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Augmented Large Language Models (LLMs) enhance the capabilities of standalone LLMs by integrating external data sources through API calls. In interactive LLM applications, efficient scheduling is crucial for maintaining low request completion times, directly impacting user engagement. However, these augmentations introduce scheduling challenges due to the need to manage limited memory for cached information (KV caches). As a result, traditional size-based scheduling algorithms, such as Shortest Job First (SJF), become less effective at minimizing completion times. Existing work focuses only on handling requests during API calls by preserving, discarding, or swapping memory without considering how to schedule requests with API calls. In this paper, we propose LAMPS, a novel LLM inference framework for augmented LLMs. LAMPS minimizes request completion time through a unified scheduling approach that considers the total length of requests and their handling strategies during API calls. Recognizing that LLM inference is memory-bound, our approach ranks requests based on their consumption of memory over time, which depends on both the output sizes and how a request is managed during its API calls. To implement our scheduling, LAMPS predicts the strategy that minimizes memory waste of a request during its API calls, aligning with but improving upon existing approaches. We also propose starvation prevention techniques and optimizations to mitigate the overhead of our scheduling. We implement LAMPS on top of vLLM and evaluate its performance against baseline LLM inference systems, demonstrating improvements in end-to-end latency by 27%-85% and reductions in TTFT by 4%-96% compared to the existing augmented-LLM system, with even greater gains over vLLM.

Efficient scheduling is crucial for interactive Large Language Model (LLM) applications, where low request completion time directly impacts user engagement. Size-based scheduling algorithms like Shortest Remaining Process Time (SRPT) aim to reduce average request completion time by leveraging known or estimated request sizes and allowing preemption by incoming jobs with shorter service times. However, two main challenges arise when applying size-based scheduling to LLM systems. First, accurately predicting output lengths from prompts is challenging and often resource-intensive, making it impractical for many systems. As a result, the state-of-the-art LLM systems default to first-come, first-served scheduling, which can lead to head-of-line blocking and reduced system efficiency. Second, preemption introduces extra memory overhead to LLM systems as they must maintain intermediate states for unfinished (preempted) requests. In this paper, we propose TRAIL, a method to obtain output predictions from the target LLM itself. After generating each output token, we recycle the embedding of its internal structure as input for a lightweight classifier that predicts the remaining length for each running request. Using these predictions, we propose a prediction-based SRPT variant with limited preemption designed to account for memory overhead in LLM systems. This variant allows preemption early in request execution when memory consumption is low but restricts preemption as requests approach completion to optimize resource utilization. On the theoretical side, we derive a closed-form formula for this SRPT variant in an M/G/1 queue model, which demonstrates its potential value. In our system, we implement this preemption policy alongside our embedding-based prediction method.

We show how to utilize machine learning approaches to improve sliding window algorithms for approximate frequency estimation problems, under the ``algorithms with predictions'' framework. In this dynamic environment, previous learning-augmented algorithms are less effective, since properties in sliding window resolution can differ significantly from the properties of the entire stream. Our focus is on the benefits of predicting and filtering out items with large next arrival times -- that is, there is a large gap until their next appearance -- from the stream, which we show improves the memory-accuracy tradeoffs significantly. We provide theorems that provide insight into how and by how much our technique can improve the sliding window algorithm, as well as experimental results using real-world data sets. Our work demonstrates that predictors can be useful in the challenging sliding window setting.

Gradient aggregation has long been identified as a major bottleneck in today's large-scale distributed machine learning training systems. One promising solution to mitigate such bottlenecks is gradient compression, directly reducing communicated gradient data volume. However, in practice, many gradient compression schemes do not achieve acceleration of the training process while also preserving accuracy. In this work, we identify several common issues in previous gradient compression systems and evaluation methods. These issues include excessive computational overheads; incompatibility with all-reduce; and inappropriate evaluation metrics, such as not using an end-to-end metric or using a 32-bit baseline instead of a 16-bit baseline. We propose several general design and evaluation techniques to address these issues and provide guidelines for future work. Our preliminary evaluation shows that our techniques enhance the system's performance and provide a clearer understanding of the end-to-end utility of gradient compression methods., Comment: 9 pages, 3 figures

Identifying heavy hitters and estimating the frequencies of flows are fundamental tasks in various network domains. Existing approaches to this challenge can broadly be categorized into two groups, hashing-based and competing-counter-based. The Count-Min sketch is a standard example of a hashing-based algorithm, and the Space Saving algorithm is an example of a competing-counter algorithm. Recent works have explored the use of machine learning to enhance algorithms for frequency estimation problems, under the algorithms with prediction framework. However, these works have focused solely on the hashing-based approach, which may not be best for identifying heavy hitters. In this paper, we present the first learned competing-counter-based algorithm, called LSS, for identifying heavy hitters, top k, and flow frequency estimation that utilizes the well-known Space Saving algorithm. We provide theoretical insights into how and to what extent our approach can improve upon Space Saving, backed by experimental results on both synthetic and real-world datasets. Our evaluation demonstrates that LSS can enhance the accuracy and efficiency of Space Saving in identifying heavy hitters, top k, and estimating flow frequencies.

