**Distribution Testing Under the Parity Trace** by Renato Ferreira Pinto Jr and Nathaniel Harms (arXiv) The featured paper considers the classic setup in distribution testing with *a twist*. To explain the results, let me introduce the framework considered in this work. Consider distributions supported over \([n]\). Suppose I want to design distribution testers where instead of obtaining samples in the usual way, I first obtain an ordered list of samples, I store it in a sequence \(S\) and only the least significant bit of each element of \(S\) is made available to your distribution testing algorithm. This is called a parity trace. Note that with this access model, suddenly a bunch of standard tasks become non-trivial. To take an example from the paper, you can no longer distinguish between the uniform distribution supported on \(\{1,2, \ldots, n\}\) and the uniform distribution supported on \(\{n+1, n+2, \ldots 2n\}\) in this access model. Nevertheless, the paper shows, you can still obtain testers which require only sublinear number of accesses for testing uniformity in this model.

Another cool feature of this *big* paper is an unexpected foray into the trace reconstruction literature from a property testing viewpoint. I wish I understood the formal connection better to describe a bit more about it. But for now, let me leave that as an appetizer which (hopefully) encourages you to take a look at the paper.

**New Lower Bounds for Adaptive Tolerant Junta Testing** by Xi Chen and Shyamal Patel (arXiv) If you are a regular here on the PTReview corner, you are probably no stranger to the *tolerant junta testing* problem. As a corollary to the main result, the paper in question proves a lower bound of \(k^{\Omega(\log k)}\) queries on any adaptive algorithm that wishes to test whether the input function \(f\) is \(\varepsilon_1\) close to being a \(k\)-junta or whether it is \(\varepsilon_2\)-far \(\left(\text{where } \varepsilon_2 \geq \varepsilon_1 + \displaystyle\frac{1}{poly(k)}\right)\). Indeed, another remarkable achievement of the paper is that it achieves a superpolynomial separation between non-tolerant versions and the tolerant versions of any natural property of boolean functions under the adaptive setting.

**Near-Optimal Degree Testing for Bayes Nets** by Vipul Arora, Arnab Bhattacharyya, Clément L. Canonne (our own!) and Joy Qiping Yang (arXiv) This paper continues a line of investigation which a subset of the authors were a part of (which we also covered in our News for April 2022). Let us remind ourselves of the setup. You are given a probability distribution \(\mathcal{P}\) supported over the Boolean Hypercube. Suppose \(\mathcal{P}\) can be generated by a Bayseian Network. You may think of a Bayesian Network as a DAG where each vertex tosses a coin (with different heads probabilities). The question seeks to test whether \(\mathcal{P}\) admits a sparse Bayesian Net (in the sense of each vertex having small in-degree). The main result of the paper gives an algorithm for this task which requires \(\Theta(2^{n/2}/\varepsilon^2)\) samples. The paper also proves an almost matching lower bound.

**A \(d^{1/2+o(1)}\) Monotonicity Tester for Boolean Functions on \(d\)-Dimensional Hypergrids** by Hadley Black, Deeparnab Chakrabarty and C. Seshadhri (again, our own!) (arXiv) Again, the problem of monotonicity testing of boolean functions hardly requires any detailing to the regular readers of PTReview. As you can see in our News from November 2022 there were two concurrent papers mulling over this problem over the hypergrid domain. The featured paper is the result of a dedicated pursuit by the authors and the key result is what the title says. Namely, you can test monotonicity with a number of (non-adaptive, one-sided) queries that has no dependence on \(n\).

For the first time in PTReview history, there is no paper to report. Nada. Zilch.

The calm before the storm…?

]]>**Dynamic \((1 + \epsilon)\)-Approximate Matching Size in Truly Sublinear Update Time** by Sayan Bhattacharya, Peter Kiss, and Thatchaphol Saranurak (arXiv). This work throws light on connections between the dynamic and query models of computation and uses them for making advances on approximating the size of a maximum cardinality matching (MCM) in a general graph. In particular, as the main technical ingredient in obtaining an improved dynamic algorithm for maintaining an approximation to the size of MCM, the authors provide a \(\pm \epsilon n\) approximation algorithm for estimating the size of MCM in a general \(n\)-vertex graph by making \(n^{2 – \Omega_{\epsilon}(1)}\) adjacency queries. Prior to this result, the state of the art (Behnezhad, Roghani & Rubinstein; STOC’23) was a \(n^{2 – \Omega(1)}\)-query algorithm for the same problem with a multiplicative approximation guarantee of \(1.5\) and an additive guarantee of \(o(n)\).

