Optimal transport (OT), as a metric, has gained significant attention in the field of machine learning in recent years due to its remarkable ability to capture geometric relationships between data distributions. It has demonstrated impressive achievements in many fields [16, 3, 13, 31]. To overcome the limitation of OT in handling data with unequal quantities, researchers introduced unbalanced optimal transport (UOT) [8] by relaxing the constraints using penalty functions. UOT has been found extensive applications in computational biology [40], machine learning [25], and deep learning domains [51]. However, compared to traditional metrics, the computational burden associated with OT, including UOT, has impeded their widespread adoption on large-scale problems. The state-of-the-art linear programming algorithms suffer from cubic computational complexity and are challenging to parallelize on GPUs [46].

Bayesian Optimization (BO) is a class of surrogate-based, sample-efficient algorithms for optimizing black-box problems with small evaluation budgets. The BO pipeline itself is highly configurable with many different design choices regarding the initial design, surrogate model, and acquisition function (AF). Unfortunately, our understanding of how to select suitable components for a problem at hand is very limited. In this work, we focus on the definition of the AF, whose main purpose is to balance the trade-off between exploring regions with high uncertainty and those with high promise for good solutions. We propose Self-Adjusting Weighted Expected Improvement (SAWEI), where we let the exploration-exploitation trade-off self-adjust in a data-driven manner, based on a convergence criterion for BO. On the noise-free black-box BBOB functions of the COCO benchmarking platform, our method exhibits a favorable any-time performance compared to handcrafted baselines and serves as a robust default choice for any problem structure. The suitability of our method also transfers to HPOBench. With SAWEI, we are a step closer to on-the-fly, data-driven, and robust BO designs that automatically adjust their sampling behavior to the problem at hand.

Robot programming tools ranging from inverse kinematics (IK) to model predictive control (MPC) are most often described as constrained optimization problems. Even though there are currently many commercially-available second-order solvers, robotics literature recently focused on efficient implementations and improvements over these solvers for real-time robotic applications. However, most often, these implementations stay problem-specific and are not easy to access or implement, or do not exploit the geometric aspect of the robotics problems. In this work, we propose to solve these problems using a fast, easy-to-implement first-order method that fully exploits the geometric constraints via Euclidean projections, called Augmented Lagrangian Spectral Projected Gradient Descent (ALSPG). We show that 1. using projections instead of full constraints and gradients improves the performance of the solver and 2. ALSPG stays competitive to the standard second-order methods such as iLQR in the unconstrained case. We showcase these results with IK and motion planning problems on simulated examples and with an MPC problem on a 7-axis manipulator experiment.

Abstract--Convex optimization is crucial in controlling legged robots, where stability and optimal control are vital. Many control problems can be formulated as convex optimization problems, with a convex cost function and constraints capturing system dynamics. Our review focuses on active balancing problems and presents a general framework for formulating them as second-order cone programming (SOCP) for robustness and efficiency with existing interior point algorithms. We then discuss some prior work around the Zero Moment Point stability criterion, Linear Quadratic Regulator Control, and then the feedback model predictive control (MPC) approach to improve prediction accuracy and reduce computational costs. Finally, these techniques are applied to stabilize the robot for jumping and landing tasks. Further research in convex optimization of legged robots can have a significant societal impact. These advancements have the potential to revolutionize industries and help humans in daily life. Control problems can be formulated as optimization problems We start with some literature and initial works on convex by defining an objective function that quantifies the optimization applications in legged robots, which lay the desired behavior of the system, and a set of constraints that foundation to the most widely used optimization methods, capture the physical limitations of the system and any other Model Predictive Control.

The design of machine learning systems often requires trading off different objectives, for example, prediction error and energy consumption for deep neural networks (DNNs). Typically, no single design performs well in all objectives; therefore, finding Pareto-optimal designs is of interest. The search for Pareto-optimal designs involves evaluating designs in an iterative process, and the measurements are used to evaluate an acquisition function that guides the search process. However, measuring different objectives incurs different costs. For example, the cost of measuring the prediction error of DNNs is orders of magnitude higher than that of measuring the energy consumption of a pre-trained DNN as it requires re-training the DNN. Current state-of-the-art methods do not consider this difference in objective evaluation cost, potentially incurring expensive evaluations of objective functions in the optimization process. In this paper, we develop a novel decoupled and cost-aware multi-objective optimization algorithm, which we call Flexible Multi-Objective Bayesian Optimization (FlexiBO) to address this issue. For evaluating each design, FlexiBO selects the objective with higher relative gain by weighting the improvement of the hypervolume of the Pareto region with the measurement cost of each objective. This strategy, therefore, balances the expense of collecting new information with the knowledge gained through objective evaluations, preventing FlexiBO from performing expensive measurements for little to no gain. We evaluate FlexiBO on seven state-of-the-art DNNs for image recognition, natural language processing (NLP), and speech-to-text translation. Our results indicate that, given the same total experimental budget, FlexiBO discovers designs with 4.8% to 12.4% lower hypervolume error than the best method in state-of-the-art multi-objective optimization.