In light of recent work on scheduling with predicted job sizes, we consider the effect of the cost of predictions in queueing systems, removing the assumption in prior research that predictions are external to the system's resources and/or cost-free. In particular, we introduce a novel approach to utilizing predictions, SkipPredict, designed to address their inherent cost. Rather than uniformly applying predictions to all jobs, we propose a tailored approach that categorizes jobs based on their prediction requirements. To achieve this, we employ one-bit "cheap predictions" to classify jobs as either short or long. SkipPredict prioritizes predicted short jobs over long jobs, and for the latter, SkipPredict applies a second round of more detailed "expensive predictions" to approximate Shortest Remaining Processing Time for these jobs. Our analysis takes into account the cost of prediction. We examine the effect of this cost for two distinct models. In the external cost model, predictions are generated by some external method without impacting job service times but incur a cost. In the server time cost model, predictions themselves require server processing time, and are scheduled on the same server as the jobs.

Quantization is a fundamental optimization for many machine-learning use cases, including compressing gradients, model weights and activations, and datasets. The most accurate form of quantization is \emph{adaptive}, where the error is minimized with respect to a given input, rather than optimizing for the worst case. However, optimal adaptive quantization methods are considered infeasible in terms of both their runtime and memory requirements. We revisit the Adaptive Vector Quantization (AVQ) problem and present algorithms that find optimal solutions with asymptotically improved time and space complexity. We also present an even faster near-optimal algorithm for large inputs. Our experiments show our algorithms may open the door to using AVQ more extensively in a variety of machine learning applications.

The randomized play-the-winner (RPW) model is a generalized P\'olya Urn process with broad applications ranging from clinical trials to molecular evolution. We derive an exact expression for the variance of the RPW model by transforming the P\'olya Urn process into a martingale, correcting an earlier result of Matthews and Rosenberger (1997). We then use this result to approximate the full probability mass function of the RPW model for certain parameter values relevant to genetic applications. Finally, we fit our model to genomic sequencing data of SARS-CoV-2, demonstrating a novel method of estimating the viral mutation rate that delivers comparable results to existing scientific literature., Comment: 27 pages, 2 figures

Deep neural networks (DNNs) are the de facto standard for essential use cases, such as image classification, computer vision, and natural language processing. As DNNs and datasets get larger, they require distributed training on increasingly larger clusters. A main bottleneck is the resulting communication overhead where workers exchange model updates (i.e., gradients) on a per-round basis. To address this bottleneck and accelerate training, a widely-deployed approach is compression. However, previous deployments often apply bi-directional compression schemes by simply using a uni-directional gradient compression scheme in each direction. This results in significant computational overheads at the parameter server and increased compression error, leading to longer training and lower accuracy. We introduce Tensor Homomorphic Compression (THC), a novel bi-directional compression framework that enables the direct aggregation of compressed values and thus eliminating the aforementioned computational overheads. Moreover, THC is compatible with in-network aggregation (INA), which allows for further acceleration. Our evaluation shows that training representative vision and language models with THC reaches target accuracy by 1.40x to 1.47x faster using INA and 1.28x to 1.33x faster using a software PS compared with state-of-the-art systems., Comment: 12 pages body, 21 pages total

We introduce Proteus, a novel self-designing approximate range filter, which configures itself based on sampled data in order to optimize its false positive rate (FPR) for a given space requirement. Proteus unifies the probabilistic and deterministic design spaces of state-of-the-art range filters to achieve robust performance across a larger variety of use cases. At the core of Proteus lies our Contextual Prefix FPR (CPFPR) model - a formal framework for the FPR of prefix-based filters across their design spaces. We empirically demonstrate the accuracy of our model and Proteus' ability to optimize over both synthetic workloads and real-world datasets. We further evaluate Proteus in RocksDB and show that it is able to improve end-to-end performance by as much as 5.3x over more brittle state-of-the-art methods such as SuRF and Rosetta. Our experiments also indicate that the cost of modeling is not significant compared to the end-to-end performance gains and that Proteus is robust to workload shifts., Comment: 14 pages, 9 figures, originally published in the Proceedings of the 2022 International Conference on Management of Data (SIGMOD'22), ISBN: 9781450392495

Distributed Mean Estimation (DME), in which $n$ clients communicate vectors to a parameter server that estimates their average, is a fundamental building block in communication-efficient federated learning. In this paper, we improve on previous DME techniques that achieve the optimal $O(1/n)$ Normalized Mean Squared Error (NMSE) guarantee by asymptotically improving the complexity for either encoding or decoding (or both). To achieve this, we formalize the problem in a novel way that allows us to use off-the-shelf mathematical solvers to design the quantization.

