Online Caching. Jiazhong Nie directed by Prof. Manfred Warmuth SOE UC, Santa Cruz. May 16, jiazhong (UCSC) May 16, / 35

Transcription

1 Online Caching Jiazhong Nie directed by Prof. Manfred Warmuth SOE UC, Santa Cruz May 16, 2012 jiazhong (UCSC) May 16, / 35

2 Table of Contents Motivation 1 Motivation 2 Caching Problem 3 Master Policies 4 Forced and Optional Policies 5 Deterministic VS Randomized 6 Combining Caches 7 Onlineness Measure jiazhong (UCSC) May 16, / 35

3 Motivation Motivations of this research How to measure onlineness of sequence? Problem My algorithm works on my data Pass to somebody else and it does not work on their data Fix: Design expensive comparator that is computed off-line Compare cheap online algorithms against comparator on given sequence jiazhong (UCSC) May 16, / 35

4 Motivation What is a fair comparator? Comparator can be based on somewhat unreasonable resources How to measure generalization performance of online algorithm Batch: split into training and test set Online: how to split? Two main examples: Disk spindown problem Caching problem jiazhong (UCSC) May 16, / 35

5 Table of Contents Caching Problem 1 Motivation 2 Caching Problem 3 Master Policies 4 Forced and Optional Policies 5 Deterministic VS Randomized 6 Combining Caches 7 Onlineness Measure jiazhong (UCSC) May 16, / 35

6 Caching Problem Caching Problem Serve requests using a cache of limited size Policy Which cached request to evict to make room for missed request 12 base policies / experts LRU, RAND, FIFO, LIFO, MRU, LFU, MFU, SIZE, GDS, GD*, GDSF, LFUDA jiazhong (UCSC) May 16, / 35

7 Caching Problem Switching between Policies State = content of cache Switch from cache A to B: - charge # of refetched requests needed to switch from A to B Partition with switching policies # of misses = 68 partition of requests switching policies SIZE GDS LRU # of refetches = 34 jiazhong (UCSC) May 16, / 35

8 Caching Problem BestRefetch Curve BestRefetch(R) = minimum miss rate achievable by any partition with refetch rate R miss rate = # of misses over # of total requests refetch rate = # of refetches over # of total requests union of baselines = miss rate of the union of 12 baseline caches L shape decrease Sign of onlineness BestFix BestRefetch Curve Union of Baselines 0.05 miss rate jiazhong (UCSC) May 16, / 35 refetch rate

9 Table of Contents Master Policies 1 Motivation 2 Caching Problem 3 Master Policies 4 Forced and Optional Policies 5 Deterministic VS Randomized 6 Combining Caches 7 Onlineness Measure jiazhong (UCSC) May 16, / 35

10 Master Policies Combining Policies in Expert Setting One weight per policy/expert Weights updated with loss and share update Problem: need to know misses of each expert Solution: simulate all policies on virtual cache - record ID and size - but not the actual data jiazhong (UCSC) May 16, / 35

11 Two Master Policies Master Policies When serving a request Hedge+ picks policy randomly according to their weights Rank-Ideal deterministically computes requests weighted rank and refetches highest ranked requests jiazhong (UCSC) May 16, / 35

12 Master Policies The Rank of a Request A policy ranks its cached requests according to the order of evicting them (low ranked requests are evicted first) Rank all of the uncached requests with zero Evict r 6 first r 1 last r 2 and r 4 not in cache Requests r 1 r 2 r 3 r 4 r 5 r 6 Rank jiazhong (UCSC) May 16, / 35

13 Master Policies The Weighted Rank of a Request The weighted rank of a request r is the sum of its ranks weighted by different policies: weighted rank(r) = N w i rank i,r i=1 where w i is the weight of policy i and rank i,r is the rank of request r by policy i jiazhong (UCSC) May 16, / 35

14 RankIdeal Master Policies At each trail, RankIdeal Refetches as many request as possible by the descending order of their weighted ranks After refetching, the cache is loaded with the Ideal Cache (initial segment of the ordered requests). jiazhong (UCSC) May 16, / 35

15 Performance masters Master Policies BestRefetch BestFix Hedge+ RankIdeal Union of Baselines miss rate refetch rate RankIdeal is better than Hedge+ which uses the same weight update. Why? jiazhong (UCSC) May 16, / 35

16 Table of Contents Forced and Optional Policies 1 Motivation 2 Caching Problem 3 Master Policies 4 Forced and Optional Policies 5 Deterministic VS Randomized 6 Combining Caches 7 Onlineness Measure jiazhong (UCSC) May 16, / 35

17 Forced and Optional Policies Forced and Optional Policies Two types of caching policies Forced: policy is forced to cache missed request Optional: policy may or may not cache missed request In the original paper, 12 baseline policies are forced Hedge+ follows the setting of baseline policies. RankIdeal is optional RankIdeal chooses items according to weighted rank If missed request ranked too low, then not cached Is this the reason why RankIdeal outperforms Hedge+? jiazhong (UCSC) May 16, / 35

18 Forced and Optional Policies Examples of Forced and Optional Base Policies Optional SIZE: only cache a request if it is smaller than the smallest cached one Optional LRU and Forced LRU are the same The missed request is the most recently used one Always cached for both variants jiazhong (UCSC) May 16, / 35

