We study supervisory switching control for partially-observed linear dynamical systems. The objective is to identify and deploy a suitable controller for the unknown system by periodically selecting among a collection of N candidate controllers, some of which may destabilize the underlying system. While classical estimator-based supervisory control guarantees asymptotic stability, it lacks quantitative finite-time performance bounds. Conversely, current non-asymptotic methods in both online learning and system identification require restrictive assumptions that are incompatible in a control setting, such as system stability, which preclude testing potentially unstable controllers. To bridge this gap, we propose a novel, non-asymptotic analysis of supervisory control that adapts multi-armed bandit algorithms to a control-theoretic setting. The proposed data-driven algorithm evaluates candidate controllers via scoring criteria that leverage system observability to isolate the effects of state history, enabling both detection of destabilizing controllers and accurate system identification. We present two algorithmic variants with dimension-free, finite-time guarantees, where each identifies the matching controller in O(Nlog2 N) steps, while simultaneously achieving finite L2-gain with respect to system disturbances.

This paper presents a task-oriented computational framework to enhance Visual-Inertial Navigation (VIN) in robots, addressing challenges such as limited time and energy resources. The framework strategically selects visual features using a Mean Square Error (MSE)-based, non-submodular objective function and a simplified dynamic anticipation model. To address the NP‐hardness of this problem, we introduce four polynomial‐time approximation algorithms: a classic greedy method with constant‐factor guarantees; a low‐rank greedy variant that significantly reduces computational complexity; a randomized greedy sampler that balances efficiency and solution quality; and a linearization‐based selector based on a first‐order Taylor expansion for near‐constant‐time execution. We establish rigorous performance bounds by leveraging submodularity ratios, curvature, and element‐wise curvature analyses. Extensive experiments on both standardized benchmarks and a custom control‐aware platform validate our theoretical results, demonstrating that these methods achieve strong approximation guarantees while enabling real‐time deployment.

The association between tie strength and network structure is a fundamental topic in the analysis of complex social systems. We study this association by analyzing tie strength using higher-order networks, an increasingly relevant model which can encode group interactions between three or more individuals. First, we introduce three measures based on algebraic topology which characterize the network context and influence of an edge. Our experimental results across 15 datasets indicate that these measures outperform standard network proxies in estimating tie strength as a function of the unweighted network topology. These measures can further replicate and explain a puzzle wherein certain bridging ties are surprisingly strong. We then consider a single centrality measure which combines the three initial measures, is highly inversely correlated with tie strength, and can be interpreted through an information exchange process that highlights ties that have access to useful information. In this sense, we are able to illuminate the information advantages of weak ties due to their network position.

Rotary Position Embeddings (RoPE) provide transformers with a fixed grid of positional frequencies, yet trained models use these frequencies highly non-uniformly. We study what determines this frequency usage and propose a data-centered explanation: RoPE frequencies are selected to match the relative-distance structure of the training data. Viewing each frequency as a positional lens, we formalize a field-resolution tradeoff and show that, for a data-induced dependency profile of width W, the optimal frequency scales as 1/W. This frequency-matching principle explains controlled observations on synthetic and text-based data, and suggests that the mid-low frequency bands observed in language models arise from the multi-scale dependency structure of natural language. We further connect frequency selection to position-interpolation-based length generalization: scaling frequencies down expands the effective field while reducing resolution. This helps when longer-context dependencies are approximate dilations of those seen during training, but can fail when relevant dependencies do not scale with context length. Empirically, we show that natural language exhibits approximate self-similarity across positional scales, explaining why test-time frequency scaling can support long-context generalization. Overall, our results identify a data-driven mechanism behind emergent RoPE frequency usage and show that long-context generalization depends on two forms of scale matching: between learned frequencies and training-time dependencies, and between frequency scaling and how those dependencies extend to longer contexts.

We study the problem of imitating an expert demonstrator in a discrete-time, continuous stateand-action space control system. We show that there exist stable dynamics (i.e. contracting exponentially quickly) and smooth, deterministic experts such that any smooth, deterministic imitator policy necessarily suffers error on execution that is exponentially larger, as a function of problem horizon, than the error under the distribution of expert training data. Our negative result applies to both behavior cloning and offline-RL algorithms, unless they produce highly improper imitator policies - those which are non-smooth, non-Markovian, or which exhibit highly state-dependent stochasticity - or unless the expert trajectory distribution is sufficiently spread. We provide preliminary evidence of the benefits of these more complex policy parameterizations, explicating the benefits of today's popular policy parameterizations in robot learning (e.g. action-chunking and diffusion-policies). We also establish a host of complementary negative and positive results for imitation in control systems.

