Group testing is a fundamental and long-studied problem in statistical inference, which was introduced in the 1940s by Dorfman [23] . The goal is to detect a set of k defective items, for a known parameter k, out of population of size p using n p tests. To achieve this, one is allowed to utilize a procedure that is able to test items in groups. Each test is returned positive if and only if at least one item in the group is defective.
As an illustrative example, group testing can be applied in the context of medical testing, enabling us to efficiently screen a population for a rare disease. In this setting, we assume that we have an estimate on the number of infected individuals (who correspond to the "defective items"), as well as a way to take a sample, of say saliva or blood, from each individual, and test it. The idea then is that the number of tests needed to identify the infected individuals can be dramatically reduced by pooling samples. Indeed, utilizing group testing for pooling samples in medical tests was the original idea of Dorfman, and it is also highly relevant nowadays due to the ongoing COVID-19 pandemic [44, 51, 56] . Besides medical testing though, group testing has found several real-world applications in a wide variety of areas, including communication protocols [6] , molecular biology and DNA library screening [16] , pattern matching [17] , databases [20] and data compression [32] .
From an information theoretic point of view, the number of tests required depends on various assumptions on the mathematical model used, and the reader is referred to the survey of Aldridge, Johnson and Scarlett [5] for details on the several models that have been studied. Here we focus on the sublinear sparse regime, i.e., the case where k scales sublinearly with p; k/p → 0, as p → +∞. A standard sublinear setting in the group testing literature is when k = p α for some constant sparsity parameter α ∈ (0, 1) [5] . This is the most interesting regime from a mathematical perspective, but also the one that is suitable for modeling the early stages of an epidemic [55] in the context of medical testing. Further, we are interested in the so-called approximate recovery task (also known as "partial" recovery), where we tolerate having some small number of incorrectly classified items. Note that, since in practical settings we only expect to have an estimation on the input number of defective items k, incorrect classifications may be unavoidable even if we are able to guarantee "exact" recovery. Note also that in several applications it might not be required for both false-positive errors (non-defective items incorrectly classified as defective) and false-negative errors (defective items incorrectly classified as non-defective) to be zero. As an example, when screening for diseases, a small number of false-positive errors might arguably be a small cost to pay compared to performing many more pooled tests.
More formally, let σ * denote the set of defective items andσ be the output of our estimator. For a set of items σ let σ := [p] \ σ denote the complement of σ. Given d ∈ N let Similar to the above definition, everywhere in this work, we say than a sequence of events (A p ) p∈N happen asymptotically almost surely (a.a.s.) as p → +∞ if lim p→+∞ Pr [A p ] = 1.
Finally, in this paper we consider the well-known and very simple to implement non-adaptive Bernoulli group testing design, where all tests are conducted in parallel (non-adaptivity) and each item is part of any certain test with some fixed probability and independently of each other item. Despite its simplicity, Bernoulli group testing is asymptotically information theoretically optimal in the context of (1 − o(1))approximate recovery in the following sense. (The first part of Theorem 1.2 is proven in [46] , while the second part in [47] .) Theorem 1.2 ( [46, 47] ). Let k, p, d ∈ N with 1 ≤ k ≤ p. We assume that k, p → +∞ with k = o(p). Fix parameters γ, η ∈ (0, 1). Under non-adaptive Bernoulli group testing in which each item participates in a test with probability ν/k, and d = γk, we have the following:
(a) With ν satisfying 1 − ν k k = 1 2 , there exists an estimator with the property that lim p→+∞ Pr d [err] = 0, provided that n > (1 + η)n * , where n * = k log 2 p k .
(b) For any test design and any estimator, in order to achieve lim p→+∞ Pr d [err] = 0 it is necessary that n > (1 − η)n * γ , where n * γ = (1 − γ)k log 2 p k = (1 − γ)n * .
Remark 1.1. Strictly speaking, the result in [46] requires ν = ln 2 for Part (a) of Theorem 1.2, which is though equal asymptotically to the value of ν as we stated it. In Lemma 1.3 we provide a slightly more general result which justifies this minor change.
Remark 1.2. It is known [19] that performing n exact tests is asymptotically a necessary and sufficient condition (information theoretically) for exact recovery in the non-adaptive group testing setting, where k = p α , α ∈ (0, 1), n exact = max k ln 2 log 2 k, n * = max α ln 2 , 1 − α p α log 2 p.
(1)
Observe that for α > 1 1+ 1 ln 2 ≈ 0.41 the first term in (1) dominates and, therefore, the approximate recovery bound promised by Theorem 1.2 provides an improvement in the number of tests required.
We remark that the estimator promised by the first part Theorem 1.2 is in principle not computationally efficient. In particular, Theorem 1.2 refers to exhaustive search towards finding a set of k items which (nearly) satisfies all the tests, namely none of the k items participate in a negative test, and at least one of them participates in each positive test. (See Lemma 1.3 for an exact statement and [46, 47] for a relevant discussion). Note also that exhaustive search in principle takes p k time. The currently best known efficient algorithm [48] for (1 − o(1))-approximate recovery under nonadaptive Bernoulli group testing requires (ln 2) −1 · n * tests asymptotically. This gap between the number of tests above which recovery is information theoretically possible, and the number of tests required by the currently best known algorithms to succeed, is usually referred to as a computational-statistical gap. The extent to which this gap is of fundamental nature has drawn significant attention in the group testing community and has been explicitly posed as one of nine open problems in the recent theoretical survey on the topic [5, Section 6, Open Problem 3] . The natural belief that simple "random-design" settings exhibit a potentially fundamental such computational-statistical gap, similar to the one observed in the infamous planted clique model (see e.g. the introduction on [30] and below), is one of the main reasons why researchers in the area of group testing have studied test designs other than Bernoulli testing (see e.g. [19, 34, 41, 54] ). Indeed, recovery in the non-adaptive Bernoulli group testing setting can be seen as equivalently solving planted Set Cover instances coming from a certain random family. (Recall that Set Cover, similar to Max Clique, is an NP-complete problem. We discuss this connection in Remark 1.5.) Notably, in the context of exact recovery via non-adaptive testing, the line of work going beyond the Bernoulli design culminated with the very recent paper of Coja-Oghlan et.al. [19] which shows that n exact is the information-theoretic/algorithmic phase transition threshold. Specifically, Coja-Oghlan et.al. provide a sophisticated test design which is inspired by recent advances in coding theory known as spatially coupled low-density parity check codes [25, 37] , which matches the information theoretical optimal bound, n exact . In addition, they propose an efficient algorithm which, given as input n exact tests (asymptotically) from their test design, successfully infers the set of defective items with high probability.
In the light of Theorem 1.2, Remarks 1.1, 1.2,the simplicity of the Bernoulli group testing design, and our discussion above, a natural and important question both from a theoretical and application point of view is the following:
Is there a computational-statistical gap under non-adaptive Bernoulli group testing for the task of (1 − o(1))-approximate recovery?
Our main contribution in this paper is to provide extensive theoretical and experimental evidence suggesting that, perhaps surprisingly, such a gap does not exist. We do so by showing that in some, potentially fundamental, geometric way the Bernoulli group testing behaves differently than other models exhibiting computational-statistical gaps. In particular, we provide evidence which indicate that the inference task at hand can be performed efficiently via a very simple local search algorithm.
We stress that, given the recent results of Coja-Ohglan et.al. [19] regarding exact recovery, it would not be surprising if a "spatial coupling"-inspired test design also succeeds in closing the computationalstatical gap in the approximate recovery task. After all, "spatial coupling"-inspired models are known to not exhibit gaps that appear in simpler random models in all the contexts they have been applied, see e.g. [2] for a discussion of this phenomenon in the context of random constraint satisfaction problems and e.g. [22, 36] in the context of compressing sensing. However, the main message of this paper is that the very simple non-adaptive Bernoullig group testing design may already be sufficient for computationally efficient (1 − o(1))-approximate recovery.
We discuss our results informally forthwith, and we present them formally in Sections 2 and 3. We remark that throughout the paper we will be interested in different values of the parameter ν, which recall that dictates the probability ν/k with which each item participates in a certain test, being precise in each of our statements.