Distributed protocols are widely used to support network functions such as clock synchronization and multicast. As the network gets larger and faster, it is increasingly challenging for these protocols to react quickly to network events. The theory community has made significant progress in developing distributed message passing algorithms with improved convergence times. With the emerging programmability at switches, it now becomes feasible to adopt and adapt these theoretical advances for networking functions. In this paper, we propose FRANCIS, a new framework for running message passing algorithms on programmable switches to enable fast reactions to network events in large networks. We introduce an execution engine with computing and communication primitives for supporting message passing algorithms in P4 switches. We exemplify the framework's usefulness by improving the resiliency and reaction times of clock synchronization and source-routed multicast. In particular, our approach allows lower clock drift than Sundial and PTP, quickly recovers from multiple failures, and reduces the time uncertainty bound by up to 5x. Compared with state-of-the-art multicast solutions, our approach uses packet headers up to 33\% smaller and has an order of magnitude faster reaction time.

Multiparty computation approaches to secure neural network inference commonly rely on garbled circuits for securely executing nonlinear activation functions. However, garbled circuits require excessive communication between server and client, impose significant storage overheads, and incur large runtime penalties. To reduce these costs, we propose an alternative to garbled circuits: Tabula, an algorithm based on secure lookup tables. Our approach precomputes lookup tables during an offline phase that contains the result of all possible nonlinear function calls. Because these tables incur exponential storage costs in the number of operands and the precision of the input values, we use quantization to reduce these storage costs to make this approach practical. This enables an online phase where securely computing the result of a nonlinear function requires just a single round of communication, with communication cost equal to twice the number of bits of the input to the nonlinear function. In practice our approach costs 2 bytes of communication per nonlinear function call in the online phase. Compared to garbled circuits with 8-bit quantized inputs, when computing individual nonlinear functions during the online phase, experiments show Tabula with 8-bit activations uses between $280$-$560 \times$ less communication, is over $100\times$ faster, and uses a comparable (within a factor of 2) amount of storage; compared against other state-of-the-art protocols Tabula achieves greater than $40\times$ communication reduction. This leads to significant performance gains over garbled circuits with quantized inputs during the online phase of secure inference of neural networks: Tabula reduces end-to-end inference communication by up to $9 \times$ and achieves an end-to-end inference speedup of up to $50 \times$, while imposing comparable storage and offline preprocessing costs.

For job scheduling systems, where jobs require some amount of processing and then leave the system, it is natural for each user to provide an estimate of their job's time requirement in order to aid the scheduler. However, if there is no incentive mechanism for truthfulness, each user will be motivated to provide estimates that give their job precedence in the schedule, so that the job completes as early as possible. We examine how to make such scheduling systems incentive compatible, without using monetary charges, under a natural queueing theory framework. In our setup, each user has an estimate of their job's running time, but it is possible for this estimate to be incorrect. We examine scheduling policies where if a job exceeds its estimate, it is with some probability "punished" and re-scheduled after other jobs, to disincentivize underestimates of job times. However, because user estimates may be incorrect (without any malicious intent), excessive punishment may incentivize users to overestimate their job times, which leads to less efficient scheduling. We describe two natural scheduling policies, BlindTrust and MeasuredTrust. We show that, for both of these policies, given the parameters of the system, we can efficiently determine the set of punishment probabilities that are incentive compatible, in that users are incentivized to provide their actual estimate of the job time. Moreover, we prove for MeasuredTrust that in the limit as estimates converge to perfect accuracy, the range of punishment probabilities that are incentive compatible converges to $[0,1]$. Our formalism establishes a framework for studying further queue-based scheduling problems where job time estimates from users are utilized, and the system needs to incentivize truthful reporting of estimates., Comment: 20 pages

