On the Complexity of Random Satisfiability Problems with Planted Solutions

Introduction

Boolean satisfiability and constraint satisfaction problems are central to complexity theory; they are canonical NP-complete problems and their approximate versions are also hard. Are they easier on average for natural distributions? An instance of random satisfiability is generated by fixing a distribution over clauses, then drawing i.i.d. clauses from this distribution. The average-case complexity of satisfiability problems is also motivated by its applications to models of disorder in physical systems, and to cryptography, which requires problems that are hard on average.

Here we study planted satisfiability, in which an assignment is fixed in advance, and clauses are selected from a distribution defined by the planted assignment. Planted satisfiability and, more generally, random models with planted solutions appear widely in several different forms such as network clustering with planted partitions (the stochastic block model and its variants), random $k$-SAT with a planted assignment, and a proposed one-way function from cryptography [Gol00].

It was noted in [BHL+02] that drawing satisfied $k$-SAT clauses uniformly at random from all those satisfied by an assignment $\sigma\in\{\pm 1\}^{n}$ often does not result in a difficult instance of satisfiability even if the number of observed clauses is relatively small. However, by changing the proportions of clauses depending on the number of satisfied literals under $\sigma$, one can create more challenging distributions over instances. Such “quiet plantings” have been further studied in [JMS05, AJM05, KZ09, KMZ14]. Algorithms for planted $3$-SAT with various relative proportions were given by Flaxman [Fla03] and Coja-Oghlan et al. [COCF10], the first of which works for $\Theta(n\log n)$ clauses but excludes distributions close to $3$-XOR-SAT, and the second of which works for all planted $3$-SAT distributions but requires $\Theta(n^{3/2}\ln^{10}n)$ clauses (note that a satisfiable $k$-XOR-SAT formula can be viewed as a satisfiable $k$-SAT formula with the same literals since XOR implies OR). As $k$ increases, the problem exhibits a larger algorithmic gap: the number of clauses required by known algorithms to efficiently identify a planted assignment is $\Omega(n^{k/2})$ while the number at which the planted assignment is the unique satisfying assignment is $O(n\log n)$.

We give a simple model for producing instances of planted $k$-SAT that generalizes and unifies past work on specific distributions for planted satisfiability. In this model, each clause $C$, a $k$-tuple of the $2n$ literals (variables and their negations), is included in the random formula with probability proportional to $Q(y)$ where $y\in\{\pm 1\}^{k}$ is the value of the literals in $C$ on the planted assignment $\sigma$. Here $Q$ can be an arbitrary probability distribution over $\{\pm 1\}^{k}$. By choosing $Q$ supported only on $k$-bit strings with at least one true value, we can ensure that only satisfiable $k$-SAT formulas will be produced, but the model is more general and allows “noisy” versions of satisfiability. We refer to an instance obtained by taking $Q$ to be uniform over $k$-bit strings with an even number of 1’s as $k$-XOR-SAT (since each clause also satisfies an XOR constraint).

Our general formulation of the planted $k$-SAT problem and the notion of distribution complexity reveal a connection between planted $k$-SAT and the problem of inverting a PRG based on Goldreich’s candidate one-way function [Gol00], for which the link between $r$-wise independence and algorithmic tractability was known before [MST06, AM09, BQ09, ABR12]. In this problem for a fixed predicate $P:\{\pm 1\}^{k}\rightarrow\{-1,1\}$, we are given access to samples from a distribution $P_{\sigma}$, for a planted assignment $\sigma\in\{\pm 1\}^{n}$. A random sample from this distribution is a randomly and uniformly chosen ordered $k$-tuple of variables (without repetition) $x_{i_{1}},\ldots,x_{i_{k}}$ together with the value $P(\sigma_{i_{1}},\ldots,\sigma_{i_{k}})$. As in the problem above, the goal is to recover $\sigma$ given $m$ random and independent samples from $P_{\sigma}$ or at least to be able to distinguish any planted distribution from one in which the value is a uniform random coin flip (in place of $P(\sigma_{i_{1}},\ldots,\sigma_{i_{k}})$). The number of evaluations of $P$ for which the problem remains hard determines the stretch of the pseudo-random generator (PRG). We note that despite the similarities between these two types of planted problems, we are not aware of any reductions between them (in Section 6 we show some relationships between these models and an even more general planted CSP model of Abbe and Montanari [AM15]).

Bogdanov and Qiao [BQ09] show that an SDP-based algorithm of Charikar and Wirth [CW04] can be used to find the input (which is the planted assignment) for any predicate that is not pairwise-independent using $m=O(n)$ such evaluations. The same approach can be used to recover the input for any $(r-1)$-wise (but not $r$-wise) independent predicate using $O(n^{r/2})$ evaluations [App16].

Another important family of algorithms for recovering the planted assignment in Goldreich’s PRG is algebraic, based on Gaussian elimination and its generalizations [MST06, AL16]. These attack algorithms are not captured by the framework of statistical algorithms we work with in this paper. While algebraic approaches also apply to planted satisfiability problems, almost all planting functions $Q$ (in a measure-theoretic sense) are resilient against such algorithms, and any planted satisfiability problem can be made resistant by adding an $\epsilon$-fraction of uniformly random constraints.

The assumption that recovering the planted assignment in this problem is hard for some predicate has been used extensively in complexity theory and cryptography [Ale11, Gol00, IKOS08, ABW10, App13], and the hardness of a decision version of this planted $k$-CSP is stated as the DCSP hypothesis in [BKS13]. Applebaum [App13] reduced the search problem (finding the planted assignment) to the decision problem (distinguishing the output from uniformly random). Our lower bounds below will be for this second, a priori easier, task.

Nearly optimal integrality gaps for LP and SDP hierarchies were recently given for this problem [OW14] (and references therein) for $\Omega(n^{r/2-\epsilon})$ evaluations of a predicate that is $(r-1)$-wise but not $r$-wise independent. Goldreich’s PRG is shown to be an $\epsilon$-biased generator in [MST06, ABR12], and lower bounds against DPLL-style algorithms are given in [CEMT09]. Applebaum and Lovett [AL16] give lower bounds against algebraic attacks in a framework based on polynomial calculus.

For a survey of these developments, see [App16].

For the planted $k$-SAT problems and the planted $k$-CSPs arising from Goldreich’s construction we address the following question: How many random constraints are needed to efficiently recover the planted assignment?

For these problems we prove unconditional lower bounds for a broad class of algorithms. Statistical (query) algorithms, defined by Kearns in the context of PAC learning [Kea98] and by Feldman et al. [FGR+12] for general problems on distributions, are algorithms that can be implemented without explicit access to random clauses, only being able to estimate expectations of functions of a random constraint to a desired accuracy. Many of the algorithmic approaches used in machine learning theory and practice have been shown to be implementable using statistical queries (e.g. [BFKV98, DV08, BDMN05, CKL+06, BF15]; see [Fel17] for a brief overview) including most standard approaches to convex optimization [FGV15]. Other common techniques such as Expectation Maximization (EM) [DLR77], MCMC optimization [TW87, GS90], (generalized) method of moments [Han12], and simulated annealing [KJV83, Č85] are also known to fit into this framework. The only known problem for which a superpolynomial separation between the complexity of statistical algorithms and the usual computational complexity is known is solving linear equations over a finite field (which can be done via Gaussian elimination).

The simplest form of algorithms that we refer to as statistical are algorithm that can be implemented using evaluations of Boolean functions on a random sample. Formally, for a distribution $D$ over some domain (in our case all $k$-clauses) 1-STAT oracle is the oracle that given any function $h:X\rightarrow\{0,1\}$ takes a random sample $x$ from $D$ and returns $h(x)$. While lower bounds for this oracle are easiest to state and interpret, the strongest form of our lower bounds is for algorithms that use VSTAT oracle defined in [FGR+12]. ${\mbox{VSTAT}}(t)$ oracle captures the information about the expectation of a given function that is obtained by estimating it on $t$ independent samples.

