In this paper, we propose an online convex optimization approach with two different levels of adaptivity. On a higher level, our approach is agnostic to the unknown types and curvatures of the online functions, while at a lower level, it can exploit the unknown niceness of the environments and attain problem-dependent guarantees. Specifically, we obtain $\mathcal{O}(\log V_T)$, $\mathcal{O}(d \log V_T)$ and $\widehat{\mathcal{O}}(\sqrt{V_T})$ regret bounds for strongly convex, exp-concave and convex loss functions, respectively, where $d$ is the dimension, $V_T$ denotes problem-dependent gradient variations and the $\widehat{\mathcal{O}}(\cdot)$-notation omits $\log V_T$ factors. Our result not only safeguards the worst-case guarantees but also directly implies the small-loss bounds in analysis. Moreover, when applied to adversarial/stochastic convex optimization and game theory problems, our result enhances the existing universal guarantees. Our approach is based on a multi-layer online ensemble framework incorporating novel ingredients, including a carefully designed optimism for unifying diverse function types and cascaded corrections for algorithmic stability. Notably, despite its multi-layer structure, our algorithm necessitates only one gradient query per round, making it favorable when the gradient evaluation is time-consuming. This is facilitated by a novel regret decomposition with carefully designed surrogate losses.

This paper presents new quadrature rules for functions in a reproducing kernel Hilbert space using nodes drawn by a sampling algorithm known as randomly pivoted Cholesky. The resulting computational procedure compares favorably to previous kernel quadrature methods, which either achieve low accuracy or require solving a computationally challenging sampling problem. Theoretical and numerical results show that randomly pivoted Cholesky is fast and achieves comparable quadrature error rates to more computationally expensive quadrature schemes based on continuous volume sampling, thinning, and recombination. Randomly pivoted Cholesky is easily adapted to complicated geometries with arbitrary kernels, unlocking new potential for kernel quadrature.

The Classification Tree (CT) is one of the most common models in interpretable machine learning. Although such models are usually built with greedy strategies, in recent years, thanks to remarkable advances in Mixer-Integer Programming (MIP) solvers, several exact formulations of the learning problem have been developed. In this paper, we argue that some of the most relevant ones among these training models can be encapsulated within a general framework, whose instances are shaped by the specification of loss functions and regularizers. Next, we introduce a novel realization of this framework: specifically, we consider the logistic loss, handled in the MIP setting by a linear piece-wise approximation, and couple it with $\ell_1$-regularization terms. The resulting Optimal Logistic Tree model numerically proves to be able to induce trees with enhanced interpretability features and competitive generalization capabilities, compared to the state-of-the-art MIP-based approaches.

Designing civil structures such as bridges, dams or buildings is a complex task requiring many synergies from several experts. Each is responsible for different parts of the process. This is often done in a sequential manner, e.g. the structural engineer makes a design under the assumption of certain material properties (e.g. the strength class of the concrete), and then the material engineer optimizes the material with these restrictions. This paper proposes a holistic optimization procedure, which combines the concrete mixture design and structural simulations in a joint, forward workflow that we ultimately seek to invert. In this manner, new mixtures beyond standard ranges can be considered. Any design effort should account for the presence of uncertainties which can be aleatoric or epistemic as when data is used to calibrate physical models or identify models that fill missing links in the workflow. Inverting the causal relations established poses several challenges especially when these involve physics-based models which most often than not do not provide derivatives/sensitivities or when design constraints are present. To this end, we advocate Variational Optimization, with proposed extensions and appropriately chosen heuristics to overcome the aforementioned challenges. The proposed methodology is illustrated using the design of a precast concrete beam with the objective to minimize the global warming potential while satisfying a number of constraints associated with its load-bearing capacity after 28days according to the Eurocode, the demoulding time as computed by a complex nonlinear Finite Element model, and the maximum temperature during the hydration.

This paper introduces a methodology designed to augment the inverse design optimization process in scenarios constrained by limited compute, through the strategic synergy of multi-fidelity evaluations, machine learning models, and optimization algorithms. The proposed methodology is analyzed on two distinct engineering inverse design problems: airfoil inverse design and the scalar field reconstruction problem. It leverages a machine learning model trained with low-fidelity simulation data, in each optimization cycle, thereby proficiently predicting a target variable and discerning whether a high-fidelity simulation is necessitated, which notably conserves computational resources. Additionally, the machine learning model is strategically deployed prior to optimization to reduce the search space, thereby further accelerating convergence toward the optimal solution. The methodology has been employed to enhance two optimization algorithms, namely Differential Evolution and Particle Swarm Optimization. Comparative analyses illustrate performance improvements across both algorithms. Notably, this method is adeptly adaptable across any inverse design application, facilitating a harmonious synergy between a representative low-fidelity machine learning model, and high-fidelity simulation, and can be seamlessly applied across any variety of population-based optimization algorithms.

