# News for March 2017

March was a relatively good month for property testing, with 3 papers — including a foray in differential privacy.

Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps, by Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri (arXiv). Testing unateness (that is, whether a function is monotone after a rotation of the hypercube) has recently received significant attention. Understanding the power of adaptivity in property testing, on the other hand, has been a long-standing question, which has sparked much interest over the past decade. In this paper, the authors obtain optimal results for testing unateness of real-valued functions $$f\colon \{0,1\}^d\to \mathbb{R}$$ and $$f\colon [n]^d\to \mathbb{R}$$ (up to the dependence on the proximty parameter $$\varepsilon$$), and show that — unlike what happens for monotonicity testing of real-valued functions — adaptivity helps. Namely, adaptive testing of unatess of functions $$f\colon \{0,1\}^d\to \mathbb{R}$$ has query complexity $$O(d/\varepsilon)$$, but non-adaptive testers require $$\Omega(d\log d)$$ queries.
This work is a merged, and significantly improved version of two papers we covered a few months back, by the same authors.

Switching from function to distribution testing, last month saw two independent samples appear online:

Near-Optimal Closeness Testing of Discrete Histogram Distributions, by Ilias Diakonikolas, Daniel M. Kane, and Vladimir Nikishkin (arXiv). Closeness testing of discrete distributions is the question of testing, given sample access to two unknown distributions $$p,q$$ over a known discrete domain $$[n] \stackrel{\rm def}{=} \{1,\dots,n\}$$, whether $$p=q$$ or $$d_{\rm TV}(p,q)> \varepsilon$$ (are the two distributions the same, or differ significantly in total variation distance?). This quite fundamental question is by now fully understood in the general case, where $$p,q$$ are allowed to be arbitrary distributions over ; but what if one has some additional knowledge about them? Specifically, what if we were guaranteed that both distributions have some “nice structure” — e.g., that they are piecewise constant with $$k$$ pieces ($$k$$-histograms)? Leveraging machinery the authors developed in a series of previous work (for the continuous case), the authors show that in this case, the sample complexity of “testing closeness under structural assumptions” is quite different than in the general case; and that the resulting (near tight) bound is a rather intricate tradeoff between the three parameters $$n,k,\varepsilon$$.

Priv’IT: Private and Sample Efficient Identity Testing, by Bryan Cai, Constantinos Daskalakis, and Gautam Kamath (arXiv). Distribution testing is a vibrant field; differential privacy (DP) is an incredibly active and topical area. What about differentially private distribution testing — a.k.a. testing the data without learning too much about any single sample? In this paper, the authors address the task of performing identity testing of distributions (“given samples from an unknown arbitrary distribution, decide whether it matches a fixed known probability distribution/model”) in the DP setting, focusing on getting the best of both worlds: how to guarantee privacy without sacrificing efficiency. Not only do they obtain asymptotically near-optimal bounds in both the testing and differential privacy parameters — they also run some experiments to validate their approach empirically. (Plus, the title is rather cute.)

Usual disclaimer: if we forgot or misrepresented your paper, please signal it in the comments — we’ll address it as quickly as we can.

# News for February 2017

This February, we had four exciting testing papers, covering domains from Boolean functions to distributions to graphs. Read on to catch up on the latest results!

Property Testing of Joint Distributions using Conditional Samples, by Rishiraj Bhattacharyya and Sourav Chakraborty (arXiv). This paper focuses on the problems of identity testing and independence testing for multivariate distributions in the conditional sampling model. Multivariate distribution testing has not previously been considered in the conditional sampling model. While previous algorithms for identity testing would generally carry over to this setting, the authors restrict themselves to algorithms where the queries are on subcubes of the domain. Interestingly, the authors show that the complexity of these problems is polynomial in the dimension (and that a polynomial dependence is necessary). That is, despite the “simple” structure of the queries, they are still powerful enough to avoid the curse of dimensionality which is present in vanilla multivariate distribution testing.

An Improved Dictatorship Test with Perfect Completeness, by Amey Bhangale, Subhash Khot, and Devanathan Thiruvenkatachari (arXiv, ECCC). A function is a dictator if it depends on exactly one variable. This paper provides an improved tester with perfect completeness for this property. In particular, they give a (non-adaptive) $$k$$-query test which never rejects a dictator function, and accepts a far-from-dictator function with probability at most $$\frac{2k +1}{2^k}$$. This improves upon the previous best result, which accepts a far-from-dictator function with probability at most $$\frac{2k+3}{2^k}$$.

An Adaptivity Hierarchy Theorem for Property Testing, by Clément L. Canonne and Tom Gur (arXiv, ECCC). This paper investigates the power of adaptivity in property testing. We most frequently think of algorithms as either adaptive or non-adaptive. In the former, an algorithm may specify each query after viewing the results of all previous queries, while in the latter, all the queries must be specified in advance. The authors focus on the natural interpolation, in which the algorithm is allowed to make queries in $$k$$ rounds, where the queries in a round may depend on the result of queries in all previous rounds. They show that the power of an algorithm can shift dramatically with just one more round of adaptivity — there exist properties which can be tested with $$\tilde O(k)$$ queries with $$k$$ rounds of adaptivity, but require $$\Omega(n)$$ queries with $$k-1$$ rounds of adaptivity.

