arXiv subjects

Coralia Cartis

Publications and source records attributed to Coralia Cartis.

At least 55 records · Page 3Linked to original sources

arXiv2020

Adaptive regularization with cubics on manifolds

Adaptive regularization with cubics (ARC) is an algorithm for unconstrained, non-convex optimization. Akin to the popular trust-region method, its iterations can be thought of as approximate, safe-guarded Newton steps. For cost functions with Lipschitz continuous Hessian, ARC has optimal iteration complexity, in the sense that it produces an iterate with gradient smaller than $\varepsilon$ in $O(1/\varepsilon^{1.5})$ iterations. For the same price, it can also guarantee a Hessian with smallest eigenvalue larger than $-\varepsilon^{1/2}$. In this paper, we study a generalization of ARC to optimization on Riemannian manifolds. In particular, we generalize the iteration complexity results to this richer framework. Our central contribution lies in the identification of appropriate manifold-specific assumptions that allow us to secure these complexity guarantees both when using the exponential map and when using a general retraction. A substantial part of the paper is devoted to studying these assumptions---relevant beyond ARC---and providing user-friendly sufficient conditions for them. Numerical experiments are encouraging.

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arXiv2020

A dimensionality reduction technique for unconstrained global optimization of functions with low effective dimensionality

We investigate the unconstrained global optimization of functions with low effective dimensionality, that are constant along certain (unknown) linear subspaces. Extending the technique of random subspace embeddings in [Wang et al., Bayesian optimization in a billion dimensions via random embeddings. JAIR, 55(1): 361--387, 2016], we study a generic Random Embeddings for Global Optimization (REGO) framework that is compatible with any global minimization algorithm. Instead of the original, potentially large-scale optimization problem, within REGO, a Gaussian random, low-dimensional problem with bound constraints is formulated and solved in a reduced space. We provide novel probabilistic bounds for the success of REGO in solving the original, low effective-dimensionality problem, which show its independence of the (potentially large) ambient dimension and its precise dependence on the dimensions of the effective and randomly embedding subspaces. These results significantly improve existing theoretical analyses by providing the exact distribution of a reduced minimizer and its Euclidean norm and by the general assumptions required on the problem. We validate our theoretical findings by extensive numerical testing of REGO with three types of global optimization solvers, illustrating the improved scalability of REGO compared to the full-dimensional application of the respective solvers.

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arXiv2020

Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions

We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An adaptive regularization algorithm is then proposed to find such approximate minimizers. Under suitable Lipschitz continuity assumptions, whenever the feasible set is convex, it is shown that using a model of degree $p$, this algorithm will find a strong approximate q-th-order minimizer in at most ${\cal O}\left(\max_{1\leq j\leq q}ε_j^{-(p+1)/(p-j+1)}\right)$ evaluations of the problem's functions and their derivatives, where $ε_j$ is the $j$-th order accuracy tolerance; this bound applies when either $q=1$ or the problem is not composite with $q \leq 2$. For general non-composite problems, even when the feasible set is nonconvex, the bound becomes ${\cal O}\left(\max_{1\leq j\leq q}ε_j^{-q(p+1)/p}\right)$ evaluations. If the problem is composite, and either $q > 1$ or the feasible set is not convex, the bound is then ${\cal O}\left(\max_{1\leq j\leq q}ε_j^{-(q+1)}\right)$ evaluations. These results not only provide, to our knowledge, the first known bound for (unconstrained or inexpensively-constrained) composite problems for optimality orders exceeding one, but also give the first sharp bounds for high-order strong approximate $q$-th order minimizers of standard (unconstrained and inexpensively constrained) smooth problems, thereby complementing known results for weak minimizers.

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arXiv2019

Escaping local minima with derivative-free methods: a numerical investigation

We apply a state-of-the-art, local derivative-free solver, Py-BOBYQA, to global optimization problems, and propose an algorithmic improvement that is beneficial in this context. Our numerical findings are illustrated on a commonly-used but small-scale test set of global optimization problems and associated noisy variants, and on hyperparameter tuning for the machine learning test set MNIST. As Py-BOBYQA is a model-based trust-region method, we compare mostly (but not exclusively) with other global optimization methods for which (global) models are important, such as Bayesian optimization and response surface methods; we also consider state-of-the-art representative deterministic and stochastic codes, such as DIRECT and CMA-ES. As a heuristic for escaping local minima, we find numerically that Py-BOBYQA is competitive with global optimization solvers for all accuracy/budget regimes, in both smooth and noisy settings. In particular, Py-BOBYQA variants are best performing for smooth and multiplicative noise problems in high-accuracy regimes. As a by-product, some preliminary conclusions can be drawn on the relative performance of the global solvers we have tested with default settings.