Fine-grained network telemetry is becoming a modern datacenter standard and is the basis of essential applications such as congestion control, load balancing, and advanced troubleshooting. As network size increases and telemetry gets more fine-grained, there is a tremendous growth in the amount of data needed to be reported from switches to collectors to enable network-wide view. As a consequence, it is progressively hard to scale data collection systems. We introduce Direct Telemetry Access (DTA), a solution optimized for aggregating and moving hundreds of millions of reports per second from switches into queryable data structures in collectors' memory. DTA is lightweight and it is able to greatly reduce overheads at collectors. DTA is built on top of RDMA, and we propose novel and expressive reporting primitives to allow easy integration with existing state-of-the-art telemetry mechanisms such as INT or Marple. We show that DTA significantly improves telemetry collection rates. For example, when used with INT, it can collect and aggregate over 400M reports per second with a single server, improving over the Atomic MultiLog by up to $16$x., Comment: As appearing in the proceedings of ACM SIGCOMM'23

Programmable switches are driving a massive increase in fine-grained measurements. This puts significant pressure on telemetry collectors that have to process reports from many switches. Past research acknowledged this problem by either improving collectors' stack performance or by limiting the amount of data sent from switches. In this paper, we take a different and radical approach: switches are responsible for directly inserting queryable telemetry data into the collectors' memory, bypassing their CPU, and thereby improving their collection scalability. We propose to use a method we call \emph{direct telemetry access}, where switches jointly write telemetry reports directly into the same collector's memory region, without coordination. Our solution, DART, is probabilistic, trading memory redundancy and query success probability for CPU resources at collectors. We prototype DART using commodity hardware such as P4 switches and RDMA NICs and show that we get high query success rates with a reasonable memory overhead. For example, we can collect INT path tracing information on a fat tree topology without a collector's CPU involvement while achieving 99.9\% query success probability and using just 300 bytes per flow., Comment: To appear in ACM HotNets 2021

We consider the problem of scheduling to minimize mean response time in M/G/1 queues where only estimated job sizes (processing times) are known to the scheduler, where a job of true size $s$ has estimated size in the interval $[\beta s, \alpha s]$ for some $\alpha \geq \beta > 0$. We evaluate each scheduling policy by its approximation ratio, which we define to be the ratio between its mean response time and that of Shortest Remaining Processing Time (SRPT), the optimal policy when true sizes are known. Our question: is there a scheduling policy that (a) has approximation ratio near 1 when $\alpha$ and $\beta$ are near 1, (b) has approximation ratio bounded by some function of $\alpha$ and $\beta$ even when they are far from 1, and (c) can be implemented without knowledge of $\alpha$ and $\beta$? We first show that naively running SRPT using estimated sizes in place of true sizes is not such a policy: its approximation ratio can be arbitrarily large for any fixed $\beta < 1$. We then provide a simple variant of SRPT for estimated sizes that satisfies criteria (a), (b), and (c). In particular, we prove its approximation ratio approaches 1 uniformly as $\alpha$ and $\beta$ approach 1. This is the first result showing this type of convergence for M/G/1 scheduling. We also study the Preemptive Shortest Job First (PSJF) policy, a cousin of SRPT. We show that, unlike SRPT, naively running PSJF using estimated sizes in place of true sizes satisfies criteria (b) and (c), as well as a weaker version of (a)., Comment: Published at ITCS 2022

Distributed Mean Estimation (DME) is a central building block in federated learning, where clients send local gradients to a parameter server for averaging and updating the model. Due to communication constraints, clients often use lossy compression techniques to compress the gradients, resulting in estimation inaccuracies. DME is more challenging when clients have diverse network conditions, such as constrained communication budgets and packet losses. In such settings, DME techniques often incur a significant increase in the estimation error leading to degraded learning performance. In this work, we propose a robust DME technique named EDEN that naturally handles heterogeneous communication budgets and packet losses. We derive appealing theoretical guarantees for EDEN and evaluate it empirically. Our results demonstrate that EDEN consistently improves over state-of-the-art DME techniques., Comment: To appear in ICML 2022

We show that aggregated model updates in federated learning may be insecure. An untrusted central server may disaggregate user updates from sums of updates across participants given repeated observations, enabling the server to recover privileged information about individual users' private training data via traditional gradient inference attacks. Our method revolves around reconstructing participant information (e.g: which rounds of training users participated in) from aggregated model updates by leveraging summary information from device analytics commonly used to monitor, debug, and manage federated learning systems. Our attack is parallelizable and we successfully disaggregate user updates on settings with up to thousands of participants. We quantitatively and qualitatively demonstrate significant improvements in the capability of various inference attacks on the disaggregated updates. Our attack enables the attribution of learned properties to individual users, violating anonymity, and shows that a determined central server may undermine the secure aggregation protocol to break individual users' data privacy in federated learning., Comment: ICML 2021

