Learning What Works Best When - Simons Institute for the ......Simons Institute, UC Berkeley Georgia...
Transcript of Learning What Works Best When - Simons Institute for the ......Simons Institute, UC Berkeley Georgia...
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Learning What Works Best When
Swati GuptaSimons Institute, UC BerkeleyGeorgia Institute of Technology
Joint work with Michel Goemans (MIT), Patrick Jaillet (MIT), Iain Dunning (Deepmind), John Silberholz (Ross School of Business)
05 + 02 + 2=01+8Mathematical and Computational Challenges in
Real-Time Decision Making
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Three fundamental questions
2Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Three fundamental questions
(a) How to compute projections?
?
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Three fundamental questions
(a) How to compute projections?
(b) How far can we move along a direction while staying feasible?
2Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Three fundamental questions
(a) How to compute projections?
(b) How far can we move along a direction while staying feasible?
(c) Can we learn which algorithm works best on an unseen instance?
2Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Three fundamental questions
(b) How far can we move along a direction while staying feasible?
(c) Can we learn which algorithm works best on an unseen instance?
(a) How to compute projections?
2Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Why are projections important? Key step in many algorithms across
Online Learning Game Theory
Machine Learning Stochastic optimization
Robust optimization …
• Problem setup • Examples • Online Mirror Descent • Projection!
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Online Learning
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Online Learning
Select an action Or a decision
Historic Data Or Prior knowledge
Incur losses or gains in a dynamic environment
Repeats over time
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Online Learning
Select an action Or a decision
Historic Data Or Prior knowledge
Incur losses or gains in a dynamic environment
Repeats over time
How to perform well compared to best fixed decision in hindsight?
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Online Learning
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pages rank
Matchings
i j
Online Learning
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Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
pages rank
Matchings
i j
Online Learning
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Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
pages rank
Matchings
i j
Online Learning
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Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
s-t paths
pages rank
Matchings
i j
[Cohen, Gupta, Kalas, Perakis, ‘16]
Online Learning
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Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
s-t paths
pages rank
Matchings
i j
Spanning Trees
[Cohen, Gupta, Kalas, Perakis, ‘16]
Online Learning
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allow convex combinations, sample at random
Decision Space Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
Online Learning
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allow convex combinations, sample at random
Decision Space Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
Online Learning Framework
- learner chooses a decision,
- a linear loss revealed,
- the loss incurred for time t:
Online Learning
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Suppose xt = (2, 3 ,1, 4)
Online Learning Framework
- learner chooses a decision,
- a linear loss revealed,
- the loss incurred for time t:
Page 1 at rank 2 Page 2 at rank 3 Page 3 at rank 1 Page 4 at rank 4
Display: Page 3 Page 1 Page 2 Page 4
Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
Example
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Suppose xt = (2, 3 ,1, 4)
Online Learning Framework
- learner chooses a decision,
- a linear loss revealed,
- the loss incurred for time t:
Page 1 at rank 2 Page 2 at rank 3 Page 3 at rank 1 Page 4 at rank 4
Display: Page 3 Page 1 Page 2 Page 4
Observe user clicks:
20% 40% 30% 10%
Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
0.40 0.30 0.20 0.10
Example
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Suppose xt = (2, 3 ,1, 4)
Example Loss Function:
Online Learning Framework
- learner chooses a decision,
- a linear loss revealed,
- the loss incurred for time t:
Page 1 at rank 2 Page 2 at rank 3 Page 3 at rank 1 Page 4 at rank 4
Display: Page 3 Page 1 Page 2 Page 4
Observe user clicks:
20% 40% 30% 10%
Penalizes if a highly desired page is put later in the ranking
Loss for xt = 2*0.40 (page 1) + 3*0.30 (page 2) + 1*0.20 (page 3) + 4*0.10 (page 4).
Permutations 1, 2, 3, 4, 2, 3, 1, 4, 3, 1, 4, 2, 4, 1, 3, 2, 2, 3, 4, 1…
0.40 0.30 0.20 0.10
Example
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Bottleneck in First-Order Projection-Based Algorithms
[Zinkevich 2003], [Nemirovski, Yudin 1983]
Optimal regret in many cases [for e.g. Srebro, Sridharan, Tewari 2010]
But Computationally Slow!
