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Researcher profile

Zhao Song

Zhao Song contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine LearningData Structures and AlgorithmsComputational ComplexityArtificial IntelligenceComputation and LanguageCryptography and SecurityDiscrete Mathematicseess.ASInformation Theorymath.ITSoundmath.COmath.FAmath.OCmath.PRphysics.soc-phPopulations and Evolutionquant-phSocial and Information Networks

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Published work

37 published item(s)

preprint2026arXiv

Strategic commitments shape collective cybersecurity under AI inequality

The growing integration of AI into cybersecurity is reshaping the balance between attackers and defenders. When access to advanced AI-enabled defence tools is uneven, resource-limited defenders may be unable to adopt effective protection, creating persistent system vulnerabilities. We study the impact of differential AI access using an evolutionary game-theoretic model in a finite population. We first show that when high-capability defence is costly, the population is driven toward low-cost, weak-defence behaviour, sustaining attacks and weakening long-run security. To address this problem, we introduce differential access to AI defence tools by allowing defenders to choose between low- and high-capability protection based on their resources. We then examine the role of a small group of committed defenders who always adopt strong defence and influence others through social learning. Although commitment increases the prevalence of strong defence, it alone cannot stabilise secure outcomes due to high defence costs. We therefore incorporate a targeted subsidy to remove the cost disadvantage from committed defenders. Our analysis shows that subsidised commitment significantly increases strong defence adoption, suppresses successful attacks, and improves overall system resilience. Simulations across a broad parameter space confirm that subsidies consistently outperform commitment alone. In addition, social-welfare analysis shows improved defender outcomes while keeping attacker gains low. These findings suggest that targeted support for key defenders can be an effective mechanism for stabilising cybersecurity in AI-driven environments and provide a theoretical bridge between cybersecurity policy, AI governance, and strategic allocation of defensive AI capabilities.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Dynamic Tensor Product Regression

In this work, we initiate the study of \emph{Dynamic Tensor Product Regression}. One has matrices $A_1\in \mathbb{R}^{n_1\times d_1},\ldots,A_q\in \mathbb{R}^{n_q\times d_q}$ and a label vector $b\in \mathbb{R}^{n_1\ldots n_q}$, and the goal is to solve the regression problem with the design matrix $A$ being the tensor product of the matrices $A_1, A_2, \dots, A_q$ i.e. $\min_{x\in \mathbb{R}^{d_1\ldots d_q}}~\|(A_1\otimes \ldots\otimes A_q)x-b\|_2$. At each time step, one matrix $A_i$ receives a sparse change, and the goal is to maintain a sketch of the tensor product $A_1\otimes\ldots \otimes A_q$ so that the regression solution can be updated quickly. Recomputing the solution from scratch for each round is very slow and so it is important to develop algorithms which can quickly update the solution with the new design matrix. Our main result is a dynamic tree data structure where any update to a single matrix can be propagated quickly throughout the tree. We show that our data structure can be used to solve dynamic versions of not only Tensor Product Regression, but also Tensor Product Spline regression (which is a generalization of ridge regression) and for maintaining Low Rank Approximations for the tensor product.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Faster Sinkhorn's Algorithm with Small Treewidth

Computing optimal transport (OT) distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. In this paper, we study the problem of approximating the general OT distance between two discrete distributions of size $n$. Given the cost matrix $C=AA^\top$ where $A \in \mathbb{R}^{n \times d}$, we proposed a faster Sinkhorn's Algorithm to approximate the OT distance when matrix $A$ has treewidth $τ$. To approximate the OT distance, our algorithm improves the state-of-the-art results [Dvurechensky, Gasnikov, and Kroshnin ICML 2018] from $\widetilde{O}(ε^{-2} n^2)$ time to $\widetilde{O}(ε^{-2} n τ)$ time.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Online Adaptive Mahalanobis Distance Estimation

Mahalanobis metrics are widely used in machine learning in conjunction with methods like $k$-nearest neighbors, $k$-means clustering, and $k$-medians clustering. Despite their importance, there has not been any prior work on applying sketching techniques to speed up algorithms for Mahalanobis metrics. In this paper, we initiate the study of dimension reduction for Mahalanobis metrics. In particular, we provide efficient data structures for solving the Approximate Distance Estimation (ADE) problem for Mahalanobis distances. We first provide a randomized Monte Carlo data structure. Then, we show how we can adapt it to provide our main data structure which can handle sequences of \textit{adaptive} queries and also online updates to both the Mahalanobis metric matrix and the data points, making it amenable to be used in conjunction with prior algorithms for online learning of Mahalanobis metrics.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Smoothed Online Combinatorial Optimization Using Imperfect Predictions

Smoothed online combinatorial optimization considers a learner who repeatedly chooses a combinatorial decision to minimize an unknown changing cost function with a penalty on switching decisions in consecutive rounds. We study smoothed online combinatorial optimization problems when an imperfect predictive model is available, where the model can forecast the future cost functions with uncertainty. We show that using predictions to plan for a finite time horizon leads to regret dependent on the total predictive uncertainty and an additional switching cost. This observation suggests choosing a suitable planning window to balance between uncertainty and switching cost, which leads to an online algorithm with guarantees on the upper and lower bounds of the cumulative regret. Empirically, our algorithm shows a significant improvement in cumulative regret compared to other baselines in synthetic online distributed streaming problems.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