We consider the problem where $n$ clients transmit $d$-dimensional real-valued vectors using $d(1+o(1))$ bits each, in a manner that allows the receiver to approximately reconstruct their mean. Such compression problems naturally arise in distributed and federated learning. We provide novel mathematical results and derive computationally efficient algorithms that are more accurate than previous compression techniques. We evaluate our methods on a collection of distributed and federated learning tasks, using a variety of datasets, and show a consistent improvement over the state of the art., Comment: Appears in NeurIPS 2021

Counters are the fundamental building block of many data sketching schemes, which hash items to a small number of counters and account for collisions to provide good approximations for frequencies and other measures. Most existing methods rely on fixed-size counters, which may be wasteful in terms of space, as counters must be large enough to eliminate any risk of overflow. Instead, some solutions use small, fixed-size counters that may overflow into secondary structures. This paper takes a different approach. We propose a simple and general method called SALSA for dynamic re-sizing of counters and show its effectiveness. SALSA starts with small counters, and overflowing counters simply merge with their neighbors. SALSA can thereby allow more counters for a given space, expanding them as necessary to represent large numbers. Our evaluation demonstrates that, at the cost of a small overhead for its merging logic, SALSA significantly improves the accuracy of popular schemes (such as Count-Min Sketch and Count Sketch) over a variety of tasks. Our code is released as open-source [1]., Comment: An extended version of the conference paper that will appear in IEEE ICDE 2021

In this paper, we provide new approximation algorithms for dynamic variations of the longest increasing subsequence (\textsf{LIS}) problem, and the complementary distance to monotonicity (\textsf{DTM}) problem. In this setting, operations of the following form arrive sequentially: (i) add an element, (ii) remove an element, or (iii) substitute an element for another. At every point in time, the algorithm has an approximation to the longest increasing subsequence (or distance to monotonicity). We present a $(1+\epsilon)$-approximation algorithm for \textsf{DTM} with polylogarithmic worst-case update time and a constant factor approximation algorithm for \textsf{LIS} with worst-case update time $\tilde O(n^\epsilon)$ for any constant $\epsilon > 0$.% $n$ in the runtime denotes the size of the array at the time the operation arrives. Our dynamic algorithm for \textsf{LIS} leads to an almost optimal algorithm for the Erd\"{o}s-Szekeres partitioning problem. Erd\"{o}s-Szekeres partitioning problem was introduced by Erd\"{o}s and Szekeres in 1935 and was known to be solvable in time $O(n^{1.5}\log n)$. Subsequent work improve the runtime to $O(n^{1.5})$ only in 1998. Our dynamic \textsf{LIS} algorithm leads to a solution for Erd\"{o}s-Szekeres partitioning problem with runtime $\tilde O_{\epsilon}(n^{1+\epsilon})$ for any constant $\epsilon > 0$.

In this note, we present a substantial improvement on the computational complexity of the Erd\"{o}s-Szekeres partitioning problem and review recent works on dynamic \textsf{LIS}.

We consider the fundamental problem of communicating an estimate of a real number $x\in[0,1]$ using a single bit. A sender that knows $x$ chooses a value $X\in\set{0,1}$ to transmit. In turn, a receiver estimates $x$ based on the value of $X$. We consider both the biased and unbiased estimation problems and aim to minimize the cost. For the biased case, the cost is the worst-case (over the choice of $x$) expected squared error, which coincides with the variance if the algorithm is required to be unbiased. We first overview common biased and unbiased estimation approaches and prove their optimality when no shared randomness is allowed. We then show how a small amount of shared randomness, which can be as low as a single bit, reduces the cost in both cases. Specifically, we derive lower bounds on the cost attainable by any algorithm with unrestricted use of shared randomness and propose near-optimal solutions that use a small number of shared random bits. Finally, we discuss open problems and future directions.

Commodity network devices support adding in-band telemetry measurements into data packets, enabling a wide range of applications, including network troubleshooting, congestion control, and path tracing. However, including such information on packets adds significant overhead that impacts both flow completion times and application-level performance. We introduce PINT, an in-band telemetry framework that bounds the amount of information added to each packet. PINT encodes the requested data on multiple packets, allowing per-packet overhead limits that can be as low as one bit. We analyze PINT and prove performance bounds, including cases when multiple queries are running simultaneously. PINT is implemented in P4 and can be deployed on network devices. Using real topologies and traffic characteristics, we show that PINT concurrently enables applications such as congestion control, path tracing, and computing tail latencies, using only sixteen bits per packet, with performance comparable to the state of the art., Comment: To appear in ACM SIGCOMM 2020

