The proliferation of saddle points, rather than poor local minima, is increasingly understood to be a primary obstacle in large-scale non-convex optimization for machine learning. Variable elimination algorithms, like Variable Projection (VarPro), have long been observed to exhibit superior convergence and robustness in practice, yet a principled understanding of why they so effectively navigate these complex energy landscapes has remained elusive. In this work, we provide a rigorous geometric explanation by comparing the optimization landscapes of the original and reduced formulations. Through a rigorous analysis based on Hessian inertia and the Schur complement, we prove that variable elimination fundamentally reshapes the critical point structure of the objective function, revealing that local maxima in the reduced landscape are created from, and correspond directly to, saddle points in the original formulation. Our findings are illustrated on the canonical problem of non-convex matrix factorization, visualized directly on two-parameter neural networks, and finally validated in training deep Residual Networks, where our approach yields dramatic improvements in stability and convergence to superior minima. This work goes beyond explaining an existing method; it establishes landscape simplification via saddle point transformation as a powerful principle that can guide the design of a new generation of more robust and efficient optimization algorithms.

In this paper, we propose a novel fairness framework grounded in the concept of happiness, a measure of the utility each group gains from decision outcomes. By capturing fairness through this intuitive lens, we not only offer a more human-centered approach, but also one that is mathematically rigorous: In order to compute the optimal, fair post-processing strategy, only a linear program needs to be solved. This makes our method both efficient and scalable with existing optimization tools. Furthermore, it unifies and extends several well-known fairness definitions, and our empirical results highlight its practical strengths across diverse scenarios.

We propose SLIM (Stochastic Learning and Inference in overidentified Models), a scalable stochastic approximation framework for nonlinear GMM. SLIM forms iterative updates from independent mini-batches of moments and their derivatives, producing unbiased directions that ensure almost-sure convergence. It requires neither a consistent initial estimator nor global convexity and accommodates both fixed-sample and random-sampling asymptotics. We further develop an optional second-order refinement achieving full-sample GMM efficiency and inference procedures based on random scaling and plug-in methods, including plug-in, debiased plug-in, and online versions of the Sargan--Hansen $J$-test tailored to stochastic learning. In Monte Carlo experiments based on a nonlinear demand system with 576 moment conditions, 380 parameters, and $n = 10^5$, SLIM solves the model in under 1.4 hours, whereas full-sample GMM in Stata on a powerful laptop converges only after 18 hours. The debiased plug-in $J$-test delivers satisfactory finite-sample inference, and SLIM scales smoothly to $n = 10^6$.

Finding lower and better-generalizing minima is crucial for deep learning. However, most existing optimizers stop searching the parameter space once they reach a local minimum. Given the complex geometric properties of the loss landscape, it is difficult to guarantee that such a point is the lowest or provides the best generalization. To address this, we propose an adaptor "E" for gradient-based optimizers. The adapted optimizer tends to continue exploring along landscape valleys (areas with low and nearly identical losses) in order to search for potentially better local minima even after reaching a local minimum. This approach increases the likelihood of finding a lower and flatter local minimum, which is often associated with better generalization. We also provide a proof of convergence for the adapted optimizers in both convex and non-convex scenarios for completeness. Finally, we demonstrate their effectiveness in an important but notoriously difficult training scenario, large-batch training, where Lamb is the benchmark optimizer. Our testing results show that the adapted Lamb, ALTO, increases the test accuracy (generalization) of the current state-of-the-art optimizer by an average of 2.5% across a variety of large-batch training tasks. This work potentially opens a new research direction in the design of optimization algorithms.

Adding entropic regularization to Optimal Transport (OT) problems has become a standard approach for designing efficient and scalable solvers. However, regularization introduces a bias from the true solution. To mitigate this bias while still benefiting from the acceleration provided by regularization, a natural solver would adaptively decrease the regularization as it approaches the solution. Although some algorithms heuristically implement this idea, their theoretical guarantees and the extent of their acceleration compared to using a fixed regularization remain largely open. In the setting of semi-discrete OT, where the source measure is continuous and the target is discrete, we prove that decreasing the regularization can indeed accelerate convergence. To this end, we introduce DRAG: Decreasing (entropic) Regularization Averaged Gradient, a stochastic gradient descent algorithm where the regularization decreases with the number of optimization steps. We provide a theoretical analysis showing that DRAG benefits from decreasing regularization compared to a fixed scheme, achieving an unbiased $\mathcal{O}(1/t)$ sample and iteration complexity for both the OT cost and the potential estimation, and a $\mathcal{O}(1/\sqrt{t})$ rate for the OT map. Our theoretical findings are supported by numerical experiments that validate the effectiveness of DRAG and highlight its practical advantages.

Federated Learning (FL) enables collaborative model training without data sharing, yet participants face a fundamental challenge, e.g., simultaneously ensuring fairness across demographic groups while protecting sensitive client data. We introduce a differentially private fair FL algorithm (\textit{FedPF}) that transforms this multi-objective optimization into a zero-sum game where fairness and privacy constraints compete against model utility. Our theoretical analysis reveals a surprising inverse relationship, i.e., stricter privacy protection fundamentally limits the system's ability to detect and correct demographic biases, creating an inherent tension between privacy and fairness. Counterintuitively, we prove that moderate fairness constraints initially improve model generalization before causing performance degradation, where a non-monotonic relationship that challenges conventional wisdom about fairness-utility tradeoffs. Experimental validation demonstrates up to 42.9 % discrimination reduction across three datasets while maintaining competitive accuracy, but more importantly, reveals that the privacy-fairness tension is unavoidable, i.e., achieving both objectives simultaneously requires carefully balanced compromises rather than optimization of either in isolation. The source code for our proposed algorithm is publicly accessible at https://github.com/szpsunkk/FedPF.

