Lower Bounds for Local Search by Quantum Arguments

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Lower Bounds for Local Search by Quantum Arguments

• Scott Aaronson (UC Berkeley)
• http//www.cs.berkeley.edu/aaronson
• August 14, 2003

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Quantum Background Needed for This Talk
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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 LOCAL SEARCH
• Open problems

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The LOCAL SEARCH Problem

• Given undirected connected graph G(V,E) and
  function
• Task Find a v?V such thatfor all neighbors w of
  v

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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?

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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 0,1n, finds minimum 0n 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

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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 0,1n to
  0,1n-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)

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FNP
TFNP
PLS
PPP
FP
FBQP
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Deterministic Query Complexity of LOCAL SEARCH

• Depends on graph G

• For an N-vertex line, ?(log N)

• Similar for complete binary tree

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Deterministic Lower Bound

• Llewellyn, Tovey, Trick ?(2n/?n) for Boolean
  hypercube 0,1n

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• 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

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Randomized Query Complexity

• for any graph with N vertices and max degree d
• Steepest descent algorithm- Choose vertices
  uniformly and query them- Let v0 be queried
  vertex with minimum f- Repeatedly let vt1 be
  minimum neighbor of vt, until local min is found
• Claim Local min is found when whp
• Proof At most vertices have smaller f-value
  than v0 whp. In that case distance from v0 to
  local min in steepest descent tree is at most

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Randomized Lower Bound

• Aldous 1983 2n/2-o(n) for hypercube
• Idea Pick random start vertex, then take random
  walk. Label each vertex with 1st hitting time

• Random walk mixes in n log n steps
• If you havent yet found a v with f(v)lt2n/2,
  intuitively the best you can do is continue
  stabbing in the dark
• Hard to prove!

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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
  v0 with minimum f-value in time
• As before, follow v0 to local min by steepest
  descent

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Ambainis Adversary MethodMost General Version
A Set of 0-inputs B Set of 1-inputs Choose a
function R(f,g)?0 For all f?A, g?B, and indices
v, let
Then quantum query complexity is ?(1/?geom) where
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Example ?(?N) for Inverting a Permutation
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
f g
?(f,2)1, but ?(g,2)2/N ?(g,6)1, but
?(f,6)2/N
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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
Compare to
Example For inverting a permutation, we get ?(N)
instead of ?(?N)
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New Lower Bounds for LOCAL SEARCH

• On Boolean hypercube 0,1n
• quantum queries
• randomized queries
• On d-dimensional cube of N vertices (d3)
• quantum queries
• randomized queries

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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 LOCAL SEARCH
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Good Snakes
Let D be a distribution over snakes (x0,,xL-1),
with xL-1h and xi1 adjacent to xi 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 DX,j be the distribution over snakes
Y(x0,,xL-1) drawn from D conditioned on xtyt
for all tgtj. Then (1) (2) For all vertices v of
G,
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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 LOCAL SEARCH on G is
, and the randomized is
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Sensitivity
x0
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xL-1yL-1h
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Sources of Trouble
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.
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Boolean Hypercube 0,1n
Instead of Aldous random walk, more convenient
to define snake distribution D using a
coordinate loop Given v?0,1n, let v(i) (v
with ith bit flipped) Let x0 h, xt1 xt
with ½ probability, xt1 xt(t mod n) with ½
probability Mixes completely in n steps Theorem
A snake drawn from D is n2/2n/2-good with
probability at least 9/10
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d-dimensional cube (d3)
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
Theorem A snake drawn from D is
(logN)/N1/2-1/d-good with probability at least
9/10
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Open Problems

• 2n/4 vs. 2n/3 gap for quantum complexity on
  0,1n
• 2n/2/n2 vs. 2n/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