Chapter 7
Meta-Learning
Another research topic in online portfolio selection (OLPS) is meta-learning,
or meta-algorithms (MAs) (Das and Banerjee 2011), which is closely related to
expert learning (Cesa-Bianchi and Lugosi 2006). This is directly applicable to the
“fund of fund” (FOF),
which delegates portfolio capital to other funds. In general,
MA defines several base experts, each of which is equipped with strategies from the
same strategy class or different classes, or even MAs. Each expert outputs a portfolio
vector, and MA combines these portfolios to form a final portfolio, which is used for
rebalance. The whole system can achieve the best performance among the experts
in hindsight, which thus is desired for some nonuniversal algorithms. MAs are sim-
ilar to Covers UP algorithm in the follow the winner approach; however, they are
proposed to handle different classes of experts, among which UP’s CRP becomes a
special case. On the one hand, MAs can be used to smooth the final performance with
respect to all experts, especially when base experts are sensitive to certain environ-
ments/parameters. On the other hand, combining universal algorithms and heuristic
algorithms, which is not easy to obtain a theoretical regret bound, can provide the
universality property for the whole system. Finally, MAs can be applied to all existing
approaches and thus have much broader areas of application.
7.1 Aggregating Algorithms
Though BCRP is optimal for an independent and identically distributed (i.i.d.) market,
which is often suspected in real markets, the optimal portfolio may not belong to CRP.
Several algorithms have been proposed to track a different set of experts. The base
experts in this category belong to a special class rather than complex experts from
multiple classes.
Vovk and Watkins (1998) applied the aggregating algorithm (AA) (Vovk 1990,
1997, 1999, 2001) to the OLPS task, of which Covers UPis a special case. The general
setting for AA is to define a countable or finite set of base experts and sequentially
allocate the resource among multiple base experts to achieve a good performance that
FOF selects portfolios on different fund managers, rather than on assets. For example, an FOF
manager may evenly split his fund, and put one part to fund A and the other to Fund B.
41
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42 META-LEARNING
is no worse than any fixed combination of underlying experts. Its portfolio update
formula (Vovk and Watkins 1998, Algorithm 1) for OLPS is
b
t+1
=
m
b
t1
i=1
(b ·x
t
)
η
P
0
(db)
m
t1
i=1
(b ·x
t
)
η
P
0
(db)
.
As a special case, Covers UP corresponds to AA with uniform prior distribution
and η = 1.
Several further algorithms have been proposed. Singer (1997) proposed the
switching portfolios (SP), which switches among a set of strategies handling dif-
ferent regimes. The author proposed two switching schemes, both of which assume
the duration of base strategies is geometrically distributed. While the first strategy
assumes a fixed distribution, the second assumes that the distribution is dynamically
changing. The authors further presented the lower bound of its logarithmic wealth
with respect to the best switching regime. Empirical evaluations show that SP can
outperform UP, EG, and BCRP.
Levina and Shafer (2008) proposed the Gaussian random walk strategy, which
switches among base experts according to a Gaussian distribution. Kozat and Singer
(2007) extended SP to piecewise fixed fraction strategies, which partitions the peri-
ods into different segments and transits among these segments. Kozat and Singer
(2008) extended Kozat and Singer (2007) to the cases of transaction costs. Kozat and
Singer (2009, 2010) further generalized to sequential decision problems. Kozat et al.
(2008) proposed another piecewise universal portfolio selection strategy via context
trees, and Kozat et al. (2011) also generalized to sequential decision problems via tree
weighting.
SP adopts the notion of regime switching (Hamilton 1994, 2008), which seems to
be more plausible than an i.i.d. market assumption. Regime switching is also applied to
some state-of-the-art trading strategies (Hardy 2001). However, existing geometrical
and Gaussian distributions do not seem to fit the market well, which leads to other
possible distributions that can fit the markets better.
7.2 Fast Universalization
Akcoglu et al. (2005) proposed fast universalization (FU), which extends Covers
(1991) UP from a parameterized CRP class to a wide class of investment strate-
gies, including trading strategies operating on a single stock and portfolio strategies
allocating wealth among the whole market. FU’s basic idea is to evenly split the
wealth among base experts, let these experts operate on their own, and finally pool
their wealth. FU’s update is the same as that of Covers UP, and it also asymptotically
achieves a growth rate that equals that of an optimal fixed convex combination of base
experts. In cases in which all experts are CRPs, FU would downgrade to Covers UP.
Formally, FU’s investment can be described as
b
t
=
W
S
t
(w)R
t
(w)dμ(w)
W
R
t
(w)dμ(w)
, (7.1)
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SUMMARY 43
where R
0
(w) = 1 for the w W. Note the mean is the same as Covers 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 Covers 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.
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