pareto optimal solutions for smoothed analystspeople.csail.mit.edu/moitra/docs/paretod.pdf ·...
TRANSCRIPT
![Page 1: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/1.jpg)
Pareto Optimal Solutionsfor Smoothed Analysts
Ankur Moitra, IAS
joint work with Ryan O’Donnell, CMU
September 26, 2011
Ankur Moitra (IAS) Pareto September 26, 2011
![Page 2: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/2.jpg)
1
vi+1
wi+1
vi
wi
vn
wn
values
weights
...
...
...
...
v W1
w
![Page 3: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/3.jpg)
1
vi+1
wi+1
vi
wi
vn
wn
values
weights
...
...
...
...
v W1
w
![Page 4: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/4.jpg)
1
vi+1
wi+1
vi
wi
vn
wn
values
weights
...
...
...
...val
ue
W − weight
v W1
w
![Page 5: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/5.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w
W − weight
vvalues
weights
...
...
...
...val
ue
![Page 6: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/6.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w
W − weight
vvalues
weights
...
...
...
...val
ue
![Page 7: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/7.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w
W − weight
vvalues
weights
...
...
...
...
Not Pareto Optimal
val
ue
![Page 8: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/8.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w
W − weight
vvalues
weights
...
...
...
...
Pareto Optimal!
val
ue
![Page 9: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/9.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w
W − weight
vvalues
weights
...
...
...
...val
ue
![Page 10: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/10.jpg)
1
vi+1
wi+1
vi
wi
vn
wn
PO(i)
1
w Wvvalues
weights
...
...
...
...val
ue
W − weight
![Page 11: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/11.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w
W − weight
vvalues
weights
...
...
...
...val
ue
![Page 12: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/12.jpg)
1
vi+1
wi+1
vi
wi
vn
wn
v
vi+1
wi+1
1
w ...
...
...val
ue
W − weight
Wvalues
weights
...
![Page 13: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/13.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w ...
...
...val
ue
W − weight
vvalues
weights
...
![Page 14: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/14.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w ...
...
...val
ue
W − weight
vvalues
weights
...
![Page 15: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/15.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w ...
...
...val
ue
W − weight
vvalues
weights
...
![Page 16: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/16.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w ...
...
...val
ue
W − weight
vvalues
weights
...
![Page 17: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/17.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w ...
...
...val
ue
W − weight
vvalues
weights
...
![Page 18: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/18.jpg)
1
vi+1
wi+1
vi
wi
vn
wnW1
w ...
...
...val
ue
W − weight
vvalues
weights
...
![Page 19: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/19.jpg)
1
vi+1
wi+1
vi
wi
vn
wn
PO(i+1)
1
w
...
...val
ue
W − weight
Wvvalues
weights
...
...
![Page 20: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/20.jpg)
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 ofPareto optimal solutions
![Page 21: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/21.jpg)
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 ofPareto optimal solutions
![Page 22: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/22.jpg)
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 ofPareto optimal solutions
![Page 23: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/23.jpg)
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 ofPareto optimal solutions
![Page 24: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/24.jpg)
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 ofPareto optimal solutions
![Page 25: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/25.jpg)
Results of Beier and Vocking (STOC 2003, STOC 2004)
Smoothed Analysis of Pareto Curves:
Polynomial bound on |PO(i)|s (in two dimensions) inthe 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
![Page 26: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/26.jpg)
Results of Beier and Vocking (STOC 2003, STOC 2004)
Smoothed Analysis of Pareto Curves:
Polynomial bound on |PO(i)|s (in two dimensions) inthe 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
![Page 27: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/27.jpg)
Results of Beier and Vocking (STOC 2003, STOC 2004)
Smoothed Analysis of Pareto Curves:
Polynomial bound on |PO(i)|s (in two dimensions) inthe 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
![Page 28: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/28.jpg)
Results of Beier and Vocking (STOC 2003, STOC 2004)
Smoothed Analysis of Pareto Curves:
Polynomial bound on |PO(i)|s (in two dimensions) inthe 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
![Page 29: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/29.jpg)
Results of Beier and Vocking (STOC 2003, STOC 2004)
Smoothed Analysis of Pareto Curves:
Polynomial bound on |PO(i)|s (in two dimensions) inthe 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
![Page 30: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/30.jpg)
Capturing Tradeoffs
Question
What if the precise objective function (of a decisionmaker) 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 competingobjectives
![Page 31: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/31.jpg)
Capturing Tradeoffs
Question
What if the precise objective function (of a decisionmaker) 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 competingobjectives
![Page 32: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/32.jpg)
Capturing Tradeoffs
Question
What if the precise objective function (of a decisionmaker) 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 competingobjectives
![Page 33: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/33.jpg)
Capturing Tradeoffs
Question
What if the precise objective function (of a decisionmaker) 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 competingobjectives
![Page 34: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/34.jpg)
Capturing Tradeoffs
Question
What if the precise objective function (of a decisionmaker) 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 competingobjectives
![Page 35: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/35.jpg)
Proposition
Only useful approach if Pareto curves are small
Confirmed empirically e.g. Muller-Hannemann, Weihe: German
train system
Question
Why should we expect Pareto curves to be small?
