arXiv subjects

Chi-Ning Chou

Publications and source records attributed to Chi-Ning Chou.

27 records · Page 2Linked to original sources

arXiv2021

Optimal Streaming Approximations for all Boolean Max-2CSPs and Max-kSAT

We prove tight upper and lower bounds on approximation ratios of all Boolean Max-2CSP problems in the streaming model. Specifically, for every type of Max-2CSP problem, we give an explicit constant $α$, s.t. for any $ε>0$ (i) there is an $(α-ε)$-streaming approximation using space $O(\log{n})$; and (ii) any $(α+ε)$-streaming approximation requires space $Ω(\sqrt{n})$. This generalizes the celebrated work of [Kapralov, Khanna, Sudan SODA 2015; Kapralov, Krachun STOC 2019], who showed that the optimal approximation ratio for Max-CUT was $1/2$. Prior to this work, the problem of determining this ratio was open for all other Max-2CSPs. Our results are quite surprising for some specific Max-2CSPs. For the Max-DCUT problem, there was a gap between an upper bound of $1/2$ and a lower bound of $2/5$ [Guruswami, Velingker, Velusamy APPROX 2017]. We show that neither of these bounds is tight, and the optimal ratio for Max-DCUT is $4/9$. We also establish that the tight approximation for Max-2SAT is $\sqrt{2}/2$, and for Exact Max-2SAT it is $3/4$. As a byproduct, our result gives a separation between space-efficient approximations for Max-2SAT and Exact Max-2SAT. This is in sharp contrast to the setting of polynomial-time algorithms with polynomial space, where the two problems are known to be equally hard to approximate. Finally, we prove that the tight streaming approximation for \mksat{} is $\sqrt{2}/2$ for every $k\geq2$.

cs.CC↗

arXiv2020

ODE-Inspired Analysis for the Biological Version of Oja's Rule in Solving Streaming PCA

Oja's rule [Oja, Journal of mathematical biology 1982] is a well-known biologically-plausible algorithm using a Hebbian-type synaptic update rule to solve streaming principal component analysis (PCA). Computational neuroscientists have known that this biological version of Oja's rule converges to the top eigenvector of the covariance matrix of the input in the limit. However, prior to this work, it was open to prove any convergence rate guarantee. In this work, we give the first convergence rate analysis for the biological version of Oja's rule in solving streaming PCA. Moreover, our convergence rate matches the information theoretical lower bound up to logarithmic factors and outperforms the state-of-the-art upper bound for streaming PCA. Furthermore, we develop a novel framework inspired by ordinary differential equations (ODE) to analyze general stochastic dynamics. The framework abandons the traditional step-by-step analysis and instead analyzes a stochastic dynamic in one-shot by giving a closed-form solution to the entire dynamic. The one-shot framework allows us to apply stopping time and martingale techniques to have a flexible and precise control on the dynamic. We believe that this general framework is powerful and should lead to effective yet simple analysis for a large class of problems with stochastic dynamics.

q-bio.NC↗

arXiv2020

Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits

The linear cross-entropy benchmark (Linear XEB) has been used as a test for procedures simulating quantum circuits. Given a quantum circuit $C$ with $n$ inputs and outputs and purported simulator whose output is distributed according to a distribution $p$ over $\{0,1\}^n$, the linear XEB fidelity of the simulator is $\mathcal{F}_{C}(p) = 2^n \mathbb{E}_{x \sim p} q_C(x) -1$ where $q_C(x)$ is the probability that $x$ is output from the distribution $C|0^n\rangle$. A trivial simulator (e.g., the uniform distribution) satisfies $\mathcal{F}_C(p)=0$, while Google's noisy quantum simulation of a 53 qubit circuit $C$ achieved a fidelity value of $(2.24\pm0.21)\times10^{-3}$ (Arute et. al., Nature'19). In this work we give a classical randomized algorithm that for a given circuit $C$ of depth $d$ with Haar random 2-qubit gates achieves in expectation a fidelity value of $Ω(\tfrac{n}{L} \cdot 15^{-d})$ in running time $\textsf{poly}(n,2^L)$. Here $L$ is the size of the \emph{light cone} of $C$: the maximum number of input bits that each output bit depends on. In particular, we obtain a polynomial-time algorithm that achieves large fidelity of $ω(1)$ for depth $O(\sqrt{\log n})$ two-dimensional circuits. To our knowledge, this is the first such result for two dimensional circuits of super-constant depth. Our results can be considered as an evidence that fooling the linear XEB test might be easier than achieving a full simulation of the quantum circuit.

