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no code implementations • 29 Mar 2022 • Annie Marsden, Vatsal Sharan, Aaron Sidford, Gregory Valiant

We show that any memory-constrained, first-order algorithm which minimizes $d$-dimensional, $1$-Lipschitz convex functions over the unit ball to $1/\mathrm{poly}(d)$ accuracy using at most $d^{1. 25 - \delta}$ bits of memory must make at least $\tilde{\Omega}(d^{1 + (4/3)\delta})$ first-order queries (for any constant $\delta \in [0, 1/4]$).

no code implementations • 8 Mar 2022 • Jonathan A. Kelner, Jerry Li, Allen Liu, Aaron Sidford, Kevin Tian

We design a new iterative method tailored to the geometry of sparse recovery which is provably robust to our semi-random model.

no code implementations • 9 Feb 2022 • Yujia Jin, Aaron Sidford, Kevin Tian

We generalize our algorithms for minimax and finite sum optimization to solve a natural family of minimax finite sum optimization problems at an accelerated rate, encapsulating both above results up to a logarithmic factor.

no code implementations • 4 Nov 2021 • Jonathan Kelner, Annie Marsden, Vatsal Sharan, Aaron Sidford, Gregory Valiant, Honglin Yuan

We consider the problem of minimizing a function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ which is implicitly decomposable as the sum of $m$ unknown non-interacting smooth, strongly convex functions and provide a method which solves this problem with a number of gradient evaluations that scales (up to logarithmic factors) as the product of the square-root of the condition numbers of the components.

no code implementations • NeurIPS 2021 • Hilal Asi, Yair Carmon, Arun Jambulapati, Yujia Jin, Aaron Sidford

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function.

no code implementations • 13 Jun 2021 • Yujia Jin, Aaron Sidford

We prove new upper and lower bounds for sample complexity of finding an $\epsilon$-optimal policy of an infinite-horizon average-reward Markov decision process (MDP) given access to a generative model.

no code implementations • 4 May 2021 • Yair Carmon, Arun Jambulapati, Yujia Jin, Aaron Sidford

We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$.

no code implementations • 14 Jan 2021 • Jan van den Brand, Yin Tat Lee, Yang P. Liu, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, Di Wang

In the special case of the minimum cost flow problem on $n$-vertex $m$-edge graphs with integer polynomially-bounded costs and capacities we obtain a randomized method which solves the problem in $\tilde{O}(m+n^{1. 5})$ time.

Data Structures and Algorithms Optimization and Control

no code implementations • 12 Nov 2020 • Michael B. Cohen, Aaron Sidford, Kevin Tian

We show that standard extragradient methods (i. e. mirror prox and dual extrapolation) recover optimal accelerated rates for first-order minimization of smooth convex functions.

no code implementations • NeurIPS 2020 • Nima Anari, Moses Charikar, Kirankumar Shiragur, Aaron Sidford

In this paper we provide a new efficient algorithm for approximately computing the profile maximum likelihood (PML) distribution, a prominent quantity in symmetric property estimation.

1 code implementation • NeurIPS 2020 • Daniel Levy, Yair Carmon, John C. Duchi, Aaron Sidford

We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $\chi^2$ divergence uncertainty sets.

no code implementations • 17 Sep 2020 • Yair Carmon, Yujia Jin, Aaron Sidford, Kevin Tian

For linear regression with an elementwise nonnegative matrix, our guarantees improve on exact gradient methods by a factor of $\sqrt{\mathrm{nnz}(A)/(m+n)}$.

no code implementations • ICML 2020 • Yujia Jin, Aaron Sidford

We present a unified framework based on primal-dual stochastic mirror descent for approximately solving infinite-horizon Markov decision processes (MDPs) given a generative model.

no code implementations • 4 Aug 2020 • Arun Jambulapati, Jerry Li, Christopher Musco, Aaron Sidford, Kevin Tian

In this paper, we revisit the decades-old problem of how to best improve $\mathbf{A}$'s condition number by left or right diagonal rescaling.

no code implementations • 6 Apr 2020 • Nima Anari, Moses Charikar, Kirankumar Shiragur, Aaron Sidford

For each problem we provide polynomial time algorithms that given $n$ i. i. d.\ samples from a discrete distribution, achieve an approximation factor of $\exp\left(-O(\sqrt{n} \log n) \right)$, improving upon the previous best-known bound achievable in polynomial time of $\exp(-O(n^{2/3} \log n))$ (Charikar, Shiragur and Sidford, 2019).