Motivated by recent work on scheduling with predicted job sizes, we consider the performance of scheduling algorithms with minimal advice, namely a single bit. Besides demonstrating the power of very limited advice, such schemes are quite natural. In the prediction setting, one bit of advice can be used to model a simple prediction as to whether a job is "large" or "small"; that is, whether a job is above or below a given threshold. Further, one-bit advice schemes can correspond to mechanisms that tell whether to put a job at the front or the back for the queue, a limitation which may be useful in many implementation settings. Finally, queues with a single bit of advice have a simple enough state that they can be analyzed in the limiting mean-field analysis framework for the power of two choices. Our work follows in the path of recent work by showing that even small amounts of even possibly inaccurate information can greatly improve scheduling performance., Comment: 16 pages, draft version, to be submitted, subject to cahnge

We introduce algorithms that use predictions from machine learning applied to the input to circumvent worst-case analysis. We aim for algorithms that have near optimal performance when these predictions are good, but recover the prediction-less worst case behavior when the predictions have large errors., Comment: survey is to appear as a chapter in Beyond the Worst-Case Analysis of Algorithms, a collection edited by Tim Roughgarden. We hope to occasionally update the survey here, with new versions that include discussions of new results and advances in the area of Algorithms with Predictions

Bloom filters are space-efficient probabilistic data structures that are used to test whether an element is a member of a set, and may return false positives. Recently, variations referred to as learned Bloom filters were developed that can provide improved performance in terms of the rate of false positives, by using a learned model for the represented set. However, previous methods for learned Bloom filters do not take full advantage of the learned model. Here we show how to frame the problem of optimal model utilization as an optimization problem, and using our framework derive algorithms that can achieve near-optimal performance in many cases. Experimental results from both simulated and real-world datasets show significant performance improvements from our optimization approach over both the original learned Bloom filter constructions and previously proposed heuristic improvements., Comment: 13 pages, 3 figures

Counters are a fundamental building block for networking applications such as load balancing, traffic engineering, and intrusion detection, which require estimating flow sizes and identifying heavy hitter flows. Existing works suggest replacing counters with shorter multiplicative error \emph{estimators} that improve the accuracy by fitting more of them within a given space. However, such estimators impose a computational overhead that degrades the measurement throughput. Instead, we propose \emph{additive} error estimators, which are simpler, faster, and more accurate when used for network measurement. Our solution is rigorously analyzed and empirically evaluated against several other measurement algorithms on real Internet traces. For a given error target, we improve the speed of the uncompressed solutions by $5\times$-$30\times$, and the space by up to $4\times$. Compared with existing state-of-the-art estimators, our solution is $ 9\times$-$35\times$ faster while being considerably more accurate., Comment: To appear in IEEE INFOCOM 2020

In this work we consider the problem of recovering $n$ discrete random variables $x_i\in \{0,\ldots,k-1\}, 1 \leq i \leq n$ (where $k$ is constant) with the smallest possible number of queries to a noisy oracle that returns for a given query pair $(x_i,x_j)$ a noisy measurement of their modulo $k$ pairwise difference, i.e., $y_{ij} = (x_i-x_j) \mod k$. This is a joint discrete alignment problem with important applications in computer vision, graph mining, and spectroscopy imaging. Our main result is a polynomial time algorithm that learns exactly with high probability the alignment (up to some unrecoverable offset) using $O(n^{1+o(1)})$ queries., Comment: Paper appeared in the 15th Workshop on Algorithms and Models for the Web Graph (WAW 2018), invited to Internet Mathematics special issue. Overlaps in text with earlier unpublished note arxiv:1609.00750. (v2 minor updates)

Suppose a customer is faced with a sequence of fluctuating prices, such as for airfare or a product sold by a large online retailer. Given distributional information about what price they might face each day, how should they choose when to purchase in order to maximize the likelihood of getting the best price in retrospect? This is related to the classical secretary problem, but with values drawn from known distributions. In their pioneering work, Gilbert and Mosteller [\textit{J. Amer. Statist. Assoc. 1966}] showed that when the values are drawn i.i.d., there is a thresholding algorithm that selects the best value with probability approximately $0.5801$. However, the more general problem with non-identical distributions has remained unsolved. In this paper we provide an algorithm for the case of non-identical distributions that selects the maximum element with probability $1/e$, and we show that this is tight. We further show that if the observations arrive in a random order, this barrier of $1/e$ can be broken using a static threshold algorithm, and we show that our success probability is the best possible for any single-threshold algorithm under random observation order. Moreover, we prove that one can achieve a strictly better success probability using more general multi-threshold algorithms, unlike the non-random-order case. Along the way, we show that the best achievable success probability for the random-order case matches that of the i.i.d.\ case, which is approximately $0.5801$, under a "no-superstars" condition that no single distribution is very likely ex ante to generate the maximum value. We also extend our results to the problem of selecting one of the $k$ best values.

