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We extend the framework of augmented distribution testing (Aliakbarpour, Indyk, Rubinfeld, and Silwal, NeurIPS 2024) to the differentially private setting. This captures scenarios where a data ana- lyst must perform hypothesis testing tasks on sensitive data, but is able to leverage prior knowledge (public, but possibly erroneous or untrusted) about the data distribution. We design private algorithms in this augmented setting for three flagship distribution testing tasks, uniformity, identity, and closeness testing, whose sample complexity smoothly scales with the claimed quality of the auxiliary information. We complement our algorithms with information- theoretic lower bounds, showing that their sample complexity is optimal (up to logarithmic factors). Keywords: distribution testing, identity testing, closeness testing, differential privacy, learning- augmented algorithms 

Consider the following stochastic matching problem. We are given a known graph 𝐺 = (𝑉 , 𝐸). An unknown subgraph 𝐺𝑝 = (𝑉 , 𝐸𝑝 ) is realized where 𝐸𝑝 includes every edge of 𝐸 independently with some probability 𝑝 ∈ (0, 1]. The goal is to query a sparse subgraph 𝐻 of 𝐺, such that the realized edges in 𝐻 include an approximate maximum matching of 𝐺𝑝 . This problem has been studied extensively over the last decade due to its applications in kidney exchange, online dating, and online labor markets. For any fixed 𝜀 > 0, [BDH STOC’20] showed that any graph 𝐺 has a subgraph 𝐻 with quasipoly(1/𝑝) = (1/𝑝)poly(log(1/𝑝 ) ) maximum degree, achieving a (1 − 𝜀)-approximation. A major open question is the best approximation achievable with poly(1/𝑝)- degree subgraphs. A long line of work has progressively improved the approximation in the poly(1/𝑝)-degree regime from .5 [BDH+ EC’15] to .501 [AKL EC’17], .656 [BHFR SODA’19], .666 [AB SOSA’19], .731 [BBD SODA’22] (bipartite graphs), and most recently to .68 [DS ’24]. In this work, we show that a poly(1/𝑝)-degree subgraph can obtain a (1 − 𝜀)-approximation for any desirably small fixed 𝜀 > 0, achieving the best of both worlds. Beyond its quantitative improvement, a key conceptual contribu- tion of our work is to connect local computation algorithms (LCAs) to the stochastic matching problem for the first time. While prior work on LCAs mainly focuses on their out-queries (the number of vertices probed to produce the output of a given vertex), our analysis also bounds the in-queries (the number of vertices that probe a given vertex). We prove that the outputs of LCAs with bounded in- and out-queries (in-n-out LCAs for short) have limited correlation, a property that our analysis crucially relies on and might find applications beyond stochastic matchings. 

Counting small subgraphs, referred to as motifs, in large graphs is a fundamental task in graph analysis, extensively studied across various contexts and computational models. In the sublinear-time regime, the relaxed problem of approximate counting has been explored within two prominent query frameworks: the standard model, which permits degree, neighbor, and pair queries, and the strictly more powerful augmented model, which additionally allows for uniform edge sampling. Currently, in the standard model, (opti- mal) results have been established only for approximately counting edges, stars, and cliques, all of which have a radius of one. This contrasts sharply with the state of affairs in the augmented model, where algorithmic results (some of which are optimal) are known for any input motif, leading to a disparity which we term the “scope gap" between the two models. In this work, we make significant progress in bridging this gap. Our approach draws inspiration from recent advancements in the augmented model and utilizes a framework centered on counting by uniform sampling, thus allowing us to establish new results in the standard model and simplify on previous results. In particular, our first, and main, contribution is a new algorithm in the standard model for approximately counting any Hamiltonian motif in sublinear time, where the complexity of the algorithm is the sum of two terms. One term equals the complexity of the known algorithms by Assadi, Kapralov, and Khanna (ITCS 2019) and Fichtenberger and Peng (ICALP 2020) in the (strictly stronger) augmented model and the other is an additional, necessary, additive overhead. Our second contribution is a variant of our algorithm that en- ables nearly uniform sampling of these motifs, a capability pre- viously limited in the standard model to edges and cliques. Our third contribution is to introduce even simpler algorithms for stars and cliques by exploiting their radius-one property. As a result, we simplify all previously known algorithms in the standard model for stars (Gonen, Ron, Shavitt (SODA 2010)), triangles (Eden, Levi, Ron Seshadhri (FOCS 2015)) and cliques (Eden, Ron, Seshadri (STOC 2018)). 