A Sublinear Adversarial Training Algorithm

Adversarial training is a widely used strategy for making neural networks resistant to adversarial perturbations. For a neural network of width $m$, $n$ input training data in $d$ dimension, it takes $Ω(mnd)$ time cost per training iteration for the forward and backward computation. In this paper we analyze the convergence guarantee of adversarial training procedure on a two-layer neural network with shifted ReLU activation, and shows that only $o(m)$ neurons will be activated for each input data per iteration. Furthermore, we develop an algorithm for adversarial training with time cost $o(m n d)$ per iteration by applying half-space reporting data structure.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Accelerating Frank-Wolfe Algorithm using Low-Dimensional and Adaptive Data Structures

In this paper, we study the problem of speeding up a type of optimization algorithms called Frank-Wolfe, a conditional gradient method. We develop and employ two novel inner product search data structures, improving the prior fastest algorithm in [Shrivastava, Song and Xu, NeurIPS 2021]. * The first data structure uses low-dimensional random projection to reduce the problem to a lower dimension, then uses efficient inner product data structure. It has preprocessing time $\tilde O(nd^{ω-1}+dn^{1+o(1)})$ and per iteration cost $\tilde O(d+n^ρ)$ for small constant $ρ$. * The second data structure leverages the recent development in adaptive inner product search data structure that can output estimations to all inner products. It has preprocessing time $\tilde O(nd)$ and per iteration cost $\tilde O(d+n)$. The first algorithm improves the state-of-the-art (with preprocessing time $\tilde O(d^2n^{1+o(1)})$ and per iteration cost $\tilde O(dn^ρ)$) in all cases, while the second one provides an even faster preprocessing time and is suitable when the number of iterations is small.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

An $O(k\log n)$ Time Fourier Set Query Algorithm

Fourier transformation is an extensively studied problem in many research fields. It has many applications in machine learning, signal processing, compressed sensing, and so on. In many real-world applications, approximated Fourier transformation is sufficient and we only need to do the Fourier transform on a subset of coordinates. Given a vector $x \in \mathbb{C}^{n}$, an approximation parameter $ε$ and a query set $S \subset [n]$ of size $k$, we propose an algorithm to compute an approximate Fourier transform result $x'$ which uses $O(ε^{-1} k \log(n/δ))$ Fourier measurements, runs in $O(ε^{-1} k \log(n/δ))$ time and outputs a vector $x'$ such that $\| ( x' - \widehat{x} )_S \|_2^2 \leq ε\| \widehat{x}_{\bar{S}} \|_2^2 + δ\| \widehat{x} \|_1^2 $ holds with probability of at least $9/10$.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

An improved quantum-inspired algorithm for linear regression

We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09, arXiv:0811.3171] for low-rank matrices [Wossnig, Zhao, and Prakash, Physical Review Letters'18, arXiv:1704.06174], when the input matrix $A$ is stored in a data structure applicable for QRAM-based state preparation. Namely, suppose we are given an $A \in \mathbb{C}^{m\times n}$ with minimum non-zero singular value $σ$ which supports certain efficient $\ell_2$-norm importance sampling queries, along with a $b \in \mathbb{C}^m$. Then, for some $x \in \mathbb{C}^n$ satisfying $\|x - A^+b\| \leq \varepsilon\|A^+b\|$, we can output a measurement of $|x\rangle$ in the computational basis and output an entry of $x$ with classical algorithms that run in $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^6}{σ^{12}\varepsilon^4}\big)$ and $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^2}{σ^8\varepsilon^4}\big)$ time, respectively. This improves on previous "quantum-inspired" algorithms in this line of research by at least a factor of $\frac{\|A\|^{16}}{σ^{16}\varepsilon^2}$ [Chia, Gilyén, Li, Lin, Tang, and Wang, STOC'20, arXiv:1910.06151]. As a consequence, we show that quantum computers can achieve at most a factor-of-12 speedup for linear regression in this QRAM data structure setting and related settings. Our work applies techniques from sketching algorithms and optimization to the quantum-inspired literature. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired settings, for comparison against future quantum computers.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Bounding the Width of Neural Networks via Coupled Initialization -- A Worst Case Analysis

A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analysis. While a similar technique has been studied for random inputs [Daniely, NeurIPS 2020], it has not been analyzed with arbitrary inputs. Using this technique, we show how to significantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with logistic loss, from roughly $γ^{-8}$ [Ji and Telgarsky, ICLR 2020] to $γ^{-2}$, where $γ$ denotes the separation margin with a Neural Tangent Kernel, as well as in the over-parameterized setting with squared loss, from roughly $n^4$ [Song and Yang, 2019] to $n^2$, implicitly also improving the recent running time bound of [Brand, Peng, Song and Weinstein, ITCS 2021]. For the under-parameterized setting we also prove new lower bounds that improve upon prior work, and that under certain assumptions, are best possible.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Fast Distance Oracles for Any Symmetric Norm

