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On the Regret of Coded Caching with Adversarial Requests

Anupam Nayak, Kota Srinivas Reddy, Nikhil Karamchandani

TL;DR

This work investigates adversarial regret in coded caching with a broadcast delivery channel within an online-learning framework. It proposes a Follow-The-Perturbed-Leader policy that updates cache placements in a restricted set of time slots and analyzes a non-linear rate expression by a careful transformation of the request vector, achieving sublinear regret $O(\sqrt{T})$ relative to a static oracle. The paper also derives upper bounds on switching costs under unrestricted and restricted switching and validates the theoretical findings through numerical experiments on real data. The results establish the first regret guarantees for coded caching under adversarial requests and provide practical guidance on balancing online learning performance with cache-update costs.

Abstract

We study the well-known coded caching problem in an online learning framework, wherein requests arrive sequentially, and an online policy can update the cache contents based on the history of requests seen thus far. We introduce a caching policy based on the Follow-The-Perturbed-Leader principle and show that for any time horizon T and any request sequence, it achieves a sub-linear regret of \mathcal{O}(\sqrt(T) ) with respect to an oracle that knows the request sequence beforehand. Our study marks the first examination of adversarial regret in the coded caching setup. Furthermore, we also address the issue of switching cost by establishing an upper bound on the expected number of cache updates made by our algorithm under unrestricted switching and also provide an upper bound on the regret under restricted switching when cache updates can only happen in a pre-specified subset of timeslots. Finally, we validate our theoretical insights with numerical results using a real-world dataset

On the Regret of Coded Caching with Adversarial Requests

TL;DR

This work investigates adversarial regret in coded caching with a broadcast delivery channel within an online-learning framework. It proposes a Follow-The-Perturbed-Leader policy that updates cache placements in a restricted set of time slots and analyzes a non-linear rate expression by a careful transformation of the request vector, achieving sublinear regret relative to a static oracle. The paper also derives upper bounds on switching costs under unrestricted and restricted switching and validates the theoretical findings through numerical experiments on real data. The results establish the first regret guarantees for coded caching under adversarial requests and provide practical guidance on balancing online learning performance with cache-update costs.

Abstract

We study the well-known coded caching problem in an online learning framework, wherein requests arrive sequentially, and an online policy can update the cache contents based on the history of requests seen thus far. We introduce a caching policy based on the Follow-The-Perturbed-Leader principle and show that for any time horizon T and any request sequence, it achieves a sub-linear regret of \mathcal{O}(\sqrt(T) ) with respect to an oracle that knows the request sequence beforehand. Our study marks the first examination of adversarial regret in the coded caching setup. Furthermore, we also address the issue of switching cost by establishing an upper bound on the expected number of cache updates made by our algorithm under unrestricted switching and also provide an upper bound on the regret under restricted switching when cache updates can only happen in a pre-specified subset of timeslots. Finally, we validate our theoretical insights with numerical results using a real-world dataset
Paper Structure (18 sections, 15 theorems, 76 equations, 2 figures, 1 algorithm)

This paper contains 18 sections, 15 theorems, 76 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Our Proposed Algorithm
  4. Main Results
  5. Unrestricted Switching
  6. Restricted Switching:
  7. Numerical Experiments
  8. Conclusions and Future Work
  9. Appendix
  10. Proof of Proposition \ref{['proposition:expected_rate']}
  11. Proof of Theorem \ref{['theorem:regret_unrestricted']}
  12. Proof of Theorem \ref{['theorem:advswitch']}
  13. Proof of Theorem \ref{['theorem:regret_restricted_switching']}
  14. Linear approximation
  15. Stochastic request setting:
  16. ...and 3 more sections

Key Result

Proposition 1

For a coded caching problem with the given cache configuration $\mathbf{s}_t$ and the request vector $\mathbf{x}_t$, the above-discussed placement and delivery policies give a transmission rate of expected length $K(\mathbf{s}_t,\mathbf{x}_t)$ given by where $\mathbf{y}_t=\min\{\mathbb{I}_N, \mathbf{x}_t\}$.

Figures (2)

  • Figure 1: An illustration of coded caching problem setup. It contains a central server with $N=5$ files, each of size $F$ bits, and $K=4$ users, connected to a separate cache of size $MF=F$ bits. Here $s=[1,0,1,0,0]$ indicates that stored files are $A$ and $C$, and unstored files are the remaining. For a request profile $\mathbf{r}=(E,A,C,E)$, it's request pattern is $x=[1,0,1,0,2]$.
  • Figure 2: We compare the performance of our proposed policy in Algorithm \ref{['alg:FTPL']} against the benchmark policies under different system parameters. The plots display the average regret per time step $R(T)/T$ against horizon $T$. The first figure compares the performance of benchmark policies against our policy for $N=10$ files, $K=6$ users, and cache size $M=3$. In the next figure, we increase the cache size to $M=4$. The last figure for the case $N=20$, $K=10$, and $M=4$. Our policy outperforms the benchmarks and has a sub linear regret which matches our theoretical findings.

Theorems & Definitions (30)

  • Proposition 1
  • proof
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • proof
  • Lemma 1
  • proof
  • ...and 20 more