Computational gaps between what existential or brute-force/information-theoretic methods promise and what computationally efficient algorithms achieve arise in the study of several "non-planted" models like random constraint satisfaction problems, see e.g. [1, 18, 28] , but also several "planted" inference algorithmic tasks, like high dimensional linear regression [29, 31] , tensor PCA [7, 43] , sparse PCA [27, 8] and, of course, the planted clique problem [33, 12, 30] . In the latter context the gaps are commonly referred to as computational-statistical gaps, as we mentioned above. Often times such gaps are explained by the presence of a geometric property in the solution space of the problem known as Overlap Gap Property (OGP), which has been repeatedly observed to appear exactly at the regime where local algorithms cannot solve efficiently the problem. Additionally, it has also been observed that at the absence of this property the problem is amenable to simple local search algorithms. OGP is a notion originated in spin glass theory and the groundbreaking work of Talagrand [52] and, in some form, it has been originally introduced in the context of computational gap for the "non-planted model" of random k-SAT [3, 40] . Recently it has been defined and analyzed also in the context of computational-statistical gaps for the high dimensional linear regression model [29, 31] , the tensor PCA model [7] , the planted clique model [30] and the sparse PCA model [8, 27] . In all such planted models, it is either proven, or suggested by evidence, that the OGP phase transition takes place exactly at the point where we observe and expect local algorithms to work. From a rigorous point of view, the existence of OGP has been proven to imply the failure of various MCMC methods [30, 27, 8] , yet a rigorous proof that at its absence local method works remains one of the important conjectures in this line of research. Interestingly, progress in the last direction has been recently made in the context of certain "non-planted" mean field spin glass systems [42, 4] where under the conjecture of the absence of a property similar to OGP the success of a certain "local" Approximate Message Passing method has been established. In this work we attempt to locate exactly the OGP phase transition (if it exists at all) in order to understand if the non-adaptive Bernoulli group testing exhibits a computational-statistical gap.
It should be noted that over the recent years researchers have approached the study of whether such gaps are fundamental from various different angles other than the OGP. For example, researchers have studied average-case reductions between various inference models (see e.g. [13, 15] and references therein), the performance of various restricted classes of inference algorithms, such as low-degree methods (see e.g. [38, 50] ), message passing algorithms such as Belief Propagation and Approximate Message Passing (see e.g. [11, Chapters 3, 4] and references therein), and statistical-query algorithms (see e.g. [24] ).
To conclude, in line with the thread of research of studying computational-statistical gaps via their geometric phase transitions, in this paper we seek to understand whether the optimization landscape of the (1 − o(1))-approximate inference task under Bernoulli group testing is smooth enough for local search algorithms to succeed. Towards that end, we report the following.
Absence of the Overlap Gap Property (OGP) For our first and main result we provide first moment evidence suggesting that the landscape of the optimization problem corresponding to the inference task does not exhibit the Overlap Gap Property accross the regime where inference is information-theoretic possible. We state our result formally in Section 2.2, where we also give details regarding its technical aspects. Here we informally discuss why we believe this is a potentially fundamental reason for computational tractability, and what we mean when referring to "first moment evidence".
A first observation towards approximate recovery is that one can remove from consideration any item that participates in a negative test, as such an item is certainly non-defective. We call the remaining items as potentially defective. We also say that a potentially defective item explains a certain positive test if it participates in it. The following key lemma informs us that finding a set of k potentially defective items whose elements explain all but a few positive tests implies approximate recovery. Its proof can be found in Appendix A.
We assume that k, p → +∞ with k = o(p). Fix parameters δ, ∈ (0, 1). Assume we observe n ≥ (1 + )k log 2 p k tests under non-adaptive Bernoulli group testing in which each item participates in a test with probability ν/k, where ν satisfies (1 − ν k ) ν = 1 2 . There exists an > 0 such that every set of size k which satisfies at least (1 − )n tests, it must contain at least (1 − δ)k defective items, a.a.s. as p → +∞.
Based on Lemma 1.3, to perform optimal approximate recovery we consider the case where ν > 0 satisfies (1 − ν k ) k = 1 2 and consider the task of minimizing the number of unexplained tests over the space of k-tuples of potentially defective items, which we denote by Ω k . In particular, we seek to understand "smoothness" properties of the underlying optimization landscape.
Informally, the Overlap Gap Property (OGP) with respect to this minimization problem says that the near-optimal solutions of this problem form two disjoint and well-separated clusters, one corresponding to sets in Ω k which are "close" to the set of defective items, and one corresponding to sets in Ω k which are "far" from it. (See Definition 2.3.) At the presence of such a disconnectivity property, one can rigorously prove that a class of natural local search algorithms fails (see e.g. [30] ). As mentioned above a highly non-rigorous, yet surprisingly accurate in many contexts computational prediction [29, 30, 31, 8] is that the OGP can be the only type of computational bottlenecks there is for local algorithms. In other words, the prediction is that, at its absence, an appropriate local search algorithm works efficiently and can solve the optimization problem to (near) optimality. In this work, we provide "first moment evidence" that the OGP never appears in the landscape of the mentioned minimization problem when inference is possible, i.e. when n ≥ (1 + )k log 2 p k for any > 0. Now in order to explain what we mean by "first moment evidence" for the absence of the OGP, let us point out that the standard way of studying the existence of OGP is by studying the monotonicity of the function φ( ), namely the minimum possible value of unexplained tests, which is minimized over the sets in Ω k that contain exactly defective items (has overlap with the set of defective items equal to ) (see e.g. [29, 30] ). This is because a necessary implication of the existence of OGP is that, roughly speaking, φ( ) is not decreasing. (See Lemma 2.4 for a formal statement.) Intuitively, such a connection holds since OGP roughly implies that the overlap values for which φ( ) is small (few unexplained sets) are either "large" (corresponding to large intersection with the true defective items -the "close" cluster) or "small" (corresponding small intersection with the true defective items -the "far" cluster). Hence, such a function cannot be decreasing with .
Towards understanding the monotonicity properties of φ( ) we need to understand the value φ( ) which is the optimal value of a restricted random combinatorial optimization problem. For this we use the moments method. In particular, we define for t, ≥ 0, the counting random variable
where recall that σ * denotes the set of defective items, and let H(σ) denote the number of unexplained tests with respect to σ. Observe that
and that, in particular, by Markov's inequality and Paley's-Zigmund's inequality we have for all t > 0 and ∈ {0, 1, 2, . . . , (1 − )k }:
Hence, if for some t 1 ,
a.s. The employment of the first moment to get an a.a.s. lower bound is called a first moment method, and the employment of the second moment to get an a.a.s. upper bound is called the second moment method.
In many cases of sparse combinatorial optimization problems it has been established in the literature that the first and second moment methods can be proven for t 1 , t 2 sufficiently close to each other, sometimes satisfying even t 2 ≤ t 1 + 2, a phenomenon known as 2-point concentration (see e.g. [14] or the more recent [10] ). Here we say that we provide first moment evidence for the OGP, because we do not check the second moment method to prove the sufficient concentration of measure, but we only use the first moment to derive a prediction for the value on which φ( ) concentrates on. Specifically, we define the function which maps to the value t = t > 0 for which the first moment satisfies E [Σ t, ] = 1, (first moment prediction) and we use this function as an approximation for the value of φ( ). Note that the first moment prediction is expected to correspond to the critical t > 0 above of which this first moment will blow up (where the second moment method can work) and below it will shrink to zero (where Markov inequality works). The first moment prediction has been proven to provide tight predictions for the associated function φ( ) in many setting similar to the group testing where OGP has been studied, such as the regression setting [29] and the planted clique setting [30] . Notably, in these setting the approximation is tight enough so that the existence or not of OGP is directly related to the monotonicity of the first moment function, and furthermore the transition point where this function becomes decreasing corresponds exactly to the point where local algorithms are expected to work. Finally, it is worth pointing out that a similar "first moment", or "annealed complexity", method has been employed for the study of the landscape of the spiked tensor PCA model [9, 45] .
In our work, we explicitly derive the first moment function for the Bernoulli group testing and prove that it remains decreasing (in the sense of Lemma 2.4) throughout the regime n ≥ (1 + )k log 2 p k for any > 0 (see Figure 1 ). Upon a conjectured tightness of the second moment method, we show how this implies the absence of the OGP when n ≥ (1 + )k log 2 p k for any > 0. We consider this notable evidence that a local search algorithm can succeed in this regime, suggesting that there is actually no computational-statistical gap. Remarkably this is in full agreement with our experiments section below. We formalize and further explain all the latter statements in Section 2.2.
Absence of bad local minima For our second theoretical contribution we study a much more strict notion of optimization landscape smoothness, namely the absence of "bad" local minima. This is a very stringent, but certainly sufficient, condition for the success of even greedy local improvements algorithms. Hence, it is naturally not expected to coincide with the OGP phase transition where a potentially more elaborate local algorithm could be needed (see [31] for a similar result in the context of regression). Yet, our evidence that the OGP does not appear for the Bernoulli group testing model when n ≥ (1 + )k log 2 p k for any > 0 , conceivably suggests that the landscape of the minimization problem could be smooth enough to not even contain bad local minima when n ≥ (1 + )k log 2 p k for some reasonably small values of > 0. We consider the same objective function as in the previous paragraph, but this time we study the space of k -tuples of potentially defective items, where k = (1 + )k for any fixed ∈ [0, 1). Crucially, we now study all the possible Bernoulli group testing designs by considering a fixed by arbitrary parameter ν (recall that for each test, each item is tested with probability ν/k). Informally, a bad local minimum σ is a k -tuple of potentially defective items that does not contain a non-negligible number of defective items, and such that every other k -tuple in Hamming distance two from σ explains at most the same number of positive tests. We give a formal definition in Section 2.3, Definition 2.11.
