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We study weighted programming, a programming paradigm for specifying mathematical models. More specifically, the weighted programs we investigate are like usual imperative programs with two additional features: (1) nondeterministic branching and (2) weighting execution traces. Weights can be numbers but also other objects like words from an alphabet, polynomials, formal power series, or cardinal numbers. We argue that weighted programming as a paradigm can be used to specify mathematical models beyond probability distributions (as is done in probabilistic programming). We develop weakest-precondition- and weakest-liberal-precondition-style calculi \`{a} la Dijkstra for reasoning about mathematical models specified by weighted programs. We present several case studies. For instance, we use weighted programming to model the ski rental problem - an optimization problem. We model not only the optimization problem itself, but also the best deterministic online algorithm for solving this problem as weighted programs. By means of weakest-precondition-style reasoning, we can determine the competitive ratio of the online algorithm on source code level.

The importance of programming education has lead to dedicated educational programming environments, where users visually arrange block-based programming constructs that typically control graphical, interactive game-like programs. The Scratch programming environment is particularly popular, with more than 70 million registered users at the time of this writing. While the block-based nature of Scratch helps learners by preventing syntactical mistakes, there nevertheless remains a need to provide feedback and support in order to implement desired functionality. To support individual learning and classroom settings, this feedback and support should ideally be provided in an automated fashion, which requires tests to enable dynamic program analysis. The Whisker framework enables automated testing of Scratch programs, but creating these automated tests for Scratch programs is challenging. In this paper, we therefore investigate how to automatically generate Whisker tests. This raises important challenges: First, game-like programs are typically randomised, leading to flaky tests. Second, Scratch programs usually consist of animations and interactions with long delays, inhibiting the application of classical test generation approaches. Evaluation on common programming exercises, a random sample of 1000 Scratch user programs, and the 1000 most popular Scratch programs demonstrates that our approach enables Whisker to reliably accelerate test executions, and even though many Scratch programs are small and easy to cover, there are many unique challenges for which advanced search-based test generation using many-objective algorithms is needed in order to achieve high coverage.

Learners are often introduced to programming via dedicated languages such as Scratch, where block-based commands are assembled visually in order to control the interactions of graphical sprites. Automated testing of such programs is an important prerequisite for supporting debugging, providing hints, or assessing learning outcomes. However, writing tests for Scratch programs can be challenging: The game-like and randomised nature of typical Scratch programs makes it difficult to identify specific timed input sequences used to control the programs. Furthermore, precise test assertions to check the resulting program states are incompatible with the fundamental principle of creative freedom in programming in Scratch, where correct program behaviour may be implemented with deviations in the graphical appearance or timing of the program. The event-driven and actor-oriented nature of Scratch programs, however, makes them a natural fit for describing program behaviour using finite state machines. In this paper, we introduce a model-based testing approach by extending Whisker, an automated testing framework for Scratch programs. The model-based extension describes expected program behaviour in terms of state machines, which makes it feasible to check the abstract behaviour of a program independent of exact timing and pixel-precise graphical details, and to automatically derive test inputs testing even challenging programs. A video demonstrating model-based testing with Whisker is available at the following URL: //youtu.be/edgCNbGSGEY

The GPU programming model is primarily designed to support the development of applications that run on one GPU. However, just a single GPU is limited in its capabilities in terms of memory capacity and compute power. To handle large problems that exceed these capabilities, one must rewrite application code to manually transfer data between GPU memory and higher-level memory and/or distribute the work across multiple GPUs, possibly in multiple nodes. This means a large engineering effort is required to scale GPU applications beyond a single GPU. We present Lightning: a framework that follows the common GPU programming paradigm, but enables scaling to larger problems. Lightning enables multi-GPU execution of GPU kernels, even across multiple nodes, and seamlessly spills data to main memory and disk when required. Existing CUDA kernels can easily be adapted for use in Lightning, with data access annotations on these kernels allowing Lightning to infer their data requirements and dependencies. Lightning efficiently distributes the work/data across GPUs and maximizes efficiency by overlapping scheduling, data movement, and work when possible. We present the design and implementation of Lightning, as well as experimental results on up to 32 GPUs for eight benchmarks and an application from geospatial clustering. Evaluation shows excellent performance on problem sizes that far exceed the memory capacity of a single GPU.

