1. Online Algorithms via Projections set cover, paging, k-server
Niv Buchbinder
Tel-Aviv
Anupam Gupta
CMU
Marco Molinaro
PUC-Rio
Seffi Naor
Technion

2. K-Server problem
Finite metric space (𝑉,𝑑) with |𝑉|=𝑛 points 𝑘= # of “servers” that algo controls Input: request sequence 𝑟 1 , 𝑟 2 , …, 𝑟 𝑡 , … Output: on seeing 𝑟 𝑡 , algo needs to have a server at 𝑟 𝑡 Min: total distance moved by servers OPT = optimal cost/solution in hindsight Goal: online ALG such that 𝔼[𝑐𝑜𝑠𝑡(𝐴𝐿𝐺)] ≤ 𝛼.𝑐𝑜𝑠𝑡(𝑂𝑃𝑇) + 𝑐′

3. (very partial) history and results
Deterministic 2𝑘−1 upper bound, 𝑘 lower bound [Koutsoupias Papadimitriou 95] Randomized Ω( log 𝑘 ) even when metric is a star [folklore 90s?] 𝑂(log 𝑘) for weighted stars [Bansal Buchbinder Naor 07] 𝑂( log 3 𝑛 log 2 ⁡𝑘) [Bansal Buchbinder Madry Naor 11] 𝑶( 𝐥𝐨𝐠 𝟔 𝒌) [Bubeck Cohen Lee Lee Madry 18] [Lee 18]
Rounding, loses 𝑶(𝟏)
𝑶( log 𝟐 𝑘 ) fractional on HSTs
If HSTs, then general metrics, loses 𝑶( log 𝟒 𝑘 )

4. (very partial) history and results
Deterministic 2𝑘−1 upper bound, 𝑘 lower bound [Koutsoupias Papadimitriou 95] Randomized Ω( log 𝑘 ) even when metric is a star [folklore 90s?] 𝑂(log 𝑘) for weighted stars [Bansal Buchbinder Naor 07] 𝑂( log 3 𝑛 log 2 ⁡𝑘) [Bansal Buchbinder Madry Naor 11] 𝑶( 𝐥𝐨𝐠 𝟔 𝒌) [Bubeck Cohen Lee Lee Madry 18]
𝑶( log 𝟐 𝑘 ) fractional on HSTs

5. 𝑂( log 2 𝑘 ) on HST
Technique: Continuous time online mirror descent - Differential inclusion gives trajectory 𝑥 𝑡 Can be discretized
𝑥 𝑡
𝑥 𝑡+1
Hopefully some progress in role of regularization in online algo

6. Our result Our result: Very coarse discretization works!
≡ projection-based algorithm works
Projection-based algorithms as a natural option for movement-based online problems? [Buchbinder Chen Naor 14]
Thm: Discrete*, projection-based algo gives 𝑂( log 2 𝑘 ) approximation for fractional k-server on HSTs
𝑥 𝑡
𝑥 𝑡+1
𝑥 𝑡
𝑥 𝑡+1
Hopefully some progress in role of regularization in online algo

7. projection-based algorithm
“Base” polytope 𝐾 based on metric (HST) and 𝑘 At time t: polytope 𝐾 𝑡 of feasible states where both alg and opt need to be in (i.e. 𝐾 + some server at the requested position 𝑟 𝑡 ) Distance is a Bregman divergence Use variants of KL divergence
𝑲 𝒕 𝑲
𝑥 𝑡
𝑲 𝒕+𝟏
Algorithm:
𝑥 𝑡+1 ← arg min 𝑥∈ 𝐾 𝑡+1 distance(𝑥, 𝑥 𝑡 )
𝑥 𝑡+1

8. rest of the talk
- Illustration on a simpler problem: Online Set Cover - Some words about generalization to k-server - Closing remarks

9. online set cover
𝑥 𝑡−1 ∈ 𝐑 + 𝑛 At time t: 𝑎 𝑡 , 𝑥 ≥1 for some 𝑎 𝑡 ∈ 0,1 𝑛 Monotonically increase 𝑥 𝑡−1 → 𝑥 𝑡 Satisfy all constraints until now Movement cost at time t: | 𝑥 𝑡 − 𝑥 𝑡−1 | 1 = 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 Min: total movement cost 𝒕 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 = 𝟏, 𝑥 𝑇
𝒙 𝒕
𝒙 𝒕−𝟏

10. the projection-based algorithm
Define the feasible states set
𝐾 𝑡 = all constraints up to time t
= 𝑥≥0 𝑎 𝑠 , 𝑥 ≥1 ∀𝑠≤𝑡}
(cannot add monotonicity 𝑥≥ 𝑥 𝑡−1 , OPT does not satisfy)
𝑲 𝒕

