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Found 54 papers, 8 papers with code
Faster Spectral Density Estimation and Sparsification in the Nuclear Norm
We provide a randomized algorithm that, with $O(n\epsilon^{-2})$ queries to a degree and neighbor oracle and in $O(n\epsilon^{-3})$ time, estimates the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 metric.
Closing the Computational-Query Depth Gap in Parallel Stochastic Convex Optimization
When combined with previous state-of-the-art methods our result closes a gap between the best-known query depth and the best-known computational depth of parallel algorithms.
Truncated Variance Reduced Value Iteration
We provide an $\tilde{O}(A_{\text{tot}}[(1 - \gamma)^{-3}\epsilon^{-2} + (1 - \gamma)^{-2}])$-time algorithm in the sampling setting, where the probability transition matrix is unknown but accessible through a generative model which can be queried in $\tilde{O}(1)$-time, and an $\tilde{O}(s + (1-\gamma)^{-2})$-time algorithm in the offline setting where the probability transition matrix is known and $s$-sparse.
A Whole New Ball Game: A Primal Accelerated Method for Matrix Games and Minimizing the Maximum of Smooth Functions
For $n>d$ and $\epsilon=1/\sqrt{n}$ this improves over all existing first-order methods.
Matrix Completion in Almost-Verification Time
In the well-studied setting where $\mathbf{M}$ has incoherent row and column spans, our algorithms complete $\mathbf{M}$ to high precision from $mr^{2+o(1)}$ observations in $mr^{3 + o(1)}$ time (omitting logarithmic factors in problem parameters), improving upon the prior state-of-the-art [JN15] which used $\approx mr^5$ samples and $\approx mr^7$ time.
Moments, Random Walks, and Limits for Spectrum Approximation
We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments.
ReSQueing Parallel and Private Stochastic Convex Optimization
We give a parallel algorithm obtaining optimization error $\epsilon_{\text{opt}}$ with $d^{1/3}\epsilon_{\text{opt}}^{-2/3}$ gradient oracle query depth and $d^{1/3}\epsilon_{\text{opt}}^{-2/3} + \epsilon_{\text{opt}}^{-2}$ gradient queries in total, assuming access to a bounded-variance stochastic gradient estimator.
On the Efficient Implementation of High Accuracy Optimality of Profile Maximum Likelihood
We provide an efficient unified plug-in approach for estimating symmetric properties of distributions given $n$ independent samples.
RECAPP: Crafting a More Efficient Catalyst for Convex Optimization
The accelerated proximal point algorithm (APPA), also known as "Catalyst", is a well-established reduction from convex optimization to approximate proximal point computation (i. e., regularized minimization).
Efficient Convex Optimization Requires Superlinear Memory
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 $\delta \in [0, 1/4]$).
Semi-Random Sparse Recovery in Nearly-Linear Time
We design a new iterative method tailored to the geometry of sparse recovery which is provably robust to our semi-random model.
Sharper Rates for Separable Minimax and Finite Sum Optimization via Primal-Dual Extragradient Methods
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.
Big-Step-Little-Step: Efficient Gradient Methods for Objectives with Multiple Scales
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.
Stochastic Bias-Reduced Gradient Methods
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.
Towards Tight Bounds on the Sample Complexity of Average-reward MDPs
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.
Thinking Inside the Ball: Near-Optimal Minimization of the Maximal Loss
We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$.
Minimum Cost Flows, MDPs, and $\ell_1$-Regression in Nearly Linear Time for Dense Instances
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
Relative Lipschitzness in Extragradient Methods and a Direct Recipe for Acceleration
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.
Instance Based Approximations to Profile Maximum Likelihood
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.
Large-Scale Methods for Distributionally Robust Optimization
We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $\chi^2$ divergence uncertainty sets.
Coordinate Methods for Matrix Games
For linear regression with an elementwise nonnegative matrix, our guarantees improve on exact gradient methods by a factor of $\sqrt{\mathrm{nnz}(A)/(m+n)}$.
