Search for: All records

Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$. 

In virtual machine (VM) allocation systems, caching repetitive and similar VM allocation requests and associated resolution rules is crucial for reducing computational costs and meeting strict latency requirements. While modern allocation systems distribute requests among multiple allocator agents and use caching to improve performance, current schedulers often neglect the cache state and latency considerations when assigning each new request to an agent. Due to the high variance in costs of cache hits and misses and the associated processing overheads of updating the caches, simple load-balancing and cache-aware mechanisms result in high latencies. We introduce Kamino, a high-performance, latencydriven and cache-aware request scheduling system aimed at minimizing end-to-end latencies. Kamino employs a novel scheduling algorithm grounded in theory which uses partial indicators from the cache state to assign each new request to the agent with the lowest estimated latency. Evaluation of Kamino using a high-fidelity simulator on large-scale production workloads shows a 42% reduction in average request latencies. Our deployment of Kamino in the control plane of a large public cloud confirms these improvements, with a 33% decrease in cache miss rates and a 17% reduction in memory usage 

Recent work has shown that Transformers trained from scratch can successfully solve various arithmetic and algorithmic tasks, such as adding numbers and computing parity. While these Transformers generalize well on unseen inputs of the same length, they struggle with length generalization, i.e., handling inputs of unseen lengths. In this work, we demonstrate that looped Transformers with an adaptive number of steps significantly improve length generalization. We focus on tasks with a known iterative solution, involving multiple iterations of a RASP-L operation—a length-generalizable operation that can be expressed by a finite-sized Transformer. We train looped Transformers using our proposed learning algorithm and observe that they learn highly length-generalizable solutions for various tasks.