arXiv subjects

Yair Carmon

Publications and source records attributed to Yair Carmon.

46 records · Page 3Linked to original sources

arXiv2019

Lower Bounds for Finding Stationary Points I

We prove lower bounds on the complexity of finding $ε$-stationary points (points $x$ such that $\|\nabla f(x)\| \le ε$) of smooth, high-dimensional, and potentially non-convex functions $f$. We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of $f$ at a query point $x$. We show that for any (potentially randomized) algorithm $\mathsf{A}$, there exists a function $f$ with Lipschitz $p$th order derivatives such that $\mathsf{A}$ requires at least $ε^{-(p+1)/p}$ queries to find an $ε$-stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton's method, and generalized $p$th order regularization are worst-case optimal within their natural function classes.

math.OC↗

arXiv2019

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. Unlike MMW, where every step requires full matrix exponentiation, our steps require only a single product of the form $e^A b$, which the Lanczos method approximates efficiently. Our key technique is to view the sketch as a $\textit{randomized mirror projection}$, and perform mirror descent analysis on the $\textit{expected projection}$. Our sketch solves the online eigenvector problem, improving the best known complexity bounds by $Ω(\log^5 n)$. We also apply this sketch to semidefinite programming in saddle-point form, yielding a simple primal-dual scheme with guarantees matching the best in the literature.

cs.LG↗

arXiv2019

Analysis of Krylov Subspace Solutions of Regularized Nonconvex Quadratic Problems

We provide convergence rates for Krylov subspace solutions to the trust-region and cubic-regularized (nonconvex) quadratic problems. Such solutions may be efficiently computed by the Lanczos method and have long been used in practice. We prove error bounds of the form $1/t^2$ and $e^{-4t/\sqrtκ}$, where $κ$ is a condition number for the problem, and $t$ is the Krylov subspace order (number of Lanczos iterations). We also provide lower bounds showing that our analysis is sharp.

math.OC↗

arXiv2017

Lower Bounds for Finding Stationary Points II: First-Order Methods

We establish lower bounds on the complexity of finding $ε$-stationary points of smooth, non-convex high-dimensional functions using first-order methods. We prove that deterministic first-order methods, even applied to arbitrarily smooth functions, cannot achieve convergence rates in $ε$ better than $ε^{-8/5}$, which is within $ε^{-1/15}\log\frac{1}ε$ of the best known rate for such methods. Moreover, for functions with Lipschitz first and second derivatives, we prove no deterministic first-order method can achieve convergence rates better than $ε^{-12/7}$, while $ε^{-2}$ is a lower bound for functions with only Lipschitz gradient. For convex functions with Lipschitz gradient, accelerated gradient descent achieves the rate $ε^{-1}\log\frac{1}ε$, showing that finding stationary points is easier given convexity.

math.OC↗

arXiv2017

"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. 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" of being non-convex. This non-convexity certificate allows us to exploit negative curvature and obtain deterministic, dimension-free acceleration of convergence for non-convex functions. For a function $f$ with Lipschitz continuous gradient and Hessian, we compute a point $x$ with $\|\nabla f(x)\| \le ε$ in $O(ε^{-7/4} \log(1/ ε) )$ gradient and function evaluations. Assuming additionally that the third derivative is Lipschitz, we require only $O(ε^{-5/3} \log(1/ ε) )$ evaluations.

math.OC↗

arXiv2017

Accelerated Methods for Non-Convex Optimization

We present an accelerated gradient method for non-convex optimization problems with Lipschitz continuous first and second derivatives. The method requires time $O(ε^{-7/4} \log(1/ ε) )$ to find an $ε$-stationary point, meaning a point $x$ such that $\|\nabla f(x)\| \le ε$. The method improves upon the $O(ε^{-2} )$ complexity of gradient descent and provides the additional second-order guarantee that $\nabla^2 f(x) \succeq -O(ε^{1/2})I$ for the computed $x$. Furthermore, our method is Hessian free, i.e. it only requires gradient computations, and is therefore suitable for large scale applications.