Beyond Talagrand Functions: New Lower Bounds for Testing Monotonicity and Unateness, by Xi Chen, Erik Waingarten, and Jinyu Xie (arXiv). The main result in this paper is an improved lower bound for the well-studied problem of testing monotonicity of Boolean functions. The authors show that any two-sided and adaptive algorithm requires $$\tilde \Omega(n^{1/3})$$ samples to test whether a function is monotone or far from monotone. This improves upon the previous result of $$\tilde \Omega(n^{1/4})$$ by Belovs and Blais, which was the first to show that adaptive monotonicity testing requires polynomially-many queries. The construction of Belovs and Blais uses Talagrand’s random DNFs, for which they also showed $$\tilde O(n^{1/4})$$ queries are sufficient. This paper analyzes an extension of this family, denoted as two-level Talagrand functions. The result leaves open the tantalizing question: does adaptivity help with monotonicity testing? Or does the complexity remain $$\tilde \Theta(\sqrt{n})$$? The authors also prove lower bounds for the problem of testing unateness, a problem of recent interest and a natural generalization of monotonicity.

# News for January 2017

2017 starts off rather slow for property testing. Though, we have an intriguing paper to report – an experimental analysis of a classic sublinear algorithm.

Evaluating a Sublinear-time Algorithm for the Minimum Spanning Tree Weight Problem, by Gabriele Santi and Leonardo De Laurentiis (arXiv). The Chazelle-Rubinfeld-Trevisan Minimum Spanning Tree algorithm is a classic in sublinear algorithms. This algorithms provides a $$(1+\varepsilon)$$ approximation to the MST in time independent of the number of vertices (although it does depend on the average degree). But how this compare with Prim’s algorithm on real instances, in a real (not theoretical) computer? This intriguing paper does a detailed experimental comparison. Having done experimental graph algorithms myself, I can attest to the various challenges: how to choose a test set of graphs? How to set error parameters? Can data structure optimization on the coding side beat asymptotic improvements? This paper does a series of experiments on synthetic graph generators (such as Erdős-Rényi, Barabási-Albert, Watts-Strogatz models). They do validate the basic CRT algorithm at scale, by showing that it is faster than Prim for graphs with more than a million edges. Their experiments suggest that the sublinear-time algorithm gives little benefits when $$\varepsilon \leq 0.2$$. The paper has many experiments for a variety of settings, and the authors do a comprehensive study of the various parameters. I’d definitely recommend to anyone interested in exploring how property testing might influence algorithms in the real world.

# News for December 2016

December was indeed a merry month for property testing, with seven new papers appearing online.* Let’s hope the trend continues!

Cube vs. Cube Low Degree Test, by Amey Bhangale, Irit Dinur, and Inbal Livni Navon (ECCC). This work provides a new and improved analysis of the “low-degree test” of Raz and Safra, tightening the dependence on the alphabet size of the soundness parameter. Specifically, the goal is as follows: given query access to the encoding of a function $$f\colon \mathbb{F}^m\to \mathbb{F}$$ as a “cube table,” decide whether $$f$$ is a low-degree polynomial (i.e., of degree at most $$d$$). With direct applications to PCP theorems, this question has a long history; here, the authors focus on a very simple and natural test, introduced by Raz–Safra and Arora–Sudan. In particular, they improve the soundness guarantee, which previously required the error to be at least $$\textrm{poly}(d)/\mathbb{F}^{1/8}$$, to obtain a dependence on the field size which is only $$\mathbb{F}^{-1/2}$$.

Robust Multiplication-based Tests for Reed-Muller Codes, by Prahladh Harsha and Srikanth Srinivasan (arXiv). Given a function $$f\colon \mathbb{F}_q^n\to\mathbb{F}_q$$ purported to be a degree-$$d$$ polynomial, deciding whether this is indeed the case is a question with applications to hardness of approximation and — as the title strongly hints— to testing of Reed—Muller codes. Here, the authors generalize and improve on a test originally due to Dinur and Guruswami, improving on the soundness: that is, they show a robust version of the soundness guarantee of this multiplication test, answering a question left open by Dinur and Guruswami.

A Note on Testing Intersection of Convex Sets in Sublinear Time, by Israela Solomon (arXiv). In this note, the author addresses the following testing question: given $$n$$ convex sets in $$\mathbb{R}^d$$, distinguish between the case where (i) the intersection is non-empty, and (ii) even after removing any $$\varepsilon n$$ sets, the intersection is still empty. Through the use of a generalization of Helly’s Theorem due Katchalski and Liu (1979), the author then provides and analyzes an algorithm for this question, with query complexity $$O(\log 1/(d\varepsilon))$$.

A Characterization of Constant-Sample Testable Properties, by Eric Blais and Yuichi Yoshida (arXiv). A lot of work in property testing has been to understand which properties can be locally tested, i.e. admit testers with constant query complexity. Here, the authors tackle — and answer — the variant of this question in the sample-based testing model, that is restricting the ability of the algorithm to only obtain the value of the function on independently and uniformly distributed locations. Namely, they provide a full characterization of the properties of Boolean-valued function which are testable with constant sample complexity, resolving the natural question of whether most properties are easily tested from samples only. As it turns out, only those properties which are (essentially) $$O(1)$$-part-symmetric have this nice testability — where a property $$\mathcal{P}$$ is $$k$$-part-symmetric if one can divide the input variables in $$k$$ blocks, such that membership to $$\mathcal{P}$$ is invariant by permutations inside each block.

The last three are all works on distribution testing, and all concerned with the high-dimensional case. Indeed, most of the distribution testing literature so far has been focusing in probability distributions over arbitrary discrete sets or the one-dimension ordered line; however, when trying to use these results in the (arbitrary) high-dimensional case on is subjected to the curse of dimensionality. Can one leverage (presumed) structure to reduce the cost, and obtain computationally and sample-efficient testers?

