More Like this

1. Minimum-Time Charging of Energy Storage in Microgrids via Approximate Conic Relaxation

   https://doi.org/10.23919/ECC51009.2020.9143992  

   Guthrie, James; Mallada, Enrique (May 2020, 2020 European Control Conference)

   null (Ed.)

   We study the problem of maximizing energy transfer to a load in a DC microgrid while respecting constraints on bus voltages and currents, and accounting for the impact of neighboring constant power loads. Both the objective and dynamics give rise to indefinite quadratic terms, resulting in a non-convex optimization problem. Through change of variables and relaxations we develop a closely related second-order cone program. The problem retains the same feasible set as the original problem but utilizes a linear approximation of the non-convex objective. We demonstrate how this can be used to design approximately optimal charging profiles for periodic pulsed loads in real time. 

   more » « less
2. Optimal Sets and Solution Paths of ReLU Networks

   Mishkin, Aaron; Pilanci, Mert (January 2023, International Conference on Machine Learning)

   We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the non-convex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the non-convex training objective. Since all stationary points of the ReLU training problem can be represented as optima of sub-sampled convex programs, our work provides a general expression for all critical points of the non-convex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks. 

   more » « less
3. Optimal sets and solution paths of ReLU networks

   Mishkin, Aaron (July 2023, ICML'23: Proceedings of the 40th International Conference on Machine Learning)

   We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the non-convex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the nonconvex training objective. Since all stationary points of the ReLU training problem can be represented as optima of sub-sampled convex programs, our work provides a general expression for all critical points of the non-convex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks. 

   more » « less
4. High-Dimensional Robust Mean Estimation via Gradient Descent

   Cheng, Yu; Diakonikolas, Ilias; Ge, Rong; Soltanolkotabi, Mahdi (July 2020, International Conference on Machine Learning 2020)

   We study the problem of high-dimensional robust mean estimation in the presence of a constant fraction of adversarial outliers. A recent line of work has provided sophisticated polynomialtime algorithms for this problem with dimension-independent error guarantees for a range of natural distribution families. In this work, we show that a natural non-convex formulation of the problem can be solved directly by gradient descent. Our approach leverages a novel structural lemma, roughly showing that any approximate stationary point of our non-convex objective gives a near-optimal solution to the underlying robust estimation task. Our work establishes an intriguing connection between algorithmic high-dimensional robust statistics and non-convex optimization, which may have broader applications to other robust estimation tasks. 

   more » « less
5. High-Dimensional Robust Mean Estimation via Gradient Descent

   Cheng, Yu; Diakonikolas, Ilias; Ge, Rong; Soltanolkotabi, Mahdi (January 2020, International Conference on Machine Learning)

   We study the problem of high-dimensional robust mean estimation in the presence of a constant fraction of adversarial outliers. A recent line of work has provided sophisticated polynomial-time algorithms for this problem with dimension-independent error guarantees for a range of natural distribution families. In this work, we show that a natural non-convex formulation of the problem can be solved directly by gradient descent. Our approach leverages a novel structural lemma, roughly showing that any approximate stationary point of our non-convex objective gives a near-optimal solution to the underlying robust estimation task. Our work establishes an intriguing connection between algorithmic high-dimensional robust statistics and non-convex optimization, which may have broader applications to other robust estimation tasks. 

   more » « less