Ban & Rudin: Big Data Newsvendor
10 Article submitted to Operations Research; manuscript no.
2.3.2. Kernel Optimization (KO) Method. Here we introduce an alternative approach
that can take features into account. We call this approach the Kernel-weights Optimization (KO)
method because it is based on Nadaraya-Watson kernel regression (Nadaraya 1964, Watson 1964).
One of the goals of nonparametric regression is to estimate the expectation of a dependent variable
(e.g. demand) conditional on independent variables taking on a particular value. That is, given past
data (x
1
, y
1
), . . . , (x
n
, y
n
) one wants to estimate
m(x
n+1
) = E[Y |x
n+1
],
where Y ∈R is the dependent variable and x
n+1
∈R
p
is a vector of new independent variables. In
1964, Nadaraya and Watson proposed to estimate this quantity by the locally weighted average
m
h
(x
n+1
) =
P
n
i=1
K
w
(x
n+1
−x
i
)y
i
P
n
i=1
K
w
(x
n+1
−x
i
)
,
where K
w
(·) is a kernel function with bandwidth w. Typical examples of the kernel function include
the uniform kernel
K(u) =
1
2
I(kuk
2
≤1)
and the Gaussian kernel
K(u) =
1
√
2π
exp
−kuk
2
2
/2
,
with K
w
(·) := K(·/w)/w. Now for an order quantity q, the feature-dependent newsvendor expected
cost after observing features x
n+1
is given by
E[C(q; D)|x
n+1
], (7)
which depends (implicitly) on the demand distribution at x
n+1
. Thus if we consider the newsvendor
cost to be the dependent variable, we can estimate (7) by the Nadaraya-Watson estimator
P
n
i=1
K
w
(x
n+1
−x
i
)C(q, d
i
)
P
n
i=1
K
w
(x
n+1
−x
i
)
.
This gives rise to a new approach to feature-data-driven newsvendor, which we call the Kernel
Optimization (KO) Method.
min
q≥0
˜
R(q; S
n
, x
n+1
) = min
q≥0
P
n
i=1
K
w
(x
n+1
−x
i
)C(q, d
i
)
P
n
i=1
K
w
(x
n+1
−x
i
)
. (NV-KO)
Note that there are no edge effects in the objective estimate if the kernel is smooth, which is the case
for the Gaussian kernel. Notice that the optimization is over the non-negative reals, and the optimal
decision implicitly depends on x
n+1
. (NV-KO) is a one-dimensional piecewise linear optimization
problem, and we can find its solution according to the following proposition.
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