## 1 Introduction  
This paper considers the problem of adaptively reconstructing a monotonically increasing function \(F^{\dagger}\) from imperfect pointwise observations of this function. In the statistical literature, the problem of estimating a monotone function is commonly known as _isotonic regression_, and it assumed that the observed data consist of noisy pointwise evaluations of \(F^{\dagger}\). However, we consider this problem under assumptions that differ from the standard formulation, and these differences motivate our algorithmic approach to the problem. To be concrete, our two motivating examples are that  
\[F^{\dagger}(x)\coloneqq\mathbb{P}_{\Xi\sim\mu}[g(\Xi)\leq x] \tag{1}\]  
is the cumulative distribution function (CDF) of a known real-valued function \(g\) of a random variable \(\Xi\) with known distribution \(\mu\), or that  
\[F^{\dagger}(x)\coloneqq\sup_{(g,\mu)\in\mathcal{A}}\mathbb{P}_{\Xi\sim\mu}[g( \Xi)\leq x] \tag{2}\]  
is the supremum of a family of such CDFs over some class \(\mathcal{A}\). We assume that we have access to a numerical optimisation routine that can, for each \(x\) and some given numerical parameters \(q\) (e.g., the number of iterations or other convergence tolerance parameters), produce a _numerical estimate_ or _observation_\(G(x,q)\) of \(F^{\dagger}(x)\); furthermore, we assume that \(G(x,q)\leq F^{\dagger}(x)\) is always true, i.e.,the numerical optimisation routine always under-estimates the true optimum value, and that the positive error \(F^{\dagger}(x)-G(x,q)\) can be controlled to some extent through the choice of the optimisation parameters \(q\), but remains essentially influenced by randomness in the optimisation algorithm for each \(x\). The assumption \(G(x,q)\leq F^{\dagger}(x)\) is for example coherent with either Equation (1), which may be approached by increasing the number of samples (say \(q\)) in a Monte Carlo simulation, or Equation (2), which is a supremum over a set that may be explored only partially by the algorithm. A single observation \(G(x,q)\) yields some limited information about \(F^{\dagger}(x)\); a key limitation is that one may not even know a priori how accurate \(G(x,q)\) is. Naturally, one may repeatedly evaluate \(G\) at \(x\), perhaps with different values of the optimisation parameters \(q\), in order to more accurately estimate \(F^{\dagger}(x)\). However, a key observation is that a _suite_ of observations \(G(x_{i},q_{i})\), \(i=1,\ldots,I\), contains much more information than simply estimates of \(F^{\dagger}(x_{i})\), \(i=1,\ldots,I\), and this information can and must be used. For example, suppose that the values \((G(x_{i},q_{i}))_{i=1}^{I}\) are not increasing, e.g., because  
\[G(x_{i},q_{i})>G(x_{i^{\prime}},q_{i^{\prime}})\quad\text{and}\quad x_{i}<x_{ i^{\prime}}.\]  
Such a suite of observations would be inconsistent with the axiomatic requirement that \(F^{\dagger}\) is an increasing function. In particular, while the observation at \(x_{i}\) may be relatively good or bad on its own merits, the observation \(G(x_{i^{\prime}},q_{i^{\prime}})\) at \(x_{i^{\prime}}\), which violates monotonicity, is in some sense \"useless\" as it gives no better lower bound on \(F^{\dagger}(x_{i^{\prime}})\) than the observation at \(x_{i}\) does. The observation at \(x_{i^{\prime}}\) is thus a good candidate for repetition with more stringent optimisation parameters \(q\)--and this is not something that could have been known without comparing it to the rest of the data set. The purpose of this article is to leverage this and similar observations to define an algorithm for the reconstruction of the function \(F^{\dagger}\), repeating old observations of insufficient quality and introducing new ones as necessary. The principal parameter in the algorithm is an \"exchange rate\" \(\mathcal{E}\) that quantifies the degree to which the algorithm prefers to have a few high-quality evaluations versus many poor-quality evaluations.