32 accepted papers
We develop optimization methods which offer new trade-offs between the number of gradient and Hessian computations needed to compute the critical point of a non-convex function. We provide a method that for a twice-differentiable $f\colon \mathbb{R}^d \rightarrow \mathbb{R}$ with $L_2$-Lipschitz Hes…
We consider the problem of minimizing a $d$-dimensional Lipschitz convex function using a stochastic gradient oracle. We introduce and motivate a setting where the noise of the stochastic gradient is isotropic in that it is bounded in every direction with high probability. We then develop an algorit…
We consider the well-studied problem of completing a rank-$r$, $\mu$-incoherent matrix $\mathbf{M} \in \mathbb{R}^{d \times d}$ from incomplete observations. We focus on this problem in the semi-random setting where each entry is independently revealed with probability at least $p = \frac{\textup{po…
We provide faster randomized algorithms for computing an $\epsilon$-optimal policy in a discounted Markov decision process with $A_{\text{tot}}$-state-action pairs, bounded rewards, and discount factor $\gamma$. We provide an $\tilde{O}(A_{\text{tot}}[(1 - \gamma)^{-3}\epsilon^{-2} + (1 - \gamma)^{-…
Minimizing a convex function with access to a first order oracle---that returns the function evaluation and (sub)gradient at a query point---is a canonical optimization problem and a fundamental primitive in machine learning. Gradient-based methods are the most popular approaches used for solving t…
We consider the parallel complexity of submodular function minimization (SFM). We provide a pair of methods which obtain two new query versus depth trade-offs a submodular function defined on subsets of $n$ elements that has integer values between $-M$ and $M$. The first method has depth $2$ an…
We give a quantum algorithm for computing an $\epsilon$-approximate Nash equilibrium of a zero-sum game in a $m \times n$ payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time $\widetilde{O}(\sqrt{m + n}\cdot \epsilon^{-2.5} +…
We consider the problem of minimizing a continuous function given given access to a natural quantum generalization of a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off whi…
We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including: Diagonal preconditioning. We give an algorithm which, given…
We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high probability, an $\widetilde{O}(n\epsilon^{-1})$-bit sketch fro…
We provide an efficient unified plug-in approach for estimating symmetric properties of distributions given $n$ independent samples. Our estimator is based on profile-maximum-likelihood (PML) and is sample optimal for estimating various symmetric properties when the estimation error $\epsilon \gg n^…
We develop a variant of the Monteiro-Svaiter (MS) acceleration framework that removes the need to solve an expensive implicit equation at every iteration. Consequently, for any $p\ge 2$ we improve the complexity of convex optimization with Lipschitz $p$th derivative by a logarithmic factor, matchin…
The accelerated proximal point method (APPA), also known as "Catalyst", is a well-established reduction from convex optimization to approximate proximal point computation (i.e., regularized minimization). This reduction is conceptually elegant and yields strong convergence rate guarantees. However,…
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 $f$. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an e…
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. When the mixing time of the probability transition matrix of all policies is at most $t_\mathrm{mi…
Consider an oracle which takes a point x and returns the minimizer of a convex function f in an l
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. When applied to an average-reward MDP with $A_{tot}$ total actions and mixing time bound $t_{mix}$ our method computes a…
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. We provide an algorithm which matches the previous best known efficient algorithms for computing approximate PML di…
We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $\chi^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters,…
In this paper we settle the sampling complexity of solving discounted two-player turn-based zero-sum stochastic games up to polylogarithmic factors. Given a stochastic game with discount factor $\gamma\in(0,1)$ we provide an algorithm that computes an $\epsilon$-optimal strategy with high-probabili…
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. We give an algorithm which solves the problem to additive $\epsilon$ accuracy with $\tilde{…
In this paper we provide a general framework for estimating symmetric properties of distributions from i.i.d. samples. For a broad class of symmetric properties we identify the {\em easy} region where empirical estimation works and the {\em difficult} region where more complex estimators are requir…
A landmark result of non-smooth convex optimization is that gradient descent is an optimal algorithm whenever the number of computed gradients is smaller than the dimension $d$. In this paper we study the extension of this result to the parallel optimization setting. Namely we consider optimization…
Given a n-by-d data matrix A, principal component projection (PCP) and principal component regression (PCR), i.e. projection and regression restricted to the top-eigenspace of A, are fundamental problems in machine learning, optimization, and numerical analysis. In this paper we provide the first al…
We present a randomized primal-dual algorithm that solves the problem min
In this paper, we obtain improved running times for regression and top eigenvector computation for numerically sparse matrices. Given a data matrix $\mat{A} \in \R^{n \times d}$ where every row $a \in \R^d$ has $\|a\|_2^2 \leq L$ and numerical sparsity $\leq s$, i.e. $\|a\|_1^2 / \|a\|_2^2 \leq s$,…
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.…
We develop and analyze a variant of Nesterov’s accelerated gradient descent (AGD) for minimization of smooth non-convex functions. We prove that one of two cases occurs: either our AGD variant converges quickly, as if the function was convex, or we produce a certificate that the function is “guilty”…
This paper considers the problem of canonical-correlation analysis (CCA) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics…
We give faster algorithms and improved sample complexities for the fundamental problem of estimating the top eigenvector. Given an explicit matrix $A \in \mathbb{R}^{n \times d}$, we show how to compute an $\epsilon$-approximate top eigenvector of $A^TA$ in time $\tilde O\left( \left[\text{nnz}(A) +…
We show how to efficiently project a vector onto the top principal components of a matrix, *without explicitly computing these components*. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoidi…
We develop a family of accelerated stochastic algorithms that optimize sums of convex functions. Our algorithms improve upon the fastest running time for empirical risk minimization (ERM), and in particular linear least-squares regression, across a wide range of problem settings. To achieve this, we…