In the Distance Oracle problem, the goal is to preprocess $n$ vectors $x_1, x_2, \cdots, x_n$ in a $d$-dimensional metric space $(\mathbb{X}^d, \| \cdot \|_l)$ into a cheap data structure, so that given a query vector $q \in \mathbb{X}^d$ and a subset $S\subseteq [n]$ of the input data points, all distances $\| q - x_i \|_l$ for $x_i\in S$ can be quickly approximated (faster than the trivial $\sim d|S|$ query time). This primitive is a basic subroutine in machine learning, data mining and similarity search applications. In the case of $\ell_p$ norms, the problem is well understood, and optimal data structures are known for most values of $p$. Our main contribution is a fast $(1+\varepsilon)$ distance oracle for any symmetric norm $\|\cdot\|_l$. This class includes $\ell_p$ norms and Orlicz norms as special cases, as well as other norms used in practice, e.g. top-$k$ norms, max-mixture and sum-mixture of $\ell_p$ norms, small-support norms and the box-norm. We propose a novel data structure with $\tilde{O}(n (d + \mathrm{mmc}(l)^2 ) )$ preprocessing time and space, and $t_q = \tilde{O}(d + |S| \cdot \mathrm{mmc}(l)^2)$ query time, for computing distances to a subset $S$ of data points, where $\mathrm{mmc}(l)$ is a complexity-measure (concentration modulus) of the symmetric norm. When $l = \ell_{p}$ , this runtime matches the aforementioned state-of-art oracles.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Federated Adversarial Learning: A Framework with Convergence Analysis

Federated learning (FL) is a trending training paradigm to utilize decentralized training data. FL allows clients to update model parameters locally for several epochs, then share them to a global model for aggregation. This training paradigm with multi-local step updating before aggregation exposes unique vulnerabilities to adversarial attacks. Adversarial training is a popular and effective method to improve the robustness of networks against adversaries. In this work, we formulate a general form of federated adversarial learning (FAL) that is adapted from adversarial learning in the centralized setting. On the client side of FL training, FAL has an inner loop to generate adversarial samples for adversarial training and an outer loop to update local model parameters. On the server side, FAL aggregates local model updates and broadcast the aggregated model. We design a global robust training loss and formulate FAL training as a min-max optimization problem. Unlike the convergence analysis in classical centralized training that relies on the gradient direction, it is significantly harder to analyze the convergence in FAL for three reasons: 1) the complexity of min-max optimization, 2) model not updating in the gradient direction due to the multi-local updates on the client-side before aggregation and 3) inter-client heterogeneity. We address these challenges by using appropriate gradient approximation and coupling techniques and present the convergence analysis in the over-parameterized regime. Our main result theoretically shows that the minimum loss under our algorithm can converge to $ε$ small with chosen learning rate and communication rounds. It is noteworthy that our analysis is feasible for non-IID clients.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Hyperbolic Concentration, Anti-concentration, and Discrepancy

Chernoff bound is a fundamental tool in theoretical computer science. It has been extensively used in randomized algorithm design and stochastic type analysis. Discrepancy theory, which deals with finding a bi-coloring of a set system such that the coloring of each set is balanced, has a huge number of applications in approximation algorithms design. Chernoff bound [Che52] implies that a random bi-coloring of any set system with $n$ sets and $n$ elements will have discrepancy $O(\sqrt{n \log n})$ with high probability, while the famous result by Spencer [Spe85] shows that there exists an $O(\sqrt{n})$ discrepancy solution. The study of hyperbolic polynomials dates back to the early 20th century when used to solve PDEs by Gårding [Går59]. In recent years, more applications are found in control theory, optimization, real algebraic geometry, and so on. In particular, the breakthrough result by Marcus, Spielman, and Srivastava [MSS15] uses the theory of hyperbolic polynomials to prove the Kadison-Singer conjecture [KS59], which is closely related to discrepancy theory. In this paper, we present a list of new results for hyperbolic polynomials: * We show two nearly optimal hyperbolic Chernoff bounds: one for Rademacher sum of arbitrary vectors and another for random vectors in the hyperbolic cone. * We show a hyperbolic anti-concentration bound. * We generalize the hyperbolic Kadison-Singer theorem [Brä18] for vectors in sub-isotropic position, and prove a hyperbolic Spencer theorem for any constant hyperbolic rank vectors. The classical matrix Chernoff and discrepancy results are based on determinant polynomial. To the best of our knowledge, this paper is the first work that shows either concentration or anti-concentration results for hyperbolic polynomials. We hope our findings provide more insights into hyperbolic and discrepancy theories.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Perfectly Balanced: Improving Transfer and Robustness of Supervised Contrastive Learning

An ideal learned representation should display transferability and robustness. Supervised contrastive learning (SupCon) is a promising method for training accurate models, but produces representations that do not capture these properties due to class collapse -- when all points in a class map to the same representation. Recent work suggests that "spreading out" these representations improves them, but the precise mechanism is poorly understood. We argue that creating spread alone is insufficient for better representations, since spread is invariant to permutations within classes. Instead, both the correct degree of spread and a mechanism for breaking this invariance are necessary. We first prove that adding a weighted class-conditional InfoNCE loss to SupCon controls the degree of spread. Next, we study three mechanisms to break permutation invariance: using a constrained encoder, adding a class-conditional autoencoder, and using data augmentation. We show that the latter two encourage clustering of latent subclasses under more realistic conditions than the former. Using these insights, we show that adding a properly-weighted class-conditional InfoNCE loss and a class-conditional autoencoder to SupCon achieves 11.1 points of lift on coarse-to-fine transfer across 5 standard datasets and 4.7 points on worst-group robustness on 3 datasets, setting state-of-the-art on CelebA by 11.5 points.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Pixelated Butterfly: Simple and Efficient Sparse training for Neural Network Models