Our result is a bound on the number of tests required for the absence of bad local minima as a function of parameters , δ, ν, where we are interested in (1 − − δ)-approximate recovery. We state the theorem informally below, and formally in Section 2.3, Theorem 2.12.
Theorem 1.4 (Informal Statement). Let k, p ∈ N with 1 ≤ k ≤ p. We assume that k, p → +∞ with k = o(p). Fix parameters ν > 0, R ∈ (0, 1), δ, ∈ [0, 1) such that δ > 0, set k = (1 + )k , and assume that we observe the outcome of n = log 2 ( p k ) R tests under non-adaptive Bernoulli group testing in which each item participates in a test with probability ν/k. If
where Q = Q(λ, ζ, ν, ) is given in (8) (due to its elaborate form), then there exists no bad local minima with respect to (1 − − δ)-approximate recovery in the space of k -tuples of items. Remark 1.3. Note that for any ν > 0, n easy = ( νe −ν ln 2 ) −1 log 2 p k is the number of tests required for (1 − o(1))-approximate recovery via the straightforward algorithms known as Combinatorial Orthogonal Matching Pursuit (COMP) and Definite Defectives (DD). COMP simply amounts to outputting every item that does not participate in a negative test. DD amounts to first discarding every items that participates in a negative test, and then outputting the set of items which have the property that they are the only item in a certain positive test (see also [5] ).
According to Remark 1.3, Theorem 1.4 quantifies the decrease in the number of tests that is possible (with respect to the requirements of straightforward algorithms like DD and COMP), while at the same time guaranteeing that the landscape remains smooth enough to not contain any bad local minima. Nonetheless, the final result includes a complicated optimization problem of some large deviation function max λ≥0 min ζ∈[0,1−δ) Q(λ, ζ, ν, ), which the bigger it is, the more tests can be saved (and of course it is always non-negative as it can be directly checked that it obtains the value 0 for λ = 0).
After proper inspection of the properties of the function Q, and exploiting the facts that in Theorem 1.4 we allow to be positive and ν to take any value of our choice, we manage, as a corollary, to (slightly) improve the state-of-the art result regarding the number of tests required for efficient approximate recovery under the restriction that no false-negatives are allowed. As we have already mentioned, such a requirement is potentially desirable in medical testing. We state the corollary informally below and formally in Section 2.3, Corollary 2.13.
. Under non-adaptive Bernoulli group testing in which each item participates in a test with probability ln(5/2)/k, there exists a greedy local search algorithm such that, asymptotically almost surely, given as input at least 1.829 log 2 p k tests, it outputs a set of 1.01k items that is guaranteed to contain every defective item.
To put Corollary 1.5 in to context, let us point out that if k = p α , α ∈ (0, 1) and we make no assumptions on the value of α ∈ (0, 1), then the best known algorithm (in terms of the number of tests it requires to succeed) for approximate recovery with the guarantee of never introducing false-negative errors is COMP [5] . Indeed, COMP succeeds given as input at least n COMP ≈ 1.883 log 2 p k tests. If α is known to be appropriately small then, to the best of our knowledge, the currently best known algorithm among the ones that never introduce false-negative errors is the so-called Separate Decoding algorithm of [48] . In particular, Separate Decoding requires as input (asymptotically) at least
tests, assuming we choose the value of ν so that it satisfies (1−ν/k) k = 1/2. (Note that ln 2 ≈ 0.693.) Thus, if α is sufficiently large, say α ≥ 0.559, then it can be easily checked that the algorithm of Corollary 1.5 outperforms both COMP and Separate Decoding on the task of approximate recovery with the guarantee of never introducing false-negative errors. As a final remark, the proof of Theorem 1.4 is conceptually straightforward but technically elaborate. We thus view it as an important first step towards a rigorous understanding of the optimization landscape. The main proof strategy is presented in Section 5, while the proofs of some of the more technical intermediate lemmas are deferred to Appendix C.
Experimental results: The Smallest Satisfying Set estimator via local search. In our previous results we provided evidence for the absence of OGP when n ≥ (1 + )k log 2 p k for every > 0 and showed that for some reasonable values of > 0 the landscape is smooth enough that greedy local search methods work. Naturally, one would like to verify that the OGP prediction is correct and that local search methods can successfully (1 − o(1))-approximately recover for any > 0. In our last set of results, we fix the Bernoulli group testing design with ν satisfying (1 − ν k ) k = 1 2 and we investigate experimentally to what extent approximate recovery is indeed amenable to local search. We discuss our findings here and, more extensively, in Section 3, where we also present the results of the simulations we conducted. The key takeaways are the following: 2. Based on this observation, we propose an approach for solving the so-called Smallest Satisfying Set (SSS) problem via local search. Solving the SSS problem is a theoretically optimal method both for exact and approximate inference, which does not require prior knowledge of the parameter k. It is typically approached in practice by being modeled as an Integer Program and solved by Branch and Bound methods and Linear Programming Relaxation techniques, as it is equivalent to solving a certain random family of instances of the NP-complete Set Cover problem. (The reader is referred to Chapter 2, Section 2 in [5] for more details on the SSS problem.) Our experiments suggest that the random family of Set Cover instances induced by the non-adaptive Bernoulli group testing problem might in fact be tractable, and this may be of practical interest.
8
To describe the algorithms we implemented we need the following definition.
(a) every positive test contains at least one item from S;
(b) no negative test contains any item from S.
In other words, a set is satisfying if it explains every positive test and none of its elements participates in a negative test. Clearly, the set of defective items is a satisfying set.
Recalling now Lemma 1.3, we see that solving the k-Satisfying Set (k-SS) problem , i.e., the problem of finding a satisfying set of size k, guarantees (1 − o(1))-approximate recovery. (In fact, Lemma 1.3 implies that even a near-optimal solution suffices, but we will not need this extra property for the purposes of this section.) Based on this observation, we now describe a simple algorithm for solving the k-SS problem which is essentially the well-known Glauber Dynamics Markov Chain Monte Carlo algorithm.
Glauber Dynamics is a simple local Markov chain that was originally used from statistical physicists to simulate the so-called Ising model (see e.g. [21] ). More generally though, Glauber Dynamics is an algorithm designed for sampling from distributions with exponentially large support which has received a lot of attention due to its simplicity and wide applicability, see e.g. [39] . Here we use it for optimization purposes, with the intention to exploit the fact that its stationary distribution assigns the bulk of its probability mass to nearly-optimal states.
Specifically, given a Bernoulli group testing instance, let PD denote the set of potentially defective items, which recall that is the set of items that do not participate in any negative test. Recall also Ω k is the state space of our algorithm, i.e., the set of all possible subsets of exactly k potentially defective items which do not belong in any negative test. Finally, for a set (k-tuple) σ ∈ Ω k let P (σ) denote the number of positive tests explained by σ. To solve the k-SS problem we will use the following simple local algorithm. for i = 1 to N do 5: Pick an item i ∈ σ uniformly at random. 6: Pick an item in j ∈ PD \ σ uniformly at random. 7: Let τ = (σ ∪ {j}) \ {i} 8: Move to τ with probability e βP (τ ) e βP (τ ) +e βP (σ)
if P (σ) = |PosTests| then
We found a k-SS 10: return σ return σ Remark 1.4. Note that β, sometimes called the inverse temperature, is a parameter to be chosen by the user. In our experiments we choose β = 5, but we have noticed that the proposed algorithm is actually quite robust with respect to the choice of β. Note also that we always run our algorithms for at most N = 20|PD| ln |PD| = O(p ln p) steps, i.e, for a nearly linear number of steps.
The main outcome of the experiments we conducted with Glauber Dynamics is that we had an almost perfect (nearly probability one) success rate with respect to solving the k-SS problem to exact optimality given at least (1 + ) log 2 p k tests as input (see Section 3) for any ≥ 0 we tried. This certainly supports the OGP prediction which, in fact, only suggest that one can solve the problem to near-optimality, since it shows that it is solvable to exact optimality. We also check the performance of the algorithm in terms of exact/approximate recovery. While, as we discussed from a theory standpoint, approximate recovery should be guaranteed for any optimal solution (any satisfying set), in our experiment we do observe a significant but not perfect success in terms of approximate recovery. We believe this is related to the naturally bounded values of k, p we consider (in the order of thousands), since we also observe that the success in the recovery task get increasingly better as we increase k, p.