We consider the centralized optimal estimation problem in spatially distributed systems. We use the setting of spatially invariant systems as an idealization for which concrete and detailed results are given. Such estimators are known to have a degree of spatial localization in the sense that the estimator gains decay in space, with the spatial decay rates serving as a proxy for how far measurements need to be shared in an optimal distributed estimator. In particular, we examine the dependence of spatial decay rates on problem specifications such as system dynamics, measurement and process noise variances, as well as their spatial autocorrelations. We propose non-dimensional parameters that characterize the decay rates as a function of problem specifications. In particular, we find an interesting matching condition between the characteristic lengthscale of the dynamics and the measurement noise correlation lengthscale for which the optimal centralized estimator is completely decentralized. A new technique - termed the Branch Point Locus - is introduced to quantify spatial decay rates in terms of analyticity regions in the complex spatial frequency plane. Our results are illustrated through two case studies of systems with dynamics modeled by diffusion and the Swift-Hohenberg equation, respectively.

In this paper, we develop a game-theoretic model of sharing decisions among users of a Twitter-like social network. Agents have a subjective prior on an unobservable real-valued state, representing their beliefs on a topic that is subject to a binary action. Agents cast their binary action, matching the sign of the state. Before the action stage, a small fraction of agents receive a piece of news which impacts their beliefs. Those who receive the news update their belief and make a decision as to whether to share the news with their followers in order to influence their beliefs, and in turn their actions, given a fixed cost for sharing. We characterize the underlying news spread as an endogenous Susceptible-Infected (SI) epidemic process and derive agents' sharing decisions as well as the size of the sharing cascade at the equilibrium of the game. We show that lower credibility news can result in a larger cascade than fully credible news provided that the network connectivity surpasses a connectivity limit. We further delineate the relationship between cascade size, network connectivity, and news credibility in terms of polarization and diversity in prior beliefs: We demonstrate that increased polarization reduces the connectivity limit whereas larger in-party diversity has a non-monotone effect on the connectivity limit, which depends on both the levels of polarization and in-party diversity. Our results provide a theoretical foundation for recent empirical observations demonstrating faster and wider spread of low-credibility and false information on social networks.

Deep recommender systems rely heavily on large embedding tables to handle high-cardinality categorical features such as user/item identifiers, and face significant memory constraints at scale. To tackle this challenge, hashing techniques are often employed to map multiple entities to the same embedding and thus reduce the size of the embedding tables. Concurrently, graph-based collaborative signals have emerged as powerful tools in recommender systems, yet their potential for optimizing embedding table reduction remains unexplored. This paper introduces GraphHash, the first graph-based approach that leverages modularity-based bipartite graph clustering on user-item interaction graphs to reduce embedding table sizes. We demonstrate that the modularity objective has a theoretical connection to message-passing, which provides a foundation for our method. By employing fast clustering algorithms, GraphHash serves as a computationally efficient proxy for message-passing during preprocessing and a plug-and-play graph-based alternative to traditional ID hashing. Extensive experiments show that GraphHash substantially outperforms diverse hashing baselines on both retrieval and click-through-rate prediction tasks. In particular, GraphHash achieves on average a 101.52% improvement in recall when reducing the embedding table size by more than 75%, highlighting the value of graph-based collaborative information for model reduction.

We study finite approximation of the Pareto front for the multi-objective linear quadratic regulator (LQR). Classical results characterize Pareto-optimal controllers through weighted sums of the objectives, but do not quantify the effect of discretizing the scalarization weights. We prove the needed sensitivity property, showing that an ε perturbation of the weights changes the full objective vector by O(ε). Consequently, computing optimal controls over a finite grid of the weight simplex uniformly approximates the Pareto front. We also extend the guarantee to unknown dynamics via certainty equivalence, showing that if the model estimates are accurate to the target accuracy, the same procedure stabilizes the true system and attains the same Pareto-front guarantee.

We study tensor completion (TC) through the lens of low-rank tensor decomposition (TD). Many TD algorithms use fast alternating minimization methods to solve highly structured linear regression problems at each step (e.g., for CP, Tucker, and tensor-train decompositions). However, such algebraic structure is often lost in TC regression problems, making direct extensions unclear. This work proposes a novel lifting method for approximately solving TC regression problems using structured TD regression algorithms as blackbox subroutines, enabling sublinear-time methods. We analyze the convergence rate of our approximate Richardson iteration-based algorithm, and our empirical study shows that it can be 100x faster than direct methods for CP completion on real-world tensors.