**Uniformity Testing over Hypergrids with Subcube Conditioning** by Xi Chen and Cassandra Marcussen (arXiv). As the name indicates, the paper studies the fundamental problem of testing uniformity of distributions supported over hypergrids \([m]^n\). The tester that they present make \(O(\text{poly}(m)\sqrt{n}/\epsilon^2)\) queries to a conditional subcube sampling oracle, which, when given a subcube of \([m]^n\), returns a point sampled from the distribution conditioned on the point belonging to the subcube. The result is a generalization of the uniformity tester for distributions supported over the \(n\)-dimensional hypercube (Canonne, Chen, Kamath, Levi and Waingarten; SODA ’21).

**Easy Testability for Posets** by Panna Timea Fekete and Gabor Kun (arXiv). This paper deals with testing properties of directed graphs in the adjacency matrix model. The main characters of the story are posets, or directed acyclic graphs (DAGs) that are transitively closed. Given a family \(\mathcal{F}\) of finite posets, let \(\mathcal{P}_\mathcal{F}\) denote the set of all finite posets that do not contain any element of \(\mathcal{F}\) as a subposet. The main result of the paper is an \(\epsilon\)-tester with query complexity \(\text{poly}(1/\epsilon)\) for \(\mathcal{P}_\mathcal{F}\). The authors obtain this result by proving a removal lemma for posets. The result is placed in the larger context of understanding what properties of graphs can be tested with query complexity that has a polynomial dependence on \(1/\epsilon\) in the adjacency matrix model.

**Compressibility-Aware Quantum Algorithms** on Strings by Daniel Gibney and Sharma V. Thankachan (arXiv). Lastly, we have a paper on quantum string algorithms that run in sublinear time. In short, the authors present quantum algorithms with optimal running times for computing the Lempel-Ziv (LZ) encoding and Burrows Wheeler Transform (BWT) of highly compressible strings. A main consequence of these results is a faster quantum algorithm for computing the longest common subsequence (LCS) of two strings when the concatenation of the strings is highly compressible. It is to be noted that sublinear-time algorithms do not exist for these problems in the classical model of computation. More details follow.

Factoring a string into disjoint substrings (factors) in an specific manner is the main step in the LZ compression algorithm. The smaller the number of factors, the more compressible the string is. This paper gives a quantum algorithm for the problem of computing the LZ factorization of a string in time \(\tilde{O}(\sqrt{nz})\), where \(z\) is the number of factors in the string. They also show that their algorithm is optimal. Using this algorithm, they obtain a fast algorithm for computing the BWT of an input string, as well as an algorithm running in time \(\tilde{O}(\sqrt{nz})\) to compute the LCS of two strings, where \(n\) is the length and \(z\) is the number of factors in the concatenation of the two strings. When \(z\) is \(o(n^{1/3})\), this algorithm gives an improvement over the previous best quantum algorithm running in time \(\tilde{O}(n^{2/3})\).

**An efficient asymmetric removal lemma and its limitations**, by Lior Gishboliner, Asaf Shapira, and Yuval Wigderson (arXiv). One of the jewels of graph property testing is the triangle removel lemma (and its many generalizations and variants), which relates the number of triangles in a dense graph to its distance from being triangle-free: namely, any graph \(\varepsilon\)-far from being triangle-free must have \(\delta(\varepsilon)n^3\) triangles, where the density \(\delta(\varepsilon)\) only depends on the distance (and not the size of the graph!). This immediately leads to constant-query testers (and even “proximity-oblivious” testers) for triangle-freness (and, more generally, pattern-freeness). Unfortunately, the dependence on \(\varepsilon\) is quite bad, essentially a tower-type function (and it is known no polynomial bound is possible). This work attempts to bypass this impossibility result by proving an asymmetric removal lemma, or the form “any graph \(\varepsilon\)-far from being triangle-free must have \(\mathrm{poly}(\varepsilon)n^5\) 5-cycles” (and generalizations beyond triangles). This seems like a very interesting direction, with potential applications to property testing, and (who knows!) efficient testers for many properties hithertho only known to be (practically) testable for constant \(\varepsilon\).

