We analyze inexact Riemannian gradient descent (RGD) where Riemannian gradients and retractions are inexactly (and cheaply) computed. Our focus is on understanding when inexact RGD converges and what is the complexity in the general nonconvex and constrained setting. We answer these questions in a general framework of tangential Block Majorization-Minimization (tBMM). We establish that tBMM converges to an $\epsilon$-stationary point within $O(\epsilon^{-2})$ iterations. Under a mild assumption, the results still hold when the subproblem is solved inexactly in each iteration provided the total optimality gap is bounded. Our general analysis applies to a wide range of classical algorithms with Riemannian constraints including inexact RGD and proximal gradient method on Stiefel manifolds. We numerically validate that tBMM shows improved performance over existing methods when applied to various problems, including nonnegative tensor decomposition with Riemannian constraints, regularized nonnegative matrix factorization, and low-rank matrix recovery problems.

Koopman operators are infinite-dimensional operators that linearize nonlinear dynamical systems, facilitating the study of their spectral properties and enabling the prediction of the time evolution of observable quantities. Recent methods have aimed to approximate Koopman operators while preserving key structures. However, approximating Koopman operators typically requires a dictionary of observables to capture the system's behavior in a finite-dimensional subspace. The selection of these functions is often heuristic, may result in the loss of spectral information, and can severely complicate structure preservation. This paper introduces Multiplicative Dynamic Mode Decomposition (MultDMD), which enforces the multiplicative structure inherent in the Koopman operator within its finite-dimensional approximation. Leveraging this multiplicative property, we guide the selection of observables and define a constrained optimization problem for the matrix approximation, which can be efficiently solved. MultDMD presents a structured approach to finite-dimensional approximations and can more accurately reflect the spectral properties of the Koopman operator. We elaborate on the theoretical framework of MultDMD, detailing its formulation, optimization strategy, and convergence properties. The efficacy of MultDMD is demonstrated through several examples, including the nonlinear pendulum, the Lorenz system, and fluid dynamics data, where we demonstrate its remarkable robustness to noise.

This paper discusses the revenue management (RM) problem to maximize revenue by pricing items or services. One challenge in this problem is that the demand distribution is unknown and varies over time in real applications such as airline and retail industries. In particular, the time-varying demand has not been well studied under scenarios of unknown demand due to the difficulty of jointly managing the remaining inventory and estimating the demand. To tackle this challenge, we first introduce an episodic generalization of the RM problem motivated by typical application scenarios. We then propose a computationally efficient algorithm based on posterior sampling, which effectively optimizes prices by solving linear programming. We derive a Bayesian regret upper bound of this algorithm for general models where demand parameters can be correlated between time periods, while also deriving a regret lower bound for generic algorithms. Our empirical study shows that the proposed algorithm performs better than other benchmark algorithms and comparably to the optimal policy in hindsight. We also propose a heuristic modification of the proposed algorithm, which further efficiently learns the pricing policy in the experiments.

In this study, we propose a safety-critical compliant control strategy designed to strictly enforce interaction force constraints during the physical interaction of robots with unknown environments. The interaction force constraint is interpreted as a new force-constrained control barrier function (FC-CBF) by exploiting the generalized contact model and the prior information of the environment, i.e., the prior stiffness and rest position, for robot kinematics. The difference between the real environment and the generalized contact model is approximated by constructing a tracking differentiator, and its estimation error is quantified based on Lyapunov theory. By interpreting strict interaction safety specification as a dynamic constraint, restricting the desired joint angular rates in kinematics, the proposed approach modifies nominal compliant controllers using quadratic programming, ensuring adherence to interaction force constraints in unknown environments. The strict force constraint and the stability of the closed-loop system are rigorously analyzed. Experimental tests using a UR3e industrial robot with different environments verify the effectiveness of the proposed method in achieving the force constraints in unknown environments.

Knowing the locations of nearby moving objects is important for a mobile robot to operate safely in a dynamic environment. Dynamic object tracking performance can be improved if robots share observations of tracked objects with nearby team members in real-time. To share observations, a robot must make up-to-date estimates of the transformation from its coordinate frame to the frame of each neighbor, which can be challenging because of odometry drift. We present Multiple Object Tracking with Localization Error Elimination (MOTLEE), a complete system for a multi-robot team to accurately estimate frame transformations and collaboratively track dynamic objects. To accomplish this, robots use open-set image-segmentation methods to build object maps of their environment and then use our Temporally Consistent Alignment of Frames Filter (TCAFF) to align maps and estimate coordinate frame transformations without any initial knowledge of neighboring robot poses. We show that our method for aligning frames enables a team of four robots to collaboratively track six pedestrians with accuracy similar to that of a system with ground truth localization in a challenging hardware demonstration. The code and hardware dataset are available at https://github.com/mit-acl/motlee.