Online Mirror Descent
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Bottleneck in First-Order Projection-Based Algorithms
constrained decision set
[Zinkevich 2003], [Nemirovski, Yudin 1983]
Optimal regret in many cases [for e.g. Srebro, Sridharan, Tewari 2010]
But Computationally Slow!
Online Mirror Descent
8Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Bottleneck in First-Order Projection-Based Algorithms
unconstrained gradient step
constrained decision set
[Zinkevich 2003], [Nemirovski, Yudin 1983]
Optimal regret in many cases [for e.g. Srebro, Sridharan, Tewari 2010]
But Computationally Slow!
Online Mirror Descent
8Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Bottleneck in First-Order Projection-Based Algorithms
project
unconstrained gradient step
constrained decision set
[Zinkevich 2003], [Nemirovski, Yudin 1983]
Optimal regret in many cases [for e.g. Srebro, Sridharan, Tewari 2010]
But Computationally Slow!
Online Mirror Descent
8Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Bottleneck in First-Order Projection-Based Algorithms
projectProjections are obtained by
minimizing a convex function (potentially in each time step)
unconstrained gradient step
constrained decision set
[Zinkevich 2003], [Nemirovski, Yudin 1983]
Optimal regret in many cases [for e.g. Srebro, Sridharan, Tewari 2010]
But Computationally Slow!
Online Mirror Descent
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1. Projections ▪ Motivation ▪ Problem setup ▪ Novel algorithm: Inc-Fix for separable convex minimization: ▪ Main Result: O(n) SFM or O(n) Line searches ▪ Exact computations, modulo solving a univariate equation
2. Line Searches ▪ Previous best known: Megiddo’s parametric search ▪ Using Newton’s Discrete Method: n2 + n log2n SFM (n6 improvement)
3. What works best when ▪ Problems with Max-Cut and QUBO heuristics comparative studies ▪ Our framework: Expanded instance library, Implementation of 37 heuristics,
Large-scale cloud computing on the cross product ▪ Hyper-heuristic: Map every instance to a feature space, learn
“performance” of heuristics
Outline 9
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10
(i) Which decision sets? Submodular Base Polytopes
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(i) Which decision sets?Permutations 1, 2, 3, 2, 3, 1, 3, 1, 2…
Submodular Base Polytopes
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(i) Which decision sets?Permutations 1, 2, 3, 2, 3, 1, 3, 1, 2…
B(f)
P(f)
(1,2,3)
(1,3,2)
(2,3,1)(3,2,1)
(3,1,2)
(2,1,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
Submodular Base Polytopes
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(i) Which decision sets?Permutations 1, 2, 3, 2, 3, 1, 3, 1, 2…
Submodular set function Captures the property of diminishing returns
B(f)
P(f)
(1,2,3)
(1,3,2)
(2,3,1)(3,2,1)
(3,1,2)
(2,1,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
Submodular Base Polytopes
Ground set E
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(i) Which decision sets?Permutations 1, 2, 3, 2, 3, 1, 3, 1, 2…
Submodular set function Captures the property of diminishing returns
B(f)
P(f)
(1,2,3)
(1,3,2)
(2,3,1)(3,2,1)
(3,1,2)
(2,1,3)
Exp!
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
Submodular Base Polytopes
Ground set E
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10
(i) Which decision sets?Permutations 1, 2, 3, 2, 3, 1, 3, 1, 2…
Submodular set function Captures the property of diminishing returns
B(f)
P(f)
(1,2,3)
(1,3,2)
(2,3,1)(3,2,1)
(3,1,2)
(2,1,3)
f(S) =
Choice of f(.) gives different structures
Exp!
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
Submodular Base Polytopes
Ground set E
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10
(i) Which decision sets?Permutations 1, 2, 3, 2, 3, 1, 3, 1, 2…
Submodular set function Captures the property of diminishing returns
B(f)
P(f)
(1,2,3)
(1,3,2)
(2,3,1)(3,2,1)
(3,1,2)
(2,1,3)
f(S) =
Choice of f(.) gives different structures
Exp!