Motivated by our success in solving to optimality the k-SS problem, and as an attempt to check our success without the knowledge of the value of k, we investigate next the performance of a local search approach for solving the SSS problem which we describe below.
As the name suggests, the SSS problem amounts to finding the smallest satisfying set without assuming prior knowledge of k. It is based on the idea that the set of defective items is a satisfying set, but since defectivity is rare, the latter is likely to be small in size compared to the rest satisfying sets. More formally, we have the following corollary of Lemma 1.3 which implies that solving the SSS is a theoretically optimal approach for approximate recovery.
We assume that k, p → +∞ with k = o(p). Fix parameters η, γ ∈ (0, 1). Assume we observe at least n = (1 + η)k log p k tests under non-adaptive Bernoulli group testing in which each item participates in a test with probability ν/k, where ν satisfies (1 − ν k ) ν = 1 2 . The smallest satisfying set contains at least (1 − γ)k defective items a.a.s. as p → +∞.
Proof. We know that the smallest satisfying set is of size at most k, since the set of defective items is satisfying and has size k. Applying Lemma 1.3 with parameters δ = γ and = η, we also know that every other satisfying set of size k contains at least (1 − γ)k defective items. So assume that the smallest satisfying set σ has k < k items, and that it contains less than (1 − γ)k defective items. Observe now that we can form a new satisfying set σ by adding k − k non-defective items to σ. By construction, σ is a satisfying set of size k that contains less than (1 − γ)k items, contradicting Lemma 1.3, and thus, concluding the proof of the corollary.
Remark 1.5. The SSS problem for a given group testing instance is equivalent to the following Set Cover instance: Consider an element for each positive test, and a set for each potentially defective item containing every positive test (element) in which it participates. Finding the smallest satisfying set amounts to finding the smallest in size set that covers every element, i.e., solving the corresponding Set Cover problem.
Given the Glauber Dynamics algorithm, our approach to solving the SSS is straightforward. We always start by eliminating every item that is contained in a negative set, so that we are left with the set of potentially defective items. We then use the Glauber Dynamics algorithm as an oracle for solving the k -SS problem for every k ∈ {1, . . . , p} and proceed by the standard technique for reducing optimization problems to feasibility problems via binary search.
In Section 3 we present results from simulations suggesting that our approach for solving the SSS problems, besides theoretically optimally in terms of recovery, is efficient and furthermore is able to outperform the other known popular algorithms for approximate inference of the set of defective items (including DD, COMP among others) when n ≥ (1 + )k log 2 ( p k ) for any > 0. As a final remark, our results are in agreement with earlier experimental findings on Markov Chain Monte Carlo algorithms for noisy group testing settings in applied contexts, such as computational biology [35, 49] and security [26] , but we note that our work is the first one to provide a concrete theoretical explanation for their strong performance in simulations, and a principled way to exploit local algorithms for implementing the SSS estimator.
The paper is organized as follows. In Section 2 we formally statement our theoretical results, Theorems 2.7, 2.9 and 2.12. In Section 3 we present our experimental results. In Section 4 we prove Theorem 2.7, Theorem 2.9 and Theorem 2.10. In Section 5 we prove Theorem 2.12. Several technical proofs are deferred to Appendices A, B, C, D.
In this section we formally present our theoretical results.
We start with formally defining the inference model of interest. There are p items, k out of them are defective and chosen uniformly at random among the total p items. We denote by θ * ∈ {0, 1} p the indicator vector of the defective items. We perform n tests, where at each test each item is independently included in the test with probability ν/k, where ν > 0 is some positive constant. In most cases we consider ν satisfying (1 − ν k ) k = 1 2 , as informed by the first part of Theorem 1.2, but in some cases we choose some other more suitable value of ν in which case we clearly state it. For i = 1, . . . , n we call X i ∈ {0, 1} p the indicator vector corresponding to the subset of the items chosen for the i-th test and Y i = 1( X i , θ * ≥ 1) the binary outcome of the i-th test. We also define the test matrix X ∈ {0, 1} n×p the matrix with rows X i , i = 1, . . . , n and the test outcomes vector Y ∈ {0, 1} n the vector with elements Y i , i = 1, . . . , n. Finally, we define by
a quantity we call the rate. With respect to scaling purposes we assume that as p → +∞, k, n → +∞ under the restrictions that R is kept equal to a fixed constant and that k = o(p).
The Bayesian task of interest is to asymptotically approximately recover the vector θ * given access to (Y, X), i.e., to construct an estimatorθ =θ(Y, X) ∈ {0, 1} p with
asymptotically almost surely (a.a.s.), with respect to the randomness of the prior on θ * and the test matrix X, as p → +∞. Here d H (·, ·) denotes the Hamming distance between two binary vectors. Note also that achieving (3) with an estimator of sparsity k implies (1 − o(1))-approximate recovery in the sense of Definition 1.1. Importantly, for our theoretical results, we work under the assumption that k is known to the statistician. As explained in the Introduction (Theorem 1.2), using the introduced notation of the rate, the recovery task of interest is known to be information-theoretic impossible if R > 1 and information-theoretic possible if R < 1. Specficially, if R < 1 one can use the Bernoulli design with ν satisfying (1 − ν k ) k = 1 2 and then use exhaustive search to find a satisfying set. Furthermore if R < νe −ν ln 2 for some ν > 0 one can recover by using the Bernoulli design with parameter ν > 0 and then run the polynomial-time algorithm COMP which simply outputs items that never participated in any negative test. (Recall Remark 1.3.) This leaves open the computational question for which rates νe −ν ln 2 < R < 1 we can recover approximately (in the sense of (3)) using a computationally efficient estimator, the Bernoulli design with parameter ν -and in particular how close we can get to R = 1. Of particular interest is, of course, the case where ν satisfies (1 − ν k ) k = 1 2 where approximate recovery in the sense of (3) is known to be information-theoretically possible for all R < 1.
As we have already mentioned, our first and main theorem provides first moment evidence for the absence of the Overlap Gap Property (OGP) asymptotically up to the information theoretic threshold R = 1 for the task of (1 − o(1))-approximate recovery. We propose this evidence as a potentially fundamental reason for computational tractability in this regime. For this section, and towards this goal, we fix the Bernoulli design for parameter ν > 0 with (1 − ν k ) k = 1 2 . Recall that the optimal estimator in this Bernoulli design, which works for all R < 1, corresponds to finding an (approximately) satisfying set.
Towards understanding the computational difficulty of finding an (approximately) satisfying set in this context we first apply, as a pre-processing step, the COMP algorithm which is known to achieve (1 − o(1))approximate recovery for all R < 1 2 and for the value of ν we chose. Indeed, we know this is true for R < νe −ν ln 2 and then notice that as k → +∞, ν = (1 + o(1)) ln 2 and therefore it holds νe −ν ln 2 = 1+o(1) 2 . For this reason when R < 1 2 the recovery goal is trivially achieved by this polynomial-time preprocessing step. Hence, in what follows we assume R to satisfy 1 2 < R < 1 (we ignore the case where R = 1 2 to avoid any unnecessary criticality issues, and focus solely on higher rates than 1 2 ). Now COMP simply removes from consideration any item that takes part in a negative test. We call any item that is not removed from consideration after this step as potentially defective. The following two lemmas estimate the number of positive tests and the number of potentially defective items. The proofs of Lemmas 2.1, 2.2 can be found in Appendix B. 
asymptotically almost surely as p → +∞.
Lemma 2.2. Suppose 1 2 < R < 1. Let PD denote the set of potentially defective items. For every constant η ∈ (0, 1):
asymptotically almost surely as p → +∞.
Remark 2.1. In the light of Lemmas 2.1, 2.2, in what follows we condition on the values of the random variables P and PD and we assume that their cardinalities, which we denote by n pos and p respectively, satisfy (4) and (5), respectively for a sufficiently small constant η of interest. We also slightly abuse the notation and denote by θ * ∈ {0, 1} p the indicator vector of the defective items, by X i ∈ {0, 1} p the row vector corresponding to the positive test with index i ∈ P, whose non-zero entries indicate the potentially defective items that participate in this test, and we switch to indexing with p .