Overparameterized neural networks generalize well but are expensive to train. Ideally, one would like to reduce their computational cost while retaining their generalization benefits. Sparse model training is a simple and promising approach to achieve this, but there remain challenges as existing methods struggle with accuracy loss, slow training runtime, or difficulty in sparsifying all model components. The core problem is that searching for a sparsity mask over a discrete set of sparse matrices is difficult and expensive. To address this, our main insight is to optimize over a continuous superset of sparse matrices with a fixed structure known as products of butterfly matrices. As butterfly matrices are not hardware efficient, we propose simple variants of butterfly (block and flat) to take advantage of modern hardware. Our method (Pixelated Butterfly) uses a simple fixed sparsity pattern based on flat block butterfly and low-rank matrices to sparsify most network layers (e.g., attention, MLP). We empirically validate that Pixelated Butterfly is 3x faster than butterfly and speeds up training to achieve favorable accuracy--efficiency tradeoffs. On the ImageNet classification and WikiText-103 language modeling tasks, our sparse models train up to 2.5x faster than the dense MLP-Mixer, Vision Transformer, and GPT-2 medium with no drop in accuracy.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Speeding Up Sparsification using Inner Product Search Data Structures

We present a general framework that utilizes different efficient data structures to improve various sparsification problems involving an iterative process. We also provide insights and characterization for different iterative process, and answer that when should we use which data structures in what type of problem. We obtain improved running time for the following problems. * For constructing linear-sized spectral sparsifier (Batson, Spielman and Srivastava, 2012), all the existing deterministic algorithms require $Ω(d^4)$ time. In this work, we provide the first deterministic algorithm that breaks that barrier which runs in $O(d^{ω+1})$ time, where $ω$ is the exponent of matrix multiplication. * For one-sided Kadison-Singer-typed discrepancy problem, we give fast algorithms for both small and large number of iterations. * For experimental design problem, we speed up a key swapping process. In the heart of our work is the design of a variety of different inner product search data structures that have efficient initialization, query and update time, compatible to dimensionality reduction and robust against adaptive adversary.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Symmetric Sparse Boolean Matrix Factorization and Applications

In this work, we study a variant of nonnegative matrix factorization where we wish to find a symmetric factorization of a given input matrix into a sparse, Boolean matrix. Formally speaking, given $\mathbf{M}\in\mathbb{Z}^{m\times m}$, we want to find $\mathbf{W}\in\{0,1\}^{m\times r}$ such that $\| \mathbf{M} - \mathbf{W}\mathbf{W}^\top \|_0$ is minimized among all $\mathbf{W}$ for which each row is $k$-sparse. This question turns out to be closely related to a number of questions like recovering a hypergraph from its line graph, as well as reconstruction attacks for private neural network training. As this problem is hard in the worst-case, we study a natural average-case variant that arises in the context of these reconstruction attacks: $\mathbf{M} = \mathbf{W}\mathbf{W}^{\top}$ for $\mathbf{W}$ a random Boolean matrix with $k$-sparse rows, and the goal is to recover $\mathbf{W}$ up to column permutation. Equivalently, this can be thought of as recovering a uniformly random $k$-uniform hypergraph from its line graph. Our main result is a polynomial-time algorithm for this problem based on bootstrapping higher-order information about $\mathbf{W}$ and then decomposing an appropriate tensor. The key ingredient in our analysis, which may be of independent interest, is to show that such a matrix $\mathbf{W}$ has full column rank with high probability as soon as $m = \widetildeΩ(r)$, which we do using tools from Littlewood-Offord theory and estimates for binary Krawtchouk polynomials.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

InstaHide: Instance-hiding Schemes for Private Distributed Learning

How can multiple distributed entities collaboratively train a shared deep net on their private data while preserving privacy? This paper introduces InstaHide, a simple encryption of training images, which can be plugged into existing distributed deep learning pipelines. The encryption is efficient and applying it during training has minor effect on test accuracy. InstaHide encrypts each training image with a "one-time secret key" which consists of mixing a number of randomly chosen images and applying a random pixel-wise mask. Other contributions of this paper include: (a) Using a large public dataset (e.g. ImageNet) for mixing during its encryption, which improves security. (b) Experimental results to show effectiveness in preserving privacy against known attacks with only minor effects on accuracy. (c) Theoretical analysis showing that successfully attacking privacy requires attackers to solve a difficult computational problem. (d) Demonstrating that use of the pixel-wise mask is important for security, since Mixup alone is shown to be insecure to some some efficient attacks. (e) Release of a challenge dataset https://github.com/Hazelsuko07/InstaHide_Challenge Our code is available at https://github.com/Hazelsuko07/InstaHide

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preprint2021arXiv

Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs

For a directed graph $G$ with $n$ vertices and a start vertex $u_{\sf start}$, we wish to (approximately) sample an $L$-step random walk over $G$ starting from $u_{\sf start}$ with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be $\tildeΘ(n \cdot L)$. Prior to our work, a better space complexity was only known with $\tilde{O}(\sqrt{L})$ passes. We settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is $\tildeΘ(n \cdot \sqrt{L})$, by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes $p$, and shows that any $p$-pass algorithm for this problem uses $\tildeΩ(n \cdot L^{1/p})$ space. In addition, we show a similar $\tildeΘ(n \cdot \sqrt{L})$ bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an $L$-step random walk from every vertex in the graph.

