execute massively parallel elementwise or pairwise operations using techniques like symbolic formula parsing and lazy evaluation.
• They employ advanced optimizations such as register reuse, shared memory tiling, and manual loop unrolling to alleviate memory bottlenecks and boost performance.
• By fusing forward and backward passes with embedded reverse-mode automatic differentiation, these kernels achieve state-of-the-art scalability for high-dimensional tensor computations.

A vectorized CUDA kernel is a GPU program designed to perform massively parallel, elementwise or pairwise operations across large datasets by leveraging device-wide hardware SIMD execution. In kernel and geometric applications, these kernels remove the bottleneck of explicit memory allocation for large $M \times N$ matrices, enabling efficient execution of “quadratic” operations, such as pairwise distance and kernel matrix computations, with memory consumption scaling linearly with input dimensions. Modern vectorized CUDA kernels, as illustrated in the KeOps library, integrate symbolic formula parsing, automatic code generation, tiling, register-level memory management, streaming-based chunking, and embedded reverse-mode automatic differentiation, yielding state-of-the-art performance and scalability for high-dimensional tensor computations (Charlier et al., 2020).

1. Symbolic Formula Architecture and Lazy Evaluation

High-level interfaces (e.g., Python/Numpy, PyTorch, Matlab, R) used in the KeOps framework abstain from allocating explicit $M \times N$ matrices. Instead, data structures called LazyTensors represent symbolic arrays whose individual entries are defined by mathematical formulas, such as $M_{ij} := F(p, x_i, y_j)$, where $F$ is user-supplied and $p$ encodes any required parameters. Construction of LazyTensors records an abstract syntax tree (AST) representing the sequence of arithmetic and reduction operators applied. This approach allows for the concise and expressive specification of operations—for example, using concise Python syntax to define and combine pairwise distances, kernels, and reduction operations such as virtual matrix-vector products. The activation of actual computation (for instance, via a matrix multiplication or reduction command) triggers:

• Parsing of the AST.
• Generation of a compact C++ meta-program implementing $F$ and its derivatives.
• Compilation and linking via nvcc into a dynamic library (with output cached on disk).
• Loading and launching of the resulting CUDA kernel.

This procedure is entirely automated, abstracting away low-level details from the user while retaining full flexibility and performance (Charlier et al., 2020).

2. Formula Decomposition and Fused Forward/Backward Code Generation

During code generation, KeOps decomposes the user-defined formula $F$ into a tree of elemental operations, such as subtraction, squaring, summing, exponentiation, multiplication, and accumulation. Each tree node is converted into a templated C++ functor with:

• An Operation method yielding inlined device code for the operation.
• A recursive DiffT structure encoding the backward-pass (reverse-mode) differentiation rule for autodiff.

These functors are composed into a generic tiled Map–Reduce CUDA kernel skeleton. The kernel fuses the computation of the forward pass (e.g., $y = F(x_i, y_j) \cdot b_j$ reductions) and backward pass (e.g., direct computation of $\partial L/\partial x$) into a single device pass over all $(i,j)$ pairs. This fusion avoids intermediate memory storage and enables further optimization via inlining and template specialization (Charlier et al., 2020).

3. CUDA Kernel Design: Tiling, Shared Memory, and Register Optimization

KeOps vectorized CUDA kernels utilize a two-dimensional grid of thread blocks, each of shape $(B_I, B_J)$ (typically $B_I=B_J=32$ or $64$), to cover the $(i,j)$ parameter space. Each block computes a sub-tile $(i \in [i_0, i_0 + B_I), j \in [j_0, j_0 + B_J))$ using the following optimizations:

• Register reuse: Threads preload $x_i$ into registers to avoid redundant global memory access.
• Shared memory tiling: In a $j$-loop unrolled over $B_J$, each $y_j$ and $b_j$ block is coalesced from global memory into shared memory for efficient access and reuse.
• Register-level accumulation: Partial outputs and intermediate gradients are retained per-thread in registers.
• Manual loop unrolling: The inner $j$-loop is unrolled for increased instruction-level parallelism and vectorization, typically with a #pragma unroll directive.
• Coalesced memory access: Aligned and consecutive memory accesses among warps further optimize memory bandwidth.
• Fused computation: Both forward and backward passes are realized in a single Map–Reduce kernel to minimize memory traffic.