Note that after the execution of COMP we have in consideration n pos positive tests and p potentially defective items. At this point, and recalling Lemma 1.3, the task of finding an (approximately) satisfying set of cardinality k corresponds to finding a set of k items out of the p which explain all but a vanishing fraction of the positive tests (note that here we used that the number of positive tests is of the same order as the number of total number of tests a.a.s. -Lemma 4). Given a k-sparse θ ∈ {0, 1} p we say that a positive test i ∈ P is unexplained by θ if X i , θ = 0, and by H(θ) we denote the number of unexplained tests with respect to θ, i.e. H(θ) := |{i ∈ P : X i , θ = 0}|. We sometimes call H(θ) the energy at θ, motivated by the statistical physics context where OGP originates from. Notice that θ corresponds to an (approximately) satisfying set if and only if it holds (H(θ) = o(n)) H(θ) = 0. Using this notation, to find an (approximately) satisfying set it suffices to solve to near-optimality the following optimization problem
We make two observations. First the optimal value of (Φ) is clearly zero and achieved by θ * . Second, again by Lemma 1.3, the level of near-optimality which suffices for approximate recovery corresponds to the θ with H(θ) = o(n).
As explained in the Introduction, inspired by statistical physics, and the successful prediction of the computational thresholds at least in the context of the sparse regression [29] and of the planted clique problems [30] , we study the presense/absence of OGP in the landscape of (Φ) to offer a heuristic understanding of the rates in R ∈ ( 1 2 , 1) for which the optimization problem can be solved to near-optimality, or equivalently θ * can be approximately recovered via local methods. As explained also in the Introduction, we provide evidence that the OGP never appears for any R ∈ ( 1 2 , 1). The OGP informally says that the space of near-optimal solution of (Φ) separate into two disjoint clusters, one corresponding to θ which are "close" to θ * (the high overlap cluster) and one corresponding to θ which are "far" to θ * (the low overlap cluster). We now formally define it for any fixed p , k.
if there exists a threshold value r p ∈ R satisfying the following properties.
(1) For any θ ∈ {0, 1} p with θ 0 = k and H(θ ) ≤ r p , it holds that either θ, θ * ≤ ζ p or θ, θ * ≥ ζ p + W p + 1.
(2) There exist θ 1 , θ 2 ∈ {0, 1} p such that θ 1 0 = θ 2 0 = k, θ 1 , θ ≤ ζ p , θ 2 , θ ≥ ζ p + W p + 1 and min{H(θ 1 ), H(θ 2 )} ≤ r p ;
(3) It holds max θ∈{0,1} p : θ 0 =k,ζ p +1≤ θ,θ * ≤ζ p +W p −1 H(θ) ≥ r p + H p .
Let us provide some intuition on the definition of the OGP, as it slightly generalizes the previous definitions used for the Overlap Gap Property in the literature [29, 30, 31] , since we define it adjusted to the task of approximate recovery. The parameter W p corresponds to the width of the OGP and the parameter H p to the height of the OGP. The first condition says that all solutions θ achieving energy H(θ) less than some threshold value r p must either have dot product (overlap) with θ * less than ζ p or bigger than ζ p + W p . This gives rise to two clusters of near-optimal solutions (in the sense of achieving energy at most r n ) whose overlap with θ * differs by at least W p . The second condition makes sure that the two clusters are non-empty. Now, the third condition implies that there is some energy level achieved by some θ with overlap with θ * strictly between ζ p and ζ p + W p and energy higher by at least the height H p > 0, as compared to all energies achieved by solutions in the two clusters. Finally, notice that the requirement ζ p ≥ k 2 p holds, as one can always achieve overlap with θ * of the order (1 + o(1)) k 2 p , by simply choosing a binary k-sparse vector at random, and therefore an OGP for smaller overlap sizes than it is not relevant for recovery, but it is definitely relevant for all overlaps bigger than k 2 p . A highly informal, yet surprising accurate in certain contexts, computational prediction is that local search algorithms attempting to solve (Φ) are able to find in polynomial in p time a θ with θ, θ * > ζ p if and only if for all width levels W p ∈ {1, . . . , k − ζ p − 1} with W p = ω(1) and height levels H p > 0 with H p = ω(log p ) the (Φ) does not exhibit the (ζ p , W p , H p )-OGP a.a.s. as p → +∞. The heuristic intuition behind this prediction is as follows. If two clusters of near-optimal solutions of (Φ) are separated by a growing width W p = ω(1) and height H p = ω(log p ) then no local search algorithm is able to either "jump across" the clusters by tuning the local search radius at Hamming distance W p from its current state (as this would take k W p p −k W p = e Ω(W p log p ) -time), or use natural MCMC methods such as Glauber dynamics with finite temperature to "jump over" the height H p (as this would normally require e H p = e ω(log p ) -time). The prediction now is that unless such an OGP appears, an appropriate local search algorithm works.
We now provide evidence that for all 1 2 < R < 1 such an OGP indeed does not take place in (Φ) when ζ p ≤ (1 − )k, for arbitrarily small fixed > 0. Such a result suggests that locals search methods can obtain overlap (1 − )k, i.e. achieve (1 − )-approximate recovery, in polynomial-time. We first define the following restricted optimization problems for = 0, 1, 2, . . . , k
with optimal value φ( ). Note that Φ( ) is simply Φ constrained on only θ satisfying θ, θ * = . Of course θ * is not known to the statistician, and Φ( ) are considered solely for analysis purposes. Trivially, as spans all possible values of θ, θ * , min =0,1,...,k φ( ) = H(θ * ) = 0. Furthermore, for R < 1 by Lemma 1.3 it holds arg min =0,1,...,k φ( ) = (1 − o(1))k a.a.s. Now we offer a necessary implication of the existence of OGP in terms of the monotonicity of φ( ), which allows us to argue for its abscence in what follows. As mentioned in the introduction, such links between OGP and the monotonicity of φ( ) have appeared in the literature but the exact lemma below is not known, to the best of our knowledge. The proof of Lemma 2.4 can be found in Appendix B. Using Lemma 2.4 we now provide evidence that for any arbitrarily small but fixed ∈ (0, 1) Φ a.a.s. as p → +∞ does not satisfy the (ζ p , W p , H p )-OGP for any choice of parameters ζ p ≤ M = (1 − )k , W p = ω(1) and H p > 0. Notice that the additional stronger condition of H p = ω(log p ) is not required for our argument and only the weaker H p > 0 suffices. To prove the abscence of such an OGP in light of Lemma 2.4 according to which it suffices to study the monotonicity of φ( ) across the different arithmetic progressions with some difference W p = ω(1).
Towards understanding the concentration properties of φ( ) we use the moments method, which we have already explained in the Introduction, but which we repeat here for completeness. We define for t, ≥ 0, the counting random variable
In particular, by Markov's inequality and Paley's-Zigmund's inequality we have for all t > 0 and ∈ {0, 1, 2, . . . , (1 − )k }:
14 Hence if for some t 1 , t 2 > 0 it holds E [Z t 1 , ] = o(1) we have φ( ) > t 1 a.a.s as p → +∞ and if
a.s. The employment of the first moment to get a a.a.s. lower bound is called a first moment method, and the employment of the second moment to get an a.a.s. upper bound is called the second moment method.
As we have already explained in the Introduction, we define the function which maps to the value t = t > 0 for which the first moment satisfies E [Z t 1 , ] = 1 (first moment prediction), which we use as our heuristic approximation for φ( ), which has been a succesful approximation in both regression [29] , the planted clique [27] and sparse PCA models [8] . Now using standard large deviation theory of the Binomial random variable we arrive at the following definition of the first moment prediction.
Definition 2.5. Suppose 1 2 < R < 1. Then for every fixed value of p , n pos satisfying Lemma 2.2 and 2.1 for sufficiently small η > 0 we denote by F p (λ), λ ∈ [0, 1) the first moment function defined implicitly by being the unique function satisfying the following two constraints for each λ ∈ [0, 1),
where α(x, y) := x ln x y + (1 − x) ln 1−x 1−y . Remark 2.2. Here α(x, y), x, y ∈ [0, 1], is the Kullback-Leibler divergence (relative entropy function) between two Bernoulli random variables with probability of success x and y, respectively. It naturally appears here using the large deviation properties of the Binomial distributions.
The non-voidness of Definition 2.5 is given in the following Proposition, along with some basic analytic properties of the first moment function. The proof of Proposition 2.6 can be found in Appendix B.
Proposition 2.6. Suppose 1 2 < R < 1. Then the first moment function F p (λ) exists, is unique and for any > 0 it is continuously differentiable for λ ∈ [0, 1 − ], a.a.s. as p → +∞ (with respect to the randomness of p , n pos satisfying Lemma 2.2 and 2.1 for sufficiently small η. ). Now we prove rigorously using the first moment method that φ( ) can be lower bounded in terms of the first moment function F p ( ).