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preprint2020arXiv

A Faster Interior Point Method for Semidefinite Programming

Semidefinite programs (SDPs) are a fundamental class of optimization problems with important recent applications in approximation algorithms, quantum complexity, robust learning, algorithmic rounding, and adversarial deep learning. This paper presents a faster interior point method to solve generic SDPs with variable size $n \times n$ and $m$ constraints in time \begin{align*} \widetilde{O}(\sqrt{n}( mn^2 + m^ω+ n^ω) \log(1 / ε) ), \end{align*} where $ω$ is the exponent of matrix multiplication and $ε$ is the relative accuracy. In the predominant case of $m \geq n$, our runtime outperforms that of the previous fastest SDP solver, which is based on the cutting plane method of Jiang, Lee, Song, and Wong [JLSW20]. Our algorithm's runtime can be naturally interpreted as follows: $\widetilde{O}(\sqrt{n} \log (1/ε))$ is the number of iterations needed for our interior point method, $mn^2$ is the input size, and $m^ω+ n^ω$ is the time to invert the Hessian and slack matrix in each iteration. These constitute natural barriers to further improving the runtime of interior point methods for solving generic SDPs.

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preprint2020arXiv

A novel route to cyclic dominance in voluntary social dilemmas

Cooperation is the backbone of modern human societies, making it a priority to understand how successful cooperation-sustaining mechanisms operate. Cyclic dominance, a non-transitive setup comprising at least three strategies wherein the first strategy overrules the second which overrules the third which, in turn, overrules the first strategy, is known to maintain bio-diversity, drive competition between bacterial strains, and preserve cooperation in social dilemmas. Here, we present a novel route to cyclic dominance in voluntary social dilemmas by adding to the traditional mix of cooperators, defectors, and loners, a fourth player type, risk-averse hedgers, who enact tit-for-tat upon paying a hedging cost to avoid being exploited. When this cost is sufficiently small, cooperators, defectors, and hedgers enter a loop of cyclic dominance that preserves cooperation even under the most adverse conditions. In contrast, when the hedging cost is large, hedgers disappear, consequently reverting to the traditional interplay of cooperators, defectors, and loners. In the interim region of hedging costs, complex evolutionary dynamics ensues, prompting transitions between states with two, three, or four competing strategies. Our results thus reveal that voluntary participation is but one pathway to sustained cooperation via cyclic dominance.

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preprint2020arXiv

An Improved Cutting Plane Method for Convex Optimization, Convex-Concave Games and its Applications

Given a separation oracle for a convex set $K \subset \mathbb{R}^n$ that is contained in a box of radius $R$, the goal is to either compute a point in $K$ or prove that $K$ does not contain a ball of radius $ε$. We propose a new cutting plane algorithm that uses an optimal $O(n \log (κ))$ evaluations of the oracle and an additional $O(n^2)$ time per evaluation, where $κ= nR/ε$. $\bullet$ This improves upon Vaidya&#39;s $O( \text{SO} \cdot n \log (κ) + n^{ω+1} \log (κ))$ time algorithm [Vaidya, FOCS 1989a] in terms of polynomial dependence on $n$, where $ω< 2.373$ is the exponent of matrix multiplication and $\text{SO}$ is the time for oracle evaluation. $\bullet$ This improves upon Lee-Sidford-Wong&#39;s $O( \text{SO} \cdot n \log (κ) + n^3 \log^{O(1)} (κ))$ time algorithm [Lee, Sidford and Wong, FOCS 2015] in terms of dependence on $κ$. For many important applications in economics, $κ= Ω(\exp(n))$ and this leads to a significant difference between $\log(κ)$ and $\mathrm{poly}(\log (κ))$. We also provide evidence that the $n^2$ time per evaluation cannot be improved and thus our running time is optimal. A bottleneck of previous cutting plane methods is to compute leverage scores, a measure of the relative importance of past constraints. Our result is achieved by a novel multi-layered data structure for leverage score maintenance, which is a sophisticated combination of diverse techniques such as random projection, batched low-rank update, inverse maintenance, polynomial interpolation, and fast rectangular matrix multiplication. Interestingly, our method requires a combination of different fast rectangular matrix multiplication algorithms.

