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In this paper we present a fully distributed, asynchronous, and general purpose optimization algorithm for Consensus Simultaneous Localization and Mapping (CSLAM). Multi-robot teams require that agents have timely and accurate solutions to their state as well as the states of the other robots in the team. To optimize this solution we develop a CSLAM back-end based on Consensus ADMM called MESA (Manifold, Edge-based, Separable ADMM). MESA is fully distributed to tolerate failures of individual robots, asynchronous to tolerate communication delays and outages, and general purpose to handle any CSLAM problem formulation. We demonstrate that MESA exhibits superior convergence rates and accuracy compare to existing state-of-the art CSLAM back-end optimizers.

Confidence bounds are an essential tool for rigorously quantifying the uncertainty of predictions. In this capacity, they can inform the exploration-exploitation trade-off and form a core component in many sequential learning and decision-making algorithms. Tighter confidence bounds give rise to algorithms with better empirical performance and better performance guarantees. In this work, we use martingale tail bounds and finite-dimensional reformulations of infinite-dimensional convex programs to establish new confidence bounds for sequential kernel regression. We prove that our new confidence bounds are always tighter than existing ones in this setting. We apply our confidence bounds to the kernel bandit problem, where future actions depend on the previous history. When our confidence bounds replace existing ones, the KernelUCB (GP-UCB) algorithm has better empirical performance, a matching worst-case performance guarantee and comparable computational cost. Our new confidence bounds can be used as a generic tool to design improved algorithms for other kernelised learning and decision-making problems.

In this paper, we study the problem of watermarking large language models (LLMs). We consider the trade-off between model distortion and detection ability and formulate it as a constrained optimization problem based on the green-red algorithm of Kirchenbauer et al. (2023a). We show that the optimal solution to the optimization problem enjoys a nice analytical property which provides a better understanding and inspires the algorithm design for the watermarking process. We develop an online dual gradient ascent watermarking algorithm in light of this optimization formulation and prove its asymptotic Pareto optimality between model distortion and detection ability. Such a result guarantees an averaged increased green list probability and henceforth detection ability explicitly (in contrast to previous results). Moreover, we provide a systematic discussion on the choice of the model distortion metrics for the watermarking problem. We justify our choice of KL divergence and present issues with the existing criteria of ``distortion-free'' and perplexity. Finally, we empirically evaluate our algorithms on extensive datasets against benchmark algorithms.

In this paper, we concentrate on decentralized optimization problems with nonconvex and nonsmooth objective functions, especially on the decentralized training of nonsmooth neural networks. We introduce a unified framework, named DSM, to analyze the global convergence of decentralized stochastic subgradient methods. We prove the global convergence of our proposed framework under mild conditions, by establishing that the generated sequence asymptotically approximates the trajectories of its associated differential inclusion. Furthermore, we establish that our proposed framework encompasses a wide range of existing efficient decentralized subgradient methods, including decentralized stochastic subgradient descent (DSGD), DSGD with gradient-tracking technique (DSGD-T), and DSGD with momentum (DSGDm). In addition, we introduce SignSGD employing the sign map to regularize the update directions in DSGDm, and show it is enclosed in our proposed framework. Consequently, our convergence results establish, for the first time, global convergence of these methods when applied to nonsmooth nonconvex objectives. Preliminary numerical experiments demonstrate that our proposed framework yields highly efficient decentralized subgradient methods with convergence guarantees in the training of nonsmooth neural networks.

In this paper, we present a novel approach for detecting the discontinuity interfaces of a discontinuous function. This approach leverages Graph-Informed Neural Networks (GINNs) and sparse grids to address discontinuity detection also in domains of dimension larger than 3. GINNs, trained to identify troubled points on sparse grids, exploit graph structures built on the grids to achieve efficient and accurate discontinuity detection performances. We also introduce a recursive algorithm for general sparse grid-based detectors, characterized by convergence properties and easy applicability. Numerical experiments on functions with dimensions n = 2 and n = 4 demonstrate the efficiency and robust generalization of GINNs in detecting discontinuity interfaces. Notably, the trained GINNs offer portability and versatility, allowing integration into various algorithms and sharing among users.