Caveat: Smoothed Analysis is not a complete explanation
![Page 36: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/36.jpg)
Proposition
Only useful approach if Pareto curves are small
Confirmed empirically e.g. Muller-Hannemann, Weihe: German
train system
Question
Why should we expect Pareto curves to be small?
Caveat: Smoothed Analysis is not a complete explanation
![Page 37: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/37.jpg)
Proposition
Only useful approach if Pareto curves are small
Confirmed empirically e.g. Muller-Hannemann, Weihe: German
train system
Question
Why should we expect Pareto curves to be small?
Caveat: Smoothed Analysis is not a complete explanation
![Page 38: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/38.jpg)
Proposition
Only useful approach if Pareto curves are small
Confirmed empirically e.g. Muller-Hannemann, Weihe: German
train system
Question
Why should we expect Pareto curves to be small?
Caveat: Smoothed Analysis is not a complete explanation
![Page 39: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/39.jpg)
The Model
Adversary chooses:
Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)
an adversarial objective function OBJ1 : Z → <+
d − 1 linear objective functions
OBJi(~x) = [w i1,w
i2, ...w
in] · ~x
... each w ij is a random variable on [−1,+1] – density
is bounded by φ
![Page 40: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/40.jpg)
The Model
Adversary chooses:
Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)
an adversarial objective function OBJ1 : Z → <+
d − 1 linear objective functions
OBJi(~x) = [w i1,w
i2, ...w
in] · ~x
... each w ij is a random variable on [−1,+1] – density
is bounded by φ
![Page 41: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/41.jpg)
The Model
Adversary chooses:
Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)
an adversarial objective function OBJ1 : Z → <+
d − 1 linear objective functions
OBJi(~x) = [w i1,w
i2, ...w
in] · ~x
... each w ij is a random variable on [−1,+1] – density
is bounded by φ
![Page 42: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/42.jpg)
The Model
Adversary chooses:
Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)
an adversarial objective function OBJ1 : Z → <+
d − 1 linear objective functions
OBJi(~x) = [w i1,w
i2, ...w
in] · ~x
... each w ij is a random variable on [−1,+1] – density
is bounded by φ
![Page 43: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/43.jpg)
The Model
Adversary chooses:
Z ⊂ {0, 1}n (e.g. spanning trees, Hamiltonian cycles)
an adversarial objective function OBJ1 : Z → <+
d − 1 linear objective functions
OBJi(~x) = [w i1,w
i2, ...w
in] · ~x
... each w ij is a random variable on [−1,+1] – density
is bounded by φ
![Page 44: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/44.jpg)
History
Let PO be the set of Pareto optimal solutions...
[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)
[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)
[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)
[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))
... where f (d) = 2d−1(d + 1)!
[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS
![Page 45: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/45.jpg)
History
Let PO be the set of Pareto optimal solutions...
[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)
[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)
[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)
[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))
... where f (d) = 2d−1(d + 1)!
[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS
![Page 46: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/46.jpg)
History
Let PO be the set of Pareto optimal solutions...
[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)
[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)
[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)
[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))
... where f (d) = 2d−1(d + 1)!
[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS
![Page 47: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/47.jpg)
History
Let PO be the set of Pareto optimal solutions...
[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)
[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)
[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)
[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))
... where f (d) = 2d−1(d + 1)!
[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS
![Page 48: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/48.jpg)
History
Let PO be the set of Pareto optimal solutions...
[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)
[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)
[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)
[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))
... where f (d) = 2d−1(d + 1)!
[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS
![Page 49: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/49.jpg)
History
Let PO be the set of Pareto optimal solutions...
