# Research

My research interest is theoretical computer science (TCS), and in particular, computational complexity. I am enthralled by problems in circuit complexity, algebraic computation, and quantum computation. In addition, I am also interested in pseudorandomness, cryptography, coding theory, and the interaction of TCS with other fields.

## Conference Publications

Approximability of all finite CSPs in the dynamic streaming setting
with Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy.
To appear in Symposium on Foundations of Computer Science (FOCS 2021)
[arxiv] [eccc] [ ] [ ]

@article{CGSV21b,
title={Approximability of all finite CSPs in the dynamic streaming setting},
author={Chou, Chi-Ning and Golovnev, Sasha and Sudan, Madhu and Velusamy, Santhoshini},
year={2021},
archivePrefix = "arXiv",
eprint = "2105.01161",
primaryClass={cs.CC}
}


A constraint satisfaction problem (CSP), Max-CSP($\mathcal{F}$), is specified by a finite set of constraints $\mathcal{F}\subseteq\{[q]^k\to\{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on n variables is given by m applications of constraints from $\mathcal{F}$ to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(\gamma,\beta)$-approximation version of the problem for parameters $0\leq\beta\leq\gamma\leq1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied.

In this work we consider the approximability of this problem in the context of streaming algorithms and give a dichotomy result in the dynamic setting, where constraints can be inserted or deleted. Specifically, for every family $\mathcal{F}$ and every $\beta<\gamma$, we show that either the approximation problem is solvable with polylogarithmic space in the dynamic setting, or not solvable with $o\sqrt{n}$ space. We also establish tight inapproximability results for a broad subclass in the streaming insertion-only setting. Our work builds on, and significantly extends previous work by the authors who consider the special case of Boolean variables ($q=2$), singleton families ($|\mathcal{F}|=1$) and where constraints may be placed on variables or their negations. Our framework extends non-trivially the previous work allowing us to appeal to richer norm estimation algorithms to get our algorithmic results. For our negative results we introduce new variants of the communication problems studied in the previous work, build new reductions for these problems, and extend the technical parts of previous works.

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Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits
with Boaz Barak and Xun Gao.
Innovations in Theoretical Computer Science Conference (ITCS 2021)
[arxiv] [conference version] [slides] [ ] [ ]

@InProceedings{BCG21,
author =	{Boaz Barak and Chi-Ning Chou and Xun Gao},
title =	{Spoofing Linear Cross-Entropy Benchmarking in Shallow Quantum Circuits},
booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages =	{30:1--30:20},
year =	{2021},
archivePrefix = "arXiv",
eprint = "2005.02421",
primaryClass={quant-ph}
}


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|0n⟩. 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±0.21)×10−3 (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 $\Omega(\frac{n}{L}\cdot15^{-d})$ in running time 𝗉𝗈𝗅𝗒(n,2L). Here L is the size of the *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 $\omega(1)$ 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.

Optimal Streaming Approximations for all Boolean Max 2-CSPs and Max k-SAT
with Alexander Golovnev, Santhoshini Velusamy.
Symposium on Foundations of Computer Science (FOCS 2020)
[arxiv] [conference version] [slides] [ ] [ ]

@InProceedings{CGV20,
title={Optimal Streaming Approximations for all Boolean Max-2CSPs and Max-ksat},
author={Chou, Chi-Ning and Golovnev, Sasha and Velusamy, Santhoshini},
booktitle={2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)},
year={2020},
pages={330-341},
doi={10.1109/FOCS46700.2020.00039},
archivePrefix = "arXiv",
eprint = "2004.11796",
primaryClass={cs.CC}
}


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 $\alpha$, s.t. for any $\epsilon>0$ (i) there is an $(\alpha-\epsilon)$-streaming approximation using space $O(\log n)$; and (ii) any $(\alpha-\epsilon)$-streaming approximation requires space $\Omega(\sqrt{n})$. 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.

