A thorough theoretical understanding of the surprising generalization ability of deep networks (and other overparameterized models) is still lacking. Here we demonstrate that simplicity bias is a major phenomenon to be reckoned with in overparameterized machine learning. In addition to explaining the outcome of simplicity bias, we also study its source: following concrete rigorous examples, we argue that (i) simplicity bias can explain generalization in overparameterized learning models such as neural networks; (ii) simplicity bias and excellent generalization are optimizer-independent, as our example shows, and although the optimizer affects training, it is not the driving force behind simplicity bias; (iii) simplicity bias in pre-training models, and subsequent posteriors, is universal and stems from the subtle fact that uniformly-at-random constructed priors are not uniformly-at-random sampled ; and (iv) in neural network models, the biasing mechanism in wide (and shallow) networks is different from the biasing mechanism in deep (and narrow) networks.

Recent advances in systems of networked sensors have set the stage for smart environments which will have wide-ranging applications from intelligent wildlife monitoring to social applications such as health and elderly care service provisioning. Perhaps the most natural problem in sensor systems is the ¿efficient¿ propagation of a sensed local event. In order to address this problem, the notion of greedy embedding was defined by Papadimitriou and Ratajczak, where the authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, the greedy embedding conjecture was proved by Leighton and Moitra. However, their algorithm does not result in a drawing that is planar and convex in the Euclidean plane for all 3-connected planar graphs. Here we give a random algorithm for embedding 3-connected planar graphs a greedy convex embedding. Our convex embedding is especially useful for the case of sensor networks, where the position assigned to each sensor is the midpoint of the positions of its neighbors.

Recent advances in networked systems and wireless communications have set the stage for applications with wide-ranging benefits. Perhaps the most natural problem in such systems is the ¿efficient¿ propagation of locally stored data. In order to address this problem, the notion of greedy embedding was defined by Papadimitriou and Ratajczak, where the authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, the greedy embedding conjecture was proved by Leighton and Moitra. However, their algorithm does not result in a drawing that is planar and convex in the Euclidean plane for all 3-connected planar graphs. Here we consider the planar convex greedy embedding conjecture and give a probabilistic proof for the existence of such embeddings. In addition, we discuss a second proof which is almost immediate in the case of an embedding into the 3-dimensional sphere.