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Online metric algorithms with untrusted predictions

Antonios Antoniadis, Christian Coester, Marek Elias, Adam Polak, Bertrand Simon

arXiv:2003.02144v3cs.DS

TL;DR

Online systems need to benefit from imperfect predictors while retaining acceptable performance when predictions are poor. The paper introduces action predictions for MTS and related online problems, proving consistency, smoothness, and robustness guarantees, with stronger error dependence for caching and empirical evidence suggesting practicality.

  • Problem

    Online decision systems need prediction-augmented performance without assuming that machine-learned predictions are always accurate, but existing prediction setups for caching do not appear applicable to general MTS.

  • Method

    The paper uses action predictions of an offline algorithm's decisions and combines them with online-algorithm techniques to design prediction-augmented algorithms for MTS and online matching on the line.

  • Results

    The resulting algorithms provide consistency, smoothness, and robustness for MTS, while caching receives a stronger logarithmic dependence on η/Off and online matching on the line achieves O(min{log n, 1 + η/Off}).

  • Takeaways & Limitations

    The action-prediction setup applies across a broad class of online problems and supports practical caching algorithms that match or outperform previous approaches in most experiments.

  • Takeaways & Limitations

    The paper notes that even low prediction error relative to an offline algorithm may not prevent the prediction-composed solution from incurring much higher cost.

Abstract

from arXiv · show

Machine-learned predictors, although achieving very good results for inputs resembling training data, cannot possibly provide perfect predictions in all situations. Still, decision-making systems that are based on such predictors need not only to benefit from good predictions but also to achieve a decent performance when the predictions are inadequate. In this paper, we propose a prediction setup for arbitrary metrical task systems (MTS) (e.g., caching, k-server and convex body chasing) and online matching on the line. We utilize results from the theory of online algorithms to show how to make the setup robust. Specifically for caching, we present an algorithm whose performance, as a function of the prediction error, is exponentially better than what is achievable for general MTS. Finally, we present an empirical evaluation of our methods on real world datasets, which suggests practicality.

1 Introduction

The paper introduces action predictions for MTS and develops algorithms that combine near-optimal performance with graceful degradation as prediction error increases. It proves general guarantees for MTS, improves error dependence for caching, extends the setup beyond MTS, and evaluates the methods empirically.

  • General guarantees: Perfect predictions yield performance close to optimal, while worsening predictions cause smooth degradation and poor predictions retain performance comparable to the best prediction-free online algorithm.These properties are termed consistency, smoothness, and robustness.
  • Prediction setup: Action predictions represent the action an offline algorithm would take, providing a prediction setup applicable to arbitrary MTS.The setup can use an optimal or near-optimal offline algorithm as the reference.
  • General guarantees: For MTS, the paper proves deterministic and randomized algorithms whose competitive performance depends on prediction error relative to an offline algorithm.The randomized guarantee is close to 1 + ǫ when the reference algorithm is near-optimal and η is much smaller than its cost.
  • Caching: Caching admits a stronger algorithm with logarithmic dependence on η/Off, improving the general MTS dependence.The algorithm, Trust&Doubt, gradually adapts its level of trust in the predictor.
  • Beyond MTS: The action-prediction approach also yields an O(min{log n, 1 + η/Off}) deterministic algorithm for online matching on the line and extends to online metric bipartite matching.The paper uses this result to demonstrate applicability beyond MTS.
  • Caching and evaluation: Precise reoccurrence-time predictions alone cannot improve weighted caching below an Ω(log k) competitive ratio, and prior theoretical guarantees cannot be directly compared because their error measures differ.The paper instead compares caching approaches experimentally and reports that its algorithms outperform or match previous methods in most settings.

2 Preliminaries

MTS models online decision-making over metric-space states with switching costs and task-dependent costs, without knowledge of future tasks. Its performance is evaluated using competitive ratio, including comparison against an optimal or specified offline algorithm; caching is a special case.

