We study sampling problems associated with potentials that lack smoothness. The potentials can be either convex or non-convex. Departing from the standard smooth setting, the potentials are only assumed to be weakly smooth or non-smooth, or the summation of multiple such functions. We develop a sampling algorithm that resembles proximal algorithms in optimization for this challenging sampling task. Our algorithm is based on a special case of Gibbs sampling known as the alternating sampling framework (ASF). The key contribution of this work is a practical realization of the ASF based on rejection sampling for both non-convex and convex potentials that are not necessarily smooth. In almost all the cases of sampling considered in this work, our proximal sampling algorithm achieves better complexity than all existing methods.

We present Asynchronous Stochastic Parallel Pose Graph Optimization (ASAPP), the first asynchronous algorithm for distributed pose graph optimization (PGO) in multi-robot simultaneous localization and mapping. By enabling robots to optimize their local trajectory estimates without synchronization, ASAPP offers resiliency against communication delays and alleviates the need to wait for stragglers in the network. Furthermore, ASAPP can be applied on the rank-restricted relaxations of PGO, a crucial class of non-convex Riemannian optimization problems that underlies recent breakthroughs on globally optimal PGO. Under bounded delay, we establish the global first-order convergence of ASAPP using a sufficiently small stepsize. The derived stepsize depends on the worst-case delay and inherent problem sparsity, and furthermore matches known result for synchronous algorithms when there is no delay. Numerical evaluations on simulated and real-world datasets demonstrate favorable performance compared to state-of-the-art synchronous approach, and show ASAPP's resilience against a wide range of delays in practice.

Recently, a new class of non-convex optimization problems motivated by the statistical problem of learning an acyclic directed graphical model from data has attracted significant interest. While existing work uses standard first-order optimization schemes to solve this problem, proving the global optimality of such approaches has proven elusive. The difficulty lies in the fact that unlike other non-convex problems in the literature, this problem is not "benign", and possesses multiple spurious solutions that standard approaches can easily get trapped in. In this paper, we prove that a simple path-following optimization scheme globally converges to the global minimum of the population loss in the bivariate setting.

In this paper, we consider the problem of learning safe policies for probabilistic-constrained reinforcement learning (RL). Specifically, a safe policy or controller is one that, with high probability, maintains the trajectory of the agent in a given safe set. We establish a connection between this probabilistic-constrained setting and the cumulative-constrained formulation that is frequently explored in the existing literature. We provide theoretical bounds elucidating that the probabilistic-constrained setting offers a better trade-off in terms of optimality and safety (constraint satisfaction). The challenge encountered when dealing with the probabilistic constraints, as explored in this work, arises from the absence of explicit expressions for their gradients. Our prior work provides such an explicit gradient expression for probabilistic constraints which we term Safe Policy Gradient-REINFORCE (SPG-REINFORCE). In this work, we provide an improved gradient SPG-Actor-Critic that leads to a lower variance than SPG-REINFORCE, which is substantiated by our theoretical results. A noteworthy aspect of both SPGs is their inherent algorithm independence, rendering them versatile for application across a range of policy-based algorithms. Furthermore, we propose a Safe Primal-Dual algorithm that can leverage both SPGs to learn safe policies. It is subsequently followed by theoretical analyses that encompass the convergence of the algorithm, as well as the near-optimality and feasibility on average. In addition, we test the proposed approaches by a series of empirical experiments. These experiments aim to examine and analyze the inherent trade-offs between the optimality and safety, and serve to substantiate the efficacy of two SPGs, as well as our theoretical contributions.

Evolutionary differential equation discovery proved to be a tool to obtain equations with less a priori assumptions than conventional approaches, such as sparse symbolic regression over the complete possible terms library. The equation discovery field contains two independent directions. The first one is purely mathematical and concerns differentiation, the object of optimization and its relation to the functional spaces and others. The second one is dedicated purely to the optimizational problem statement. Both topics are worth investigating to improve the algorithm's ability to handle experimental data a more artificial intelligence way, without significant pre-processing and a priori knowledge of their nature. In the paper, we consider the prevalence of either single-objective optimization, which considers only the discrepancy between selected terms in the equation, or multi-objective optimization, which additionally takes into account the complexity of the obtained equation. The proposed comparison approach is shown on classical model examples -- Burgers equation, wave equation, and Korteweg - de Vries equation.

Feature transformation aims to reconstruct an effective representation space by mathematically refining the existing features. It serves as a pivotal approach to combat the curse of dimensionality, enhance model generalization, mitigate data sparsity, and extend the applicability of classical models. Existing research predominantly focuses on domain knowledge-based feature engineering or learning latent representations. However, these methods, while insightful, lack full automation and fail to yield a traceable and optimal representation space. An indispensable question arises: Can we concurrently address these limitations when reconstructing a feature space for a machine-learning task? Our initial work took a pioneering step towards this challenge by introducing a novel self-optimizing framework. This framework leverages the power of three cascading reinforced agents to automatically select candidate features and operations for generating improved feature transformation combinations. Despite the impressive strides made, there was room for enhancing its effectiveness and generalization capability. In this extended journal version, we advance our initial work from two distinct yet interconnected perspectives: 1) We propose a refinement of the original framework, which integrates a graph-based state representation method to capture the feature interactions more effectively and develop different Q-learning strategies to alleviate Q-value overestimation further. 2) We utilize a new optimization technique (actor-critic) to train the entire self-optimizing framework in order to accelerate the model convergence and improve the feature transformation performance. Finally, to validate the improved effectiveness and generalization capability of our framework, we perform extensive experiments and conduct comprehensive analyses.