This oracle is based on the well-known statistical query oracle defined by Kearns [Kea98] that uses the same tolerance $\tau$ for all query functions. The ${\mbox{VSTAT}}(t)$ oracle corresponds more tightly to access to $t$ samples and allows us to prove upper and lower bounds that closely correspond to known algorithmic bounds.

More formally, for a clause distribution $Q$ and an assignment $\sigma$ let $Q_{\sigma}$ denote the distribution over clauses proportional to $Q$ for the planted assignment $\sigma$ (see Section 2 for a formal definition). Let $U_{k}$ denote the uniform distribution over $k$-clauses.

Let $Q$ be a distribution over $k$-clauses of complexity $r$. Then any (randomized) statistical algorithm that, given access to a distribution $D$ that equals $U_{k}$ with probability $1/2$ and equals $Q_{\sigma}$ with probability $1/2$ for a randomly and uniformly chosen $\sigma\in\{\pm 1\}^{n}$, decides correctly whether $D=Q_{\sigma}$ or $D=U_{k}$ with probability at least $2/3$ needs either:

$\Omega(q)$ calls to ${\mbox{VSTAT}}(\frac{n^{r}}{(\log q)^{r}})$ for any $q\geq 1$, or,

$\Omega((\frac{n}{\log n})^{r})$ calls to 1-STAT.

Let $Q$ be a clause distribution of distribution complexity $r$. Then there exists an algorithm that uses $O(n^{r/2}\log^{2}n)$ calls to ${\mbox{1-MSTAT}}(n^{\lceil r/2\rceil})$ and time linear in the number of oracle calls to identify the planted assignment with probability $1-o(1)$.

We prove this bound by showing that the algorithm from [FPV14] based on a subsampled power iteration can be implemented using statistical query oracles. The same upper bound holds for Goldreich’s planted $k$-CSP.

In addition to providing a matching lower bound, the algorithm gives an example of statistical query algorithm for performing power iteration to compute eigenvectors or singular vectors. Spectral algorithms are among the most commonly used for problems with planted solutions (including Flaxman’s algorithm [Fla03] for planted satisfiability) and our lower bounds can be used to derive lower bounds against such algorithms. The alternative approach for solving planted constraint satisfaction problems with $O(n^{r/2})$ samples is to use an SDP solver as shown in [BQ09] (with the “birthday paradox” as shown in [OW14]; see also [App16]). This approach can also be implemented using statistical queries, although a direct implementation using a generic SDP solver such as the one we describe in Section 4 will require quadratically more samples and will not give a non-trivial statistical algorithm for the problem (since solving using $O(n^{r})$ clauses is trivial).

We now briefly mention some of the corollaries and applications of our results.

A closely related problem is refuting the satisfiability of a random $k$-SAT formula (with no planting), a problem conjectured to be hard by Feige [Fei02]. A refutation algorithm takes a $k$-SAT formula $\Phi$ as an input and returns either SAT or UNSAT. If $\Phi$ is satisfiable, the algorithm always returns SAT and for $\Phi$ drawn uniformly at random from all $k$-SAT formulas of $n$ variables and $m$ clauses the algorithm must return UNSAT with probability at least $2/3$. For this refutation problem, an instance becomes unsatisfiable w.h.p. after $O(n)$ clauses, but algorithmic bounds are as high as those for finding a planted assignment under the noisy XOR distribution: $O(n^{k/2})$ clauses suffice[FGK05, COGLS04, HPS09, GL03, FO04, AOW15].

To relate this problem to our lower bounds we define an equivalent distributional version of the problem. In this version the input formula is obtained by sampling $m$ i.i.d. clauses from some unknown distribution $D$ over clauses. The goal is to say UNSAT (with probability at least $2/3$) when clauses are sampled from the uniform distribution and to say SAT for every distribution supported on simultaneously satisfiable clauses.

In the distributional setting, an immediate consequence of Theorem 1.2 is that Feige’s hypothesis holds for the class of statistical query algorithms. The proof (see Theorem 3.8) follows from the fact that our decision problem (distinguishing between a planted $k$-SAT instance and the uniform $k$-SAT instance) is a special case of the distributional refutation problem.

1.2 Hard instances of k𝑘k-SAT:

Finding distributions of planted $k$-SAT instances that are algorithmically intractable has been a pursuit of researchers in both computer science and physics. The distribution complexity parameter defined here generalizes the notion of “quiet plantings” studied in physics [BHL+02, JMS05, KZ09, KMZ14] to an entire hierarchy of “quietness”. In particular, there are easy to generate distributions of satisfiable $k$-SAT instances with distribution complexity as high as $k-1$ ($r=k$ can be achieved using XOR constraints but these instances are solvable by Gaussian elimination). These instances can also serve as strong tests of industrial SAT solvers as well as the underlying hard instances in cryptographic applications. In recent work, Blocki et al. extended our lower bounds from the Boolean setting to ${\mathcal{Z}}_{d}$ and applied them to show the security of a class of humanly computable password protocols [BBDV14].

1.3 Lower bounds for convex programs:

We note that conditions on the value of the convex program that we make are weaker than the standard conditions that a convex relaxation must satisfy. Specifically, it is usually assumed that a convex relaxation does not increase the value of the objective (for example, the value for a satisfiable instance must be 0) and also that the minimum of the objective function for all “bad” instances will be noticeably larger than that of the “good” instances. In Section 4 we also prove lower bounds against convex programs in exponentially high dimension as long as the appropriate norms of points in the domain and gradients are not too large. We are not aware of this form of lower bounds against convex programs for planted satisfiability stated before. We also remark that our lower bounds are incomparable to lower bounds for programs given in [OW14] since they analyze a specific SDP for which the mapping $\cal M$ maps to functions over an $O(n^{k})$-dimensional set $K$ is defined using a high level of the Sherali-Adams or Lovász-Schrijver hierarchies. Further details are given in Section 4.

2 Overview of the technique

Our proof of the lower bound builds on the notion of statistical dimension given in [FGR+12] which itself is based on ideas developed in a line of work on statistical query learning [Kea98, BFJ+94, Fel12].

Our primary technical contribution is a new, stronger notion of statistical dimension and its analysis for planted $k$-CSP problems. The statistical dimension in [FGR+12] is based on upper-bounding average or maximum pairwise correlations between appropriately defined density functions. While these dimensions can be used for our problem (and, indeed, were a starting point for this work) they do not lead to the tight bounds we seek. Specifically, at best they give lower bounds for ${\mbox{VSTAT}}(n^{r/2})$, whereas we will prove lower bounds for ${\mbox{VSTAT}}(n^{r})$ to match the current best upper bounds.

Our stronger notion directly examines a natural operator, which, for a given function, evaluates how well the expectation of the function discriminates between different distributions. We show that a norm of this operator for large sets of input distributions gives a lower bound on the complexity of any statistical query algorithm for the problem. Its analysis for our problem is fairly involved and a key element of the proof is the use of concentration of polynomials on $\{\pm 1\}^{n}$ (derived from the hypercontractivity results of Bonami and Beckner [Bon70, Bec75]).

We remark that the $k$-XOR-SAT problem is equivalent to PAC learning of (general) parity functions from random $k$-sparse examples. The latter is the classic problem addressed by Kearns’ original lower bound [Kea98]. While superficially the planted setting is similar to learning of $k$-sparse parities from random uniform examples for which optimal statistical query lower bounds are well-known and easy to derive, the problems, techniques and the resulting bounds are qualitatively different. One significant difference is that the correlation between parity functions on the uniform distribution is 0, whereas in our setting the distributions are not uniform and pairwise correlations between them can be relatively large. Moreover, as mentioned earlier, the techniques based on pairwise correlations do not suffice for the strong lower bounds we give.

Our stronger technique gives further insight into the complexity of statistical algorithms and has a natural interpretation in terms of the geometry of the space of all planted assignments with a metric defined (between pairs of assignments) to capture properties of statistical algorithms. The fraction of solutions that are at distance greater than some threshold from a fixed assignment goes up sharply from exponentially small to a polynomial fraction as the distance threshold increases. We call this a ‘paring’ transition as a large number of distributions become amenable to being separated from the planted solution and discarded.

