A recent paper of Braverman, Cohen, and Garg (STOC'18) introduced the concept of a weighted pseudorandom generator (WPRG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with (potentially negative) real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of WPRGs for ordered branching programs whose seed length has a better dependence on the error parameter s than the classic PRG construction of Nisan (STOC'90 and Combinatorica 1992). In this article we give an explicit construction of WPRGs that achieve parameters that are impossible to achieve by a PRG. In particular, we construct a WPRG for ordered permutation branching programs of unbounded width with a single accept state that has seed length O(log(3/2) n) for error parameter epsilon = 1/poly(n), where n is the input length. In contrast, recent work of Hoza et al. (ITCS'21) shows that any PRG for this model requires seed length Omega(log(2) =) to achieve error epsilon= 1/poly(=). As a corollary, we obtain explicit WPRGs with seed length O(log(3/2 )=) and error s = 1/poly(=) for ordered permutation branching programs of width F = poly(=) with an arbitrary number of accept states. Previously, seed length o(log(2 )=) was only known when both the width and the reciprocal of the error are subpolynomial, i. e., omega = n(0(1)) and epsilon = 1/n(0(1)) (Braverman, Rao, Raz, Yehudayoff, FOCS'10 and SICOMP 2014). The starting point for our work was the recent family of space-efficient algorithms for estimating random-walk probabilities in directed graphs by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford, and Vadhan (FOCS'20), which are based on spectral graph theory and space-efficient Laplacian solvers. We interpret these algorithms as giving WPRGs with large seed length, which we then derandomize to obtain our results. We also note that this approach gives a simpler proof of the original result of Braverman, Cohen, and Garg, as independently discovered by Cohen, Doron, Renard, Sberlo, and Ta-Shma (CCC'21).
Goldwasser et al. (ITCS'21) proposed the setting of PAC verification, where a hypothesis (machine learning model) that purportedly satisfies the agnostic PAC learning objective is verified using an interactive proof. In this paper we develop this notion further in a number of ways. First, we prove a lower bound of Omega(root d/epsilon(2)) i.i.d. samples for PAC verification of hypothesis classes with VC dimension d. Second, we present a protocol for PAC verification of unions of intervals over [IR that improves upon their proposed protocol for that task, and matches our lower bound's dependence on d. Third, we introduce a natural generalization of their definition to verification of general statistical algorithms, which is applicable to a wider variety of settings beyond agnostic PAC learning. Showcasing our proposed definition, our final result is a protocol for the verification of statistical query algorithms that satisfy a combinatorial constraint on their queries. In particular, this protocol can be used for verification of adaptive data analysis.
The Stabbing Planes proof system [8] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations mod2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [25] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.
A major proof complexity problem is to prove a superpolynomial lower bound on the length of Frege proofs of arbitrary depth. A more general question is to prove an Extended Frege lower bound. Surprisingly, proving such bounds turns out to be much easier in the algebraic setting. In this paper, we study a proof system that can simulate Extended Frege: an extension of the Polynomial Calculus proof system where we can take a square root and introduce new variables that are equivalent to arbitrary depth algebraic circuits. We prove that an instance of the subset-sum principle, the binary value principle 1 + x 1 + 2 x 2 + ... + 2 n −1 x n = 0 (BVP n ), requires refutations of exponential bit size over Q in this system. Part and Tzameret [18] proved an exponential lower bound on the size of Res-Lin (Resolution over linear equations [22]) refutations of BVP n . We show that our system p-simulates Res-Lin and thus we get an alternative exponential lower bound for the size of Res-Lin refutations of BVP n .
We propose a combinatorial hypothesis regarding a subspace vs. subspace agreement test, and prove that if correct it leads to a proof of the 2-to-1 Games Conjecture, albeit with imperfect completeness. This paper presents the second installment in a line of work by various subsets of the authors (with additional contributions by Barak, Kothari, and Steurer (ITCS'19)), which led to a proof of the 2-to-2 Games Conjecture.
