Results 1 - 10
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18
Beating the adaptive bandit with high probability
, 2009
"... We provide a principled way of proving Õ( √ T) high-probability guarantees for partial-information (bandit) problems over convex decision sets. First, we prove a regret guarantee for the full-information problem in terms of “local ” norms, both for entropy and self-concordant barrier regularization, ..."
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Cited by 16 (5 self)
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We provide a principled way of proving Õ( √ T) high-probability guarantees for partial-information (bandit) problems over convex decision sets. First, we prove a regret guarantee for the full-information problem in terms of “local ” norms, both for entropy and self-concordant barrier regularization, unifying these methods. Given one of such algorithms as a black-box, we can convert a bandit problem into a full-information problem using a sampling scheme. The main result states that a high-probability Õ ( √ T) bound holds whenever the black-box, the sampling scheme, and the estimates of missing information satisfy a number of conditions, which are relatively easy to check. At the heart of the method is a construction of linear upper bounds on confidence intervals. As applications of the main result, we provide the first known efficient algorithm for the sphere with an Õ( √ T) high-probability bound. We also derive the result for the n-simplex, improving the O ( √ nT log(nT)) bound of Auer et al [3] by replacing the log T term with log log T and closing the gap to the lower bound of Ω ( √ nT). The guarantees we obtain hold for adaptive adversaries (unlike the in-expectation results of [1]) and the algorithms are efficient, given that the linear upper bounds on confidence can be computed. 1
Interior-Point Methods for Full-Information and Bandit Online Learning
"... Abstract—We study the problem of predicting individual sequences with linear loss with full and partial (or bandit) feedback. Our main contribution is the first efficient algorithm for the problem of online linear optimization in the bandit setting which achieves the optimal regret. In addition, for ..."
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Cited by 9 (4 self)
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Abstract—We study the problem of predicting individual sequences with linear loss with full and partial (or bandit) feedback. Our main contribution is the first efficient algorithm for the problem of online linear optimization in the bandit setting which achieves the optimal regret. In addition, for the full-information setting, we give a novel regret minimization algorithm. These results are made possible by the introduction of interior-point methods for convex optimization to online learning. Index Terms—Bandit feedback, interior-point methods, online convex optimization, online learning. I.
Composite Self-Concordant Minimization
"... We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function endowed with a computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on ..."
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Cited by 7 (5 self)
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We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function endowed with a computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting large-scale applications and demonstrate them numerically on both synthetic and real data.
Online learning with predictable sequences.
- In COLT,
, 2013
"... Abstract We present methods for online linear optimization that take advantage of benign (as opposed to worst-case) sequences. Specifically if the sequence encountered by the learner is described well by a known "predictable process", the algorithms presented enjoy tighter bounds as compa ..."
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Cited by 6 (1 self)
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Abstract We present methods for online linear optimization that take advantage of benign (as opposed to worst-case) sequences. Specifically if the sequence encountered by the learner is described well by a known "predictable process", the algorithms presented enjoy tighter bounds as compared to the typical worst case bounds. Additionally, the methods achieve the usual worst-case regret bounds if the sequence is not benign. Our approach can be seen as a way of adding prior knowledge about the sequence within the paradigm of online learning. The setting is shown to encompass partial and side information. Variance and path-length bounds Hazan and Kale We further extend our methods to include competing with a set of possible predictable processes (models), that is "learning" the predictable process itself concurrently with using it to obtain better regret guarantees. We show that such model selection is possible under various assumptions on the available feedback.
The homogeneous interior-point algorithm: Nonsymmetric cones, . . .
, 2013
"... The overall topic of this thesis is convex conic optimization, a sub-field of mathe-matical optimization that attacks optimization problem with a certain geometric structure. These problems allow for modelling of an extremely wide range of real-world problems, but the availability of solution algori ..."
