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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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