We conjecture that our lower bounds hold for all algorithms with the exception of those based on Gaussian elimination. Formalizing “based on Gaussian elimination” requires substantial care. Indeed, in an earlier version of this work we excluded Gaussian elimination by only excluding density functions of low algebraic degree. (Here algebraic degree refers to the degree of the polynomial over ${\mathcal{Z}}_{2}^{k}$ required to represent the function. For example, the parity function equals to $x_{1}+x_{2}+\cdots+x_{k}$ and therefore has algebraic degree $1$). This resulted in a conjecture that was subsequently disproved by Applebaum and Lovett [AL16] using an algorithm that combines Gaussian elimination with fixing some of the variables. An alternative approach to excluding Gaussian elimination-based methods is to exploit their fragility to even low rates of random noise. Here random noise would correspond to mixing in of random and uniform constraints to the distribution. In other words for $\alpha\in$, $Q$ becomes $Q^{\alpha}=(1-\alpha)Q+\alpha U_{k}$. Observe that for all constant $\alpha<1$, the complexity of $Q^{\alpha}$ is the same as the complexity of $Q$.

We conjecture that an analogous statement also holds for Goldreich’s $k$-CSP. Note that in this case mixing in an $\alpha$-fraction of random and uniform constraints can be equivalently seen as flipping the given value of the predicate with probability $\alpha/2$ randomly and independently for each constraint.

3 Other related work

Hypergraph Partitioning. Another closely related model to planted satisfiability is random hypergraph partitioning, in which a partition of the vertex set is fixed, then $k$-uniform hyperedges added with probabilities that depend on their overlap with the partition. To obtain a planted satisfiability model from a planted hypergraph, let the vertex set be the set of $2n$ literals, with the partition given by the planted assignment $\sigma$. A $k$-clause is then a $k$-uniform hyperedge. The two models are not exactly equivalent, as in planted satisfiability we have the extra information that pairs of literals corresponding to the same variable must receive different assignments; however, to the best of our knowledge all the known algorithmic approaches to planted satisfiability work for planted hypergraph partitioning as well. The Goldreich CSP model is closely related to a hypergraph version of the censored block model [AM15, ABBS14] in which random hyperedges are labeled with values that depend on how the edges overlap with a planted partition.

The case $k=2$ of $k$-uniform hypergraph partitioning is called the stochastic block model. The input is a random graph with different edge probabilities within and across an unknown partition of the vertices, and the algorithmic task is to recover partial or complete information about the partition given the resulting graph. Work on this model includes Bopanna [Bop87], McSherry’s general-purpose spectral algorithm [McS01], and Coja-Oghlan’s algorithm that works for graphs of constant average degree [CO06].

Shattering and paring. Random satisfiability problems (without a planted solution) such as $k$-SAT and $k$-coloring random graphs exhibit a shattering phenomenon in the solution space for large enough $k$ [KMRT+07, ACO08]: as the density of constraints increases, the set of all solutions evolves from a large connected cluster to a exponentially large set of well-separated clusters. The shattering threshold empirically coincides with the threshold for algorithmic tractability (while this is the case for large $k$, for $k=3,4$ there is some evidence that the survey propagation algorithm may succeed beyond the shattering threshold [MPRT16]). Shattering has also been used to prove that certain algorithms fail at high enough densities [GS14].

Both the shattering and paring phenomena give an explanation for the failure of known algorithms on random instances. Both capture properties of local algorithms, in the sense that in both cases, the performance of Gaussian elimination, an inherently global algorithm, is unaffected by the geometry of the solution space: both random $k$-XOR-SAT and random planted $k$-XOR-SAT are solvable at all densities despite exhibiting shattering and paring respectively.

The paring phenomenon differs from shattering in several significant ways. As the paring transition is a geometric property of a carefully chosen metric, there is a direct and provable link between paring and algorithmic tractability, as opposed to the empirical coincidence of shattering and algorithmic failure. In addition, while shattering is known to hold only for large enough $k$, the paring phenomenon holds for all $k$, and already gives strong lower bounds for $3$-uniform constraints.

One direction for future work would be to show that the paring phenomenon exhibits a sharp threshold; in other words, improve the analysis of the statistical dimension of planted satisfiability in Section 5 to remove the logarithmic gap between the upper and lower bounds. An application of such an improvement would be to apply the lower bound framework to the planted coloring conjecture from [DKMZ11]; as the gap between impossibility and efficient recovery is only a constant factor there, the paring transition would need to be located more precisely.

Definitions

We now define a general model for planted satisfiability problems that unifies various previous ways to produce a random $k$-SAT formula where the relative probability that a clause is included in the formula depends on the number of satisfied literals in the clause [Fla03, JMS05, AJM05, KV06b, KZ09, COCF10, KMZ14].

Fix an assignment $\sigma\in\{\pm 1\}^{n}$. We represent a $k$-clause by an ordered $k$-tuple of literals from $x_{1},\dots x_{n},\overline{x}_{1},\dots\overline{x}_{n}$ with no repetition of variables and let $X_{k}$ be the set of all such $k$-clauses. For a $k$-clause $C=(l_{1},\ldots,l_{k})$ let $\sigma(C)\in\{\pm 1\}^{k}$ be the $k$-bit string of values assigned by $\sigma$ to literals in $C$, that is $\sigma(l_{1}),\dots,\sigma(l_{k})$, where $\sigma(l_{i})$ is the value of literal $l_{i}$ in assignment $\sigma$ with $-1$ corresponding to TRUE and 1 to FALSE. In a planted model, we draw clauses with probabilities that depend on the value of $\sigma(C)$.

To generate a random formula, $F(Q,\sigma,m)$ we draw $m$ i.i.d. $k$-clauses according to the probability distribution $Q_{\sigma}$, where

Problems. The algorithmic problems studied in this paper can be stated as follows: Given the function $Q$ and a sample of $m$ independent clauses drawn according to $Q_{\sigma}$, recover $\sigma$, or some $\tau$ correlated with $\sigma$. Note that since unsatisfiable clauses are allowed to have non-zero weight, for some distributions the problem is effectively satisfiability with random noise. Our lower bounds are for the potentially easier problem of distinguishing a randomly and uniformly chosen planted distribution from the uniform distribution over $k$-clauses. Namely, let ${\mathcal{D}}_{Q}$ denote the set of all distributions $Q_{\sigma}$, where $\sigma\in\{\pm 1\}^{k}$ and $U_{k}$ be the uniform distribution over $k$-clauses. Let ${\mathcal{B}}({\mathcal{D}}_{Q},U_{k})$ denote the decision problem in which given samples from an unknown input distribution $D\in{\mathcal{D}}_{Q}\cup\{U_{k}\}$ the goal is to output $1$ if $D\in{\mathcal{D}}_{Q}$ and 0 if $D=U_{k}$.

As in the problem above, the goal is to recover $\sigma$ given $m$ random and independent samples from $P_{\sigma}$ or at least to be able to distinguish any planted distribution from one in which the value is a uniform random coin flip (or, equivalently, the distribution obtained when the function $P\equiv 0$). Our goal is to understand the smallest number $m$ of $k$-clauses that suffice to find the planted assignment or at least to distinguish a planted distribution from a uniform one.

For a clause distribution $Q$, we define its distribution complexity $r(Q)$ as the smallest integer $r\geq 1$ for which there exists a set $S\subseteq[k]$ of size $r$ and

To see the difference between a hard and easy distribution $Q$, first consider planted uniform $k$-SAT: $Q(0)=0$, $Q(i)=1/(2^{k}-1)$ for $i\geq 1$. The distribution complexity of $Q$ is $r=1$.

Next, consider the noisy parity distribution (or noisy planted $k$-XOR-SAT) with $Q(l)=\delta/2^{k-1}$ for $l$ even, and $Q(l)=(2-\delta)/2^{k-1}$ for $l$ odd, for some $\delta\neq 1$. In this case, we have $\hat{Q}(l)=0$ for $1\leq l\leq k-1$, and so the distribution complexity of $Q$ is $r=k$. We will see that such parity-type distributions are in fact the hardest for statistical algorithms to detect.