MANY MANY MORE INTERESTING EXAMPLES!!
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
Submodular Base Polytopes
Ground set E
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(ii) Minimize what?▪ Bregman Divergences
Convex, non-negative, not symmetric
11
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(ii) Minimize what?▪ Bregman Divergences
Convex, non-negative, not symmetric
y
11
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(ii) Minimize what?▪ Bregman Divergences
Convex, non-negative, not symmetric
y
D(x, y)
x
11
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(ii) Minimize what?▪ Bregman Divergences
Convex, non-negative, not symmetric
y
D(x, y)
x
11
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(ii) Minimize what?▪ Bregman Divergences
Convex, non-negative, not symmetric
y
D(x, y)
x
11
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(ii) Minimize what?▪ Bregman Divergences
Convex, non-negative, not symmetric
12
Separable Strictly Convex Functions
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(ii) Minimize what?▪ Bregman Divergences
Convex, non-negative, not symmetric
12
Separable Strictly Convex Functions
Why do we need different divergences: convergence, regret bounds
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Algorithm Inc-Fix For Separable Strictly Convex Minimization Over Base Polytopes:
(b). Minimizing separable convex fns (sq. Euclidean distance, KL-divergence, ...)
(a). Which decision sets? Submodular Base Polytopes: B(f) (Permutations, k-subsets..)
13
13
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(1,3,2)
(2,3,1)
(3,2,1)
(3,1,2)
(2,1,3)(1,2,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
yB(f)
e3
e2
e1
-4
-1.4
-1
0.3
= x e
-ye
Project: y = (1.4, 4, 1)T
under Euclidean distance
“greedy in gradient space” – proof from first-order optimality conditions
14Inc-Fix Algorithm
[Gupta, Goemans, Jaillet]14
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(1,3,2)
(2,3,1)
(3,2,1)
(3,1,2)
(2,1,3)(1,2,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
yB(f)
e3
e2
e1
inc -4
-1.4
-1
0.3
= x e
-ye
Project: y = (1.4, 4, 1)T
under Euclidean distance
“greedy in gradient space” – proof from first-order optimality conditions
14Inc-Fix Algorithm
[Gupta, Goemans, Jaillet]14
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(1,3,2)
(2,3,1)
(3,2,1)
(3,1,2)
(2,1,3)(1,2,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
yB(f)
e3
e2
e1
inc -4
-1.4
-1
0.3
= x e
-ye
Project: y = (1.4, 4, 1)T
under Euclidean distance
“greedy in gradient space” – proof from first-order optimality conditions
14Inc-Fix Algorithm
[Gupta, Goemans, Jaillet]14
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(1,3,2)
(2,3,1)
(3,2,1)
(3,1,2)
(2,1,3)(1,2,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
yB(f)
e3
e2
e1
inc
inc
-4
-1.4
-1
0.3
= x e
-ye
Project: y = (1.4, 4, 1)T
under Euclidean distance
“greedy in gradient space” – proof from first-order optimality conditions
14Inc-Fix Algorithm
[Gupta, Goemans, Jaillet]14
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
![Page 47: Learning What Works Best When - Simons Institute for the ......Simons Institute, UC Berkeley Georgia Institute of Technology Joint work with Michel Goemans (MIT), Patrick Jaillet (MIT),](https://reader035.fdocuments.net/reader035/viewer/2022071110/5fe4eef2f6fd0176d55dfd0b/html5/thumbnails/47.jpg)
(1,3,2)
(2,3,1)
(3,2,1)
(3,1,2)
(2,1,3)(1,2,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
yB(f)
e3
e2
e1
inc
incfix
-4
-1.4
-1
0.3
= x e
-ye
Project: y = (1.4, 4, 1)T
under Euclidean distance
“greedy in gradient space” – proof from first-order optimality conditions
14Inc-Fix Algorithm
[Gupta, Goemans, Jaillet]14
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(1,3,2)
(2,3,1)
(3,2,1)
(3,1,2)
(2,1,3)(1,2,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
yB(f)inc,
e3
e2
e1
inc
incfix
-4
-1.4
-1
0.3
= x e
-ye
Project: y = (1.4, 4, 1)T
under Euclidean distance
“greedy in gradient space” – proof from first-order optimality conditions
14Inc-Fix Algorithm
[Gupta, Goemans, Jaillet]14
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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(1,3,2)
