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Learning to Cache With No Regrets
Georgios S. Paschos, Apostolos Destounis, Luigi Vigneri, George Iosifidis
TL;DR
Caching policies with guarantees often rely on assumptions about request popularity, although request models can be unknown or time-varying. This paper frames caching as Online Linear Optimization and shows that OGA attains the optimal single-cache regret rate.
Problem
Existing caching policies rely on request-model assumptions, while the underlying model can be unknown or time-varying.
Source: A. Background and Related Work · p. 1Method
The paper formulates caching as Online Linear Optimization, measuring loss against the best static cache configuration in hindsight.
Source: §II · p. 3Results
The online caching regret is Θ(√CT), and OGA is universally optimal for a cache of capacity C among N files.
Source: B. Methodology and Contributions · p. 2Takeaways & Limitations
The OLO framework supports principled analysis and redesign of dynamic policies for single caches and cache networks.
Source: §VI · p. 9
Abstract
from arXiv · showhide
This paper introduces a novel caching analysis that, contrary to prior work, makes no modeling assumptions for the file request sequence. We cast the caching problem in the framework of Online Linear Optimization (OLO), and introduce a class of minimum regret caching policies, which minimize the losses with respect to the best static configuration in hindsight when the request model is unknown. These policies are very important since they are robust to popularity deviations in the sense that they learn to adjust their caching decisions when the popularity model changes. We first prove a novel lower bound for the regret of any caching policy, improving existing OLO bounds for our setting. Then we show that the Online Gradient Ascent (OGA) policy guarantees a regret that matches the lower bound, hence it is universally optimal. Finally, we shift our attention to a network of caches arranged to form a bipartite graph, and show that the Bipartite Subgradient Algorithm (BSA) has no regret
I. INTRODUCTION
Caching policies often depend on a known request model, although that model may be unknown or time-varying. The paper uses regret and Online Linear Optimization to develop policies for single caches and bipartite cache networks, with OGA achieving the single-cache lower bound.
A. Background and Related Work
- Prior policy performance depends on the request model, which may be unknown and time-varying, leaving a need for policies with guarantees under general popularity conditions.For caching networks, the cited prior joint routing and caching approaches assume stationary popularity.Source: A. Background and Related Work · pp. 1–2
B. Methodology and Contributions
- The paper formulates caching as Online Linear Optimization, minimizing utility loss over horizon T against the best cache configuration in hindsight.The utility can reflect hit ratio, delay, or preferential treatment of files.Source: B. Methodology and Contributions · p. 2
- LRU and LFU have Ω(T) regret, so some request patterns make their losses grow over time.The authors contrast these model-specific policies with OGA’s sublinear regret guarantee.Source: B. Methodology and Contributions · p. 2
- The online caching regret is Θ(√CT), and OGA is universally optimal for a cache of capacity C among N files.The paper combines a lower bound for any policy with OGA’s upper bound.Source: B. Methodology and Contributions · p. 2
- For bipartite cache networks, BSA jointly handles caching and routing and achieves regret O(√deg JCT).Here J is the number of caches and deg is the maximum network node degree.Source: B. Methodology and Contributions · p. 2
- The OLO perspective connects to LRU/LFU-type policies and provides directions for analyzing and redesigning them.The authors present this as a new use of OLO for caching optimality bounds.Source: B. Methodology and Contributions · p. 2
II. SYSTEM MODEL
The model allows arbitrary, possibly unknown request distributions and frames caching decisions as online optimization against a best static configuration in hindsight. It uses regret to assess whether a policy learns a good configuration over time.
II. SYSTEM MODEL
- Requests may follow arbitrary, unknown popularity distributions, including stationary, non-stationary, and adversarial patterns.The request distribution may also change over time.Source: §II · p. 2
- Regret compares a policy’s losses with those of the best static configuration chosen in hindsight, which knows the realized request sequence.This benchmark helps distinguish policies that learn good configurations from those that fail to do so.Source: §II · p. 3
- Sublinear regret makes average losses vanish relative to the best static configuration in hindsight, without prior knowledge of the request distribution.The model treats arbitrary request distributions as adversarially selected linear utility functions, with the caching decision made before the request.Source: §II · p. 3
- Under periodic requests cycling through C + 1 files, LRU and LFU have Ω(T) regret because a static cache can outperform them over the horizon.After the first C requests, LRU misses each request; the hindsight cache stores the first C files. The same reasoning applies to LFU.Source: §II · p. 3
III. REGRET LOWER BOUND
The section establishes a regret lower bound for caching, tailored to the capped-simplex configuration set. It relates the bound to Gaussian order statistics and shows it is tighter than classical OLO bounds for this setting.