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arXiv2018

Universal regularization methods - varying the power, the smoothness and the accuracy

Adaptive cubic regularization methods have emerged as a credible alternative to linesearch and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general/new class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size and applied to convexly-constrained optimization problems. We investigate the worst-case evaluation complexity/global rate of convergence of these algorithms, when the level of sufficient smoothness of the objective may be unknown or may even be absent. We find that the methods accurately reflect in their complexity the degree of smoothness of the objective and satisfy increasingly better bounds with improving accuracy of the models. The bounds vary continuously and robustly with respect to the regularization power and accuracy of the model and the degree of smoothness of the objective.

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arXiv2018

Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints

We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e.\ problems where the cost of evaluating/enforcing of the (possibly nonconvex or even disconnected) constraints, if any, is negligible compared to that of evaluating the objective function. These bounds unify, extend or improve all known upper and lower complexity bounds for unconstrained and convexly-constrained problems. It is shown that, given an accuracy level $ε$, a degree of highest available Lipschitz continuous derivatives $p$ and a desired optimality order $q$ between one and $p$, a conceptual regularization algorithm requires no more than $O(ε^{-\frac{p+1}{p-q+1}})$ evaluations of the objective function and its derivatives to compute a suitably approximate $q$-th order minimizer. With an appropriate choice of the regularization, a similar result also holds if the $p$-th derivative is merely Hölder rather than Lipschitz continuous. We provide an example that shows that the above complexity bound is sharp for unconstrained and a wide class of constrained problems, we also give reasons for the optimality of regularization methods from a worst-case complexity point of view, within a large class of algorithms that use the same derivative information.

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arXiv2018

Convergence Rate Analysis of a Stochastic Trust Region Method via Submartingales

We propose a novel framework for analyzing convergence rates of stochastic optimization algorithms with adaptive step sizes. This framework is based on analyzing properties of an underlying generic stochastic process, in particular by deriving a bound on the expected stopping time of this process. We utilize this framework to analyze the bounds on expected global convergence rates of a stochastic variant of a traditional trust region method, introduced in \cite{ChenMenickellyScheinberg2014}. While traditional trust region methods rely on exact computations of the gradient, Hessian and values of the objective function, this method assumes that these values are available up to some dynamically adjusted accuracy. Moreover, this accuracy is assumed to hold only with some sufficiently large, but fixed, probability, without any additional restrictions on the variance of the errors. This setting applies, for example, to standard stochastic optimization and machine learning formulations. Improving upon the analysis in \cite{ChenMenickellyScheinberg2014}, we show that the stochastic process defined by the algorithm satisfies the assumptions of our proposed general framework, with the stopping time defined as reaching accuracy $\|\nabla f(x)\|\leq ε$. The resulting bound for this stopping time is $O(ε^{-2})$, under the assumption of sufficiently accurate stochastic gradient, and is the first global complexity bound for a stochastic trust-region method. Finally, we apply the same framework to derive second order complexity bound under some additional assumptions.

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arXiv2018

Improving the Flexibility and Robustness of Model-Based Derivative-Free Optimization Solvers

We present DFO-LS, a software package for derivative-free optimization (DFO) for nonlinear Least-Squares (LS) problems, with optional bound constraints. Inspired by the Gauss-Newton method, DFO-LS constructs simplified linear regression models for the residuals. DFO-LS allows flexible initialization for expensive problems, whereby it can begin making progress from as few as two objective evaluations. Numerical results show DFO-LS can gain reasonable progress on some medium-scale problems with fewer objective evaluations than is needed for one gradient evaluation. DFO-LS has improved robustness to noise, allowing sample averaging, the construction of regression-based models, and multiple restart strategies together with an auto-detection mechanism. Our extensive numerical experimentation shows that restarting the solver when stagnation is detected is a cheap and effective mechanism for achieving robustness, with superior performance over both sampling and regression techniques. We also present our package Py-BOBYQA, a Python implementation of BOBYQA (Powell, 2009), which also implements robustness to noise strategies. Our numerical experiments show that Py-BOBYQA is comparable to or better than existing general DFO solvers for noisy problems. In our comparisons, we introduce a new adaptive measure of accuracy for the data profiles of noisy functions that strikes a balance between measuring the true and the noisy objective improvement.