2 Statistical algorithms

We can define planted satisfiability as the problem of identifying an unknown distribution $D$ on a domain $X$ given $m$ independent samples from $D$. For us, $X$ is the set of all possible $k$-clauses or $k$-hyperedges, and each partition or assignment $\sigma$ defines a unique distribution $D_{\sigma}$ over $X$.

Extending the work of Kearns [Kea98] in learning theory, Feldman et al. [FGR+12] defined statistical query algorithms for problems over distributions. Roughly speaking, these are algorithms that do not see samples from the distribution but instead have access to estimates of the expectation of any bounded function of a sample from the distribution. More formally, a statistical algorithm can access the input distribution via one of the following oracles.

Let $D$ be the input distribution over the domain $X$. Given any function $h:X\rightarrow\{0,1,\ldots,L-1\}$, ${\mbox{1-MSTAT}}(L)$ takes a random sample $x$ from $D$ and returns $h(x)$.

This oracle is a generalization of the 1-STAT oracle from [FGR+12] and was first defined by Ben-David and Dichterman in the context of PAC learning [BD98]. It was also more recently studied in [SD15, SVW15]. For the planted SAT problem this oracle allows an algorithm to evaluate a multi-valued function on a random clause. By repeating the query, the algorithm can estimate the expectation of the function as its average on independent samples. Being able to output one of multiple possible values gives the algorithm considerable flexibility, e.g., each value could correspond to whether a clause has a certain pattern on a subset of literals. With $L=n^{k}$, the algorithm can identify the random clause. We will therefore be interested in the trade-off between $L$ and the number of queries needed to solve the problem.

The definition of $\tau$ means that ${\mbox{VSTAT}}(t)$ can return any value $v$ for which the distribution $B(t,v)$ (outcomes of $t$ independent Bernoulli variables with bias $v$) is close to $B(t,E[h])$ in total variation distance [FGR+12]. In most cases $p>1/t$ and then $\tau$ also corresponds to returning the expectation of a function to within the standard deviation error of averaging the function over $t$ samples. However, it is important to note that within this constraint on the error, the oracle can return any value, possibly in an adversarial way.

In this paper, we also define the following generalizationFor simplicity, this definition generalizes VSTAT only for Boolean query functions. of the VSTAT oracle to multi-valued functions.

where $p_{Z}=\Pr_{D}[h(x)\in Z]$. The query cost of such a query is $|\mathcal{S}|$.

We note that ${\mbox{VSTAT}}(t)$ is equivalent to ${\mbox{MVSTAT}}(2,t)$ (the latter only allows Boolean queries but that is not an essential difference) and any query to ${\mbox{MVSTAT}}(L,t)$ can be easily answered using $L$ queries to ${\mbox{VSTAT}}(4\cdot Lt)$ (see Theorem 7.2 for a proof). The additional strength of this oracle comes from allowing the sets in $\mathcal{S}$ to depend on the unknown distribution $D$ and, in particular, be fixed but unknown to the algorithm. This is useful for ensuring that potential functions of our discrete power iteration algorithm for planted SAT behave in the same way as if the algorithm were executed on true samples (see Section 8.4). Another useful way to think of $L$-valued oracles in the context of vector-based algorithms is as a vector of $L$ Boolean functions which are non-zero on disjoint parts of the domain. This view also allows to extend MVSTAT to bounded-range (non-Boolean) functions.

An important property of every one of these oracles is that it can be easily simulated using $t$ samples (in the case of VSTAT/MVSTAT the success probability is a positive constant but it can be amplified to $1-\delta$ using $O(t\log(1/\delta))$ samples). The goal of our generalization of oracles to $L>2$ was to show that even nearly optimal sample complexity can be achieved by a statistical algorithm using an oracle for which a nearly matching lower bound applies.

Results

We state our upper and lower bounds for the planted satisfiability problem. Identical upper and lower bounds apply to Goldreich’s planted $k$-CSPs with $r$ being the degree of lowest-degree non-zero Fourier coefficient of $P$. For brevity, we omit the repetitive definitions and statements in this section. In Section 6 we give the extension of our lower bounds to this problem and also make the connections between the two problems explicit.

We begin with lower bounds for any statistical algorithm. For a clause distribution $Q$ let ${\mathcal{B}}({\mathcal{D}}_{Q},U_{k})$ denote the decision problem of distinguishing whether the input distribution is one of the planted distributions or is uniform.

For an assignment $\sigma\in\{\pm 1\}^{n}$, let $Q_{\sigma}$ be a distribution over $k$-clauses of complexity $r$ and ${\mathcal{D}}_{Q}$ be this family of distributions. Assume that the input distribution $D$ is $U_{k}$ with probability $1/2$ and with the remaining probability $1/2$ it is $Q_{\sigma}$ for a uniform random $\sigma\in\{\pm 1\}^{n}$. Then any (randomized) statistical algorithm that decides correctly whether $D\in{\mathcal{D}}_{Q}$ or $D=U_{k}$ with probability at least $2/3$ (over the choice of $D$ and randomness of the algorithm) needs either

$m$ calls to the ${\mbox{1-MSTAT}}(L)$ oracle with $m\cdot L\geq c_{1}\left(\frac{n}{\log n}\right)^{r}$ for a constant $c_{1}=\Omega_{k}(1)$, OR

$q$ queries to ${\mbox{MVSTAT}}\left(L,\frac{c_{2}}{L}\cdot\frac{n^{r}}{(\log q)^{r}}\right)$ for a constant $c_{2}=\Omega_{k}(1)$ and any $q\geq L$.

The first part of the theorem exhibits the trade-off between the number of queries $m$ and the number of values the query can take $L$. It might be helpful to think of the latter as evaluating $L$ disjoint functions on a random sample, a task that would have complexity growing with $L$. The second part of the theorem is a superpolynomial lower bound (in $n$, for any fixed $r$) if the parameter $t$ (recall the oracle is allowed only an error equal to the standard deviation of averaging over $t$ random samples) is less than $n^{r}/(\log n)^{2r}$.

2 Algorithms

We next turn to our algorithmic results, motivated by two considerations. First, the $O(n^{r/2})$-clause algorithm implicit in [BQ09] does not appear to lead to a non-trivial statistical algorithm. Second, much of the literature on upper bounds for planted problems uses spectral methods, and so we aim to implement such spectral algorithms statistically.

The algorithm we present is statistical and nearly matches the lower bound. It can be viewed as a discrete rounding of the power iteration algorithm for a suitable matrix constructed from the clauses of the input.

Let ${\mathcal{Z}}_{Q}$ be a planted satisfiability problem with clause distribution $Q$ having distribution complexity $r$. Then there exists an algorithm to solve ${\mathcal{Z}}_{Q}$ using $O(n^{r/2}\log n)$ random clauses and time linear in this number. This algorithm can be implemented statistically in any of the following ways.

Using $O(n^{r/2}\log^{2}n)$ calls to ${\mbox{1-MSTAT}}(n^{\lceil r/2\rceil})$;

For even $r$: using $O(\log n)$ calls to ${\mbox{MVSTAT}}(n^{r/2},n^{r/2}\log\log n)$;

For odd $r$: using $O(\log n)$ calls to ${\mbox{MVSTAT}}(O(n^{\lceil r/2\rceil}),O(n^{r/2}\log n))$;

Thus for any $r$, the upper bound matches the lower bound up to logarithmic factors for sample size parameter $t=n^{r/2}$, with $L=n^{\lceil r/2\rceil}$ being only slightly higher in the odd case than the $L=n^{r/2}$ that the lower bound implies for such $t$. The algorithm is a discretized variant of the algorithm based on power iteration with subsampling from [FPV14]. The upper bound holds for the problem of finding the planted assignment exactly, except in the case $r=1$. Here $\Omega(n\log n)$ clauses are required for complete identification since that many clauses are needed for each variable to appear at least once in the formula. In this case $O(n^{1/2}\log n)$ samples suffice to find an assignment with non-trivial correlation with the planted assignment, i.e. one that agrees with the planted assignment on $n/2+t\sqrt{n}$ variables for an arbitrary constant $t$.