We consider the following problem, which is useful in applications such as joint image and shape alignment. The goal is to recover $n$ discrete variables $g_i \in \{0, \ldots, k-1\}$ (up to some global offset) given noisy observations of a set of their pairwise differences $\{(g_i - g_j) \bmod k\}$; specifically, with probability $\frac{1}{k}+\delta$ for some $\delta > 0$ one obtains the correct answer, and with the remaining probability one obtains a uniformly random incorrect answer. We consider a learning-based formulation where one can perform a query to observe a pairwise difference, and the goal is to perform as few queries as possible while obtaining the exact joint alignment. We provide an easy-to-implement, time efficient algorithm that performs $O\big(\frac{n \lg n}{k \delta^2}\big)$ queries, and recovers the joint alignment with high probability. We also show that our algorithm is optimal by proving a general lower bound that holds for all non-adaptive algorithms. Our work improves significantly recent work by Chen and Cand\'{e}s \cite{chen2016projected}, who view the problem as a constrained principal components analysis problem that can be solved using the power method. Specifically, our approach is simpler both in the algorithm and the analysis, and provides additional insights into the problem structure., Comment: 10 pages

The supermarket model refers to a system with a large number of queues, where new customers choose d queues at random and join the one with the fewest customers. This model demonstrates the power of even small amounts of choice, as compared to simply joining a queue chosen uniformly at random, for load balancing systems. In this work we perform simulation-based studies to consider variations where service times for a customer are predicted, as might be done in modern settings using machine learning techniques or related mechanisms. Our primary takeaway is that using even seemingly weak predictions of service times can yield significant benefits over blind First In First Out queueing in this context. However, some care must be taken when using predicted service time information to both choose a queue and order elements for service within a queue; while in many cases using the information for both choosing and ordering is beneficial, in many of our simulation settings we find that simply using the number of jobs to choose a queue is better when using predicted service times to order jobs in a queue. In our simulations, we evaluate both synthetic and real-world workloads--in the latter, service times are predicted by machine learning. Our results provide practical guidance for the design of real-world systems; moreover, we leave many natural theoretical open questions for future work, validating their relevance to real-world situations., Comment: Revision, 16 pages, 22 figures. Published at IEEE TPDS

In a $k$-party communication problem, the $k$ players with inputs $x_1, x_2, \ldots, x_k$, respectively, want to evaluate a function $f(x_1, x_2, \ldots, x_k)$ using as little communication as possible. We consider the message-passing model, in which the inputs are partitioned in an arbitrary, possibly worst-case manner, among a smaller number $t$ of players ($t

In many traditional job scheduling settings, it is assumed that one knows the time it will take for a job to complete service. In such cases, strategies such as shortest job first can be used to improve performance in terms of measures such as the average time a job waits in the system. We consider the setting where the service time is not known, but is predicted by for example a machine learning algorithm. Our main result is the derivation, under natural assumptions, of formulae for the performance of several strategies for queueing systems that use predictions for service times in order to schedule jobs. As part of our analysis, we suggest the framework of the "price of misprediction," which offers a measure of the cost of using predicted information., Comment: 14 pages, submitted; version 2 has updated references, some corrections

We consider online variations of the Pandora's box problem (Weitzman. 1979), a standard model for understanding issues related to the cost of acquiring information for decision-making. Our problem generalizes both the classic Pandora's box problem and the prophet inequality framework. Boxes are presented online, each with a random value and cost drew jointly from some known distribution. Pandora chooses online whether to open each box given its cost, and then chooses irrevocably whether to keep the revealed prize or pass on it. We aim for approximation algorithms against adversaries that can choose the largest prize over any opened box, and use optimal offline policies to decide which boxes to open (without knowledge of the value inside). We consider variations where Pandora can collect multiple prizes subject to feasibility constraints, such as cardinality, matroid, or knapsack constraints. We also consider variations related to classic multi-armed bandit problems from reinforcement learning. Our results use a reduction-based framework where we separate the issues of the cost of acquiring information from the online decision process of which prizes to keep. Our work shows that in many scenarios, Pandora can achieve a good approximation to the best possible performance.

