Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}(\log n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or $\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.
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We propose quantum subroutines for the simplex method that avoid classical computation of the basis inverse. We show how to quantize all steps of the simplex algorithm, including checking optimality, unboundedness, and identifying a pivot (i.e., pricing the columns and performing the ratio test) according to Dantzig's rule or the steepest edge rule. The quantized subroutines obtain a polynomial speedup in the dimension of the problem, but have worse dependence on other numerical parameters. For example, for a problem with $m$ constraints, $n$ variables, at most $d_c$ nonzero elements per column of the costraint matrix, at most $d$ nonzero elements per column or row of the basis, basis condition number $\kappa$, and optimality tolerance $\epsilon$, pricing can be performed in $\tilde{O}(\frac{1}{\epsilon}\kappa d \sqrt{n}(d_c n + d m))$ time, where the $\tilde{O}$ notation hides polylogarithmic factors; classically, pricing requires $O(d_c^{0.7} m^{1.9} + m^{2 + o(1)} + d_c n)$ time in the worst case using the fastest known algorithm for sparse matrix multiplication. For well-conditioned sparse problems the quantum subroutines scale better in $m$ and $n$, and may therefore have an advantage for very large problems. The running time of the quantum subroutines can be improved if the constraint matrix admits an efficient algorithmic description, or if quantum RAM is available.
We aim to make stochastic gradient descent (SGD) adaptive to (i) the noise $\sigma^2$ in the stochastic gradients and (ii) problem-dependent constants. When minimizing smooth, strongly-convex functions with condition number $\kappa$, we prove that $T$ iterations of SGD with exponentially decreasing step-sizes and knowledge of the smoothness can achieve an $\tilde{O} \left(\exp \left( \frac{-T}{\kappa} \right) + \frac{\sigma^2}{T} \right)$ rate, without knowing $\sigma^2$. In order to be adaptive to the smoothness, we use a stochastic line-search (SLS) and show (via upper and lower-bounds) that SGD with SLS converges at the desired rate, but only to a neighbourhood of the solution. On the other hand, we prove that SGD with an offline estimate of the smoothness converges to the minimizer. However, its rate is slowed down proportional to the estimation error. Next, we prove that SGD with Nesterov acceleration and exponential step-sizes (referred to as ASGD) can achieve the near-optimal $\tilde{O} \left(\exp \left( \frac{-T}{\sqrt{\kappa}} \right) + \frac{\sigma^2}{T} \right)$ rate, without knowledge of $\sigma^2$. When used with offline estimates of the smoothness and strong-convexity, ASGD still converges to the solution, albeit at a slower rate. We empirically demonstrate the effectiveness of exponential step-sizes coupled with a novel variant of SLS.
We introduce a family of stochastic optimization methods based on the Runge-Kutta-Chebyshev (RKC) schemes. The RKC methods are explicit methods originally designed for solving stiff ordinary differential equations by ensuring that their stability regions are of maximal size.In the optimization context, this allows for larger step sizes (learning rates) and better robustness compared to e.g. the popular stochastic gradient descent method. Our main contribution is a convergence proof for essentially all stochastic Runge-Kutta optimization methods. This shows convergence in expectation with an optimal sublinear rate under standard assumptions of strong convexity and Lipschitz-continuous gradients. For non-convex objectives, we get convergence to zero in expectation of the gradients. The proof requires certain natural conditions on the Runge-Kutta coefficients, and we further demonstrate that the RKC schemes satisfy these. Finally, we illustrate the improved stability properties of the methods in practice by performing numerical experiments on both a small-scale test example and on a problem arising from an image classification application in machine learning.
We propose the homotopic policy mirror descent (HPMD) method for solving discounted, infinite horizon MDPs with finite state and action space, and study its policy convergence. We report three properties that seem to be new in the literature of policy gradient methods: (1) HPMD exhibits global linear convergence of the value optimality gap, and local superlinear convergence of the policy to the set of optimal policies with order $\gamma^{-2}$. The superlinear convergence of the policy takes effect after no more than $\mathcal{O}(\log(1/\Delta^*))$ number of iterations, where $\Delta^*$ is defined via a gap quantity associated with the optimal state-action value function; (2) HPMD also exhibits last-iterate convergence of the policy, with the limiting policy corresponding exactly to the optimal policy with the maximal entropy for every state. No regularization is added to the optimization objective and hence the second observation arises solely as an algorithmic property of the homotopic policy gradient method. (3) For the stochastic HPMD method, we further demonstrate a better than $\mathcal{O}(|\mathcal{S}| |\mathcal{A}| / \epsilon^2)$ sample complexity for small optimality gap $\epsilon$, when assuming a generative model for policy evaluation.