3 Statistical dimension for decision problems

For a domain $X$, let ${\mathcal{D}}$ be a set of distributions over $X$ and let $D$ be a distribution over $X$ which is not in ${\mathcal{D}}$. For $t>0$, the distributional decision problem ${\mathcal{B}}({\mathcal{D}},D)$ using $t$ samples is to decide, given access to $t$ random samples from an arbitrary unknown distribution $D^{\prime}\in{\mathcal{D}}\cup\{D\}$, whether $D^{\prime}\in{\mathcal{D}}$ or $D^{\prime}=D$. Lower bounds on the complexity of statistical algorithms use the notion of statistical dimension introduced in [FGR+12], based on ideas from [BFJ+94, Fel12].

Our concept of statistical dimension is essentially the same as in [FGR+12] but uses $\kappa_{2}({\mathcal{D}}^{\prime},D)$ instead of average correlations.

The dimension is equal to (at least) $d$ if there exists a reference distribution $D$ and a “hard” subset of distributions ${\mathcal{D}}_{D}$, such that no large subset of ${\mathcal{D}}_{D}$ has discrimination norm larger than $\kappa$ (and, consequently, cannot be distinguished from $D$ using a single query to ${\mbox{VSTAT}}(1/(3\kappa^{2}))$). Here large subset means at least $1/d$ fraction of distributions in ${\mathcal{D}}_{D}$. We remark that this statistical dimension can be easily extended to general search problems as in [FGR+12] (the extension can be found in an earlier version of this work [FPV13, v5]). A detailed treatment and additional approaches to proving statistical query lower bounds for search problems can be found in a subsequent work of Feldman [Fel16].

The statistical dimension with discrimination norm $\kappa$ of a problem over distributions gives a lower bound on the complexity of any statistical algorithm.

Any randomized statistical algorithm that solves ${\mathcal{B}}({\mathcal{D}},D)$ with probability $\geq 2/3$ over the randomness in the algorithm requires $\Omega(d/L)$ calls to ${\mbox{MVSTAT}}(L,1/(12\cdot\kappa^{2}\cdot L))$.

Any randomized statistical algorithm that solves ${\mathcal{B}}({\mathcal{D}},D)$ with probability $\geq 2/3$ over the randomness in the algorithm requires at least $m$ calls to ${\mbox{1-MSTAT}}(L)$ for $m=\Omega\left(\min\left\{d,1/\kappa^{2}\right\}/L\right)$.

Further, the lower bound also holds when the input distribution $D^{\prime}$ is chosen randomly as follows: $D^{\prime}=D$ with probability $1/2$ and $D^{\prime}$ equals a random and uniform element of ${\mathcal{D}}_{D}$ with probability $1/2$, where ${\mathcal{D}}_{D}$ is the set of distributions for which the value of $d$ is attained.

We prove this theorem in a slightly more general form in Section 7. Our proof relies on techniques from [FGR+12] and simulations of MVSTAT and 1-MSTAT using VSTAT and 1-STAT, respectively.

In our setting the domain $X_{k}$ is the set of all clauses of $k$ ordered literals (without variable repetition); the class of distributions ${\mathcal{D}}_{Q}$ is the set of all distributions $Q_{\sigma}$ where $\sigma$ ranges over all $2^{n}$ assignments; the distribution $D$ is the uniform distribution over $X_{k}$ referred to as $U_{k}$.

In the Section 5 we prove the following bound on the statistical dimension with discrimination norm of planted satisfiability.

For any distribution $Q$ over $k$-clauses of distributional complexity $r$, there exists a constant $c>0$ (that depends on $Q$) such that for any $q\geq 1$,

In Section 6.1 we prove the same lower bound for the generalized planted $k$-CSP problem. Our proof is based on a reduction showing that any statistical query for an instance of the $k$-CSP problem of complexity $r$ can be converted to a query for a planted $k$-SAT instance of distribution complexity $r$. The reduction ensures that the resulting query is essentially as informative in distinguishing the planted distribution from the reference one as the original query. As a result it reduces a bound on $\kappa_{2}$ of a planted $k$-CSP problem to an almost equivalent bound on $\kappa_{2}$ of the corresponding planted $k$-SAT problem.

4 Corollaries and applications

Finding distributions of planted $k$-SAT instances that are algorithmically intractable has been a pursuit of researchers in both computer science and physics. It was recognized in [BHL+02, JMS05] that uniform planted $k$-SAT is easy algorithmically due to the bias towards true literals, and so they proposed distributions in which true and false literals under the planted assignment appear in equal proportion. Such distributions have complexity $r\geq 2$ in our terminology. These distributions have been termed ‘quiet plantings’ since evidence of the planting is suppressed.

Further refinement of the analysis of quiet plantings was given in [KMZ14], in which the authors analyze belief propagation equations and give predicted densities at which quiet plantings transition from intractable to tractable. Their criteria for a quiet planting is exactly the equation that characterizes distribution complexity $r\geq 2$, and the conditions under which the tractability density diverges to infinity corresponds to distribution complexity $r\geq 3$.

The distribution complexity parameter defined here generalizes quiet plantings to an entire hierarchy of quietness. In particular, there are distributions of satisfiable $k$-SAT instances with distribution complexity as high as $k-1$ ($r=k$ can be achieved using XOR constraints but these instances are solvable by Gaussian elimination). Our main results show that for distributions with complexity $r\geq 3$, the number of clauses required to recover the planted assignment is super-linear (for statistical algorithms with $L\leq n^{r/2}$).

4.2 Feige’s Hypothesis

As a second application of our main result, we show that Feige’s $3$-SAT hypothesis [Fei02] holds for the class of statistical algorithms. A refutation algorithm takes a $k$-SAT formula $\Phi$ as an input and returns either SAT or UNSAT. The algorithm must satisfy the following:

If $\Phi$ is satisfiable, the algorithm always returns SAT.

If $\Phi$ is drawn uniformly at random from all $k$-SAT formulas of $n$ variables and $m$ clauses, where $m/n$ is above the satisfiability threshold (the clause density at which the formula become unsatisfiable with high probability), then the algorithm must return UNSAT with probability at least $2/3$ (or some other arbitrary constant).

As with planted satisfiability, the larger $m$ is the easier refutation becomes, and so the challenge becomes finding efficient refutation algorithms that succeed on the sparsest possible instances. Efficient $3$-SAT refutation algorithms are known for $m=\Omega(n^{3/2})$ [COGL04, FO04]. Feige hypothesized 1) that no polynomial-time algorithm can refute formulas with $m\leq\Delta n$ clauses for any constant $\Delta$ and 2) for every $\epsilon>0$ and large enough constant $\Delta$, there is no polynomial-time algorithm that answers UNSAT on most 3-SAT formulas but answers SAT on all formulas that have assignments satisfying $(1-\epsilon)$-fraction of constraints. Hypothesis 2 is strictly weaker than hypothesis 1. Based on these hypotheses he derived hardness-of-approximation results for several fundamental combinatorial optimization problems.

To apply our bounds we need to first define a distributional version of the problem.

In the distributional $k$-SAT refutation problem the input formula is obtained by sampling $m$ i.i.d. clauses from some unknown distribution $D$ over clauses. An algorithm successfully solves the distributional problem if:

The algorithm returns SAT for every distribution supported on simultaneously satisfiable clauses.

The algorithm returns UNSAT with probability at least $2/3$ when clauses are sampled from the uniform distribution and $m/n$ is above the satisfiability threshold.

The original refutation problem and distributional refutation problem are equivalent: a refutation algorithm for the original problem solves the distributional version and vice versa.

