Why are learned indexes so effective? Paolo Fabrizio Giorgio - - PowerPoint PPT Presentation

Why are learned indexes so effective?

Paolo Ferragina1 Fabrizio Lillo2 Giorgio Vinciguerra1

1University of Pisa 2University of Bologna

A classical problem in computer science

• Given a set of 𝑜 sorted input keys (e.g. integers)
• Implement membership and predecessor queries
• Range queries in databases, conjunctive queries in search

engines, IP lookup in routers…

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𝑛𝑓𝑛𝑐𝑓𝑠 36 = True 𝑞𝑠𝑓𝑒𝑓𝑑𝑓𝑡𝑡𝑝𝑠 50 = 48

2 11 13 15 18 23 24 29 31 34 36 44 47 48 55 59 60 71 73 74 76 88 95

1 𝑜

Indexes

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2 11 13 15 18 23 24 29 31 34 36 44 47 48 55 59 60 71 73 74 76 88 95

1 𝑜 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜 𝑙𝑓𝑧

B-tree

Input data as pairs (𝑙𝑓𝑧, 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜)

4 positions keys

2 11 13 15 18 23 24 29 31 34 36 44 47 48 55 59 60 71 73 74 76 88 95

1 𝑜

Input data as pairs (𝑙𝑓𝑧, 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜)

5 positions keys

2 11 13 15 18 23 24 29 31 34 36 44 47 48 55 59 60 71 73 74 76 88 95

1 𝑜 2 3 4

2 11 13 15 1 2 3 4

Learned indexes

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𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜 𝑙𝑓𝑧 Black-box trained on a dataset of pairs (key, pos) 𝒠 = { 2,1 , 11,2 , … , (95, 𝑜)} Binary search in 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜 − 𝜁, 𝑞𝑝𝑡𝑗𝑢𝑗𝑝𝑜 + 𝜁 (approximate)

positions keys

2 11 13 15 18 23 24 29 31 34 36 44 47 48 55 59 60 71 73 74 76 88 95

1 𝑜 e.g. 𝜁 is of the order of 100–1000

The knowledge gap in learned indexes

Practice Same query time of traditional tree-based indexes Space improvements of

• rders of magnitude,

from GBs to few MBs Theory Same asymptotic query time of traditional tree-based indexes Same asymptotic space

• ccupancy of traditional

tree-based indexes

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👏

vs

👎

vs

PGM-index: An optimal learned index

• 1. Fix a max error 𝜁, e.g. so that keys in [𝑞𝑝𝑡 − 𝜁,𝑞𝑝𝑡 + 𝜁]fit a cache-line
• 2. Find the smallest Piecewise Linear 𝜁-Approximation (PLA)
• 3. Store triples (𝑔𝑗𝑠𝑡𝑢𝑙𝑓𝑧, 𝑡𝑚𝑝𝑞𝑓, 𝑗𝑜𝑢𝑓𝑠𝑑𝑓𝑞𝑢) for each segment

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https://pgm.di.unipi.it

positions keys

1 3 8 11 12 19 22 23 24 28 29 33 38 47 48 53 55 56 57

𝑞𝑝𝑡 − 𝜁, 𝑞𝑝𝑡 + 𝜁

24 8

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[Ferragina and Vinciguerra, PVLDB 2020]

What is the space of learned indexes?

• Space occupancy ∝ Number segments
• The number of segments depends on
• The size of the input dataset
• How the points (𝑙𝑓𝑧, 𝑞𝑝𝑡) map to the plane
• The value 𝜁, i.e. how much the approximation is precise

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positions keys positions keys positions keys

𝜁! 𝜁" ≪ 𝜁!

Model and assumptions

• Consider gaps 𝑕! = 𝑙!"# − 𝑙! between consecutive input keys
• Model the gaps as positive iid rvs that follow a distribution with

finite mean 𝜈 and variance 𝜏$

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positions keys

𝑕( 𝑕) 𝑕* 𝑕+

1 2 3 4 5 𝑙" 𝑙# 𝑙$ 𝑙% 𝑙&

The main result

• Theorem. If 𝜁 is sufficiently larger than 𝜏/𝜈, the expected number
• f keys covered by a segment with maximum error 𝜁 is

𝐿 = 𝜈$ 𝜏$ 𝜁$ and the number of segments on a dataset of size 𝑜 is 𝑜 𝐿 with high probability.

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The main consequence

The PGM-index achieves the same asymptotic query performance of a traditional 𝜁-way tree-based index while improving its space from 𝜤(𝒐/𝜻) to 𝑷(𝒐/𝜻𝟑)

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Learned indexes are pr provably better than traditional indexes

(note that 𝜁 is of the order of 100-1000)

Sketch of the proof

• 1. Consider a segment on the stream of random gaps and the two

parallel lines at distance 𝜁

• 2. How many steps before a new segment is needed?

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Start a new segment from here

positions keys

𝜁 𝜁

Sketch of the proof (2)

3. A discrete-time random walk, iid increments with mean 𝜈 4. Compute the expectation of 𝑗∗ = min 𝑗 ∈ ℕ 𝑙I, 𝑗 is outside the red strip i.e. the Mean Exit Time (MET) of the random walk 5. Show that the slope 𝑛 = 1/𝜈 maximises 𝐹[𝑗∗], giving 𝐹[𝑗∗] = 𝜈J/𝜏J 𝜁J

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positions keys

𝜁 𝜁

Start a new segment from here

(𝑙𝑓𝑧!∗, 𝑗∗)

random walker location time

Start a new segment from here

𝑗∗

𝜁 𝑛 𝜁 𝑛

Simulations

• 1. Generate 107 random streams of gaps according to

several probability distributions

• 2. Compute and average

I. The length of a segment found by the algorithm that computes the smallest PLA, adopted in the PGM-index II. The exit time of the random walk

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Simulations of (𝜈*/𝜏*)𝜁*

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250 50 100 150 200 250 ε Pareto k = 3, α = 3

OPT MET Thm 1 (3.0ε2)

0.5 1 1.5 Mean segment length ·106 Lognormal µ = 1, σ = 0.5

OPT MET Thm 1 (3.521ε2)

250 50 100 150 200 250 ε

More distributions in the paper

OPT = Average segment length in a PGM-index MET = Mean exit time of the random walk

Both OPT and MET agree on the slope 1/µ, but OPT is more robust

Stress test of “𝜁 sufficiently larger than 𝜏/𝜈”

50 100 150 200 250 0.1 0.2 ε Relative error σ/µ = 0.15

Pareto k = 10, α = 7.741 Gamma θ = 5, k = 44.444 Lognormal µ = 2, σ = 0.149 44.444ε2

50 100 150 200 250 0.2 0.4 0.6 0.8 ε σ/µ = 1.5

Pareto k = 10, α = 2.202 Gamma θ = 5, k = 0.444 Lognormal µ = 2, σ = 1.086 0.444ε2

50 100 150 200 250 0.5 1 ε σ/µ = 15

Pareto k = 10, α = 2.002 Gamma θ = 5, k = 0.004 Lognormal µ = 2, σ = 2.328 0.004ε2

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Conclusions

• No theoretical grounds for the efficiency of learned indexes was known
• We have shown that on data with iid gaps, the mean segment length is Θ(𝜁J)
• The PGM-index takes O(𝑜/𝜁J) space w.h.p., a quadratic improvement in 𝜁
• ver traditional indexes (𝜁 is usually of the order of 100–1000)
• Open problems:

1. Do the results still hold without the iid assumption on the gaps? 2. Is the segment found by the optimal algorithm adopted in the PGM-index a constant factor longer than the one found by the random walker?

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