Scenario Min-Max Optimization
and
the Risk of Empirical Costs
Algo Carè
University of Brescia, Italy
University of Melbourne, Australia
MTA SZTAKI, Budapest, Hungary
a
with S. Garatti and M.C. Campi
a
Politecnico di Milano
b
University of Brescia
b
Scenario Min-Max Optimization
and
the Risk of Empirical Costs
Algo Carè
University of Brescia, Italy
University of Melbourne, Australia
MTA SZTAKI, Budapest, Hungary
a
with S. Garatti and M.C. Campi
a
Politecnico di Milano
b
University of Brescia
b
Scenario Min-Max Optimization
and
the Risk of Empirical Costs
Algo Carè
University of Brescia, Italy
University of Melbourne, Australia
MTA SZTAKI, Budapest, Hungary
a
with S. Garatti and M.C. Campi
a
Politecnico di Milano
b
University of Brescia
b
Scenario Min-Max Optimization
and
the Risk of Empirical Costs
Algo Carè
University of Brescia, Italy
University of Melbourne, Australia
MTA SZTAKI, Budapest, Hungary
a
with S. Garatti and M.C. Campi
a
Politecnico di Milano
b
University of Brescia
b
Convex problem
Convex problem
design parameter(s)
Convex problem
design parameter(s)
Convex problem
design parameter(s)
uncertain parameter
Convex problem
design parameter(s)
uncertain parameter
Uncertain problem!
The Scenario
Approach
Scenario Approach
Solution:
Scenario Approach
Solution:
Scenario Approach
Solution:
unknown
Scenario Approach
Solution:
unknown
Scenario Approach
unknown
Scenario Approach
unknown
Scenario Approach
unknown
Scenario Approach
unknown
Scenario Approach
unknown
Scenario Approach
unknown
Scenario Approach
unknown
Scenario solution:
Empirical distribution of
Empirical distribution of
Empirical distribution of
Empirical distribution of
Ordered values of
Empirical distribution of
Ordered values of
Empirical distribution of
Probability distribution of
Probability distribution of
PDF
Probability distribution of
PDF
CDF
Probability distribution of
PDF
CDF
Probability distribution of
PDF
CDF
Probability distribution of
PDF
CDF
Probability distribution of
PDF
CDF
?
Probability distribution of
?
TAKE-HOME MESSAGE:
It is possible to “reconstruct”
the distribution of the cost
by using the sole scenarios
that have been used to compute .
Without using
any specific knowledge of
and of .
TAKE-HOME MESSAGE:
It is possible to “reconstruct”
the distribution of the cost
by using the sole scenarios
that have been used to compute .
Without using
any specific knowledge of
and of .
TAKE-HOME MESSAGE:
It is possible to “reconstruct”
the distribution of the cost
by using the sole scenarios
that have been used to compute .
Without using any new observation
nor any specific knowledge of .
HOW?
HOW?
By combining a-posteriori knowledge
(the empirical distribution of the cost)
with distribution-free theorems
(invariant properties of convex problems)
example (and a few technical
details) following...
Empirical distribution of
HOW?
By combining a-posteriori knowledge
(the empirical distribution of the cost)
with distribution-free theorems
(invariant properties of convex problems)
example (and a few technical
details) following...
HOW?
By combining a-posteriori knowledge
(the empirical distribution of the cost)
with distribution-free theorems
(invariant properties of convex problems)
example (and a few technical
details) following...
HOW?
By combining a-posteriori knowledge
(the empirical distribution of the cost)
with distribution-free theorems
(invariant properties of convex problems)
example (and a few technical
details) following...
Channel equalization example
Channel equalization example
Channel equalization example
For details: see paper
Channel equalization example
Channel equalization example
Channel equalization example
N and d
are all that
matters
N and d
are all that
matters
N and d
are all that
matters
with confidence 1-10
-6
Risk
Cost
Risk
Cost
Risk
Cost
Risk
Cost
Risk
Cost
Risk
Cost
Risk
Cost
Risk
Cost
Risk
Cost
Risk
Cost
Zoomed detail
Risk
Zoomed detail
Risk
Zoomed detail
Risk
Zoomed detail
Risk
Risk
Cost
1-Risk
Cost
Probability distribution of
We have reconstructed (wrapped)
the real distribution of the
cost from data ( scenarios).
No specific knowledge of
and has been used.
We have reconstructed (wrapped)
the real distribution of the
cost from data ( scenarios).
No specific knowledge of
and has been used.
Thank you!
REFERENCE
A. Carè, S. Garatti, and M.C. Campi,
Scenario Min-Max Optimization and the Risk of Empirical Costs.
SIAM Journal on Optimization, 25(4):2061-2080, 2015.
[email protected]
Ordered Dirichlet C.D.F.
“Fully” vs “non-fully” supported