Reuse of data in adaptive workflows poses challenges regarding overfitting and the statistical validity of results. Previous work has demonstrated that interacting with data via differentially private algorithms can mitigate overfitting, achieving worstcase generalization guarantees with asymptotically optimal data requirements. However, such past work assumes data is static and cannot accommodate situations where data grows over time. In this paper we address this gap, presenting the first generalization bounds for adaptive analysis on dynamic data. We allow the analyst to adaptively schedule their queries conditioned on the current size of the data, in addition to previous queries and responses. We also incorporate time-varying empirical accuracy bounds and mechanisms, allowing for tighter guarantees as data accumulates. In a batched query setting, the asymptotic data requirements of our bound grows with the square-root of the number of adaptive queries, matching prior works' improvement over data splitting for the static setting. We instantiate our bound for statistical queries with the clipped Gaussian mechanism, where it empirically outperforms baselines composed from static bounds.

We study an abstract framework for interactive learning called interactive estimation in which the goal is to estimate a target from its "similarity" to points queried by the learner. We introduce a combinatorial measure called dissimilarity dimension which is used to derive learnability bounds in our model. We present a simple, general, and broadly-applicable algorithm, for which we obtain both regret and PAC generalization bounds that are polynomial in the new dimension. We show that our framework subsumes and thereby unifies two classic learning models: statistical-query learning and structured bandits. We also delineate how the dissimilarity dimension is related to well-known parameters for both frameworks, in some cases yielding significantly improved analyses.

The $k$-sparse parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the $k$-sparse parity problem with sign stochastic gradient descent, a variant of stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that this approach can efficiently solve the $k$-sparse parity problem on a $d$-dimensional hypercube ($k\le O(\sqrt{d})$) with a sample complexity of $\tilde{O}(d^{k-1})$ using $2^{\Theta(k)}$ neurons, matching the established $\Omega(d^{k})$ lower bounds of Statistical Query (SQ) models. Our theoretical analysis begins by constructing a good neural network capable of correctly solving the $k$-parity problem. We then demonstrate how a trained neural network with sign SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. To the best of our knowledge, this is the first result that matches the SQ lower bound for solving $k$-sparse parity problem using gradient-based methods.

The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$--potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.

Neural Information Processing Systems http://nips.cc/

A recent line of work initiated by Dworket al. [9] posed the question: Can we designgeneralpurpose algorithms for ensuring generalization in the presence of post-selection?