[Beier, Vocking STOC 2003] (d = 2), E [|PO|] = O(n5φ)
[Beier, Vocking STOC 2004] (d = 2), E [|PO|] = O(n4φ)
[Beier, Roglin, Vocking IPCO 2007] (d = 2), E [|PO|] = O(n2φ)(tight)
[Roglin, Teng, FOCS 2009] E [|PO|] = O((n√φ)f (d))
... where f (d) = 2d−1(d + 1)!
[Dughmi, Roughgarden, FOCS 2010] any FPTAS can betransformed to a truthful in expectation FPTAS
![Page 50: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/50.jpg)
Our Results
Theorem
E [|PO|] ≤ 2 · (4φd)d(d−1)/2n2d−2
...answers a conjecture of Teng
[Bently et al, JACM 1978]: 2n points sampled from a d-dimensionalGaussian,
E [|PO|] = Θ(nd−1)
square factor difference necessary for d = 2
![Page 51: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/51.jpg)
Our Results
Theorem
E [|PO|] ≤ 2 · (4φd)d(d−1)/2n2d−2
...answers a conjecture of Teng
[Bently et al, JACM 1978]: 2n points sampled from a d-dimensionalGaussian,
E [|PO|] = Θ(nd−1)
square factor difference necessary for d = 2
![Page 52: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/52.jpg)
Our Results
Theorem
E [|PO|] ≤ 2 · (4φd)d(d−1)/2n2d−2
...answers a conjecture of Teng
[Bently et al, JACM 1978]: 2n points sampled from a d-dimensionalGaussian,
E [|PO|] = Θ(nd−1)
square factor difference necessary for d = 2
![Page 53: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/53.jpg)
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!
![Page 54: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/54.jpg)
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!
![Page 55: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/55.jpg)
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!
![Page 56: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/56.jpg)
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!
![Page 57: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/57.jpg)
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!
![Page 58: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/58.jpg)
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!
![Page 59: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/59.jpg)
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
![Page 60: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/60.jpg)
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
![Page 61: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/61.jpg)
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
![Page 62: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/62.jpg)
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
![Page 63: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/63.jpg)
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
![Page 64: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/64.jpg)
An Alternative Condition
time OBJ
OBJ2
1
![Page 65: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/65.jpg)
An Alternative Condition
time OBJ
OBJ2
1
![Page 66: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/66.jpg)
An Alternative Condition
time OBJ
OBJ2
1
![Page 67: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/67.jpg)
An Alternative Condition
time OBJ
OBJ2
1
![Page 68: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/68.jpg)
An Alternative Condition
time1OBJ
OBJ2
![Page 69: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/69.jpg)
An Alternative Condition
time1OBJ
OBJ2
![Page 70: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/70.jpg)
An Alternative Condition
time1OBJ
OBJ2
![Page 71: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/71.jpg)
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) eventoccurs
Then bound the expected number of events
The key is in the definition
![Page 72: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/72.jpg)
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) eventoccurs
Then bound the expected number of events
The key is in the definition
![Page 73: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/73.jpg)
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) eventoccurs
Then bound the expected number of events
The key is in the definition
![Page 74: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/74.jpg)
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) eventoccurs
Then bound the expected number of events
The key is in the definition
![Page 75: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/75.jpg)
The Common Case (Beier, Roglin, Vocking)
time
x
OBJ
OBJ2
1
![Page 76: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/76.jpg)
The Common Case (Beier, Roglin, Vocking)
time
x
y
1OBJ
OBJ2
![Page 77: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/77.jpg)
The Common Case (Beier, Roglin, Vocking)
x
yEMPTY
time OBJ
OBJ2
1
![Page 78: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/78.jpg)
The Common Case (Beier, Roglin, Vocking)
x
yEMPTY
i
OBJ
OBJ2
1time
![Page 79: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/79.jpg)
The Common Case (Beier, Roglin, Vocking)
x
y = 0i
yEMPTY
i
OBJ
OBJ2
1time
i
x = 1
![Page 80: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/80.jpg)
The Common Case (Beier, Roglin, Vocking)
x
y = 0i
yEMPTY
i
OBJ
OBJ2
1time
EMPTY
i
x = 1
![Page 81: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/81.jpg)
Complete Family of Events
