Pareto Optimal Solutions for Smoothed Analysts
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1 Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan O Donnell, CMU September 26, 2011 Ankur Moitra (IAS) Pareto September 26, 2011
2 values weights v1 w v i w i v i+1 w i v n w n W
3 values weights v1 w v i w i v i+1 w i v n w n W
4 values weights v1 w v i w i v i+1 w i v n w n W value W weight
5 values weights v w v i w i v i+1 w i v n w n W value W weight
6 values weights v w v i w i v i+1 w i v n w n W value W weight
7 values weights v w v i w i v i+1 w i v n w n W Not Pareto Optimal value W weight
8 values weights v w v i w i v i+1 w i v n w n W Pareto Optimal! value W weight
9 values weights v w v i w i v i+1 w i v n w n W value W weight
10 values weights v w v i w i v i+1 w i v n w n W value PO(i) W weight
11 values weights v w v i w i v i+1 w i v n w n W value W weight
12 values weights v1... w... 1 v i w i v i+1 w i v n w n W w i+1 v i+1 value W weight
13 values weights v1... w... 1 v i w i v i+1 w i v n w n W value W weight
14 values weights v1... w... 1 v i w i v i+1 w i v n w n W value W weight
15 values weights v1... w... 1 v i w i v i+1 w i v n w n W value W weight
16 values weights v1... w... 1 v i w i v i+1 w i v n w n W value W weight
17 values weights v1... w... 1 v i w i v i+1 w i v n w n W value W weight
18 values weights v1... w... 1 v i w i v i+1 w i v n w n W value W weight
19 values weights v w v i w i v i+1 w i v n w n W value PO(i+1) W weight
20 Dynamic Programming with Lists (Nemhauser, Ullmann):
21 Dynamic Programming with Lists (Nemhauser, Ullmann): PO(i + 1) can be computed from PO(i)...
22 Dynamic Programming with Lists (Nemhauser, Ullmann): PO(i + 1) can be computed from PO(i) in linear (in PO(i) ) time
23 Dynamic Programming with Lists (Nemhauser, Ullmann): PO(i + 1) can be computed from PO(i) in linear (in PO(i) ) time an optimal solution can be computed from PO(n)
24 Dynamic Programming with Lists (Nemhauser, Ullmann): PO(i + 1) can be computed from PO(i) in linear (in PO(i) ) time an optimal solution can be computed from PO(n) Bottleneck in many algorithms: enumerate the set of Pareto optimal solutions
25 Results of Beier and Vöcking (STOC 2003, STOC 2004) Smoothed Analysis of Pareto Curves:
26 Results of Beier and Vöcking (STOC 2003, STOC 2004) Smoothed Analysis of Pareto Curves: Polynomial bound on PO(i) s (in two dimensions) in the framework of Smoothed Analysis (Spielman, Teng)
27 Results of Beier and Vöcking (STOC 2003, STOC 2004) Smoothed Analysis of Pareto Curves: Polynomial bound on PO(i) s (in two dimensions) in the framework of Smoothed Analysis (Spielman, Teng) Knapsack has polynomial smoothed complexity:
28 Results of Beier and Vöcking (STOC 2003, STOC 2004) Smoothed Analysis of Pareto Curves: Polynomial bound on PO(i) s (in two dimensions) in the framework of Smoothed Analysis (Spielman, Teng) Knapsack has polynomial smoothed complexity: first NP-hard problem that is (smoothed) easy
29 Results of Beier and Vöcking (STOC 2003, STOC 2004) Smoothed Analysis of Pareto Curves: Polynomial bound on PO(i) s (in two dimensions) in the framework of Smoothed Analysis (Spielman, Teng) Knapsack has polynomial smoothed complexity: first NP-hard problem that is (smoothed) easy generalizes long line of results on random instances
30 Capturing Tradeoffs
31 Capturing Tradeoffs Question What if the precise objective function (of a decision maker) is unknown?
32 Capturing Tradeoffs Question What if the precise objective function (of a decision maker) is unknown? e.g. travel planning: prefer lower fare, shorter trip, fewer transfers
33 Capturing Tradeoffs Question What if the precise objective function (of a decision maker) is unknown? e.g. travel planning: prefer lower fare, shorter trip, fewer transfers Question Can we algorithmically help a decision maker?