Offline optimization is an emerging problem in many experimental engineering domains including protein, drug or aircraft design, where online experimentation to collect evaluation data is too expensive or dangerous. To avoid that, one has to optimize an unknown function given only its offline evaluation at a fixed set of inputs. A naive solution to this problem is to learn a surrogate model of the unknown function and optimize this surrogate instead. However, such a naive optimizer is prone to erroneous overestimation of the surrogate (possibly due to over-fitting on a biased sample of function evaluation) on inputs outside the offline dataset. Prior approaches addressing this challenge have primarily focused on learning robust surrogate models. However, their search strategies are derived from the surrogate model rather than the actual offline data. To fill this important gap, we introduce a new learning-to-search perspective for offline optimization by reformulating it as an offline reinforcement learning problem. Our proposed policy-guided gradient search approach explicitly learns the best policy for a given surrogate model created from the offline data. Our empirical results on multiple benchmarks demonstrate that the learned optimization policy can be combined with existing offline surrogates to significantly improve the optimization performance.

Localization in already mapped environments is a critical component in many robotics and automotive applications, where previously acquired information can be exploited along with sensor fusion to provide robust and accurate localization estimates. In this work, we offer a new perspective on map-based localization by reusing prior topological and metric information. Thus, we reformulate this long-studied problem to go beyond the mere use of metric maps. Our framework seamlessly integrates LiDAR, iner\-tial and GNSS measurements, and scan-to-map registrations in a sliding window graph fashion, which allows to accommodate the uncertainty of each observation. The modularity of our framework allows it to work with different sensor configurations (\textit{e.g.,} LiDAR resolutions, GNSS denial) and environmental conditions (\textit{e.g.,} map-less regions, large environments). We have conducted different validation experiments, including deployment in a real-world automotive application, demonstrating the accuracy, efficiency, and versatility of our system in online localization.

Instrumental variables (IVs) are a popular and powerful tool for estimating causal effects in the presence of unobserved confounding. However, classical approaches rely on strong assumptions such as the $\textit{exclusion criterion}$, which states that instrumental effects must be entirely mediated by treatments. This assumption often fails in practice. When IV methods are improperly applied to data that do not meet the exclusion criterion, estimated causal effects may be badly biased. In this work, we propose a novel solution that provides $\textit{partial}$ identification in linear systems given a set of $\textit{leaky instruments}$, which are allowed to violate the exclusion criterion to some limited degree. We derive a convex optimization objective that provides provably sharp bounds on the average treatment effect under some common forms of information leakage, and implement inference procedures to quantify the uncertainty of resulting estimates. We demonstrate our method in a set of experiments with simulated data, where it performs favorably against the state of the art. An accompanying $\texttt{R}$ package, $\texttt{leakyIV}$, is available from $\texttt{CRAN}$.

Recent progress of deep generative models in the vision and language domain has stimulated significant interest in more structured data generation such as molecules. However, beyond generating new random molecules, efficient exploration and a comprehensive understanding of the vast chemical space are of great importance to molecular science and applications in drug design and materials discovery. In this paper, we propose a new framework, ChemFlow, to traverse chemical space through navigating the latent space learned by molecule generative models through flows. We introduce a dynamical system perspective that formulates the problem as learning a vector field that transports the mass of the molecular distribution to the region with desired molecular properties or structure diversity. Under this framework, we unify previous approaches on molecule latent space traversal and optimization and propose alternative competing methods incorporating different physical priors.

Evolution Strategies (ES) are effective gradient-free optimization methods that can be competitive with gradient-based approaches for policy search. ES only rely on the total episodic scores of solutions in their population, from which they estimate fitness gradients for their update with no access to true gradient information. However this makes them sensitive to deceptive fitness landscapes, and they tend to only explore one way to solve a problem. Quality-Diversity methods such as MAP-Elites introduced additional information with behavior descriptors (BD) to return a population of diverse solutions, which helps exploration but leads to a large part of the evaluation budget not being focused on finding the best performing solution. Here we show that behavior information can also be leveraged to find the best policy by identifying promising search areas which can then be efficiently explored with ES. We introduce the framework of Quality with Just Enough Diversity (JEDi) which learns the relationship between behavior and fitness to focus evaluations on solutions that matter. When trying to reach higher fitness values, JEDi outperforms both QD and ES methods on hard exploration tasks like mazes and on complex control problems with large policies.