Testing Ising Models, by Costis Daskalakis, Nishanth Dikkala, and Gautam Kamath (arXiv). In this paper, the authors take on the above question in the context of Markov Random Fields, and more specifically in the Ising model. Modeling the (unknown) high-dimensional distributions as Ising models with some promise on the parameters, they tackle the two questions of identity and independence testing; respectively, “is the unknown Ising model equal to a fixed, known reference distribution?” and “is the high-dimensional, a priori complex distribution a product distribution?”. They obtain both testing algorithms and lower bounds for these two problems, where the soundness guarantee sought is in terms of the symmetrized Kullback—Leibler divergence (a notion of distance more stringent that the statistical distance).

Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing, by Costis Daskalakis and Qinxuan Pan (arXiv). Another natural model to study structured high-dimensional distribution is that of Bayesian networks, that is directed graphs providing a succinct description of the dependencies between coordinates. This paper studies the closeness testing problem (testing if two unknown distributions are equal or far, with regard to the usual statistical distance) for Bayesian networks, parameterized by the dimension, alphabet size, and (promise on the) maximum in-degree of the unknown Bayes net. At its core is a new inequality relating the (squared) Hellinger distance of two Bayesian networks to the sum of the (squared) Hellinger distances between their marginals.

Testing Bayesian Networks, by Clément Canonne, Ilias Diakonikolas, Daniel Kane, and Alistair Stewart (arXiv). Tackling the testing of Bayesian networks as well, this second paper considers two of the most standard testing questions — identity (one-unknown) and closeness (two-unknown) testing — under various assumptions, in order to pinpoint the exact sample complexity in each case. Specifically the goal is to see when (and under which natural restrictions) does testing become easier than learning for Bayesian networks, focusing on the dimension and maximum in-degree as parameters.

* As usual, if we forgot one or you find imprecisions in our review of a paper, please let us now in the comments below.

# News for November 2016

November was quite eventful for property testing, with six exciting new results for you to peruse.

Alice and Bob Show Distribution Testing Lower Bounds (They don’t talk to each other anymore.), by Eric Blais, Clément L. Canonne, and Tom Gur (ECCC). The authors examine distribution testing lower bounds through the lens of communication complexity, a-la Blais, Brody, and Matulef, who previously showed such a connection for property testing lower bounds in the Boolean function setting. In this work, the authors’ main result involves testing identity to a specific distribution $$p$$. While Valiant and Valiant showed tight bounds involving the $$\ell_{2/3}$$-quasinorm of $$p$$, this paper gives tight bounds using a different quantity, namely Peetre’s $$K$$-functional. Their techniques also give lower bounds for several other properties (some old and some new), including monotonicity, sparse symmetric support, and $$k$$-juntas in the PAIRCOND model.

Fast property testing and metrics for permutations, by Jacob Fox and Fan Wei (arXiv). This paper proves a general testing result for permutations. In particular, it shows that any hereditary property of permutations is two-sided testable with respect to the rectangular distance with a constant number of queries. While in many such testing results on combinatorial objects (such as graphs), a “constant number of queries” may be exorbitantly large (due to complexities arising from an application of the strong regularity lemma), surprisingly, the complexity obtained in this paper is polynomial in $$1/\varepsilon$$.

A Unified Maximum Likelihood Approach for Optimal Distribution Property Estimation, by Jayadev Acharya, Hirakendu Das, Alon Orlitsky, and Ananda Theertha Suresh (arXiv, ECCC). There has been a considerable deal of work recently on estimating several symmetric distribution properties, namely support size, support coverage, entropy, and distance to uniformity. One drawback of these results is that, despite the similarities between these properties, seemingly different techniques are required to obtain optimal rates for each. This paper shows that one concept, pattern maximum likelihood (PML), unifies them all. A PML distribution of a multiset of samples is any distribution which maximizes the likelihood of observing the multiplicities of the multiset, after discarding the labels on the elements. This can behave quite differently from the sequence maximum likelihood (SML), or empirical distribution. In particular, if a multiset of samples on support $$\{x, y\}$$ is $$\{x, y, x\}$$, then the SML is $$(2/3, 1/3)$$, while the PML is $$(1/2, 1/2)$$. The main result of this paper is, if one can approximate PML, then applying the plug-in estimator gives the optimal sample complexity for all of the aforementioned properties. The one catch is that efficient approximation of the PML is currently open. Consider the gauntlet thrown to all our readers!

Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures, by Ilias Diakonikolas, Daniel M. Kane, and Alistair Stewart (arXiv, ECCC). While the main focus of this work is on lower bounds for distribution estimation in the statistical query (SQ) model, this paper also has some interesting lower bounds for multivariate testing problems. Namely, they show that it is impossible to achieve a sample complexity which is significantly sublinear in the dimension for either of the following two problems:

• Given samples from an $$n$$-dimensional distribution $$D$$, distinguish whether $$D = \mathcal{N}(0,I)$$ or $$D$$ is $$\varepsilon/100$$-close to any $$\mathcal{N}(\mu, I)$$ where $$\|\mu\|_2 \geq \varepsilon$$.
• Given samples from an $$n$$-dimensional distribution $$D$$, distinguish whether $$D = \mathcal{N}(0,I)$$ or $$D$$ is a mixture of $$k$$ Gaussians with almost non-overlapping components.

Collision-based Testers are Optimal for Uniformity and Closeness, by Ilias Diakonikolas, Themis Gouleakis, John Peebles, and Eric Price (arXiv, ECCC). In the TCS community, the seminal results in the field of distribution testing include the papers of Goldreich and Ron and Batu et al., which study uniformity testing and $$\ell_2$$-closeness testing (respectively) using collision based testers. While these testers appeared to be lossy, subsequent results have attained tight upper and lower bounds for these problems. As suggested by the title, this paper shows that collision-based testers actually achieve the optimal sample complexities for uniformity and $$\ell_2$$-closeness testing.