III. REGRET LOWER BOUND
- The lower bound is tailored to caching’s capped-simplex action set, for which existing OLO bounds do not apply.The authors note the bound is tighter than the classical Ω(√(log N)T) OLO bound and that their lemmas also support regret analysis of capped-simplex sets.Source: §III · pp. 3–4
- The theorem bounds every caching policy’s regret using the expected sum of the C largest coordinates of a Gaussian vector.The proof studies centered cumulative file utilities and their order statistics.Source: §III · pp. 3–4
- For uniform file utilities and C < N/2, the paper gives a specialized lower bound for hit-rate maximization.The corollary fixes γ = C/N and assumes equal utility w across files.Source: §III · pp. 3–4
- The lower bound and the later O(√(CT)) upper bound establish Θ(√(CT)) regret for online caching.The paper states this conclusion after introducing the upper bound.Source: §III · p. 4
IV. ONLINE GRADIENT ASCENT
The paper uses a gradient-based online algorithm for caching and reports that it achieves the best possible regret.
IV. ONLINE GRADIENT ASCENT
- Online Gradient Ascent achieves the best possible regret despite its simplicity.The authors note gradient-based algorithms’ scalability properties in resource allocation.Source: §IV · p. 5
A. Algorithm Design and Properties
OGA updates the cache by ascending along the latest utility gradient and projecting onto the admissible configurations. It is a causal policy using the current configuration and latest request, and achieves the optimal regret rate under the stated uniform-utility setting.
A. Algorithm Design and Properties
- OGA updates the cache by stepping in the utility-gradient direction and projecting the result onto the admissible configuration set.The policy uses the current cache configuration and most recent request, without storing the full history.Source: A. Algorithm Design and Properties · p. 5
- The regret analysis bounds OGA’s regret using the configuration-set diameter and an upper bound on the utility-gradient norm.The derivation uses a constant stepsize and telescoping terms over the horizon.Source: A. Algorithm Design and Properties · p. 5
- For C/N = γ, equal file utilities, and C < N/2, OGA achieves the best possible regret rate.The paper concludes that OGA has the fastest learning rate under these conditions.Source: A. Algorithm Design and Properties · p. 5
B. Projection Algorithm
The projection algorithm computes OGA’s feasible caching configuration efficiently by partitioning files according to the projection’s KKT conditions. Its update favors the requested file while reducing other cached files symmetrically.
B. Projection Algorithm
- KKT conditions partition files into M1 at 1, M2 at adjusted intermediate values, and M3 at 0.A candidate partition can be checked in linear time; only the unique optimal partition satisfies all KKT conditions.Source: B. Projection Algorithm · p. 6
- The proposed projection algorithm runs in O(N log N) steps, improving on exhaustive O(N^2) search over ordered partitions.It uses the fact that at most one projected value can exceed 1, so |M1| is 0 or 1.Source: B. Projection Algorithm · p. 6
- After a request, OGA raises that file’s caching variable by the stepsize and symmetrically lowers other positive variables until the capacity constraint holds.Source: B. Projection Algorithm · p. 6
C. Performance and Relation to Other Policies
OGA performs close to the best of LRU and LFU across four request models, while its decisions relate to both policies. Its convergence speed and accuracy vary with the stepsize.
C. Performance and Relation to Other Policies
- Larger stepsizes yield faster but less accurate convergence, while smaller stepsizes converge more slowly but more accurately.Source: C. Performance and Relation to Other… · p. 6, Figure 3 · p. 6
- For C = 1000, w = 1, and T = 2 · 10^5, the horizon-optimal stepsize is η* = 0.1, which the experiments verify.Source: C. Performance and Relation to Other… · p. 6
- OGA can be viewed as regularized utility-LFU: it favors files with the largest cumulative frequencies while smoothing decisions with a Euclidean regularizer.The unregularized Follow-the-Leader policy in this setting is identical to LFU.Source: C. Performance and Relation to Other… · p. 6
- OGA’s caching decisions are strongly correlated with LRU’s, but it lowers weights for recent files that are infrequently requested.Unlike LRU’s one-shot eviction, OGA evicts files gradually, chunk by chunk.Source: C. Performance and Relation to Other… · p. 6
- OGA is always close to the better of LRU and LFU across four request models, with the second-best policy’s hit advantage reaching 16% over LRU and 20% over LFU.The models are i.i.d. Zipf, Poisson shot noise, university web browsing, and random replacement.Source: C. Performance and Relation to Other… · p. 7, Figure 4 · p. 7
V. BIPARTITE ONLINE CACHING
The paper extends its caching analysis to networks in which users reach caches through a weighted bipartite graph.
