Presentation is loading. Please wait.

Presentation is loading. Please wait.

Lower Bounds for Local Search by Quantum Arguments Scott Aaronson (UC Berkeley) August 14, 2003.

Similar presentations

Presentation on theme: "Lower Bounds for Local Search by Quantum Arguments Scott Aaronson (UC Berkeley) August 14, 2003."— Presentation transcript:

1 Lower Bounds for Local Search by Quantum Arguments Scott Aaronson (UC Berkeley) August 14, 2003

2 Quantum Background Needed for This Talk

3 Outline Problem: Find a local minimum of a function using as few function evaluations (queries) as possible Relational adversary method: A quantum method for proving quantum and classical lower bounds on query complexity (only other example: Kerenidis and de Wolf 2003) Applying the method to L OCAL S EARCH Open problems

4 The L OCAL S EARCH Problem Given: undirected connected graph G=(V,E) and function Task: Find a v V such that for all neighbors w of v 3 3 4 3 3 3 2

5 Motivation Why do local search algorithms work so well in practice? Conventional wisdom: Because finding a local optimum is intrinsically not that hard We show this is falseeven for quantum computers Raises a question: Why do exponentially long chains of descending values, as used for lower bounds, almost never occur in real-world problems?

6 Motivation #2 Quantum adiabatic algorithm (Farhi et al.): Quantum analogue of simulated annealing Can sometimes tunnel through barriers to reach global instead of local optima Further strange feature: For function f(x)=|x| on Boolean hypercube {0,1} n, finds minimum 0 n in O(1) queries, instead of O(n) classically We give first example where adiabatic algorithm is provably only polynomially faster than simulated annealing at finding local optima

7 Motivation #3 Megiddo and Papadimitriou defined a complexity class TFNP, of NP search problems for which we know a solution exists Example: Given a circuit that maps {0,1} n to {0,1} n-1, find two inputs that map to same output Papadimitriou: Are TFNP problems good candidates for fast quantum algorithms? My answer: Probably not –Collision lower bound (A 2002): PPP FBQP relative to an oracle (PPP = Polynomial Pigeonhole Principle, FBQP = Function Bounded-Error Quantum Polytime) –This work: PLS FBQP relative to an oracle (PLS = Polynomial Local Search)


9 Deterministic Query Complexity of L OCAL S EARCH Depends on graph G For an N-vertex line, (log N) 5479 6 Similar for complete binary tree

10 Deterministic Lower Bound Oracle returns decreasing values of f(v), until the set of queried vertices cuts G into 2 pieces Then oracle restricts the problem to largest piece Cuttability tightly characterizes query complexity 8 7 5 6 24 3 1 Llewellyn, Tovey, Trick : (2 n / n) for Boolean hypercube {0,1} n

11 Randomized Query Complexity for any graph with N vertices and max degree d Steepest descent algorithm: - Choosevertices uniformly and query them - Let v 0 be queried vertex with minimum f - Repeatedly let v t+1 be minimum neighbor of v t, until local min is found Claim: Local min is found when whp Proof: At most vertices have smaller f- value than v 0 whp. In that case distance from v 0 to local min in steepest descent tree is at most

12 Randomized Lower Bound Random walk mixes in n log n steps If you havent yet found a v with f(v)<2 n/2, intuitively the best you can do is continue stabbing in the dark Hard to prove! 12 3 56 8 1213 Aldous 1983: 2 n/2-o(n) for hypercube Idea: Pick random start vertex, then take random walk. Label each vertex with 1 st hitting time

13 Quantum Query Complexity O((Nd) 1/3 ) for any graph with N vertices and max degree d Choose (Nd) 2/3 vertices uniformly at random Use Grovers quantum search algorithm to find the v 0 with minimum f-value in time As before, follow v 0 to local min by steepest descent

14 A: Set of 0-inputsB: Set of 1-inputs Choose a function R(f,g) 0 For all f A, g B, and indices v, let Ambainis Adversary Method Most General Version Then quantum query complexity is (1/ geom ) where

15 Example: ( N) for Inverting a Permutation 315624345621315624345621 Let A = set of permutations of {1,…,N} with 1 on left half, B = set with 1 on right half R(f,g)=1 if g obtained from f by swapping the 1, R(f,g)=0 otherwise fgfg (f,2)=1, but (g,2)=2/N (g,6)=1, but (f,6)=2/N

16 Compare to Relational Adversary Method Let A, B, R(f,g), (f,v), (g,v) be as before Then classical randomized query complexity is (1/ min ) where Example: For inverting a permutation, we get (N) instead of ( N)

17 New Lower Bounds for L OCAL S EARCH On Boolean hypercube {0,1} n : quantum queries randomized queries On d-dimensional cube of N vertices (d3): quantum queries randomized queries

18 Modified Problem Starting from the head, follow a snake of L N descending values to the unique local minimum of f, then return an answer bit found there. Clearly a lower bound for this problem implies an equivalent lower bound for L OCAL S EARCH 6 5 4 3 2 1 7 7 8 8 8 9 9 10 11 9 910 11 12 13 10 (Known) Snake Head Snake Tail (contains binary answer) 10 11 12 G

19 Let D be a distribution over snakes (x 0,…,x L-1 ), with x L-1 =h and x i+1 adjacent to x i for all i We say an X drawn from D is -good if the following holds. Choose j uniformly from {0,…,L-1}, and let D X,j be the distribution over snakes Y=(x 0,…,x L-1 ) drawn from D conditioned on x t =y t for all t>j. Then (1) (2) For all vertices v of G, Good Snakes

20 Theorem: Suppose theres a snake distribution D, such that a snake drawn from D is -good with probability at least 9/10. Then the quantum query complexity of L OCAL S EARCH on G is, and the randomized is

21 Sensitivity 4 3 2 1 5 6 7 8 9 10 11 j x0x0 5 4 3 2 1 6 7 8 9 10 11 y0y0 Large (f X,v) but small (f Y,v) Large (f Y,v) but small (f X,v) x L-1 =y L-1 =h

22 Sources of Trouble 2 1 3 4 5 6 7 8 Bunched-Up Snake 2 1 Snake Tails Intersect 3 4 3 2 1 Idea: Just remove inputs that cause trouble! Lemma: Suppose a graph G has average degree k. Then G has an induced subgraph with minimum degree at least k/2.

23 Instead of Aldous random walk, more convenient to define snake distribution D using a coordinate loop Given v {0,1} n, let v (i) = (v with i th bit flipped) Let x 0 = h, x t+1 = x t with ½ probability, x t+1 = x t (t mod n) with ½ probability Mixes completely in n steps Theorem: A snake drawn from D is n 2 /2 n/2 -good with probability at least 9/10 Boolean Hypercube {0,1} n

24 Drawbacks of random walk become more serious: mixing time is too long, too many self-intersections Instead define D by struts of randomly chosen lengths connected at endpoints d-dimensional cube (d3) Theorem: A snake drawn from D is (logN)/N 1/2-1/d - good with probability at least 9/10

25 Open Problems 2 n/4 vs. 2 n/3 gap for quantum complexity on {0,1} n 2 n/2 /n 2 vs. 2 n/2 n gap for randomized complexity 2D square grid Conjecture: Deterministic, randomized, and quantum query complexities are polynomially related for every family of graphs Apply relational adversary method to other problems

Download ppt "Lower Bounds for Local Search by Quantum Arguments Scott Aaronson (UC Berkeley) August 14, 2003."

Similar presentations

Ads by Google