Testing submodularity and other properties of valuation functions, by Eric Blais, and Abhinav Bommireddi (arXiv). This paper studies the query complexity of several properties which have been studied in the context of valuation functions in algorithmic game theory. These properties are real-valued functions over the Boolean hypercube, and include submodularity, additivity, unit-demand, and much more. The authors show that, for constant $$\varepsilon$$ and any $$p \geq 1$$, these properties are constant-sample testable. Their results are obtained via an extension of the testing by implicit learning method of Diakonikolas et al.

# News for October 2016

Alas, a rather dry month for property testing. We did find one quantum computing result based on the classic linearity testing theorem.

Robust Self-Testing of Many-Qubit States, by Anand Natarajan and Thomas Vidick (arXiv). (Frankly, my understanding of quantum computation is poor, and this summary may reflect that. Then again, some searching on Google and Wikipedia have definitely broadened my horizons.) One of the key concepts in quantum computation is the notion of entanglement. This allows for correlations between (qu)bits of information beyond what can be classically achieved. Given some device with supposed quantum properties (such as sets of entangled bits), is there a way of verification by measuring various outcomes of the device? This is referred to as self-testing of quantum states. This paper proves such a result for a set of $$n$$ EPR pairs, which one can think of $$n$$ pairs of “entangled” qubits. The interest to us property testers is the application of a quantum version of the seminar Blum, Luby, Rubinfeld linearity test.

# News for September 2016

September has just concluded, and with it the rise of Fall in property testing: no less than 5 papers* this month.

Removal Lemmas for Matrices, by Noga Alon and Omri Ben-Eliezer (arXiv). The graph removal lemma, and its many variants and generalizations, is a cornerstone in graph property testing (among others), and is a fundamental ingredient in many graph testing results. In its simplest form, it states that  for any fixed $$h$$-vertex pattern $$H$$ and $$\varepsilon > 0$$,  there is some constant $$\delta=\delta(\varepsilon)$$ such that any graph $$G$$ on $$n$$ vertices containing at most $$\delta n^h$$ copies of $$H$$ can be “fixed” (made $$H$$-free) by removing at most $$\varepsilon n^2$$ edges.
The authors introduce and establish several analogues of this result, in the context of matrices. These matrix removal lemmas have direct implications in property testing (discussed in Section 1.2); for instance, they imply constant-query testing of $$F$$-freeness of (binary) matrices, where $$F$$ is any family of binary matrices closed under row permutations.

Improving and extending the testing of distributions for shape-restricted properties, by Eldar Fischer, Oded Lachish, and Yadu Vasudev (arXiv). Testing membership to a class (family) is a very natural question in distribution testing, and one which has received significant attention lately. In this paper, the authors build on, improve, and generalize the techniques of Canonne, Diakonikolas, Gouleakis, and Rubinfeld (2016) to obtain a generic, one-size-fits-all algorithm for this question. They also consider the same question in the “conditional oracle” setting, for which their ideas yield significantly more sample-efficient algorithms. At the core of their approach is a better and simpler learning procedure for families of distributions that enjoy some decomposability property.

The Bradley―Terry condition is $$L_1$$–testable, by Agelos Georgakopoulos and  Konstantinos Tyros (arXiv). An incursion in probability models: say that the directed weighted graph $$((V,E),w)$$ on $$n$$ vertices, with $$w\colon E_n\to[0,1]$$ such that $$w(x,y)=w(y,x)$$ for all $$(x,y)\in V^2$$, satisfies the Bradley—Terry condition if there exists an assignment of probabilities $$(a_x)_{x\in V}$$ such that

$$\qquad\displaystyle \frac{w(x,y)}{w(y,x)} = \frac{a_x}{a_y}$$

for all $$(x,y)\in V^2$$. This condition captures whether such a graph corresponds to a Bradley—Terry tournament.
This paper considers this characterization from a property testing point of view: that is, it addresses the task of testing, in the $$L_1$$-sense of Berman, Raskhodnikova, and Yaroslavtsev (2012), if a $$((V,E),w)$$ satisfies the Bradley—Terry condition (or is far from any such graph).

On SZK and PP, by Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, and Prashant Nalini Vasudevan (arXiv, ECCC). Another (unexpected) connection! This work proves query and communication complexity separations between two complexity classes, $$\textsf{NISZK}$$ (non-interactive statistical zero knowledge) and $$\textsf{UPP}$$ (a generalization of $$\textsf{PP}$$ with unbounded coin flips, and thus arbitrary gap in the error probability). While I probably cannot even begin to list all that goes over my head in this paper, the authors show in Section 2.3.2 some implications of their results for property testing, for instance in distribution testing. In more detail, their results yield some lower bounds on the sample complexity of (tolerant) property testers with success probability arbitrarily close to $$1/2$$ (as opposed to the “usual” setting of $$1/2+\Omega(1)$$).

Quantum conditional query complexity, by Imdad S. B. Sardharwalla, Sergii Strelchuk, and Richard Jozsa (arXiv). Finally, our last paper of the list deals with both quantum algorithms and distribution testing, by introducing a new distribution testing model in a quantum setting, the quantum conditional oracle. This model, the quantum analogue of the conditional query oracle of Chakraborty et al. and Canonne et al., allows the (quantum) testing algorithm to get samples from the probability distribution to test, conditioning on chosen subsets of the domain.
In this setting, they obtain speedups over both (i)  quantum “standard” oracle and (ii)  “classical” conditional query testing for many distribution testing problems; and also consider applications to quantum testing of Boolean functions, namely balancedness.

