Venkatesan Guruswami

Joshua Brakensiek, Venkatesan Guruswami

The seminal work of Benczúr and Karger demonstrated cut sparsifiers of near-linear size. Subsequent extensions have yielded sparsifiers for hypergraph cuts and more recently linear codes over Abelian groups. A decade ago, Kogan and Krauthgamer asked about the sparsifiability of arbitrary constraint satisfaction problems (CSPs). For this question, a trivial lower bound is the size of a non-redundant CSP instance, which admits, for each constraint, an assignment satisfying only that constraint (so that no constraint can be dropped by the sparsifier). For instance, for graph cuts, spanning trees are non-redundant instances. Our main result is that redundant clauses are sufficient for sparsification: for any CSP predicate R, every unweighted instance of CSP(R) has a sparsifier of size at most its non-redundancy (up to polylog and $1/ε$ factors). For weighted instances, we similarly pin down the sparsifiability to the so-called chain length of the predicate. These results precisely determine the extent to which any CSP can be sparsified. Our result is established in the general setting of non-linear codes, or equivalently set families, yielding a VC-type theorem for multiplicative error approximation. A key technical ingredient in our work is a novel application of the entropy method from Gilmer's recent breakthrough on the union-closed sets conjecture. As an immediate consequence of our main theorem, a number of results in the non-redundancy literature immediately extend to CSP sparsification. We also contribute new techniques for understanding the non-redundancy of CSP predicates. By adapting methods from the matching vector codes literature in coding theory, we are able to construct an explicit predicate whose non-redundancy lies between $Ω(n^{1.5})$ and $\widetilde{O}(n^{1.6})$, the first example with a provably non-integral exponent.

Venkatesan Guruswami, Rishi Saket

In recent years the framework of learning from label proportions (LLP) has been gaining importance in machine learning. In this setting, the training examples are aggregated into subsets or bags and only the average label per bag is available for learning an example-level predictor. This generalizes traditional PAC learning which is the special case of unit-sized bags. The computational learning aspects of LLP were studied in recent works (Saket, NeurIPS'21; Saket, NeurIPS'22) which showed algorithms and hardness for learning halfspaces in the LLP setting. In this work we focus on the intractability of LLP learning Boolean functions. Our first result shows that given a collection of bags of size at most $2$ which are consistent with an OR function, it is NP-hard to find a CNF of constantly many clauses which satisfies any constant-fraction of the bags. This is in contrast with the work of (Saket, NeurIPS'21) which gave a $(2/5)$-approximation for learning ORs using a halfspace. Thus, our result provides a separation between constant clause CNFs and halfspaces as hypotheses for LLP learning ORs. Next, we prove the hardness of satisfying more than $1/2 + o(1)$ fraction of such bags using a $t$-DNF (i.e. DNF where each term has $\leq t$ literals) for any constant $t$. In usual PAC learning such a hardness was known (Khot-Saket, FOCS'08) only for learning noisy ORs. We also study the learnability of parities and show that it is NP-hard to satisfy more than $(q/2^{q-1} + o(1))$-fraction of $q$-sized bags which are consistent with a parity using a parity, while a random parity based algorithm achieves a $(1/2^{q-2})$-approximation.

Vipul Arora, Arnab Bhattacharyya, Mathews Boban et al.

We study the problem of robust multivariate polynomial regression: let $p\colon\mathbb{R}^n\to\mathbb{R}$ be an unknown $n$-variate polynomial of degree at most $d$ in each variable. We are given as input a set of random samples $(\mathbf{x}_i,y_i) \in [-1,1]^n \times \mathbb{R}$ that are noisy versions of $(\mathbf{x}_i,p(\mathbf{x}_i))$. More precisely, each $\mathbf{x}_i$ is sampled independently from some distribution $χ$ on $[-1,1]^n$, and for each $i$ independently, $y_i$ is arbitrary (i.e., an outlier) with probability at most $ρ< 1/2$, and otherwise satisfies $|y_i-p(\mathbf{x}_i)|\leqσ$. The goal is to output a polynomial $\hat{p}$, of degree at most $d$ in each variable, within an $\ell_\infty$-distance of at most $O(σ)$ from $p$. Kane, Karmalkar, and Price [FOCS'17] solved this problem for $n=1$. We generalize their results to the $n$-variate setting, showing an algorithm that achieves a sample complexity of $O_n(d^n\log d)$, where the hidden constant depends on $n$, if $χ$ is the $n$-dimensional Chebyshev distribution. The sample complexity is $O_n(d^{2n}\log d)$, if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most $O(σ)$, and the run-time depends on $\log(1/σ)$. In the setting where each $\mathbf{x}_i$ and $y_i$ are known up to $N$ bits of precision, the run-time's dependence on $N$ is linear. We also show that our sample complexities are optimal in terms of $d^n$. Furthermore, we show that it is possible to have the run-time be independent of $1/σ$, at the cost of a higher sample complexity.

Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami et al.

Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length $\ell$ among any $N \leq 2^\ell$ players such that for a threshold parameter $t$, (i) the knowledge of any $t$ shares does not reveal any information about the secret and, (ii) any choice of $t+1$ shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length $\ell$, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any $t+g$ shares, for a positive integer gap parameter $g$. Ramp secret sharing scheme necessarily requires shares of length $\ell/g$. Other than the bound related to secret length $\ell$, the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio $g/N$. In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio $g/N$, where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any $τ$ fraction of observed shares, and reconstruction from any $ρ$ fraction of shares, for any choices of $0 \leq τ< ρ\leq 1$. Our construction achieves secret length $N(ρ-τ-o(1))$, which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length $Ω(N(ρ-τ))$.

Mahdi Cheraghchi, Venkatesan Guruswami

Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error-detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Dziembowski et al. show existence of non-malleable codes for any class of tampering functions of bounded size. We consider constructions of coding schemes against two well-studied classes of tampering functions: bit-wise tampering functions (where the adversary tampers each bit of the encoding independently) and split-state adversaries (where two independent adversaries arbitrarily tamper each half of the encoded sequence). 1. For bit-tampering, we obtain explicit and efficiently encodable and decodable codes of length $n$ achieving rate $1-o(1)$ and error (security) $\exp(-\tildeΩ(n^{1/7}))$. We improve the error to $\exp(-\tildeΩ(n))$ at the cost of making the construction Monte Carlo with success probability $1-\exp(-Ω(n))$. Previously, the best known construction of bit-tampering codes was the Monte Carlo construction of Dziembowski et al. (ICS 2010) achieving rate ~.1887. 2. We initiate the study of seedless non-malleable extractors as a variation of non-malleable extractors introduced by Dodis and Wichs (STOC 2009). We show that construction of non-malleable codes for the split-state model reduces to construction of non-malleable two-source extractors. We prove existence of such extractors, which implies that codes obtained from our reduction can achieve rates arbitrarily close to 1/5 and exponentially small error. Currently, the best known explicit construction of split-state coding schemes is due to Aggarwal, Dodis and Lovett (ECCC TR13-081) which only achieves vanishing (polynomially small) rate.

Mahdi Cheraghchi, Venkatesan Guruswami

Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), encode messages $s$ in a manner so that tampering the codeword causes the decoder to either output $s$ or a message that is independent of $s$. While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly non-malleable coding becomes possible against every fixed family $F$ of tampering functions that is not too large (for instance, when $|F| \le \exp(2^{αn})$ for some $α\in [0, 1)$ where $n$ is the number of bits in a codeword). In this work, we study the "capacity of non-malleable coding", and establish optimal bounds on the achievable rate as a function of the family size, answering an open problem from Dziembowski et al. (ICS 2010). Specifically, 1. We prove that for every family $F$ with $|F| \le \exp(2^{αn})$, there exist non-malleable codes against $F$ with rate arbitrarily close to $1-α$ (this is achieved w.h.p. by a randomized construction). 2. We show the existence of families of size $\exp(n^{O(1)} 2^{αn})$ against which there is no non-malleable code of rate $1-α$ (in fact this is the case w.h.p for a random family of this size). 3. We also show that $1-α$ is the best achievable rate for the family of functions which are only allowed to tamper the first $αn$ bits of the codeword, which is of special interest. As a corollary, this implies that the capacity of non-malleable coding in the split-state model (where the tampering function acts independently but arbitrarily on the two halves of the codeword) equals 1/2. We also give an efficient Monte Carlo construction of codes of rate close to 1 with polynomial time encoding and decoding that is non-malleable against any fixed $c > 0$ and family $F$ of size $\exp(n^c)$, in particular tampering functions with, say, cubic size circuits.