We study the problem of finding an optimal kernel from a prescribed convex set of kernels K for learning a real-valued function by regularization. We establish for a wide variety of regularization functionals that this leads to a convex optimization problem and, for square loss regularization, we characterize the solution of this problem. We show that, although K may be an uncountable set, the optimal kernel is always obtained as a convex combination of at most m+2 basic kernels, where m is the number of data examples. In particular, our results apply to learning the optimal radial kernel or the optimal dot product kernel. 1.

A robot is deployed by a Web search engine in order to maintain the currency of its data base of Web pages. This paper studies robot scheduling policies that minimize the fractions r i of time pages spend out-of-date, assuming independent Poisson page-change processes, and a general distribution for the page access time X . We show that, if X is decreased in the increasing-convex ordering sense, then r i is decreased for all i under any scheduling policy, and that, in order to minimize expected total obsolescence time of any page, the accesses to that page should be as evenly spaced in time as possible. We then investigate the problem of scheduling to minimize the cost function P c i r i ; where the c i are given weights proportional to the page-change rates ¯ i . We give a tight bound on the performance of such a policy and prove that the optimal frequency at which the robot should access page i is proportional to ln(h i ) \Gamma1 , where h i := Ee \Gamma¯ i X : Note that this...

In this paper, we develop a filtering theory for deterministic traffic regulation and service guarantees under the (min; +)-algebra. We show that traffic regulators that generate f-upper constrained outputs can be implemented optimally by a linear time invariant filter with the impulse response f under the (min; +)-algebra, where f is the subadditive closure defined in the paper. Analogous to the classical filtering theory, there is an associate calculus, including feedback, concatenation, "filter bank summation" and performance bounds. The calculus is also applicable to the recently developed concept of service curves that can be used for deriving deterministic service guarantees. Our filtering approach not only yields easier proofs for more general results than those in the literature, but also allows us to design traffic regulators via systematic methods such as concatenation, filter bank summation, linear system realization, and FIR-IIR realization. We illustrate the use of ...

This paper develops a variance reduction technique for Monte Carlo simulations of path-dependent options driven by high-dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables which maximizes the product of probability and payoff---the most important path. The directions used for stratified sampling are optimal for a quadratic approximation to the integrand or payoff function. Indeed, under differentiability assumptions our importance sampling method eliminates variability due to the linear part of the payoff function, and stratification eliminates much of the variability due to the quadratic part of the payoff. The two parts of the method are linked because the asymptotically optimal drift vector frequently provides a particularly effective direction for stratification. We illustrate the use of the method with path-dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of their payoffs. KEY WORDS: Monte Carlo methods, variance reduction, large deviations, Laplace principle 1. INTRODUCTION This paper develops a variance reduction technique for Monte Carlo simulations driven by high-dimensional Gaussian vectors, with particular emphasis on the pricing of pathdependent options. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to...

Interference avoidance has been shown to reduce total square correlation (TSC) for given ensembles of user signature waveforms (codewords) in a synchronous CDMA system. In all experiments we have conducted, sequential application of interference avoidance produces an optimal codeword set when starting from randomly chosen initial codewords. Here we provide the rst formal proof of convergence to optimal codeword ensembles for greedy interference avoidance algorithms augmented by a technique called \class warfare" whereby users which reside in more heavily loaded areas of the signal space purposely interfere with (attack) the reception of users in less crowded areas. Coordination of deliberate interference by a complete class of aggrieved user is also sometimes necessary. Such \attacks" and subsequent codeword adjustment by attacked users are shown to strictly decrease TSC. Along the way we also show, using linear algebra and a variant of stochastic ordering, equivalence between minimiz...

Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interior-point methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap. We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.

Abstract—Motivated by the emergence of programmable radios, we seek to understand a new class of communication system where pairs of transmitters and receivers can adapt their modulation/demodulation method in the presence of interference to achieve better performance. Using signal to interference ratio as a metric and a general signal space approach, we present a class of iterative distributed algorithms for synchronous systems which results in an ensemble of optimal waveforms for multiple users connected to a common receiver (or colocated independent receivers). That is, the waveform ensemble meets the Welch Bound with equality and, therefore, achieves minimum average interference over the ensemble of signature waveforms. We derive fixed points for a number of scenarios, provide examples, look briefly at ensemble stability under user addition and deletion as well as provide a simplistic comparison to synchronous code-division multiple-access. We close with suggestions for future work. Index Terms—Adaptive modulation, code-division multiple-access systems, codeword optimization, interference avoidance, multiuser

In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A secondorder analysis is also given. Second-order necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed. Key words: Semidefinite programming, cone constraints, convex programming, duality, second-order optimality conditions, tangent cones, optimal value function, sensitivity analysis. AMS subject classification: 90C25, 90C30, 90C31 1 Introduction In this paper we consider the following optimization problem (P ) min x2IR m f(x) subject to G(x) 0: Here G : IR m ! S n is a mapping from IR m into the space S n of n \Theta n symmetric matrices and, for A; B 2 S n , the notation A B (the notation A B) means that the matrix A \Gamma B is positive semidefinite (negative semidefinite). Consider the cone K ae S n of positive semidefinite matrices. Then the co...

Abstract—We consider the capacity of a narrowband point to point communication system employing multiple-element antenna arrays at both the transmitter and the receiver with covariance feedback. Under covariance feedback the receiver is assumed to have perfect Channel State Information (CSI) while at the transmitter the channel matrix is modeled as consisting of zero mean complex jointly Gaussian random variables with known covariances. Specifically we assume a channel matrix with i.i.d. rows and correlated columns, a common model for downlink transmission. We determine the optimal transmit precoding strategy to maximize the Shannon capacity of such a system. We also derive closed form necessary and sufficient conditions on the spatial covariance for when the maximum capacity is achieved by beamforming. The conditions for optimality of beamforming agree with the notion of waterfilling over multiple degrees of freedom. I.

The burstiness of compressed video complicates the provisioning of network resources for emerging multimedia services. For stored video applications, the server can smooth the variable-bit-rate stream by prefetching frames into the client playback buffer in advance of each burst. Drawing on a priori knowledge of the frame lengths and client buffer size, such bandwidth smoothing techniques can minimize the peak and variability of the rate requirements while avoiding underflow and overflow of the playback buffer. However, in an internetworking environment, a single service provider typically does not control the entire path from the stored-video server to the client buffer. To develop efficient techniques for transmitting variable-bit-rate video across a portion of the route, we investigate bandwidth smoothing across a tandem of nodes, which may or may not include the server and client sites. We show that it is possible to compute an optimal transmission schedule for the tandem system by...