In this paper, we study the individual preference (IP) stability, which is an notion capturing individual fairness and stability in clustering. Within this setting, a clustering is $\alpha$-IP stable when each data point's average distance to its cluster is no more than $\alpha$ times its average distance to any other cluster. In this paper, we study the natural local search algorithm for IP stable clustering. Our analysis confirms a $O(\log n)$-IP stability guarantee for this algorithm, where $n$ denotes the number of points in the input. Furthermore, by refining the local search approach, we show it runs in an almost linear time, $\tilde{O}(nk)$.

In this note we highlight some connections of UMAP to the basic principles of Information Geometry. Originally, UMAP was derived from Category Theory observations. However, we posit that it also has a natural geometric interpretation.

In this paper, we propose two deep joint source and channel coding (DJSCC) structures with attention modules for the multi-input multi-output (MIMO) channel, including a serial structure and a parallel structure. With singular value decomposition (SVD)-based precoding scheme, the MIMO channel can be decomposed into various sub-channels, and the feature outputs will experience sub-channels with different channel qualities. In the serial structure, one single network is used at both the transmitter and the receiver to jointly process data streams of all MIMO subchannels, while data steams of different MIMO subchannels are processed independently via multiple sub-networks in the parallel structure. The attention modules in both serial and parallel architectures enable the system to adapt to varying channel qualities and adjust the quantity of information outputs in accordance with the channel qualities. Experimental results demonstrate the proposed DJSCC structures have improved image transmission performance, and reveal the phenomenon via non-parameter entropy estimation that the learned DJSCC transceivers tend to transmit more information over better sub-channels.

In this paper, we propose a complexity measure for exchangeable graphs by considering the graph-generating mechanism. Exchangeability for graphs implies distributional invariance under node permutations, making it a suitable default model for a wide range of graph data. For this well-studied class of graphs, we quantify complexity using graphon entropy. Graphon entropy is a graph property, meaning it is invariant under graph isomorphisms. Therefore, we focus on estimating the entropy of the generating mechanism for a graph realization, rather than selecting a specific graph feature. This approach allows us to consider the global properties of a graph, capturing its important graph-theoretic and topological characteristics, such as sparsity, symmetry, and connectedness. We introduce a consistent graphon entropy estimator that achieves the nonparametric rate for any arbitrary exchangeable graph with a smooth graphon representative. Additionally, we develop tailored entropy estimators for situations where more information about the underlying graphon is available, specifically for widely studied random graph models such as Erd\H{o}s-R\'enyi, Configuration Model and Stochastic Block Model. We determine their large-sample properties by providing a Central Limit Theorem for the first two, and a convergence rate for the third model. We also conduct a simulation study to illustrate our theoretical findings and demonstrate the connection between graphon entropy and graph structure. Finally, we investigate the role of our entropy estimator as a complexity measure for characterizing real-world graphs.

In this paper, we propose a conceptually simple and geometrically interpretable objective function, i.e. additive margin Softmax (AM-Softmax), for deep face verification. In general, the face verification task can be viewed as a metric learning problem, so learning large-margin face features whose intra-class variation is small and inter-class difference is large is of great importance in order to achieve good performance. Recently, Large-margin Softmax and Angular Softmax have been proposed to incorporate the angular margin in a multiplicative manner. In this work, we introduce a novel additive angular margin for the Softmax loss, which is intuitively appealing and more interpretable than the existing works. We also emphasize and discuss the importance of feature normalization in the paper. Most importantly, our experiments on LFW BLUFR and MegaFace show that our additive margin softmax loss consistently performs better than the current state-of-the-art methods using the same network architecture and training dataset. Our code has also been made available at //github.com/happynear/AMSoftmax