Online Algorithms with Limited Data Retention

TL;DR

This work introduces a prescriptive framework for online algorithms constrained by limited data retention, where each data point must be removed after at most $m$ rounds and the algorithm can store only a subset of recent data. It develops SGD-inspired subsampling methods for mean estimation and linear regression that achieve near-optimal accuracy with memory $m$ that scales poly(d, log(d/\epsilon)) or similar, dramatically improving over naive baselines that retain more data. The analysis leverages multidimensional random subset sum results to simulate SGD under adversarial noise and provides matching lower bounds, illustrating fundamental tradeoffs between retention length, dimensionality, and statistical accuracy. The findings have implications for data-deletion policies, privacy, and the design of retention-aware learning systems, and open avenues for extending the framework to other tasks and non-stationary environments.

Abstract

We introduce a model of online algorithms subject to strict constraints on data retention. An online learning algorithm encounters a stream of data points, one per round, generated by some stationary process. Crucially, each data point can request that it be removed from memory $m$ rounds after it arrives. To model the impact of removal, we do not allow the algorithm to store any information or calculations between rounds other than a subset of the data points (subject to the retention constraints). At the conclusion of the stream, the algorithm answers a statistical query about the full dataset. We ask: what level of performance can be guaranteed as a function of $m$? We illustrate this framework for multidimensional mean estimation and linear regression problems. We show it is possible to obtain an exponential improvement over a baseline algorithm that retains all data as long as possible. Specifically, we show that $m = \textsc{Poly}(d, \log(1/ε))$ retention suffices to achieve mean squared error $ε$ after observing $O(1/ε)$ $d$-dimensional data points. This matches the error bound of the optimal, yet infeasible, algorithm that retains all data forever. We also show a nearly matching lower bound on the retention required to guarantee error $ε$. One implication of our results is that data retention laws are insufficient to guarantee the right to be forgotten even in a non-adversarial world in which firms merely strive to (approximately) optimize the performance of their algorithms. Our approach makes use of recent developments in the multidimensional random subset sum problem to simulate the progression of stochastic gradient descent under a model of adversarial noise, which may be of independent interest.

Theorems & Definitions (26)

• Proposition 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3: Based on Corollary 24 of becchetti2022multidimensional
• Proposition 3.4
• Theorem 3.5: Theorem 2.4 of Lueker:1998
• Theorem 3.6
• Theorem 4.1
• Lemma 4.2
• Proposition 1: \ref{['prop.batch']}