We extend the framework of augmented distribution testing (Aliakbarpour, Indyk, Rubinfeld, and Silwal, NeurIPS 2024) to the differentially private setting. This captures scenarios where a data analyst must perform hypothesis testing tasks on sensitive data, but is able to leverage prior knowledge (public, but possibly erroneous or untrusted) about the data distribution. We design private algorithms in this augmented setting for three flagship distribution testing tasks, uniformity, identity, and closeness testing, whose sample complexity smoothly scales with the claimed quality of the auxiliary information. We complement our algorithms with information-theoretic lower bounds, showing that their sample complexity is optimal (up to logarithmic factors). 

The online list update problem is defined as follows: we are given a list of items and the cost to access any particular item is its position from the start of the list. A sequence of item accesses come online, and our goal is to dynamically reorder the list so that the aggregate access cost is small. We study the stochastic version of the problem where the items are accessed i.i.d. from an unknown distribution p. The study of the stochastic version goes back at least 60 years to McCabe. In this paper, we first consider the simple online algorithm which swaps an accessed item with the item right before it, unless it is at the very front. This algorithm is known as the Transposition rule. Wetheoretically analyze the stationary behavior of Transposition and prove that its performance is within 1 + o(1) factor of the optimal offline algorithm for access sequences sampled from heavy-tailed distributions, proving a conjecture of Rivest from 1976. While the stationary behavior of the Transposition rule is theoretically optimal in the aforemen tioned i.i.d setting, it can catastrophically fail under adversarial access sequences where only the last and second to last items are repeatedly accessed. A desirable outcome would be a policy that performs well under both circumstances. To achieve this, we use reinforcement learning to design an adaptive policy that performs well for both the i.i.d. setting and the above-mentioned adversarial access. Unsurprisingly, the learned policy appears to be an interpolation between Move-to-Front and Transposition with its behavior closer to Move-to-Front for adversarial access sequences and closer to Transposition for sequences sampled from heavy tailed distributions suggesting that the policy is adaptive and capable of responding to patterns in the access sequence. 

The online list update problem is defined as follows: we are given a list of items and the cost to access any particular item is its position from the start of the list. A sequence of item accesses come online, and our goal is to dynamically reorder the list so that the aggregate access cost is small. We study the stochastic version of the problem where the items are accessed i.i.d. from an unknown distribution p. The study of the stochastic version goes back at least 60 years to McCabe. In this paper, we first consider the simple online algorithm which swaps an accessed item with the item right before it, unless it is at the very front. This algorithm is known as the Transposition rule. We theoretically analyze the stationary behavior of Transposition and prove that its performance is within 1+o(1) factor of the optimal offline algorithm for access sequences sampled from heavy-tailed distributions, proving a conjecture of Rivest from 1976. While the stationary behavior of the Transposition rule is theoretically optimal in the aforementioned i.i.d setting, it can catastrophically fail under adversarial access sequences where only the last and second to last items are repeatedly accessed. A desirable outcome would be a policy that performs well under both circumstances. To achieve this, we use reinforcement learning to design an adaptive policy that performs well for both the i.i.d. setting and the above-mentioned adversarial access. Unsurprisingly, the learned policy appears to be an interpolation between Move-to-Front and Transposition with its behavior closer to Move-to-Front for adversarial access sequences and closer to Transposition for sequences sampled from heavy tailed distributions suggesting that the policy is adaptive and capable of responding to patterns in the access sequence. 

We consider the problem of hypothesis testing for discrete distributions. In the standard model, where we have sample access to an underlying distribution p, extensive research has established optimal bounds for uniformity testing, identity testing (goodness of fit), and closeness testing (equivalence or two-sample testing). We explore these problems in a setting where a predicted data distribution, possibly derived from historical data or predictive machine learning models, is available. We demonstrate that such a predictor can indeed reduce the number of samples required for all three property testing tasks. The reduction in sample complexity depends directly on the predictor’s quality, measured by its total variation distance from p. A key advantage of our algorithms is their adaptability to the precision of the prediction. Specifically, our algorithms can self-adjust their sample complexity based on the accuracy of the available prediction, operating without any prior knowledge of the estimation’s accuracy (i.e. they are consistent). Additionally, we never use more samples than the standard approaches require, even if the predictions provide no meaningful information (i.e. they are also robust). We provide lower bounds to indicate that the improvements in sample complexity achieved by our algorithms are information-theoretically optimal. Furthermore, experimental results show that the performance of our algorithms on real data significantly exceeds our worst-case guarantees for sample complexity, demonstrating the practicality of our approach. 