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preprint2020arXiv

Average Case Column Subset Selection for Entrywise $\ell_1$-Norm Loss

We study the column subset selection problem with respect to the entrywise $\ell_1$-norm loss. It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $n^{Ω(1)}$ number of columns to obtain a $(1+ε)$-approximation to the best entrywise $\ell_1$-norm low rank approximation of an $n \times n$ matrix. Nevertheless, we show that under certain minimal and realistic distributional settings, it is possible to obtain a $(1+ε)$-approximation with a nearly linear running time and poly$(k/ε)+O(k\log n)$ columns. Namely, we show that if the input matrix $A$ has the form $A = B + E$, where $B$ is an arbitrary rank-$k$ matrix, and $E$ is a matrix with i.i.d. entries drawn from any distribution $μ$ for which the $(1+γ)$-th moment exists, for an arbitrarily small constant $γ> 0$, then it is possible to obtain a $(1+ε)$-approximate column subset selection to the entrywise $\ell_1$-norm in nearly linear time. Conversely we show that if the first moment does not exist, then it is not possible to obtain a $(1+ε)$-approximate subset selection algorithm even if one chooses any $n^{o(1)}$ columns. This is the first algorithm of any kind for achieving a $(1+ε)$-approximation for entrywise $\ell_1$-norm loss low rank approximation.

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preprint2020arXiv

Faster Dynamic Matrix Inverse for Faster LPs

Motivated by recent Linear Programming solvers, we design dynamic data structures for maintaining the inverse of an $n\times n$ real matrix under $\textit{low-rank}$ updates, with polynomially faster amortized running time. Our data structure is based on a recursive application of the Woodbury-Morrison identity for implementing $\textit{cascading}$ low-rank updates, combined with recent sketching technology. Our techniques and amortized analysis of multi-level partial updates, may be of broader interest to dynamic matrix problems. This data structure leads to the fastest known LP solver for general (dense) linear programs, improving the running time of the recent algorithms of (Cohen et al.&#39;19, Lee et al.&#39;19, Brand&#39;20) from $O^*(n^{2+ \max\{\frac{1}{6}, ω-2, \frac{1-α}{2}\}})$ to $O^*(n^{2+\max\{\frac{1}{18}, ω-2, \frac{1-α}{2}\}})$, where $ω$ and $α$ are the fast matrix multiplication exponent and its dual. Hence, under the common belief that $ω\approx 2$ and $α\approx 1$, our LP solver runs in $O^*(n^{2.055})$ time instead of $O^*(n^{2.16})$.

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preprint2020arXiv

Four Deviations Suffice for Rank 1 Matrices

We prove a matrix discrepancy bound that strengthens the famous Kadison-Singer result of Marcus, Spielman, and Srivastava. Consider any independent scalar random variables $ξ_1, \ldots, ξ_n$ with finite support, e.g. $\{ \pm 1 \}$ or $\{ 0,1 \}$-valued random variables, or some combination thereof. Let $u_1, \dots, u_n \in \mathbb{C}^m$ and $$ σ^2 = \left\| \sum_{i=1}^n \text{Var}[ ξ_i ] (u_i u_i^{*})^2 \right\|. $$ Then there exists a choice of outcomes $\varepsilon_1,\ldots,\varepsilon_n$ in the support of $ξ_1, \ldots, ξ_n$ s.t. $$ \left \|\sum_{i=1}^n \mathbb{E} [ ξ_i] u_i u_i^* - \sum_{i=1}^n \varepsilon_i u_i u_i^* \right \| \leq 4 σ. $$ A simple consequence of our result is an improvement of a Lyapunov-type theorem of Akemann and Weaver.

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preprint2020arXiv

Generalized Leverage Score Sampling for Neural Networks

Leverage score sampling is a powerful technique that originates from theoretical computer science, which can be used to speed up a large number of fundamental questions, e.g. linear regression, linear programming, semi-definite programming, cutting plane method, graph sparsification, maximum matching and max-flow. Recently, it has been shown that leverage score sampling helps to accelerate kernel methods [Avron, Kapralov, Musco, Musco, Velingker and Zandieh 17]. In this work, we generalize the results in [Avron, Kapralov, Musco, Musco, Velingker and Zandieh 17] to a broader class of kernels. We further bring the leverage score sampling into the field of deep learning theory. $\bullet$ We show the connection between the initialization for neural network training and approximating the neural tangent kernel with random features. $\bullet$ We prove the equivalence between regularized neural network and neural tangent kernel ridge regression under the initialization of both classical random Gaussian and leverage score sampling.