The first direction is immediate: assume that we have a refutation algorithm $A$ for a fixed formula. We run the refutation algorithm on the $m$ clauses sampled from the input distribution and output the algorithm’s answer. By definition, if the input distribution is uniform then the sampled clauses will give a random formula from this distribution. So $A$ will return UNSAT with probability at least $2/3$. If the clauses in the support of the input distribution can be satisfied then the formula sampled from it will be necessarily satisfiable and $A$ must return SAT.

In the other direction, we again run the distributional refutation algorithm $A$ on the $m$ clauses of $\Phi$ and output its answer (each clause is used as a new sample consecutively). If $\Phi$ was sampled from the uniform distribution above the satisfiability threshold, then the samples we produced are distributed according to the uniform distribution. Therefore, with probability at least $2/3$ $A$ returns UNSAT. If $\Phi$ is satisfiable then consider the distribution $D_{\Phi}$ which is uniform over the $m$ clauses of $\Phi$. $\Phi$ has non-zero probability to be the outcome of $m$ i.i.d. clauses sampled from $D_{\Phi}$. Therefore $A$ must output SAT on it since otherwise it would violate its guarantees. Therefore the output of our algorithm will be SAT for $\Phi$. ∎

In the distributional setting, an immediate consequence of Theorem 3.1 is that Feige’s hypothesis holds for the class of statistical algorithms.

Any (randomized) statistical algorithm that solves the distributional $k$-SAT refutation problem requires:

$m$ calls to the ${\mbox{1-MSTAT}}(L)$ oracle with $m\cdot L\geq c_{1}\left(\frac{n}{\log n}\right)^{k}$ for a constant $c_{1}=\Omega_{k}(1)$.

$q$ queries to ${\mbox{MVSTAT}}\left(L,\frac{c_{2}}{L}\cdot\frac{n^{k}}{(\log q)^{k}}\right)$ for a constant $c_{2}=\Omega_{k}(1)$ and any $q\geq L$.

The decision problem in Theorem 3.1 is a special case of the distributional refutation problem. Specifically, say there is such a refutation algorithm. Let $Q$ be a fully satisfiable clause distribution with distribution complexity $k$. Then consider a distribution $D$ so that either $D=U_{k}$ or $D=Q_{\sigma}\in{\mathcal{D}}_{Q}$ for a uniformly chosen $\sigma\in\{\pm 1\}^{n}$. Then run the refutation algorithm on $D$. If $D\in{\mathcal{D}}_{Q}$, then the algorithm must output SAT, and so we conclude $D\in{\mathcal{D}}_{Q}$. If $D=U_{k}$, then with probability $2/3$ the algorithm must output UNSAT in which case we conclude that $D=U_{k}$. This gives an algorithm for distinguishing $U_{k}$ from ${\mathcal{D}}_{Q}$ with probability at least $2/3$, a contradiction to Theorem 3.1. ∎

We note that the only distributions with $r=k$ are noisy $k$-XOR-SAT distributions. Such distributions generate satisfiable formulas only when the noise rate is 0 and then formulas are refutable via Gaussian elimination. Therefore if one excludes the easy (noiseless) $k$-XOR-SAT distribution then we obtain only the stronger form of Feige’s conjecture ($\epsilon>0$) with $r=k=3$.

4.3 Hardness of approximation

We note finally that optimal inapproximability results can be derived from Theorem 3.1 as well, including the fact that pairwise independent predicates (as studied in [AM09]) are approximation-resistant for the class of statistical algorithms.

Our work provides a means to generate candidate distributions of hard instances for approximation algorithms for CSP’s: find a distribution $Q$ on $\{\pm 1\}^{k}$ supported only on vectors that satisfy the CSP predicate with high distribution complexity (as in the example of $4$-SAT above). Then statistical algorithms cannot efficiently distinguish the planted distribution (all constraints satisfied) from the uniformly random distribution (eg. ($1-2^{-k}$)-fraction of constraints satisfied in the case of $k$-SAT).

Convex Programs and SQ Algorithms for Solving CSPs

In this section we show how our lower bounds for planted $k$-SAT together with general statistical query algorithms for solving stochastic convex programs from [FGV15] imply lower bounds on convex programs that can be used to solve planted $k$-SAT (analogous results also hold for Goldreich’s $k$-CSP but we omit them for brevity). At a high level we observe that a convex relaxation can be viewed as a reduction from our planted constraint satisfaction problem to a stochastic convex optimization problem. Existence of such a reduction together with a statistical query algorithm for the corresponding stochastic convex program would violate the lower bounds that we prove. Hence, as a contrapositive, we rule out existence of several types of convex relaxations for the planted CSPs.

We first describe several standard ways in which Boolean constraint satisfaction problemsAs usual in this literature, constraint satisfaction also refers to the problem of maximizing the number of satisfied constraints. are relaxed to an LP or an SDP.

More generally, the canonical LP relaxation of a $k$-CSP with $m$ constraints results in a program of the following type (see [O’D11] for a textbook version or [Rag08, BKS13, OW14] for some applications):

subject to $\bar{x}\in K$. Here, $C_{i}$ denotes the $k$-ary Boolean predicate of the $i$-th constraint, $V_{i}$ denotes the $k$-tuple of variables of $i$-th constraint and $x_{V_{i},y}$ is the variable that tells whether variables in $V_{i}$ are assigned values $y$ (its constrained to be in $$and interpreted as probability). The set$K$is an$O_{k}(n^{k})$-dimensional convex set that makes sure that$x_{V_{i},y}$’s are consistent in a natural sense (with additional PSD cone constraints in the case of SDPs).

2 Statistical Query Algorithms for Stochastic Convex Optimization

We now describe several results from [FGV15] giving upper bounds on solving various stochastic convex programs by statistical query algorithms. Bounds for a number of additional types of convex programs are given in [FGV15] and can be applied in this context in a similar way. We start by defining the problem of distribution-independent stochastic convex optimization formally.

For general convex functions with range scaled to $$, Feldman et al. [FGV15] describe two statistical query algorithms that both use${\mbox{VSTAT}}(O(N^{2}/\epsilon^{2}))$to find an$\epsilon$-approximate solution to the stochastic convex program. The first algorithm is based on the random walk approach from [KV06a, LV06]. The second algorithm is based on the classic center-of-gravity method [Lev65] and requires fewer queries.

The theorem above ignores computational considerations since those do not play any role in our information-theoretic lower bounds. An efficient version of this algorithm is also given in [FGV15].

Let $p\in$, $L,R>0$, and $K\subseteq{\mathcal{B}}_{p}^{N}(R)$ be a convex body. Let $\mathcal{F}$ be the set of all functions $f$ that satisfy, for all $x\in K$, $\|\nabla f(x)\|_{q}\leq L$ for $q=1-(1/p)$. Then there is an algorithm that solves ${\mbox{Opt}}(K,\mathcal{F},\epsilon)$ using $O\left(N\log N\cdot(LR/\epsilon)^{2}\right)$ queries to ${\mbox{VSTAT}}(O((\log N\cdot LR/\epsilon)^{2}))$.

3 Corollaries for Planted k𝑘k-CSPs

Now observe that a convex relaxation (of the type that we defined) is just a mapping from Boolean constraints to convex functions in some class of functions $\mathcal{F}$ over a convex body $K$. Such mapping allows to implement a statistical query oracle for the distribution over convex functions given a statistical query oracle for the input distribution over Boolean constraints. In particular, it allows us to run a statistical query algorithm for ${\mbox{Opt}}(K,\mathcal{F},\epsilon)$ on the stochastic convex program that corresponds to the input distribution over $k$-clauses. Now assume that the value of the solution for the stochastic convex program corresponding to a planted $k$-CSP is smaller than the value of the solution for the the stochastic convex program corresponding to the uniform distribution over constraints by at least $\epsilon$. Then a statistical query algorithm for ${\mbox{Opt}}(K,\mathcal{F},\epsilon)$ solves the decision version of our planted $k$-CSP. Hence, if ${\mbox{Opt}}(K,\mathcal{F},\epsilon)$ can be solved using some number of queries to a statistical oracle that violates our lower bound then a convex relaxation that satisfies these properties cannot exist. We now make these statements formally.

