Transformers have emerged as a preferred model for many tasks in natural langugage processing and vision. Recent efforts on training and deploying Transformers more efficiently have identified many strategies to approximate the self-attention matrix, a key module in a Transformer architecture. Effective ideas include various prespecified sparsity patterns, low-rank basis expansions and combinations thereof. In this paper, we revisit classical Multiresolution Analysis (MRA) concepts such as Wavelets, whose potential value in this setting remains underexplored thus far. We show that simple approximations based on empirical feedback and design choices informed by modern hardware and implementation challenges, eventually yield a MRA-based approach for self-attention with an excellent performance profile across most criteria of interest. We undertake an extensive set of experiments and demonstrate that this multi-resolution scheme outperforms most efficient self-attention proposals and is favorable for both short and long sequences. Code is available at \url{https://github.com/mlpen/mra-attention}.

User agent (UA) strings transmitted during HTTP transactions convey client system configuration details to ensure that content returned by a server is appropriate for the requesting host. As such, analysis of UA strings and their structure offers a unique perspective on active client systems in the Internet and when tracked longitudinally, offers a perspective on the nature of system and configuration dynamics. In this paper, we describe our study of UA string characteristics. Our work is based on analyzing a unique corpus of over 1B UA strings collected over a period of 2 years by comScore. We begin by analyzing the general characteristics of UA strings, focusing on the most prevalent strings and dynamic behaviors. We identify the top 10 most popular User Agents, which account for 26% of total daily volume. These strings describe the expected instances of popular platforms such as Microsoft, Apple and Google. We then report on the characteristics of low-volume UA strings, which has important implications for unique device identification. We show that this class of user agent generates the overwhelming majority of traffic, with between 2M and 10M instances observed each day. We show that the distribution of UA strings has temporal dependence and we show the distribution measured depends on the type of content served. Finally, we report on two large-scale UA anomalies characterized by web browsers sending false and misleading UAs in their web requests.

Identifying anomalies rapidly and accurately is critical to the efficient operation of large computer networks. Accurately characterizing important classes of anomalies greatly facilitates their identification; however, the subtleties and complexities of anomalous traffic can easily confound this process. In this paper we report results of signal analysis of four classes of network traffic anomalies: outages, flash crowds, attacks and measurement failures. Data for this study consists of IP flow and SNMP measurements collected over a six month period at the border router of a large university. Our results show that wavelet filters are quite effective at exposing the details of both ambient and anomalous traffic. Specifically, we show that a pseudo-spline filter tuned at specific aggregation levels will expose distinct characteristics of each class of anomaly. We show that an effective way of exposing anomalies is via the detection of a sharp increase in the local variance of the filtered data. We evaluate traffic anomaly signals at different points within a network based on topological distance from the anomaly source or destination. We show that anomalies can be exposed effectively even when aggregated with a large amount of additional traffic. We also compare the difference between the same traffic anomaly signals as seen in SNMP and IP flow data, and show that the more coarse-grained SNMP data can also be used to expose anomalies effectively.