Random Walks and Evolving Sets: Faster Convergences and Limitations

Transcription

1 Random Walks and Evolving Sets: Faster Convergences and Limitations Siu On Chan 1 Tsz Chiu Kwok 2 Lap Chi Lau 2 1 The Chinese University of Hong Kong 2 University of Waterloo Jan 18, /23

2 Finding small clusters Given 1-regular graph with edge weights w (Edge) expansion φ(s) w(s, V \ S) S Task: compute φ δ (G) min S δn φ(s) 2/23

3 Algorithm 1: Random walk For finding small clusters [Spielman Teng 04, Andersen Chung Lang 06, ] Theorem ([Kwok Lau 12]) Random walk returns S with φ(s) = O( φ(s )/ε) and S = O( S 1+ε ) in poly time, when start from nice vertices in S When ε = 1/ log S φ(s) = O( φ(s ) log S ) and S = O( S ) 3/23

4 Analysis: Lovász Simonovits [ 90] Measure progress of lazy random walk mixing with a curve Large φ(g) implies fast mixing (lazy = self-loop of weight 1/2 at every vertex) More on this later 4/23

5 Evolving Set Process [Morris 02] Yields strong bounds on mixing times [Morris Peres 05] Evolving Set Process is Markov Chain {S t } on subsets of V Given S t, choose U uniformly from [0, 1] S t+1 {v V w(s t, v) U} small U S t large U S t+1 5/23

6 Algorithm 2: Evolving Set Process Can find small clusters [Andersen Peres 09, Oveis Gharan Trevisan 12] Theorem ([Oveis Gharan Trevisan 12, AOPT 16]) Evolving Set Process returns S with φ(s) = O( φ(s )/ε) and S = O( S 1+ε ) in poly time, when start from nice vertices in S 6/23

7 Algorithm 2: Evolving Set Process Can find small clusters [Andersen Peres 09, Oveis Gharan Trevisan 12] Theorem ([Oveis Gharan Trevisan 12, AOPT 16]) Evolving Set Process returns S with φ(s) = O( φ(s )/ε) and S = O( S 1+ε ) in poly time, when start from nice vertices in S Conjecture [Oveis Gharan 13, AOPT 16] With non-trivial probability, in fact S = O( S ) If true, will refute Small-Set-Expansion Hypothesis that φ δ (G) is hard to approximate [Raghavendra Steurer 10] (cousin of Unique-Games Conjecture) 6/23

8 Our results: Generalized Lovász Simonovits analysis Also measure random walk mixing with LS curve 1. Large combinatorial gap ϕ(g) implies fast mixing 2. Large robust vertex expansion φ V (G) + laziness imply fast mixing 7/23

9 Our results: Combinatorial gap Large combinatorial gap ϕ(g) implies fast mixing ϕ(g) w(s, T) min 1 S = T n/2 S 1. Similar definition as φ(g) (which only allows T = S) 2. ϕ(g) = φ(g) for lazy graphs 3. ϕ(g) small if G has near-bipartite component Corollary (Expansion of graph powers) φ δ/4 (G t ) = min{ω( tφ δ (G)), 1/20} without laziness assumption of [Kwok Lau 14] 8/23

10 Our results: Vertex expansion Robust vertex expansion φ V defined by [Kannan Lovász Montenegro 06] Theorem Evolving Set Process returns S with φ(s) = O(φ(S )/(εφ V (S))) and S = O( S 1+ε ) in poly time, when start from nice vertices in S Compare: φ(s) = O( φ(s )/ε) in [APOT 16] 9/23

11 Our results: Vertex expansion Robust vertex expansion φ V defined by [Kannan Lovász Montenegro 06] Theorem Evolving Set Process returns S with φ(s) = O(φ(S )/(εφ V (S))) and S = O( S 1+ε ) in poly time, when start from nice vertices in S Compare: φ(s) = O( φ(s )/ε) in [APOT 16] Implies constant factor approximation when φ V (G) min S n/2 φv (S) is Ω(1) Evolving Set Process analog of spectral partitioning result in [Kwok Lau Lee 16] 9/23

12 Our results: Hard instances Theorem For arbitrarily small δ, some graphs have a hidden small cluster S φ(s ) ε and S = δn but Evolving Set Process never returns S with φ(s) 1 ε and S δ ε n Refute the conjecture in [Oveis Gharan 13, AOPT 16] Also limitation of random walk, PageRank, etc 10/23

13 Details: Generalized Lovász Simonovits Analysis 11/23

14 Lovász Simonovits Let p be probability vector Measure random walk mixing with curve C(p, x) C(p, x) sum of first x largest elements in p for integer x C(p, x) C(Ap, x) 12/23

15 Lovász Simonovits Let p be probability vector Measure random walk mixing with curve C(p, x) C(p, x) sum of first x largest elements in p for integer x C(p, x) C(Ap, x) x x(1 φ(g)) x(1 + φ(g)) When lazy C(Ap, x) 1 (C(p, x(1 φ(g))) + C(p, x(1 + φ(g)))) 2 12/23

16 Lovász Simonovits Let p be probability vector Measure random walk mixing with curve C(p, x) C(p, x) sum of first x largest elements in p for integer x C(p, x) C(Ap, x) x x(1 φ(g)) x(1 + φ(g)) When lazy C(Ap, x) 1 (C(p, x(1 φ(g))) + C(p, x(1 + φ(g)))) 2 By induction: C(A t p, x) x n + x (1 φ(g)2 8 ) t 12/23