V. BIPARTITE ONLINE CACHING
- The network model uses a weighted bipartite graph to connect users and caches, with link weights representing delay, cost, or other quality-of-service metrics.Source: §V · p. 7
A. Bipartite Caching Model
The bipartite model couples fractional caching with routing from user locations to reachable caches or the origin. The paper establishes concavity of the resulting utility objective.
A. Bipartite Caching Model
- The network comprises user locations and caches with individual capacities; reachability links determine which caches can serve each location, alongside an origin server holding the full library.Source: A. Bipartite Caching Model · p. 7
- MDS coding lets caches store fractions of files as random linear combinations, which complement one another for decoding with high probability.Source: A. Bipartite Caching Model · p. 7
- Routing decisions must respect reachability and cache capacity, fully route each request, and are chosen after the request arrives and after the caching policy acts.The origin route is unconstrained, ensuring feasible routing for every request.Source: A. Bipartite Caching Model · p. 7
- The utility weights represent the benefit of serving a file fraction from a cache rather than the origin, including location-specific cache value.Regret is measured against the best static caching configuration in hindsight, accounting for its associated routing.Source: A. Bipartite Caching Model · pp. 7–8
- The per-slot objective is concave in the caching configuration; the proof mixes the optimal routings for two configurations to obtain feasible routing for their convex combination.Source: A. Bipartite Caching Model · p. 8
B. Bipartite Subgradient Algorithm (BSA)
BSA extends Online Gradient Ascent to bipartite caching by using a supergradient to update cache decisions and projecting onto the feasible set. Its regret is bounded under the stated capacity condition, and simulations show it outperforming lazy-LRU in the representative scenario.
B. Bipartite Subgradient Algorithm (BSA)
- The vector g is a supergradient of the concave utility function f at y, providing the direction used to extend OGA to bipartite caching.The supergradient property is stated over all y′ in Y_J.Source: B. Bipartite Subgradient Algorithm (BSA) · p. 8
- BSA updates caching decisions using a supergradient, then projects the result onto the feasible set Y_J.The step size is η_t, the supergradient comes from Lemma 3, and the projection is Euclidean.Source: B. Bipartite Subgradient Algorithm (BSA) · p. 8
- BSA’s regret bound applies when each cache has capacity C < N/2 and depends on the maximum right degree and maximum utility.The proof extends the OGA analysis by replacing the gradient with a supergradient; L bounds the supergradient norm.Source: B. Bipartite Subgradient Algorithm (BSA) · pp. 8–9
- For a requested file, BSA increases its allocation among reachable caches using a subgradient-based estimate of hypothetical utility gain, then locally projects to satisfy cache capacity.Other positive allocations at each cache are decreased as needed to meet the local constraint.Source: B. Bipartite Subgradient Algorithm (BSA) · p. 9
- BSA outperforms lazy-LRU by 45.8% in a simulated bipartite network with 3 caches and 4 user locations.The comparison uses stationary Zipf requests over 100 files, with requests arriving uniformly across locations; BSA learns popular files and allocates them to the high-utility cache.Source: B. Bipartite Subgradient Algorithm (BSA) · p. 9, Figure 5 · p. 9
VI. CONCLUSIONS
The paper connects established caching policies to Online Linear Optimization and uses that framework to analyze and redesign them. It reports universally optimal performance for online gradient caching and extends the approach to bipartite networks with routing decisions.
VI. CONCLUSIONS
- The paper connects policies such as LRU and LFU to Online Linear Optimization, enabling their performance characterization and redesign.It also reports universally optimal performance for online gradient caching and applies a similar approach to bipartite networks that include routing decisions.Source: §VI · p. 9