11. the projection-based algorithm
Define the feasible states set
𝐾 𝑡 = all constraints up to time t
= 𝑥≥0 𝑎 𝑠 , 𝑥 ≥1 ∀𝑠≤𝑡}
(cannot add monotonicity 𝑥≥ 𝑥 𝑡−1 , OPT does not satisfy)
𝑲 𝒕
Algorithm:
𝑥 0 = 1/n
𝑥 𝑡 ← arg min 𝑥∈ 𝐾 𝑡 D(𝑥| 𝑥 𝑡−1 )
𝐷(𝑝 𝑞 = 𝑖 𝑝 𝑖 log 𝑝 𝑖 𝑞 𝑖 − 𝑝 𝑖 + 𝑞 𝑖
unnormalized KL divergence

12. Guarantee of Proj-based algo
Obs: The algo is feasible: 𝑥 𝑡 are monotonically increasing
Thm: ALG ≤ log 𝑛 ⋅OPT+ 1
Not new [Alon et al. 03], [Buchbinder, Chen, Naor 14], ...
Comparing our fractional solution 𝑥 𝑡 to integral OPT 𝑦 𝑡

13. Analysis: cost
Cost: 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1
Algo: 𝑥 0 = 1/n
𝑥 𝑡 ← arg min 𝑥∈ 𝐾 𝑡 D(𝑥| 𝑥 𝑡−1 )
𝐷(𝑝 𝑞 = 𝑖 𝑝 𝑖 log 𝑝 𝑖 𝑞 𝑖 − 𝑝 𝑖 + 𝑞 𝑖
Main property of divergence: Reverse Pythagorean inequality
𝐷(𝑦 𝑥 ≥ 𝐷(𝑦| 𝑥 𝑝𝑟𝑜𝑗 ) + 𝐷( 𝑥 𝑝𝑟𝑜𝑗 |𝑥)
𝑥 𝑝𝑟𝑜𝑗 𝑥 ≥𝟎
⇒𝐷 𝑦 𝑥 𝑝𝑟𝑜𝑗 −𝐷(𝑦 𝑥 ≤0 𝑦

14. Analysis: cost
Cost: 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1
Algo: 𝑥 0 = 1/n
𝑥 𝑡 ← arg min 𝑥∈ 𝐾 𝑡 D(𝑥| 𝑥 𝑡−1 )
𝐷(𝑝 𝑞 = 𝑖 𝑝 𝑖 log 𝑝 𝑖 𝑞 𝑖 − 𝑝 𝑖 + 𝑞 𝑖
Main property of divergence: Reverse Pythagorean inequality
𝑥 𝑡
𝑥 𝑡−1
𝐾 𝑡
⇒𝐷 𝑦 𝑡 𝑥 𝑡 −𝐷( 𝑦 𝑡 𝑥 𝑡−1 ≤0
𝑦 𝑡

15. Analysis: cost
Cost: 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1
Algo: 𝑥 0 = 1/n
𝑥 𝑡 ← arg min 𝑥∈ 𝐾 𝑡 D(𝑥| 𝑥 𝑡−1 )
𝐷(𝑝 𝑞 = 𝑖 𝑝 𝑖 log 𝑝 𝑖 𝑞 𝑖 − 𝑝 𝑖 + 𝑞 𝑖
Main property of divergence: Reverse Pythagorean inequality
Φ(𝑦 𝑥 =𝐷 𝑦 𝑥 + 𝟏,𝑦 − 𝟏,𝑥
𝑥 𝑡
𝑥 𝑡−1
𝐾 𝑡
𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡 𝑥 𝑡−1 ≤0
⇒𝐷 𝑦 𝑡 𝑥 𝑡 −𝐷( 𝑦 𝑡 𝑥 𝑡−1 ≤0
𝑦 𝑡

16. Analysis: cost
Cost: 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1
Algo: 𝑥 0 = 1/n
𝑥 𝑡 ← arg min 𝑥∈ 𝐾 𝑡 D(𝑥| 𝑥 𝑡−1 )
𝐷(𝑝 𝑞 = 𝑖 𝑝 𝑖 log 𝑝 𝑖 𝑞 𝑖 − 𝑝 𝑖 + 𝑞 𝑖
Main property of divergence: Reverse Pythagorean inequality
Φ(𝑦 𝑥 =𝐷 𝑦 𝑥 + 𝟏,𝑦 − 𝟏,𝑥
𝑥 𝑡−1
𝑥 𝑡
𝑦 𝑡
𝐾 𝑡
𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡−1 𝑥 𝑡−1 ≤Φ( 𝑦 𝑡 𝑥 𝑡−1 −Φ( 𝑦 𝑡−1 𝑥 𝑡−1
𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡 𝑥 𝑡−1 ≤0
ALGs cost
change in potential
≈ OPTs cost ?