Transformer-based models demonstrate a remarkable ability for in-context learning (ICL), where they can adapt to unseen tasks from a few prompt examples without parameter updates. Recent research has illuminated how Transformers perform ICL, showing that the optimal linear self-attention (LSA) mechanism can implement one step of gradient descent for linear least-squares objectives when trained on random linear regression tasks. Building on this, we investigate ICL for nonlinear function classes. We first prove that LSA is inherently incapable of outperforming linear predictors on nonlinear tasks, underscoring why prior solutions cannot readily extend to these problems. To overcome this limitation, we analyze a Transformer block consisting of LSA and feed-forward layers inspired by the gated linear units (GLU), which is a standard component of modern Transformers. We show that this block achieves nonlinear ICL by implementing one step of gradient descent on a polynomial kernel regression loss. Furthermore, our analysis reveals that the expressivity of a single block is inherently limited by its dimensions. We then show that a deep Transformer can overcome this bottleneck by distributing the computation of richer kernel functions across multiple blocks, performing block-coordinate descent in a high-dimensional feature space that a single block cannot represent. Our findings highlight that the feed-forward layers provide a crucial and scalable mechanism by which Transformers can express nonlinear representations for ICL.

Recent studies have revealed various manifestations of position bias in transformer architectures, from the "lost-in-the-middle" phenomenon to attention sinks, yet a comprehensive theoretical understanding of how attention masks and positional encodings shape these biases remains elusive. This paper presents a graph-theoretic framework for analyzing position bias in multi-layer attention. Modeling attention masks as directed graphs, we quantify how tokens interact with contextual information based on their sequential positions. We uncover two key insights: First, causal masking inherently biases attention toward earlier positions, as tokens in deeper layers attend to increasingly more contextualized representations of earlier tokens. Second, we characterize the competing effects of the causal mask and relative positional encodings, such as the decay mask and rotary positional encoding (RoPE): while both mechanisms introduce distance-based decay within individual attention maps, their aggregate effect across multiple attention layers—coupled with the causal mask—leads to a trade-off between the long-term decay effects and the cumulative importance of early sequence positions. Through controlled numerical experiments, we not only validate our theoretical findings but also reproduce position biases observed in real-world LLMs. Our framework offers a principled foundation for understanding positional biases in transformers, shedding light on the complex interplay of attention mechanism components and guiding more informed architectural design.

The dynamics of random transitive delegations on a graph are of particular interest when viewed through the lens of an emerging voting paradigm: liquid democracy. This paradigm allows voters to choose between directly voting and transitively delegating their votes to other voters so that those selected cast a vote weighted by the number of delegations that they received. In the epistemic setting, where voters decide on a binary issue for which there is a ground truth, previous work showed that a few voters may amass such a large amount of influence that liquid democracy is less likely to identify the ground truth than direct voting. We quantify the amount of permissible concentration of power and examine more realistic delegation models, showing that they behave well by ensuring that (with high probability) there is a permissible limit on the maximum number of delegations received. Our theoretical results demonstrate that the delegation process is similar to well-known processes on random graphs that are sufficiently bounded for our purposes. Along the way, we prove new bounds on the size of the largest component in an infinite Pólya urn process, which may be of independent interest. In addition, we empirically validate the theoretical results, running six experiments (for a total of N = 168 participants, 62 delegation graphs, and over 11,000 votes collected). We find that empirical delegation behaviors meet the conditions for our positive theoretical guarantees. Overall, our work alleviates concerns raised about liquid democracy and bolsters the case for the applicability of this emerging paradigm. This paper was accepted by Martin Bichler, market design, platform, and demand analytics. Funding: This work was supported by the Michael Hammer Fellowship, the Office of Naval Research [2016 Vannevar Bush Faculty Fellowship, 2020 ONR Vannevar Bush Faculty Fellowship, and Grant N00014-20-1-2488], the Office of Secretary of Defense [Grants ARO MURI W911NF-19-0217 and ARO W911NF-17-1-0592], Simons [Investigator Award 622132], the Open Philanthropy Foundation, and the National Science Foundation [Grants IIS-178108, CCF-1733556, CCF-1918421, CCF-2007080, IIS-1703846, and IIS-2024287]. J. Y. Halpern was supported by a grant from the Cooperative AI Foundation [ARO Grant W911NF-22-1-0061]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.02470 .

This article introduces a leave-one-out regression adjustment estimator (LOORA) for estimating average treatment effects in randomized controlled trials. The method removes the finite-sample bias of conventional regression adjustment and provides exact variance expressions for LOORA versions of the Horvitz-Thompson and difference-in-means estimators under simple and complete random assignment. Ridge regularization limits the influence of high-leverage observations, improving stability and precision in small samples. In large samples, LOORA attains the asymptotic efficiency of regression-adjusted estimator as characterized by Lin (2013, Annals of Applied Statistics), while remaining exactly unbiased. To construct confidence intervals, we rely on asymptotic variance estimates that treat the estimator as a two-step procedure, accounting for both the regression adjustment and the random assignment stages. Two within-subject experimental applications that provide realistic joint distributions of potential outcomes as ground truth show that LOORA eliminates substantial biases and achieves close-to-nominal confidence interval coverage.

