Source-linked AI summary
Algorithms with Predictions
Michael Mitzenmacher, Sergei Vassilvitskii
TL;DR
Algorithms with predictions use machine-learning forecasts to obtain performance that improves with prediction quality while preserving worst-case-style guarantees when forecasts are poor. The paper develops this framework across algorithmic examples, formalizes consistency and robustness, and shows prediction-aware designs can outperform naive use of predictions. Its central message is that prediction quality should guide performance without making algorithms fragile to large errors.
Problem
Worst-case analysis can be overly pessimistic, but prediction-based algorithms must improve with accurate forecasts without becoming fragile when forecasts are badly wrong.
Method
The paper abstracts a predictor as a function h and designs algorithms whose guarantees depend on prediction error, using careful mechanisms to balance consistency and robustness.
Results
The framework yields improved prediction-dependent performance while recovering bounded behavior under large errors, with caching summarized as improving over naive prediction use and remaining within a constant factor of marking.
Takeaways & Limitations
Prediction quality can be incorporated as an explicit performance parameter, provided algorithms use forecasts carefully rather than treating them as truth.
Takeaways & Limitations
Consistency and robustness are often in tension, so the desired trade-off depends on confidence in the predictions and the chosen parameter.
Abstract
from arXiv · showhide
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.
1 Introduction
Algorithms with predictions use machine-learning forecasts to obtain performance tied to prediction quality while retaining guarantees under large errors. Binary search and ski rental illustrate both the benefits and the consistency–robustness trade-off.
- Motivation: The framework replaces fixed input assumptions with predictions produced by a machine-learning function applied to each input.It is intended to connect algorithm performance to prediction quality rather than rely on a specific randomness model.
- Challenges: Prediction-based algorithms should provide formal performance guarantees corresponding to the predictor’s observed error.Choosing what quantity to predict is problem-specific and affects both performance and analytical bounds.
- Binary search: Binary search can start near the predicted position, expand exponentially until bracketing the query, and then search the resulting interval.The method uses h(q) as a predicted position and applies binary search after locating an interval guaranteed to contain q, if it exists.
- Binary search: 2 log ηq bounds the search cost when ηq = |h(q) − t(q)|, tying running time directly to prediction error.Because ηq is at most n, even a very poor predictor cannot asymptotically degrade performance.
- Ski rental: Treating a ski-rental prediction as truth achieves OPT + η but can spend arbitrarily more than the standard deterministic strategy when the prediction is wrong.A tunable algorithm instead buys on days ⌈λb⌉ or ⌈b/λ⌉, achieving a ratio no more than 1 + λ at zero error and no worse than 1 + 1/λ for large errors.
- Model: Consistency targets near-optimal performance with accurate predictions, while robustness bounds performance under arbitrarily bad predictions.These goals can conflict: smaller λ favors consistency, whereas larger λ limits the cost of incorrect predictions.
2 Counting Sketches
Counting sketches use predictions about frequent items to allocate extra space selectively. This preserves accuracy for predicted heavy hitters and reduces harmful collisions, with formal improvements reported for heavy-tailed streams.
- Counting sketches: A counting sketch saves memory by replacing per-item counters with approximate counts that may fail for individual items.The Count-Min sketch is introduced for streaming items such as URLs or IP addresses.
- Prediction use: Predicting the most frequent items allows the sketch to keep individual counters for likely heavy hitters without storing counters for every item.This provides accuracy for correctly predicted heavy hitters while reserving space selectively.
- Prediction use: Separating predicted heavy hitters reduces the chance that a small item’s counters all collide with a large item’s counters.The benefit is especially relevant for skewed or heavy-tailed frequency distributions.
- Results: Formal results for Count-Min sketch and Count-Sketch under Zipfian frequencies improve the space/performance tradeoff over sketches without predictions.The cited works also report that the improvement holds in practice.
3 Learned Bloom Filters
Learned Bloom filters combine a learned membership function with a standard Bloom-filter backup to reduce false positives without allowing false negatives. Their space–false-positive advantage depends on predictor quality and representative test queries.
- Standard Bloom filters: A standard Bloom filter represents a set with an m-bit array and k hash functions, with false-positive probability decreasing exponentially in bits per item.The setup assumes perfectly random hash functions and uses m/n bits per item.
- Learned Bloom filters: A learned Bloom filter trains a function f to recognize set members and thresholds its output to make membership decisions.The function ideally outputs 1 for members and 0 for nonmembers, but practical predictions are imperfect.
- Threshold choice: Using τ = minx∈S f(x) prevents false negatives but can produce too many false positives when the predictor is imperfect.The threshold guarantees that every set element is accepted by the learned function, at the cost of admitting more nonmembers.
- Learned Bloom filters: A larger threshold uses the learned function as a prefilter, while a backup Bloom filter stores incorrectly rejected set elements and prevents false negatives.The backup filter can still produce false positives, but it corrects the learned function’s false negatives.
- Space–accuracy tradeoff: A learned function correctly identifying half the set can make the backup filter roughly half the size of a full Bloom filter at roughly the same false-positive rate.The combination is advantageous when the learned function’s representation is smaller than the corresponding savings.
- Limitations and variants: Choosing the threshold empirically requires representative test queries; otherwise future false-positive rates may exceed expectations.Sandwiched learned Bloom filters place the learned filter between two standard Bloom filters and can yield better performance.
