However,whenlargeamounts ofannotated dataareavailable, models thatallow end-to-end training such as neural networks are often preferred.

While low-variance reparameterization gradients of a continuous relaxation can provide an effective solution, a continuous relaxation is not always available or tractable.

B.7 Additional Details on the Running Time In this section, we provide additional details on the running time of the algorithms.

Graph neural networks (GNNs) have emerged as a powerful tool for learning softwareengineering tasksincluding codecompletion, bugfinding,andprogram repair. The IPA-GNN can be seen either as a continuous relaxation of the RNN model or as a GNN variant more tailored to execution.

Learning to control the structure of sentences is a challenging problem in text generation. Existing work either relies on simple deterministic approaches or RL-based hard structures. We explore the use of structured variational autoencoders to infer latent templates for sentence generation using a soft, continuous relaxation in order to utilize reparameterization for training. Specifically, we propose a Gumbel-CRF, a continuous relaxation of the CRF sampling algorithm using a relaxed Forward-Filtering Backward-Sampling (FFBS) approach. As a reparameterized gradient estimator, the Gumbel-CRF gives more stable gradients than score-function based estimators. As a structured inference network, we show that it learns interpretable templates during training, which allows us to control the decoder during testing. We demonstrate the effectiveness of our methods with experiments on data-to-text generation and unsupervised paraphrase generation.

Similarity-based Hierarchical Clustering (HC) is a classical unsupervised machine learning algorithm that has traditionally been solved with heuristic algorithms like Average-Linkage. Recently, Dasgupta reframed HC as a discrete optimization problem by introducing a global cost function measuring the quality of a given tree. In this work, we provide the first continuous relaxation of Dasgupta's discrete optimization problem with provable quality guarantees. The key idea of our method, HypHC, is showing a direct correspondence from discrete trees to continuous representations (via the hyperbolic embeddings of their leaf nodes) and back (via a decoding algorithm that maps leaf embeddings to a dendrogram), allowing us to search the space of discrete binary trees with continuous optimization. Building on analogies between trees and hyperbolic space, we derive a continuous analogue for the notion of lowest common ancestor, which leads to a continuous relaxation of Dasgupta's discrete objective. We can show that after decoding, the global minimizer of our continuous relaxation yields a discrete tree with a (1+eps)-factor approximation for Dasgupta's optimal tree, where eps can be made arbitrarily small and controls optimization challenges. We experimentally evaluate HypHC on a variety of HC benchmarks and find that even approximate solutions found with gradient descent have superior clustering quality than agglomerative heuristics or other gradient based algorithms. Finally, we highlight the flexibility of HypHC using end-to-end training in a downstream classification task.

The integration of algorithmic components into neural architectures has gained increased attention recently, as it allows training neural networks with new forms of supervision such as ordering constraints or silhouettes instead of using ground truth labels. Many approaches in the field focus on the continuous relaxation of a specific task and show promising results in this context. But the focus on single tasks also limits the applicability of the proposed concepts to a narrow range of applications. In this work, we build on those ideas to propose an approach that allows to integrate algorithms into end-to-end trainable neural network architectures based on a general approximation of discrete conditions. To this end, we relax these conditions in control structures such as conditional statements, loops, and indexing, so that resulting algorithms are smoothly differentiable. To obtain meaningful gradients, each relevant variable is perturbed via logistic distributions and the expectation value under this perturbation is approximated. We evaluate the proposed continuous relaxation model on four challenging tasks and show that it can keep up with relaxations specifically designed for each individual task.

We revisit the classical problem of optimal experimental design (OED) under a new mathematical model grounded in a geometric motivation. Specifically, we introduce models based on elementary symmetric polynomials; these polynomials capture "partial volumes" and offer a graded interpolation between the widely used A-optimal design and D-optimal design models, obtaining each of them as special cases. We analyze properties of our models, and derive both greedy and convex-relaxation algorithms for computing the associated designs. Our analysis establishes approximation guarantees on these algorithms, while our empirical results substantiate our claims and demonstrate a curious phenomenon concerning our greedy method. Finally, as a byproduct, we obtain new results on the theory of elementary symmetric polynomials that may be of independent interest.