arXiv subjects

Kevin A. Lai

Publications and source records attributed to Kevin A. Lai.

9 recordsLinked to original sources

arXiv2020

Fast Convergence of Fictitious Play for Diagonal Payoff Matrices

Fictitious Play (FP) is a simple and natural dynamic for repeated play in zero-sum games. Proposed by Brown in 1949, FP was shown to converge to a Nash Equilibrium by Robinson in 1951, albeit at a slow rate that may depend on the dimension of the problem. In 1959, Karlin conjectured that FP converges at the more natural rate of $O(1/\sqrt{t})$. However, Daskalakis and Pan disproved a version of this conjecture in 2014, showing that a slow rate can occur, although their result relies on adversarial tie-breaking. In this paper, we show that Karlin's conjecture is indeed correct for the class of diagonal payoff matrices, as long as ties are broken lexicographically. Specifically, we show that FP converges at a $O(1/\sqrt{t})$ rate in the case when the payoff matrix is diagonal. We also prove this bound is tight by showing a matching lower bound in the identity payoff case under the lexicographic tie-breaking assumption.

cs.GT↗

arXiv2020

Higher-order methods for convex-concave min-max optimization and monotone variational inequalities

We provide improved convergence rates for constrained convex-concave min-max problems and monotone variational inequalities with higher-order smoothness. In min-max settings where the $p^{th}$-order derivatives are Lipschitz continuous, we give an algorithm HigherOrderMirrorProx that achieves an iteration complexity of $O(1/T^{\frac{p+1}{2}})$ when given access to an oracle for finding a fixed point of a $p^{th}$-order equation. We give analogous rates for the weak monotone variational inequality problem. For $p>2$, our results improve upon the iteration complexity of the first-order Mirror Prox method of Nemirovski [2004] and the second-order method of Monteiro and Svaiter [2012]. We further instantiate our entire algorithm in the unconstrained $p=2$ case.

math.OC↗

arXiv2019

Last-iterate convergence rates for min-max optimization

While classic work in convex-concave min-max optimization relies on average-iterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because many natural algorithms, such as Simultaneous Gradient Descent/Ascent, provably diverge or cycle even in simple convex-concave min-max settings, and previous work on global last-iterate convergence rates has been limited to the bilinear and convex-strongly concave settings. In this work, we show that the Hamiltonian Gradient Descent (HGD) algorithm achieves linear convergence in a variety of more general settings, including convex-concave problems that satisfy a "sufficiently bilinear" condition. We also prove similar convergence rates for the Consensus Optimization (CO) algorithm of [MNG17] for some parameter settings of CO.

math.OC↗

arXiv2018

$\ell_1$ Regression using Lewis Weights Preconditioning and Stochastic Gradient Descent

We present preconditioned stochastic gradient descent (SGD) algorithms for the $\ell_1$ minimization problem $\min_{x}\|A x - b\|_1$ in the overdetermined case, where there are far more constraints than variables. Specifically, we have $A \in \mathbb{R}^{n \times d}$ for $n \gg d$. Commonly known as the Least Absolute Deviations problem, $\ell_1$ regression can be used to solve many important combinatorial problems, such as minimum cut and shortest path. SGD-based algorithms are appealing for their simplicity and practical efficiency. Our primary insight is that careful preprocessing can yield preconditioned matrices $\tilde{A}$ with strong properties (besides good condition number and low-dimension) that allow for faster convergence of gradient descent. In particular, we precondition using Lewis weights to obtain an isotropic matrix with fewer rows and strong upper bounds on all row norms. We leverage these conditions to find a good initialization, which we use along with recent smoothing reductions and accelerated stochastic gradient descent algorithms to achieve $ε$ relative error in $\tilde{O}(nnz(A) + d^{2.5} ε^{-2})$ time with high probability, where $nnz(A)$ is the number of non-zeros in $A$. This improves over the previous best result using gradient descent for $\ell_1$ regression. We also match the best known running times for interior point methods in several settings. Finally, we also show that if our original matrix $A$ is approximately isotropic and the row norms are approximately equal, we can give an algorithm that avoids using fast matrix multiplication and obtains a running time of $\tilde{O}(nnz(A) + s d^{1.5}ε^{-2} + d^2ε^{-2})$, where $s$ is the maximum number of non-zeros in a row of $A$. In this setting, we beat the best interior point methods for certain parameter regimes.