ODE-Inspired Analysis for the Biological Version of Oja’s Rule in Solving Streaming PCA
with Mien Brabeeba Wang.
Conference on Learning Theory (COLT 2020).
[arxiv] [conference version (extended abstract)] [slides] [ ] [ ]

@InProceedings{CW20,
title = 	 {ODE-Inspired Analysis for the Biological Version of Oja{'}s Rule in Solving Streaming PCA},
author = 	 {Chou, Chi-Ning and Wang, Mien Brabeeba},
booktitle = 	 {Proceedings of Thirty Third Conference on Learning Theory (COLT 2020)},
pages = 	 {1339--1343},
year = 	 {2020},
editor = 	 {Abernethy, Jacob and Agarwal, Shivani},
volume = 	 {125},
series = 	 {Proceedings of Machine Learning Research},
month = 	 {09--12 Jul},
publisher = 	 {PMLR},
pdf = 	 {http://proceedings.mlr.press/v125/chou20a/chou20a.pdf},
url = 	 {http://proceedings.mlr.press/v125/chou20a.html},
archivePrefix = "arXiv",
eprint = "1911.02363",
primaryClass={q-bio.NC}
}


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.

(Nearly) Efficient Algorithms for the Graph Matching Problem on Correlated Random Graphs
with Boaz Barak, Zhixian Lei, Tselil Schramm, Yueqi Sheng
Advances in Neural Information Processing Systems (NeurIPS 2019).
[arxiv] [conference version] [slides by Tselil] [poster] [ ] [ ]

@InProceedings{BCLSS19,
title = {(Nearly) Efficient Algorithms for the Graph Matching Problem on Correlated Random Graphs},
author = {Barak, Boaz and Chou, Chi-Ning and Lei, Zhixian and Schramm, Tselil and Sheng, Yueqi},
booktitle = {Advances in Neural Information Processing Systems 32 (NeurIPS 2019)},
pages = {9186--9194},
year = {2019},
url = {http://papers.nips.cc/paper/9118-nearly-efficient-algorithms-for-the-graph-matching-problem-on-correlated-random-graphs.pdf},
archivePrefix = "arXiv",
eprint = "1805.02349",
primaryClass={cs.DS}
}


We consider the graph matching/similarity problem of determining how similar two given graphs $G_0,G_1$ are and recovering the permutation $\pi$ on the vertices of $G_1$ that minimizes the symmetric difference between the edges of $G_0$ and $\pi(G_1)$. Graph matching/similarity has applications for pattern matching, computer vision, social network anonymization, malware analysis, and more. We give the first efficient algorithms proven to succeed in the correlated Erdös-Rényi model (Pedarsani and Grossglauser, 2011).

Specifically, we give a polynomial time algorithm for the graph similarity/hypothesis testing task which works for every constant level of correlation between the two graphs that can be arbitrarily close to zero. We also give a quasi-polynomial ($n^{O(\log n)}$ time) algorithm for the graph matching task of recovering the permutation minimizing the symmetric difference in this model. This is the first algorithm to do so without requiring as additional input a "seed" of the values of the ground truth permutation on at least $n^{\Omega(1)}$ vertices. Our algorithms follow a general framework of counting the occurrences of subgraphs from a particular family of graphs allowing for tradeoffs between efficiency and accuracy.

Tracking the $\ell_2$ Norm with Constant Update Time
with Zhixian Lei, Preetum Nakkiran
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019).
[arxiv] [conference version] [slides] [ ] [ ]

@InProceedings{CLN19,
author =	{Chi-Ning Chou and Zhixian Lei and Preetum Nakkiran},
title =	{Tracking the l_2 Norm with Constant Update Time},
booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages =	{2:1--2:15},
ISSN =	{1868-8969},
year =	{2019},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11217},
URN =		{urn:nbn:de:0030-drops-112175},
annote =	{Keywords: Streaming algorithms, Sketching algorithms, Tracking, CountSketch},
archivePrefix = "arXiv",
eprint = "1807.06479",
primaryClass={cs.DS}
}


The $\ell_2$ 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 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 $O(\epsilon^{-2}\log(1/\delta))$ words and $O(\epsilon^{-2}\log(1/\delta))$ update time to obtain an $\epsilon$-accurate estimate with probability at least $1-\delta$.