Residual connections and normalization layers have become standard design choices for graph neural networks (GNNs), and were proposed as solutions to the mitigate the oversmoothing problem in GNNs. However, how exactly these methods help alleviate the oversmoothing problem from a theoretical perspective is not well understood. In this work, we provide a formal and precise characterization of (linearized) GNNs with residual connections and normalization layers. We establish that (a) for residual connections, the incorporation of the initial features at each layer can prevent the signal from becoming too smooth, and determines the subspace of possible node representations; (b) batch normalization prevents a complete collapse of the output embedding space to a one-dimensional subspace through the individual rescaling of each column of the feature matrix. This results in the convergence of node representations to the top-k eigenspace of the message-passing operator; (c) moreover, we show that the centering step of a normalization layer — which can be understood as a projection — alters the graph signal in message-passing in such a way that relevant information can become harder to extract. Building on the last theoretical insight, we introduce GraphNormv2, a novel and principled normalization layer. GraphNormv2 features a learnable centering step designed to preserve the integrity of the original graph signal. Experimental results corroborate the effectiveness of our method, demonstrating improved performance across various GNN architectures and tasks.

We study the inductive biases of diffusion models with a conditioning-variable, which have seen widespread application as both text-conditioned generative image models and observation-conditioned continuous control policies. We observe that when these models are queried conditionally, their generations consistently deviate from the idealized "denoising" process upon which diffusion models are formulated, inducing disagreement between popular sampling algorithms (e.g. DDPM, DDIM). We introduce *Schedule Deviation*, a rigorous measure which captures the rate of deviation from a standard denoising process, and provide a methodology to compute it. Crucially, we demonstrate that the deviation from an idealized denoising process occurs irrespective of the model capacity or amount of training data. We posit that this phenomenon occurs due to the difficulty of bridging distinct denoising flows across different parts of the conditioning space and show theoretically how such a phenomenon can arise through an inductive bias towards smoothness.

Social pressure is a key factor affecting the evolution of opinions on networks in many types of settings, pushing people to conform to their neighbors' opinions. To study this, the interacting Pólya urn model was introduced by Jadbabaie et al. [1], in which each agent has two kinds of opinion: inherent beliefs, which are hidden from the other agents and fixed; and declared opinions, which are randomly sampled at each step from a distribution which depends on the agent's inherent belief and her neighbors' past declared opinions (the social pressure component), and which is then communicated to their neighbors. Each agent also has a bias parameter denoting her level of resistance to social pressure. At every step, each agent updates her declared opinion (simultaneously with all other agents) according to her neighbors' aggregate past declared opinions, her inherent belief, and her bias parameter. We study the asymptotic behavior of this opinion dynamics model and show that agents' declaration probabilities converge almost surely in the limit using Lyapunov theory and stochastic approximation techniques. We also derive a sufficient condition for the agents to approach consensus on their declared opinions.

It is commonly accepted that some phenomena are social: for example, individuals' smoking habits often correlate with those of their peers. Such correlations can have a variety of explanations, such as direct contagion or shared socioeconomic circumstances. The network linear-in-means model is a workhorse statistical model which incorporates these peer effects by including average neighborhood characteristics as regressors. Although the model's parameters are identifiable under mild structural conditions on the network, it remains unclear whether identification ensures reliable estimation in the "infill" asymptotic setting, where a single network grows in size. We show that when covariates are i.i.d. and the average network degree of nodes increases with the population size, standard estimators suffer from bias or slow convergence rates due to asymptotic collinearity induced by network averaging. As an alternative, we demonstrate that linear-in-sums models, which are based on aggregate rather than average neighborhood characteristics, do not exhibit such issues as long as the network degrees have some nontrivial variation, a condition satisfied by most network models.

Surety bonds are financial agreements between a contractor (principal) and obligee (project owner) to complete a project. However, most large-scale projects involve multiple contractors, creating a network and introducing the possibility of incomplete obligations to propagate and result in project failures. Typical models for risk assessment assume independent failure probabilities within each contractor. However, we take a network approach, modeling the contractor network as a directed graph where nodes represent contractors and project owners and edges represent contractual obligations with associated financial records. To understand risk propagation throughout the contractor network, we extend the celebrated Friedkin-Johnsen model and introduce a stochastic process to simulate principal failures across the network. From a theoretical perspective, we show that under natural monotonicity conditions on the contractor network, incorporating network effects leads to increases in both the average risk and the tail probability mass of the loss distribution (i.e. larger right-tail risk) for the surety organization. We further use data from a partnering insurance company to validate our findings, estimating an approximately 2