1 code implementation • NeurIPS 2019 • Moses Charikar, Kirankumar Shiragur, Aaron Sidford

In this paper we provide a general framework for estimating symmetric properties of distributions from i. i. d.

no code implementations • NeurIPS 2019 • Arun Jambulapati, Aaron Sidford, Kevin Tian

Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two $n$-dimensional distributions, is a fundamental primitive which arises in many learning and statistical settings.

no code implementations • NeurIPS 2019 • Yujia Jin, Aaron Sidford

Given a data matrix $\mathbf{A} \in \mathbb{R}^{n \times d}$, principal component projection (PCP) and principal component regression (PCR), i. e. projection and regression restricted to the top-eigenspace of $\mathbf{A}$, are fundamental problems in machine learning, optimization, and numerical analysis.

no code implementations • 29 Aug 2019 • Aaron Sidford, Mengdi Wang, Lin F. Yang, Yinyu Ye

In this paper, we settle the sampling complexity of solving discounted two-player turn-based zero-sum stochastic games up to polylogarithmic factors.

no code implementations • NeurIPS 2019 • Yair Carmon, Yujia Jin, Aaron Sidford, Kevin Tian

We present a randomized primal-dual algorithm that solves the problem $\min_{x} \max_{y} y^\top A x$ to additive error $\epsilon$ in time $\mathrm{nnz}(A) + \sqrt{\mathrm{nnz}(A)n}/\epsilon$, for matrix $A$ with larger dimension $n$ and $\mathrm{nnz}(A)$ nonzero entries.

1 code implementation • 27 Jun 2019 • Oliver Hinder, Aaron Sidford, Nimit S. Sohoni

This function class, which we call the class of smooth quasar-convex functions, is parameterized by a constant $\gamma \in (0, 1]$, where $\gamma = 1$ encompasses the classes of smooth convex and star-convex functions, and smaller values of $\gamma$ indicate that the function can be "more nonconvex."

no code implementations • NeurIPS 2019 • Sébastien Bubeck, Qijia Jiang, Yin Tat Lee, Yuanzhi Li, Aaron Sidford

Namely we consider optimization algorithms interacting with a highly parallel gradient oracle, that is one that can answer $\mathrm{poly}(d)$ gradient queries in parallel.

no code implementations • 3 Jun 2019 • Arun Jambulapati, Aaron Sidford, Kevin Tian

Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two distributions, is a fundamental primitive which arises in many learning and statistical settings.

no code implementations • 21 May 2019 • Moses Charikar, Kirankumar Shiragur, Aaron Sidford

Generalizing work of Acharya et al. 2016 on the utility of approximate PML we show that our algorithm provides a nearly linear time universal plug-in estimator for all symmetric functions up to accuracy $\epsilon = \Omega(n^{-0. 166})$.

no code implementations • 18 Apr 2019 • Vatsal Sharan, Aaron Sidford, Gregory Valiant

We consider the problem of performing linear regression over a stream of $d$-dimensional examples, and show that any algorithm that uses a subquadratic amount of memory exhibits a slower rate of convergence than can be achieved without memory constraints.

no code implementations • 7 Mar 2019 • Yair Carmon, John C. Duchi, Aaron Sidford, Kevin Tian

We show that a simple randomized sketch of the matrix multiplicative weight (MMW) update enjoys (in expectation) the same regret bounds as MMW, up to a small constant factor.

no code implementations • NeurIPS 2018 • Aaron Sidford, Mengdi Wang, Xian Wu, Lin Yang, Yinyu Ye

In this paper we consider the problem of computing an $\epsilon$-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in $O(1)$ time.

no code implementations • NeurIPS 2018 • Neha Gupta, Aaron Sidford

This running time improves upon the previous best unaccelerated running time of $\tilde{O}(nd + n L d / \mu)$.

1 code implementation • 5 Jun 2018 • Aaron Sidford, Mengdi Wang, Xian Wu, Lin F. Yang, Yinyu Ye

In this paper we consider the problem of computing an $\epsilon$-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in $O(1)$ time.

Optimization and Control

no code implementations • 22 Nov 2017 • Naman Agarwal, Sham Kakade, Rahul Kidambi, Yin Tat Lee, Praneeth Netrapalli, Aaron Sidford

Given a matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ and a vector $b \in\mathbb{R}^{d}$, we show how to compute an $\epsilon$-approximate solution to the regression problem $ \min_{x\in\mathbb{R}^{d}}\frac{1}{2} \|\mathbf{A} x - b\|_{2}^{2} $ in time $ \tilde{O} ((n+\sqrt{d\cdot\kappa_{\text{sum}}})\cdot s\cdot\log\epsilon^{-1}) $ where $\kappa_{\text{sum}}=\mathrm{tr}\left(\mathbf{A}^{\top}\mathbf{A}\right)/\lambda_{\min}(\mathbf{A}^{T}\mathbf{A})$ and $s$ is the maximum number of non-zero entries in a row of $\mathbf{A}$.