quant-ph↗

arXiv2019

Tracking the $\ell_2$ Norm with Constant Update Time

The \emph{$\ell_2$ tracking problem} is the task of obtaining a streaming algorithm that, given access to a stream of items $a_1,a_2,a_3,\ldots$ from a universe $[n]$, outputs at each time $t$ an estimate to the $\ell_2$ norm of the \textit{frequency vector} $f^{(t)}\in \mathbb{R}^n$ (where $f^{(t)}_i$ is the number of occurrences of item $i$ in the stream up to time $t$). The previous work [Braverman-Chestnut-Ivkin-Nelson-Wang-Woodruff, PODS 2017] gave an streaming algorithm with (the optimal) space using $O(ε^{-2}\log(1/δ))$ words and $O(ε^{-2}\log(1/δ))$ update time to obtain an $ε$-accurate estimate with probability at least $1-δ$. We give the first algorithm that achieves update time of $O(\log 1/δ)$ which is independent of the accuracy parameter $ε$, together with the nearly optimal space using $O(ε^{-2}\log(1/δ))$ words. Our algorithm is obtained using the \textsf{CountSketch} of [Charilkar-Chen-Farach-Colton, ICALP 2002].

cs.DS↗

arXiv2019

Closure of VP under taking factors: a short and simple proof

In this note, we give a short, simple and almost completely self contained proof of a classical result of Kaltofen [Kal86, Kal87, Kal89] which shows that if an $n$ variate degree $d$ polynomial $f$ can be computed by an arithmetic circuit of size $s$, then each of its factors can be computed by an arithmetic circuit of size at most $\textsf{poly}\left(s, n, d\right)$. However, unlike Kaltofen's argument, our proof does not directly give an efficient algorithm for computing the circuits for the factors of $f$.

cs.CC↗

arXiv2019

(Nearly) Efficient Algorithms for the Graph Matching Problem on Correlated Random Graphs

We give a quasipolynomial time algorithm for the graph matching problem (also known as noisy or robust graph isomorphism) on correlated random graphs. Specifically, for every $γ>0$, we give a $n^{O(\log n)}$ time algorithm that given a pair of $γ$-correlated $G(n,p)$ graphs $G_0,G_1$ with average degree between $n^{\varepsilon}$ and $n^{1/153}$ for $\varepsilon = o(1)$, recovers the "ground truth" permutation $π\in S_n$ that matches the vertices of $G_0$ to the vertices of $G_n$ in the way that minimizes the number of mismatched edges. We also give a recovery algorithm for a denser regime, and a polynomial-time algorithm for distinguishing between correlated and uncorrelated graphs. Prior work showed that recovery is information-theoretically possible in this model as long the average degree was at least $\log n$, but sub-exponential time algorithms were only known in the dense case (i.e., for $p > n^{-o(1)}$). Moreover, "Percolation Graph Matching", which is the most common heuristic for this problem, has been shown to require knowledge of $n^{Ω(1)}$ "seeds" (i.e., input/output pairs of the permutation $π$) to succeed in this regime. In contrast our algorithms require no seed and succeed for $p$ which is as low as $n^{o(1)-1}$.

