multiple gradient descent algorithm
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Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Multiple Gradient Descent Algorithm
Outi Wilppu
March 1, 2013
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
1 Introduction
2 Multiple Gradient Descent AlgorithmSteepest Descent MethodTheoretical backroundAlgorithm
3 MGDA for non-smooth convex problemsTheoretical backroundAlgorithmExampleFuture work
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Motivation
Many real life problems have multiobjective nature
• Problem has several goals such that they should all be asgood as possible
• Goals are conflicting and have to make compromises
• For example in economics and engineering
• Many problems have also non-smooth nature
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Problem
Definition
We consider a multiobjective problem of form
min f1(x), . . . , fk(x) (1)
s. t. x ∈ S
where
• Objective functions fi (x) : Rn −→ R• Set S ⊆ Rn is a set of feasible solutions
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Pareto optimality
Definition
A solution x∗ ∈ S is Pareto optimal if there is no other solutionx ∈ S such that fi (x) ≤ fi (x
∗) for all i = 1, . . . , k andfi (x) < fi (x
∗) for at least one index i .
• Usually there exists a lot of Pareto optimal solutions, calledPareto optimal set
• All Pareto optimal solution are mathematically equally good
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Steepest Descent Method
• A method to solve smooth unconstrained optimizationproblem with single objective function
min f (x), s. t. x ∈ Rn
• Search direction d = − ∇f (x)
‖ ∇f (x) ‖, which is the steepest
descent direction at the point x
• Stepsize λ can be found with line search
• A new point is of the form x+ = x + λd, wheref (x+) < f (x) until the optimum is reached
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
About MGDA
• Assume that in problem (1) we have
– All objective functions fi are continiously differentiable– Problem has no constrains and feasible set is Rn
• The main idea is to find a common descent direction for allobjective functions by defining the convex hull of gradients ofobjective functions and finding the minimum norm element ofthe convex hull
• With this method we obtain one Pareto stationary solution
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Pareto stationary
Definition
Objective functions fi are said to be Pareto stationary at the pointx∗ if
k∑i=1
αi∇fi (x∗) = 0, αi ≥ 0 ∀i ,
k∑i=1
αi = 1.
• Pareto stationarity is a necessary condition for Paretooptimality
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Theorem
Let objective funtions fi , i = 1, . . . , k be continiously differentiableand
U = {w ∈ Rn | w =k∑
i=1
αi∇fi (x), αi ≥ 0 ∀ i ,k∑
i=1
αi = 1}
Let vector w∗ be a minimum norm element of convex hull of U i.e.w∗ = argmin{‖ w ‖ |w ∈ conv U}. Then
1 either w∗ = 0 and objective functions fi , i = 1, . . . , k arePareto stationary at the point x∗
2 or w∗ 6= 0 and −w∗ is common descent direction for allobjective functions. Additionally, if w∗ ∈ U thenuTw∗ =‖ w∗ ‖2 for all u ∈ conv U.
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Search direction
• To find the vector w we need to solve the problem
min ‖k∑
i=1
αi∇fi (x)‖2
s. t. αi ≥ 0∀ i ,k∑
i=1
αi = 1.
• In case of two objective functions we have
α1 =
‖v‖2−uTv
‖u‖2+‖v‖2−2uTv, if uTv < min{‖u‖2, ‖v‖2}
0, if min{‖u‖, ‖v‖} = ‖v‖1, if min{‖u‖, ‖v‖} = ‖u‖
where u = ∇f1(x), v = ∇f2(x) and α2 = 1− α1.
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Stepsize
• Stepsize λ is the largest strictly positive real number such thatall the functions gi (t) = fi (x− tw) are monotonicallydecreasing over the interval [0, t].