34 Capturing Tradeoffs Question What if the precise objective function (of a decision maker) is unknown? e.g. travel planning: prefer lower fare, shorter trip, fewer transfers Question Can we algorithmically help a decision maker? Pareto curves capture tradeoffs among competing objectives
35 Proposition Only useful approach if Pareto curves are small
36 Proposition Only useful approach if Pareto curves are small Confirmed empirically e.g. Müller-Hannemann, Weihe: German train system
37 Proposition Only useful approach if Pareto curves are small Confirmed empirically e.g. Müller-Hannemann, Weihe: German train system Question Why should we expect Pareto curves to be small?
38 Proposition Only useful approach if Pareto curves are small Confirmed empirically e.g. Müller-Hannemann, Weihe: German train system Question Why should we expect Pareto curves to be small? Caveat: Smoothed Analysis is not a complete explanation
39 The Model Adversary chooses:
40 The Model Adversary chooses: Z {0, 1} n (e.g. spanning trees, Hamiltonian cycles)
41 The Model Adversary chooses: Z {0, 1} n (e.g. spanning trees, Hamiltonian cycles) an adversarial objective function OBJ 1 : Z R +
42 The Model Adversary chooses: Z {0, 1} n (e.g. spanning trees, Hamiltonian cycles) an adversarial objective function OBJ 1 : Z R + d 1 linear objective functions OBJ i ( x) = [w1, i w2, i...wn] i x
43 The Model Adversary chooses: Z {0, 1} n (e.g. spanning trees, Hamiltonian cycles) an adversarial objective function OBJ 1 : Z R + d 1 linear objective functions OBJ i ( x) = [w i 1, w i 2,...w i n] x... each w i j is a random variable on [ 1, +1] density is bounded by φ
44 History Let PO be the set of Pareto optimal solutions...
45 History Let PO be the set of Pareto optimal solutions... [Beier, Vöcking STOC 2003] (d = 2), E[ PO ] = O(n 5 φ)
46 History Let PO be the set of Pareto optimal solutions... [Beier, Vöcking STOC 2003] (d = 2), E[ PO ] = O(n 5 φ) [Beier, Vöcking STOC 2004] (d = 2), E[ PO ] = O(n 4 φ)
47 History Let PO be the set of Pareto optimal solutions... [Beier, Vöcking STOC 2003] (d = 2), E[ PO ] = O(n 5 φ) [Beier, Vöcking STOC 2004] (d = 2), E[ PO ] = O(n 4 φ) [Beier, Röglin, Vöcking IPCO 2007] (d = 2), E[ PO ] = O(n 2 φ) (tight)
48 History Let PO be the set of Pareto optimal solutions... [Beier, Vöcking STOC 2003] (d = 2), E[ PO ] = O(n 5 φ) [Beier, Vöcking STOC 2004] (d = 2), E[ PO ] = O(n 4 φ) [Beier, Röglin, Vöcking IPCO 2007] (d = 2), E[ PO ] = O(n 2 φ) (tight) [Röglin, Teng, FOCS 2009] E[ PO ] = O((n φ) f (d) )... where f (d) = 2 d 1 (d + 1)!
