Digital Signal Processing Reference
In-Depth Information
Our model relies on Bregman matrix divergences that we have compared with
others that have been previously defined elsewhere. In the general case, not restricted
to allocation (SPD) matrices, our definition presents the interest to split the divergence
between a separable divergence, and terms that can be non-zero when the argument
matrices are not symmetric, or do not share the same transition matrices.
We have also defined Bregman matrix divergences that rely on functional compo-
sition of generators, and obtained a generalization of Bregman matrix divergences
for
q
-norms used elsewhere [
13
]. We have shown that properties of the usual
q
-norm
Bregman divergences can be generalized to our so-called Bregman-Schatten diver-
gences. We have also proposed an on-line learning algorithm to track efficient portfo-
lios in our matrix mean-divergence model with Bregman-Schatten divergences. The
algorithm has been devised and analyzed in the setting of symmetric positive def-
inite matrices for allocations. The algorithm generalizes conventional vector-based
q
-norm algorithms. Theoretical bounds for risk premia exhibit penalties that have
the same flavor as those already known in the framework of supervised learning [
15
].
Like most of the bounds in the supervised learning literature, they are not directly
applicable: in particular, we have to know
ν
∗
beforehand for Theorem 2 to be applica-
ν
−
1
ν
◦
◦
ble, or at least a lowerbound
1).
From a learning standpoint, rather than finding prescient and non adaptive strate-
gies like in constant rebalanced portfolio selection [
10
], on-line learning in the mean-
divergence model rather aims at finding non prescient and adaptive strategies yielding
efficient portfolios. This, we think, may constitute an original starting point for fur-
ther works on efficient portfolio selection, with new challenging problems to solve,
chief among them learning about investor's risk aversion parameters.
(hence, we would typically fix
Acknowledgments
The authors wish to thank the reviewers for useful comments, and gratefully
acknowledge the support of grant ANR-07-BLAN-0328-01.
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