• A new point x+ is x+ = x− λw• Algorithm stops when w = 0
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
About MGDA for non-smooth convex problems
• Assume that in the problem (1)
– Objective functions fi are convex functions, not necessarilydifferentiable
– Problem has no constrains and the feasible set is Rn
– The subdifferentials of objective functions fi are known
• The basic idea is the same as before but now instead ofgradients we use the steepest descent subgradients
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Subdifferential and subgradient
Definition
Let function f be convex. A subdifferential of f at the point x is aset
∂f (x) = {ξ ∈ Rn | f ′(x;d) ≥ ξTd}
for all d ∈ Rn. A vector ξ ∈ ∂f (x) is called subgradient offunction f at the point x.
Theorem
Let function f : Rn −→ R be convex and differentiable at the pointx ∈ Rn. Then
∂f (x) = {∇f (x)}.
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Steepest descent direction
• To determine the steepest descent direction we need to solvethe problem
min f ′(x;d) s. t. ‖d‖ = 1 (2)
• The problem (2) can be written
min ‖ξ‖ s. t. ξ ∈ ∂f (x)
• Thus the steepest descent direction at the point x is thesmallest norm subgradient in the subdifferential ∂f (x)
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Algorithm
• To obtain a search direction we need to define the steepestdescent subgradient for every objective function at the pointx. After that we can solve problem
min ‖k∑
i=1
αiξi‖2
s. t. αi ≥ 0 ∀ i ,k∑
i=1
αi = 1
where ξi ∈ ∂fi (x) is the steepest descent subgradient.
• Stepsize and new point are determined as in smooth case
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
• With this method we obtain one Pareto stationary solutionsince
k∑i=1
αiξi = 0, αi ≥ 0 and
k∑i=1
αi = 1.
• In convex case Pareto stationarity is necessary and sufficientcondition to Pareto optimality
⇒ We obtain one Pareto optimal solution
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Example
Consider the following problem:
min {f1(x), f2(x)}s. t. x ∈ R2
where
f1(x) = max{x21 + (x2 − 1)2, (x1 + 1)2}
f2(x) = max{2x1 − x2, x21 + x2}
and starting point x0 = (1, 0.5).
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
The steepest descent subgradients at the point x0:
– ξ1 = (4, 0)
– To determine ξ2 we need to find ρ ∈ [0, 1] such that we findthe minimum norm of subdifferential
∂f2(x0) = ρ(2,−1) + (1− ρ)(2, 1) = (2, 1− 2ρ).
To obtain the steepest descent subgradient ρ =1
2and
ξ2 = (2, 0).
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
A common descent search direction:
– w = αξ1 + (1− α)ξ2
– We obtain α from formula
α =
‖ξ2‖2−ξT1 ξ2
‖ξ1‖2+‖ξ2‖2−2ξT1 ξ2, if ξT1 ξ2 < min{‖ξ1‖2, ‖ξ2‖2}
0, if min{‖ξ1‖, ‖ξ2‖} = ‖ξ2‖1, if min{‖ξ1‖, ‖ξ2‖} = ‖ξ1‖
– Thus w = (2, 0)
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Stepsize:
– Function g1(t) = f1(x0 − tw) is descent if t ∈ [0, 0.6875]
– Function g2(t) = f2(x0 − tw) is descent if t ∈ [0, 0.5]
⇒ Stepsize λ = 0.5
New point:
– x1 = x0 − λw = (0, 0.5)
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
k ξ1 ξ2 α w λ xk+1
0 (4, 0) (2, 0) 0 (2, 0) 0.5 (0, 0.5)
1 (2, 0) (0, 1) 0.2 (0.4, 0.8) 0.403 (−0.161, 0.178)
2 (1.678, 0) (−0.322, 1) 0.329 (0.336; 0.671)
Solution: x∗ = (−0.161, 0.178), f1(x∗) ≈ 0.704 andf2(x∗) ≈ 0.203
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Future work
• The biggest drawback is that we need to know wholesubdifferential
• The aim is to find an effective way to get a common descentdirection for all objective functions
• We do not need more than one arbitrary subgradient offunction
Outi Wilppu Multiple Gradient Descent Algorithm
Outline Introduction Multiple Gradient Descent Algorithm MGDA for non-smooth convex problems
Thank you for your attention!
Outi Wilppu Multiple Gradient Descent Algorithm
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