Let $Q$ be a distribution over $k$-clauses of complexity $r$. Assume that there exists a mapping that maps each $k$-clause $C\in X_{k}$ to a convex function $f_{C}:K\rightarrow$ over some convex $N$-dimensional set $K$ that for some $\epsilon>0$ and $\alpha$ satisfies:

Then for every $q\geq 1$, solving ${\mbox{Opt}}(K,\mathcal{F},\epsilon)$ using ${\mbox{VSTAT}}\left(\frac{n^{r}}{(\log q)^{r}}\right)$ requires $\Omega(q)$ queries.

For instances of planted satisfiability there is a constant gap between the fraction of clauses that can be satisfied in a formula sampled from $U_{k}$ and the fraction of clauses that can be satisfied in a formula sampled from $Q_{\sigma}$. Thus, for convex relaxations that satisfy the conditions of Corollary 4.5 the lower bounds imply a large integrality gap.

Note that these corollaries give concrete lower bounds on the dimension and other structural properties of convex programs that can be used to solve an average-case $k$-CSP without any assumptions about how the convex program is solved. In particular, it does not need to be solved via a statistical algorithm or even computationally efficiently. As far as we know, this approach to obtaining lower bounds for convex relaxations from convex optimization algorithms and statistical query lower bounds is new.

We observe that standard lift-and-project procedures (Sherali-Adams, Lovász-Schrijver, Lasserre) for strengthening LP/SDP formulations do not affect the analysis above. While these procedures add a large number of auxiliary variables and constraints the resulting program is still a convex optimization problem in the same dimension (although implementation of the separation oracle becomes more computationally intensive). Hence the use of such procedures does not necessarily affect the bounds on the number of queries and tolerance we gave above.

At a more conceptual level, the primary difference between the commonly considered hierarchies of LP/SDP relaxations and our approach is as follows. The expected objective value of the stochastic convex programs corresponding to these hierarchies of relaxations captures the expected objective of the original Boolean $k$-CSP. Yet, solving stochastic convex programs corresponding to these relaxations for all distributions requires $\Omega(n^{r/2})$ samples information theoretically (e.g. [FGV15]). Lower bounds against such relaxations effectively prove that this number of samples is necessary even for the uniform distribution over the clauses: given fewer samples the optimum of the objective based on the given random samples will have a much lower value than the optimum of the expected objective (a phenomenon that is referred to as overfitting). In contrast, our approach rules out relaxations for which the resulting stochastic convex program can be solved by a statistical query algorithm using $q$ queries to ${\mbox{VSTAT}}\left(\frac{n^{r}}{(\log q)^{r}}\right)$. In particular there is no overfitting. However, such relaxations end up not being sufficiently expressive: the optimum of the expected objective of the relaxation does not differentiate between the planted distributions and the uniform one. This difference makes our lower bounds incomparable and, in a way, complementary to existing work on lower bounds for specific hierarchies of convex relaxations.

Statistical Dimension of Planted Satisfiability

In this section, we prove our lower bound on the statistical dimension with discrimination norm of the planted satisfiability problem (Theorem 3.5). Recall that the theorem states that for any distribution $Q$ over $k$-clauses of distributional complexity $r$, there exists a constant $c>0$ such that for any $q\geq 1$,

where $\chi_{S}(x)=\prod_{i\in S}x_{i}$ is a parity or Walsh basis function, and the Fourier coefficient of the set $S$ is defined as:

where $\hat{Q}(\emptyset)=2^{-k}$ follows from $Q$ being a distribution over $\{\pm 1\}^{k}$. Plugging this into eq.(2) we obtain

In addition we will use the following simple way to convert strong concentration to a bound on expectation over subsets of assignments.

The set $\mathcal{S}$ contains $1/d$ fraction of points in $\{\pm 1\}^{n}$ and therefore

Let $\mathcal{S}\subseteq\{\pm 1\}^{n}$ be a set of assignments for which $d=2^{n}/|\mathcal{S}|$. Then

We are now ready to bound the discrimination norm.

Let $Q$ be a clause distribution of the distributional complexity $r=r(Q)$, let ${\mathcal{D}}^{\prime}\subseteq\{Q_{\sigma}\}_{\sigma\in\{\pm 1\}^{n}}$ be a set of distributions over clauses and $d=2^{n}/|{\mathcal{D}}^{\prime}|$. Then $\kappa_{2}({\mathcal{D}}^{\prime},U_{k})=O_{k}\left((\ln d/n)^{r/2}\right)$.

By plugging this into eq.(3) and using the fact that $\ln d<n$ we get,

By the definition of $\kappa_{2}({\mathcal{D}}^{\prime},U_{k})$ we obtain the claim. ∎

(of Theorem 3.5) Our reference distribution is the uniform distribution $U_{k}$ and the set of distributions ${\mathcal{D}}={\mathcal{D}}_{Q}=\{Q_{\sigma}\}_{\sigma\in\{\pm 1\}^{n}}$ is the set of distributions for all possible assignments. Let ${\mathcal{D}}^{\prime}\subseteq{\mathcal{D}}$ be a set of distributions of size $|{\mathcal{D}}|/q$ and $\mathcal{S}=\{\sigma\ |\ Q_{\sigma}\in{\mathcal{D}}^{\prime}\}$. Then, by Lemma 5.7, we get

Planted k𝑘k-CSPs

Before going into the proof of the lower bound for this model we show two additional connections between this model and our planted satisfiability model. First we show that planted satisfiability can be easily reduced to the planted $k$-CSP above while preserving the complexity parameter (we remark that the reduction will always produce a non-boolean $P$ and hence requires our generalization). The second connection is that both of these models can be seen as special cases of a more general model of planted constraint satisfaction introduced by Abbe and Montanari [AM15].

There exists an algorithm that for every distribution $Q$ over $\{\pm 1\}^{k}$ of complexity $r$, and any $\sigma\in\{\pm 1\}^{n}$, given a random sample distributed according to $Q_{\sigma}$ outputs a random sample distributed according to $P_{\sigma}$, where $P\equiv Q-2^{-k}$. Further, $r(Q)=r(P)$.

Given a random clause $C$ the algorithm outputs the tuple of variables $v(C)$ together with a bit $b$ chosen according to the following rule. With probability $1/2$: if $s(C)={\bf 1}_{k}$ then output 1, otherwise $-1$; with probability $1/2-2^{-k-1}$ output 1 and $-1$ with probability $2^{-k-1}$.

Let us analyze the resulting distribution. First we note that the output distribution is uniform over $Y_{k}$. This follows from the fact that for every $u\in Y_{k}$, $\sum_{v(C)=u}Q_{\sigma}(C)=\sum_{y\in\{\pm 1\}^{k}}Q(y)=1$. We now evaluate the expectation of the bit $b$ produced by our reduction as a function of $\sigma(u)$ (the values assigned by $\sigma$ to variables in $u$). From the definition of $Q_{\sigma}$, for every $u\in Y_{k}$ and $z\in\{\pm 1\}^{k}$,

where we use $\circ$ to denote the element-wise product of two vectors. In particular, $\Pr_{C\sim Q_{\sigma}}[s(C)={\bf 1}_{k}\ |\ v(C)=u]=Q(\sigma(u))$. This means that

This means that the reduction produces a random sample from $P_{\sigma}$ for $P(y)\equiv Q(y)-2^{-k}$. Note that $\hat{P}(\emptyset)=0$ and hence this reduction satisfies $r(Q)=r(P)$.

We now show how both of these models can be seen as special cases of the model in [AM15]. The model is specified by a collection of distributions $\{\Phi(\cdot\ |\ y)\}_{y\in\{\pm 1\}^{k}}$ over some output alphabet $Z$. For a planted assignment $\sigma\in\{\pm 1\}^{n}$ (their model allows a more general alphabet for each variable but $\{\pm 1\}$ suffices to subsume the models discussed in this paper) the planted distribution $\Phi_{\sigma}$ is defined as follows. A random sample from this distribution is a randomly and uniformly chosen ordered $k$-tuple of variables $u\in Y_{k}$ together with value $z$ chosen randomly and independently according to $\Phi(\cdot\ |\ \sigma(u))$. We first observe that for any $P:\{\pm 1\}^{k}\rightarrow$, setting $Z=\{\pm 1\}$ and having $\Phi(b\ |\ y)=(1+b\cdot P(y))/2$ recovers exactly the generalized Goldreich’s planted $k$-CSP for function $P$.

