Graph convolution networks (GCNs) have become effective models for graph classification. Similar to many deep networks, GCNs are vulnerable to adversarial attacks on graph topology and node attributes. Recently, a number of effective attack and defense algorithms have been designed, but no certificate of robustness has been developed for GCN-based graph classification under topological perturbations with both local and global budgets. In this paper, we propose the first certificate for this problem. Our method is based on Lagrange dualization and convex envelope, which result in tight approximation bounds that are efficiently computable by dynamic programming. When used in conjunction with robust training, it allows an increased number of graphs to be certified as robust.

Aligning large language models (LLMs) with human preferences is critical for real-world deployment, yet existing methods like RLHF face computational and stability challenges. While DPO establishes an offline paradigm with single hyperparameter $\beta$, subsequent methods like SimPO reintroduce complexity through dual parameters ($\beta$, $\gamma$). We propose {ReLU-based Preference Optimization (RePO)}, a streamlined algorithm that eliminates $\beta$ via two advances: (1) retaining SimPO's reference-free margins but removing $\beta$ through gradient analysis, and (2) adopting a ReLU-based max-margin loss that naturally filters trivial pairs. Theoretically, RePO is characterized as SimPO's limiting case ($\beta \to \infty$), where the logistic weighting collapses to binary thresholding, forming a convex envelope of the 0-1 loss. Empirical results on AlpacaEval 2 and Arena-Hard show that RePO outperforms DPO and SimPO across multiple base models, requiring only one hyperparameter to tune.

The non-convex nature of trained neural networks has created significant obstacles in their incorporation into optimization models. Considering the wide array of applications that this embedding has, the optimization and deep learning communities have dedicated significant efforts to the convexification of trained neural networks. Many approaches to date have considered obtaining convex relaxations for each non-linear activation in isolation, which poses limitations in the tightness of the relaxations. Anderson et al. (2020) strengthened these relaxations and provided a framework to obtain the convex hull of the graph of a piecewise linear convex activation composed with an affine function; this effectively convexifies activations such as the ReLU together with the affine transformation that precedes it. In this article, we contribute to this line of work by developing a recursive formula that yields a tight convexification for the composition of an activation with an affine function for a wide scope of activation functions, namely, convex or ``S-shaped". Our approach can be used to efficiently compute separating hyperplanes or determine that none exists in various settings, including non-polyhedral cases. We provide computational experiments to test the empirical benefits of these convex approximations.

We study the problem of learning a tensor from a set of linear measurements. A prominent methodology for this problem is based on a generalization of trace norm regularization, which has been used extensively for learning low rank matrices, to the tensor setting. In this paper, we highlight some limitations of this approach and propose an alternative convex relaxation on the Euclidean ball. We then describe a technique to solve the associated regularization problem, which builds upon the alternating direction method of multipliers. Experiments on one synthetic dataset and two real datasets indicate that the proposed method improves significantly over tensor trace norm regularization in terms of estimation error, while remaining computationally tractable.

Oberman gave a stochastic control formulation of the problem of estimating the convex envelope of a non-convex function. Based on this, we develop a reinforcement learning scheme to approximate the convex envelope, using a variant of Q-learning for controlled optimal stopping. It shows very promising results on a standard library of test problems.

This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the L1-norm. In this paper, we investigate more general set-functions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nondecreasing submodular set-functions, the corresponding convex envelope can be obtained from its Lovasz extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or high-dimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rank-statistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as non-factorial priors for supervised learning.

Systems for relative localization in multi-robot systems based on ultra-wideband (UWB) ranging have recently emerged as robust solutions for GNSS-denied environments. Scalability remains one of the key challenges, particularly in ad-hoc deployments. Recent solutions include dynamic allocation of active and passive localization modes for different robots or nodes in the system. With larger-scale systems becoming more distributed, key research questions arise in the areas of security and trustability of such localization systems. This paper studies the potential integration of collaborative-decision making processes with distributed ledger technologies. Specifically, we investigate the design and implementation of a methodology for running an UWB role allocation algorithm within smart contracts in a blockchain. In previous works, we have separately studied the integration of ROS2 with the Hyperledger Fabric blockchain, and introduced a new algorithm for scalable UWB-based localization. In this paper, we extend these works by (i) running experiments with larger number of mobile robots switching between different spatial configurations and (ii) integrating the dynamic UWB role allocation algorithm into Fabric smart contracts for distributed decision-making in a system of multiple mobile robots. This enables us to deliver the same functionality within a secure and trustable process, with enhanced identity and data access management. Our results show the effectiveness of the UWB role allocation for continuously varying spatial formations of six autonomous mobile robots, while demonstrating a low impact on latency and computational resources of adding the blockchain layer that does not affect the localization process.

This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the L1-norm. In this paper, we investigate more general set-functions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nondecreasing submodular set-functions, the corresponding convex envelope can be obtained from its Lovasz extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or high-dimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rank-statistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as non-factorial priors for supervised learning. Papers published at the Neural Information Processing Systems Conference.

Dropout and its extensions (eg. DropBlock and DropConnect) are popular heuristics for training neural networks, which have been shown to improve generalization performance in practice. However, a theoretical understanding of their optimization and regularization properties remains elusive. Recent work shows that in the case of single hidden-layer linear networks, Dropout is a stochastic gradient descent method for minimizing a regularized loss, and that the regularizer induces solutions that are low-rank and balanced. In this work we show that for single hidden-layer linear networks, DropBlock induces spectral k-support norm regularization, and promotes solutions that are low-rank and have factors with equal norm. We also show that the global minimizer for DropBlock can be computed in closed form, and that DropConnect is equivalent to Dropout. We then show that some of these results can be extended to a general class of Dropout-strategies, and, with some assumptions, to deep non-linear networks when Dropout is applied to the last layer. We verify our theoretical claims and assumptions experimentally with commonly used network architectures.

We give a formal and complete characterization of the explicit regularizer induced by dropout in deep linear networks with squared loss. We show that (a) the explicit regularizer is composed of an $\ell_2$-path regularizer and other terms that are also re-scaling invariant, (b) the convex envelope of the induced regularizer is the squared nuclear norm of the network map, and (c) for a sufficiently large dropout rate, we characterize the global optima of the dropout objective. We validate our theoretical findings with empirical results.