Partial Differential Equations (PDEs) play a ubiquitous role in scientific computing and engineering. While numerical methods make solving PDEs tractable, these numerical solvers encounter several issues, particularly for hyperbolic PDEs. These issues arise from multiple sources including the PDE’s physical model, which can lead to effects like shock wave formation, and the PDE solver’s inherent approximations, which can introduce spurious numerical artifacts. These issues can cause the solver’s program execution to crash (due to overflow) or return results with unacceptable levels of inaccuracy (due to spurious oscillations or dissipation). Moreover, these challenges are compounded by the nonlinear nature of many of these PDEs. In addition, PDE solvers must obey numerical invariants like the CFL condition. Hence there exists a critical need to apply program analysis to PDE solvers to certify such problems do not arise and that invariants are always satisfied. As a solution, we develop Phocus, which is the first abstract interpretation of hyperbolic PDE solvers. Phocus can certify precise bounds on nonlinear PDE solutions and certify key invariants such as the CFL condition and a solution’s total variation bound. Hence Phocus can verify the absence of shock formation, the stability of the solver, and bounds on the amount of spurious numerical effects. To enable effective abstract interpretation of hyperbolic PDE solvers, Phocus uses a novel optimization-based procedure to synthesize precise abstract transformers for multiple finite difference schemes. To evaluate Phocus, we develop a new set of PDE benchmark programs and use them to perform an extensive experimental evaluation which demonstrates Phocus’s significant precision benefits and scalability to several thousand mesh points.

As large quantities of stochastic data are processed onboard tiny edge devices, these systems must constantly make decisions under uncertainty. This challenge necessitates principled embedded compiler support for time- and energy-efficient probabilistic inference. However, compiling probabilistic inference to run on the edge is significantly understudied, and the existing research is limited to computationally expensive MCMC algorithms. Hence, these works cannot leverage faster variational inference algorithms which can better scale to larger data sizes that are representative of realistic workloads in the edge setting. However, naively writing code for differentiable inference on resource-constrained edge devices is challenging due to the need for expensive floating point computations. Even when using reduced precision, a developer still faces the challenge of choosing the right quantization scheme, as gradients can be notoriously unstable in the face of low-precision. To address these challenges, we propose ViX which is the first compiler for low-precision probabilistic programming with variational inference. ViX generates optimized variational inference code in reduced precision by automatically exploiting Bayesian domain knowledge and analytical mathematical properties to ensure that low-precision gradients can still be effectively used. ViX can scale inference to much larger data-sets than previous compilers for resource-constrained probabilistic programming while attaining both high accuracy and significant speedup. Our evaluation of ViX across 7 benchmarks shows that ViX-generated code is up to 8.15× faster than performing the same variational inference in 32-bit floating point and also up to 22.67× faster than performing the variational inference in 64-bit double precision, all with minimal accuracy loss. Further, on a subset of our benchmarks, ViX can scale inference to data sizes between 16–80× larger than the existing state-of-the-art tool Statheros.

We present Pasado, a technique for synthesizing precise static analyzers for Automatic Differentiation. Our technique allows one to automatically construct a static analyzer specialized for the Chain Rule, Product Rule, and Quotient Rule computations for Automatic Differentiation in a way that abstracts all of the nonlinear operations of each respective rule simultaneously. By directly synthesizing an abstract transformer for the composite expressions of these 3 most common rules of AD, we are able to obtain significant precision improvement compared to prior works which compose standard abstract transformers together suboptimally. We prove our synthesized static analyzers sound and additionally demonstrate the generality of our approach by instantiating these AD static analyzers with different nonlinear functions, different abstract domains (both intervals and zonotopes) and both forward-mode and reverse-mode AD. We evaluate Pasado on multiple case studies, namely soundly computing bounds on a neural network’s local Lipschitz constant, soundly bounding the sensitivities of financial models, certifying monotonicity, and lastly, bounding sensitivities of the solutions of differential equations from climate science and chemistry for verified ranges of initial conditions and parameters. The local Lipschitz constants computed by Pasado on our largest CNN are up to 2750× more precise compared to the existing state-of-the-art zonotope analysis. The bounds obtained on the sensitivities of the climate, chemical, and financial differential equation solutions are between 1.31 − 2.81× more precise (on average) compared to a state-of-the-art zonotope analysis.