Recent work has suggested enhancing Bloom filters by using a pre-filter, based on applying machine learning to determine a function that models the data set the Bloom filter is meant to represent. Here we model such learned Bloom filters,, with the following outcomes: (1) we clarify what guarantees can and cannot be associated with such a structure; (2) we show how to estimate what size the learning function must obtain in order to obtain improved performance; (3) we provide a simple method, sandwiching, for optimizing learned Bloom filters; and (4) we propose a design and analysis approach for a learned Bloomier filter, based on our modeling approach., Comment: 12 pages; the complete version of the paper that appears in NIPS 2018, including addendum on learned Bloomier filters. arXiv admin note: substantial text overlap with arXiv:1802.00884, arXiv:1803.01474

We initiate the theoretical study of directory reconciliation, a generalization of document exchange, in which Alice and Bob each have different versions of a set of documents that they wish to synchronize. This problem is designed to capture the setting of synchronizing different versions of file directories, while allowing for changes of file names and locations without significant expense. We present protocols for efficiently solving directory reconciliation based on a reduction to document exchange under edit distance with block moves, as well as protocols combining techniques for reconciling sets of sets with document exchange protocols. Along the way, we develop a new protocol for document exchange under edit distance with block moves inspired by noisy binary search in graphs, which uses only $O(k \log n)$ bits of communication at the expense of $O(k \log n)$ rounds of communication.

We consider variations of set reconciliation problems where two parties, Alice and Bob, each hold a set of points in a metric space, and the goal is for Bob to conclude with a set of points that is close to Alice's set of points in a well-defined way. This setting has been referred to as robust set reconciliation. More specifically, in one variation we examine the goal is for Bob to end with a set of points that is close to Alice's in earth mover's distance, and in another the goal is for Bob to have a point that is close to each of Alice's. The first problem has been studied before; our results scale better with the dimension of the space. The second problem appears new. Our primary novelty is utilizing Invertible Bloom Lookup Tables in combination with locality sensitive hashing. This combination allows us to cope with the geometric setting in a communication-efficient manner.

In this work we provide a new technique to design fast approximation algorithms for graph problems where the points of the graph lie in a metric space. Specifically, we present a sampling approach for such metric graphs that, using a sublinear number of edge weight queries, provides a {\em linear sampling}, where each edge is (roughly speaking) sampled proportionally to its weight. For several natural problems, such as densest subgraph and max cut among others, we show that by sparsifying the graph using this sampling process, we can run a suitable approximation algorithm on the sparsified graph and the result remains a good approximation for the original problem. Our results have several interesting implications, such as providing the first sublinear time approximation algorithm for densest subgraph in a metric space, and improving the running time of estimating the average distance., Comment: FOCS 2018

We provide a simple method for improving the performance of the recently introduced learned Bloom filters, by showing that they perform better when the learned function is sandwiched between two Bloom filters., Comment: Short note. Related to arXiv:1802.00884 ; will probably combine them into a larger submission at some point in the future

Recent work has suggested enhancing Bloom filters by using a pre-filter, based on applying machine learning to model the data set the Bloom filter is meant to represent. Here we model such learned Bloom filters, clarifying what guarantees can and cannot be associated with such a structure., Comment: 5 pages, commentary on the "Learned Index Structures" paper

In many data structure settings, it has been shown that using "double hashing" in place of standard hashing, by which we mean choosing multiple hash values according to an arithmetic progression instead of choosing each hash value independently, has asymptotically negligible difference in performance. We attempt to extend these ideas beyond data structure settings by considering how threshold arguments based on second moment methods can be generalized to "arithmetic progression" versions of problems. With this motivation, we define a novel "quasi-random" hypergraph model, random arithmetic progression (AP) hypergraphs, which is based on edges that form arithmetic progressions and unifies many previous problems. Our main result is to show that second moment arguments for 3-NAE-SAT and 2-coloring of 3-regular hypergraphs extend to the double hashing setting. We leave several open problems related to these quasi-random hypergraphs and the thresholds of associated problem variations., Comment: 10 pages, to be submitted

The large memory requirements of deep neural networks limit their deployment and adoption on many devices. Model compression methods effectively reduce the memory requirements of these models, usually through applying transformations such as weight pruning or quantization. In this paper, we present a novel scheme for lossy weight encoding which complements conventional compression techniques. The encoding is based on the Bloomier filter, a probabilistic data structure that can save space at the cost of introducing random errors. Leveraging the ability of neural networks to tolerate these imperfections and by re-training around the errors, the proposed technique, Weightless, can compress DNN weights by up to 496x with the same model accuracy. This results in up to a 1.51x improvement over the state-of-the-art.