Fair cake-cutting is a mathematical subfield that studies the problem of fairly dividing a resource among a number of participants. The so-called ``cake,'' as an object, represents any resource that can be distributed among players. This concept is connected to supervised multi-label classification: any dataset can be thought of as a cake that needs to be distributed, where each label is a player that receives its share of the dataset. In particular, any efficient cake-cutting solution for the dataset is equivalent to an optimal decision function. Although we are not the first to demonstrate this connection, the important ramifications of this parallel seem to have been partially forgotten. We revisit these classical results and demonstrate how this connection can be prolifically used for fairness in machine learning problems. Understanding the set of achievable fair decisions is a fundamental step in finding optimal fair solutions and satisfying fairness requirements. By employing the tools of cake-cutting theory, we have been able to describe the behavior of optimal fair decisions, which, counterintuitively, often exhibit quite unfair properties. Specifically, in order to satisfy fairness constraints, it is sometimes preferable, in the name of optimality, to purposefully make mistakes and deny giving the positive label to deserving individuals in a community in favor of less worthy individuals within the same community. This practice is known in the literature as cherry-picking and has been described as ``blatantly unfair.''

When machine learning (ML) models are used in applications that involve humans (e.g., online recommendation, school admission, hiring, lending), the model itself may trigger changes in the distribution of targeted data it aims to predict. Performative prediction (PP) is a framework that explicitly considers such model-dependent distribution shifts when learning ML models. While significant efforts have been devoted to finding performative stable (PS) solutions in PP for system robustness, their societal implications are less explored and it is unclear whether PS solutions are aligned with social norms such as fairness. In this paper, we set out to examine the fairness property of PS solutions in performative prediction. We first show that PS solutions can incur severe polarization effects and group-wise loss disparity. Although existing fairness mechanisms commonly used in literature can help mitigate unfairness, they may fail and disrupt the stability under model-dependent distribution shifts. We thus propose novel fairness intervention mechanisms that can simultaneously achieve both stability and fairness in PP settings. Both theoretical analysis and experiments are provided to validate the proposed method.

In this paper, we consider reinforcement learning of nonlinear systems with continuous state and action spaces. We present an episodic learning algorithm, where we for each episode use convex optimization to find a two-layer neural network approximation of the optimal $Q$-function. The convex optimization approach guarantees that the weights calculated at each episode are optimal, with respect to the given sampled states and actions of the current episode. For stable nonlinear systems, we show that the algorithm converges and that the converging parameters of the trained neural network can be made arbitrarily close to the optimal neural network parameters. In particular, if the regularization parameter in the training phase is given by $\rho$, then the parameters of the trained neural network converge to $w$, where the distance between $w$ and the optimal parameters $w^\star$ is bounded by $\mathcal{O}(\rho)$. That is, when the number of episodes goes to infinity, there exists a constant $C$ such that \[ \|w-w^\star\| \le C\rho. \] In particular, our algorithm converges arbitrarily close to the optimal neural network parameters as the regularization parameter goes to zero. As a consequence, our algorithm converges fast due to the polynomial-time convergence of convex optimization algorithms.

We study the problem of Trajectory Optimization (TO) for a general class of stiff and constrained dynamic systems. We establish a set of mild assumptions, under which we show that TO converges numerically stably to a locally optimal and feasible solution up to arbitrary user-specified error tolerance. Our key observation is that all prior works use SQP as a black-box solver, where a TO problem is formulated as a Nonlinear Program (NLP) and the underlying SQP solver is not allowed to modify the NLP. Instead, we propose a white-box TO solver, where the SQP solver is informed with characteristics of the objective function and the dynamic system. It then uses these characteristics to derive approximate dynamic systems and customize the discretization schemes.