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arXiv2018

Global rates of convergence for nonconvex optimization on manifolds

We consider the minimization of a cost function $f$ on a manifold $M$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance $\varepsilon$. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of $f$ to the tangent spaces of $M$, both of these algorithms produce points with Riemannian gradient smaller than $\varepsilon$ in $O(1/\varepsilon^2)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian's least eigenvalue is larger than $-\varepsilon$ in $O(1/\varepsilon^3)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy $\varepsilon$ (up to constants) and hence are sharp in that sense. These are the first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of $\mathbb{R}^n$, under simpler assumptions.

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arXiv2017

A Derivative-Free Gauss-Newton Method

We present DFO-GN, a derivative-free version of the Gauss-Newton method for solving nonlinear least-squares problems. As is common in derivative-free optimization, DFO-GN uses interpolation of function values to build a model of the objective, which is then used within a trust-region framework to give a globally-convergent algorithm requiring $O(ε^{-2})$ iterations to reach approximate first-order criticality within tolerance $ε$. This algorithm is a simplification of the method from [H. Zhang, A. R. Conn, and K. Scheinberg, A Derivative-Free Algorithm for Least-Squares Minimization, SIAM J. Optim., 20 (2010), pp. 3555-3576], where we replace quadratic models for each residual with linear models. We demonstrate that DFO-GN performs comparably to the method of Zhang et al. in terms of objective evaluations, as well as having a substantially faster runtime and improved scalability.

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arXiv2017

Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization

We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould and Toint (2010,2011). To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton's method as well as of their linesearch variants. For each method in this class and arbitrary accuracy threshold $ε\in (0,1)$, we exhibit a smooth objective function with bounded range, whose gradient is globally Lipschitz continuous and whose Hessian is $α-$Hölder continuous (for given $α\in [0,1]$), for which the method in question takes at least $\lfloorε^{-(2+α)/(1+α)}\rfloor$ function evaluations to generate a first iterate whose gradient is smaller than $ε$ in norm. Moreover, we also construct another function on which Newton's takes $\lfloorε^{-2}\rfloor$ evaluations, but whose Hessian is Lipschitz continuous on the path of iterates. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Furthermore, for $α=1$, this lower bound is of the same order in $ε$ as the upper bound on the worst-case evaluation complexity of the cubic and other methods in a class of methods proposed in Curtis, Robinson and samadi (2017) or in Royer and Wright (2017), thus implying that these methods have optimal worst-case evaluation complexity within a wider class of second-order methods, and that Newton's method is suboptimal.

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arXiv2017

Improved second-order evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

The unconstrained minimization of a sufficiently smooth objective function $f(x)$ is considered, for which derivatives up to order $p$, $p\geq 2$, are assumed to be available. An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order $p$ and that is guaranteed to find a first- and second-order critical point in at most $O \left(\max\left( ε_1^{-\frac{p+1}{p}}, ε_2^{-\frac{p+1}{p-1}} \right) \right)$ function and derivatives evaluations, where $ε_1$ and $ε_2 >0$ are prescribed first- and second-order optimality tolerances. Our approach extends the method in Birgin et al. (2016) to finding second-order critical points, and establishes the novel complexity bound for second-order criticality under identical problem assumptions as for first-order, namely, that the $p$-th derivative tensor is Lipschitz continuous and that $f(x)$ is bounded from below. The evaluation-complexity bound for second-order criticality improves on all such known existing results.

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arXiv2017

Evaluation complexity bounds for smooth constrained nonlinear optimisation using scaled KKT conditions, high-order models and the criticality measure $χ$

Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of $O(ε^{-3/2})$ proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an $ε$-approximate first-order critical point can be obtained under significantly weaker assumptions. Moreover, the result is generalized to the case where high-order derivatives are used, resulting in a bound of $O(ε^{-(p+1)/p})$ evaluations whenever derivatives of order $p$ are available. It is also shown that the bound of $O(ε_P^{-1/2}ε_D^{-3/2})$ evaluations ($ε_P$ and $ε_D$ being primal and dual accuracy thresholds) suggested by Cartis, Gould and Toint (SINUM, 2015) for the general nonconvex case involving both equality and inequality constraints can be generalized to a bound of $O(ε_P^{-1/p}ε_D^{-(p+1)/p})$ evaluations under similarly weakened assumptions. This paper is variant of a companion report (NTR-11-2015, University of Namur, Belgium) which uses a different first-order criticality measure to obtain the same complexity bounds.