We consider the question of orienting the edges in a graph G such that every vertex has bounded out-degree. For graphs of arboricity α, there is an orientation in which every vertex has out-degree at most α and, moreover, the best possible maximum out-degree of an orientation is at least α - 1. We are thus interested in algorithms that can achieve a maximum out-degree of close to α. A widely studied approach for this problem in the distributed algorithms setting is a "peeling algorithm" that provides an orientation with maximum out-degree α(2+ε) in a logarithmic number of iterations. We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form "What is the orientation of edge (u,v)?" by probing the input graph. When the peeling algorithm is executed in the LCA setting by applying standard techniques, e.g., the Parnas-Ron paradigm, it requires Ω(n) probes per query on an n-vertex graph. In the case where G has unbounded degree, we show that any LCA that orients its edges to yield maximum out-degree r must use Ω(√ n/r) probes to G per query in the worst case, even if G is known to be a forest (that is, α = 1). We also show several algorithms with sublinear probe complexity when G has unbounded degree. When G is a tree such that the maximum degree Δ of G is bounded, we demonstrate an algorithm that uses Δ n^{1-log_Δ r + o(1)} probes to G per query. To obtain this result, we develop an edge-coloring approach that ultimately yields a graph-shattering-like result. We also use this shattering-like approach to demonstrate an LCA which 4-colors any tree using sublinear probes per query. 

Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results. For (1+ε)-approximation of job scheduling of n jobs on a single machine, we develop a fully dynamic algorithm with O(lognε) update and O(logn) query worst-case time. Further, we design a local computation algorithm that uses only O(logNε) queries when all jobs are length at least 1 and have starting/ending times within [0,N]. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we design a fully dynamic deterministic algorithm whose worst-case update and query time are poly(logn,1ε). Equivalently, this is the first algorithm that maintains a (1+ε)-approximation of the maximum independent set of a collection of weighted intervals in poly(logn,1ε) time updates/queries. This is an exponential improvement in 1/ε over the running time of a randomized algorithm of Henzinger, Neumann, and Wiese ~[SoCG, 2020], while also removing all dependence on the values of the jobs' starting/ending times and rewards, as well as removing the need for any randomness. We also extend our approaches for interval scheduling on a single machine to examine the setting with M machines. 

There are many important high dimensional function classes that have fast agnostic learning algorithms when strong assumptions on the distribution of examples can be made, such as Gaussianity or uniformity over the domain. But how can one be sufficiently confident that the data indeed satisfies the distributional assumption, so that one can trust in the output quality of the agnostic learning algorithm? We propose a model by which to systematically study the design of tester-learner pairs (A,T), such that if the distribution on examples in the data passes the tester T then one can safely trust the output of the agnostic learner A on the data. To demonstrate the power of the model, we apply it to the classical problem of agnostically learning halfspaces under the standard Gaussian distribution and present a tester-learner pair with a combined run-time of nÕ(1/є4). This qualitatively matches that of the best known ordinary agnostic learning algorithms for this task. In contrast, finite sample Gaussian distribution testers do not exist for the L1 and EMD distance measures. Previously it was known that half-spaces are well-approximated with low-degree polynomials relative to the Gaussian distribution. A key step in our analysis is showing that this is the case even relative to distributions whose low-degree moments approximately match those of a Gaussian. We also go beyond spherically-symmetric distributions, and give a tester-learner pair for halfspaces under the uniform distribution on {0,1}n with combined run-time of nÕ(1/є4). This is achieved using polynomial approximation theory and critical index machinery of [Diakonikolas, Gopalan, Jaiswal, Servedio, and Viola 2009]. Can one design agnostic learning algorithms under distributional assumptions and count on future technical work to produce, as a matter of course, tester-learner pairs with similar run-time? Our answer is a resounding no, as we show there exist some well-studied settings for which 2Õ(√n) run-time agnostic learning algorithms are available, yet the combined run-times of tester-learner pairs must be as high as 2Ω(n). On that account, the design of tester-learner pairs is a research direction in its own right independent of standard agnostic learning. To be specific, our lower bounds apply to the problems of agnostically learning convex sets under the Gaussian distribution and for monotone Boolean functions under the uniform distribution over {0,1}n.