  • MTS receives a task at each time step and chooses a state without knowing future tasks, paying movement distance plus the selected state's task cost.
  • MTS differs from the experts problem through metric switching costs and one-step lookahead before incurring the task cost.
  • Competitive ratio measures an online algorithm's worst-case cost relative to an offline benchmark, with expectation over the online algorithm's randomness.
  • Against a specific offline algorithm Off, the competitive ratio compares the online algorithm's cost with Off's cost rather than necessarily with the optimum.
  • Caching is an MTS special case whose states are k-page subsets, whose distance counts differing pages, and whose task costs encode whether the requested page is present.

3 Logarithmic Error Dependence for Caching

Trust&Doubt adapts its reliance on lazy action predictions through alternating Trust and Doubt intervals, achieving logarithmic dependence on prediction error for caching while retaining an O(log k) worst-case guarantee.

  • Algorithm and guarantees: Trust&Doubt achieves a competitive ratio logarithmic in prediction error while retaining the optimal worst-case guarantee O(log k).The algorithm is designed for unweighted caching and improves the general MTS dependence on error.
  • Algorithm and guarantees: Caching requests are divided into phases containing k distinct pages, with marked pages requested in the current phase and unmarked pages remaining unrequested.The first request to a page in a phase is its arrival.
  • Algorithm and guarantees: Within each phase, Trust&Doubt alternates Trust intervals that match the predicted cache with Doubt intervals that randomize evictions from unmarked pages.Doubt intervals end after 2^i−1 arrivals in the ith interval, after which the algorithm restores the predicted cache.
  • Analysis: 54? No quantitative value is supplied in the excerpt beyond the bound that the expected cache misses in phase ℓ are 1 + O(d_ℓ).Here d_ℓ denotes the number of Doubt intervals in phase ℓ.
  • Analysis: Error intervals grow geometrically, so the number of such intervals is constrained by the prediction error, while non-error intervals are constrained by the offline algorithm’s cost.The analysis combines these bounds with phase-level cost arguments to derive the competitive guarantee.
  • Optimality: The O(log k) bound matches the known lower bound for randomized online caching without predictions, and the logarithmic dependence on error is asymptotically tight.The lower bound states f(x) = Ω(log x) as x →∞ for arbitrary k with action-prediction error at most η.

4 Robust Algorithms for MTS

The section first presents Follow the Prediction (FtP), which follows action predictions while allowing cheaper nearby states, then robustifies it by combining algorithms. The resulting bounds are tight for general MTS on uniform metrics, while caching avoids the general lower-bound behavior because its requests impose more restricted cost functions.

  • Follow the Prediction: FtP follows predictions unless moving to a nearby state with substantially lower service cost is beneficial.It can move to the alternative state, pay its service cost, and return to the predicted state.
  • Follow the Prediction: 1 + 4η is FtP’s competitive ratio against any offline algorithm, where η is the prediction error relative to that algorithm.The bound follows by comparing FtP with an offline algorithm and summing per-time-step error contributions.
  • Combining online algorithms: Robust combinations select among multiple online algorithms while paying switching overhead, yielding guarantees close to the best constituent algorithm.The framework applies to MTS and, when simultaneous simulation and switching bounds are available, to problems beyond MTS.
  • Combining online algorithms: (2γ^m)/(γ−1)+1 becomes 9 for m = 2 and is bounded by 2em for larger m in the deterministic combination.Combined with FtP, this gives the deterministic robust MTS result.
  • Lower bounds: Theorem 20 gives a lower bound of Ω(min{α_n, 1 + η̄}) for MTS on an n-point uniform metric, matching the general upper bounds up to constants.Here α_n is Θ(n) for deterministic algorithms and Θ(log n) for randomized algorithms without predictions.
  • Lower bounds: Caching can achieve logarithmic rather than linear dependence on prediction error because caching requests do not impose infinite costs at several states simultaneously.The lower-bound construction relies on cost functions with infinite values at multiple points, unlike one-page-per-step caching requests.

5 Beyond Metrical Task Systems

The paper extends action predictions to online matching on the line, where predicted configurations are handled through virtual switching and potential-based analysis. This yields a prediction-sensitive deterministic algorithm, while randomized combination is unhelpful when switching costs can be as large as the optimum.