Specifically, earlier works focus a scene representation. The new representation enables on tracking using various scene representations like interactive-time reconstruction and photo-realistic rendering feature point clouds [15, 26, 40], surfels [53, 71], depth of real-world and synthetic scenes. We propose novel maps [43, 58], or implicit representations [14, 42, 44]. Later strategies for seeding and optimizing Gaussian splats to works focused more on the map quality and density. With extend their use from multiview offline scenarios to sequential the advent of powerful neural scene representations like monocular RGBD input data setups. In addition, we neural radiance fields [38] that allow for high fidelity viewsynthesis, extend Gaussian splats to encode geometry and experiment a rapidly growing body of dense neural SLAM with tracking against this scene representation. Our methods [19, 34, 51, 60, 62, 64, 81, 84] has been developed.

We present a framework that enables to write a family of convex approximations of complex contact models. Within this framework, we show that we can incorporate well established and experimentally validated contact models such as the Hunt & Crossley model. Moreover, we show how to incorporate Coulomb's law and the principle of maximum dissipation using a regularized model of friction. Contrary to common wisdom that favors the use of rigid contact models, our convex formulation is robust and performant even at high stiffness values far beyond that of materials such as steel. Therefore, the same formulation enables the modeling of compliant surfaces such as rubber gripper pads or robot feet as well as hard objects. We characterize and evaluate our approximations in a number of tests cases. We report their properties and highlight limitations. Finally, we demonstrate robust simulation of robotic tasks at interactive rates, with accurately resolved stiction and contact transitions, as required for meaningful sim-to-real transfer. Our method is implemented in the open source robotics toolkit Drake.

Learning in games is a well-studied framework in which agents iteratively refine their strategies through repeated interactions with their environment. One natural way for agents to iteratively refine their strategies is by best-responding. This idea can be applied in many forms, the simplest and earliest instance of which was fictitious play (FP) [Brown, 1951]. This algorithm involves the agent observing the strategies played by the opponent and then playing a strategy that corresponds to the best response to the average of the observed strategies. This algorithm was shown to converge [Robinson, 1951], but its convergence rate can, in the worst case, scale quite poorly with the number of actions available to each player [Daskalakis and Pan, 2014]. It is then natural to ask what are the best convergence guarantees that can be obtained for the computation of Nash equilibria in two-player zero-sum games or coarse correlated equilibria in multiplayer games when agents are learning through a best-response oracle. In the online learning community, methods based only on best-response oracles are special cases of methods based on a linear minimization oracle (LMO), which can be queried for points that minimize a linear objective over the feasible set. Such methods are known as projection-free methods because they avoid potentially expensive projections onto the feasible set. Projection-free online learning algorithms might perform multiple LMO calls per iteration, so our paper and related literature are concerned not only with the number of iterations T of online learning but also the total number of LMO calls, which we will denote by N. Because LMOs for polyhedral decision sets essentially correspond to a best-response oracle (BRO), we will use these two terms interchangeably.

Decision-making can be challenging due to the stochastic nature of real-world processes. This complexity is evident in various contexts, such as manufacturing, where uncertain processing times make it challenging to meet strict customer deadlines. Formulating Constrained Optimization (CO) models for these problems is common, but unknown parameter values during decision-making add challenges, because wrong estimates of the parameters can lead to infeasibilities. In practice, such infeasibilities are repaired when reality unfolds. For instance, in a manufacturing system, tasks may be postponed due to delays in earlier stages to maintain the factory's flow. Various repair policies and schedule definitions are used across different contexts. Historical data, represented as scenarios of unknown parameters like task duration, is often available. Simple averaging of these scenarios is a common yet naive approach that ignores uncertainty.

Regularization is a critical component in deep learning training, with weight decay being a commonly used approach. It applies a constant penalty coefficient uniformly across all parameters. This may be unnecessarily restrictive for some parameters, while insufficiently restricting others. To dynamically adjust penalty coefficients for different parameter groups, we present constrained parameter regularization (CPR) as an alternative to traditional weight decay. Instead of applying a single constant penalty to all parameters, we enforce an upper bound on a statistical measure (e.g., the L$_2$-norm) of parameter groups. Consequently, learning becomes a constraint optimization problem, which we address by an adaptation of the augmented Lagrangian method. CPR only requires two hyperparameters and incurs no measurable runtime overhead. Additionally, we propose a simple but efficient mechanism to adapt the upper bounds during the optimization. We provide empirical evidence of CPR's efficacy in experiments on the "grokking" phenomenon, computer vision, and language modeling tasks. Our results demonstrate that CPR counteracts the effects of grokking and consistently matches or outperforms traditional weight decay.