Efficiently Solving MDPs with Stochastic Mirror Descent
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.
Fast and Near-Optimal Diagonal Preconditioning
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.
The Bethe and Sinkhorn Permanents of Low Rank Matrices and Implications for Profile Maximum Likelihood
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).
A General Framework for Symmetric Property Estimation
In this paper we provide a general framework for estimating symmetric properties of distributions from i. i. d.
A Direct tilde{O}(1/epsilon) Iteration Parallel Algorithm for Optimal Transport
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.
Principal Component Projection and Regression in Nearly Linear Time through Asymmetric SVRG
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.
Solving Discounted Stochastic Two-Player Games with Near-Optimal Time and Sample Complexity
In this paper, we settle the sampling complexity of solving discounted two-player turn-based zero-sum stochastic games up to polylogarithmic factors.
Variance Reduction for Matrix Games
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.
Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond
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."
Complexity of Highly Parallel Non-Smooth Convex Optimization
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.
A Direct $\tilde{O}(1/ε)$ Iteration Parallel Algorithm for Optimal Transport
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.
Efficient Profile Maximum Likelihood for Universal Symmetric Property Estimation
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 $\epsilon = \Omega(n^{-0. 166})$.
Memory-Sample Tradeoffs for Linear Regression with Small Error
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.
A Rank-1 Sketch for Matrix Multiplicative Weights
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.
Near-Optimal Time and Sample Complexities for Solving Markov Decision Processes with a Generative Model
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.
Exploiting Numerical Sparsity for Efficient Learning : Faster Eigenvector Computation and Regression
This running time improves upon the previous best unaccelerated running time of $\tilde{O}(nd + n L d / \mu)$.
Near-Optimal Time and Sample Complexities for Solving Discounted Markov Decision Process with a Generative Model
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
Leverage Score Sampling for Faster Accelerated Regression and ERM
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 $\mathbf{A}$.
Variance Reduced Value Iteration and Faster Algorithms for Solving Markov Decision Processes
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) ~ .
A Markov Chain Theory Approach to Characterizing the Minimax Optimality of Stochastic Gradient Descent (for Least Squares)
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.
Stability of the Lanczos Method for Matrix Function Approximation
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
“Convex Until Proven Guilty”: Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions
We develop and analyze a variant of Nesterov’s accelerated gradient descent (AGD) for minimization of smooth non-convex functions.
Accelerating Stochastic Gradient Descent For Least Squares Regression
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.
Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness
We thus effectively compute a histogram of the spectrum, which can stand in for the true singular values in many applications.
Parallelizing Stochastic Gradient Descent for Least Squares Regression: mini-batching, averaging, and model misspecification
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.
Faster Eigenvector Computation via Shift-and-Invert Preconditioning
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 $\tilde O([\frac{nnz(A)^{3/4} (d*sr(A))^{1/4}}{\sqrt{gap}} ] * \log 1/\epsilon )$.
Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis
Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor.
Principal Component Projection Without Principal Component Analysis
To achieve our results, we first observe that ridge regression can be used to obtain a "smooth projection" onto the top principal components.
Streaming PCA: Matching Matrix Bernstein and Near-Optimal Finite Sample Guarantees for Oja's Algorithm
This work provides improved guarantees for streaming principle component analysis (PCA).
Robust Shift-and-Invert Preconditioning: Faster and More Sample Efficient Algorithms for Eigenvector Computation
Combining our algorithm with previous work to initialize $x_0$, we obtain a number of improved sample complexity and runtime results.
Un-regularizing: approximate proximal point and faster stochastic algorithms for empirical risk minimization
We develop a family of accelerated stochastic algorithms that minimize sums of convex functions.
Competing with the Empirical Risk Minimizer in a Single Pass
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.
Uniform Sampling for Matrix Approximation
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.