Theorem 2.7. Suppose 1 2 < R < 1. For every > 0, there exists a constant C = C( ) such that a.a.s. as p → +∞ for every integer ∈ {0, 1, . . . , (1 − )k } we have:
Using the second moment method to establish concentration, we conjecture that the Theorem 2.7 can be strengthened to prove the following result (as explained in the literature, similar results have been proven to be tight in multiple cases where OGP has been studied [29, 10, 30, 8] . One can see that theF remains strictly decreasing throught the difference choices of R < 1 (in accordance with Theorem 2.9) and it converges to the zero function as R tends to 1.
Given Conjecture 2.8 and our desire to study the monotonicity of φ( ) it seems natural to study the monotonicity of the implicitly defined, but potentially easier to handle first moment function F p ( ). We are able to analyze the monotonicity of the first moment function F p ( k ) and prove that it is indeed strictly decreasing as a function of for any R with 1 2 < R < 1. (See also Figure 1 .) The result can be cast as elementary but in fact requires the careful use of various multivariate calculus and asymptotics tools to get around the rather complicated definition of the function F p (λ). Theorem 2.9. Let arbitrary 1 2 < R < 1. For every > 0 there exists a constant D = D( ) > 0 such that a.a.s. as p → +∞ for every ∈ {0, . . . , (1 − )k } we have:
In particular, F p ( ) is strictly decreasing as a function of ∈ {0, 1, 2, . . . , (1 − )k }, a.a.s. as p → +∞.
This monotonicity result and Lemma 2.4 suggests that possibly φ( ) is decreasing as a function of implying indeed the absence of the OGP for all R ∈ ( 1 2 , 1). Yet, to establish this we have to tolerate the error naturally appearing in Theorem 2.7 and Conjecture 2.8. We are able to show that indeed such a error tolerance is possible and we obtain the following result.
Theorem 2.10. Let arbitrary 1 2 < R < 1 and k ≤ p 1−c for some c > 0. Suppose Conjecture 2.8 holds. Then for any > 0 the following is true asymptotically almost surely as p → +∞. For any ζ p ,
Note that in Theorem 2.10 we require the very weak assumption that k ≤ p 1−c for some c > 0 which holds throughout the sublinear regime k = o(p). We consider this assumption to be of technical nature.
Motivated by the evidence for the absence of the OGP for all rates R < 1 in the landscape of Bernoulli group testing with ν with (1 − ν k ) k = 1 2 , which is described in the previous section, we now turn our attention to a much more strict notion of optimization landscape smoothness, namely the absence of "bad" local minima. The fact that OGP may hold for all R < 1, suggests that the absence of bad local minima may hold for reasonable values of R near 1. If true, this has clear rigorous algorithmic implications such as the certain success of greedy local search methods. Furthermore, partial motivation for studying this notion of optimization complexity is a rigorous understanding of the performance of Glauber Dynamics, the algorithm we described in the Introduction, and which we implemented for our experimental results (and which achieves an almost perfect success rate in solving the corresponding optimization problem for all R < 1, see Section 3). Indeed, the absence of bad local minima guarantees the success of the Glauber dynamics algorithm for large values of β and, in particular, for β = +∞.
Recall that PD denotes the set of potentially defective items, that for every integer k ∈ {1, 2 . . . , p}, Ω k denotes the set of all possible subsets of exactly k items which do not participate in any negative test and that, for a state (k -tuple) σ ∈ Ω k , P (σ) denotes the number of positive tests explained by state σ. We consider the above greedy local search algorithm (which can be seen as an "aggressive" version of Glauber dynamics).
In the second theorem of the present work we study the optimization landscape of the [approximate] k -SS problem for k ≥ k. One potential motivation for studying values of k that are larger (but still close to) k is to obtain algorithms for approximate recovery which never introduce false-negatives errors. Indeed, 1: procedure GREEDY LOCAL SEARCH(k , PD, PosTests) 2: σ 0 ← an arbitrary state from Ω k 3:
while P (σ t ) < |PosTests| do 5: For all possible pairs (i, j) ∈ σ t × (PD \ σ t ), let τ ij := (σ t ∪ {j}) \ {i}. 6 :
Move to the state τ ∈ T σt ∪ {σ t } with the highest P -value (solving ties uniformly at random). 8: if σ t = σ t+1 then 9: return σ t 10:
t ← t + 1 return σ t we will show that utilizing our theorem we will be able to slightly improve upon the state-of-the-art results for this task.
To formally state our theorem, we need the following definition.
Definition 2.11. Let k, p ∈ N with 1 ≤ k ≤ p. Fix parameters δ, ∈ [0, 1), and set k = (1 + )k . We say that a state σ ∈ Ω k is a (δ, )-bad local minimum if it contains less than (1 − δ)k defective items and there exists no state τ ∈ T σ such that P (τ ) ≥ P (σ) + 1.
Our second theoretical contribution is a sufficient condition for the absence of (δ, )-bad local minima.
Theorem 2.12. Let k, p ∈ N with 1 ≤ k ≤ p. We assume that k, p → +∞ with k = o(p). Fix parameters ν > 0, R ∈ (0, 1), δ, ∈ [0, 1) such that δ > 0, set k = (1 + )k , and assume that we observe the outcome of n = log 2 ( p k ) R tests under non-adaptive Bernoulli group testing in which each item participates in a test with probability ν/k. If
then there exists no (δ, )-bad local minimum in Ω k .
A concrete consequence of Theorem 2.12 is the following corollary. Its proof can be found in Appendix D.
Corollary 2.13. Let k, p ∈ N with 1 ≤ k ≤ p. We assume that k, p → +∞ with k = o(p). Assume that we observe the outcome of n = log 2 ( p k ) R tests under non-adaptive Bernoulli group testing in which each item participates in a test with probability ln(5/2)/k. If R < 0.5468, then GREEDY LOCAL SEARCH with input k = (1.01)k terminates in at most n steps almost surely, and furthermore outputs a k -tuple that contains the k defective items asymptotically almost surely as p → +∞. 
In this section we present experimental results that provide further evidence suggesting the absence of a computational-statistical gap in the Bernoulli group testing problem with ν such that (1 − ν k ) k = 1 2 and the tractability of finding a satisfying set for all R ≤ 1. In particular, our experiments are in agreement with the prediction from the absence of the OGP for all R < 1. The key insights from our experimental results can be summarized as follows.
1. The k-SS problem is efficiently solvable to exact optimality via a simple local search algorithm even at rate R = 1.
2. Solving the SSS problem via a local search method is efficient and outperforms other popular algorithms in terms of exact and approximate recovery.
We give more details forthwith. We note that all our experiments were performed in a MacBookPro with a 2.3 GHz 8-Core Intel Core i9 Processor and 16 GB 2667 MHz DDR4 RAM.
In Figure 2 we demonstrate the performance of Glauber Dynamics for solving the k-SS problem on instances where k = p 1/3 and ν satisfies (1 − ν/k) k = 1/2. While the choice of ν should already be wellmotivated, the choice of k is chosen because k = p 1/3 corresponds the largest power of p for which, not only approximate, but even exact recovery is theoretically possible for any R < 1 by the Bernoulli design of choice [5] . Now each data point that corresponds to a particular value of the inverse rate R −1 is computed by creating 100 Bernoulli group testing instances and counting how many times the algorithm succeeds (where success refers to solving the optimization, exact-and approximate-recovery task, respectively). The gray lines, which support our first point above, correspond to the probability that Glauber Dynamics successfully solves the k-SS problem to exact optimality (i.e. finds a satisfying set). Remarkably, we observe an almost perfect success rate in this task (> 99.7%). The orange lines correspond to the probability of successful exact recovery, while the blue lines correspond to the probability of successful 90%-approximate recovery. Note that in terms of exact/approximate recovery the algorithm is not succeeding with high probability when R is close to 1, albeit always outputing a satisfying set, seemingly opposing the theoretical results (Lemma 1.3). We naturally consider this an artifact of considering finite p, which is supported by the second diagram of Figure 3 , where as p increases, the success probability of the approximate recovery task (of solving the harder SSS problem, which does not know the value of k) at a given fixed rate R < 1 also increases, as predicted by Lemma 1.3.
In Figure 3 we demonstrate how our proposed algorithm for solving now the SSS problem (solving the k -SS problem for k = 0, 1, . . . , p and performing binary search) compares to a series of known popular algorithms for 90%-approximate recovery. Again, we choose k = p 1/3 , ν that satisfies (1 − ν/k) k = 1/2, and we compute each data point by creating 100 Bernoulli group testing instances and counting how many times the algorithm succeeds in terms of the 90%-approximate-recovery task. The algorithms we compare our approach are the following. (Some of them we have already discussed, but we include their description here again for completeness. The reader is also referred to [5] for more details.)