(2,3,1)
(3,2,1)
(3,1,2)
(2,1,3)(1,2,3)
(3,0,2)
(3, 0, 0)
(0,0,3)
(3,2,0)
yB(f)inc,
e3
e2
e1
inc
inc
fix
fix
-4
-1.4
-1
0.3
= x e
-ye
Project: y = (1.4, 4, 1)T
under Euclidean distance
x(*)
“greedy in gradient space” – proof from first-order optimality conditions
14Inc-Fix Algorithm
[Gupta, Goemans, Jaillet]14
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Running timeSquared Euclidean Distance, KL-Divergence: Movement along lines
In general: Piecewise smooth movement
y15
15Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Running timeSquared Euclidean Distance, KL-Divergence: Movement along lines
In general: Piecewise smooth movement
y15
15Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Running timeSquared Euclidean Distance, KL-Divergence: Movement along lines
In general: Piecewise smooth movement
y
Details: How to do this movement?
15
15Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Running timeSquared Euclidean Distance, KL-Divergence: Movement along lines
In general: Piecewise smooth movement
y
Details: How to do this movement? O(n) Line Searches + non-linear equations in a single variable
15
15Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Running timeSquared Euclidean Distance, KL-Divergence: Movement along lines
In general: Piecewise smooth movement
y
Details: How to do this movement?
Using structural properties, we show Inc-Fix can be implemented in, in general,
O(n) Submodular Function Minimizations*
O(n) Line Searches + non-linear equations in a single variable
Running time?
LSW’15: CLSW’16:*Require maximal minimizers, note that checking for feasibility itself requires a SFM.
15
15Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Setup: 100 Elements, Fixed cardinality-based submodular function, Projections of 10 randomly generated points
Frank-Wolfe
Inc-Fix
0 20 40 60 80 100 Elements
0 20 40 60 80 100 Elements
Gra
die
nt V
alu
es
Gra
die
nt V
alu
es
Inc-Fix Frank-WolfeGap from optimality
Computations for cardinality-based f(.)
For cardinality-based functions, Inc-Fix takes for exact, while vanilla FW takes for -approx. (O(n (log n + k)) for simplex, k-subsets, k-truncated-permutations)
16
16Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Setup: 100 Elements, Fixed cardinality-based submodular function, Projections of 10 randomly generated points
Frank-Wolfe
Inc-Fix
0 20 40 60 80 100 Elements
0 20 40 60 80 100 Elements
Gra
die
nt V
alu
es
Gra
die
nt V
alu
es
Inc-Fix Frank-WolfeGap from optimality
Computations for cardinality-based f(.)
For cardinality-based functions, Inc-Fix takes for exact, while vanilla FW takes for -approx. (O(n (log n + k)) for simplex, k-subsets, k-truncated-permutations)
16
16Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Setup: 100 Elements, Fixed cardinality-based submodular function, Projections of 10 randomly generated points
Frank-Wolfe
Inc-Fix
0 20 40 60 80 100 Elements
0 20 40 60 80 100 Elements
Gra
die
nt V
alu
es
Gra
die
nt V
alu
es
Inc-Fix Frank-WolfeGap from optimality
Computations for cardinality-based f(.)
For cardinality-based functions, Inc-Fix takes for exact, while vanilla FW takes for -approx. (O(n (log n + k)) for simplex, k-subsets, k-truncated-permutations)
16
16Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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1. Projections ▪ Motivation ▪ Problem setup ▪ Novel algorithm: Inc-Fix for separable convex minimization: ▪ Main Result: O(n) SFM or O(n) Line searches ▪ Exact computations, modulo solving a univariate equation
2. Line Searches ▪ Previous best known: Megiddo’s parametric search ▪ Using Newton’s Discrete Method: n2 + n log2n SFM (n6 improvement)
3. What works best when ▪ Problems with Max-Cut and QUBO heuristics comparative studies ▪ Our framework: Expanded instance library, Implementation of 37 heuristics,
Large-scale cloud computing on the cross product ▪ Hyper-heuristic: Map every instance to a feature space, learn
“performance” of heuristics
Outline 17
17Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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3. Feasibility along a Line
How much to move in a direction while staying feasible?