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preprint2020arXiv

Low Rank Approximation with Entrywise $\ell_1$-Norm Error

We study the $\ell_1$-low rank approximation problem, where for a given $n \times d$ matrix $A$ and approximation factor $α\geq 1$, the goal is to output a rank-$k$ matrix $\widehat{A}$ for which $$\|A-\widehat{A}\|_1 \leq α\cdot \min_{\textrm{rank-}k\textrm{ matrices}~A&#39;}\|A-A&#39;\|_1,$$ where for an $n \times d$ matrix $C$, we let $\|C\|_1 = \sum_{i=1}^n \sum_{j=1}^d |C_{i,j}|$. This error measure is known to be more robust than the Frobenius norm in the presence of outliers and is indicated in models where Gaussian assumptions on the noise may not apply. The problem was shown to be NP-hard by Gillis and Vavasis and a number of heuristics have been proposed. It was asked in multiple places if there are any approximation algorithms. We give the first provable approximation algorithms for $\ell_1$-low rank approximation, showing that it is possible to achieve approximation factor $α= (\log d) \cdot \mathrm{poly}(k)$ in $\mathrm{nnz}(A) + (n+d) \mathrm{poly}(k)$ time, where $\mathrm{nnz}(A)$ denotes the number of non-zero entries of $A$. If $k$ is constant, we further improve the approximation ratio to $O(1)$ with a $\mathrm{poly}(nd)$-time algorithm. Under the Exponential Time Hypothesis, we show there is no $\mathrm{poly}(nd)$-time algorithm achieving a $(1+\frac{1}{\log^{1+γ}(nd)})$-approximation, for $γ> 0$ an arbitrarily small constant, even when $k = 1$. We give a number of additional results for $\ell_1$-low rank approximation: nearly tight upper and lower bounds for column subset selection, CUR decompositions, extensions to low rank approximation with respect to $\ell_p$-norms for $1 \leq p < 2$ and earthmover distance, low-communication distributed protocols and low-memory streaming algorithms, algorithms with limited randomness, and bicriteria algorithms. We also give a preliminary empirical evaluation.

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preprint2020arXiv

Meta-learning for mixed linear regression

In modern supervised learning, there are a large number of tasks, but many of them are associated with only a small amount of labeled data. These include data from medical image processing and robotic interaction. Even though each individual task cannot be meaningfully trained in isolation, one seeks to meta-learn across the tasks from past experiences by exploiting some similarities. We study a fundamental question of interest: When can abundant tasks with small data compensate for lack of tasks with big data? We focus on a canonical scenario where each task is drawn from a mixture of $k$ linear regressions, and identify sufficient conditions for such a graceful exchange to hold; The total number of examples necessary with only small data tasks scales similarly as when big data tasks are available. To this end, we introduce a novel spectral approach and show that we can efficiently utilize small data tasks with the help of $\tildeΩ(k^{3/2})$ medium data tasks each with $\tildeΩ(k^{1/2})$ examples.

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preprint2020arXiv

Non-Autoregressive Neural Text-to-Speech

In this work, we propose ParaNet, a non-autoregressive seq2seq model that converts text to spectrogram. It is fully convolutional and brings 46.7 times speed-up over the lightweight Deep Voice 3 at synthesis, while obtaining reasonably good speech quality. ParaNet also produces stable alignment between text and speech on the challenging test sentences by iteratively improving the attention in a layer-by-layer manner. Furthermore, we build the parallel text-to-speech system and test various parallel neural vocoders, which can synthesize speech from text through a single feed-forward pass. We also explore a novel VAE-based approach to train the inverse autoregressive flow (IAF) based parallel vocoder from scratch, which avoids the need for distillation from a separately trained WaveNet as previous work.

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preprint2020arXiv

Over-parameterized Adversarial Training: An Analysis Overcoming the Curse of Dimensionality

Adversarial training is a popular method to give neural nets robustness against adversarial perturbations. In practice adversarial training leads to low robust training loss. However, a rigorous explanation for why this happens under natural conditions is still missing. Recently a convergence theory for standard (non-adversarial) supervised training was developed by various groups for {\em very overparametrized} nets. It is unclear how to extend these results to adversarial training because of the min-max objective. Recently, a first step towards this direction was made by Gao et al. using tools from online learning, but they require the width of the net to be \emph{exponential} in input dimension $d$, and with an unnatural activation function. Our work proves convergence to low robust training loss for \emph{polynomial} width instead of exponential, under natural assumptions and with the ReLU activation. Key element of our proof is showing that ReLU networks near initialization can approximate the step function, which may be of independent interest.

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preprint2020arXiv

Privacy-preserving Learning via Deep Net Pruning

This paper attempts to answer the question whether neural network pruning can be used as a tool to achieve differential privacy without losing much data utility. As a first step towards understanding the relationship between neural network pruning and differential privacy, this paper proves that pruning a given layer of the neural network is equivalent to adding a certain amount of differentially private noise to its hidden-layer activations. The paper also presents experimental results to show the practical implications of the theoretical finding and the key parameter values in a simple practical setting. These results show that neural network pruning can be a more effective alternative to adding differentially private noise for neural networks.

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preprint2020arXiv

Sketching Transformed Matrices with Applications to Natural Language Processing

Suppose we are given a large matrix $A=(a_{i,j})$ that cannot be stored in memory but is in a disk or is presented in a data stream. However, we need to compute a matrix decomposition of the entry-wisely transformed matrix, $f(A):=(f(a_{i,j}))$ for some function $f$. Is it possible to do it in a space efficient way? Many machine learning applications indeed need to deal with such large transformed matrices, for example word embedding method in NLP needs to work with the pointwise mutual information (PMI) matrix, while the entrywise transformation makes it difficult to apply known linear algebraic tools. Existing approaches for this problem either need to store the whole matrix and perform the entry-wise transformation afterwards, which is space consuming or infeasible, or need to redesign the learning method, which is application specific and requires substantial remodeling. In this paper, we first propose a space-efficient sketching algorithm for computing the product of a given small matrix with the transformed matrix. It works for a general family of transformations with provable small error bounds and thus can be used as a primitive in downstream learning tasks. We then apply this primitive to a concrete application: low-rank approximation. We show that our approach obtains small error and is efficient in both space and time. We complement our theoretical results with experiments on synthetic and real data.