Related (more removal lemmata!), a different work on this topic:

**The Minimum Degree Removal Lemma Thresholds**, by Lior Gishboliner, Zhihan Jin, and Benny Sudakov (arXiv). As mentioned above, removal lemmata relate the distance \(\varepsilon\) from being \(H\)-free (for a given subgraph \(H\)) to the density \(\delta(\varepsilon)\) of occurrences of \(H\) in the graph. Sadly, it is known that this density will be superpolynomial (in the distance) unless \(H\) is bipartite… which, while technically still yielding testing algorithms (query complexity independent of the size of the graph!), yields very inefficient testers (very bad dependence on \(\varepsilon\)!). This paper studies one direction to bypass this sad state of affairs: under which additional assumption on the underlying graph (specifically, bounds on its minimum degree) can we obtain a polynomial bound on \(\delta(\varepsilon)\)? And a linear bound? The authors give a tight degree condition for \(\delta(\varepsilon)\) to be polynomial when \(H\) is an odd cycle, and their results for the linear-dependence case establishes a separation between the two. Put differently: obtaining polynomial-query testers via removal lemmas is possible for a strictly larger class of graphs than linear-query ones!

And now, for something completely different: testing binary matrices!

**A Note on Property Testing of the Binary Rank**, by Nader H. Bshouty (arXiv). The binary rank of a matrix \(M\in\{0,1\}^{n\times m}\) is the smallest \(d\) such that there exist \(A\in\{0,1\}^{n\times d}\) and \(B\in\{0,1\}^{d\times m}\) with \(M=AB\); this can also be seen as the minimal number of bipartite cliques needed to partition the edges of a bipartite graph represented by \(M\). One can also define the relaxed notion of \(s\)-binary rank, if one enforces that each edge of the bipartite graph is covered by at most \(s\) bipartite cliques. The property testing question is then to decide, given inputs \(s,d,\varepsilon\), if \(M\) has \(s\)-binary rank at most \(d\), or is \(\varepsilon\)-far from it. The main result of this note is to give one-sided testers (one adaptive, and one non-adaptive) for \(s\)-binary rank with query complexity \(\tilde{O}(2^d)\) (for constant \(s\)), improving on the previous algorithms by a factor \(2^d\).

Into the quantum realm!

**Testing quantum satisfiability,** by Ashley Montanaro, Changpeng Shao, and Dominic Verdon (arXiv). Classically, one can study the property version of \(k\)-SAT, which asks to decide whether a given instance is satisfiable or far from being so. And people (namely, Alon and Shapira, in 2003) did! Quantumly, one can define an analogue of \(k\)-SAT, “quantum \(k\)-SAT”: and people (namely, Bravyi, in 2011) did! But what about property testing of quantum \(k\)-SAT? Well, now, people (namely, the authors of this paper) just did! Showing (Corollary 1.10) that one can efficiently distinguish between (1) the quantum \(k\)-SAT is satisfiable, and (2) it is far from satisfiable by a product state. This, effectively, extends the result of Alon–Shapira’03 to the quantum realm.

And to conclude, a foray into reinforcement learning via distribution testing…

**Lower Bounds for Learning in Revealing POMDPs**, by Fan Chen, Huan Wang, Caiming Xiong, Song Mei, and Yu Bai (arXiv). Alright, I’m even more out of my depth than usual here, so I’ll just copy (part of) the abstract, for fear I don’t do justice to the authors’ work: “This paper studies the fundamental limits of reinforcement learning (RL) in the challenging *partially observable* setting. While it is well-established that learning in Partially Observable Markov Decision Processes (POMDPs) requires exponentially many samples in the worst case, a surge of recent work shows that polynomial sample complexities are achievable under the *revealing condition* — A natural condition that requires the observables to reveal some information about the unobserved latent states. However, the fundamental limits for learning in revealing POMDPs are much less understood, with existing lower bounds being rather preliminary and having substantial gaps from the current best upper bounds. We establish strong PAC and regret lower bounds for learning in revealing POMDPs. […] **Technically, our hard instance construction adapts techniques in distribution testing, which is new to the RL literature and may be of independent interest.**” (Emphasis mine)

