--- title: Prior-Guided Caching Theorem url: https://www.emergentmind.com/topics/prior-guided-caching-theorem type: topic --- # Prior-Guided Caching Theorem Searching arXiv for the cited papers and closely related work. arXiv search query: 2202.04262 Parsimonious Learning-Augmented Caching The expression **Prior-Guided Caching Theorem** is used in several arXiv contexts to denote caching results in which a prior, a predictor, or an access-frequency model shapes cache placement, delivery, or eviction decisions. In **parsimonious learning-augmented caching**, the prior consists of predicted next-arrival times that are queried only selectively, yielding competitive ratios that depend on both cache size and queried prediction error [2202.04262]. In **information-theoretic caching for computing**, the prior is a request distribution $p_Y$ that determines the optimal cached description through a single-letter rate region [1504.00553]. In **single-user average-case coded caching**, the prior ranks files by request probability, and caching the most requested files is optimal [1601.05690]. In **premise-erasure caching for derivation-based reasoning**, the prior is an access frequency over queries, and caching becomes optimal above a critical frequency threshold, with a derivation penalty that approaches $1/\epsilon$ under erasure [2603.00930]. ## 1. Terminological scope and formal settings The four uses of the expression share a common design principle: cache resources are allocated using statistical or predictive information about future demand. The formal objects, however, are different. In the online learning-augmented setting, the objective is to minimize cache misses against the offline optimum under adversarial request sequences. In the information-theoretic settings, the objective is to characterize optimal memory–rate regions or average delivery rates under a stochastic request law. In the derivation-based setting, the objective is to compare amortized storage cost against per-access derivation cost under erasure. | Setting | Guidance signal | Canonical guarantee | |---|---|---| | Learning-augmented online caching | Predicted next request times for queried pages | $CR \le O(\min\{\log_{b+1} k + E[\eta]/opt,\log k\})$ | | Information-theoretic caching for computing | Request prior $p_Y$ | $R_c \ge I(X;V|Y),\ R_u \ge H(f(X,Y)|V,Y)$ | | Single-user multiple-request caching | Popularity prior $p=(p_1,\dots,p_N)$ | Top-$M$ most probable files are optimal | | Premise-erasure caching | Access frequency $f(q)$ and erasure rate $\epsilon$ | Cache if $f(q)\cdot Dd(q\mid B)\ge \rho_s\cdot \sigma^*(q)$ | This suggests that the term is best read as a family resemblance rather than a single canonical theorem. Across the four formulations, “prior-guided” refers respectively to adaptive prediction querying, source-coding optimization under a demand law, popularity-aware file selection, and thresholded caching under access-frequency and reliability models [2202.04262] [1504.00553] [1601.05690] [2603.00930]. ## 2. Parsimonious learning-augmented caching In "Parsimonious Learning-Augmented Caching" [2202.04262], the request sequence is $\Gamma=\langle p_1,p_2,\dots,p_T\rangle$ over a universe $U$, the cache capacity is $k$, and the objective is to minimize the total number of misses. The offline benchmark is Belady’s furthest-in-future algorithm, with $opt := cost_\Gamma(\mathrm{FiF})$. A randomized online algorithm $A$ is $c$-competitive if $$ E[cost_\Gamma(A)] \le c\cdot opt + b, $$ where $b\ge 0$ is a $\Gamma$-independent constant. The analysis uses the standard phase partition into maximal contiguous blocks requesting at most $k$ distinct pages, together with marking. If $\ell_h$ is the number of distinct clean pages in phase $h$, then $$ \frac{1}{2}\sum_h \ell_h \le opt \le \sum_h \ell_h, $$ which serves as a phase-wise proxy for $opt$ [2202.04262]. The distinctive feature is **parsimonious querying**. At time $t$, for any page $p$ currently in cache, an oracle may return a prediction $\tau_{p,t}$ of the next request time, with true next arrival $a_{p,t}\ge t$. Rather than querying all pages in cache, the algorithm limits itself to at most $b$ queried pages per miss. On a miss, it samples $b$ unmarked pages uniformly at random without replacement, queries their predicted next-arrival times, and evicts the sampled page with the furthest predicted arrival. The total number of queries is $m$, with $$ m \le b\cdot cost(\mathrm{Alg}), $$ and therefore, for fixed $b$, $m=o(T)$ whenever $opt=o(T)$. The main algorithm, **AdaptiveQuery-$b$**, combines marking with an eviction-chain decomposition. For each clean page $f_i$ in a phase, the associated chain is $$ C_i=\langle q_{i,0}:=f_i, q_{i,1},\dots,q_{i,M_i}\rangle, $$ where $q_{i,j+1}$ is the stale page evicted when serving the next request to $q_{i,j}$. If the current chain length satisfies $j\le \lfloor \log k\rfloor$, the algorithm uses predicted guidance on a sample of size $b$; if $j>\lfloor \log k\rfloor$, it switches to evicting an unmarked page uniformly at random, using no queries. The query budget per miss is therefore at most $b$ in the predicted portion of a chain and zero thereafter. The error model is defined only on queried predictions. The total $\ell_1$ error is $$ \eta := \sum_{(p,t)\in Q} |\tau_{p,t}-a_{p,t}|, $$ where $Q$ is the set of queried page–time pairs. The analysis also counts **inversions**, namely queried pairs whose predicted order reverses the true order of next arrivals. If $I$ is the number of such inversions, then $$ \eta \ge \frac{1}{2}I. $$ This stability relation allows bad evictions to be charged to prediction errors on the pages actually queried. Under perfect predictions, the theorem states that for any integer $b>0$, AdaptiveQuery-$b$ is $$ 2(\log_{b+1} k + 3)\text{-competitive} $$ and makes at most $$ 2b(\log_{b+1} k + 3)\cdot opt $$ queries in expectation. Under noisy predictions, with the switch to randomized marking after chain length exceeds $\lfloor \log k\rfloor$, the competitive ratio satisfies $$ CR(\mathrm{Alg}) \le O\!\left(\min\left\{\log_{b+1} k + \frac{E[\eta]}{opt},\ \log k\right\}\right), $$ while still making at most $b$ queries per cache miss. The proof is based on stale-page ranks, geometric rank decay under sampling, and an additive error term. If $S$ is a uniform sample of $b$ pages from $r$ unmarked stale pages, then the minimum sampled rank satisfies $$ E[\min S]\le \frac{r}{b+1}, $$ which yields expected chain length at most $\log_{b+1} k+3$ in the perfect-prediction case. The lower bound is near-tight. For any integer $c\le \ln k$, any $(c+4)$-competitive algorithm under perfect predictions must make at least $$ \frac{1}{12\ln(k+1)}\cdot c\cdot k^{1/c}\cdot opt $$ queries in expectation. This implies that constant-competitive behavior with sublinear querying cannot be sustained when the miss rate $opt/T$ is $\Omega(1)$. The work is also positioned against learning-augmented caching schemes of Lykouris–Vassilvitskii, Rohatgi, and Wei: those schemes assume predictions for all pages in cache at each step, whereas the parsimonious algorithm queries only $b$ pages per miss while retaining quantitatively similar guarantees [2202.04262]. ## 3. Information-theoretic prior-guided caching for computing In "Information-Theoretic Caching: Sequential Coding for Computing" by Wang, Lim, and Gastpar [1504.00553], the prior-guided theorem is formulated for lossless computation from a discrete memoryless source $\langle X,Y\rangle$ with finite alphabets and joint pmf $p_{X,Y}$. The request prior is $P(Y)=p_Y$. The per-letter desired data is an element-wise