• Combinatorial orthogonal matching pursuit (COMP): In the COMP algorithm we remove every item that takes part in a negative test, and output the rest. Each item in the output of COMP is called potentially defective (PD).
• Definite defective (DD): In the DD algorithm we output the set of PD items, which have the property that they are the only PD item in a certain positive test. Each item in the output of DD is called definite defective (DD).
• Sequential COMP (SCOMP): The SCOMP algorithm is defined as follows. (Recalling Remark 1.5, SCOMP is essentially Chvatal's greedy approximation algorithm for Set Cover.)
1. Initialize S to be the set of DD items. 2. Say that a positive test is unexplained if it does not contain any items from S. Add to S the PD item not in S that is in the most unexplained tests, and mark the corresponding tests as no longer unexplained. (Ties may be broken arbitrarily).
Step 2 until no tests remain unexplained. The estimate of SCOMP is S.
• Max Degree (MD) : The MD algorithm sorts the PD items in decreasing order with respect to how many positive tests each such item takes part in. It then outputs the k first PD items in this order.
In this section we prove Theorems 2.7, 2.9 and 2.10 . We will find helpful the following technical results regarding the relative entropy function α(x, y) := x ln x y + (1 − x) ln 1−x 1−y , x, y ∈ [0, 1]. Lemma 4.1. Let x, y ∈ (0, 1 2 ] with x < y be fixed and N → ∞. We have Pr[Bin(N, y) ≤ xN ] = e −N α(x,y)+O(ln N ) .
Lemma 4.2. We will find useful the partial derivatives of α(x, y) with respect to x and y: For any δ ∈ (0, 1), and c 0 = c 0 (δ) > 0 it holds that for all x 0 , x, y > 0 such that x < (1−δ)y, x 0 < x, y ≤ 1 2 , we have:
As a consequence: 
Recall that for any θ ∈ {0, 1} p with θ 0 = k we denote by H(θ) the number of unexplained tests with respect to θ. Denote also by θ * the unknown binary k-sparse vector supported on the true defective items and notice that H(θ * ) = 0. Finally, recall Remark 2.1. Now for t, ≥ 0, let
In particular, by Markov's inequality, for all t > 0 and ∈ {0, 1, 2, . . . , (1 − )k }:
Therefore, in order to prove Theorem 2.7 it suffices to show that there exists an appropriate large constant
To see this notice that combining (9) and (10) implies that for all ≤ (1 − )k it holds φ( ) ≥ n pos F p k − C ln k asymptotically almost surely, which is our claim. Towards that end, fix ∈ {0, 1, 2, . . . , (1 − )k } and θ ∈ {0, 1} p such that θ 0 = k and θ, θ * = , and observe that by the linearity of expectation: Proof. Recall that each item takes part in a certain positive test independently of the other items and tests and that we have conditioned on the value of the random variable P, namely the set of indices of the positive tests. For any fixed i ∈ P:
concluding the proof. (Note that in the above calculation Pr[·] denotes the probability with respect to the original, unconditional probability space.)
Combining (11), Lemma 4.4, and Lemma 4.1 (with N = n pos , x = t npos and y = 1 − 2 −(1− k ) ) we obtain:
.
For large enough p, and therefore large enough n pos according to Lemma 2.1, we can apply Lemma 4.3 with
where for the second inequality we used the definition of the first moment function F p and, in particular, (6) . Overall,
which indeed tends to zero for sufficiently large C since, from our conditioning, n pos = O(n) = O(k ln(p/k)) and k = p α , α ∈ (0, 1), concluding the proof.
We start by showing the following technical lemmas in Appendix B.
Lemma 4.5. Suppose 1 2 < R < 1. There exists δ = δ( ) > 0 such that a.a.s. as p → +∞ it holds for all λ ∈ [0, 1 − ]:
Lemma 4.6. There exists a sufficiently small constant η > 0 such that for all λ ∈ [0, 1 − ] it holds
a.a.s. as p → +∞.
All the constants (included for example in the asymptotic notations) in this proof may depend on the value of .
Let us now fix some ∈ {0, 1, 2, . . . , (1 − )k }. Using (6) and elementary algebra with binomial coefficients we obtain that almost surely:
Let δ be the constant promised by Lemma 4.5 and let us define the compact convex set
for which we have for all λ ∈ [0, 1 − ], (F p (λ), 1 − 2 λ−1 ) ∈ T, a.a.s. as p → +∞. Combining (12) with an application of the two dimensional mean value theorem on T (e.g. by restricting α on the line segment connecting
or equivalently
Now from Proposition 2.6, F p is continuously differentiable in [0, 1 − ]. Therefore, for constants possibly dependent on , we have that it necessarily holds
Furthermore since ≤ (1 − )k we also have uniformly over all such ,
Hence, applying Lemma 4.2 we get:
For similar reasons for constants possibly dependent only on > 0,
Combining (14), (15) allows us to conclude that
Combing now (13) and (16) we obtain:
Now, using Lemma 4.3 and the definition of T we get that for the constant c 0 = ln 2 > 0,
which therefore using (17) shows that it suffices to prove that for some constant D 0 = D 0 ( ) it holds
a.a.s. as p → +∞ and therefore it suffices to show that for some constant D 0 = D 0 ( ) it holds
Now using Lemma 4.6 and simple rearrangement of the terms, we obtain for some constant η > 0,
where for the second inequality we used that k p−k
, which in turn follows by the well-
, which is true for any pair of integers a ≥ b.
Hence, by elementary algebra it holds,
where D 0 is an appropriately small positive constant, since k = o(p), a.a.s. as p → +∞ n pos = Θ(k ln p k ) and ≤ (1 − )k. This completes the proof of the theorem.
Notice that if for any ζ p , W p ∈ {0, 1, . . . , k}, H p > 0 with ζ p ≤ (1 − )k, ζ p + W p ≤ k, W p = ω(1) and H p > 0 the (ζ p , W p , H p )-OGP holds then we can apply Lemma 2.4 for ζ = ζ p , W = W p , H = H p and finally M = (1 − )k (by redefining W p by min{W p , (1 − )k} if necessary) to deduce that for some u = 0, 1, . . . , W p − 1, the function φ( W p + u) needs to be not non-increasing as a function of = 0, 1, . . . ,
Hence it suffices to show that for any > 0 a.a.s. as p → +∞ any W p = ω(1) and any u < W p the function the function φ( W p + u) is decreasing as a function of = 0, 1, . . . , (1 − )k/W p . Notice that now to prove this, it actually suffices to prove that for any > 0 a.a.s. as p → +∞ there exists a constant Q( ) > 0 such that for any 2 
Towards that goal, let us fix some Q = Q( ) that we will take sufficiently large for our needs. Notice that from Conjecture 2.8 for some constant C = C( ), a.a.s. it holds for any such 2 , 1
Now using Theorem 2.9 we also have for some constant D = D( ) > 0 a.a.s. for any such 2 , 1 by telescopic summation,
Combining (21) and (22) we conclude that a.a.s. for any such 2 , 1
Using that we condition on the a.a.s. event that for some sufficiently small η < 1 − 1 2R Lemma 2.2 hold we have a.a.s.
Hence we conclude that a.a.s. for any such 2 , 1 ,
Using now that k ≤ p 1−c for some c > 0 we have that ln p k = Ω(ln k). Hence, indeed we can choose a constant Q( ) > 0 so that as p → +∞ it holds
Combining (24) and (25) completes the proof.
In this section we prove Theorem 2.12. We will assume without loss of generality that R ≥ νe −ν ln 2 . (Indeed, if R < νe −ν ln 2 , then we can utilize only the first log 2 p k /( νe −ν ln 2 ) tests and ignore the rest.) We start the analysis by estimating the number of possibly defective items which are actually nondefective by proving Lemma 5.1. Its proof can be found in Appendix C.
Lemma 5.1. Let q denote the number of possibly defective items which are actually non-defective. For every constant η ∈ (0, 1):
asymptotically almost surely.
We call q the number of possible defective items that are actually non-defective and we condition on its value satisfying the condition of Lemma 5.1 in what follows.