49 History Let PO be the set of Pareto optimal solutions... [Beier, Vöcking STOC 2003] (d = 2), E[ PO ] = O(n 5 φ) [Beier, Vöcking STOC 2004] (d = 2), E[ PO ] = O(n 4 φ) [Beier, Röglin, Vöcking IPCO 2007] (d = 2), E[ PO ] = O(n 2 φ) (tight) [Röglin, Teng, FOCS 2009] E[ PO ] = O((n φ) f (d) )... where f (d) = 2 d 1 (d + 1)! [Dughmi, Roughgarden, FOCS 2010] any FPTAS can be transformed to a truthful in expectation FPTAS
50 Our Results Theorem E[ PO ] 2 (4φd) d(d 1)/2 n 2d 2
51 Our Results Theorem E[ PO ] 2 (4φd) d(d 1)/2 n 2d 2...answers a conjecture of Teng
52 Our Results Theorem E[ PO ] 2 (4φd) d(d 1)/2 n 2d 2...answers a conjecture of Teng [Bently et al, JACM 1978]: 2 n points sampled from a d-dimensional Gaussian, E[ PO ] = Θ(n d 1 ) square factor difference necessary for d = 2
53 On Smoothed Analysis
54 On Smoothed Analysis Bounds are notoriously pessimistic (e.g. Simplex)
55 On Smoothed Analysis Bounds are notoriously pessimistic (e.g. Simplex) Method of Analysis:
56 On Smoothed Analysis Bounds are notoriously pessimistic (e.g. Simplex) Method of Analysis: Define a bad event...that you can blame if your algorithm runs slowly
57 On Smoothed Analysis Bounds are notoriously pessimistic (e.g. Simplex) Method of Analysis: Define a bad event...that you can blame if your algorithm runs slowly Prove this event is rare
58 On Smoothed Analysis Bounds are notoriously pessimistic (e.g. Simplex) Method of Analysis: Define a bad event...that you can blame if your algorithm runs slowly Prove this event is rare Proposition Randomness is your friend!
59 Our Approach
60 Our Approach Goal Define bad events using as little randomness as possible
61 Our Approach Goal Define bad events using as little randomness as possible... conserve randomness, save for analysis
62 Our Approach Goal Define bad events using as little randomness as possible... conserve randomness, save for analysis Challenge Events become convoluted (to say the least!)
63 Our Approach Goal Define bad events using as little randomness as possible... conserve randomness, save for analysis Challenge Events become convoluted (to say the least!) We give an algorithm to define these events
64 An Alternative Condition OBJ 2 time OBJ1
65 An Alternative Condition OBJ 2 time OBJ1
66 An Alternative Condition OBJ 2 time OBJ1
67 An Alternative Condition OBJ 2 time OBJ1
68 An Alternative Condition OBJ 2 time OBJ 1
69 An Alternative Condition OBJ 2 time OBJ 1
70 An Alternative Condition OBJ 2 time OBJ 1
71 Method of Proof Goal Count the (expected) number of Pareto optimal solutions
72 Method of Proof Goal Count the (expected) number of Pareto optimal solutions Define a complete family of events... for each Pareto optimal solution at least one (unique) event occurs
73 Method of Proof Goal Count the (expected) number of Pareto optimal solutions Define a complete family of events... for each Pareto optimal solution at least one (unique) event occurs Then bound the expected number of events
74 Method of Proof Goal Count the (expected) number of Pareto optimal solutions Define a complete family of events... for each Pareto optimal solution at least one (unique) event occurs Then bound the expected number of events The key is in the definition
75 The Common Case (Beier, Röglin, Vöcking) OBJ 2 x time OBJ1
76 The Common Case (Beier, Röglin, Vöcking) OBJ 2 y x time OBJ 1
77 The Common Case (Beier, Röglin, Vöcking) OBJ 2 y EMPTY x time OBJ1
78 The Common Case (Beier, Röglin, Vöcking) OBJ 2 y EMPTY i x time OBJ1
79 The Common Case (Beier, Röglin, Vöcking) OBJ 2 y y i = 0 EMPTY i x x i = 1 time OBJ1
80 The Common Case (Beier, Röglin, Vöcking) OBJ 2 y y i = 0 EMPTY i EMPTY x x i = 1 time OBJ1