* As usual, in case we missed or misrepresented any, please signal it in the comments.

# News for August 2016

This summer is a very exciting one for property testing, with seven (!) papers uploaded during the month of August. In particular, there have been a number of results on testing unateness.

An $$\tilde O(n)$$ Queries Adaptive Tester for Unateness, by Subhash Khot and Igor Shinkar (arXivECCC). A Boolean function is unate if, for each $$i \in [n]$$, it is monotone non-increasing or monotone non-decreasing in coordinate $$i$$. This is a generalization of monotonicity, one of the most studied problems in the property testing literature. This paper gives an adaptive algorithm for testing if a Boolean function is unate. The algorithm requires $$\tilde O(n)$$ queries, improving on the $$O(n^{1.5})$$ query tester of Goldreich et al. from 2000.

A $$\tilde O(n)$$ Non-Adaptive Tester for Unateness, by Deeparnab Chakrabarty and C. Seshadhri (arXivECCC). The unateness testing continues! This paper also gives a $$\tilde O(n)$$ algorithm for this problem, but improves upon the previous work by making non-adaptive queries. The algorithm and analysis are very clean and elegant — with a theorem by Chakrabarty et al. in place, they take up just a single page!

Testing Unateness of Real-Valued Functions, by Roksana Baleshzar, Meiram Murzabulatov, Ramesh Krishnan S. Pallavoor, and Sofya Raskhodnikova (arXiv). The final paper this month on testing unateness, this work studies real-valued functions on the hypergrid $$[n]^d$$ (as opposed to Boolean functions on the hypercube). They give an adaptive algorithm requiring $$O\left(\frac{d \log (\max (d,n))}{\varepsilon}\right)$$ queries, generalizing the above result by Khot and Shinkar, which works only for Boolean functions on the hypercube and requires $$O\left(\frac{d \log d}{\varepsilon}\right)$$ queries. Additionally, they prove lower bounds for this setting (of $$\Omega(\min(d, |R|^2))$$, where $$R$$ is the range of the function) and the Boolean hypercube setting (of $$\Omega(\sqrt{d}/\varepsilon)$$). The latter lower bound leaves open a tantalizing question: does a $$O(\sqrt{d})$$ query algorithm exist? In other words, is testing unateness no harder than testing monotonicity?

Testing k-Monotonicity, by Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, Karl Wimmer (arXivECCC). And now, a different generalization of monotonicity! This paper studies testing $$k$$-monotonicity, where a Boolean function over a poset $$\mathcal{D}$$ is $$k$$-monotone if it alternates between the values $$0$$ and $$1$$ at most $$k$$ times on any ascending chain in $$\mathcal{D}$$. Classical monotonicity is then $$1$$-monotone in this definition. On the hypercube domain, the authors prove a separation between testing monotonicity and testing $$k$$-monotonicity and a separation between testing and learning. They also present a tolerant test for $$k$$-monotonicity over the hypergrid $$[n]^d$$ with complexity independent of $$n$$.

The Dictionary Testing Problem, by Siddharth Barman, Arnab Bhattacharyya, and Suprovat Ghoshal (arXiv). The dictionary learning problem has enjoyed extensive study in computer science. In this problem, given a matrix $$Y$$, one wishes to construct a decomposition $$Y = AX$$ such that each column of $$X$$ is $$k$$-sparse. In the settings typically studied, such a decomposition is guaranteed to exist. This paper initiates the dictionary testing problem, in which one wishes to test whether $$Y$$ admits such a decomposition, or is far from it. The authors provide a very elegant characterization — it admits a decomposition iff the columns of $$Y$$ have a sufficiently narrow Gaussian width.

The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs, by Graham Cormode, Hossein Jowhari, Morteza Monemizadeh, and S. Muthukrishnan (arXiv). This paper provides streaming algorithms for estimating the size of the maximum matching in graphs with some underlying sparsity. They give a one-pass algorithm for insert-only and dynamic streams with bounded arboricity, and a two-pass algorithm for random order streams with bounded degree. Interestingly, for the latter result, they use local exploration techniques, which have seen previous use in the property testing literature.

Testing Assignments to Constraint Satisfaction Problems, by Hubie Chen, Matt Valeriote, and Yuichi Yoshida (arXiv). Given a CSP from a finite relational structure $$A$$ and query access to an assignment, is the CSP satisfied? This paper shows a dichotomy theorem which characterizes which $$A$$ admit a constant-query test. They go on to show a trichotomy theorem for when instances may include existentially quantified variables, classifying which $$A$$ admit a constant-query test, which admit only a sublinear-query test, and which require a linear number of queries.

Minimizing Quadratic Functions in Constant Time, by Kohei Hayashi and Yuichi Yoshida (arXiv). This paper investigates the problem of (approximately) minimizing quadratic functions in high-dimensions. Prior approaches to this problem scale poorly with the dimension — at least linearly, if not worse. This work uses a sampling-based method to avoid paying this cost: the resulting time complexity is completely independent of the dimension!

# Distribution Testing: a short non-survey

The past years have witnessed a great deal of activity in the field of distribution testing— so much, in fact, that it has become a bit of a challenge to even keep track of what’s happening and what has already happened. While this blog post is not going to be a comprehensive (comprehensible, maybe?) survey or summary, it hopefully will at least shed some light on what distribution testing is — and where to look at if interested in it.

Obligatory XKCD comic.