18
18Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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3. Feasibility along a Line
How much to move in a direction while staying feasible?
18
18Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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3. Feasibility along a Line
How much to move in a direction while staying feasible?
18
18Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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3. Feasibility along a Line
How much to move in a direction while staying feasible?
Sub-problem in many methods: • Inc-Fix, of course
• Frank-Wolfe [Frank, Wolfe, Jaggi, Lacoste-Julien, Freund, Grigas, …]
• Caratheodory’s Theorem
18
18Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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3. Feasibility along a Line
How much to move in a direction while staying feasible?
Sub-problem in many methods: • Inc-Fix, of course
• Frank-Wolfe [Frank, Wolfe, Jaggi, Lacoste-Julien, Freund, Grigas, …]
• Caratheodory’s Theorem
Others: ▪ Densest sub-graphs
[Nagano et al. 2011]
▪ Minimum Ratio Problems [Cunningham 1985]
18
18Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Discrete Newton Method for Parametric Line Search
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Discrete Newton Method for Parametric Line Search
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Discrete Newton Method for Parametric Line Search
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Discrete Newton Method for Parametric Line Search
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line SearchInc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Discrete Newton Method for Parametric Line Search
Open question to bound the no. of iterations!
Recall19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Line Search
We show a quadratic bound on the number of Newton’s iterations: <= n2 + o(n log2n) SFM (n6 improvement) [Goemans, Gupta, Jaillet, IPCO 2017]
Inc-Fix uses only positive directions (well-understood) General: Megiddo’s parametric search: Õ(n8) SFM [Nagano 2011]
Discrete Newton Method for Parametric Line Search
Open question to bound the no. of iterations!
Recall
Strongly Polynomial
19
19Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Sequence of subsetsConsider any submodular function a sequence of sets
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We show a quadratic bound on the number of Newton’s iterations: <= n2 + o(n log2n) SFM (n6 improvement) [Goemans, Gupta, Jaillet, IPCO 2017]
20Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Sequence of subsetsConsider any submodular function a sequence of sets How large
can q be?
20
We show a quadratic bound on the number of Newton’s iterations: <= n2 + o(n log2n) SFM (n6 improvement) [Goemans, Gupta, Jaillet, IPCO 2017]
20Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Sequence of subsetsConsider any submodular function a sequence of sets How large
can q be?
using submodularity of
20
We show a quadratic bound on the number of Newton’s iterations: <= n2 + o(n log2n) SFM (n6 improvement) [Goemans, Gupta, Jaillet, IPCO 2017]
20Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Sequence of subsetsConsider any submodular function a sequence of sets How large
can q be?
using submodularity of
using Birkhoff’s representation theorem
20
We show a quadratic bound on the number of Newton’s iterations: <= n2 + o(n log2n) SFM (n6 improvement) [Goemans, Gupta, Jaillet, IPCO 2017]
20Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Sequence of subsetsConsider any submodular function a sequence of sets How large
can q be?
using submodularity of
using Birkhoff’s representation theorem
20
We show a quadratic bound on the number of Newton’s iterations: <= n2 + o(n log2n) SFM (n6 improvement) [Goemans, Gupta, Jaillet, IPCO 2017]
20Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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1. Projections ▪ Motivation ▪ Problem setup ▪ Novel algorithm: Inc-Fix for separable convex minimization: ▪ Main Result: O(n) SFM or O(n) Line searches ▪ Exact computations, modulo solving a univariate equation
2. Line Searches ▪ Previous best known: Megiddo’s parametric search ▪ Using Newton’s Discrete Method: n2 + n log2n SFM (n6 improvement)
3. What works best when ▪ Problems with Max-Cut and QUBO heuristics comparative studies ▪ Our framework: Expanded instance library, Implementation of 37 heuristics,
Large-scale cloud computing on the cross product ▪ Hyper-heuristic: Map every instance to a feature space, learn
“performance” of heuristics.