81 Complete Family of Events Let I = (a, b) be an interval of [ n, +n] of width ɛ
82 Complete Family of Events Let I = (a, b) be an interval of [ n, +n] of width ɛ Let E be the event that there is an x PO:
83 Complete Family of Events Let I = (a, b) be an interval of [ n, +n] of width ɛ Let E be the event that there is an x PO: OBJ 2 ( x) I
84 Complete Family of Events Let I = (a, b) be an interval of [ n, +n] of width ɛ Let E be the event that there is an x PO: OBJ 2 ( x) I y have the smallest OBJ 1 ( y) among all points for which OBJ 2 ( y) > b
85 Complete Family of Events Let I = (a, b) be an interval of [ n, +n] of width ɛ Let E be the event that there is an x PO: OBJ 2 ( x) I y have the smallest OBJ 1 ( y) among all points for which OBJ 2 ( y) > b y i = 0, x i = 1
86 Complete Family of Events Let I = (a, b) be an interval of [ n, +n] of width ɛ Let E be the event that there is an x PO: OBJ 2 ( x) I y have the smallest OBJ 1 ( y) among all points for which OBJ 2 ( y) > b y i = 0, x i = 1 Claim Pr[E] ɛφ
87 Analysis OBJ1: arbitrary OBJ 2 : [??,??,...??...?? ] x E OBJ 2 Z time OBJ1
88 Analysis OBJ1: arbitrary OBJ 2 : [??,??,...??...?? ] x E OBJ 2 Z I time OBJ1
89 Analysis OBJ 2 OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i Z I time OBJ1
90 Analysis OBJ 2 OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i Z I x z time OBJ1
91 Analysis OBJ 2 I OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x = 1 i Z x = 0 i Z i Z i x z time OBJ1
92 Analysis OBJ 2 I y OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x = 1 i Z x = 0 i Z i Z i x z time OBJ1
93 Analysis OBJ 2 I y OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x = 1 i Z x = 0 i Z i Z i x z T OBJ (x)>obj (y) 1 1 time OBJ1
94 Analysis OBJ 2 I y OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x z x = 1 i Z x = 0 i Z i Z i T OBJ (x)>obj (y) 1 1 time OBJ1
95 Analysis OBJ 2 I y OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x same on index i z x = 1 i Z x = 0 i Z i Z i T OBJ (x)>obj (y) 1 1 time OBJ1
96 Analysis OBJ 2 I y OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x = 1 i Z x = 0 i Z i Z i x z T OBJ (x)>obj (y) 1 1 time OBJ1
97 Analysis OBJ 2 I y OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x z x = 1 i Z T i Z x = 0 i Z i OBJ (x)>obj (y) 1 1 time OBJ1
98 Analysis OBJ 2 I y OBJ1: arbitrary OBJ [ w,, ] E 2: 1 w 2?? w n x index i x = 1 i x and y = 0 i z x = 1 i Z Z i Z i T x = 0 i OBJ (x)>obj (y) 1 1 time OBJ1
99 A Re-interpretation
100 A Re-interpretation Implicitly defined a Transcription Algorithm:
101 A Re-interpretation Implicitly defined a Transcription Algorithm: Question Given a Pareto optimal solution x, which event should we blame?
102 A Re-interpretation Implicitly defined a Transcription Algorithm: Question Given a Pareto optimal solution x, which event should we blame? Blame event E: interval I, index i, x i and y i
103 A Re-interpretation Implicitly defined a Transcription Algorithm: Question Given a Pareto optimal solution x, which event should we blame? Blame event E: interval I, index i, x i and y i Transcription Algorithm Input: values of r.v.s (and Pareto optimal x) Output: interval I, index i, x i and y i
104 A Re-interpretation Suppose output is event E: interval I, index i, x i and y i
105 A Re-interpretation Suppose output is event E: interval I, index i, x i and y i Question Which solution x caused this event to occur?
106 A Re-interpretation Suppose output is event E: interval I, index i, x i and y i Question Which solution x caused this event to occur? We do not need to know w i to determine the identity of x!