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arXiv2017

Global convergence rate analysis of unconstrained optimization methods based on probabilistic models

We present global convergence rates for a line-search method which is based on random first-order models and directions whose quality is ensured only with certain probability. We show that in terms of the order of the accuracy, the evaluation complexity of such a method is the same as its counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. We particularize and improve these results in the convex and strongly convex case. We also analyze a probabilistic cubic regularization variant that allows approximate probabilistic second-order models and show improved complexity bounds compared to probabilistic first-order methods; again, as a function of the accuracy, the probabilistic cubic regularization bounds are of the same (optimal) order as for the deterministic case.

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arXiv2016

Quantitative recovery conditions for tree-based compressed sensing

As shown in [Blumensath and Davies 2009, Baraniuk et al. 2010], signals whose wavelet coefficients exhibit a rooted tree structure can be recovered using specially-adapted compressed sensing algorithms from just n=O(k) measurements, where k is the sparsity of the signal. Motivated by these results, we introduce a simplified proportional-dimensional asymptotic framework which enables the quantitative evaluation of recovery guarantees for tree-based compressed sensing. In the context of Gaussian matrices, we apply this framework to existing worst-case analysis of the Iterative Tree Projection (ITP) algorithm which makes use of the tree-based Restricted Isometry Property (RIP). Within the same framework, we then obtain quantitative results based on a new method of analysis, recently introduced in [Cartis and Thompson, 2015], which considers the fixed points of the algorithm. By exploiting the realistic average-case assumption that the measurements are statistically independent of the signal, we obtain significant quantitative improvements when compared to the tree-based RIP analysis. Our results have a refreshingly simple interpretation, explicitly determining a bound on the number of measurements that are required as a multiple of the sparsity. For example we prove that exact recovery of binary tree-based signals from noiseless Gaussian measurements is asymptotically guaranteed for ITP with constant stepsize provided n>50k. All our results extend to the more realistic case in which measurements are corrupted by noise.

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arXiv2015

Active-set prediction for interior point methods using controlled perturbations

We propose the use of controlled perturbations to address the challenging question of optimal active-set prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close to the central path of the perturbed problem. We also find that a primal-dual path-following algorithm applied to the perturbed problem is able to accurately predict the optimal active set of the original problem when the duality gap for the perturbed problem is not too small; furthermore, depending on problem conditioning, this prediction can happen sooner than predicting the active set for the perturbed problem or when the original one is solved. Encouraging preliminary numerical experience is reported when comparing activity prediction for the perturbed and unperturbed problem formulations.

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arXiv2014

A new and improved quantitative recovery analysis for iterative hard thresholding algorithms in compressed sensing

We present a new recovery analysis for a standard compressed sensing algorithm, Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2008), which considers the fixed points of the algorithm. In the context of arbitrary measurement matrices, we derive a sufficient condition for convergence of IHT to a fixed point and a necessary condition for the existence of fixed points. These conditions allow us to perform a sparse signal recovery analysis in the deterministic noiseless case by implying that the original sparse signal is the unique fixed point and limit point of IHT, and in the case of Gaussian measurement matrices and noise by generating a bound on the approximation error of the IHT limit as a multiple of the noise level. By generalizing the notion of fixed points, we extend our analysis to the variable stepsize Normalised IHT (N-IHT) (Blumensath and Davies, 2010). For both stepsize schemes, we obtain lower bounds on asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. Exploiting the reasonable average-case assumption that the underlying signal and measurement matrix are independent, comparison with previous results within this framework shows a substantial quantitative improvement.

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arXiv2013

An exact tree projection algorithm for wavelets

We propose a dynamic programming algorithm for projection onto wavelet tree structures. In contrast to other recently proposed algorithms which only give approximate tree projections for a given sparsity, our algorithm is guaranteed to calculate the projection exactly. We also prove that our algorithm has O(Nk) complexity, where N is the signal dimension and k is the sparsity of the tree approximation.

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