We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex nonsmooth function, while the denominator part is a concave or convex function. This problem is difficult to solve since it is nonconvex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. One is applied to the original fractional function, the other is based on the associated parametric problem. The proposed methods iteratively solve a one-dimensional subproblem \textit{globally}, and they are guaranteed to converge to coordinate-wise stationary points. In the case of a convex denominator, we prove that the proposed CD methods using sequential nonconvex approximation find stronger stationary points than existing methods. Under suitable conditions, CD methods with an appropriate initialization converge linearly to the optimal point (also the coordinate-wise stationary point). In the case of a concave denominator, we show that the resulting problem is quasi-convex, and any critical point is a global minimum. We prove that the algorithms converge to the global optimal solution with a sublinear convergence rate. We demonstrate the applicability of the proposed methods to some machine learning and signal processing models. Our experiments on real-world data have shown that our method significantly and consistently outperforms existing methods in terms of accuracy.
Stochastic gradient descent (SGD) undergoes complicated multiplicative noise for the mean-square loss. We use this property of SGD noise to derive a stochastic differential equation (SDE) with simpler additive noise by performing a random time change. Using this formalism, we show that the log loss barrier $\Delta\log L=\log[L(\theta^s)/L(\theta^*)]$ between a local minimum $\theta^*$ and a saddle $\theta^s$ determines the escape rate of SGD from the local minimum, contrary to the previous results borrowing from physics that the linear loss barrier $\Delta L=L(\theta^s)-L(\theta^*)$ decides the escape rate. Our escape-rate formula strongly depends on the typical magnitude $h^*$ and the number $n$ of the outlier eigenvalues of the Hessian. This result explains an empirical fact that SGD prefers flat minima with low effective dimensions, giving an insight into implicit biases of SGD.
We study the problem of group testing with non-identical, independent priors. So far, the pooling strategies that have been proposed in the literature take the following approach: a hand-crafted test design along with a decoding strategy is proposed, and guarantees are provided on how many tests are sufficient in order to identify all infections in a population. In this paper, we take a different, yet perhaps more practical, approach: we fix the decoder and the number of tests, and we ask, given these, what is the best test design one could use? We explore this question for the Definite Non-Defectives (DND) decoder. We formulate a (non-convex) optimization problem, where the objective function is the expected number of errors for a particular design. We find approximate solutions via gradient descent, which we further optimize with informed initialization. We illustrate through simulations that our method can achieve significant performance improvement over traditional approaches.
Second-order optimizers are thought to hold the potential to speed up neural network training, but due to the enormous size of the curvature matrix, they typically require approximations to be computationally tractable. The most successful family of approximations are Kronecker-Factored, block-diagonal curvature estimates (KFAC). Here, we combine tools from prior work to evaluate exact second-order updates with careful ablations to establish a surprising result: Due to its approximations, KFAC is not closely related to second-order updates, and in particular, it significantly outperforms true second-order updates. This challenges widely held believes and immediately raises the question why KFAC performs so well. We answer this question by showing that KFAC approximates a first-order algorithm, which performs gradient descent on neurons rather than weights. Finally, we show that this optimizer often improves over KFAC in terms of computational cost and data-efficiency.
This paper studies the accelerated gradient descent for general nonconvex problems under the gradient Lipschitz and Hessian Lipschitz assumptions. We establish that a simple restarted accelerated gradient descent (AGD) finds an $\epsilon$-approximate first-order stationary point in $O(\epsilon^{-7/4})$ gradient computations with simple proofs. Our complexity does not hide any polylogarithmic factors, and thus it improves over the state-of-the-art one by the $O(\log\frac{1}{\epsilon})$ factor. Our simple algorithm only consists of Nesterov's classical AGD and a restart mechanism, and it does not need the negative curvature exploitation or the optimization of regularized surrogate functions. Technically, our simple proof does not invoke the analysis for the strongly convex AGD, which is crucial to remove the $O(\log\frac{1}{\epsilon})$ factor.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.
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