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The WIDROW-HOFF (a.k.a. LEAST MEAN SQUARED (LMS) )
algorithm is the most well-understood and principled approach to
training a linear system to minimize this squared error loss [Widrow60] . It does this by making a
small move (scaled by the parameter $\eta$) in the direction of the
gradient of error. This gradient is defined exactly by the derivative of
Equation (FOAref) with respect to the document vector: \Delta
\mathbf{q} = - 2 \eta (\mathbf{q} \cdot \mathbf{d} - R_{\mathbf{d}})
\mathbf{d}
It is also important to remember that changes made to a
single document in response to a single query can make no guarantees
about improved performance with respect to other documents and other
queries. For example, two documents might both be moved closer to a
query (as proposed by Brauen/Roccio) while their relative rankings are
not changed at all!
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Widrow-Hoff