For an AI system to be reliable, the confidence it expresses in its decisions must match its accuracy. To assess the degree of match, examples are typically binned by confidence and the per-bin mean confidence and accuracy are compared. Most research in calibration focuses on techniques to reduce this empirical measure of calibration error, ECE_bin. We instead focus on assessing statistical bias in this empirical measure, and we identify better estimators. We propose a framework through which we can compute the bias of a particular estimator for an evaluation data set of a given size. The framework involves synthesizing model outputs that have the same statistics as common neural architectures on popular data sets. We find that binning-based estimators with bins of equal mass (number of instances) have lower bias than estimators with bins of equal width. Our results indicate two reliable calibration-error estimators: the debiased estimator (Brocker, 2012; Ferro and Fricker, 2012) and a method we propose, ECE_sweep, which uses equal-mass bins and chooses the number of bins to be as large as possible while preserving monotonicity in the calibration function. With these estimators, we observe improvements in the effectiveness of recalibration methods and in the detection of model miscalibration.

Meta-learning extracts the common knowledge acquired from learning different tasks and uses it for unseen tasks. It demonstrates a clear advantage on tasks that have insufficient training data, e.g., few-shot learning. In most meta-learning methods, tasks are implicitly related via the shared model or optimizer. In this paper, we show that a meta-learner that explicitly relates tasks on a graph describing the relations of their output dimensions (e.g., classes) can significantly improve the performance of few-shot learning. This type of graph is usually free or cheap to obtain but has rarely been explored in previous works. We study the prototype based few-shot classification, in which a prototype is generated for each class, such that the nearest neighbor search between the prototypes produces an accurate classification. We introduce "Gated Propagation Network (GPN)", which learns to propagate messages between prototypes of different classes on the graph, so that learning the prototype of each class benefits from the data of other related classes. In GPN, an attention mechanism is used for the aggregation of messages from neighboring classes, and a gate is deployed to choose between the aggregated messages and the message from the class itself. GPN is trained on a sequence of tasks from many-shot to few-shot generated by subgraph sampling. During training, it is able to reuse and update previously achieved prototypes from the memory in a life-long learning cycle. In experiments, we change the training-test discrepancy and test task generation settings for thorough evaluations. GPN outperforms recent meta-learning methods on two benchmark datasets in all studied cases.

Graph Neural Networks (GNNs) are based on repeated aggregations of information across nodes' neighbors in a graph. However, because common neighbors are shared between different nodes, this leads to repeated and inefficient computations. We propose Hierarchically Aggregated computation Graphs (HAGs), a new GNN graph representation that explicitly avoids redundancy by managing intermediate aggregation results hierarchically, eliminating repeated computations and unnecessary data transfers in GNN training and inference. We introduce an accurate cost function to quantitatively evaluate the runtime performance of different HAGs and use a novel HAG search algorithm to find optimized HAGs. Experiments show that the HAG representation significantly outperforms the standard GNN graph representation by increasing the end-to-end training throughput by up to 2.8x and reducing the aggregations and data transfers in GNN training by up to 6.3x and 5.6x, while maintaining the original model accuracy.

In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve strictly a wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms to theoretically demonstrate that the most powerful GNN can learn approximation algorithms for the minimum dominating set problem and the minimum vertex cover problem with some approximation ratios and that no GNN can perform better than with these ratios. This paper is the first to elucidate approximation ratios of GNNs for combinatorial problems. Furthermore, we prove that adding coloring or weak-coloring to each node feature improves these approximation ratios. This indicates that preprocessing and feature engineering theoretically strengthen model capabilities.

Automatic neural architecture design has shown its potential in discovering powerful neural network architectures. Existing methods, no matter based on reinforcement learning or evolutionary algorithms (EA), conduct architecture search in a discrete space, which is highly inefficient. In this paper, we propose a simple and efficient method to automatic neural architecture design based on continuous optimization. We call this new approach neural architecture optimization (NAO). There are three key components in our proposed approach: (1) An encoder embeds/maps neural network architectures into a continuous space. (2) A predictor takes the continuous representation of a network as input and predicts its accuracy. (3) A decoder maps a continuous representation of a network back to its architecture. The performance predictor and the encoder enable us to perform gradient based optimization in the continuous space to find the embedding of a new architecture with potentially better accuracy. Such a better embedding is then decoded to a network by the decoder. Experiments show that the architecture discovered by our method is very competitive for image classification task on CIFAR-10 and language modeling task on PTB, outperforming or on par with the best results of previous architecture search methods with a significantly reduction of computational resources. Specifically we obtain $2.07\%$ test set error rate for CIFAR-10 image classification task and $55.9$ test set perplexity of PTB language modeling task. The best discovered architectures on both tasks are successfully transferred to other tasks such as CIFAR-100 and WikiText-2.

This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.