4 Caching with Predictions
Caching with predictions uses predicted future arrivals to improve eviction decisions while retaining guarantees under inaccurate predictions. PredictiveMarker combines prediction-guided eviction for clean elements with randomized marking for stale elements, yielding error-dependent analysis and practical gains over standard methods.
- What to Predict?: Paging algorithms predict each requested element’s next arrival time to emulate Furthest-In-Future online.Furthest-In-Future evicts the element that returns latest, making next-arrival time a useful prediction target.
- What to Predict?: PFIF is optimal with perfect predictions, but its competitive ratio grows linearly with η(h)/opt.The error is normalized by the optimum cost because raw error scales improperly with sequence length.
- What to Predict?: A cache of size 2 yields an arbitrarily poor PFIF outcome when one element is predicted at time 0 despite appearing only at the end.PFIF retains the mispredicted element and incurs nearly continual misses, while the optimum has only constantly many misses.
- Marking Algorithms: PredictiveMarker evicts the furthest-predicted unmarked clean element, but evicts a uniformly random unmarked stale element.This modifies the marking framework while preserving randomized handling for stale requests.
- Marking Algorithms: Clean-element evictions can be charged to prediction error, while stale-element eviction chains have expected length O(log k).The analysis follows eviction chains to a clean element and bounds total chain growth using inversion and ℓ1-error arguments.
- Summary of Caching: PredictiveMarker improves competitive performance over naïve prediction use and remains within a constant factor of standard marking under very large errors.Off-the-shelf predictors also made PredictiveMarker consistently outperform methods such as Least Recently Used.
5 Scheduling with Predictions
Scheduling with predictions replaces exact service times with predicted information, analyzing how prediction quality affects scheduling and queueing performance. The analysis introduces the price of misprediction and shows that even imperfect predictions can substantially improve queue performance under high load.
- Scheduling with predicted service times: Shortest Job First is optimal when job times are known, motivating predicted variants for settings where service times are uncertain.The predicted variants include SPJF, PSPJF, and SPRPT for non-preemptive, preemptive, and remaining-processing-time scheduling.
- Price of misprediction: The price of misprediction is the ratio between a metric with exact information and the corresponding metric using predicted information.Its denominator need not be an optimal algorithm; it is the same algorithm supplied with exact information.
- General service-time model: The general model represents actual and predicted service times jointly through a density g(x, y), allowing prediction noise to depend on the actual service time.Shortest predicted job first orders jobs by predicted service time, while waiting-time calculations condition on predicted ordering and actual work.
- General service-time model: 4/3 is the price of misprediction when service times have mean-1 exponential distributions and predictions are exponential with mean equal to the actual service time.The expression for this ratio can be numerically evaluated from the joint density g(x, y).
- Scheduling queues: Under high load, simulations show that fairly weak predictors significantly improve queue performance because they usually keep long jobs behind short jobs.At λ = 0.95, multiplicative-error predictions improve substantially over no predictions for both exponential and heavier-tailed Weibull service distributions.
- Scheduling queues: Performance degrades gracefully as multiplicative prediction error α increases, while no-prediction steady-state average system times are 20 for exponential and 58 for Weibull service.The simulations average 1000 trials over 1 million time units and vary α from 0 to 0.9.
6 Bibliographic Notes
The bibliographic notes position learning-based predictions within earlier work on advice, online algorithms, learned data structures, and queueing with inexact information. They distinguish realistic prediction quality from advice-bit complexity and identify related theoretical and empirical results.
- Advice and learning-based predictions: Earlier advice work minimized advice bits from omniscient sources, whereas learning-based prediction work targets competitive-ratio improvements from realistic advice.This distinction aligns the newer model more closely with machine-learning use in practice.
- Related prediction-based algorithms: Prior studies used predictions or samples for online matching and auction revenue, while later work formalized learning with predictions through consistency and robustness.The cited framework also analyzed caching and explicit trade-offs in ski rental and online scheduling.
- Queueing references: Harchol-Balter provides a general queueing-theory reference, including derivations for SJF and SRPT with exact information.The chapter’s queueing analysis extends this background to predicted service times.
- Queueing with imperfect information: Related queueing work studies stale load information, inexact job sizes, and empirical effects of imperfect information in single- and multiple-queue settings.The cited single-queue analysis bounds performance gaps using bounds on estimate inaccuracy.
- Learned data structures: Count-Min Sketch, Count-Sketch, Bloom filters, and learned Bloom filters are cited as established data structures connected to prediction-based improvements.Learned Bloom filters have proposed variants such as sandwiched designs that combine learned and standard filters.
Exercises
The exercises ask readers to prove prediction-based competitive guarantees, design a robust caching algorithm, verify a queueing price of misprediction, and simulate prediction effects in algorithms or data structures.
- Proof and design exercises: Exercise 1 asks for a proof of the competitive-ratio bound for ski rental with predictions.The target is the bound given in equation 1.
- Proof and design exercises: Exercise 2 asks for a caching algorithm with competitive ratio O(min(a(x), b(x))) on every input x.The two candidate eviction algorithms have input-dependent ratios a(x) and b(x).
- Queueing exercises: Exercise 3 asks readers to show that the price of misprediction is 4/3 in the exponential service-time model.The suggested approaches are numerical evaluation or integration of equation 3.
- Simulation exercise: Exercise 4 asks for a simulation examining how service distributions and prediction quality affect queue waiting time or counting-sketch accuracy.Suggested projects include predicted-service-time queues and predictors for heavy hitters.