107 A Re-interpretation Suppose output is event E: interval I, index i, x i and y i Question Which solution x caused this event to occur? We do not need to know w i to determine the identity of x! Reverse Algorithm Find y
108 A Re-interpretation Suppose output is event E: interval I, index i, x i and y i Question Which solution x caused this event to occur? We do not need to know w i to determine the identity of x! Reverse Algorithm Find y Then find x
109 A Re-interpretation Reverse Algorithm: Find x (without looking at w i )
110 A Re-interpretation Reverse Algorithm: Find x (without looking at w i ) Question Does x fall into the interval I?
111 A Re-interpretation Reverse Algorithm: Find x (without looking at w i ) Question Does x fall into the interval I? Hidden random variable w i must fall into some small range
112 A Re-interpretation Reverse Algorithm: Find x (without looking at w i ) Question Does x fall into the interval I? Hidden random variable w i must fall into some small range Proposition We can deduce a missing input to Transcription Algorithm, from just some of the inputs and outputs
113 A Re-interpretation Reverse Algorithm: Find x (without looking at w i ) Question Does x fall into the interval I? Hidden random variable w i must fall into some small range Proposition We can deduce a missing input to Transcription Algorithm, from just some of the inputs and outputs Hence each output is unlikely
114 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear, Z Output: (, ), OBJ2
115 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear, Z Output: (, ), OBJ2
116 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear ), Z Output: (,, OBJ2
117 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear, Z Output: (, ), OBJ2
118 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear x, Z Output: (, ), x OBJ 2
119 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ3 : linear x y, Z Output: (, ) y, x OBJ 2
120 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ3 : linear x y, Z Output: (, ) y, x OBJ 2
121 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ3 : linear x y index i, Z Output: (i, ) y, x OBJ 2
122 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear x y index i, Z Output: (i, ) y index i 0, x 1 OBJ 2
123 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear x y index i, Z Output: (i, ) y index i 0, x 1 OBJ 2
124 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear x y index i, Z Output: (i, ) y index i 0, x 1 OBJ 2
125 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear x y index i x i = 0, Z i Z x i = 1 Z i Output: (i, ), y x index i 0 1 OBJ 2
126 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear x y index i x i = 0, Z i Z x i = 1 Z i Output: (i, ) y index i 0 1, x time OBJ 2
127 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear z x y index i x i = 0, Z i Z x i = 1 Z i Output: (i, ), y x index i 0 1 time OBJ 2
128 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear z x y index i x i = 0, Z i Z x i = 1 Z i Output: (i, ), y z x index i time OBJ 2
129 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear z x y index i x i = 0, Z i Z x i = 1 Z i Output: (i, ), y z x index i OBJ 2
130 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear z index j x y index i x i = 0, Z i Z x i = 1 Z i Output: (i, ), y z x index i OBJ 2
131 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear z index j x y index i Z x = 0 i x = 1 i Output: (i,j) index i index j y 0 1, Z i Z i z , x OBJ 2
132 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear z index j x y index i Z x = 0 i x = 1 i Output: (i,j) index i index j y 0 1, Z i Z i z , x OBJ 2
133 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial OBJ 2, OBJ 3 : linear z index j x y index i Z x = 0 i x = 1 i Output: (i,j) index i index j y 0 1, Z i Z i z , x OBJ 2
134 Transcription for d = 3 OBJ 3 OBJ 1 : adversarial x = 0 OBJ 2, OBJ3 : linear Z z y index j x index i index i index j i Z i x = 1 Z Output: (i,j), i i y z x,, OBJ 2
135 Future Directions?
136 Future Directions? Open Question Are there additional applications of randomness conservation in Smoothed Analysis?
137 Future Directions? Open Question Are there additional applications of randomness conservation in Smoothed Analysis? e.g. simplex algorithm
138 Future Directions? Open Question Are there additional applications of randomness conservation in Smoothed Analysis? e.g. simplex algorithm Open Question Are there perturbation models that make sense for non-linear objectives?
139 Future Directions? Open Question Are there additional applications of randomness conservation in Smoothed Analysis? e.g. simplex algorithm Open Question Are there perturbation models that make sense for non-linear objectives? e.g. submodular functions
140 Questions?
141 Thanks!
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