Previous rigorous approaches for this problem rely on dynamic programming (DP) and, while sample efficient, have running time quadratic in the sample size. As our main contribution, we provide new sample near-linear time algorithms for the problem that – while not being minimax optimal – achieve a significantly better sample-time tradeoff on large datasets compared to the DP approach. Our experimental evaluation shows that, compared with the DP approach, our algorithms provide a convergence rate that is only off by a factor of 2 to 4, while achieving speedups of three orders of magnitude.
The most common return measure, also referred to as the return on investment, or ROI, is calculated by dividing the difference between the cost of the investment and the gain on the investment by the cost of the investment. It is the most generic way to calculate return and is the basic formula used to calculate other return measures. For example, if an investor pays $100,000 for real estate and then sells it for $110,000, the return is calculated by taking the difference between $100,000 and $110,000, and then dividing that number by the cost of the investment, or $100,000. The calculation is $10,000 divided by $100,000, or 10%.