17. Analysis: cost 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡 𝑥 𝑡−1 ≤0
Algo: 𝑥 0 = 1/n
𝑥 𝑡 ← arg min 𝑥∈ 𝐾 𝑡 D(𝑥| 𝑥 𝑡−1 )
𝐷(𝑝 𝑞 = 𝑖 𝑝 𝑖 log 𝑝 𝑖 𝑞 𝑖 − 𝑝 𝑖 + 𝑞 𝑖
Φ(𝑦 𝑥 =𝐷 𝑦 𝑥 + 𝟏,𝑦 − 𝟏,𝑥
𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡 𝑥 𝑡−1 ≤0
𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡−1 𝑥 𝑡−1 ≤Φ( 𝑦 𝑡 𝑥 𝑡−1 −Φ( 𝑦 𝑡−1 𝑥 𝑡−1
⇒𝐷 𝑦 𝑡 𝑥 𝑡 −𝐷( 𝑦 𝑡 𝑥 𝑡−1 ≤0
≈ OPTs cost ?
Lemma: RHS ≤ log 𝑛 ∙ OPTs cost
Pf: If OPT increases 𝑦 𝑖 𝑡−1 =0 → 𝑦 𝑖 𝑡 =1
OPTs cost = +1
ΔΦ=1 log 1 𝑥 𝑖 𝑡−1 −0 log 0 𝑥 𝑖 𝑡−1 ≤ log 𝑛 (because 𝑥 𝑖 𝑡−1 ≥ 𝑥 𝑖 0 ≥ 1 𝑛 )

18. Analysis: cost 𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡 𝑥 𝑡−1 ≤0
Algo: 𝑥 0 = 1/n
𝑥 𝑡 ← arg min 𝑥∈ 𝐾 𝑡 D(𝑥| 𝑥 𝑡−1 )
𝐷(𝑝 𝑞 = 𝑖 𝑝 𝑖 log 𝑝 𝑖 𝑞 𝑖 − 𝑝 𝑖 + 𝑞 𝑖
Φ(𝑦 𝑥 =𝐷 𝑦 𝑥 + 𝟏,𝑦 − 𝟏,𝑥
𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡 𝑥 𝑡−1 ≤0
𝟏, 𝑥 𝑡 − 𝑥 𝑡−1 +Φ( 𝑦 𝑡 𝑥 𝑡 − Φ( 𝑦 𝑡−1 𝑥 𝑡−1 ≤Φ( 𝑦 𝑡 𝑥 𝑡−1 −Φ( 𝑦 𝑡−1 𝑥 𝑡−1
⇒𝐷 𝑦 𝑡 𝑥 𝑡 −𝐷( 𝑦 𝑡 𝑥 𝑡−1 ≤0
log 𝑛 ∙ OPTs cost
Adding over all times t:
ALGs total cost ≤ log⁡𝑛∙ OPTs total cost + Φ( 𝑦 0 | 𝑥 0 )
≤ log⁡𝑛∙ OPTs total cost + 1

19. rest of the talk
- Illustration on a simpler problem: Online Set Cover - Some words about generalization to k-server - Closing remarks

20. k-server polytope and distance
Use the non-trivial LP from Bubeck Cohen Lee Lee Madry 18
(~unary encoding of # servers + …)
𝐾 = { 𝑥 : 𝑥 𝑢,𝑗 ∈[0,1]
𝑥 root, 𝑗 ≤ 𝟏 𝑗≤𝑘 ∀ 𝑗∈𝑛
𝑗≤|𝑆| 𝑥 𝑝 𝑆 , 𝑗 ≥ 𝑣,ℓ ∈𝑆 𝑥 𝑣,ℓ }
(actual polytope: “anti-server” polytope)
𝐾 𝑡 =𝐾∩ 𝑥 𝑟 𝑡 ≥ 1
Also use divergence from Bubeck Cohen Lee Lee Madry 18
𝐷(𝑝 𝑞 = ∑ 𝑢 𝑤 𝑢 ( ∑ 𝑗 𝑝 𝑢,𝑗 log 𝑝 𝑢,𝑗 𝑞 𝑢,𝑗 − 𝑝 𝑢,𝑗 + 𝑞 𝑢,𝑗 )
∀ S with common parent

21. proof elements Very inspired in proof of Bubeck Cohen Lee Lee Madry 18
𝐾 𝑡 = { 𝑥 : 𝑥 𝑢,𝑗 ∈[0,1]
𝑥 root, 𝑗 ≥ 𝟏 𝑗>𝑘 ∀ 𝑗∈𝑛
𝑗≤|𝑆| 𝑥 𝑝 𝑆 , 𝑗 ≥ 𝑣,ℓ ∈𝑆 𝑥 𝑣,ℓ ∀ S with common parent
𝑥 𝑟 𝑡 , ≤ 𝛿 }
Very inspired in proof of Bubeck Cohen Lee Lee Madry 18
Simplification and KKT
Potential: D + linear terms
Cost function not linear anymore (||∙| ​ 1 -type but no monotonicity)
Stronger version of Reverse Pythagorean ineq, relates to duals
To avoid dependence on height, additional potential, delicate

22. closing remarks
Discrete projection-based algorithms for k-server (and paging) 𝑂( log 2 𝑘) -competitive for fractional k-server on HSTs Matches results of Bubeck et al. 18 Show tight 𝑂( log 𝑘) result k-server on HSTs, and on general graphs? Other problems? What are crucial properties of the LP? Right divergence?

23. Thank you!