Let I = (a, b) be an interval of [−n,+n] of width ε
Let E be the event that there is an ~x ∈ PO:
OBJ2(~x) ∈ I
~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b
~yi = 0, ~xi = 1
Claim
Pr [E ] ≤ εφ
![Page 82: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/82.jpg)
Complete Family of Events
Let I = (a, b) be an interval of [−n,+n] of width ε
Let E be the event that there is an ~x ∈ PO:
OBJ2(~x) ∈ I
~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b
~yi = 0, ~xi = 1
Claim
Pr [E ] ≤ εφ
![Page 83: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/83.jpg)
Complete Family of Events
Let I = (a, b) be an interval of [−n,+n] of width ε
Let E be the event that there is an ~x ∈ PO:
OBJ2(~x) ∈ I
~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b
~yi = 0, ~xi = 1
Claim
Pr [E ] ≤ εφ
![Page 84: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/84.jpg)
Complete Family of Events
Let I = (a, b) be an interval of [−n,+n] of width ε
Let E be the event that there is an ~x ∈ PO:
OBJ2(~x) ∈ I
~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b
~yi = 0, ~xi = 1
Claim
Pr [E ] ≤ εφ
![Page 85: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/85.jpg)
Complete Family of Events
Let I = (a, b) be an interval of [−n,+n] of width ε
Let E be the event that there is an ~x ∈ PO:
OBJ2(~x) ∈ I
~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b
~yi = 0, ~xi = 1
Claim
Pr [E ] ≤ εφ
![Page 86: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/86.jpg)
Complete Family of Events
Let I = (a, b) be an interval of [−n,+n] of width ε
Let E be the event that there is an ~x ∈ PO:
OBJ2(~x) ∈ I
~y have the smallest OBJ1(~y) among all points for whichOBJ2(~y) > b
~yi = 0, ~xi = 1
Claim
Pr [E ] ≤ εφ
![Page 87: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/87.jpg)
Analysis
[ , , ... .... ] EOBJ2 x
OBJ
OBJ2
1
: : arbitrary
time
??
Z
?? ??
OBJ
??1
![Page 88: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/88.jpg)
Analysis
[ , , ... .... ] EOBJ2 x
OBJ
OBJ2
1
: : arbitrary
time
??
I
Z
?? ??
OBJ
??1
![Page 89: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/89.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
OBJ
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
wn
Z
1
OBJ
![Page 90: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/90.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
z
1
OBJ
index i
??
I
wn
Z
x
OBJ
OBJ
OBJ2
1
: : arbitrary
time
![Page 91: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/91.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
x = 0ix = 1i
z
1
OBJ n
i −iZ
Z
Z
x
OBJ
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
w
![Page 92: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/92.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
x = 0ix = 1iy
z
1
OBJ n
i −iZ
Z
Z
x
OBJ
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
w
![Page 93: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/93.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
x = 0ix = 1i
OBJ (x)>OBJ (y)1 1
y
w1
OBJ
OBJ
n
i −iZ
Z
Z
Tx
z
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
![Page 94: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/94.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
x = 0ix = 1i
OBJ (x)>OBJ (y)1 1
y
w1
OBJ
OBJ
n
i −iZ
Z
Z
T
x
z
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
![Page 95: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/95.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
x = 0ix = 1i
OBJ (x)>OBJ (y)1 1
y same on index i
1
OBJ n
i −iZ
Z
Z
T
OBJ
x
z
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
w
![Page 96: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/96.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
x = 0ix = 1i
OBJ (x)>OBJ (y)1 1
y
w1
OBJ
OBJ
n
i −iZ
Z
Z
Tx
z
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
![Page 97: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/97.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
i
1
OBJ
OBJ
−iZ
Z
Z
T
x
z
x = 0ix = 1i
OBJ (x)>OBJ (y)1 1
y
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
wn
![Page 98: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/98.jpg)
Analysis
[ , , ... .... ] E2 xw1 w2
i
1
OBJ
Z
Z
T
x
z
and y = 0
OBJ
x = 0ix = 1i
OBJ (x)>OBJ (y)1 1
x = 1i
y
OBJ
OBJ2
1
: : arbitrary
time
index i
??
I
wn
i −iZ
![Page 99: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/99.jpg)
A Re-interpretation
Implicitly defined a Transcription Algorithm:
Question
Given a Pareto optimal solution x, which event should weblame?
Blame event E : interval I , index i , xi and yi
Transcription Algorithm
Input: values of r.v.s (and Pareto optimal x)
Output: interval I , index i , xi and yi
![Page 100: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/100.jpg)
A Re-interpretation
Implicitly defined a Transcription Algorithm:
Question
Given a Pareto optimal solution x, which event should weblame?
Blame event E : interval I , index i , xi and yi
Transcription Algorithm
Input: values of r.v.s (and Pareto optimal x)
Output: interval I , index i , xi and yi
![Page 101: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/101.jpg)
A Re-interpretation
Implicitly defined a Transcription Algorithm:
Question
Given a Pareto optimal solution x, which event should weblame?