cs.DS↗

arXiv2018

Faster Rates for Convex-Concave Games

We consider the use of no-regret algorithms to compute equilibria for particular classes of convex-concave games. While standard regret bounds would lead to convergence rates on the order of $O(T^{-1/2})$, recent work \citep{RS13,SALS15} has established $O(1/T)$ rates by taking advantage of a particular class of optimistic prediction algorithms. In this work we go further, showing that for a particular class of games one achieves a $O(1/T^2)$ rate, and we show how this applies to the Frank-Wolfe method and recovers a similar bound \citep{D15}. We also show that such no-regret techniques can even achieve a linear rate, $O(\exp(-T))$, for equilibrium computation under additional curvature assumptions.

cs.LG↗

arXiv2018

Differential Privacy for Growing Databases

We study the design of differentially private algorithms for adaptive analysis of dynamically growing databases, where a database accumulates new data entries while the analysis is ongoing. We provide a collection of tools for machine learning and other types of data analysis that guarantee differential privacy and accuracy as the underlying databases grow arbitrarily large. We give both a general technique and a specific algorithm for adaptive analysis of dynamically growing databases. Our general technique is illustrated by two algorithms that schedule black box access to some algorithm that operates on a fixed database to generically transform private and accurate algorithms for static databases into private and accurate algorithms for dynamically growing databases. These results show that almost any private and accurate algorithm can be rerun at appropriate points of data growth with minimal loss of accuracy, even when data growth is unbounded. Our specific algorithm directly adapts the private multiplicative weights algorithm to the dynamic setting, maintaining the accuracy guarantee of the static setting through unbounded data growth. Along the way, we develop extensions of several other differentially private algorithms to the dynamic setting, which may be of independent interest for future work on the design of differentially private algorithms for growing databases.

cs.DS↗

arXiv2016

Agnostic Estimation of Mean and Covariance

We consider the problem of estimating the mean and covariance of a distribution from iid samples in $\mathbb{R}^n$, in the presence of an $η$ fraction of malicious noise; this is in contrast to much recent work where the noise itself is assumed to be from a distribution of known type. The agnostic problem includes many interesting special cases, e.g., learning the parameters of a single Gaussian (or finding the best-fit Gaussian) when $η$ fraction of data is adversarially corrupted, agnostically learning a mixture of Gaussians, agnostic ICA, etc. We present polynomial-time algorithms to estimate the mean and covariance with error guarantees in terms of information-theoretic lower bounds. As a corollary, we also obtain an agnostic algorithm for Singular Value Decomposition.

cs.DS↗

arXiv2015

Label optimal regret bounds for online local learning

We resolve an open question from (Christiano, 2014b) posed in COLT'14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework the algorithm is shown a pair of items at each step, chosen from a set of $n$ items. The learner then predicts a label for each item, from a label set of size $L$ and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as collaborative filtering, online gambling, and online max cut among others. (Christiano, 2014a) designed an efficient online learning algorithm for this problem achieving a regret of $O(\sqrt{nL^3T})$, where $T$ is the number of rounds. Information theoretically, one can achieve a regret of $O(\sqrt{n \log L T})$. One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of (Christiano, 2014a), in fact achieves a regret of $O(\sqrt{nLT})$. Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well.

cs.LG↗

arXiv2015

Amortized Rotation Cost in AVL Trees

An AVL tree is the original type of balanced binary search tree. An insertion in an $n$-node AVL tree takes at most two rotations, but a deletion in an $n$-node AVL tree can take $Θ(\log n)$. A natural question is whether deletions can take many rotations not only in the worst case but in the amortized case as well. A sequence of $n$ successive deletions in an $n$-node tree takes $O(n)$ rotations, but what happens when insertions are intermixed with deletions? Heaupler, Sen, and Tarjan conjectured that alternating insertions and deletions in an $n$-node AVL tree can cause each deletion to do $Ω(\log n)$ rotations, but they provided no construction to justify their claim. We provide such a construction: we show that, for infinitely many $n$, there is a set $E$ of {\it expensive} $n$-node AVL trees with the property that, given any tree in $E$, deleting a certain leaf and then reinserting it produces a tree in $E$, with the deletion having done $Θ(\log n)$ rotations. One can do an arbitrary number of such expensive deletion-insertion pairs. The difficulty in obtaining such a construction is that in general the tree produced by an expensive deletion-insertion pair is not the original tree. Indeed, if the trees in $E$ have even height $k$, $2^{k/2}$ deletion-insertion pairs are required to reproduce the original tree.

cs.DS↗