  • Prediction setup: Online matching on the line receives a predicted server configuration for each prefix, with error measured by minimum matching distance to an offline configuration.Predicted configurations need not be nested across rounds.
  • Combining algorithms: The deterministic matching algorithm combines Follow-The-Prediction with Raghvendra’s online algorithm by virtually simulating both algorithms.This extends the combination approach despite online matching’s irrevocable assignments and lack of known MTS structure.
  • Combining algorithms: Switching costs between the two matching algorithms can reach Θ(Opt), making randomized combination uninteresting for some instances.This also makes the metric-space-diameter perspective costly in this problem.
  • Prediction setup: The algorithm maintains the distance between actual and predicted configurations as a potential and uses minimum-cost matchings to track their discrepancy.The potential is Φ_i = dist(S_i, P_i), with shared servers matched to themselves at zero cost.
  • Follow-The-Prediction: Follow-The-Prediction achieves total matching cost at most Off + 2η against any offline algorithm.Its analysis pays for virtual configuration switching through changes in the potential.

6 Experiments

The experiments evaluate prediction-augmented algorithms for caching and the ice cream MTS problem on real-world and synthetic-prediction settings. Trust&Doubt generally matches or outperforms baselines, while the general MTS combinations retain classical online guarantees.

  • Experimental setup: Experiments use BrightKite and CitiBike traces for caching, plus BrightKite-derived instances of the ice cream problem.Caching uses cache sizes k = 10 on BK and k = 100 on Citi; the ice cream instances use the same 100 BrightKite users.
  • Experimental setup: Synthetic predictions perturb exact reoccurrence times with lognormal noise, while PLECO, POPU, and LRU provide additional caching predictors.Reoccurrence-time predictions are converted into action predictions by simulating the algorithm that evicts the page predicted to appear furthest in the future.
  • Experimental setup: The evaluation compares classical online algorithms, prior prediction-augmented methods, and action-prediction methods including FtP, RobustFtP, and Trust&Doubt.The reported competitive ratios divide total page faults by the optimal number of page faults.
  • Caching results: On BK synthetic predictions, Trust&Doubt outperforms previous prediction-based algorithms and LRU, while on Citi, FtP performs well with noisy predictions and RobustFtP surpasses other guaranteed algorithms.LNonMarker performs better than the other algorithms for noisy predictions.
  • Caching results: With PLECO, POPU, and LRU predictions, Trust&Doubt is similar or better than LRU in both datasets, whereas POPU enables prediction-augmented algorithms to significantly improve over classical online algorithms.PLECO predictions are not accurate enough for previously known algorithms to improve over Marker; using LRU as predictor yields performance comparable to bare LRU.
  • Ice cream results: For the ice cream problem, the general MTS algorithms benefit from good predictions while retaining the worst-case guarantee of the classical online algorithm.Deterministic and randomized combinations are comparable to the better constituent algorithm and improve when both constituent algorithms perform similarly.

7 Conclusion

The paper introduces a general prediction setup for MTS and develops a caching algorithm with stronger prediction-error dependence. It reports empirical performance that is often better than or comparable to prior approaches, while identifying several directions for extending the theory and evaluation.

  • Contributions: The proposed prediction setup supports prediction-augmented algorithms for a large class of problems encompassing MTS.For caching, it requires less information from the predictor than previously studied setups because previous predictions can be converted into action predictions.
  • Contributions: For caching, Trust&Doubt offers guarantees with a logarithmic dependence on prediction error and performs better in most experiments than prior algorithms.The conclusion describes its guarantees as having a similar flavor to previous caching algorithms despite the more general setup.
  • Future directions: The paper conjectures that logarithmic smoothness guarantees may also be attainable for other MTS problems with request structure.It suggests studying improved prediction-error dependence under restricted request sets or specific metric structures.
  • Future directions: Online metric matching remains largely unexplored in this learning-augmentation perspective, and evaluating the resource-augmented matching algorithm in real-life settings is proposed as future work.The paper also identifies more sophisticated predictors for caching and other problems as another research direction.

