Session: Recent Advances in Conic Optimization and Machine Learning
Chair: Godai Azuma
Cluster: Conic and Semidefinite Optimization
Talk 1: Procrustean Differential Privacy: A Parameter-Scalable Method for Privacy-Preserving Collaborative Learning
Speaker: Keiyu Nosaka
Abstract: Privacy-preserving collaborative learning enables multiple parties to jointly train machine learning models without directly sharing sensitive data. While approaches such as Federated Learning and Homomorphic Encryption offer robust privacy guarantees, they often incur significant computational and communication overhead as model complexity increases. In contrast, multiplicative perturbation techniques promise enhanced efficiency; however, they are typically hampered by increased privacy risks from collusion or by a reliance on extensive deep learning-based training to achieve satisfactory performance. In this work, we introduce a novel framework that bridges these extremes by integrating the analytic Gaussian mechanism of differential privacy with the Generalized Orthogonal Procrustes Problem. Our method delivers adjustable privacy–performance trade-offs through tunable differential privacy parameters, allowing practitioners to balance protection and efficiency according to specific application needs. We substantiate our approach with theoretical guarantees and numerical analyses that evaluate its performance across varying privacy levels, data dimensions, and numbers of collaborating parties.
Talk 2: Facial Structure of Copositive and Completely Positive Cones over a Second-Order Cone
Speaker: Mitsuhiro Nishijima
Abstract: A copositive cone over a second-order cone is the closed convex cone of real symmetric matrices whose associated quadratic forms are nonnegative over the given second-order cone. In this talk, we classify the faces of those copositive cones and their duals (i.e., completely positive cones over a second-order cone), and investigate their dimension and exposedness properties. Then we compute two parameters related to chains of faces of both cones. At the end, we discuss some possible extensions of the results with a view toward analyzing the facial structure of general copositive and completely positive cones.
Talk 3: Exact Semidefinite Relaxations for Safety Verification of Neural Network
Speaker: Godai Azuma
Abstract: We study the accuracy of DeepSDP which is proposed as a semidefinite programming (SDP) based method to measure the safety and the robustness of given neural networks by guaranteeing the bounds of their outputs. The dual problem of the DeepSDP is in fact an SDP relaxation of quadratic constraints representing ReLU activation functions. In this talk, we investigate the exactness of the DeepSDP by using exactness conditions for the general SDP relaxation so that the estimated robustness is improved on. We also discuss our assumptions and some results on the accuracy.