19 Forced and Optional Policies Performance with Forced and Optional Base Policies 0.06 Forced BestRefetch BestFix Hedge+ RankIdeal Union of Baselines Optional refetch rate jiazhong (UCSC) May 16, / 35

20 Forced and Optional Policies Observation Under optional setting Rank-Ideal is still better than Hedge+. Forced/Optional is not the reason of Rank-Ideal s superiority. New goal Curves move down, i.e. lower overall miss rates BestRefetch curve gets flatter Use BestRefetch curves to design online algorithms Find a set of experts so that curves become flat or increasing jiazhong (UCSC) May 16, / 35

21 Table of Contents Deterministic VS Randomized 1 Motivation 2 Caching Problem 3 Master Policies 4 Forced and Optional Policies 5 Deterministic VS Randomized 6 Combining Caches 7 Onlineness Measure jiazhong (UCSC) May 16, / 35

22 Deterministic VS Randomized Deterministic vs Randomized Algorithms Why RankIdeal is better than Hedge+? Rank-Ideal is a deterministic algorithm while Hedge+ is a randomized one. (Deterministic version of Hedge+)MaxHedge+: Same update as Hedge+ Predicts with the policy of maximum weight jiazhong (UCSC) May 16, / 35

23 Deterministic VS Randomized Performance MaxHedge miss rate BestRefetch Best Fixed Hedge+ RankIdeal MaxHedge+ Union of Baselines refetch rate MaxHedge+ and RankIdeal perform similarly and outperform Hedge+ Deterministic algorithms are better on this data. Randomized algorithm should be better on adversary data jiazhong (UCSC) May 16, / 35

24 Table of Contents Combining Caches 1 Motivation 2 Caching Problem 3 Master Policies 4 Forced and Optional Policies 5 Deterministic VS Randomized 6 Combining Caches 7 Onlineness Measure jiazhong (UCSC) May 16, / 35

25 Combining Caches Improve the Combining of the Caches Does RankIdeal combines the caches well? it is unfaire in some sense. Recall that Rank Ideal combines caches with the rank of requests. Cache of 10 requests gives total rank of 10 i=1 i = 55 Cache of 5 requests only gives 15 Denote the score of the i-th request in a cache of n requests by score(i, n). To be fair: n score(i, n) should be a constant not depending on n i=1 jiazhong (UCSC) May 16, / 35

26 Defining score(i, n) Combining Caches For any function f (x) : [0, 1] R, the following definition of score(i, n) is fair: score(i, n) = 1 n f ( i n ) The fairness can be showed by the fact that for any n: n score(i, n) i=1 1 0 f (x)dx jiazhong (UCSC) May 16, / 35

27 Choice of f (x) Combining Caches Non-negative monotone increase functions: f (x) = x φ (φ 0) y=x^0.1 y=x^0.5 y=x y=x^2 y=x^ jiazhong (UCSC) May 16, / 35

28 Results Combining Caches For φ 1, score functions give slightly better results than RankIdeal Rank score(i,n) miss rate /9 1/7 1/5 1/ phi What is the best way for combining while ignoring the amount of refetching? jiazhong (UCSC) May 16, / 35

29 Table of Contents Onlineness Measure 1 Motivation 2 Caching Problem 3 Master Policies 4 Forced and Optional Policies 5 Deterministic VS Randomized 6 Combining Caches 7 Onlineness Measure jiazhong (UCSC) May 16, / 35

30 Onlineness Measure Problem: BestRefetch Curve Overfits Random permuted data should not have any onlineness Curve should not go down miss rate Week Data Set with Random Permutation Week Data Set refetch rate jiazhong (UCSC) May 16, / 35

31 Onlineness Measure Avoiding overfitting Reason: best partition chosen on entire request sequence - i.e. training = test set Batch learning: split into training and testing set jiazhong (UCSC) May 16, / 35

32 Onlineness Measure Online learning: split into even/odd requests Pair1 Pair2 Pair3 Pair4 Pair5 Requests: R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 Training Testing Alternative (not worth it?) Per pair, randomly choose one for training/testing Pair1 Pair2 Pair3 Pair4 Pair5 Requests: R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 Training Testing jiazhong (UCSC) May 16, / 35

33 Onlineness Measure New measure of onlineness Run policies on training requests only For each R, find best partition with refetch rate less than R Run policies on testing requests only Report missrate of the best training partitions on testing requests Decided by BestRefetch(R) on training requests SIZE GDS LRU Training R1 R3 R5 R7 R9 R11 R12 R13 R14 R15 Testing R2 R4 R6 R8 R10 R12 R14 R16 R18 R20 Measure onlineness by misses on testing requests jiazhong (UCSC) May 16, / 35

34 Onlineness Measure BestRefetch with Splitting How does splitting influence the BestRefetch curve? Miss rate increases since the obligatory miss rate increases BestRefetch Odd Requests All VC Odd Requests BestRefetch All Requests All VC All Requests miss rate refetch rate jiazhong (UCSC) May 16, / 35

35 Onlineness Measure Miss Rate of Testing Requests No overfitting to random data: testing miss rate goes up immediately random permuted data train random permuted data test original data train original data test 0.05 miss rate refetch rate jiazhong (UCSC) May 16, / 35