1 code implementation • 27 Oct 2017 • Aaron Sidford, Mengdi Wang, Xian Wu, Yinyu Ye

Given a discounted Markov Decision Process (DMDP) with $|S|$ states, $|A|$ actions, discount factor $\gamma\in(0, 1)$, and rewards in the range $[-M, M]$, we show how to compute an $\epsilon$-optimal policy, with probability $1 - \delta$ in time \[ \tilde{O}\left( \left(|S|^2 |A| + \frac{|S| |A|}{(1 - \gamma)^3} \right) \log\left( \frac{M}{\epsilon} \right) \log\left( \frac{1}{\delta} \right) \right) ~ .

no code implementations • 25 Oct 2017 • Prateek Jain, Sham M. Kakade, Rahul Kidambi, Praneeth Netrapalli, Venkata Krishna Pillutla, Aaron Sidford

This work provides a simplified proof of the statistical minimax optimality of (iterate averaged) stochastic gradient descent (SGD), for the special case of least squares.

1 code implementation • 25 Aug 2017 • Cameron Musco, Christopher Musco, Aaron Sidford

In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[\lambda_{min}(A), \lambda_{max}(A)]$.

Data Structures and Algorithms Numerical Analysis

no code implementations • ICML 2017 • Yair Carmon, John C. Duchi, Oliver Hinder, Aaron Sidford

We develop and analyze a variant of Nesterov’s accelerated gradient descent (AGD) for minimization of smooth non-convex functions.

no code implementations • 26 Apr 2017 • Prateek Jain, Sham M. Kakade, Rahul Kidambi, Praneeth Netrapalli, Aaron Sidford

There is widespread sentiment that it is not possible to effectively utilize fast gradient methods (e. g. Nesterov's acceleration, conjugate gradient, heavy ball) for the purposes of stochastic optimization due to their instability and error accumulation, a notion made precise in d'Aspremont 2008 and Devolder, Glineur, and Nesterov 2014.

no code implementations • 13 Apr 2017 • Cameron Musco, Praneeth Netrapalli, Aaron Sidford, Shashanka Ubaru, David P. Woodruff

We thus effectively compute a histogram of the spectrum, which can stand in for the true singular values in many applications.

1 code implementation • 12 Oct 2016 • Prateek Jain, Sham M. Kakade, Rahul Kidambi, Praneeth Netrapalli, Aaron Sidford

In particular, this work provides a sharp analysis of: (1) mini-batching, a method of averaging many samples of a stochastic gradient to both reduce the variance of the stochastic gradient estimate and for parallelizing SGD and (2) tail-averaging, a method involving averaging the final few iterates of SGD to decrease the variance in SGD's final iterate.

no code implementations • 26 May 2016 • Dan Garber, Elad Hazan, Chi Jin, Sham M. Kakade, Cameron Musco, Praneeth Netrapalli, Aaron Sidford

We give faster algorithms and improved sample complexities for estimating the top eigenvector of a matrix $\Sigma$ -- i. e. computing a unit vector $x$ such that $x^T \Sigma x \ge (1-\epsilon)\lambda_1(\Sigma)$: Offline Eigenvector Estimation: Given an explicit $A \in \mathbb{R}^{n \times d}$ with $\Sigma = A^TA$, we show how to compute an $\epsilon$ approximate top eigenvector in time $\tilde O([nnz(A) + \frac{d*sr(A)}{gap^2} ]* \log 1/\epsilon )$ and $\tilde O([\frac{nnz(A)^{3/4} (d*sr(A))^{1/4}}{\sqrt{gap}} ] * \log 1/\epsilon )$.

no code implementations • 13 Apr 2016 • Rong Ge, Chi Jin, Sham M. Kakade, Praneeth Netrapalli, Aaron Sidford

Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor.

no code implementations • 22 Feb 2016 • Prateek Jain, Chi Jin, Sham M. Kakade, Praneeth Netrapalli, Aaron Sidford

This work provides improved guarantees for streaming principle component analysis (PCA).

no code implementations • 22 Feb 2016 • Roy Frostig, Cameron Musco, Christopher Musco, Aaron Sidford

To achieve our results, we first observe that ridge regression can be used to obtain a "smooth projection" onto the top principal components.

no code implementations • 29 Oct 2015 • Chi Jin, Sham M. Kakade, Cameron Musco, Praneeth Netrapalli, Aaron Sidford

Combining our algorithm with previous work to initialize $x_0$, we obtain a number of improved sample complexity and runtime results.

no code implementations • 24 Jun 2015 • Roy Frostig, Rong Ge, Sham M. Kakade, Aaron Sidford

We develop a family of accelerated stochastic algorithms that minimize sums of convex functions.

no code implementations • 20 Dec 2014 • Roy Frostig, Rong Ge, Sham M. Kakade, Aaron Sidford

In the absence of computational constraints, the minimizer of a sample average of observed data -- commonly referred to as either the empirical risk minimizer (ERM) or the $M$-estimator -- is widely regarded as the estimation strategy of choice due to its desirable statistical convergence properties.

no code implementations • 21 Aug 2014 • Michael B. Cohen, Yin Tat Lee, Cameron Musco, Christopher Musco, Richard Peng, Aaron Sidford

In addition to an improved understanding of uniform sampling, our main proof introduces a structural result of independent interest: we show that every matrix can be made to have low coherence by reweighting a small subset of its rows.

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