**Testing in the bounded-degree graph model with degree bound two** by Oded Goldreich and Laliv Tauber (ECCC). One of great, central results in graph property testing is that all monotone properties are testable (with query complexity independent on graph size) on dense graphs. The sparse graph universe is far, far, more complicated and interesting. Even for graphs with degree bound 3, natural graph properties can have anywhere from constant to linear (in \(n\)) query complexity. This note shows that when considering graphs with degree bound at most 2, the landscape is quite plain. The paper shows that all properties are testable in \(poly(\varepsilon^{-1})\). Any graph with degree at most 2 is a collection of paths and cycles. In \(poly(\varepsilon^{-1})\) queries, one can approximately learn the graph. (After which the testing problem is trivial.) The paper gives a simple \(O(\varepsilon^{-4})\) query algorithm, which is improved to the nearly optimal \(\widetilde{O}(\varepsilon^{-2})\) bound.

**On the power of nonstandard quantum oracles** by Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha (arXiv). This paper is on the power of oracles in quantum computation. An important question in quantum complexity theory is whether \(QCMA\) is a strict subset of \(QMA\). The former consists of languages decided by Merlin-Arthur quantum protocols with a classical witness (the string that Merlin provides). The latter class allows Merlin to be a quantum witness. This paper shows a property testing problem where such a separation is shown. The property is essentially graph non-expansion (does there exist a set of low conductance?). The input graph should be thought of as an even (bounded) degree with “exponentially many” vertices. So it has \(N = 2^n\) vertices. The graph is represented through a special “graph-coded” function. The paper shows that there is a \(poly(n)\)-sized quantum witness for non-expansion that can be verified in \(poly(n)\) time, which includes queries to the graph-coded function. On the other hand, there is no classic \(poly(n)\)-sized witness that can be verified in \(poly(n)\) queries to the graph-coded function. (Informally speaking, any \(QCMA\) protocol needs exponentially many queries to the graph.)

Our own Clément Canonne has written a beautiful survey which is now available in FnT book format from now publishers. This appears to be a very promising read — especially for the Distribution Testers among you. Today’s post is a mere advertisement for this beautiful survey/book which is clearly the result of a dedicated pursuit.

Let me now dig into this survey a teeny tiny bit. One among the many cool features of this survey is that it uses one central example (testing goodness-of-fit) to give a unified treatment to the diverse tools and techniques used in distribution testing. Another plus for me is the historical notes section that accompanies every chapter. In particular, I really liked jumping into the informative history section at the end of Chapter 2 which has an almost story like feel to it. If the above points do not catch your fancy, then please try opening the survey. You will be hardpressed to find a book that is typeset in such an aesthetically pleasing way with colored fonts to emphasize various parameters in several intricate proofs. Happy Reading!

]]>**Sublinear Time Algorithms and Complexity of Approximate Maximum Matching** by Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein (arXiv) This paper makes significantly advances our understanding of the maximum matching problem in the sublinear regime. Your goal is to estimate the size of the maximum matching and you may assume that you have query access to the adjacency list of your graph. Our posts from Dec 2021 and June 2022 reported some impressive progress on this problem. The upshot from these works essentially said that you can beat greedy matching and obtain a \(\frac{1}{2} + \Omega(1)\) approximate maximum matching in sublinear time. Let me first go over the algorithmic results from the current paper. The paper shows the following two algorithmic results:

(1) An algorithm that runs in time \(n^{2 – \Omega_{\varepsilon}(1)}\) and returns a \(2/3 – \varepsilon\) approximation to maximum matching in general graphs, and

(2) An algorithm that runs in time \(n^{2 – \Omega_{\varepsilon}(1)}\) and returns a \(2/3 + \varepsilon\) approximation to maximum matching size in *bipartite* graphs.

The question remained — can we show a lower bound that grows superlinearly with \(n\). The current work achieves this and shows that *even on bipartite graphs*, you must make at least \(n^{1.2 – o(1)}\) *queries *to the adjacency list to get a better than \(2/3 + \Omega(1)\) approximation. (An aside: A concurrent work by Bhattacharya-Kiss-Saranurak from December also obtains similar algorithmic results for approximating the maximum matching size in general graphs).

**Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an \(\widetilde{O}(n \sqrt d)\) Monotonicity Tester** by Hadley Black, Deeparnab Chakrabarty, C. Seshadhri (arXiv) Boolean Monotonicity testing is as classic as classic gets in property testing. Encouraged by the success of isoperimetric theorems over the hypercube domain and the monotonicity testers powered by these isoperimetries (over the hypercube), one may wish to obtain efficient monotonicity testers for the hypergrid \([n]^d\). Indeed, the same gang of authors as above showed in a previous work that a Margulis style directed isoperimetry can be extended from the lowly hypercube to the hypergrid. This resulted in a tester with \(\widetilde{O}(d^{5/6})\) queries. The more intricate task of proving a directed Talagrand style isoperimetry that underlies the Khot-Minzer-Safra breakthrough was a challenge. Was. The featured work extends this isoperimetry from the hypercube to the hypergrid and this gives a tester with query complexity \(\widetilde{O}(n \sqrt d)\) which is an improvement over the \(d^{5/6}\) bound for domains where \(n\) is (say) some small constant. But as they say, when it rains, it pours. This brings us to a concurrent paper with the same result.

**Improved Monotonicity Testers via Hypercube Embeddings** by Mark Braverman, Subhash Khot, Guy Kindler, Dor Minzer (arXiv) Similar to the paper above, this paper also obtains monotonicity testers over the hypergrid domain, \([n]^d\), with \(\widetilde{O}(n^3 \sqrt d)\) queries. This paper also presents monotonicity testers over the standard hypercube domain — \(\{0,1\}^d\) in the \(p\)-biased setting. In particular, their tester issues \(\widetilde{O}(\sqrt d)\) queries to successfully test monotonicity on the \(p\)-biased cube. Coolly enough, this paper also proves directed Talagrand style isoperimetric inequalities both over the hypergrid and the \(p\)-biased hypercube domains.

**Toeplitz Low-Rank Approximation with Sublinear Query Complexity** by Michael Kapralov, Hannah Lawrence, Mikhail Makarov, Cameron Musco, Kshiteej Sheth (arXiv) Another intriguing paper for the holiday month. So, take a Toeplitz matrix. Did you know that any *psd* Toeplitz matrix admits a (near-optimal in the Frobenius norm) low-rank approximation which is itself Toeplitz? This is a remarkable statement. The featured paper proves this result and uses it to get more algorithmic mileage. In particular, suppose you are given a \(d \times d\) Toeplitz matrix \(T\). Armed with the techniques from the paper you get algorithms that return a Toeplitz matrix \(\widetilde{T}\) with rank slightly bigger than \(rank(T)\) which is a very good approximation to \(T\) in the Frobenius norm. Moreover, the algorithm only issues a number of queries sublinear in the size of \(T\).

**Sampling an Edge in Sublinear Time Exactly and Optimally** by Talya Eden, Shyam Narayanan and Jakub Tětek (arXiv) Regular readers of PTReview are no strangers to the fundamental task of sampling a random edge from a graph which you can access via query access to its vertices. Of course, you don’t have direct access to the edges of this graph. This paper considers the task of sampling a truly uniform edge from the graph \(G = (V,E)\) with \(|V| = n, |E| = m\). In STOC 22, Tětek and Thorup presented an algorithm for a relaxation of this problem where you want an \(\varepsilon\)-approximately unifrom edge. This algorithm runs in time \(O\left(\frac{n}{\sqrt{m}} \cdot \log(1/\varepsilon) \right)\). The featured paper presents an algorithm that samples an honest to goodness uniform edge in expected time \(O(n/\sqrt{m})\). This closes the problem as we already know a matching lower bound. Indeed, just consider a graph with \(O(\sqrt m)\) vertices which induce a clique and all the remaining components are singletons. You need to sample at least \(\Omega(n/\sqrt m)\) vertices before you see any edge.

