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In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply our results to obtain a lower bound on the smallest eigenvalue of the associahedron graph, and we prove that this bound gives the correct order of magnitude of this eigenvalue. We also survey what is known regarding the second-largest eigenvalue of the associahedron graph., Comment: 12 pages, 1 figure

Travel-time prediction constitutes a task of high importance in transportation networks, with web mapping services like Google Maps regularly serving vast quantities of travel time queries from users and enterprises alike. Further, such a task requires accounting for complex spatiotemporal interactions (modelling both the topological properties of the road network and anticipating events -- such as rush hours -- that may occur in the future). Hence, it is an ideal target for graph representation learning at scale. Here we present a graph neural network estimator for estimated time of arrival (ETA) which we have deployed in production at Google Maps. While our main architecture consists of standard GNN building blocks, we further detail the usage of training schedule methods such as MetaGradients in order to make our model robust and production-ready. We also provide prescriptive studies: ablating on various architectural decisions and training regimes, and qualitative analyses on real-world situations where our model provides a competitive edge. Our GNN proved powerful when deployed, significantly reducing negative ETA outcomes in several regions compared to the previous production baseline (40+% in cities like Sydney)., Comment: To appear at CIKM 2021 (Applied Research Track). 10 pages, 4 figures

Motivated by the poor performance of cross-validation in settings where data are scarce, we propose a novel estimator of the out-of-sample performance of a policy in data-driven optimization.Our approach exploits the optimization problem's sensitivity analysis to estimate the gradient of the optimal objective value with respect to the amount of noise in the data and uses the estimated gradient to debias the policy's in-sample performance. Unlike cross-validation techniques, our approach avoids sacrificing data for a test set, utilizes all data when training and, hence, is well-suited to settings where data are scarce. We prove bounds on the bias and variance of our estimator for optimization problems with uncertain linear objectives but known, potentially non-convex, feasible regions. For more specialized optimization problems where the feasible region is "weakly-coupled" in a certain sense, we prove stronger results. Specifically, we provide explicit high-probability bounds on the error of our estimator that hold uniformly over a policy class and depends on the problem's dimension and policy class's complexity. Our bounds show that under mild conditions, the error of our estimator vanishes as the dimension of the optimization problem grows, even if the amount of available data remains small and constant. Said differently, we prove our estimator performs well in the small-data, large-scale regime. Finally, we numerically compare our proposed method to state-of-the-art approaches through a case-study on dispatching emergency medical response services using real data. Our method provides more accurate estimates of out-of-sample performance and learns better-performing policies.

High-performance branch target buffers (BTBs) and the L1I cache are key to high-performance front-end. Modern branch predictors are highly accurate, but with an increase in code footprint in modern-day server workloads, BTB and L1I misses are still frequent. Recent industry trend shows usage of large BTBs (100s of KB per core) that provide performance closer to the ideal BTB along with a decoupled front-end that provides efficient fetch-directed L1I instruction prefetching. On the other hand, techniques proposed by academia, like BTB prefetching and using retire order stream for learning, fail to provide significant performance with modern-day processor cores that are deeper and wider. We solve the problem fundamentally by increasing the storage density of the last-level BTB. We observe that not all branch instructions require a full branch target address. Instead, we can store the branch target as a branch offset, relative to the branch instruction. Using branch offset enables the BTB to store multiple branches per entry. We reduce the BTB storage in half, but we observe that it increases skewness in the BTB. We propose a skewed indexed and compressed last-level BTB design called MicroBTB (MBTB) that stores multiple branches per BTB entry. We evaluate MBTB on 100 industry-provided server workloads. A 4K-entry MBTB provides 17.61% performance improvement compared to an 8K-entry baseline BTB design with a storage savings of 47.5KB per core., Comment: 9 pages