Mathieu Huot

Mathieu Huot, Sam Staton, Matthijs Vákár

We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher-order language with algebraic data types and we characterise it as the unique structure-preserving macro given a choice of derivatives for basic operations. We describe a rich semantics for differentiable programming based on diffeological spaces. We show that it interprets our language and we phrase what it means for the AD method to be correct with respect to this semantics. We show that our characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces. We explain how this is in essence a logical relations argument. Throughout we show how the analysis extends to AD methods for computing higher-order derivatives using a Taylor approximation.

Amir Shaikhha, Mathieu Huot, Shabnam Ghasemirad et al.

Automatic differentiation (AD) is a technique for computing the derivative of a function represented by a program. This technique is considered as the de-facto standard for computing the differentiation in many machine learning and optimisation software tools. Despite the practicality of this technique, the performance of the differentiated programs, especially for functional languages and in the presence of vectors, is suboptimal. We present an AD system for a higher-order functional array-processing language. The core functional language underlying this system simultaneously supports both source-to-source forward-mode AD and global optimisations such as loop transformations. In combination, gradient computation with forward-mode AD can be as efficient as reverse mode, and the Jacobian matrices required for numerical algorithms such as Gauss-Newton and Levenberg-Marquardt can be efficiently computed.

Mathieu Huot, Alexander K. Lew, Vikash K. Mansinghka et al.

We introduce a new setting, the category of $ω$PAP spaces, for reasoning denotationally about expressive differentiable and probabilistic programming languages. Our semantics is general enough to assign meanings to most practical probabilistic and differentiable programs, including those that use general recursion, higher-order functions, discontinuous primitives, and both discrete and continuous sampling. But crucially, it is also specific enough to exclude many pathological denotations, enabling us to establish new results about both deterministic differentiable programs and probabilistic programs. In the deterministic setting, we prove very general correctness theorems for automatic differentiation and its use within gradient descent. In the probabilistic setting, we establish the almost-everywhere differentiability of probabilistic programs' trace density functions, and the existence of convenient base measures for density computation in Monte Carlo inference. In some cases these results were previously known, but required detailed proofs with an operational flavor; by contrast, all our proofs work directly with programs' denotations.

Eric Li, Arijit Dasgupta, Yoni Friedman et al.

Human visual perception offers valuable insights for understanding computational principles of motion-based scene interpretation. Humans robustly detect and segment moving entities that constitute independently moveable chunks of matter, whether observing sparse moving dots, textured surfaces, or naturalistic scenes. In contrast, existing computer vision systems lack a unified approach that works across these diverse settings. Inspired by principles of human perception, we propose a generative model that hierarchically groups low-level motion cues and high-level appearance features into particles (small Gaussians representing local matter), and groups particles into clusters capturing coherently and independently moveable physical entities. We develop a hardware-accelerated inference algorithm based on parallelized block Gibbs sampling to recover stable particle motion and groupings. Our model operates on different kinds of inputs (random dots, stylized textures, or naturalistic RGB video), enabling it to work across settings where biological vision succeeds but existing computer vision approaches do not. We validate this unified framework across three domains: on 2D random dot kinematograms, our approach captures human object perception including graded uncertainty across ambiguous conditions; on a Gestalt-inspired dataset of camouflaged rotating objects, our approach recovers correct 3D structure from motion and thereby accurate 2D object segmentation; and on naturalistic RGB videos, our model tracks the moving 3D matter that makes up deforming objects, enabling robust object-level scene understanding. This work thus establishes a general framework for motion-based perception grounded in principles of human vision.

Gaurav Arya, Ruben Seyer, Frank Schäfer et al.

We develop an algorithm for automatic differentiation of Metropolis-Hastings samplers, allowing us to differentiate through probabilistic inference, even if the model has discrete components within it. Our approach fuses recent advances in stochastic automatic differentiation with traditional Markov chain coupling schemes, providing an unbiased and low-variance gradient estimator. This allows us to apply gradient-based optimization to objectives expressed as expectations over intractable target densities. We demonstrate our approach by finding an ambiguous observation in a Gaussian mixture model and by maximizing the specific heat in an Ising model.

Amir Shaikhha, Mathieu Huot, Shideh Hashemian

Sparse tensors are prevalent in many data-intensive applications, yet existing differentiable programming frameworks are tailored towards dense tensors. This presents a significant challenge for efficiently computing gradients through sparse tensor operations, as their irregular sparsity patterns can result in substantial memory and computational overheads. In this work, we introduce a novel framework that enables the efficient and automatic differentiation of sparse tensors, addressing this fundamental issue. Our experiments demonstrate the effectiveness of the proposed framework in terms of performance and scalability, outperforming state-of-the-art frameworks across a range of synthetic and real-world datasets. Our approach offers a promising direction for enabling efficient and scalable differentiable programming with sparse tensors, which has significant implications for numerous applications in machine learning, natural language processing, and scientific computing.

McCoy R. Becker, Alexander K. Lew, Xiaoyan Wang et al.

Compared to the wide array of advanced Monte Carlo methods supported by modern probabilistic programming languages (PPLs), PPL support for variational inference (VI) is less developed: users are typically limited to a predefined selection of variational objectives and gradient estimators, which are implemented monolithically (and without formal correctness arguments) in PPL backends. In this paper, we propose a more modular approach to supporting variational inference in PPLs, based on compositional program transformation. In our approach, variational objectives are expressed as programs, that may employ first-class constructs for computing densities of and expected values under user-defined models and variational families. We then transform these programs systematically into unbiased gradient estimators for optimizing the objectives they define. Our design enables modular reasoning about many interacting concerns, including automatic differentiation, density accumulation, tracing, and the application of unbiased gradient estimation strategies. Additionally, relative to existing support for VI in PPLs, our design increases expressiveness along three axes: (1) it supports an open-ended set of user-defined variational objectives, rather than a fixed menu of options; (2) it supports a combinatorial space of gradient estimation strategies, many not automated by today's PPLs; and (3) it supports a broader class of models and variational families, because it supports constructs for approximate marginalization and normalization (previously introduced only for Monte Carlo inference). We implement our approach in an extension to the Gen probabilistic programming system (genjax.vi, implemented in JAX), and evaluate on several deep generative modeling tasks, showing minimal performance overhead vs. hand-coded implementations and performance competitive with well-established open-source PPLs.

Amir Shaikhha, Mathieu Huot, Jaclyn Smith et al.

This paper introduces semi-ring dictionaries, a powerful class of compositional and purely functional collections that subsume other collection types such as sets, multisets, arrays, vectors, and matrices. We developed SDQL, a statically typed language that can express relational algebra with aggregations, linear algebra, and functional collections over data such as relations and matrices using semi-ring dictionaries. Furthermore, thanks to the algebraic structure behind these dictionaries, SDQL unifies a wide range of optimizations commonly used in databases (DB) and linear algebra (LA). As a result, SDQL enables efficient processing of hybrid DB and LA workloads, by putting together optimizations that are otherwise confined to either DB systems or LA frameworks. We show experimentally that a handful of DB and LA workloads can take advantage of the SDQL language and optimizations. SDQL can be competitive with or outperforms a host of systems that are state of the art in their own domain: in-memory DB systems Typer and Tectorwise for (flat, not nested) relational data; SciPy for LA workloads; sparse tensor compiler taco; the Trance nested relational engine; and the in-database machine learning engines LMFAO and Morpheus for hybrid DB/LA workloads over relational data.