The notion of replicable algorithms was introduced by Impagliazzo, Lei, Pitassi, and Sorrell (STOC’22) to describe randomized algorithms that are stable under the resampling of their inputs. More precisely, a replicable algorithm gives the same output with high probability when its randomness is fixed and it is run on a new i.i.d. sample drawn from the same distribution. Using replicable algorithms for data analysis can facilitate the verification of published results by ensuring that the results of an analysis will be the same with high probability, even when that analysis is performed on a new data set.

We introduce the notion of a reproducible algorithm in the context of learning. A reproducible learning algorithm is resilient to variations in its samples — with high probability, it returns the exact same output when run on two samples from the same underlying distribution. We begin by unpacking the definition, clarifying how randomness is instrumental in balancing accuracy and reproducibility. We initiate a theory of reproducible algorithms, showing how reproducibility implies desirable properties such as data reuse and efficient testability. Despite the exceedingly strong demand of reproducibility, there are efficient reproducible algorithms for several fundamental problems in statistics and learning. First, we show that any statistical query algorithm can be made reproducible with a modest increase in sample complexity, and we use this to construct reproducible algorithms for finding approximate heavy-hitters and medians. Using these ideas, we give the first reproducible algorithm for learning halfspaces via a reproducible weak learner and a reproducible boosting algorithm. Interestingly, we utilize a connection to foams as a higher-dimension randomized rounding scheme. Finally, we initiate the study of lower bounds and inherent tradeoffs for reproducible algorithms, giving nearly tight sample complexity upper and lower bounds for reproducible versus nonreproducible SQ algorithms.

We study the problem of boosting the accuracy of a weak learner in the (distribution-independent) PAC model with Massart noise. In the Massart noise model, the label of each example $x$ is independently misclassified with probability $\eta(x) \leq \eta$, where $\eta<1/2$. The Massart model lies between the random classification noise model and the agnostic model. Our main positive result is the first computationally efficient boosting algorithm in the presence of Massart noise that achieves misclassification error arbitrarily close to $\eta$. Prior to our work, no non-trivial booster was known in this setting. Moreover, we show that this error upper bound is best possible for polynomial-time black-box boosters, under standard cryptographic assumptions. Our upper and lower bounds characterize the complexity of boosting in the distribution-independent PAC model with Massart noise. As a simple application of our positive result, we give the first efficient Massart learner for unions of high-dimensional rectangles.