Suppose Est is a randomized estimation algorithm that uses n random bits and outputs values in R d . We show how to execute Est on k adaptively chosen inputs using only n + O ( k log ( d + 1 )) random bits instead of the trivial n k (at the cost of mild increases in the error and failure probability). Our algorithm combines a variant of the INW pseudorandom generator (STOC '94) with a new scheme for shifting and rounding the outputs of Est . We prove that modifying the outputs of Est is necessary in this setting, and furthermore, our algorithm's randomness complexity is near-optimal in the case d O (1) . As an application, we give a randomness-efficient version of the Goldreich-Levin algorithm; our algorithm finds all Fourier coefficients with absolute value at least of a function F : 0 1 n − 1 1 using O ( n log n ) pol y (1 ) queries to F and O ( n ) random bits (independent of ), improving previous work by Bshouty et al. (JCSS '04).