Many optimization problems are inherently multi-objective. To address them, we formalize Jacobian descent (JD), a direct generalization of gradient descent for vector-valued functions. Each step of this algorithm relies on a Jacobian matrix consisting of one gradient per objective. The aggregator, responsible for reducing this matrix into an update vector, characterizes JD. While the multi-task learning literature already contains a variety of aggregators, they often lack some natural properties. In particular, the update should not conflict with any objective and should scale proportionally to the norm of each gradient. We propose a new aggregator specifically designed to satisfy this. Emphasizing conflict between objectives, we then highlight direct applications for our methods. Most notably, we introduce instance-wise risk minimization (IWRM), a learning paradigm in which the loss of each training example is considered a separate objective. On simple image classification tasks, IWRM exhibits promising results compared to the direct minimization of the average loss. The performance of our aggregator in those experiments also corroborates our theoretical findings. Lastly, as speed is the main limitation of JD, we provide a path towards a more efficient implementation.

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

This paper presents a novel approach to visual simultaneous localization and mapping (SLAM) using multiple RGB-D cameras. The proposed method, Multicam-SLAM, significantly enhances the robustness and accuracy of SLAM systems by capturing more comprehensive spatial information from various perspectives. This method enables the accurate determination of pose relationships among multiple cameras without the need for overlapping fields of view. The proposed Muticam-SLAM includes a unique multi-camera model, a multi-keyframes structure, and several parallel SLAM threads. The multi-camera model allows for the integration of data from multiple cameras, while the multi-keyframes and parallel SLAM threads ensure efficient and accurate pose estimation and mapping. Extensive experiments in various environments demonstrate the superior accuracy and robustness of the proposed method compared to conventional single-camera SLAM systems. The results highlight the potential of the proposed Multicam-SLAM for more complex and challenging applications. Code is available at \url{https://github.com/AlterPang/Multi_ORB_SLAM}.

Even though Non-rigid Structure-from-Motion (NRSfM) has been extensively studied and great progress has been made, there are still key challenges that hinder their broad real-world applications: 1) the inherent motion/rotation ambiguity requires either explicit camera motion recovery with extra constraint or complex Procrustean Alignment; 2) existing low-rank modeling of the global shape can over-penalize drastic deformations in the 3D shape sequence. This paper proposes to resolve the above issues from a spatial-temporal modeling perspective. First, we propose a novel Temporally-smooth Procrustean Alignment module that estimates 3D deforming shapes and adjusts the camera motion by aligning the 3D shape sequence consecutively. Our new alignment module remedies the requirement of complex reference 3D shape during alignment, which is more conductive to non-isotropic deformation modeling. Second, we propose a spatial-weighted approach to enforce the low-rank constraint adaptively at different locations to accommodate drastic spatially-variant deformation reconstruction better. Our modeling outperform existing low-rank based methods, and extensive experiments across different datasets validate the effectiveness of our method.

Tail-sitters, with the advantages of both the fixed-wing unmanned aerial vehicles (UAVs) and vertical take-off and landing UAVs, have been widely designed and researched in recent years. With the change in modern UAV application scenarios, it is required that UAVs have fast maneuverable three-dimensional flight capabilities. Due to the highly nonlinear aerodynamics produced by the fuselage and wings of the tail-sitter, how to quickly generate a smooth and executable trajectory is a problem that needs to be solved urgently. We constrain the speed of the tail-sitter, eliminate the differential dynamics constraints in the trajectory generation process of the tail-sitter through differential flatness, and allocate the time variable of the trajectory through the state-of-the-art trajectory generation method named MINCO. By discretizing the trajectory in time, we convert the speed constraint on the vehicle into a soft constraint, thereby achieving the time-optimal trajectory for the tail-sitter to fly through any given waypoints.

High-dimensional robot dynamic trajectory planning poses many challenges for traditional planning algorithms. Existing planning methods suffer from issues such as long computation times, limited capacity to address intricate obstacle models, and lack of consideration for external disturbances and measurement inaccuracies in these high-dimensional systems. To tackle these challenges, this paper proposes a novel trajectory planning approach that combines Dynamic Control Barrier Function (DCBF) with a disturbance observer to create a Robust Dynamic Control Barrier Function (RDCBF) planner. This approach successfully plans trajectories in environments with complex dynamic obstacles while accounting for external disturbances and measurement uncertainties, ensuring system safety and enabling precise obstacle avoidance. Experimental results on a mobile manipulator demonstrate outstanding performance of the proposed approach.