Isoperimetric Inequalities for Real-Valued Functions with Applications to Monotonicity Testing

We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra (SICOMP 2018) for Boolean functions to the case of real-valued functions f:{0,1}^d ? ?. Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real-valued function f over an arbitrary partially ordered domain as a collection of Boolean functions over the same domain, roughly capturing the distance of f to monotonicity and the structure of violations of f to monotonicity. We apply our generalized isoperimetric inequality to improve algorithms for testing monotonicity and approximating the distance to monotonicity for real-valued functions. Our tester for monotonicity has query complexity O?(min(r ?d,d)), where r is the size of the image of the input function. (The best previously known tester makes O(d) queries, as shown by Chakrabarty and Seshadhri (STOC 2013).) Our tester is nonadaptive and has 1-sided error. We prove a matching lower bound for nonadaptive, 1-sided error testers for monotonicity. We also show that the distance to monotonicity of real-valued functions that are ?-far from monotone can be approximated nonadaptively within a factor of O(?{d log d}) with query complexity polynomial in 1/? and the dimension d. This query complexity is known to be nearly optimal for nonadaptive algorithms even for the special case of Boolean functions. (The best previously known distance approximation algorithm for real-valued functions, by Fattal and Ron (TALG 2010) achieves O(d log r)-approximation.

Sublinear-Time Computation in the Presence of Online Erasures

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase $t$ input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm's access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant $t$ with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of $t$, showing that the query complexity is $\Theta(\log t)$. In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for $t=1$. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties. We also consider implications of our results for algorithms that are resilient to online adversarial corruptions instead of erasures

Differentially Private Conditional Independence Testing

Conditional independence (CI) tests are widely used in statistical data analysis, e.g., they are the building block of many algorithms for causal graph discovery. The goal of a CI test is to accept or reject the null hypothesis that $X \perp \!\!\! \perp Y \mid Z$, where $X \in \mathbb{R}, Y \in \mathbb{R}, Z \in \mathbb{R}^d$. In this work, we investigate conditional independence testing under the constraint of differential privacy. We design two private CI testing procedures: one based on the generalized covariance measure of Shah and Peters (2020) and another based on the conditional randomization test of Cand\`es et al. (2016) (under the model-X assumption). We provide theoretical guarantees on the performance of our tests and validate them empirically. These are the first private CI tests with rigorous theoretical guarantees that work for the general case when $Z$ is continuous

Counting Distinct Elements in the Turnstile Model with Differential Privacy under Continual Observation

Privacy is a central challenge for systems that learn from sensitive data sets, especially when a system's outputs must be continuously updated to reflect changing data. We consider the achievable error for differentially private continual release of a basic statistic -- the number of distinct items -- in a stream where items may be both inserted and deleted (the turnstile model). With only insertions, existing algorithms have additive error just polylogarithmic in the length of the stream $T$. We uncover a much richer landscape in the turnstile model, even without considering memory restrictions. We show that every differentially private mechanism that handles insertions and deletions has worst-case additive error at least $T^{1/4}$ even under a relatively weak, event-level privacy definition. Then, we identify a parameter of the input stream, its maximum flippancy, that is low for natural data streams and for which we give tight parameterized error guarantees. Specifically, the maximum flippancy is the largest number of times that the contribution of a single item to the distinct elements count changes over the course of the stream. We present an item-level differentially private mechanism that, for all turnstile streams with maximum flippancy $w$, continually outputs the number of distinct elements with an $O(\sqrt{w} \cdot poly\log T)$ additive error, without requiring prior knowledge of $w$. We prove that this is the best achievable error bound that depends only on $w$, for a large range of values of $w$. When $w$ is small, the error of our mechanism is similar to the polylogarithmic in $T$ error in the insertion-only setting, bypassing the hardness in the turnstile model

Feedback Vertex Sets and Cycle Packings in Subcubic Planar Graphs

For a graph $G$, let $\tau(G)$ denote the minimum size of a set of vertices intersecting every cycle in $G$. Let \vu(G) denote the maximum size of a collection of vertex-disjoint cycles of $G$. Erd\"{o}s and P\'{o}sa~\cite{erdos65} showed that \tau(G) = O(\vu \log \vu(G)) for general graphs, and that the bound is tight. Kloks et al.~\cite{kloks} showed that for planar graphs \tau(G) \leq 5\vu(G) and conjectured that \tau(G) \leq 2\vu(G) for any planar graph $G$. The coefficient 5 has been improved to 3 independently in ~\cite{ma, chappell,chen}. However, the conjecture remains open even for subcubic graphs, which are graphs with maximum degree at most 3. We show that for any planar subcubic graph $G$, \tau(G) \leq \frac{5}{2}\vu(G). We also study the connectivity and girth of a vertex-minimal counterexample to the conjecture of Kloks et al. for subcubic graphs. In the end we present a list of reducible configurations, which are graphs $H$, such that if $G$ is a vertex-minimal counterexample to the conjecture of Kloks et al. for planar subcubic graphs, then $G$ cannot contain $H$ as a subgraph

Sublinear-Time Computation in the Presence of Online Erasures

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase $t$ input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm's access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant $t$ with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of $t$, showing that the query complexity is $\Theta(\log t)$. In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for $t=1$. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties. We also consider implications of our results for algorithms that are resilient to online adversarial corruptions instead of erasures