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Register a new userComputational Lower Bounds for Community Detection on Random Graphs
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Jiaming Xu
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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This paper studies the problem of detecting the presence of a small dense community planted in a large ErdH{o}s-Renyi random graph $mathcal{G}(N,q)$, where the edge probability within the community exceeds $q$ by a constant factor. Assuming the hardness of the planted clique detection problem, we show that the computational complexity of detecting the community exhibits the following phase transition phenomenon: As the graph size $N$ grows and the graph becomes sparser according to $q=N^{-alpha}$, there exists a critical value of $alpha = frac{2}{3}$, below which there exists a computationally intensive procedure that can detect far smaller communities than any computationally efficient procedure, and above which a linear-time procedure is statistically optimal. The results also lead to the average-case hardness results for recovering the dense community and approximating the densest $K$-subgraph.
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We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an $O(m^{1/2-varepsilon})$ time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.
We give lower bounds on the performance of two of the most popular sampling methods in practice, the Metropolis-adjusted Langevin algorithm (MALA) and multi-step Hamiltonian Monte Carlo (HMC) with a leapfrog integrator, when applied to well-conditioned distributions. Our main result is a nearly-tight lower bound of $widetilde{Omega}(kappa d)$ on the mixing time of MALA from an exponentially warm start, matching a line of algorithmic results up to logarithmic factors and answering an open question of Chewi et. al. We also show that a polynomial dependence on dimension is necessary for the relaxation time of HMC under any number of leapfrog steps, and bound the gains achievable by changing the step count. Our HMC analysis draws upon a novel connection between leapfrog integration and Chebyshev polynomials, which may be of independent interest.
In this work, we initiate a formal study of probably approximately correct (PAC) learning under evasion attacks, where the adversarys goal is to emph{misclassify} the adversarially perturbed sample point $widetilde{x}$, i.e., $h(widetilde{x}) eq c(widetilde{x})$, where $c$ is the ground truth concept and $h$ is the learned hypothesis. Previous work on PAC learning of adversarial examples have all modeled adversarial examples as corrupted inputs in which the goal of the adversary is to achieve $h(widetilde{x}) eq c(x)$, where $x$ is the original untampered instance. These two definitions of adversarial risk coincide for many natural distributions, such as images, but are incomparable in general. We first prove that for many theoretically natural input spaces of high dimension $n$ (e.g., isotropic Gaussian in dimension $n$ under $ell_2$ perturbations), if the adversary is allowed to apply up to a sublinear $o(||x||)$ amount of perturbations on the test instances, PAC learning requires sample complexity that is exponential in $n$. This is in contrast with results proved using the corrupted-input framework, in which the sample complexity of robust learning is only polynomially more. We then formalize hybrid attacks in which the evasion attack is preceded by a poisoning attack. This is perhaps reminiscent of trapdoor attacks in which a poisoning phase is involved as well, but the evasion phase here uses the error-region definition of risk that aims at misclassifying the perturbed instances. In this case, we show PAC learning is sometimes impossible all together, even when it is possible without the attack (e.g., due to the bounded VC dimension).
We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower bounded by a constant. If we want error <1/2^N then we need T=Omega(N) queries. We apply this to show that a quantum computer cannot do much better than a classical computer when amplifying the success probability of an RP-machine. A classical computer can achieve error <=1/2^k using k applications of the RP-machine, a quantum computer still needs at least ck applications for this (when treating the machine as a black-box), where c>0 is a constant independent of k. Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N) queries for quantum bounded-error search of an ordered list of N items.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
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