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- W2033166687 abstract "Random convex programs are convex optimization problems that are robust with respect to a finite number of randomly sampled instances of an uncertain variable $delta$. This paper studies random convex programs in which there is uncertainty in the objective function. Specifically, let $L(x,delta)$ be a loss function that is convex in $x$, the optimization variable, while it has an arbitrary dependence on the random variable $delta$ representing uncertainty in the optimization problem. After sampling $N$ instances $delta^{(1)}, delta^{(2)}, ldots, delta^{(N)}$ of the random variable $delta$, the random convex program can be written as follows: $min_x max_i L(x,delta^{(i)})$. The fundamental feature of this program is that its value $L^ast_N = max_i L(x^ast_N,delta^{(i)})$, where $x^ast_N$ is the solution, remains guaranteed when $x^ast_N$ is applied to the vast majority of the other unseen instances of $delta$; that is, $L(x^ast_N,delta) leq L^ast_N$ holds with high probability with respect to the uncertain variable $delta$. This generalization property has justified a systematic and rigorous use of randomization in robust optimization. In this paper, we introduce $L_1$-regularization in random convex programs and show that $L_1$-regularization boosts the above generalization property so that generalization is achieved with significantly fewer samples than in the standard convex program given above. Explicit bounds are derived that allow a rigorous and easy implementation of the method." @default.
- W2033166687 created "2016-06-24" @default.
- W2033166687 creator A5041504448 @default.
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- W2033166687 date "2013-01-01" @default.
- W2033166687 modified "2023-09-28" @default.
- W2033166687 title "Random Convex Programs with $L_1$-Regularization: Sparsity and Generalization" @default.
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- W2033166687 doi "https://doi.org/10.1137/110856204" @default.
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