math.OC↗

arXiv2016

No bad local minima: Data independent training error guarantees for multilayer neural networks

We use smoothed analysis techniques to provide guarantees on the training loss of Multilayer Neural Networks (MNNs) at differentiable local minima. Specifically, we examine MNNs with piecewise linear activation functions, quadratic loss and a single output, under mild over-parametrization. We prove that for a MNN with one hidden layer, the training error is zero at every differentiable local minimum, for almost every dataset and dropout-like noise realization. We then extend these results to the case of more than one hidden layer. Our theoretical guarantees assume essentially nothing on the training data, and are verified numerically. These results suggest why the highly non-convex loss of such MNNs can be easily optimized using local updates (e.g., stochastic gradient descent), as observed empirically.

stat.ML↗

arXiv2014

Comparison of the Achievable Rates in OFDM and Single Carrier Modulation with I.I.D. Inputs

We compare the maximum achievable rates in single-carrier and OFDM modulation schemes, under the practical assumptions of i.i.d. finite alphabet inputs and linear ISI with additive Gaussian noise. We show that the Shamai-Laroia approximation serves as a bridge between the two rates: while it is well known that this approximation is often a lower bound on the single-carrier achievable rate, it is revealed to also essentially upper bound the OFDM achievable rate. We apply Information-Estimation relations in order to rigorously establish this result for both general input distributions and to sharpen it for commonly used PAM and QAM constellations. To this end, novel bounds on MMSE estimation of PAM inputs to a scalar Gaussian channel are derived, which may be of general interest. Our results show that, under reasonable assumptions, optimal single-carrier schemes may offer spectral efficiency significantly superior to that of OFDM, motivating further research of such systems.

cs.IT↗

arXiv2014

Lower Bounds and Approximations for the Information Rate of the ISI Channel

We consider the discrete-time intersymbol interference (ISI) channel model, with additive Gaussian noise and fixed i.i.d. inputs. In this setting, we investigate the expression put forth by Shamai and Laroia as a conjectured lower bound for the input-output mutual information after application of a MMSE-DFE receiver. A low-SNR expansion is used to prove that the conjectured bound does not hold under general conditions, and to characterize inputs for which it is particularly ill-suited. One such input is used to construct a counterexample, indicating that the Shamai-Laroia expression does not always bound even the achievable rate of the channel, thus excluding a natural relaxation of the original conjectured bound. However, this relaxed bound is then shown to hold for any finite entropy input and ISI channel, when the SNR is sufficiently high. Finally, new simple bounds for the achievable rate are proven, and compared to other known bounds. Information-Estimation relations and estimation-theoretic bounds play a key role in establishing our results.

cs.IT↗

arXiv2013

Information, Estimation, and Lookahead in the Gaussian channel

We consider mean squared estimation with lookahead of a continuous-time signal corrupted by additive white Gaussian noise. We show that the mutual information rate function, i.e., the mutual information rate as function of the signal-to-noise ratio (SNR), does not, in general, determine the minimum mean squared error (MMSE) with fixed finite lookahead, in contrast to the special cases with 0 and infinite lookahead (filtering and smoothing errors), respectively, which were previously established in the literature. We also establish a new expectation identity under a generalized observation model where the Gaussian channel has an SNR jump at $t=0$, capturing the tradeoff between lookahead and SNR. Further, we study the class of continuous-time stationary Gauss-Markov processes (Ornstein-Uhlenbeck processes) as channel inputs, and explicitly characterize the behavior of the minimum mean squared error (MMSE) with finite lookahead and signal-to-noise ratio (SNR). The MMSE with lookahead is shown to converge exponentially rapidly to the non-causal error, with the exponent being the reciprocal of the non-causal error. We extend our results to mixtures of Ornstein-Uhlenbeck processes, and use the insight gained to present lower and upper bounds on the MMSE with lookahead for a class of stationary Gaussian input processes, whose spectrum can be expressed as a mixture of Ornstein-Uhlenbeck spectra.

cs.IT↗