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preprint2020arXiv

Towards a Zero-One Law for Column Subset Selection

There are a number of approximation algorithms for NP-hard versions of low rank approximation, such as finding a rank-$k$ matrix $B$ minimizing the sum of absolute values of differences to a given $n$-by-$n$ matrix $A$, $\min_{\textrm{rank-}k~B}\|A-B\|_1$, or more generally finding a rank-$k$ matrix $B$ which minimizes the sum of $p$-th powers of absolute values of differences, $\min_{\textrm{rank-}k~B}\|A-B\|_p^p$. Many of these algorithms are linear time columns subset selection algorithms, returning a subset of $\mathrm{poly}(k \log n)$ columns whose cost is no more than a $\mathrm{poly}(k)$ factor larger than the cost of the best rank-$k$ matrix. The above error measures are special cases of the following general entrywise low rank approximation problem: given an arbitrary function $g:\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$, find a rank-$k$ matrix $B$ which minimizes $\|A-B\|_g = \sum_{i,j}g(A_{i,j}-B_{i,j})$. A natural question is which functions $g$ admit efficient approximation algorithms? Indeed, this is a central question of recent work studying generalized low rank models. In this work we give approximation algorithms for $\textit{every}$ function $g$ which is approximately monotone and satisfies an approximate triangle inequality, and we show both of these conditions are necessary. Further, our algorithm is efficient if the function $g$ admits an efficient approximate regression algorithm. Our approximation algorithms handle functions which are not even scale-invariant, such as the Huber loss function, which we show have very different structural properties than $\ell_p$-norms, e.g., one can show the lack of scale-invariance causes any column subset selection algorithm to provably require a $\sqrt{\log n}$ factor larger number of columns than $\ell_p$-norms; nevertheless we design the first efficient column subset selection algorithms for such error measures.

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preprint2020arXiv

WaveFlow: A Compact Flow-based Model for Raw Audio

In this work, we propose WaveFlow, a small-footprint generative flow for raw audio, which is directly trained with maximum likelihood. It handles the long-range structure of 1-D waveform with a dilated 2-D convolutional architecture, while modeling the local variations using expressive autoregressive functions. WaveFlow provides a unified view of likelihood-based models for 1-D data, including WaveNet and WaveGlow as special cases. It generates high-fidelity speech as WaveNet, while synthesizing several orders of magnitude faster as it only requires a few sequential steps to generate very long waveforms with hundreds of thousands of time-steps. Furthermore, it can significantly reduce the likelihood gap that has existed between autoregressive models and flow-based models for efficient synthesis. Finally, our small-footprint WaveFlow has only 5.91M parameters, which is 15$\times$ smaller than WaveGlow. It can generate 22.05 kHz high-fidelity audio 42.6$\times$ faster than real-time (at a rate of 939.3 kHz) on a V100 GPU without engineered inference kernels.

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preprint2013arXiv

A Max-Product EM Algorithm for Reconstructing Markov-tree Sparse Signals from Compressive Samples

We propose a Bayesian expectation-maximization (EM) algorithm for reconstructing Markov-tree sparse signals via belief propagation. The measurements follow an underdetermined linear model where the regression-coefficient vector is the sum of an unknown approximately sparse signal and a zero-mean white Gaussian noise with an unknown variance. The signal is composed of large- and small-magnitude components identified by binary state variables whose probabilistic dependence structure is described by a Markov tree. Gaussian priors are assigned to the signal coefficients given their state variables and the Jeffreys&#39; noninformative prior is assigned to the noise variance. Our signal reconstruction scheme is based on an EM iteration that aims at maximizing the posterior distribution of the signal and its state variables given the noise variance. We construct the missing data for the EM iteration so that the complete-data posterior distribution corresponds to a hidden Markov tree (HMT) probabilistic graphical model that contains no loops and implement its maximization (M) step via a max-product algorithm. This EM algorithm estimates the vector of state variables as well as solves iteratively a linear system of equations to obtain the corresponding signal estimate. We select the noise variance so that the corresponding estimated signal and state variables obtained upon convergence of the EM iteration have the largest marginal posterior distribution. We compare the proposed and existing state-of-the-art reconstruction methods via signal and image reconstruction experiments.

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preprint2013arXiv

PREMIER - PRobabilistic Error-correction using Markov Inference in Errored Reads

In this work we present a flexible, probabilistic and reference-free method of error correction for high throughput DNA sequencing data. The key is to exploit the high coverage of sequencing data and model short sequence outputs as independent realizations of a Hidden Markov Model (HMM). We pose the problem of error correction of reads as one of maximum likelihood sequence detection over this HMM. While time and memory considerations rule out an implementation of the optimal Baum-Welch algorithm (for parameter estimation) and the optimal Viterbi algorithm (for error correction), we propose low-complexity approximate versions of both. Specifically, we propose an approximate Viterbi and a sequential decoding based algorithm for the error correction. Our results show that when compared with Reptile, a state-of-the-art error correction method, our methods consistently achieve superior performances on both simulated and real data sets.

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