We present a novel abstraction for bounding the Clarke Jacobian of a Lipschitz continuous, but not necessarily differentiable function over a local input region. To do so, we leverage a novel abstract domain built upon dual numbers, adapted to soundly over-approximate all first derivatives needed to compute the Clarke Jacobian. We formally prove that our novel forward-mode dual interval evaluation produces a sound, interval domain-based over-approximation of the true Clarke Jacobian for a given input region. Due to the generality of our formalism, we can compute and analyze interval Clarke Jacobians for a broader class of functions than previous works supported – specifically, arbitrary compositions of neural networks with Lipschitz, but non-differentiable perturbations. We implement our technique in a tool called DeepJ and evaluate it on multiple deep neural networks and non-differentiable input perturbations to showcase both the generality and scalability of our analysis. Concretely, we can obtain interval Clarke Jacobians to analyze Lipschitz robustness and local optimization landscapes of both fully-connected and convolutional neural networks for rotational, contrast variation, and haze perturbations, as well as their compositions.

We present a novel, general construction to abstractly interpret higher-order automatic differentiation (AD). Our construction allows one to instantiate an abstract interpreter for computing derivatives up to a chosen order. Furthermore, since our construction reduces the problem of abstractly reasoning about derivatives to abstractly reasoning about real-valued straight-line programs, it can be instantiated with almost any numerical abstract domain, both relational and non-relational. We formally establish the soundness of this construction. We implement our technique by instantiating our construction with both the non-relational interval domain and the relational zonotope domain to compute both first and higher-order derivatives. In the latter case, we are the first to apply a relational domain to automatic differentiation for abstracting higher-order derivatives, and hence we are also the first abstract interpretation work to track correlations across not only different variables, but different orders of derivatives. We evaluate these instantiations on multiple case studies, namely robustly explaining a neural network and more precisely computing a neural network’s Lipschitz constant. For robust interpretation, first and second derivatives computed via zonotope AD are up to 4.76× and 6.98× more precise, respectively, compared to interval AD. For Lipschitz certification, we obtain bounds that are up to 11,850× more precise with zonotopes, compared to the state-of-the-art interval-based tool.

As Edge and IoT computing devices process noisy data or make decisions in uncertain environments, they require frameworks for inexpensive, yet accurate probabilistic inference. Probabilistic programming has emerged as a powerful way for developers to write high-level programs, while abstracting away the implementation details of inference. However, the existing algorithms are slow and often assumed to require precise calculations. We present Statheros, the first compiler for low-level, fixed-point approximation of probabilistic programming. Statheros compiles programs to fixed-point inference procedures and is able to determine the optimal fixed-point type to use. We evaluate Statheros on 13 benchmarks and three embedded platforms. The results show that Statheros-generated code is 11. 5x (Arduino), 3. 8x (PocketBeagle), and 2. 2x (Raspberry Pi) faster than single-precision floating-point computation, with minimal accuracy loss.

Abstract Probabilistic Programming offers a concise way to represent stochastic models and perform automated statistical inference. However, many real-world models have discrete or hybrid discrete-continuous distributions, for which existing tools may suffer non-trivial limitations. Inference and parameter estimation can be exceedingly slow for these models because many inference algorithms compute results faster (or exclusively) when the distributions being inferred are continuous. To address this discrepancy, this paper presents Leios. Leios is the first approach for systematically approximating arbitrary probabilistic programs that have discrete, or hybrid discrete-continuous random variables. The approximate programs have all their variables fully continualized. We show that once we have the fully continuous approximate program, we can perform inference and parameter estimation faster by exploiting the existing support that many languages offer for continuous distributions. Furthermore, we show that the estimates obtained when performing inference and parameter estimation on the continuous approximation are still comparably close to both the true parameter values and the estimates obtained when performing inference on the original model.

Recent years have witnessed a resurgence of interest in video summarization. However, one of the main obstacles to the research on video summarization is the user subjectivity - users have various preferences over the summaries. The subjectiveness causes at least two problems. First, no single video summarizer fits all users unless it interacts with and adapts to the individual users. Second, it is very challenging to evaluate the performance of a video summarizer. To tackle the first problem, we explore the recently proposed query-focused video summarization which introduces user preferences in the form of text queries about the video into the summarization process. We propose a memory network parameterized sequential determinantal point process in order to attend the user query onto different video frames and shots. To address the second challenge, we contend that a good evaluation metric for video summarization should focus on the semantic information that humans can perceive rather than the visual features or temporal overlaps. To this end, we collect dense per-video-shot concept annotations, compile a new dataset, and suggest an efficient evaluation method defined upon the concept annotations. We conduct extensive experiments contrasting our video summarizer to existing ones and present detailed analyses about the dataset and the new evaluation method.