Differentiable Electricity-Market Clearing for Gradient-Based Planning

Planning a large data center is difficult because a facility big enough to matter changes the electricity prices it will pay. Those prices are set by market clearing, a constrained optimization problem solved anew in every operating condition. However, simulating the market tells a planner how a candidate plan performs but not how to improve it. Here we treat market clearing as a differentiable optimization layer: each forward pass solves the market, and reverse-mode automatic differentiation propagates the planning cost back through the cleared prices to the plan. After validating these gradients against finite differences, we apply them to a concrete problem: allocating 50 MW of data-center load across six candidate buses in two synthetic networks, under a fixed cost per active site, evaluated over 36 operating states. Judged against exhaustive enumeration of all site combinations, gradient optimization recovers the continuous allocations almost exactly, with worst-case objective gaps of 2.3\% and 8.5\% of the cost difference between the best and worst single site. Its one systematic error is instructive: near the costs at which a site should close, the smooth relaxation of the discrete site count shrinks the site rather than closing it, so discrete transitions arrive late. Differentiable market clearing thus turns market-aware planning into a problem gradients can search.

$S^3$: A Smooth Simulation Surrogate for Optimizing Discrete Abstractions of Dynamical Systems

Intelligent systems are increasingly deployed in safety-critical settings with black-box controllers, including neural networks. The properties and behaviors of these end-to-end systems can be studied with abstraction-based methods that replace them with simpler finite models. Constructing such abstractions requires balancing the soundness of over-approximating the dynamical system against conservatism, which manifests as spurious or excessive nondeterministic behaviors. Bi-simulation theory provides principled metrics for characterizing these relationships, but does not prescribe how to construct sound abstractions with minimal conservatism. We fill this gap with a smooth simulation surrogate ($S^3$) --- a differentiable objective that approximates the reverse simulation metric used to quantify conservatism. Combined with Taylor model-based reachability, $S^3$ enables gradient-based optimization of abstraction parameters while preserving soundness by construction. We evaluate this optimization pipeline on three case studies. Our results show that $S^3$ is strongly correlated with the reverse simulation metric, is computationally faster, and serves as an effective objective for reducing abstraction conservatism.

Robot Learning to Communicate through Projected Visual Abstractions

Humans routinely communicate through abstractions of their bodies, including shadows, silhouettes, and reflections. Yet robots remain largely confined to expressing themselves through their physical morphology. Enabling robots to communicate through such projected visual abstractions requires reasoning not only about bodily motion but also about how that motion is transformed into an external representation perceived by an observer. Among these abstractions, shadows provide a particularly compelling example because they emerge directly from the robot's embodiment while remaining visually distinct from the body itself. Here, we present a robotic system capable of dynamic shadow expression using a 21-degree-of-freedom dexterous hand with compliant soft skin and a learned shadow self-model. The soft-skinned embodiment reduces light leakage to produce visually continuous silhouettes, while the differentiable self-model learns the mapping between hand configurations and projected shadow appearance through task-agnostic self-exploration. Given a target shadow image or video, the robot optimizes its hand configurations through gradient-based search over 1 the learned self-model and refines the solution through collision-aware simulation to obtain physically feasible motions. For dynamic shadow performance, we further introduce expressive-region objectives, temporal smoothness regularization, and keyframe-based optimization to preserve visually important motion cues while reducing optimization complexity. We demonstrate robotic shadow expression across sign-language gestures, hand-shadow puppetry, and animal motion imitation in both simulation and physical experiments. These results establish a framework for enabling robots to manipulate projected visual abstractions of themselves for communication and visual storytelling.

COVAriance-Induced Fairness Gap Penalty for Subgroup-Fair Clustering

Fair clustering aims to make cluster assignments independent of sensitive attributes, but this goal becomes challenging when multiple sensitive attributes jointly define many subgroups. In such settings, directly extending existing fair clustering algorithms is computationally expensive or numerically unstable, especially when the number of subgroups grows exponentially and some subgroups contain only a few instances. To address these challenges, we define a subgroup-fairness gap for clustering and derive a covariance-based surrogate that exactly matches this gap. We then introduce a continuous relaxation of the surrogate, enabling efficient gradient-based optimization and yielding our proposed algorithm, COVA-FC. We also show that subgroup fairness alone does not imply marginal fairness, and extend our framework to capture a subgroup-marginal-fairness gap. Experiments on benchmark datasets show that COVA-FC achieves competitive cost-fairness trade-offs and improves computational efficiency over existing baselines in both subgroup and higher-order marginal settings.

Local-Minima-Preserving Continuous Relaxation of Ising Problems

The generalized Ising problem captures a broad spectrum of hard combinatorial problems, including MAX-CUT, Number Partitioning (NPP), and Maximum Independent Set. In this work, we consider the notion of one-flip local minima for this problem. We construct a polynomial relaxation and prove the landscape equivalence theorem: there exists a one-to-one correspondence between the local minima of the relaxation and the one-flip minima of the original Ising problem. This guarantee reduces the Ising problem to finding the local minima of a smooth function, allowing us to leverage gradient-based optimizers such as ADAM. We demonstrate that our method is scalable and it achieves strong performance across challenging benchmarks, including spin-glass models, MAX-CUT, and NPP.