We give the first algorithm that achieves update time of $O(\log 1/\delta)$ which is independent of the accuracy parameter $\epsilon$, together with the nearly optimal space using $O(\epsilon^{-2}\log(1/\delta))$ words. Our algorithm is obtained using the Count Sketch of [Charilkar-Chen-Farach-Colton, ICALP 2002].

On the Algorithmic Power of Spiking Neural Networks
with Kai-Min Chung, Chi-Jen Lu
Innovations in Theoretical Computer Science (ITCS 2019).
[arxiv] [conference version] [slides] [poster] [ ] [ ]

@InProceedings{CCL19,
author ={Chi-Ning Chou and Kai-Min Chung and Chi-Jen Lu},
title ={On the Algorithmic Power of Spiking Neural Networks},
booktitle ={10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
pages ={26:1--26:20},
year ={2019},
URL ={http://drops.dagstuhl.de/opus/volltexte/2018/10119},
archivePrefix = "arXiv",
eprint = "1803.10375",
primaryClass={cs.NE}
}


Photo by Pennstatenews.

Spiking neural networks (SNNs) are mathematical models for biological neural networks such as our brain. In this work, we study SNNs through the lens of algorithms. In particular, we show that the firing rate of the integrate-and-fire SNN can efficiently solve the non-negative least squares problem and $\ell_1$ minimization problem. Further, our proof gives new interpretations on the integrate-and-fire SNN where the external charging and spiking effects can be viewed as gradient and projection respectively in the dual space.

Personalized Difficulty Adjustment for Countering the Double-Spending Attack in Proof-of-Work Consensus Protocols
with Yu-Jing Lin, Ren Chen, Hsiu-Yao Chang, I-Ping Tu, Shih-wei Liao
IEEE International Conference on Blockchain (Blockchain 2018).
[arxiv] [conference version] [ ] [ ]

@InProceedings{CLCCTL18,
title={Personalized Difficulty Adjustment for Countering the Double-Spending Attack in Proof-of-Work Consensus Protocols},
author={Chou, Chi-Ning and Lin, Yu-Jing and Chen, Ren and Chang, Hsiu-Yao and Tu, I-Ping and Liao, Shih-wei},
booktitle={2018 IEEE Conference on Blockchain (Blockchain 2018)},
pages={1457-1462},
year={2018}
}

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.

Hardness vs Randomness for Bounded Depth Arithmetic Circuits
with Mrinal Kumar, Noam Solomon
Computational Complexity Conference (CCC 2018).
[arxiv] [eccc] [conference version] [slides] [ ] [ ]

@InProceedings{CKS18,
author ={Chi-Ning Chou and Mrinal Kumar and Noam Solomon},
title ={Hardness vs Randomness for Bounded Depth Arithmetic Circuits},
booktitle ={33rd Computational Complexity Conference (CCC 2018)},
pages ={13:1--13:17},
ISSN ={1868-8969},
year ={2018},
URL ={http://drops.dagstuhl.de/opus/volltexte/2018/8876},
URN ={urn:nbn:de:0030-drops-88765},
annote ={Keywords: Algebraic Complexity, Polynomial Factorization Circuit Lower Bounds, Polynomial Identity Testing}
}


In this paper, we study the question of hardness-randomness tradeoffs for bounded depth arithmetic circuits. We show that if there is a family of explicit polynomials $\{f_n\}$, where $f_n$ is of degree $O(\log^2n/\log^2\log n)$ in $n$ variables such that $f_n$ cannot be computed by a depth $\Delta$ arithmetic circuits of size $\poly(n)$, then there is a deterministic sub-exponential time algorithm for polynomial identity testing of arithmetic circuits of depth $\Delta-5$.