function $f(x,y)$, and the goal is to reconstruct the sequence $(f(X_i,Y_i): i\in[k])$. Caching is decomposed into a placement phase, in which a cache encoder observing $X^k$ sends a cache message $M_c$ at rate $R_c$, and a delivery phase, in which an update encoder observing $(X^k,Y^k)$ sends an update message $M_u$ at rate $R_u$. The single-user single-letter characterization is: $$ R_c \ge I(X;V|Y),\qquad R_u \ge H(f(X,Y)|V,Y), $$ for some conditional pmf $p_{V|X}$ with $|\mathcal{V}| \le |\mathcal{X}|+1$. The cached description $V$ is therefore chosen to minimize the expected delivery uncertainty about $f(X,Y)$ given $Y$, subject to the cache-rate budget. The prior $p_Y$ enters directly through the optimization over $p_{V|X}$. Several boundary statements are explicit. The minimum update rate without caching is $$ R_u^\star = H(f(X,Y)|Y), $$ and the minimum cache rate without update is $$ R_c^\star = \min I(X;V|Y) $$ over $p_{V|X}$ such that $H(f(X,Y)|V,Y)=0$. The sum-rate lower bound is $$ R_c + R_u \ge H(f(X,Y)|Y). $$ If the function is partially invertible, in the sense that $H(X|f(X,Y),Y)=0$, then $$ R_c^\star = R_u^\star = H(X|Y). $$ The paper gives closed-form solutions in several special cases. For independent source components with $\mathcal{Y}=[N]$ and $p_Y(1)\ge p_Y(2)\ge \cdots \ge p_Y(N)$, the optimal region is $$ R_c \ge r,\qquad R_u \ge \sum_{n=1}^N [p_Y(n)-p_Y(n+1)]\left(\sum_{j=1}^n H(X^{(j)})-r\right)^+, $$ with $p_Y(N+1)=0$. The interpretation is that a popularity-first caching scheme is optimal for independent components. For nested components satisfying $H(X^{(n)}|X^{(n+1)})=0$, the region becomes $$ R_c \ge r,\qquad R_u \ge \sum_{n=1}^N p_Y(n)(H(X^{(n)})-r)^+, $$ corresponding to a refinement-first strategy. For arbitrarily correlated components under uniform requests, with $\overline{X}=(X^{(1)},\dots,X^{(N)})$ and conditional total correlation $$ \Gamma(\overline{X}|V) := \left[\sum_{n=1}^N H(X^{(n)}|V)\right] - H(X^{(1)},\dots,X^{(N)}|V), $$ all boundary points satisfy $$ R_c=r,\qquad R_u=\frac{1}{N}\left[H(\overline{X})-r+\min_{p_{V|\overline{X}}: I(\overline{X};V)=r}\Gamma(\overline{X}|V)\right]. $$ If the cache rate is large enough to eliminate conditional total correlation, then for $R_c\in[R_{\mathrm{crit}},H(\overline{X})]$ the boundary is the straight line $$ R_c + N R_u = H(\overline{X}), $$ where $$ R_{\mathrm{crit}} := \min_{p_{V|\overline{X}}: \Gamma(\overline{X}|V)=0} I(\overline{X};V). $$ For $N=2$, $R_{\mathrm{crit}}$ equals Wyner’s common information. The same paper develops two-user extensions with single-letter characterizations linked to Gray–Wyner and distributed successive refinement. The private-update-aided Gray–Wyner system introduces a common cache $V_c$ and private caches $V_1,V_2$; the common-cache-aided Gray–Wyner system introduces a common update variable $V_u$; and the sequential successive-refinement system derives a two-stage compute-and-refine region. The paper also compares this framework with the Maddah-Ali–Niesen model. In the single-user average case, the adaptive static-request region $$ R_c \ge I(X;V),\qquad R_u \ge H(f(X,Y)|V,Y) $$ matches the single-user information-theoretic region, whereas a two-user example shows that coding across i.i.d. request blocks and coding within a single static block need not yield the same tradeoff [1504.00553]. ## 4. Popularity-guided optimality in single-user multiple-request caching "A New Converse Bound for Coded Caching" by Wang, Lim, and Gastpar [1601.05690] contains a distinct prior-guided theorem for the single-user average case with multiple requests. The model has $N$ equal-sized files, a single user cache of size $M$ files, and $L$ file requests drawn independently with replacement according to a popularity distribution $p=(p_1,\dots,p_N)$. Let $$ Y=\{D^{(1)},\ldots,D^{(L)}\}\subseteq[N] $$ be the set of distinct requested files, and let $$ s_n := \mathbb{P}(n\in Y)=1-(1-p_n)^L. $$ After rearranging so that $s_1\ge s_2\ge \cdots \ge s_N$, the optimal average delivery rate is $$ R_{\mathrm{avg}}^\star(M)=\sum_{n=1}^N (s_n-s_{n+1})(n-M)^+,\qquad s_{N+1}:=0. $$ It is achieved by caching the $M$ most probable files, equivalently the files with the largest $s_n$. For integer cache size $M=m$, the theorem simplifies to $$ R_{\mathrm{avg}}^\star(m)=\sum_{n=m+1}^N s_n =\sum_{n=m+1}^N \bigl(1-(1-p_n)^L\bigr). $$ The significance of the result is that optimality holds among all feasible placements, including coded placements; in this single-user multiple-request average-case setting, uncoded top-$M$ file caching is optimal. The converse is formulated via a single-letter problem: $$ R_{\mathrm{avg}}^\star(M)=\min_{p_{V|B}: V-B-Y,\ I(B;V)\le M} H(B_Y|V,Y), $$ where $B=(B_1,\dots,B_N)$ is the binary representation of the files and $V$ is a possibly coded cache content. By conditional entropy inequalities, data processing, Fano’s inequality, and the memory constraint, the paper derives the closed-form lower bound that matches the achievable top-$M$ policy. This theorem also feeds into a broader converse for multi-user coded caching. For a $K$-user system with request prior $p_D$, define $$ s_n(k):=1-(1-p_D(n))^k,\qquad s_{N+1}(k)=0. $$ Then $$ R_{\mathrm{avg}}^\star(M)\ge \max_{k\in[K]} \sum_{n=1}^N (s_n(k)-s_{n+1}(k))(n-kM)^+. $$ Under uniform demand, $$ R_{\mathrm{uniform}}^\star(M)\ge \max_{k\in[K]}\left(1-\left(1-\frac{1}{N}\right)^k\right)(N-kM)^+. $$ Compared with the decentralized Maddah-Ali–Niesen scheme, the paper establishes a multiplicative gap smaller than $4.7$ for both the worst case and the uniform-demand average case, improving previous analytical gaps of $12$ and $72$, respectively. The paper explicitly notes that the “most-popular is optimal” theorem does not directly generalize to the multi-user coded caching setting, where coded multicasting gains become central [1601.05690]. ## 5. Prior-guided caching under premise erasure and derivation constraints "The Derivation Penalty in Premise-Erasure Caching: Capacity, Strong Converse, and Dispersion Dichotomy" [2603.00930] develops a prior-guided theorem in a different direction, namely caching for derivation-based reasoning engines under independent premise erasure. The premise base $B$ has size $m:=|B|\ge 2$, each base fact is erased independently with probability $\epsilon\in(0,1)$, and a query $q$ has derivation depth $d:=Dd(q\mid B)$ and dependency count $$ \kappa(q,B):=|V_0(G(q,B))|. $$ Two decoder models are compared. In the coded scheme, the cache is an arbitrary bit string. In the derivation-constrained scheme, the cache consists of logical facts and decoding must certify $q\in Cn(\widetilde{B}\cup S)$. The paper states four coding theorems. The first establishes a universal per-step information content: $$ I_{\mathrm{step}}(B)=\log m, $$ with $$ K(q\mid \langle B\rangle)=(1+o(1))\kappa \log m $$ for generic queries in the information-rich regime. The second compares two Datalog architectures at equal depth. For the chain architecture, $$ \kappa_{\mathrm{chain}}(d)=k+d-1,\qquad C_{\mathrm{chain}}(d,B)=(k+d-1)\log m, $$ while for the balanced-merge architecture, $$ \kappa_{\mathrm{merge}}(d)=k\cdot 2^d-1,\qquad C_{\mathrm{merge}}(d,B)=k\cdot 2^d-1\log m. $$ Hence $$ \frac{C_{\mathrm{merge}}(d,B)}{C_{\mathrm{chain}}(d,B)}=\frac{k\cdot 2^d-1}{k+d-1}\to \infty $$ as $d\to\infty$. The third theorem introduces a critical access frequency. With storage price per bit per access $\rho_s>0$ and derivation cost $1$ per step, the clean-base break-even frequency is $$ f_c=\Theta(\rho_s\cdot \log(m+d)). $$ The decision rule is to cache if $f\ge f_c$ and derive on demand if $f