Blame event E : interval I , index i , xi and yi
Transcription Algorithm
Input: values of r.v.s (and Pareto optimal x)
Output: interval I , index i , xi and yi
![Page 102: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/102.jpg)
A Re-interpretation
Implicitly defined a Transcription Algorithm:
Question
Given a Pareto optimal solution x, which event should weblame?
Blame event E : interval I , index i , xi and yi
Transcription Algorithm
Input: values of r.v.s (and Pareto optimal x)
Output: interval I , index i , xi and yi
![Page 103: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/103.jpg)
A Re-interpretation
Implicitly defined a Transcription Algorithm:
Question
Given a Pareto optimal solution x, which event should weblame?
Blame event E : interval I , index i , xi and yi
Transcription Algorithm
Input: values of r.v.s (and Pareto optimal x)
Output: interval I , index i , xi and yi
![Page 104: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/104.jpg)
A Re-interpretation
Suppose output is event E : interval I , index i , xi and yi
Question
Which solution x caused this event to occur?
We do not need to know wi to determine the identity of x!
Reverse Algorithm
Find y
Then find x
![Page 105: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/105.jpg)
A Re-interpretation
Suppose output is event E : interval I , index i , xi and yi
Question
Which solution x caused this event to occur?
We do not need to know wi to determine the identity of x!
Reverse Algorithm
Find y
Then find x
![Page 106: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/106.jpg)
A Re-interpretation
Suppose output is event E : interval I , index i , xi and yi
Question
Which solution x caused this event to occur?
We do not need to know wi to determine the identity of x!
Reverse Algorithm
Find y
Then find x
![Page 107: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/107.jpg)
A Re-interpretation
Suppose output is event E : interval I , index i , xi and yi
Question
Which solution x caused this event to occur?
We do not need to know wi to determine the identity of x!
Reverse Algorithm
Find y
Then find x
![Page 108: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/108.jpg)
A Re-interpretation
Suppose output is event E : interval I , index i , xi and yi
Question
Which solution x caused this event to occur?
We do not need to know wi to determine the identity of x!
Reverse Algorithm
Find y
Then find x
![Page 109: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/109.jpg)
A Re-interpretation
Reverse Algorithm: Find x (without looking at wi)
Question
Does x fall into the interval I ?
Hidden random variable wi must fall into some small range
Proposition
We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs
Hence each output is unlikely
![Page 110: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/110.jpg)
A Re-interpretation
Reverse Algorithm: Find x (without looking at wi)
Question
Does x fall into the interval I ?
Hidden random variable wi must fall into some small range
Proposition
We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs
Hence each output is unlikely
![Page 111: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/111.jpg)
A Re-interpretation
Reverse Algorithm: Find x (without looking at wi)
Question
Does x fall into the interval I ?
Hidden random variable wi must fall into some small range
Proposition
We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs
Hence each output is unlikely
![Page 112: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/112.jpg)
A Re-interpretation
Reverse Algorithm: Find x (without looking at wi)
Question
Does x fall into the interval I ?
Hidden random variable wi must fall into some small range
Proposition
We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs
Hence each output is unlikely
![Page 113: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/113.jpg)
A Re-interpretation
Reverse Algorithm: Find x (without looking at wi)
Question
Does x fall into the interval I ?
