  Top-k maximum inner product search (MIPS) is a central task in many machine
learning applications. This paper extends top-k MIPS with a budgeted setting,
that asks for the best approximate top-k MIPS given a limit of B computational
operations. We investigate recent advanced sampling algorithms, including wedge
and diamond sampling to solve it. Though the design of these sampling schemes
naturally supports budgeted top-k MIPS, they suffer from the linear cost from
scanning all data points to retrieve top-k results and the performance
degradation for handling negative inputs.
  This paper makes two main contributions. First, we show that diamond sampling
is essentially a combination between wedge sampling and basic sampling for
top-k MIPS. Our theoretical analysis and empirical evaluation show that wedge
is competitive (often superior) to diamond on approximating top-k MIPS
regarding both efficiency and accuracy. Second, we propose a series of
algorithmic engineering techniques to deploy wedge sampling on budgeted top-k
MIPS. Our novel deterministic wedge-based algorithm runs significantly faster
than the state-of-the-art methods for budgeted and exact top-k MIPS while
maintaining the top-5 precision at least 80% on standard recommender system
data sets.
