Efficient tensor computation is a cornerstone of modern deep learning (DL) workloads, yet existing approaches struggle to achieve flexible and performant design and implementation of tensor layouts -- mappings between logical tensors and hardware resources. The increasing complexity of DL algorithms and hardware demands a generic and systematic approach to handling tensor layouts. In this work, we introduce Linear Layouts, a novel approach that models tensor layouts using linear algebra over F2. By representing tensor layouts as binary matrices acting on the bits of the hardware representation, our approach enables a generic layout definition—as opposed to the classical case-by-case approach -- and allows for generic layout-to-layout conversions, eliminating the quadratic explosion that plagues existing solutions. We integrate linear layouts with Triton and demonstrate their effectiveness in optimizing individual Triton operators as well as kernels written in Triton. We also show that linear layouts reduce engineering effort in the compiler backend while fixing several bugs in Triton's legacy layout system.

This paper introduces two extensions to the popular PyTorch machine learning framework, TorchDynamo and TorchInductor, which implement the torch.compile feature released in PyTorch 2. TorchDynamo is a Python-level just-in-time (JIT) compiler that enables graph compilation in PyTorch programs without sacrificing the flexibility of Python. It achieves this by dynamically modifying Python bytecode before execution and extracting sequences of PyTorch operations into an FX graph, which is then JIT compiled using one of many extensible backends. TorchInductor is the default compiler backend for TorchDynamo, which translates PyTorch programs into OpenAI's Triton for GPUs and C++ for CPUs. Results show that TorchDynamo is able to capture graphs more robustly than prior approaches while adding minimal overhead, and TorchInductor is able to provide a 2.27× inference and 1.41× training geometric mean speedup on an NVIDIA A100 GPU across 180+ real-world models, which outperforms six other compilers. These extensions provide a new way to apply optimizations through compilers in eager mode frameworks like PyTorch.

We introduce a novel approach to perform first-order optimization with orthogonal and unitary constraints. This approach is based on a parametrization stemming from Lie group theory through the exponential map. The parametrization transforms the constrained optimization problem into an unconstrained one over a Euclidean space, for which common first-order optimization methods can be used. The theoretical results presented are general enough to cover the special orthogonal group, the unitary group and, in general, any connected compact Lie group. We discuss how this and other parametrizations can be computed efficiently through an implementation trick, making numerically complex parametrizations usable at a negligible runtime cost in neural networks. In particular, we apply our results to RNNs with orthogonal recurrent weights, yielding a new architecture called expRNN. We demonstrate how our method constitutes a more robust approach to optimization with orthogonal constraints, showing faster, accurate, and more stable convergence in several tasks designed to test RNNs.

We introduce a framework to study the transformation of problems with manifold constraints into unconstrained problems through parametrizations in terms of a Euclidean space. We call these parametrizations trivializations. We prove conditions under which a trivialization is sound in the context of gradient-based optimization and we show how two large families of trivializations have overall favorable properties, but also suffer from a performance issue. We then introduce dynamic trivializations, which solve this problem, and we show how these form a family of optimization methods that lie between trivializations and Riemannian gradient descent, and combine the benefits of both of them. We then show how to implement these two families of trivializations in practice for different matrix manifolds. To this end, we prove a formula for the gradient of the exponential of matrices, which can be of practical interest on its own. Finally, we show how dynamic trivializations improve the performance of existing methods on standard tasks designed to test long-term memory within neural networks.