cs.DS↗

arXiv2018

On the Algorithmic Power of Spiking Neural Networks

Spiking Neural Networks (SNN) are mathematical models in neuroscience to describe the dynamics among a set of neurons that interact with each other by firing instantaneous signals, a.k.a., spikes. Interestingly, a recent advance in neuroscience [Barrett-Denève-Machens, NIPS 2013] showed that the neurons' firing rate, i.e., the average number of spikes fired per unit of time, can be characterized by the optimal solution of a quadratic program defined by the parameters of the dynamics. This indicated that SNN potentially has the computational power to solve non-trivial quadratic programs. However, the results were justified empirically without rigorous analysis. We put this into the context of natural algorithms and aim to investigate the algorithmic power of SNN. Especially, we emphasize on giving rigorous asymptotic analysis on the performance of SNN in solving optimization problems. To enforce a theoretical study, we first identify a simplified SNN model that is tractable for analysis. Next, we confirm the empirical observation in the work of Barrett et al. by giving an upper bound on the convergence rate of SNN in solving the quadratic program. Further, we observe that in the case where there are infinitely many optimal solutions, SNN tends to converge to the one with smaller l1 norm. We give an affirmative answer to our finding by showing that SNN can solve the l1 minimization problem under some regular conditions. Our main technical insight is a dual view of the SNN dynamics, under which SNN can be viewed as a new natural primal-dual algorithm for the l1 minimization problem. We believe that the dual view is of independent interest and may potentially find interesting interpretation in neuroscience.

cs.NE↗

arXiv2018

Personalized Difficulty Adjustment for Countering the Double-Spending Attack in Proof-of-Work Consensus Protocols

Bitcoin is the first secure decentralized electronic currency system. However, it is known to be inefficient due to its proof-of-work (PoW) consensus algorithm and has the potential hazard of double spending. In this paper, we aim to reduce the probability of double spending by decreasing the probability of consecutive winning. We first formalize a PoW-based decentralized secure network model in order to present a quantitative analysis. Next, to resolve the risk of double spending, we propose the personalized difficulty adjustment (PDA) mechanism which modifies the difficulty of each participant such that those who win more blocks in the past few rounds have a smaller probability to win in the next round. To analyze the performance of the PDA mechanism, we observe that the system can be modeled by a high-order Markov chain. Finally, we show that PDA effectively decreases the probability of consecutive winning and results in a more trustworthy PoW-based system.

cs.CR↗

arXiv2018

Some Closure Results for Polynomial Factorization and Applications

In a sequence of seminal results in the 80's, Kaltofen showed that the complexity class VP is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded depth arithmetic circuits or the class VNP, are closed under taking factors. In this paper, we show that all factors of degree at most $\log^a n$ of polynomials with poly(n) size depth $k$ circuits have poly(n) size circuits of depth at most $O(k + a)$. This partially answers a question of Shpilka-Yehudayoff and has applications to hardness-randomness tradeoffs for bounded depth arithmetic circuits. More precisely, this shows that a superpolynomial lower bound for bounded depth arithmetic circuits, for a family of explicit polynomials of degree poly$(\log n)$ implies deterministic sub-exponential time algorithms for polynomial identity testing (PIT) for bounded depth arithmetic circuits. This is incomparable to a beautiful result of Dvir et al., where they showed that super-polynomial lower bounds for constant depth arithmetic circuits for any explicit family of polynomials (of potentially high degree) implies sub-exponential time deterministic PIT for bounded depth circuits of bounded individual degree. Thus, we remove the "bounded individual degree" condition in [DSY09] at the cost of strengthening the hardness assumption to hold for polynomials of low degree. As direct applications of our techniques, we also show that the complexity class VNP is closed under taking factors, thereby confirming a conjecture of Bürgisser and get an alternate proof of the fact (first shown by Dutta et al.) that if a polynomial $Q$ of degree at most $d$ divides a polynomial $P$ computable by a formula of size $s$, then $Q$ has a formula of size at most poly$(s, d^{\log d}, deg(P))$.

cs.CC↗