Byte-Pair Encoding (BPE) is a popular algorithm used for tokenizing data in NLP, despite being devised initially as a compression method. BPE appears to be a greedy algorithm at face value, but the underlying optimization problem that BPE seeks to solve has not yet been laid down. We formalize BPE as a combinatorial optimization problem. Via submodular functions, we prove that the iterative greedy version is a $\frac{1}{{\sigma(\boldsymbol{\mu}^\star)}}(1-e^{-{\sigma(\boldsymbol{\mu}^\star)}})$-approximation of an optimal merge sequence, where ${\sigma(\boldsymbol{\mu}^\star)}$ is the total backward curvature with respect to the optimal merge sequence $\boldsymbol{\mu}^\star$. Empirically the lower bound of the approximation is $\approx 0.37$. We provide a faster implementation of BPE which improves the runtime complexity from $\mathcal{O}\left(N M\right)$ to $\mathcal{O}\left(N \log M\right)$, where $N$ is the sequence length and $M$ is the merge count. Finally, we optimize the brute-force algorithm for optimal BPE using memoization.

Perception algorithms that provide estimates of their uncertainty are crucial to the development of autonomous robots that can operate in challenging and uncontrolled environments. Such perception algorithms provide the means for having risk-aware robots that reason about the probability of successfully completing a task when planning. There exist perception algorithms that come with models of their uncertainty; however, these models are often developed with assumptions, such as perfect data associations, that do not hold in the real world. Hence the resultant estimated uncertainty is a weak lower bound. To tackle this problem we present introspective perception - a novel approach for predicting accurate estimates of the uncertainty of perception algorithms deployed on mobile robots. By exploiting sensing redundancy and consistency constraints naturally present in the data collected by a mobile robot, introspective perception learns an empirical model of the error distribution of perception algorithms in the deployment environment and in an autonomously supervised manner. In this paper, we present the general theory of introspective perception and demonstrate successful implementations for two different perception tasks. We provide empirical results on challenging real-robot data for introspective stereo depth estimation and introspective visual simultaneous localization and mapping and show that they learn to predict their uncertainty with high accuracy and leverage this information to significantly reduce state estimation errors for an autonomous mobile robot.

--V ertical Federated Learning (VFL) attracts increasing attention because it empowers multiple parties to jointly train a privacy-preserving model over vertically partitioned data. Recent research has shown that applying zeroth-order optimization (ZOO) has many advantages in building a practical VFL algorithm. However, a vital problem with the ZOO-based VFL is its slow convergence rate, which limits its application in handling modern large models. T o address this problem, we propose a cascaded hybrid optimization method in VFL. In this method, the downstream models (clients) are trained with ZOO to protect privacy and ensure that no internal information is shared. Meanwhile, the upstream model (server) is updated with first-order optimization (FOO) locally, which significantly improves the convergence rate, making it feasible to train the large models without compromising privacy and security. We theoretically prove that our VFL framework converges faster than the ZOO-based VFL, as the convergence of our framework is not limited by the size of the server model, making it effective for training large models with the major part on the server . Extensive experiments demonstrate that our method achieves faster convergence than the ZOO-based VFL framework, while maintaining an equivalent level of privacy protection. Moreover, we show that the convergence of our VFL is comparable to the unsafe FOO-based VFL baseline. Additionally, we demonstrate that our method makes the training of a large model feasible. Data availability is essential for machine learning, however, privacy concerns often prevent the direct sharing of data among different parties. This approach allows multiple parties to leverage their data while adhering to the privacy protection measure and the government regulation, such as the General Data Protection Regulation (GDPR) [4]. Bin Gu is with Department of machine learning, Mohamed Bin Za-yed University of Artificial Intelligence, Abu Dhabi, UAE (e-mail: jsgu-bin@gmail.com). Charles Ling, Boyu Wang, Xiang Li, Ganyu Wang is with Department of Computer Science of Western University, London, Ontario, Canada. Qingsong Zhang is with School of Electronic Engineering, Xidian University, Xi'an, China (email: qszhang1995@gmail.com).

Reconfigurable robot swarms are capable of connecting with each other to form complex structures. Current mechanical or magnetic connection mechanisms can be complicated to manufacture, consume high power, have a limited load-bearing capacity, or can only form rigid structures. In this paper, we present our low-cost soft anchor design that enables flexible coupling and decoupling between robots. Our asymmetric anchor requires minimal force to be pushed into the opening of another robot while having a strong pulling force so that the connection between robots can be secured. To maintain this flexible coupling mechanism as an assembled structure, we present our Model Predictive Control (MPC) frameworks with polygon constraints to model the geometric relationship between robots. We conducted experiments on the soft anchor to obtain its force profile, which informed the three-bar linkage model of the anchor in the simulations. We show that the proposed mechanism and MPC frameworks enable the robots to couple, decouple, and perform various behaviors in both the simulation environment and hardware platform. Our code is available at https://github.com/ZoomLabCMU/puzzlebot_anchor . Video is available at https://www.youtube.com/watch?v=R3gFplorCJg .