Pseudocode Sketch (condensed):

template<class FwdOp, class BwdOp, int B_I, int B_J> __global__ void map_reduce_kernel( const float* __restrict__ x, // M×D const float* __restrict__ y, // N×D const float* __restrict__ b, // N float* __restrict__ a, // M (output) float* __restrict__ dx // M×D (grad x) ) { __shared__ float y_s[B_J][D]; __shared__ float b_s[B_J]; // block and thread indexing code omitted float acc = 0.f; float grad_x_reg[D] = {0}; // load x[i] into x_reg for (int jb = jblock; jb < N; jb += gridDim.y*B_J) { // load y_j, b_j to shared memory __syncthreads(); #pragma unroll for (int u = 0; u < B_J; u++) { if (jb+u < N) { float diff[D], sqsum=0; // compute diff and sqsum float K = expf(- sqsum * inv2sigma2) * b_s[u]; acc += K; float scale = K * (-inv_sigma2); // accumulate grad_x_reg } } __syncthreads(); } // write back a[i], dx[i,*] }

Host code organizes blocks and streams, launching (⌈M/B_I⌉, ⌈N/B_J⌉) blocks with a (B_I, B_J) thread block and leveraging chunked processing and CUDA streams for memory management (Charlier et al., 2020).

4. Out-of-Core Execution and Chunked Streaming for Large Datasets

Since dense $M \times N$ kernel or distance matrices are never explicitly instantiated, device memory requirements are $O(M \cdot D + N \cdot D + \mathrm{block\ size}^2)$ rather than $O(M \cdot N)$. For even larger $N$ or $M$ values—where input arrays cannot reside entirely on the device—the computation is divided into chunks along the $j$ (or $i$) axis. Each chunk is:

• Transferred asynchronously to the device using CUDA streams.
• Processed in sequence by the Map–Reduce kernel.
• Combined via sum-reduction of output contributions across chunks.

This approach allows for overlap between data transfer and computation and keeps the overall footprint linear in $M + N$, regardless of the “virtual” quadratic output size (Charlier et al., 2020).

5. Embedded Reverse-Mode Automatic Differentiation

KeOps kernels provide reverse-mode automatic differentiation natively in the GPU kernel. For a loss of the form $L = \sum_{i=1}^{M} g(a_i)$, invocation of a backward pass:

• Traverses the formula’s AST in reverse to instantiate gradient (DiffT) functors.
• Generates a fused CUDA kernel that computes $\partial L / \partial p$, $\partial L / \partial x_i$, and $\partial L / \partial y_j$.
• Accumulates partial gradients within registers and shared memory, writing back final values to global GPU memory in coalesced form.

Because all differentiation formulas are known and inlined at compile time through C++ templates, intermediate buffers are optimized away by the compiler, yielding high efficiency and eliminating the need for intermediate arrays (Charlier et al., 2020).

6. Key Formulas and Performance Benchmarks

Numerical operations frequently employ standard expressions such as:

• Squared Euclidean distance: $‖x_i - y_j‖^2 = \sum_{d=1}^D (x_{i,d} - y_{j,d})^2$
• Gaussian kernel: $$k(x_i, y_j) = \exp\left(-‖x_i - y_j‖^2 / (2 \sigma^2)\right)$$
• Matrix-vector reduction: $a_i = \sum_{j=1}^N k(x_i, y_j) \cdot b_j$
• Backpropagation: $$\partial L/\partial x_i = \sum_{j=1}^N (\partial k/\partial x_i)(x_i, y_j) \cdot b_j \cdot (dL/da_i)$$

Performance analyses on a 2080 Ti GPU demonstrate that KeOps achieves substantial speed and memory advantages relative to standard libraries, as summarized:

$N$	PyTorch	TF-XLA	Halide	TVM	PyKeOps	KeOps C++
10k	9 ms	13 ms	1 ms	3.8 ms	0.7 ms	0.4 ms
100k	OOM	89 ms	34 ms	37 ms	15 ms	14.6 ms
1M	OOM	OOM	3.8 s	2.8 s	1.4 s	1.38 s

OOM: out of memory

KeOps achieves "graphics-like" throughput owing to the fusion of map and reduce phases, register-level blocking, loop unrolling, shared-memory tiling, streaming over data chunks, and compile-time specialization for both forward and backward passes (Charlier et al., 2020).

7. Principles and Broader Impact

The core principles of vectorized CUDA kernel design, as codified in the KeOps framework, include lazy formula parsing, compile-time code generation via templated functors and inline autodiff logic, Map–Reduce kernel tiling, aggressive use of shared memory and registers, chunked out-of-core streaming, and full support for reverse-mode differentiation on the GPU. By adhering to these practices, it is possible to develop high-performance, scalable CUDA libraries suitable for all $O(N^2)$ kernel and distance computations relevant to large-scale machine learning and computational geometry applications (Charlier et al., 2020).