Dynamic offloading of Machine Learning (ML) model partitions across different resource orchestration services, such as Function-as-a-Service (FaaS) and Infrastructure-as-a-Service (IaaS), can balance processing and transmission delays while minimizing costs of adaptive inference applications. However, prior work often overlooks real-world factors, such as Virtual Machine (VM) cold starts, requests under long-tail service time distributions, etc. To tackle these limitations, we model each ML query (request) as traversing an acyclic sequence of stages, wherein each stage constitutes a contiguous block of sparse model parameters ending in an internal or final classifier where requests may exit. Since input-dependent exit rates vary, no single resource configuration suits all query distributions. IaaS-based VMs become underutilized when many requests exit early, yet rapidly scaling to handle request bursts reaching deep layers is impractical. SERFLOW addresses this challenge by leveraging FaaS-based serverless functions (containers) and using stage-specific resource provisioning that accounts for the fraction of requests exiting at each stage. By integrating this provisioning with adaptive load balancing across VMs and serverless functions based on request ingestion, SERFLOW reduces cloud costs by over $23\%$ while efficiently adapting to dynamic workloads.

Database Management Systems (DBMSs) are fundamental for managing large-scale and heterogeneous data, and their performance is critically influenced by configuration parameters. Effective tuning of these parameters is essential for adapting to diverse workloads and maximizing throughput while minimizing latency. Recent research has focused on automated configuration optimization using machine learning; however, existing approaches still exhibit several key limitations. Most tuning frameworks disregard the dependencies among parameters, assuming that each operates independently. This simplification prevents optimizers from leveraging relational effects across parameters, limiting their capacity to capture performancesensitive interactions. Moreover, to reduce the complexity of the high-dimensional search space, prior work often selects only the top few parameters for optimization, overlooking others that contribute meaningfully to performance. Bayesian Optimization (BO), the most common method for automatic tuning, is also constrained by its reliance on surrogate models, which can lead to unstable predictions and inefficient exploration. To overcome these limitations, we propose RelTune, a novel framework that represents parameter dependencies as a Relational Graph and learns GNN-based latent embeddings that encode performancerelevant semantics. RelTune further introduces Hybrid-Score-Guided Bayesian Optimization (HBO), which combines surrogate predictions with an Affinity Score measuring proximity to previously high-performing configurations. Experimental results on multiple DBMSs and workloads demonstrate that RelTune achieves faster convergence and higher optimization efficiency than conventional BO-based methods, achieving state-of-the-art performance across all evaluated scenarios.

Heterogeneous robot teams operating in realistic settings often must accomplish complex missions requiring collaboration and adaptation to information acquired online. Because robot teams frequently operate in unstructured environments -- uncertain, open-world settings without prior maps -- subtasks must be grounded in robot capabilities and the physical world. While heterogeneous teams have typically been designed for fixed specifications, generative intelligence opens the possibility of teams that can accomplish a wide range of missions described in natural language. However, current large language model (LLM)-enabled teaming methods typically assume well-structured and known environments, limiting deployment in unstructured environments. We present SPINE-HT, a framework that addresses these limitations by grounding the reasoning abilities of LLMs in the context of a heterogeneous robot team through a three-stage process. Given language specifications describing mission goals and team capabilities, an LLM generates grounded subtasks which are validated for feasibility. Subtasks are then assigned to robots based on capabilities such as traversability or perception and refined given feedback collected during online operation. In simulation experiments with closed-loop perception and control, our framework achieves nearly twice the success rate compared to prior LLM-enabled heterogeneous teaming approaches. In real-world experiments with a Clearpath Jackal, a Clearpath Husky, a Boston Dynamics Spot, and a high-altitude UAV, our method achieves an 87\% success rate in missions requiring reasoning about robot capabilities and refining subtasks with online feedback. More information is provided at https://zacravichandran.github.io/SPINE-HT.

This paper presents a physics-aware cyberphysical resilience framework for radial microgrids under coordinated cyberattacks. The proposed approach models the attacker through a hypergraph neural network (HGNN) enhanced with model agnostic metalearning (MAML) to rapidly adapt to evolving defense strategies and predict high-impact contingencies. The defender is modeled via a bi-level Stackelberg game, where the upper level selects optimal tie-line switching and distributed energy resource (DER) dispatch using an Alternating Direction Method of Multipliers (ADMM) coordinator embedded within the Non-dominated Sorting Genetic Algorithm II (NSGA-II). The framework simultaneously optimizes load served, operational cost, and voltage stability, ensuring all post-defense states satisfy network physics constraints. The methodology is first validated on the IEEE 69-bus distribution test system with 12 DERs, 8 critical loads, and 5 tie-lines, and then extended to higher bus systems including the IEEE 123-bus feeder and a synthetic 300-bus distribution system. Results show that the proposed defense strategy restores nearly full service for 90% of top-ranked attacks, mitigates voltage violations, and identifies Feeder 2 as the principal vulnerability corridor. Actionable operating rules are derived, recommending pre-arming of specific tie-lines to enhance resilience, while higher bus system studies confirm scalability of the framework on the IEEE 123-bus and 300-bus systems.