This is incomparable to a beautiful result of Dvir et al.[SICOMP, 2009], where they showed that super-polynomial lower bounds for depth $\Delta$ circuits for any explicit family of polynomials (of potentially high degree) implies sub-exponential time deterministic PIT for depth $\Delta-5$ circuits of bounded individual degree. Thus, we remove the bounded individual degree" condition in the work of Dvir et al. at the cost of strengthening the hardness assumption to hold for polynomials of low degree.

The key technical ingredient of our proof is the following property of roots of polynomials computable by a bounded depth arithmetic circuit : if $f(x_1, x_2, \ldots, x_n)$ and $P(x_1, x_2, \ldots, x_n, y)$ are polynomials of degree $d$ and $r$ respectively, such that $P$ can be computed by a circuit of size $s$ and depth $\Delta$ and $P(x_1, x_2, \ldots, x_n, f) \equiv 0$, then, $f$ can be computed by a circuit of size $\poly(n, s, r, d^{O(\sqrt{d})})$ and depth $\Delta + 3$. In comparison, Dvir et al. showed that $f$ can be computed by a circuit of depth $\Delta + 3$ and size $\poly(n, s, r, d^{t})$, where $t$ is the degree of $P$ in $y$. Thus, the size upper bound in the work of Dvir et al. is non-trivial when $t$ is small but $d$ could be large, whereas our size upper bound is non-trivial when $d$ is small, but $t$ could be large.

## Journal Publications

Closure Results for Polynomial Factorization
with Mrinal Kumar, Noam Solomon
Theory of Computing.
[journal version] [slides] [video] [ ] [ ]

@journal{CKS19J,
author = {Chou, Chi-Ning and Kumar, Mrinal and Solomon, Noam},
title = {Closure Results for Polynomial Factorization},
year = {2019},
pages = {1--34},
doi = {10.4086/toc.2019.v015a013},
publisher = {Theory of Computing},
journal = {Theory of Computing},
volume = {15},
number = {13},
URL = {http://www.theoryofcomputing.org/articles/v015a013},
archivePrefix = "arXiv",
eprint = "1803.05933",
primaryClass={cs.CC}
}


In a sequence of fundamental results in the 1980s, Kaltofen (SICOMP 1985, STOC'86, STOC'87, RANDOM'89) showed that factors of multivariate polynomials with small arithmetic circuits have small arithmetic circuits. In other words, 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 $\log^a n$ of polynomials with $\poly(n)$-size depth-$k$ circuits have $\poly(n)$-size circuits of depth $O(k + a)$. This partially answers a question of Shpilka--Yehudayoff (Found. Trends in TCS, 2010) and has applications to hardness-randomness tradeoffs for bounded-depth arithmetic circuits.

As direct applications of our techniques, we also obtain simple proofs of the following results.

$\bullet$ The complexity class $\VNP$ is closed under taking factors. This confirms Conjecture~2.1 in Bürgisser's monograph (2000) and improves upon a recent result of Dutta, Saxena and Sinhababu (STOC'18) who showed a quasipolynomial upper bound on the number of auxiliary variables and the complexity of the verifier circuit of factors of polynomials in $\VNP$.

$\bullet$ A factor of degree $d$ of a polynomial $P$ which can be computed by an arithmetic formula (or an algebraic branching program) of size $s$ has a formula (an algebraic branching program, resp.) of size $\poly(s, d^{\log d}, \deg(P))$. This result was first shown by Dutta et al. (STOC'18) and we obtain a slightly different proof as an easy consequence of our techniques.

Our proofs rely on a combination of the ideas, based on Hensel lifting, developed in the polynomial factoring literature, and the depth-reduction results for arithmetic circuits, and hold over fields of characteristic zero or of sufficiently large characteristic.

