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Near-Optimal Bounds for Online Caching with Machine Learned Advice
Dhruv Rohatgi
TL;DR
Online caching with next-arrival predictions has an open question about how prediction error affects competitive-ratio improvements. This paper develops improved upper and lower bounds, showing stronger guarantees in some error regimes while leaving a significant intermediate gap.
Problem
The dependence of competitive ratio on relative prediction error η/OPT remains open, including how accurate predictions must be for improvement.
Method
The paper combines an improved caching algorithm with a lower-bound construction using phase-based input distributions whose parameter trades prediction error against competitive ratio.
Results
For η/OPT = k/log k, the prior algorithm gives competitive ratio O(log k), whereas this paper shows O(1) is possible; the bounds are asymptotically tight in two regimes.
Takeaways & Limitations
The results show that η/OPT = o(k) suffices for a non-trivial competitive ratio, while η/OPT ≤ k^(1+o(1)) is necessary.
Takeaways & Limitations
A significant gap remains when k ≤ η/OPT ≤ k log k, where neither bound is non-trivial.
Abstract
from arXiv · showhide
In the model of online caching with machine learned advice, introduced by Lykouris and Vassilvitskii, the goal is to solve the caching problem with an online algorithm that has access to next-arrival predictions: when each input element arrives, the algorithm is given a prediction of the next time when the element will reappear. The traditional model for online caching suffers from an $Ω(\log k)$ competitive ratio lower bound (on a cache of size $k$). In contrast, the augmented model admits algorithms which beat this lower bound when the predictions have low error, and asymptotically match the lower bound when the predictions have high error, even if the algorithms are oblivious to the prediction error. In particular, Lykouris and Vassilvitskii showed that there is a prediction-augmented caching algorithm with a competitive ratio of $O(1+\min(\sqrt{η/OPT}, \log k))$ when the overall $\ell_1$ prediction error is bounded by $η$, and $OPT$ is the cost of the optimal offline algorithm. The dependence on $k$ in the competitive ratio is optimal, but the dependence on $η/OPT$ may be far from optimal. In this work, we make progress towards closing this gap. Our contributions are twofold. First, we provide an improved algorithm with a competitive ratio of $O(1 + \min((η/OPT)/k, 1) \log k)$. Second, we provide a lower bound of $Ω(\log \min((η/OPT)/(k \log k), k))$.
1 Introduction
Online caching studies how to minimize cache misses relative to an offline optimum, while learned advice augments the online algorithm with predictions about future arrivals. This paper improves prediction-dependent guarantees and supplies a lower bound, although a gap remains between the bounds in an intermediate error regime.
- Problem setting: Online caching maintains a size-k cache over arriving elements and seeks a low competitive ratio against an optimal offline algorithm.The traditional randomized optimum is Θ(log k).
- Problem setting: Machine learned advice provides predictions about future data, but the algorithm must remain robust because the oracle may be inaccurate and its error bound is unknown.For caching, prior work predicts each element’s next arrival and measures prediction error in ℓ1 distance.
- Contributions: The paper provides an improved caching-with-predictions algorithm, building on prior techniques to strengthen the competitive-ratio guarantee.The introduction identifies this algorithm as the paper’s first principal contribution.
- Contributions: The results match prior work when η/opt ≥ k and are strictly better when η/opt < k; at η/opt = k/log k, the ratio improves from O(log k) to O(1).This example illustrates the improved dependence on relative prediction error.
- Contributions: The paper also provides a lower bound as a function of η/opt and k, but the upper and lower bounds remain non-trivial in disjoint regimes.Neither bound is non-trivial when k ≤ η/opt ≤ k log k.
- Roadmap: The paper is organized around preliminaries, a warm-up algorithm, a final algorithm, and a lower-bound proof.The warm-up algorithm is described as simpler and potentially more practical than the final algorithm.
2 Preliminaries
The paper formalizes caching costs, prediction error, and competitive analysis, then introduces marker-based phases and eviction chains as core analytical structures. Clean elements connect phase behavior to the offline optimum, while eviction chains account for cache misses.
- Traditional caching: A caching input is a sequence of arrivals handled by a cache of size k, where a miss requires adding the element and evicting a cached element.The algorithm’s cost is the number of cache misses, or expected misses when randomized.
- Traditional caching: The offline optimum is the minimum miss cost with the full sequence known in advance, and an online algorithm is competitive when its cost is bounded by a multiple of opt plus a constant.Known baselines are k-competitive deterministically and O(log k)-competitive randomly, both optimally.
- ML advice: With learned advice, the oracle predicts each arriving element’s next occurrence, and prediction error is the ℓ1 distance between predicted and actual next arrivals.The analysis bounds cost as a function of η/opt rather than absolute error η.
- Marker-based algorithms: Marker-based algorithms divide execution into phases, mark elements after cache hits, and choose an unmarked element for eviction after a miss.Their traditional randomized version evicts a random unmarked element and achieves O(log k)-competitiveness.
- Marker-based algorithms: A clean element appears in the current phase but not the previous one, whereas a stale element appeared in both phases.The phase structure and clean/stale classification do not depend on the exact marker-based algorithm.
- Marker-based algorithms: If L is the number of clean elements, then L/2 ≤ opt ≤ L, linking phase structure to the offline optimum.This relation supports using clean-element counts to analyze competitive ratios.
- Eviction chains: Eviction chains begin with clean arrivals and trace successive evictions, while all misses in a phase are accounted for by the total lengths of these chains.Each chain continues until an evicted element does not reappear in the phase.