Outline 21
21
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What works best when
Image from: https://www.dreamstime.com/stock-illustration-octopus-tools-illustration-image47507762
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What works best when
Image from: https://www.dreamstime.com/stock-illustration-octopus-tools-illustration-image47507762
Encounter problem in practice
Find out what isknown
Run the “best” knownalgorithm/heuristic
for the data
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What works best when
Image from: https://www.dreamstime.com/stock-illustration-octopus-tools-illustration-image47507762
Encounter problem in practice
Find out what isknown
Run the “best” knownalgorithm/heuristic
for the data
From learning decisions, to learning performance of algorithms
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Max-Cut: An NP-Hard Problem
Given an edge-weighted graph, partition nodes into two sets to maximize the weight of the edges between the sets
Equivalence with Quadratic Unconstrained Binary Optimization Problem (QUBO)
A lot of applications, and a lot of research! ◇ >32 published papers since 2010.
Computational experiments key to heuristic evaluation!
But hard to find which heuristic works best when
23Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Max-Cut: An NP-Hard Problem
Given an edge-weighted graph, partition nodes into two sets to maximize the weight of the edges between the sets
Equivalence with Quadratic Unconstrained Binary Optimization Problem (QUBO)
A lot of applications, and a lot of research! ◇ >32 published papers since 2010.
Computational experiments key to heuristic evaluation!
But hard to find which heuristic works best when
23Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Max-Cut: An NP-Hard Problem
Given an edge-weighted graph, partition nodes into two sets to maximize the weight of the edges between the sets
Equivalence with Quadratic Unconstrained Binary Optimization Problem (QUBO)
A lot of applications, and a lot of research! ◇ >32 published papers since 2010.
Computational experiments key to heuristic evaluation!
But hard to find which heuristic works best when
23Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Max-Cut: An NP-Hard Problem
Given an edge-weighted graph, partition nodes into two sets to maximize the weight of the edges between the sets
Equivalence with Quadratic Unconstrained Binary Optimization Problem (QUBO)
A lot of applications, and a lot of research! ◇ >32 published papers since 2010.
Computational experiments key to heuristic evaluation!
But hard to find which heuristic works best when
23Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Problems with standard testbedHomogeneous Test Bed: Max-Cut (105 graphs), QUBO (126 matrices)
Max-Cut Instances QUBO instances
24Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Problems with standard testbedHomogeneous Test Bed: Max-Cut (105 graphs), QUBO (126 matrices)
Max-Cut Instances QUBO instances
Which Max-Cut heuristic works best for high density graphs? Which QUBO heuristic works best for sparse matrices?
24Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Problems with status-quo
Same runtime limit?
Same hardware?
No Yes
No 55% 4%
Yes 31% 10%
◇ few published source code ◇ reimplementation uncommon ◇ different testing criteria ◇ comparison with small no. of heuristics…
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Our Approach
26Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Our Approach
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Expanded Instance Library◇ Heterogeneous instances, capture instances in real instances ◇ Real World Instances (tsplib, steinlib, dimacs, road networks, …) ◇ Network science generators (ER, NWS, BA, …) ◇ Sampled weights from 65 prob. distributions (uniform, weibull, …)
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Heterogeneity: 58 Metrics
◇ 10 global metrics: ◇ nodes, edges, 1st and 2nd eigenvalues of Laplacian, chromatic number, …
◇ 48 local metrics from summary statistics of edge/node attributes: ◇ degree, avg. neighbor degree, clustering coefficient, core…
◇ Fast to compute – at most
◇ Coverage (for normalized metrics in [0,1]): union over all instances of a small interval around the metric value for each instance ◇ average metric coverage for new test bed: 0.88 (interval +-0.05) ◇ 0.31 for 95 std Max-Cut v/s 0.71 (0.69-0.77) for ~ 95 random new ◇ 0.38 for 56 std QUBO v/s 0.64 (0.59-0.68) for ~56 random new
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Our Approach
30Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Our Approach
30Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Implementation + Evaluation