In this work, we revisit the problem of solving large-scale semidefinite programs using randomized first-order methods and stochastic smoothing. We introduce two oblivious stochastic mirror descent algorithms based on a complementary composite setting. One algorithm is designed for non-smooth objectives, while an accelerated version is tailored for smooth objectives. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. For the non-smooth case with $\mathcal{M}-$bounded oracles, we prove a convergence rate of $ O( {\mathcal{M}}/{\sqrt{T}} ) $. For the $L$-smooth case with a feasible set bounded by $D$, we derive a convergence rate of $ O( {L^2 D^2}/{(T^{2}\sqrt{T})} + {(D_0^2+\sigma^2)}/{\sqrt{T}} )$, where $D_0$ is the starting distance to an optimal solution, and $ \sigma^2$ is the stochastic oracle variance. These rates had only been obtained so far by either assuming prior knowledge of the Lipschitz constant or the starting distance to an optimal solution. We further show how to extend our framework to relative scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.

Safety is a central requirement for autonomous system operation across domains. Hamilton-Jacobi (HJ) reachability analysis can be used to construct "least-restrictive" safety filters that result in infrequent, but often extreme, control overrides. In contrast, control barrier function (CBF) methods apply smooth control corrections to guard the system against an often conservative safety boundary. This paper provides an online scheme to construct an implicit CBF through HJ reach-avoid differential dynamic programming in a receding-horizon framework, enabling smooth safety filtering with infinite-time safety guarantees. Simulations with the Dubins car and 5D bicycle dynamics demonstrate the scheme's ability to preserve safety smoothly without the conservativeness of handcrafted CBFs.

Effective decision-making requires the use of knowledge. This has been a clear, and long-standing principle in AI research, as reflected, for example, in the seminal early work on knowledge and AI--summarized by Brachman and Levesque (1985)--and the thriving Knowledge Representation and Reasoning and the Uncertainty in AI communities. However, the message has been somewhat diluted as data-driven statistical learning has become increasingly pervasive across AI. Nevertheless, the idea that reasoning and learning need to work together (Khardon and Roth, 1996; Roth, 1996) and that knowledge representation is a crucial bridge between them has not been lost. One area where the link between learning, representation, and reasoning has been shown to be essential and has been studied extensively is Natural Language Processing (NLP), and in particular, the area of Structured Output Prediction within NLP. In structured problems, there is a need to assign values to multiple random variables that are interrelated. Examples include extracting multiple relations among entities in a document, where a the two arguments for a relation such as born-in cannot refer to people, or co-reference resolution, where gender agreement must be maintained when determining that a specific pronoun refers to a given entity. In these, and many other such problems, it is natural to represent knowledge as Boolean functions over propositional variables. These functions would express knowledge, for example, of the form "if the relation between two entities is born-in, then its arguments must be a person and a location" (formalized as functions such as x

We present a method for solving general nonconvex-strongly-convex bilevel optimization problems. Our method -- the \emph{Restarted Accelerated HyperGradient Descent} (\texttt{RAHGD}) method -- finds an $\epsilon$-first-order stationary point of the objective with $\tilde{\mathcal{O}}(\kappa^{3.25}\epsilon^{-1.75})$ oracle complexity, where $\kappa$ is the condition number of the lower-level objective and $\epsilon$ is the desired accuracy. We also propose a perturbed variant of \texttt{RAHGD} for finding an $\big(\epsilon,\mathcal{O}(\kappa^{2.5}\sqrt{\epsilon}\,)\big)$-second-order stationary point within the same order of oracle complexity. Our results achieve the best-known theoretical guarantees for finding stationary points in bilevel optimization and also improve upon the existing upper complexity bound for finding second-order stationary points in nonconvex-strongly-concave minimax optimization problems, setting a new state-of-the-art benchmark. Empirical studies are conducted to validate the theoretical results in this paper.

Black-box zero-th order optimization is a central primitive for applications in fields as diverse as finance, physics, and engineering. In a common formulation of this problem, a designer sequentially attempts candidate solutions, receiving noisy feedback on the value of each attempt from the system. In this paper, we study scenarios in which feedback is also provided on the safety of the attempted solution, and the optimizer is constrained to limit the number of unsafe solutions that are tried throughout the optimization process. Focusing on methods based on Bayesian optimization (BO), prior art has introduced an optimization scheme -- referred to as SAFEOPT -- that is guaranteed not to select any unsafe solution with a controllable probability over feedback noise as long as strict assumptions on the safety constraint function are met. In this paper, a novel BO-based approach is introduced that satisfies safety requirements irrespective of properties of the constraint function. This strong theoretical guarantee is obtained at the cost of allowing for an arbitrary, controllable but non-zero, rate of violation of the safety constraint. The proposed method, referred to as SAFE-BOCP, builds on online conformal prediction (CP) and is specialized to the cases in which feedback on the safety constraint is either noiseless or noisy. Experimental results on synthetic and real-world data validate the advantages and flexibility of the proposed SAFE-BOCP.