Recall that in the “vanilla” property testing setting, one is provided with access (typically query access) to an object $$\mathcal{O}$$ from some universe $$\Omega$$ (typically the space of Boolean functions over $$\{0,1\}^n$$ or the set of $$n$$-vertex graphs), a distance measure $$\textrm{d}(\cdot,\cdot)$$ between such objects (usually the Hamming distance), a property $$\mathcal{P}\subseteq \Omega$$ (the subset of “good objects”) and a parameter $$\varepsilon \in (0,1]$$. The goal is to design a randomized algorithm that (i) accepts with high probability if $$\mathcal{O}\in\mathcal{P}$$; and (ii) rejects with high probability if $$d(\mathcal{O},\mathcal{O}’)>\varepsilon$$ for every single object $$\mathcal{O}’\in\mathcal{P}$$.

The distribution testing setting is very much like this one, with of course crucial twists. The object is now a probability distribution $$D$$ over a (discrete) set, typically $$[n]=\{1,\dots,n\}$$; the access model is usually a sample oracle, providing independent samples from $$D$$ on demand; the distance is the total variation distance between distributions (or, equivalently, the $$\ell_1$$ norm between the probability mass functions).

$$\textrm{d}_{\rm TV}(D,D’) = \sup_{S\subseteq [n]} \left( D(S) – D'(S) \right) = \frac{1}{2} \sum_{i\in[n]} \left|D(i)-D'(i) \right|$$

This brings in some quite important distinctions: for instance, randomness is now inherent: even taking a number of samples beyond any reasonable bound will not bring the probability of failure of the algorithm to zero, this time.

## A Brief History

The field made its debut in TCS with a work of Goldreich and Ron [5], who considered the question of testing if an expander graph had mixed — that is, if the distribution induced on its nodes was uniform. However, it is only a tad later, in a work of Batu, Fortnow, Rubinfeld, Smith, and White [6] that the setting was formally defined and studied, for the specific problem of testing whether two unknown random variables have the same distribution (closeness, or equivalence testing).

This started a line of work, which analyzed the sample complexity of testing uniformity (is my random variable uniformly distributed?), identity (is it distributed according to this nice distribution I fully know in advance?), independence (are my random variables independent of each other?), monotonicity (is the probability density function increasing?), and many more. Over the course of the past 16 years, many breakthroughs, new techniques, new questions, and new answers to old problems were made and found. With surprising twists (if your domain size is n, then surely $$\Theta(n)$$ samples are enough and necessary to learn an arbitrary distribution; but who would have thought that $$\Theta(n/\log n)$$ was the right answer to anything? [7]). And game changers (general theorems that apply to many questions or blackbox lemmas are really nice to have in one’s toolbox).

And that’s barely the tip of the iceberg! For more in-depth or better written prose, the interested Internet wanderer may want to consult one of the following (usual non-exhaustivity caveats apply):

• Ronitt Rubinfeld’s introduction [1]
• Dana Ron’s survey on Property testing [2] (Section 11.4)
• Oded Goldreich’s upcoming book [3] (Chapter 11)
• or my own survey [4] on distribution testing (but I cannot promise better written prose for this one).

For the more video-inclined, there are also quite a few survey talks available online, the most recent I know of being this talk by Ronitt Rubinfeld at COLT’16.

## A Brief Comparison

Note: this section is taken verbatim from my survey [4] (Section 1.2).

It is natural to wonder how the above approach to distribution testing compares to classic methods and formulations, as studied in Statistics. While the following will not be a thorough comparison, it may help shed some light on the difference.

• Null and alternative hypotheses. The standard take on hypothesis testing, simple hypothesis testing, relies on defining two classes of distributions, the null and alternative hypotheses $$\mathcal{H}_0$$ and $$\mathcal{H}_1$$. A test statistic is then tailored specifically to these two classes, in order to optimally distinguish between $$\mathcal{H}_0$$ and $$\mathcal{H}_1$$— that is, under the underlying assumption that the unknown distribution $$D\in\mathcal{H}_0\cup\mathcal{H}_1$$.

Something like that.

The test then rejects the null hypothesis $$\mathcal{H}_0$$ if statistical evidence is obtained that $$D\notin\mathcal{H}_0$$. In this view, the distribution testing formulation would be to set $$\mathcal{H}_0$$ to be the property $$\mathcal{P}$$ to be tested, and define the alternative hypothesis as “everything far from $$\mathcal{P}$$.” In this sense, the latter captures a much more adversarial setting, where almost no structure is assumed on the alternative hypothesis— setting known in Statistics as composite hypothesis testing.
• Small-sample regime. A second and fundamental difference resides in the emphasis given to the question. Traditionally, statisticians tend to focus on asymptotic analysis, characterizing— often exactly— the rate of convergence of the statistical tests under the alternative hypothesis, as the number of samples $$m$$ grows to infinity. Specifically, the goal is to pinpoint the error exponent $$\rho$$ such that the probability of error (failing to reject the null hypothesis) asymptotically decays as $$e^{-\rho m}$$. However, this asymptotic behavior will generally only hold for values of $$m$$ greater than the size of the domain (“alphabet”). In contrast, the computer science focus is on the small-sample regime, where the number of samples available is small with regard to the domain size, and one aims at achieving a fixed probability of error.
• Algorithmic flavor. At a more practical level, a third point on which the two approaches deviate is the set of techniques used in order to tackle the question. Namely, the Statistics literature very often relies on relatively simple-looking and “natural” tests and estimators, which need not be computationally efficient. (This is for instance the case for the generalized likelihood ratio test that requires to compute the maximum likelihood of the sequence of samples obtained under the two hypotheses $$\mathcal{H}_0$$ and $$\mathcal{H}_1$$; which is not in general tractable.) On the other hand, works in distribution testing are predominantly algorithmic, with a computational emphasis on the testing algorithms thus obtained.