A Deterministic combination of a collection of algorithms

This appendix analyzes MIN det, a deterministic method for combining m online algorithms under switching-cost assumptions. Its bound follows from cycling through the algorithms with geometrically increasing cost thresholds.

  • Requirements: MIN det combines m online algorithms that start in the same state and have switching costs bounded by the algorithms’ costs.These are the two structural requirements used by the theorem.
  • Guarantee: Theorem 24 bounds MIN det for 1 < γ ≤ 2 on instances with Opt_I ≥ 1.The lower-bound assumption handles instances with very small positive optimum cost; costs can be scaled when Opt_I ≥ c for c < 1.
  • Operation: The algorithm cycles through the component algorithms, switching from A_i′ to A_i and running the selected algorithm until its cost reaches the next geometric threshold.During cycle ℓ, the selected algorithm’s cost grows to γ^ℓ.
  • Guarantee: Choosing γ = m/(m−1) yields a bound equal to 9 for m = 2 and bounded by 2em.The appendix derives this parameter choice after considering the final cycle and separate cases based on the minimum component cost.

B Comparison between Trust&Doubt and the best marking algorithm

The appendix compares Trust&Doubt with the best marking algorithm under perfect predictions. Trust&Doubt can outperform marking algorithms on some instances but is outperformed on others.

  • Comparison: With perfect predictions, Trust&Doubt can outperform the best marking algorithm on one caching instance and lose to a marking algorithm on another.The comparison therefore does not establish uniform dominance even under perfect predictions.
  • Comparison: Trust&Doubt’s non-marking design enables better performance than all marking algorithms on some instances but prevents it from always matching the best marking algorithm.The paper explicitly attributes this trade-off to Trust&Doubt not belonging to the broad class of marking algorithms.
  • Constructed examples: In the first constructed example, a cache of size 3 and sequence 1, 2, 3; 4, 5, 6; 1, 2, 3 give Trust&Doubt seven cache misses while the best marking algorithm cannot keep up.The instance uses optimal-algorithm predictions that evict the page arriving furthest in the future.
  • Constructed examples: In the second constructed example, the best marking algorithm suffers eight cache misses and retains an advantage over Trust&Doubt on the final request.The cache size is 3 and the request sequence is 1, 2, 3; 4, 5, 6, 5, 6; 7, 1, 4.

C Limitations of the reoccurrence time predictions

Perfect predictions of next reoccurrence times do not improve weighted caching beyond the existing Θ(log k) competitive ratio. The paper establishes this with a randomized semi-online lower bound using a superincreasing metric and carefully constructed request sequences.

  • Lower bound: Perfect next-reoccurrence predictions cannot improve weighted caching beyond an Ω(log k) competitive ratio.The lower bound applies to every randomized semi-online algorithm receiving the currently requested page’s error-free reoccurrence time.
  • Lower bound: The hard instance uses k + 1 pages with exponentially increasing weights, equivalently represented by a weighted star whose leaf distances grow as τ^i.Page i has weight 2τ^i, while the equivalent star places leaf i at distance τ^i from the root.
  • Adversarial construction: The construction defines random request sequences σ_h inductively so that arbitrary semi-online algorithms incur much larger expected cost than offline optimum.The sequences preserve a prefix and suffix ordering of leaves, enabling the induction over h.
  • Adversarial construction: For h ≥ 2, σ_h concatenates type 1 iterations with sequence (0, 1, ..., h) and type 2 iterations containing repeated σ_{h−1} subsequences.The number of repetitions depends on τ and h, and the construction later assigns arrival times within a prescribed interval.
  • Why predictions fail: The lower-bound analysis compares the algorithm’s expected iteration costs with an offline strategy whose type 1 costs become negligible relative to type 2 costs as τ → ∞.The proof derives the second inequality through the offline strategy and concludes the asymptotic separation from the induction hypothesis.
  • Why predictions fail: The request timing is chosen so that the next-reoccurrence information is independent of whether the next iteration is type 1 or type 2.Thus, even with exact reoccurrence times, the semi-online algorithm cannot identify which hidden iteration type will occur next.
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