Constructing Kites to Integrate Mathematics and Arts Concepts

This article describes a tetrahedral kite activity that was implemented with grade 9 students (age 14-15). We detail how the three-part lesson provided opportunities to integrate mathematics and art concepts, with potential to also weave in science and engineering ideas. The first part primed students to consider tetrahedral kites, their cultural and historical significance, and the materials needed for constructing the kite. The second part had students create a prototype using nets of tissue paper decorated with mark making techniques. The third part had students create a tetrahedron kite containing cultural and geographical mark making techniques on the tissue paper sides before flying the kites at a community event. We conclude the article with recommendations to help other teachers integrate mathematics and visual arts topics through tetrahedral kites

Genuinely Distributed Byzantine Machine Learning

Machine Learning (ML) solutions are nowadays distributed, according to the so-called server/worker architecture. One server holds the model parameters while several workers train the model. Clearly, such architecture is prone to various types of component failures, which can be all encompassed within the spectrum of a Byzantine behavior. Several approaches have been proposed recently to tolerate Byzantine workers. Yet all require trusting a central parameter server. We initiate in this paper the study of the ``general'' Byzantine-resilient distributed machine learning problem where no individual component is trusted. We show that this problem can be solved in an asynchronous system, despite the presence of $\frac{1}{3}$ Byzantine parameter servers and $\frac{1}{3}$ Byzantine workers (which is optimal). We present a new algorithm, ByzSGD, which solves the general Byzantine-resilient distributed machine learning problem by relying on three major schemes. The first, Scatter/Gather, is a communication scheme whose goal is to bound the maximum drift among models on correct servers. The second, Distributed Median Contraction (DMC), leverages the geometric properties of the median in high dimensional spaces to bring parameters within the correct servers back close to each other, ensuring learning convergence. The third, Minimum-Diameter Averaging (MDA), is a statistically-robust gradient aggregation rule whose goal is to tolerate Byzantine workers. MDA requires loose bound on the variance of non-Byzantine gradient estimates, compared to existing alternatives (e.g., Krum). Interestingly, ByzSGD ensures Byzantine resilience without adding communication rounds (on a normal path), compared to vanilla non-Byzantine alternatives. ByzSGD requires, however, a larger number of messages which, we show, can be reduced if we assume synchrony.Comment: This is a merge of arXiv:1905.03853 and arXiv:1911.07537; arXiv:1911.07537 will be retracte

Mitigating the Performance Impact of Network Failures in Public Clouds

Some faults in data center networks require hours to days to repair because they may need reboots, re-imaging, or manual work by technicians. To reduce traffic impact, cloud providers \textit{mitigate} the effect of faults, for example, by steering traffic to alternate paths. The state-of-art in automatic network mitigations uses simple safety checks and proxy metrics to determine mitigations. SWARM, the approach described in this paper, can pick orders of magnitude better mitigations by estimating end-to-end connection-level performance (CLP) metrics. At its core is a scalable CLP estimator that quickly ranks mitigations with high fidelity and, on failures observed at a large cloud provider, outperforms the state-of-the-art by over 700$\times$ in some cases