**Support Size Estimation: The Power of Conditioning** by Diptarka Chakraborty, Gunjan Kumar, Kuldeep S. Meel (arXiv) This work considers the classic problem of support size estimation with a slight twist. You are given access to a stronger (conditioning based) sampling oracle. Let me highlight one of the results from this paper. So, you are given a distribution \(D\) where \(supp(D) \subseteq [n]\). You want to obtain an estimate to \(supp(D)\) that lies within \(supp(D) \pm \varepsilon n\) with high probability. Suppose you are also given access to the following sampling oracle. You may choose any subset \(S \subseteq [n]\) and you may request a sample \(x \sim D\vert_S\). An element \(x \in S\) is returned with probability \(D\vert_S(x) = D(x)/D(S)\) (for simplicity of this post, let us assume \(D(S) > 0\)). In addition, this oracle also reveals for you the value \(D(x)\). The paper shows that the algorithmic task of obtaining a high probability estimate to the support size (to within \(\pm \varepsilon n\)) with this sampling oracle admits a lower bound of \(\Omega(\log (\log n)\) calls to the sampling oracle.

**Computing (1+epsilon)-Approximate Degeneracy in Sublinear Time** by Valerie King, Alex Thomo, Quinton Yong (arXiv) Degeneracy is one of the important graph parameters which is relevant to several problems in algorithmic graph theory. A graph \(G = (V,E)\) is \(\delta\)-degenerate if all induced subgraphs of \(G\) contain a vertex with degree at most \(\delta\). The featured paper presents algorithms for a \((1 + \varepsilon)\)-approximation to degeneracy of \(G\) where you are given access to \(G\) via its adjacency list.

**Learning and Testing Latent-Tree Ising Models Efficiently** by Davin Choo, Yuval Dagan, Constantinos Daskalakis, Anthimos Vardis Kandiros (arXiv) Ising models are emerging as a rich and fertile frontier for Property Testing and Learning Theory researchers (at least to the uninitiated ones like me). This paper considers latent-tree ising models. These are ising models that can only be observed at their leaf nodes. One of the results in this paper gives an algorithm for testing whether the leaf distributions attached to two latent-tree ising models are close or far in the TV distance.

**A constant lower bound for the union-closed sets conjecture** by Justin Gilmer (arXiv) The union-closed sets conjecture of Frankl states that for any union closed set system \(\mathcal{F} \subseteq 2^{[n]}\), it holds that there is a mysterious element \(i \in [n]\) that shows up in at least \(c = 1/2\) of the sets in \(\mathcal{F}\). Gilmer took a first swipe on this problem and gave a constant lower bound of \(c = 0.01\). This has already been improved by at least four different groups to \(\frac{3-\sqrt{5}}{2}\), a bound which is the limit of Gilmer’s method (which takes all of only 9 pages!).

The key lemma Gilmer proves is the following. Suppose you sample two sets: \(A, B \sim \mathcal{D}_n\) *(iid)* from some distribution \(\mathcal{D}_n\) over the subsets of \([n]\). Suppose for every index \(i \in [n]\), it holds that the probability that the element \(i\) shows up in the random set \(A\) is at most $0.01$. Then you have \(H(A \cup B) \geq 1.26 H(A)\). This is all you need to finish Gilmer’s proof (of \(c = 0.01\)). The remaining argument is as follows. Suppose, by the way of contradiction, that no element shows up in at least \(0.01\) fraction of sets in the union closed family \(\mathcal{F}\). An application of the key lemma would then give \(H(A \cup B) > H(A)\) which is a contradiction if \(A,B\) are chosen uniformly from \(\mathcal{F}\). The proof of the key lemma is also fairly slick and uses pretty simple information theoretic tools.

**Gaussian Mean Testing Made Simple**, by Ilias Diakonikolas, Daniel Kane and Ankit Pensia (arXiv). Consider an unknown distribution distribution \(p\) over \(\mathbb{R}^d\) that we have sample access to. The paper studies the problem of determining whether \(p\) is a standard Gaussian with zero mean or whether it is a Gaussian with large mean. More formally, the task is to distinguish between the case that \(p\) is \(\mathcal{N}(0, I_d)\) and the case that \(p\) is a Gaussian of the form \(\mathcal{N}(\mu, \Sigma)\), where \(||\mu||_2 \geq \epsilon\) and \(\Sigma\) is an unknown covariance matrix. Canonne, Chen, Kamath, Levi and Weingarten (2021) gave a sample-optimal algorithm for this problem with sample complexity \(\Theta(\sqrt{d}/\epsilon^2)\) sample complexity. The current paper gives another sample-optimal algorithm for the same problem with a simpler analysis. In addition to being sample-optimal, the algorithm in the current paper also runs in time linear in the total sample size, which is an improvement over the work of Canonne et al.

