The MDS Queue: Analysing the Latency Performance of Erasure Codes

In order to scale economically, data centers are increasingly evolving their data storage methods from the use of simple data replication to the use of more powerful erasure codes, which provide the same level of reliability as replication but at a significantly lower storage cost. In particular, it is well known that Maximum-Distance-Separable (MDS) codes, such as Reed-Solomon codes, provide the maximum storage efficiency. While the use of codes for providing improved reliability in archival storage systems, where the data is less frequently accessed (or so-called "cold data"), is well understood, the role of codes in the storage of more frequently accessed and active "hot data", where latency is the key metric, is less clear. In this paper, we study data storage systems based on MDS codes through the lens of queueing theory, and term this the "MDS queue." We analytically characterize the (average) latency performance of MDS queues, for which we present insightful scheduling policies that form upper and lower bounds to performance, and are observed to be quite tight. Extensive simulations are also provided and used to validate our theoretical analysis. We also employ the framework of the MDS queue to analyse different methods of performing so-called degraded reads (reading of partial data) in distributed data storage

When Do Redundant Requests Reduce Latency ?

Several systems possess the flexibility to serve requests in more than one way. For instance, a distributed storage system storing multiple replicas of the data can serve a request from any of the multiple servers that store the requested data, or a computational task may be performed in a compute-cluster by any one of multiple processors. In such systems, the latency of serving the requests may potentially be reduced by sending "redundant requests": a request may be sent to more servers than needed, and it is deemed served when the requisite number of servers complete service. Such a mechanism trades off the possibility of faster execution of at least one copy of the request with the increase in the delay due to an increased load on the system. Due to this tradeoff, it is unclear when redundant requests may actually help. Several recent works empirically evaluate the latency performance of redundant requests in diverse settings. This work aims at an analytical study of the latency performance of redundant requests, with the primary goals of characterizing under what scenarios sending redundant requests will help (and under what scenarios they will not help), as well as designing optimal redundant-requesting policies. We first present a model that captures the key features of such systems. We show that when service times are i.i.d. memoryless or "heavier", and when the additional copies of already-completed jobs can be removed instantly, redundant requests reduce the average latency. On the other hand, when service times are "lighter" or when service times are memoryless and removal of jobs is not instantaneous, then not having any redundancy in the requests is optimal under high loads. Our results hold for arbitrary arrival processes.Comment: Extended version of paper presented at Allerton Conference 201

The Expressive Power of Low-Rank Adaptation

Low-Rank Adaptation (LoRA), a parameter-efficient fine-tuning method that leverages low-rank adaptation of weight matrices, has emerged as a prevalent technique for fine-tuning pre-trained models such as large language models and diffusion models. Despite its huge success in practice, the theoretical underpinnings of LoRA have largely remained unexplored. This paper takes the first step to bridge this gap by theoretically analyzing the expressive power of LoRA. We prove that, for fully connected neural networks, LoRA can adapt any model $f$ to accurately represent any smaller target model $\overline{f}$ if LoRA-rank $\geq(\text{width of }f) \times \frac{\text{depth of }\overline{f}}{\text{depth of }f}$. We also quantify the approximation error when LoRA-rank is lower than the threshold. For Transformer networks, we show any model can be adapted to a target model of the same size with rank-$(\frac{\text{embedding size}}{2})$ LoRA adapters.Comment: 40 pages, 5 figure