## Manuscripts

Limitations of Local Quantum Algorithms on Maximum Cuts of Sparse Hypergraphs and Beyond
with Peter J. Love, Juspreet Singh Sandhu, Jonathan Shi.
[arxiv] [ ] [ ]

@article{CLSS21,
title={Limitations of Local Quantum Algorithms on Maximum Cuts of Sparse Hypergraphs and Beyond},
author={Chou, Chi-Ning and Love, J. Peter and Sandhu, Juspreet Singh and Shi, Jonathan},
eprint={2108.06049},
archivePrefix={arXiv},
primaryClass={quant-ph}
}


In this work, we study the limitations of the Quantum Approximate Optimization Algorithm (QAOA) through the lens of statistical physics and show that there exists $\epsilon > 0$, such that $\epsilon \log(n)$ depth QAOA cannot arbitrarily-well approximate the ground state energy of random diluted $k$-spin glasses when $k \geq 4$ is even. This is equivalent to the weak approximation resistance of logarithmic depth QAOA to the Max-$k$-XOR problem. We further extend the limitation to other boolean constraint satisfaction problems as long as the problem satisfies a combinatorial property called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. As a consequence of our techniques, we confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth —-- in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. Our results provide a new way to study the power and limit of QAOA through statistical physics methods and combinatorial properties.

Quantum Meets the Minimum Circuit Size Problem
with Nai-Hui Chia, Jiayu Zhang, Ruizhe Zhang.
[arxiv] [eccc] [ ] [ ]

@article{CCZZ21,
title={Quantum Meets the Minimum Circuit Size Problem},
author={Chia, Nai-Hui and Chou, Chi-Ning and Zhang, Jiayu and Zhang, Ruizhe},
eprint={2108.03171},
archivePrefix={arXiv},
primaryClass={quant-ph}
}


In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory---its hardness is mysterious, and a better understanding of its hardness can have surprising implications to many fields in computer science.

We first define and investigate the basic complexity-theoretic properties of minimum quantum circuit size problems for three natural objects: Boolean functions, unitaries, and quantum states. We show that these problems are not trivially in NP but in QCMA (or have QCMA protocols). Next, we explore the relations between the three quantum MCSPs and their variants. We discover that some reductions that are not known for classical MCSP exist for quantum MCSPs for unitaries and states, e.g., search-to-decision reduction and self-reduction. Finally, we systematically generalize results known for classical MCSP to the quantum setting (including quantum cryptography, quantum learning theory, quantum circuit lower bounds, and quantum fine-grained complexity) and also find new connections to tomography and quantum gravity. Due to the fundamental differences between classical and quantum circuits, most of our results require extra care and reveal properties and phenomena unique to the quantum setting. Our findings could be of interest for future studies, and we post several open problems for further exploration along this direction.

Linear Space Streaming Lower Bounds for Approximating CSPs
with Alexander Golovnev, Madhu Sudan, Ameya Velingker Santhoshini Velusamy.
[arxiv] [eccc] [ ] [ ]

@article{CGSVV21,
title={Linear Space Streaming Lower Bounds for Approximating CSPs},
author={Chou, Chi-Ning and Golovnev, Sasha and Sudan, Madhu and Velingker, Ameya and Velusamy, Santhoshini},
year={2021},
archivePrefix = "arXiv",
eprint = "2106.13078",
primaryClass={cs.CC}
}


We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on n variables taking values in $\{0,\dots,q−1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $\Omega(n)$ space even on instances with $O(n)$ constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires $\Omega(n)$ space. The key technical core is an optimal, $q−(k−1)$-inapproximability for the case where every constraint is given by a system of $k−1$ linear equations mod $q$ over $k$ variables. Prior to our work, no such hardness was known for an approximation factor less than $1/2$ for any CSP. Our work builds on and extends the work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the max cut in graphs. This corresponds roughly to the case of Max $k$-LIN-modq with $k=q=2$. Each one of the extensions provides non-trivial technical challenges that we overcome in this work.