3 Our techniques and related work
The paper develops progressively stronger ways to use next-arrival predictions in caching, improving error dependence while retaining robustness goals. Its lower-bound construction trades prediction accuracy against competitive-ratio guarantees.
- Motivation: Perfectly trusting next-arrival predictions can match the offline optimum, but even small errors can make performance unbounded.The paper therefore balances prediction use with provable competitiveness.
- Prior work: The prior marker-based approach balances trusting predictions against ignoring them through eviction chains.Eviction chains partition cache misses, and their number is tied to clean arrivals and OPT.
- First algorithm: O(1 + min(log(η/opt), log k)) improves the prior dependence on η/opt by trusting predictions only once per eviction chain.The analysis charges chain lengths through prediction errors and avoids double-counting errors across chains.
- Improved algorithm: The final algorithm departs from marker-based caching and may evict marked elements to terminate chains at length O(1) when η/opt < k.This improves on the O(log(η/opt)) chain-length behavior but complicates the analysis.
- Lower bound: The lower bound uses predictions independent of future inputs and phases with t clean elements, trading smaller prediction error for a smaller competitive-ratio lower bound.Increasing t makes predictions more accurate while weakening the lower bound.
4 Marker-based predictive algorithm
The marker-based algorithm combines predicted and random evictions within phases and analyzes eviction-chain lengths through inversions in predicted arrival order. This yields a competitive ratio logarithmic in relative prediction error, capped by log k.
- Algorithm: The algorithm trusts predictions on clean arrivals and randomly evicts unmarked elements on stale arrivals.This combines prediction use with randomized marker-style behavior.
- Chain analysis: Each clean arrival can start an eviction chain whose expected length is O(1 + log N_t(e(t))).N_t(e(t)) counts relevant stale elements after the evicted element’s arrival time.
- Error charging: Prediction inversions among stale arrivals lower-bound the total ℓ1 prediction error through η_{r-1} ≥ N/2.The proof relates chain lengths to inversions and then to prediction error.
- Cost bound: The expected number of misses is O(L + L log(2η/L)) and alternatively O(L + L log k).These bounds use Jensen’s inequality and the fact that N_c ≤ k.
- Guarantee: O(1 + min(log(η/opt), log k)) is the resulting competitive ratio for lmarker.The exact-constant form is 4 + 2H_min(2η/opt, k).
5 Improved algorithm
The improved non-marker algorithm uses prediction-guided and random evictions across phases, then charges its additional chains to prediction error. It obtains stronger error dependence and a robust worst-case guarantee via black-box simulation.
- Setup: The algorithm defines phases with at most k distinct elements and distinguishes initial from non-initial arrivals.These definitions support the analysis of chains and cache misses.
- Eviction chains: Non-initial arrivals initiate eviction chains, and the chains partition cache misses even though some misses occur without arrivals.Every phase has exactly k arrivals, and every non-initial arrival causes a miss.
- Algorithm: The non-marker strategy evicts the unmarked element with the highest predicted arrival time for non-initial misses while maintaining cache markings.At least one unmarked element exists at every cache miss, so the strategy is well-defined.
- Caveat: The analysis has an O(k^2) edge-case adjustment from final phases, which does not affect the competitive ratio.The adjustment replaces η by η + O(k^2).
- Error charging: η ≥ k(C − L)/2 charges the number of non-initial elements beyond clean elements to total prediction error.C is the number of non-initial elements and L the number of clean elements.
- Robustness: The improved algorithm itself lacks a proven robustness guarantee as η/opt tends to infinity, so black-box simulation supplies an O(log k) worst-case guarantee.The simulator switches between the improved algorithm and a prediction-ignoring algorithm.
- Guarantee: As η/opt tends to zero, lnonmarker approaches competitive ratio 4, while the black-box algorithm approaches 36.The black-box robustness transformation incurs a constant-factor loss.
6 Lower bound
The section constructs a distribution of inputs and predictions showing that predictive caching algorithms still incur substantial cache-miss costs at controlled relative prediction error. The resulting lower bound applies to randomized algorithms via Yao’s minimax principle.
- Indistinguishability: Each prefix determines the algorithm’s state, but future inputs retain enough randomness that unseen elements must effectively be guessed.This prevents an algorithm from systematically identifying which unseen elements should occupy the cache.
- Input distribution: The lower-bound construction uses phases containing clean elements absent from the previous phase and stale elements drawn from earlier phases.Conditioned on the clean elements, the stale set is uniformly random, while phase requests are independently uniform over the clean and stale elements.
- Prediction error: Predictions are defined so that the first phase portion points to the subsequent timestep, while final-copy predictions point to the end of the next phase.The construction’s overall prediction error is O(nk^2 log k).
- Probabilistic lemma: Lemma 26 lower-bounds the expected number of times that the random phase sequence contains exactly j distinct elements from the relevant set.For 0 ≤ j < l, the bound is E[T_j] ≥ k/(l − j) − 1/k.
- Cache-miss lower bound: Theorem 27 shows that every deterministic algorithm incurs at least Ω(t log k/t) expected cache misses in each phase.The argument averages over equivalence classes of prefixes and uses the random choice of stale elements to bound cache hits.
- Randomized lower bound: Choosing t = (k^2 log k)/ε yields relative prediction error η/opt ≤ ε, while Yao’s minimax principle transfers the deterministic lower bound to randomized algorithms.The resulting theorem rules out a better competitive ratio for randomized caching algorithms on inputs satisfying η/opt ≤ ε.