We investigate the parameters in terms of which the complexity of sublinear-time algorithms should be expressed. Our goal is to find input parameters that are tailored to the combinatorics of the specific problem being studied and design algorithms that run faster when these parameters are small. This direction enables us to surpass the (worst-case) lower bounds, expressed in terms of the input size, for several problems. Our aim is to develop a similar level of understanding of the complexity of sublinear-time algorithms to the one that was enabled by research in parameterized complexity for classical algorithms.

Specifically, we focus on testing properties of functions. By parameterizing the query complexity in terms of the size $r$ of the image of the input function, we obtain testers for monotonicity and convexity of functions of the form $f:[n]\to \mathbb{R}$ with query complexity $O(\log r),$ with no dependence on $n$. The result for monotonicity circumvents the $\Omega(\log n)$ lower bound by Fischer (Inf. Comput., 2004) for this problem. We present several other parameterized testers, providing compelling evidence that expressing the query complexity of property testers in terms of the input size is not always the best choice.