**Superpolynomial lower bounds for decision tree learning and testing**, by Caleb Koch, Carmen Strassle and Li-Yang Tan (arXiv). Roughly speaking, the paper studies the problems of testing if a function has a low-depth decision tree and learning a low-depth decision tree approximating a function (provided that one such tree exists). In what follows, we summarize the testing results in the paper. Given an explicit representation of a function \(f:\{0,1\}^n \to \{0,1\}\) and access to samples from a known distribution \(\mathcal{D}\) over \(\{0,1\}^n\), one can aim to determine, with probability at least \(2/3\), if \(f\) has a decision tree of depth at most \(d\) or whether \(f\) is \(\epsilon\)-far from having a decision tree of depth at most \(d\log d\), where the distance is measured with respect to \(\mathcal{D}\). The paper shows that, under the randomized exponential time hypothesis, this problem cannot be solved in time \(\exp(d^{\Omega(1)})\). An immediate corollary is that the same lower bound holds for the problem of distribution-free testing of the property of having depth-\(d\) decision trees. The bound in the current paper is an improvement over the recent work of Blais, Ferreira Pinto Jr., and Harms (2021), who give a lower bound of \(\tilde{\Omega}(2^d)\) on the query complexity of testers for the same problem. However, the advantage of the latter result is that it is unconditional, as opposed to the result in the current paper.

**On Interactive Proofs of Proximity with Proof-Oblivious Queries**, by Oded Goldreich, Guy Rothblum, and Tal Skverer (ECCC). Interactive Proofs of Proximity (IPPs) are the “interactive” version of property testers, where the algorithm can both query the input and interact with an all-knowing (but untrusted) prover. In this work, the authors study the power of a specific and natural type of “adaptivity” for IPPs, asking what happens when the choice of queries and the interaction with the prover are independent, or restricted. That is, what happens when these two aspects of the IPP algorithm are in separate “modules”? Can we still test various properties as efficiently? The paper proves various results in under several models (=restrictions between the two “modules”), focusing on the intermediate restriction where the two modules (queries to the input and interaction with the prover) are separate (no interaction), but have access to shared randomness.

**Training Overparametrized Neural Networks in Sublinear Time** by Hang Hu, Zhao Song, Omri Weinstein, Danyang Zhuo (arXiv). Think of a classification problem where the inputs are in \(\mathbb{R}^d\). We have \(n\) such points (with their true labels, as training data) and wish to train a Neural Network. A two layer Rectified Linear Unit (ReLU) Neural Network (NN) works as follows. The first layer has \(m\) vertices, where each vertex has vector weight \(\vec{w}_i \in \mathbb{R}^d\). The second “hidden layer” has \(m\) vertices, each with a scalar weight \(a_1, a_2, \ldots, a_m\). This network is called overparametrized when \(m \gg n\). The output of this NN on input vector \(\vec{x}\) is (up to scaling) \(\sum_{i \leq m} a_i \phi(\vec{w_i} \cdot \vec{x})\) (where \(\phi\) is a thresholded linear function). Observe that to compute the value on a single input takes \(O(md)\) time, so the total time to compute all values on \(n\) training inputs takes \(O(mnd)\) time. The training is done by gradient descent methods; given a particular setting of weights, we compute the total loss, and then modify the weights along the gradient. Previous work showed how a single iteration can be done in time \(O(mnd + n^3)\). When \(m \gg n^2\), this can be thought of as linear in computing the loss function (which requires evaluating the NN on all the \(n\) points). This paper shows how to implement a single iteration in \(O(m^{1-\alpha}nd + n^3)\) time, for some \(\alpha > 0\). Hence, the time for an iteration is sublinear in the trivial computation. The techniques used are sparse recovery methods and random projections.