◇ We did what one would expect◇ thorough lit review (810 papers)◇ selected 95 papers (new heuristics)◇ implemented 37 heuristics from 19 highly cited papers
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Implementation + Evaluation
◇ We did what one would expect◇ thorough lit review (810 papers)◇ selected 95 papers (new heuristics)◇ implemented 37 heuristics from 19 highly cited papers
◇ Minor modifications to standardize: ◇ added random restarts◇ shared common code – data structures and subroutines◇ no parameter tuning
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Implementation + Evaluation
◇ We did what one would expect◇ thorough lit review (810 papers)◇ selected 95 papers (new heuristics)◇ implemented 37 heuristics from 19 highly cited papers
◇ Minor modifications to standardize: ◇ added random restarts◇ shared common code – data structures and subroutines◇ no parameter tuning
◇ Cloud Computing – Amazon EC2◇ Instance specific runtime limit computation◇ too low: miss performance ◇ too high: waste computational budget
◇ any new heuristic can be tested for $32.5 (20.6 CPU days/heuristic)
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https://github.com/MQLib/MQLibOpen Source Code available at
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Our Approach
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Our Approach
33Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Results
◇ No heuristic dominated all the others ◇ 30/37 heuristics strictly best on at least one
instance ◇ No heuristic matched the best performance on
more than 22.9% of the testbed
◇ Standard test beds do not capture performance ◇ Example: GLS heuristic (Merz, Freisleben 1999) ◇ Strictly best on no instances in the std test bed ◇ Sole best-performing on 6.9% expanded test bed
instances!
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Can we predict which heuristic would work best on an unseen data instance?
36
“the algorithm selection problem is to learn the mapping from instance features
to the best algorithm to run on an instance”
— Rice (1976)
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37
… Phase transitions (Cheeseman et al. 1991,
Hartman and Weigt, 2006)
… Landscape analysis(Stadler and Schnabl
1992, Krzkakala et al. 2004, Hartman and Weigt
2003, Gent and Walsh 1996,
Smith-miles et al. 2010, Wang et al. 2013… )
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Interpreting Heuristic Performance
38Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
[… “algorithmic footprints” Smith-Miles et al. 2014]
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Interpreting Heuristic Performance
38
Conjecture Generation?
Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
[… “algorithmic footprints” Smith-Miles et al. 2014]
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Comparing Heuristic Performance
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Heuristic Class Performance
40Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
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Our Approach
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Our Approach
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Algorithm Portfolio or Hyper-heuristic
◇ Random Forest Model for each heuristic ◇ Predicts if it will obtain the best solution using 58 features ◇ Final heuristic selected has maximum predicted probability ◇ Small fraction of runtime budget to select heuristic and then run the
selected heuristic on remaining time
◇ Represents state-of-the-art Max-Cut and QUBO heuristic! ◇ Improves significantly over best single heuristic (BUR02): ◇ Probability of obtaining best solution: increased from 15% to 37% ◇ Avg. deviation from best solution reduced from 0.34% to 0.09% ◇ Running 8 heuristics in parallel: 48% best solution, 0.05% avg. dev.
42Learning What Works Best When | Swati Gupta | Research Fellow, Simons Institute
(joint work with Iain Dunning, John Silberholz. INFORMS Journal on Computing, 2017)
[… SAT solvers (Xu et al 2008), constrained prog (O’ Mahoney et al. 2008)]
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1. Projections ▪ Motivation ▪ Problem setup ▪ Novel algorithm: Inc-Fix for separable convex minimization: ▪ Main Result: O(n) SFM or O(n) Line searches ▪ Exact computations, modulo solving a univariate equation
2. Line Searches ▪ Previous best known: Megiddo’s parametric search ▪ Using Newton’s Discrete Method: n2 + n log2n SFM (n6 improvement)
3. What works best when ▪ Problems with Max-Cut and QUBO heuristics comparative studies ▪ Our framework: Expanded instance library, Implementation of 37 heuristics,
Large-scale cloud computing on the cross product ▪ Hyper-heuristic: Map every instance to a feature space, learn
“performance” of heuristics
Outline 43
[email protected] swatigupta.tech
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