## Exciting times!

And while the above has hinted at all that has been done so far in the area, there is still plenty ahead — many new questions to be posed, answered, revisited under another light, tackled with new insights and techniques, and connections to be made. As a small sample (sic), we list below a few of the recent developments or trends that herald these exciting times.

### New tools

•  Appeared in the 1800’s thanks to Pearson, testers based on (some tailored variant of) a $$\chi^2$$-based statistic are now aplenty. Yielding simple, concise, very often optimal testing algorithms… [8], [9], [10], [11]
•  Information theory. Techniques and results from this related area have started to permeate distribution testing— mostly to prove lower bounds in an easier or conceptually simpler way. [12], [13,14]
• They were there from the beginning. Now, the $$\ell_2$$-testing subroutines strike back. (For neat, elegant testing algorithms that use $$\ell_2$$ as a proxy “in the right way.”) [5,6,15][14]

### New models

What if modeling the access to the data as a source of i.i.d. samples was not enough? Because we can— the situation allows it— or because we want— what we are trying to achieve is best modeled a bit differently? If we get more ways to query it, can we do more, and by how much?

This recent line of work [20,21,22,23,…] offers many challenges, ranging from the right way to ask (what to model, and how?) to the right way to answer (intuition if often off the hook, in these uncharted waters?). New tools to design, intuition to build, and problems to consider…

### New questions

And of course, even if many of the “classic” problems have by now been fully understood, there are many more that await… from class testing [16,11,17,18] to twists on old questions (uneven sample size for closeness?) to new questions entirely [19], there is no shortage. Not to forget there is a stronger connection with practice to build, and asking the “real-worlders” what makes sense to consider is also unlikely to stop producing challenges.

[1] Ronitt Rubinfeld. Taming Big Probability DistributionsXRDS, 19(1):24-28, 2012. http://xrds.acm.org/article.cfm?aid=2331052
[2] Dana Ron. Algorithmic and Analysis Techniques in Property Testing. Foundations and Trends in Theoretical Computer Science: Vol. 5: No. 2, pp 73-205. 2010. http://dx.doi.org/10.1561/0400000029
[3] Oded Goldreich. Introduction to Property Testing. (In preparation) http://www.wisdom.weizmann.ac.il/~oded/pt-intro.html
[4] Clément Canonne. A Survey on Distribution Testing: Your Data is Big. But is it Blue? 2015. http://eccc.hpi-web.de/report/2015/063/
[5] Oded Goldreich and Dana Ron. On testing expansion in bounded-degree graphs. Electronic Colloquium on Computational Complexity (ECCC), 7:20, 2000.
[6] Tuğkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing that distributions are close. In Proceedings of FOCS, pages 189-197, 2000.
[7] Gregory Valiant and Paul Valiant. The power of linear estimators. In Proceedings of FOCS, pages 403-412, October 2011.
[8] Siu-On Chan, Ilias Diakonikolas, Gregory Valiant, and Paul Valiant. Optimal algorithms for testing closeness of discrete distributions. In Proceedings of SODA, pages 1193-1203. Society for Industrial and Applied Mathematics (SIAM), 2014.
[9] Gregory Valiant and Paul Valiant. An automatic inequality prover and instance optimal identity testing. In Proceedings of FOCS, 2014.
[10] Ilias Diakonikolas, Daniel M. Kane, and Vladimir Nikishkin. Testing Identity of Structured Distributions. In Proceedings of SODA, pages 1841-1854. Society for Industrial and Applied Mathematics (SIAM), 2015.
[11] Jayadev Acharya, Constantinos Daskalakis, Gautam Kamath. Optimal Testing for Properties of Distributions. In Advances in Neural Information Processing Systems (NIPS), 2015.
[12] Yihong Wu and Pengkun Yang. Minimax rates of entropy estimation on large alphabets via best polynomial approximation. ArXiV, abs/1407.0381, 2014.
[13] Yihong Wu and Pengkun Yang. Chebyshev polynomials, moment matching, and optimal estimation of the unseen. ArXiV, abs/1504.01227, 2015.
[14] Ilias Diakonikolas and Daniel Kane. A New Approach for Testing Properties of Discrete Distributions. In Proceedings of FOCS, 2016. (To appear. Also available as abs/1601.05557)
[15] Tuğkan Batu, Eldar Fischer, Lance Fortnow, Ravi Kumar, Ronitt Rubinfeld, and Patrick White. Testing random variables for independence and identity. InProceedings of FOCS, pages 442-451, 2001.
[16] Tuğkan Batu, Ravi Kumar, and Ronitt Rubinfeld. Sublinear algorithms for testing monotone and unimodal distributions. In Proceedings of STOC, pages 381-390, New York, NY, USA, 2004. ACM.
[17] Clément L. Canonne, Ilias Diakonikolas, Themis Gouleakis, and Ronitt Rubinfeld. Testing Shape Restrictions of Discrete Distributions, In Proceedings of STACS, pages 25:1–25:14, 2016.
[18] MIT Theory Blog. Distribution Testing: Do It With Class! https://mittheory.wordpress.com/2015/08/09/distribution-testing-do-it-with-class/
[19] Maryam Aliakbarpour, Eric Blais, Ronitt Rubinfeld. Learning and Testing Junta Distributions. In Proceedings of COLT, 2016. pp. 19–46, 2016
[20] Reut Levi, Dana Ron, and Ronitt Rubinfeld. Testing properties of collections of distributions. Theory of Computing, 9:295-347, 2013.
[21] Sourav Chakraborty, Eldar Fischer, Yonatan Goldhirsh, and Arie Matsliah. On the power of conditional samples in distribution testing. In Proceedings of ITCS, pages 561-580, New York, NY, USA, 2013. ACM.
[22] Clément L. Canonne, Dana Ron, and Rocco A. Servedio. Testing probability distributions using conditional samples. SIAM Journal on Computing, 2015.
[23] Clément L. Canonne and Ronitt Rubinfeld. Testing probability distributions underlying aggregated data. In Proceedings of ICALP, pages 283-295, 2014.