17 Generalizing Lovász Simonovits x x(1 ϕ(g)) x(1 + ϕ(g)) C(Ap, x) 1 (C(p, x(1 ϕ(g))) + C(p, x(1 + ϕ(g)))) 2 ϕ(g) in place of φ(g) laziness not required ϕ(g) = φ(g) for lazy graphs generalizing LS to non-lazy More intuitive analysis 13/23

18 Sketch of main ideas C(Ap, S ) 1 (C(p, S (1 ϕ(g))) + C(p, S (1 + ϕ(g)))) 2 Sort vertices by decreasing d S (i) w(i, S) (Ap)(S) = 0 i n d S (i)p(i) = (d S (i) d S (i + 1)) p(j) 0 i n 1 j i } {{ } C(p,i) 14/23

19 Sketch of main ideas C(Ap, S ) 1 (C(p, S (1 ϕ(g))) + C(p, S (1 + ϕ(g)))) 2 Sort vertices by decreasing d S (i) w(i, S) (Ap)(S) = 0 i n d S (i)p(i) = (d S (i) d S (i + 1)) p(j) 0 i n 1 j i } {{ } C(p,i) 14/23

20 Sketch of main ideas C(Ap, S ) 1 (C(p, S (1 ϕ(g))) + C(p, S (1 + ϕ(g)))) 2 Sort vertices by decreasing d S (i) w(i, S) (Ap)(S) = 0 i n d S (i)p(i) = (d S (i) d S (i + 1)) p(j) 0 i n 1 j i } {{ } C(p,i) 1 d S (i) 14/23

21 Sketch of main ideas C(Ap, S ) 1 (C(p, S (1 ϕ(g))) + C(p, S (1 + ϕ(g)))) 2 Sort vertices by decreasing d S (i) w(i, S) (Ap)(S) = 0 i n d S (i)p(i) = (d S (i) d S (i + 1)) p(j) 0 i n 1 j i } {{ } C(p,i) 1 d S (i) d S (i) d S (i + 1) 14/23

22 Sketch of main ideas C(Ap, S ) 1 (C(p, S (1 ϕ(g))) + C(p, S (1 + ϕ(g)))) 2 From earlier (Ap)(S) (d S (i) d S (i + 1))C(p, i) 0 i n 1 d S (i) w(i, S) 1 2 Upper Area 1 S (1 ϕ(g)) 2 15/23

23 Sketch of main ideas C(Ap, S ) 1 (C(p, S (1 ϕ(g))) + C(p, S (1 + ϕ(g)))) 2 From earlier (Ap)(S) (d S (i) d S (i + 1))C(p, i) 0 i n 1 d S (i) w(i, S) 1 2 Upper Area i T d S (i) 1 S (1 ϕ(g)) 2 15/23

24 Details: Hard instances for Evolving Set Process 16/23

25 Hard instances: Noisy hypercube Noisy k-ary hypercube (k = 1/δ) k d vertices, each represented by a string of length d over [k] Transition probability from x { to y: x i When x = x 1 x 2... x d, y i = uniformly from [k] Coordinate cut S = {x x 1 = 0} satisfies φ(s) ε and S = δn prob ε prob 1 ε 17/23

26 Why local algorithms fail Lets start from S 0 = { 0} Evolving Set Process treats all vertices with the same Hamming weight equally The process only explores sets S t that are symmetric under coordinate permutations (in fact, Hamming balls) Small Hamming balls on noisy k-ary hypercube are expanding [Chan Mossel Neeman 14] 18/23

27 Hamming balls B expand B = {x [k] d x r} φ(b) = Pr x y[x, y B] Pr x [x B] 19/23

28 Hamming balls B expand B = {x [k] d x r} φ(b) = Pr x y[x, y B] Pr x [x B] Pr[x B] = E x [1 x xd 0 r] x Pr[g r ] Gaussian g Pr [x, y B] = E x,y 1 xi 0 r and 1 yi 0 r x y 1 i d Pr[g r and h r ] 1 i d Gaussians g, h 19/23

29 Hamming balls B expand B = {x [k] d x r} φ(b) = Pr x y[x, y B] Pr x [x B] 1 i d Pr g,h[g, h r ] Pr g [g r ] Pr[x B] = E x [1 x xd 0 r] x Pr[g r ] Gaussian g Pr [x, y B] = E x,y 1 xi 0 r and 1 yi 0 r x y Pr[g r and h r ] 1 i d Gaussians g, h 19/23

30 Expansion in Gaussian space B = {x [k] d x r} φ(b) = Pr x y[x, y B] Pr x [x B] Pr g,h[g, h r ] Pr g [g r ] r 0 1 r 0 20/23

31 Symmetry Obstacles to all local clustering algorithms Evolving Set Process, random walk, PageRank, etc All fall for the symmetry Folded noisy hypercubes are hard instances for SDP Folding necessary to rule out sparse cuts in those instances In our situation, we cannot fold Our instances are easy for Lasserre/sum-of-squares 21/23

32 Summary New analysis of Lovász Simonovits curve Faster convergence with large combinatorial gap ϕ(g) or robust vertex expansion φ V (G) x x(1 ϕ(g)) x(1 + ϕ(g)) Limitations of all local clustering algorithms: Coordinate cuts vs Hamming balls under symmetry 22/23

33 Open problems Ω( φ(s ) log S ) lower bound for Evolving Set Process and random walk? Thank you 23/23