We proceed now towards proving Theorem 2.12. For every ∈ {0, . . . , k − 1} let Ω k ⊆ Ω k denote the set of potentially defective k -tuples that contain exactly defective items. Fix now an integer ∈ {0, . . . , k − 1} and, for each σ ∈ Ω k , let B σ denote the bad event that there exists no state τ ij ∈ T σ such that P (τ ij ) > P (σ) and i, j are a non-defective and a defective item, respectively. Recalling Definition 2.11, to show our result it suffices to prove that
We first upper bound max σ∈Ω k Pr[B σ ] since, by a union bound and the elementary observation that |Ω k | = k q k − , in order to show (27) it suffices to show that
To that end, fix an ∈ {0, 1, . . . , (1 − δ)k − 1} and an arbitrary state σ ∈ Ω k . Let also T * σ ⊆ T σ denote the set of states τ ij such that i is a non-defective item in σ, a set of items we denote by ND σ , and j is a defective item which does not belong in σ, a set of items we denote by D \σ . For each τ ij ∈ T * σ let ∆(τ ij ) = P (τ ij ) − P (σ) denote the random variable that equals the difference between the number of positive tests explained by τ ij and the number of positive tests explained by σ. Let Θ j denote the number of positive tests that contain j and do not contain any item from σ. Let also Θij denote the number of positive tests in which (i) i is the only item in σ that participates in the test and; (ii) j does not participate in the test. Using our notation, it is an elementary observation that
To see this, at first observe that any positive test that contains at least two items from σ is explained both by the set of items in σ and by the set of items of every state in T σ . This is because our algorithm removes (and, therefore, also inserts to) at most one item from σ. Thus, such tests do not contribute to ∆(τ ij ). Therefore it suffices to focus on the contribution of tests with either one or zero items from σ. Now observe that the only tests that contribute positively to ∆(τ ij ) (and increase its value by one) among the ones that contain at most one item from σ, are exactly the ones that contain j and no item of σ. Analogously, the only tests that contribute negatively to ∆(τ ij ) (by decreasing its value by one) among the ones that contain at most one item from σ are the ones in which i is the only item that is contained in σ, and j is not contained in σ. The identity (29) follows.
Using (29) we now bound Pr[B σ ] as follows.
We now prove the following equality in distribution. The proof of Lemma 5.2 can be found in Appendix C.
Lemma 5.2. For every = 0, 1, 2, . . . , k − 1 and σ ∈ Ω k : i∈NDσ,j∈D \σ
where C ( ) q are independently and identically distributed random variables whose distribution samples with
samples from a (scaled) binomial −1 · Bin(k − , 1 − ν k ) and it is 0, otherwise.
A direct corollary of Lemma 5.2 is the following. Its proof can also be found in in Appendix C. 1 Here we mean that the result of the binomial is multiplied by a fixed number, namely, (k − ) in this case.
where T ( ) q are independently and identically distributed random variables whose distribution samples with probability (1 − ν k ) k from a (scaled) binomial random variable k − k− Bin(k − , ν k ), equals to −1 with probability ν k k − k 1 − ν k k −1 1 − 1 − ν k k− , and it is 0, otherwise.
Overall, combining (30) with Corollary 5.3, and using the standard strategy for proving large deviation bounds, we obtain:
Now (31) implies that in order to prove (28) it suffices to find λ ≥ 0 such that
or it suffices to have
or, equivalently, by taking logarithms
It is a straightforward observation using the elementary m 1 m 2 ≤ (m 1 e/m 2 ) m 2 and Lemma 5.1 with a parameter η > 0 to be defined later, that it holds for some constants C 0 , C 1 > 0,
where in deriving the second inequality we used the fact that δ > 0, i.e., at least one of , δ is positive.
Using the last inequality and the fact that n ≤ log 2 ( p k ) R , to show (32) it suffices to show that
where,
. Now using first ln p k ≤ k ln(p/k) + k and second that since T ≤ ln E e λ = λ, it suffices to show that
where,
Now, since k = o(p) and also λ ≥ 0, R > 0 are fixed constants, it suffices to show that
which is equivalent to
Now since it suffices that (33) holds for some η > 0 our condition by continuity is equivalent with
We now prove the following technical lemma in Appendix C. (1 + − ζ) ln 2 .
Combining Lemma 5.4 with (34), it suffices to have
By direct computation for every ζ ∈ [0, 1 − δ) it holds
where µ = µ(λ, , ζ, k) = λ (1+ )k − ζk k− ζk . Since k → +∞ as p → +∞ we conclude by using elementary calculus that
Combining (35) and (37) Q(λ, ζ, ν, ). (38) where recall that
As it suffices that this condition is satisfied by some λ ≥ 0, by optimizing (38) over λ ≥ 0 it suffices to have
concluding the proof.
A Proof of Lemma 1.3
Let us consider some fixed > 0 which will be chosen sufficiently small given the needs of the proof. For = 0, 1, . . . , (1 − δ)k let Z , , be the counting random variable of the number of sets of size k which satisfy at least (1 − )n tests and contain exactly defective items (with overlap ). By two union bounds, first over and then over the different sets of overlap and by fixing for each , an arbitrary set θ with overlap we have 
which now using standard asymptotics n αn ≤ e Cα log 1 α n for α > 0 and some universal constant C > 0 and Choosing δ = η, concludes the proof, since N neg ≥ (1 − δ )n(1 − ν k ) k is chosen arbitrarily, and we have already established that n neg ≥ (1 − δ )n(1 − ν k ) k asymptotically almost surely.
We proceed by analyzing the (clearly i.i.d.) impact of each test to the sum i∈NDσ,j∈D \σ (Θ j − Θ ij ). Note that each test can either contribute positively to the sum i∈NDσ,j∈D \σ (Θ j −Θ ij ), by being counted by variables in the family {Θ j } j∈D \σ , or negatively, by being counted by variables in the family {Θ ij } i∈NDσ,j∈D \σ , or not at all. Below we fix a certain test and estimate its contribution which we denote by C ( ) q . Clearly C ( ) q are i.i.d. random variables and i∈NDσ,j∈D \σ (Θ j − Θ ij ) d = n q=1 C ( ) q . To be counted by the variables in the family {Θ j } j∈D \σ , the test should not contain any item of σ, which happens with probability (1 − ν k ) k . Conditional on this event, the positive contribution of the test to the sum i∈NDσ,j∈D \σ (Θ j − Θ ij ) equals simply to |ND σ | · X = (k − ) · X, where X is the random variable equal to the number of defective items in D \σ tht participate in the test. Clearly, X ∼ Bin(k − , ν k ). For each non-defective item i ∈ ND σ , let E i denote the event that i is the only element in σ that participates in the test, and that the test is positive. Observe that events E i are mutually-exclusive by definition and, therefore, the negative contribution of the test to the sum i∈NDσ,j∈D \σ (Θ j − Θ ij ) is given by
counts the number of items in D \σ that do not participate in the test. Now it is not hard to see that
and, therefore
Combining the findings in the above two paragraphs, we conclude that the random variable C q indeed follows the distribution as described in the statement of the lemma, which completes the proof.
We use the fact that Bin(k − , 1 − ν k ) ≤ k − with probability 1 to obtain that each of the random variables C ( ) q , q ∈ {1, . . . , n} in Lemma 5.2 is stochastically dominated by a random variable D ( ) q , whose distribution samples with probability (1 − ν k ) k from a (scaled) binomial random variable (k − )Bin(k − , ν k ), it is equal to −(k − ) with probability ν k k − k 1 − ν k k −1 1 − 1 − ν k k− , and it is 0, otherwise.
We then observe that we can couple each D and, therefore, we obtain:
concluding the proof.
To prove the lemma we show that the derivative of Q(ζ) is positive. We define f (ζ) = ln e −ν(1+ ) (e ν(1−ζ)(e − λ(1+ −ζ)
1−ζ −1) − 1) + (1 + − ζ)νe −ν(1+ ) (1 − e −ν(1−ζ) )(e λ − 1) + 1 , g(ζ) = (1 + − ζ) ln 2, so that Q(ζ) = −f (ζ)/g(ζ) and, therefore,
We need to show that the righthand side of (63) is positive and, to that end, we need to calculate ∂f (ζ) ∂ζ and ∂g(ζ) ∂ζ . The latter is trivial to calculate: and, thus, ∂h(ζ) ∂ζ = ∂a(ζ) ∂ζ + ∂b(ζ) ∂ζ . We start by calculating the derivative of a(ζ), and by direct calculation we see that 
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In particular, we have:
Combining (66) and (68) we obtain:
We continue by computing the derivative of b(ζ):
We are now ready to prove that the righthand side (63) is negative. Using the notation we have introduced, this amounts to showing that (1 + − ζ) ∂h(ζ) ∂ζ 1 h(ζ) + ln (h(ζ)) < 0,
for every ζ ∈ [0, 1). Since ν = ln(5/2), λ = ln(100/83) and = 1/100 we have: It can be verified computationally that, for every ζ ∈ [0, 1), we have (i) h(ζ) ∈ (0.97, 1); (ii) ∂h(ζ) ∂ζ = ∂a(ζ) ∂ζ + ∂b(ζ) ∂ζ < 0. Therefore, each summand in (71) is strictly negative, concluding the proof. 46