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Cited by 4 (0 self)
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The overall topic of this thesis is convex conic optimization, a sub-field of mathe-matical optimization that attacks optimization problem with a certain geometric structure. These problems allow for modelling of an extremely wide range of real-world problems, but the availability of solution algorithms for these prob-lems is still limited. The goal of this thesis is to investigate and shed light on two computational aspects of homogeneous interior-point algorithms for convex conic optimization: The first part studies the possibility of devising a homogeneous interior-point method aimed at solving problems involving constraints that require nonsym-metric cones in their formulation. The second part studies the possibility of warmstarting the homogeneous interior-point algorithm for conic problems. The main outcome of the first part is the introduction of a completely new homogeneous interior-point algorithm designed to solve nonsymmetric convex conic optimization problems. The algorithm is presented in detail and then ana-lyzed. We prove its convergence and complexity. From a theoretical viewpoint,
Improved Regret Guarantees for Online Smooth Convex Optimization with Bandit Feedback
"... The study of online convex optimization in the bandit setting was initiated by Kleinberg (2004) and Flaxman et al. (2005). Such a setting models a decision maker that has to make decisions in the face of adversarially chosen convex loss functions. Moreover, the only information the decision maker re ..."
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Cited by 3 (0 self)
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The study of online convex optimization in the bandit setting was initiated by Kleinberg (2004) and Flaxman et al. (2005). Such a setting models a decision maker that has to make decisions in the face of adversarially chosen convex loss functions. Moreover, the only information the decision maker receives are the losses. The identities of the loss functions themselves are not revealed. In this setting, we reduce the gap between the best known lower and upper bounds for the class of smooth convex functions, i.e. convex functions with a Lipschitz continuous gradient. Building upon existing work on selfconcordant regularizers and one-point gradient estimation, we give the first algorithm whose expected regret is O(T 2/3), ignoring constant and logarithmic factors. 1
A primal-dual algorithmic framework for constrained convex minimization
, 2014
"... Abstract We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective ..."
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Cited by 3 (2 self)
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Abstract We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov's excessive gap technique in a structured fashion and unifies it with smoothing and primal-dual methods. For instance, through the choices of a dual smoothing strategy and a center point, our framework subsumes decomposition algorithms, augmented Lagrangian as well as the alternating direction method-of-multipliers methods as its special cases, and provides optimal convergence rates on the primal objective residual as well as the primal feasibility gap of the iterates for all.
An inexact proximal path-following algorithm for constrained convex minimization
, 2014
"... Many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constra ..."
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Cited by 3 (3 self)
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Many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constraint set affords a self-concordant barrier. We provide a new joint treatment of proximal and self-concordant barrier concepts and illustrate that such problems can be efficiently solved, without the need of lifting the problem dimensions, as in disciplined convex optimization approach. We propose an inexact path-following algorithmic framework and theoretically charac-terize the worst-case analytical complexity of this framework when the proximal subproblems are solved inexactly. To show the merits of our framework, we apply its instances to both synthetic and real-world applications, where it shows advantages over standard interior point methods. As a by-product, we describe how our framework can obtain points on the Pareto frontier of regularized problems with self-concordant objectives in a tuning free fashion.
CONSTRAINED POLYNOMIAL OPTIMIZATION PROBLEMS WITH NONCOMMUTING VARIABLES
"... Abstract. In this paper we study constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball. Our three main results are as follows: (1) an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition; (2) ( ..."
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Cited by 1 (0 self)
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Abstract. In this paper we study constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball. Our three main results are as follows: (1) an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition; (2) (eigenvalue) optima for nc polynomials can be computed using a single semidefinite program (SDP) – this sharply contrasts the commutative case where sequences of SDPs are needed; (3) the dual solution to this “single ” SDP can be exploited to extract eigenvalue optimizers with an algorithm based on two ingredients: • solution to a truncated nc moment problem via flat extensions; • Gelfand-Naimark-Segal (GNS) construction. The implementation of these procedures in our computer algebra system NCSOStools is presented and several examples pertaining to matrix inequalities are given to illustrate our results. 1.