# News for July 2016

After a few slow months, property testing is back with a bang! This month we have a wide range on results ranging from classic problems like junta testing to new models of property testing.

Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism, by Eric Blais, Clément L. Canonne, Talya Eden, Amit Levi, and Dana Ron (arXiv). The problem of junta testing requires little introduction. Given a boolean function $$f:\{-1,+1\}^n \mapsto \{-1,+1\}$$, a $$k$$-junta only depends on $$k$$ of the input variables. A classic problem has been that of property testing $$k$$-juntas, and a rich line of work (almost) resolves the complexity to be $$\Theta(k/\epsilon)$$. But what of tolerant testing, where we want to tester to accept functions close to being a junta? Previous work has shown that existing testers are tolerant, but with extremely weak parameters. This paper proves: there is a $$poly(k/\epsilon)$$-query tester that accepts every function $$\epsilon/16$$-close to a $$k$$-junta and rejects every function $$\epsilon$$-far from being a $$4k$$-junta. Note that the “gap” between the junta classes is a factor of 4, and this is the first result to achieve a constant gap. The paper also gives testers when we wish to reject functions far from being a $$k$$-junta (exactly matching the definition of tolerant testng), but the tester has an exponential dependence on $$k$$. The results have intriguing connections with isomorphism testing, and there is a neat use of constrained submodular minimization.

Testing Pattern-Freeness, by Simon Korman and Daniel Reichman (arXiv). Consider a string $$I$$ of length $$n$$ and a “pattern” $$J$$ of length $$k$$, both over some alphabet $$\Sigma$$. Our aim is to property test if $$I$$ is $$J$$-free, meaning that $$I$$ does not contains $$J$$ as a substring. This problem can also be generalized to the 2-dimensional setting, where $$I$$ and $$J$$ are matrices with entries in $$\Sigma$$. This formulation is relevant in image testing, where we need to search for a template pattern in a large image. The main results show that these testing problems can be solved in time $$O(1/\epsilon)$$, with an intriguing caveat. If the pattern $$J$$ has at least $$3$$ distinct symbols, the result holds. If $$J$$ is truly binary, then $$J$$ is not allowed to be in a specified set of forbidden patterns. The main tool is a modification lemma that shows how to “kill” a specified occurrence of $$J$$ without introducing a new one. This lemma is not true for the forbidden patterns, resulting in the dichotomy of the results.

Erasure-Resilient Property Testing, by Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma (arXiv). Property testing begins with a query model for $$f: \mathcal{D} \mapsto \mathcal{R}$$, so we can access any $$f(x)$$. But what if an adversary corrupted some fraction of the input? Consider monotonicity testing on the line, so $$f: [n] \mapsto \mathbb{R}$$. We wish to test if $$f$$ is $$\epsilon$$-far from being monotone, but the adversary corrupts/hides an $$\alpha$$-fraction of the values. When we query such a position, we receive a null value. We wish to accept if there exists some “filling” of the corrupted values that makes $$f$$ monotone, and reject if all “fillings” keep $$f$$ far from monotone. It turns out the standard set of property testers are not erasure resilient, and fail on this problem. This papers gives erasure resilient testers for properties like monotonicity, Lipschitz continuity, and convexity. The heart of the techniques involves a randomized binary tree tester for the line that can avoid the corrupted points.

Partial Sublinear Time Approximation and Inapproximation for Maximum Coverage, by Bin Fu (arXiv). Consider the classic maximum coverage problem. We have $$m$$ sets $$A_1, A_2, \ldots, A_m$$ and wish to pick the $$k$$ of them with the largest union size. The query model allows for membership queries, cardinality queries, and generation of random elements from a set. Note that the size of the input can be thought of as $$\sum_i |A_i|$$. Can one approximate this problem without reading the full input? This paper gives a $$(1-1/e)$$-factor approximation that runs in time $$poly(k)m\log m$$, and is thus sublinear in the input size. The algorithm essentially implements a greedy approximation algorithm in sublinear time. It is shown the the linear dependence on $$m$$ is necessary: there is no constant factor approximation that runs in $$q(n) m^{1-\delta}$$ (where $$n$$ denotes the maximum cardinality, $$q(\cdot)$$ is an arbitrary function, and $$\delta > 0$$).

Local Testing for Membership in Lattices, by Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, and Elena Grigorescu (arXiv). Inspired by the theory of locally testable codes, this paper introduces local testing in lattices. Given a set of basis vectors $$b_1, b_2, \ldots$$ in $$\mathbb{Z}^n$$, the lattice $$L$$ is the set of all integer linear combinations of the basic vectors. This is the natural analogue of linear error correcting codes (where everything is done over finite fields). Given some input $$t \in \mathbb{Z}^n$$, we wish to determine if $$t \in L$$, or is far (defined through some norm) from all vectors in $$L$$, by querying a constant number of coordinates in $$t$$. We assume that $$L$$ is fixed, so it can be preprocessed arbitrarily. This opens up a rich source of questions, and this work might be seen as only the first step in this direction. The papers shows a number of results. Firstly, there is a family of “code formula” lattices for which testers exists (with almost matching lower bounds). Furthermore, with high probability over random lattices, testers do not exist. Analogous to testing codes, if a constant query lattice tester exists, then there exists a canonical constant query tester.