To recover the planted satisfiability model for distribution $Q$, we let $Z=\{\pm 1\}^{k}$ and then define $\Phi(z\ |\ y)=Q(y\circ z)$. Here the output alphabet represents the negation signs of variables. A $k$-tuple of variables $u\in Y_{k}$ with $k$ negation signs $z\in\{\pm 1\}^{k}$ uniquely describes a clause $C\in X_{k}$ such that $v(C)=u$ and $s(C)=z$. Further, by Eqn. (4), we get that $\Phi_{\sigma}$ for $\Phi$ defined as above is exactly $Q_{\sigma}$. It is not hard to see that the techniques in this work can also be applied to characterize the SQ complexity of solving planted $k$-CSPs in this more general model.

We prove the analogue of Theorem 3.5 for the planted $k$-CPS, which, in turn, immediately implies that the lower bounds stated in Theorem 3.1 apply to this problem verbatim. We first note that the reduction in Lemma 6.1 implies the desired lower bound for all functions $P$ such that $P\equiv Q-2^{-k}$ for some distribution $Q$ over $\{\pm 1\}^{k}$. Unfortunately, this is not sufficient to obtain a lower bound for all functions $P:\{\pm 1\}^{-k}\rightarrow$. Indeed, this does not give a lower bound for any Boolean $P$. At the same time, we show that the reduction in Lemma 6.1 can be used to reduce bounds on the discrimination norm of the planted $k$-CSP problem to the bounds on the discrimination norm for planted satisfiability that we gave in Section 5A direct proof of this bound can be found in an earlier version of this work [FPV13, v5].. We are not aware of similar reductions in the literature and our technique might be useful for relating the complexity of other problems for which standard reductions are not known.

We now give the formal details. Let $P:\{\pm 1\}^{k}\rightarrow$ be a function on $k$-bits. Let ${\mathcal{D}}_{P}$ denote the set of all distributions $P_{\sigma}$, where $\sigma\in\{\pm 1\}^{n}$ and $U^{\prime}_{k}$ be the uniform distribution over $X^{\prime}_{k}=Y_{k}\times\{-1,1\}$. Let ${\mathcal{B}}({\mathcal{D}}_{P},U^{\prime}_{k})$ denote the decision problem in which given samples from an unknown input distribution $D\in{\mathcal{D}}_{P}\cup\{U^{\prime}_{k}\}$ the goal is to output $1$ if $D\in{\mathcal{D}}_{P}$ and 0 if $D=U^{\prime}_{k}$. Our goal is to prove the following results.

For any function $P:\{\pm 1\}^{k}\rightarrow$ of complexity $r$, there exist a constant $c>0$ (that depends on $P$) such that for any $q\geq 1$,

As in the case of Theorem 3.5, it suffices to prove the following analogue of Lemma 5.7.

Let $P:\{\pm 1\}^{k}\rightarrow$ be any function of complexity $r=r(P)$, let ${\mathcal{D}}^{\prime}\subseteq\{P_{\sigma}\}_{\sigma\in\{\pm 1\}^{n}}$ be a set of distributions over clauses and $d=2^{n}/|{\mathcal{D}}^{\prime}|$. Then $\kappa_{2}({\mathcal{D}}^{\prime},U^{\prime}_{k})=O_{k}\left((\ln d/n)^{r/2}\right)$.

where $Q\equiv(P+1)/2^{k}$. Note that $Q$ defined in this way is a distribution since for all $y\in\{\pm 1\}^{k}$, $Q(y)\geq 0$ and $\sum_{y\in\{\pm 1\}^{k}}Q(y)=2^{k}\cdot\hat{P}(\emptyset)+1=1$.

Distributions $Q_{\sigma}$,$P_{\sigma}$ $U_{k}$ and $U^{\prime}_{k}$ are uniform over $k$-tuples of variables and therefore to prove eq. (5), it suffices to prove that for every $u\in Y_{k}$,

The left hand side of this equality is equal to

By equation (4), the right side of eq. 6 is equal to

where we used the fact that $\sum_{y\in\{\pm 1\}^{k}}P(y)=2^{k}\cdot\hat{P}(\emptyset)=0$ to obtain the equality of the second line to the third one.

Now all we need to bound $\kappa_{2}({\mathcal{D}}^{\prime},U^{\prime}_{k})$ is an upper bound on $\|h\|_{U_{k}}$. First, note that by our assumption,

Using this bound on the norm and eq. (5) we can now bound $\kappa_{2}({\mathcal{D}}^{\prime},U^{\prime}_{k})$ as follows. Let ${\mathcal{D}}^{\prime}_{Q}\doteq\{Q_{\sigma}\ |\ \sigma\in\mathcal{S}\}$.

where we used Lemma 5.7 to obtain the last bound. ∎

Lower Bounds using Statistical Dimension

We first prove an analogue of lower-bound for VSTAT from [FGR+12] but using the statistical dimension based on discrimination norm instead of the average correlation. It is not hard to see that discrimination norm is upper-bounded by the square root of average correlation and therefore our result subsumes the one in [FGR+12].

We prove our lower bound for any deterministic statistical algorithm and the claim for randomized algorithms follows from the fact that the success probability of a randomized algorithm is just the expectation of its success probability for a random fixing of its coins.

Let the set ${\mathcal{D}}^{+}\subseteq{\mathcal{D}}_{D}$ be the set of distributions on which ${\mathcal{A}}$ is successful (that is outputs $b=1$) and we denote these distributions by $\{D_{1},D_{2},\ldots,D_{m}\}$. We recall that, crucially, for ${\mathcal{A}}$ to be considered successful it needs to be successful for any valid responses of VSTAT to ${\mathcal{A}}$’s queries. We note that the success probability of ${\mathcal{A}}$ is $\frac{1}{2}+\frac{1}{2}\frac{m}{|{\mathcal{D}}_{D}|}$ and therefore $m\geq(2\gamma-1)|{\mathcal{D}}_{D}|$.

For every $k\leq q$, let $A_{k}$ be the set of all distributions $D_{i}$ such that

for every $k$, $|A_{k}|\leq|{\mathcal{D}}_{D}|/d$.

Combining these two implies that $q\geq d\cdot m/|{\mathcal{D}}_{D}|$ and therefore $q\geq(2\gamma-1)d$ giving the desired lower bound.

By our assumption for $D_{i}\in A_{k}$, $|p_{i,k}-p_{k}|>\tau_{i,k}=\max\{1/t,\sqrt{p_{i,k}(1-p_{i,k})/t}\}$. If $p_{i,k}\geq 2p_{k}/3$ then

Otherwise (when $p_{i,k}<2p_{k}/3$), $p_{k}-p_{i,k}>p_{k}-2p_{k}/3=p_{k}/3$. We also know that $|p_{i,k}-p_{k}|>\tau_{i,k}\geq 1/t$ and therefore $|p_{i,k}-p_{k}|>\sqrt{\frac{p_{k}}{3t}}$. Substituting this into eq. (8) we get that

Now, by the definition of discrimination norm and its linearity we have that

2 Lower bounds for MVSTAT and 1-MSTAT

We now describe the extension of our lower bound to MVSTAT and ${\mbox{1-MSTAT}}(L)$ oracles. For simplicity we state them for the worst case search problems but all these results are based on a direct simulation of an oracle using a VSTAT oracle and therefore they equivalently apply to the average-case versions of the problem defined in Theorem 7.1.

Given the lower bound VSTAT we can obtain our lower bound for MVSTAT via the following simple simulation. For conciseness we use $L_{0}$ to denote $\{0,1,\ldots,L-1\}$.