SUMMARY 43
where R
0
(w) = 1 for the w ∈ W. Note the mean is the same as Cover’s UP, which
equally splits the money among different strategies and lets them run.
Besides the universalization in the continuous parameter space, various dis-
crete buy-and-hold combinations have been adopted by various existing algorithms.
Rewriting Cover’s UP in its discrete form, the update can be straightforwardly
obtained. For example, Borodin et al. (2004) adopted the BAH strategy to com-
bine Anticor experts with respect to a finite number of window sizes (or parameters).
Moreover, all pattern matching–based approaches adopted BAH to combine their
underlying experts, also with a finite number of window sizes (or parameters).
7.3 Online Gradient and Newton Updates
Das and Banerjee (2011) proposed two meta-optimization algorithms, named online
gradient update (OGU) and online Newton update (ONU), which are extended
from exponential gradient (EG) and online Newton step (ONS), respectively. Since
their updates and proofs are similar to their precedents, we ignore their updates.
Theoretically, OGU and ONU can achieve the same growth rate as the optimal con-
vex combination of underlying experts. Particularly, if any base expert is universal,
then the final system enjoys the universality property. This property is useful, as an
MA can combine a heuristic algorithm and a universal algorithm, and the final system
can enjoy both superior heuristic performance and the universality property.
7.4 Follow the Leading History
Hazan and Seshadhri (2009) proposed the follow the leading history (FLH) algorithm
for changing environments. FLH can incorporate various universal base experts, such
as the ONS algorithm. Its basic idea is to maintain a working set of finite experts, which
are dynamically added in and dropped out, andallocatetheweightsamongsomeactive
working experts with an MA, for example, the Herbster–Warmuth algorithm (Herbster
and Warmuth 1998). Different from other MAs with all experts operating from the
beginning, FLH adopts experts starting from different periods. Theoretically, FLH
based on universal algorithms is also universal, and empirically, FLH equipped with
ONS can significantly outperform ONS.
7.5 Summary
Meta-learning is another widely discussed principle in the research of online portfolio
selection (OLPS). It derives from base algorithms but treats these experts as the
underlying assets. Thus, from this aspect, meta-algorithms (MAs) can be widely
applied to all strategies discussed in previous chapters. We are interested in this
principle because practical trading systems usually contain multiple strategies, and
meta-learning can be used to combine these strategies in an effective way.
T&F Cat #K23731 — K23731_C007 — page 43 — 9/28/2015 — 21:15