Approximability of all Boolean CSPs in the Dynamic Streaming Setting
with Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy.
[arxiv] [eccc] [ ] [ ]

@article{CGSV21,
title={Approximability of all Boolean CSPs in the dynamic streaming setting},
author={Chou, Chi-Ning and Golovnev, Sasha and Sudan, Madhu and Velusamy, Santhoshini},
year={2021},
archivePrefix = "arXiv",
eprint = "2102.12351",
primaryClass={cs.CC}
}


A Boolean constraint satisfaction problem (CSP), $\textsf{Max-CSP}(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the $n$~variables. In the $(\gamma,\beta)$-approximation version of the problem for parameters $\gamma \geq \beta \in [0,1]$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied.

In this work we consider the approximability of $\textsf{Max-CSP}(f)$ in the (dynamic) streaming setting, where constraints are inserted (and may also be deleted in the dynamic setting) one at a time. We completely characterize the approximability of all Boolean CSPs in the dynamic streaming setting. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of $\textsf{Max-CSP}(f)$ has a probabilistic dynamic streaming algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of $\textsf{Max-CSP}(f)$ requires $\Omega(\sqrt{n})$ space for probabilistic dynamic streaming algorithms. We also extend previously known results in the insertion-only setting to a wide variety of cases, and in particular the case of $k=2$ where we get a dichotomy and the case when the satisfying assignments of $f$ support a distribution on $\{-1,1\}^k$ with uniform marginals.

Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy, APPROX 2017]] and [Chou-Golovnev-Velusamy, FOCS 2020] by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [Kapralov-Khannan-Sudan, SODA 2015], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results.

A General Framework for Analyzing Stochastic Dynamics in Learning Algorithms
with Juspreet Singh Sandhu, Mien Brabeeba Wang, Tiancheng Yu.
[arxiv][ ] [ ]

@article{CSWY21,
title={A General Framework for Analyzing Stochastic Dynamics in Learning Algorithms},
author={Chou, Chi-Ning and Sandhu, Juspreet Singh and Wang, Mien Brabeeba and Yu, Tiancheng},
eprint={2006.06171},
archivePrefix={arXiv},
primaryClass={math.OC}
}


One of the challenges in analyzing a learning algorithm is the circular entanglement between the objective value and the stochastic noise. This is also known as the "chicken and egg" phenomenon. Traditionally, people tackle this issue with the special structure of the problem and hence the analysis is difficult to generalize.

In this paper, we present a general framework for analyzing high-probability bounds for stochastic dynamics in learning algorithms. Our framework composes standard techniques from probability theory to give a streamlined three-step recipe with a general and flexible principle to tackle the "chicken and egg" problem. We demonstrate the power and the flexibility of our framework by giving unifying analysis for three very different learning problems with both the last iterate and the strong uniform high probability convergence guarantee. The problems are stochastic gradient descent for strongly convex functions, streaming principal component analysis and linear bandit with stochastic gradient descent updates. We either improve or match the state-of-the-art bounds on all three dynamics.

Closure of $\VP$ under taking factors: a short and simple proof
with Mrinal Kumar, Noam Solomon.
[arxiv] [eccc] [ ] [ ]

@unpublished{CKS19,
title={Closure of VP under taking factors: a short and simple proof},
author={Chou, Chi-Ning and Kumar, Mrinal and Solomon, Noam},
note={In submission to Theory of Computing},
eprint={1903.02366},
archivePrefix={arXiv},
primaryClass={cs.CC}
}


In this note, we give a short, simple and almost completely self contained proof of a classical result of Kaltofen 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 $\poly\left(s, n, d\right)$.

However, unlike Kaltofen's argument, our proof does not directly give an efficient algorithm for computing the circuits for the factors of $f$.