Hidden random variable wi must fall into some small range
Proposition
We can deduce a missing input to TranscriptionAlgorithm, from just some of the inputs and outputs
Hence each output is unlikely
![Page 114: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/114.jpg)
Transcription for d = 3
2
OBJ2
( , )
OBJ
OBJ3
: adversarial
: linear,
OBJ1
OBJ3
,Output: ,
Z
![Page 115: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/115.jpg)
Transcription for d = 3
2
OBJ2
( , )
Z
OBJ
OBJ3
: adversarial
: linear,
OBJ1
OBJ3
,Output: ,
![Page 116: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/116.jpg)
Transcription for d = 3
2
OBJ2
( , ),Z
OBJ
OBJ3
: adversarial
: linear,
OBJ1
OBJ3
,Output:
![Page 117: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/117.jpg)
Transcription for d = 3
2
OBJ2
( , )Output: ,Z
OBJ
OBJ3
: adversarial
: linear,
OBJ1
OBJ3
,
![Page 118: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/118.jpg)
Transcription for d = 3
x
OBJ2
OBJ2
x
( , )
,Output: ,
Z
OBJ3
: adversarial
: linear,
OBJ1
OBJ3
![Page 119: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/119.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
( , )
3
,Output: ,
Z
OBJ3
: adversarial
: linear,
OBJ1
OBJ
![Page 120: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/120.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
( , )
3
,Output: ,
Z
OBJ3
: adversarial
: linear,
OBJ1
OBJ
![Page 121: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/121.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
Z3
,Output: ,(i, )
OBJ3
index i
: adversarial
: linear,
OBJ1
OBJ
![Page 122: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/122.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
Z
10
Output: ,(i, )OBJ
3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i
![Page 123: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/123.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
Z
10
Output: ,(i, )OBJ
3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i
![Page 124: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/124.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
Z
10
Output: ,(i, )OBJ
3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i
![Page 125: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/125.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
x = 0i
x = 1i
−i
(i, )
ZZ
Z i
OBJ3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i 10
Output: ,
![Page 126: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/126.jpg)
Transcription for d = 3
x
OBJ2
OBJ2
y x
x = 0i
x = 1i
−i
y
time
ZZ
Z i
OBJ3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i 10
Output: ,(i, )
![Page 127: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/127.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
y x
x = 0i
x = 1i
−i
z
(i, )
ZZ
Z i
OBJ
time
3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i 10
Output: ,
![Page 128: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/128.jpg)
Transcription for d = 3
y
x
OBJ2
OBJ2
zy x
x = 0i
x = 1i
−i
z
(i, )
ZZ
Z i
OBJ
time
3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i 1 10
Output: ,
![Page 129: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/129.jpg)
Transcription for d = 3
z y
x
OBJ2
OBJ2
zy x
x = 0i
x = 1i
−i
(i, )
ZZ
Z i
OBJ3
index i
: adversarial
: linear,
OBJ1
OBJ3
,index i 1 10
Output: ,
![Page 130: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/130.jpg)
Transcription for d = 3
z y
x
OBJ2
OBJ2
zy x
x = 0i
x = 1i
−i
(i, )
ZZ
Z i
OBJ3
index j index i
: adversarial
: linear,
OBJ1
OBJ3
,index i 1 10
Output: ,
![Page 131: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/131.jpg)
Transcription for d = 3
z y
x
OBJ2
OBJ2
zy x
x = 0i
x = 1i
−i
,Z
Z
Z i
OBJ3
index j index i
: adversarial
: linear,
OBJ1
OBJ3
,index i
index j
1 1
1
0
0 1
Output: (i,j)
![Page 132: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/132.jpg)
Transcription for d = 3
z y
x
OBJ2
OBJ2
zy x
x = 0i
x = 1i
−i
,Z
Z
Z i
OBJ3
index j index i
: adversarial
: linear,
OBJ1
OBJ3
,index i
index j
1 1
1
0
0 1
Output: (i,j)
![Page 133: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/133.jpg)
Transcription for d = 3
z y
x
OBJ2
OBJ2
zy x
x = 0i
x = 1i
−i
,Z
Z
Z i
OBJ3
index j index i
: adversarial
: linear,
OBJ1
OBJ3
,index i
index j
1 1
1
0
0 1
Output: (i,j)
![Page 134: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/134.jpg)
Transcription for d = 3
z y
x
zy x
index i
index j
1 1
1
0
0 1
Output: (i,j),
,,
OBJ2
OBJ2
x = 0i
x = 1i
−i3 Z
Z
Z i
OBJ3
index j index i
: adversarial
: linear,
OBJ1
OBJ
![Page 135: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/135.jpg)
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 fornon-linear objectives?
e.g. submodular functions
![Page 136: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/136.jpg)
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 fornon-linear objectives?
e.g. submodular functions
![Page 137: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/137.jpg)
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 fornon-linear objectives?
e.g. submodular functions
![Page 138: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/138.jpg)
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 fornon-linear objectives?
e.g. submodular functions
![Page 139: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/139.jpg)
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 fornon-linear objectives?
e.g. submodular functions
![Page 140: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/140.jpg)
Questions?
![Page 141: Pareto Optimal Solutions for Smoothed Analystspeople.csail.mit.edu/moitra/docs/paretod.pdf · Pareto Optimal Solutions for Smoothed Analysts Ankur Moitra, IAS joint work with Ryan](https://reader034.vdocuments.mx/reader034/viewer/2022051908/5ffc380f04201447ce40bc02/html5/thumbnails/141.jpg)
Thanks!