While there now exists a large literature on policy evaluation and learning, much of prior work assumes that the treatment assignment of one unit does not affect the outcome of another unit. Unfortunately, ignoring interference may lead to biased policy evaluation and ineffective learned policies. For example, treating influential individuals who have many friends can generate positive spillover effects, thereby improving the overall performance of an individualized treatment rule (ITR). We consider the problem of evaluating and learning an optimal ITR under clustered network interference (also known as partial interference) where clusters of units are sampled from a population and units may influence one another within each cluster. Unlike previous methods that impose strong restrictions on spillover effects, the proposed methodology only assumes a semiparametric structural model where each unit's outcome is an additive function of individual treatments within the cluster. Under this model, we propose an estimator that can be used to evaluate the empirical performance of an ITR. We show that this estimator is substantially more efficient than the standard inverse probability weighting estimator, which does not impose any assumption about spillover effects. We derive the finite-sample regret bound for a learned ITR, showing that the use of our efficient evaluation estimator leads to the improved performance of learned policies. Finally, we conduct simulation and empirical studies to illustrate the advantages of the proposed methodology.

Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.

Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), as they significantly tighten the dual bounds and improve the solving performance. A key problem for cuts is when to stop cuts generation, which is important for the efficiency of solving MILPs. However, many modern MILP solvers employ hard-coded heuristics to tackle this problem, which tends to neglect underlying patterns among MILPs from certain applications. To address this challenge, we formulate the cuts generation stopping problem as a reinforcement learning problem and propose a novel hybrid graph representation model (HYGRO) to learn effective stopping strategies. An appealing feature of HYGRO is that it can effectively capture both the dynamic and static features of MILPs, enabling dynamic decision-making for the stopping strategies. To the best of our knowledge, HYGRO is the first data-driven method to tackle the cuts generation stopping problem. By integrating our approach with modern solvers, experiments demonstrate that HYGRO significantly improves the efficiency of solving MILPs compared to competitive baselines, achieving up to 31% improvement.

Alan Turing proposed in 1950 a framework called an imitation game to decide if a machine could think. Using mathematics developed largely after Turing -- category theory -- we analyze a broader class of universal imitation games (UIGs), which includes static, dynamic, and evolutionary games. In static games, the participants are in a steady state. In dynamic UIGs, "learner" participants are trying to imitate "teacher" participants over the long run. In evolutionary UIGs, the participants are competing against each other in an evolutionary game, and participants can go extinct and be replaced by others with higher fitness. We use the framework of category theory -- in particular, two influential results by Yoneda -- to characterize each type of imitation game. Universal properties in categories are defined by initial and final objects. We characterize dynamic UIGs where participants are learning by inductive inference as initial algebras over well-founded sets, and contrast them with participants learning by conductive inference over the final coalgebra of non-well-founded sets. We briefly discuss the extension of our categorical framework for UIGs to imitation games on quantum computers.

How to solve high-dimensional linear programs (LPs) efficiently is a fundamental question. Recently, there has been a surge of interest in reducing LP sizes using \textit{random projections}, which can accelerate solving LPs independently of improving LP solvers. In this paper, we explore a new direction of \emph{data-driven projections}, which use projection matrices learned from data instead of random projection matrices. Given data of past $n$-dimensional LPs, we learn an $n\times k$ projection matrix such that $n > k$. When addressing a future LP instance, we reduce its dimensionality from $n$ to $k$ via the learned projection matrix, solve the resulting LP to obtain a $k$-dimensional solution, and apply the learned matrix to it to recover an $n$-dimensional solution. On the theoretical side, a natural question is: how much data is sufficient to ensure the quality of recovered solutions? We address this question based on the framework of \textit{data-driven algorithm design}, which connects the amount of data sufficient for establishing generalization bounds to the \textit{pseudo-dimension} of performance metrics. We obtain an $\tilde{\mathrm{O}}(nk^2)$ upper bound on the pseudo-dimension, where $\tilde{\mathrm{O}}$ compresses logarithmic factors. We also provide an $\Omega(nk)$ lower bound, implying our result is tight up to an $\tilde{\mathrm{O}}(k)$ factor. On the practical side, we explore two natural methods for learning projection matrices: PCA- and gradient-based methods. While the former is simple and efficient, the latter can sometimes lead to better solution quality. Our experiments confirm the practical benefit of learning projection matrices from data, achieving significantly higher solution quality than the existing random projection while greatly reducing the time for solving LPs.

We explore a novel methodology for constructing confidence regions for parameters of linear models, using predictions from any arbitrary predictor. Our framework requires minimal assumptions on the noise and can be extended to functions deviating from strict linearity up to some adjustable threshold, thereby accommodating a comprehensive and pragmatically relevant set of functions. The derived confidence regions can be cast as constraints within a Mixed Integer Linear Programming framework, enabling optimisation of linear objectives. This representation enables robust optimization and the extraction of confidence intervals for specific parameter coordinates. Unlike previous methods, the confidence region can be empty, which can be used for hypothesis testing. Finally, we validate the empirical applicability of our method on synthetic data.

The SINDy algorithm has been successfully used to identify the governing equations of dynamical systems from time series data. However, SINDy assumes the user has prior knowledge of the variables in the system and of a function library that can act as a basis for the system. In this paper, we demonstrate on real world data how the Augmented SINDy algorithm outperforms SINDy in the presence of system variable uncertainty. We then show SINDy can be further augmented to perform robustly when both kinds of uncertainty are present.

This paper presents a {\delta}-PI algorithm which is based on damped Newton method for the H{\infty} tracking control problem of unknown continuous-time nonlinear system. A discounted performance function and an augmented system are used to get the tracking Hamilton-Jacobi-Isaac (HJI) equation. Tracking HJI equation is a nonlinear partial differential equation, traditional reinforcement learning methods for solving the tracking HJI equation are mostly based on the Newton method, which usually only satisfies local convergence and needs a good initial guess. Based upon the damped Newton iteration operator equation, a generalized tracking Bellman equation is derived firstly. The {\delta}-PI algorithm can seek the optimal solution of the tracking HJI equation by iteratively solving the generalized tracking Bellman equation. On-policy learning and off-policy learning {\delta}-PI reinforcement learning methods are provided, respectively. Off-policy version {\delta}-PI algorithm is a model-free algorithm which can be performed without making use of a priori knowledge of the system dynamics. NN-based implementation scheme for the off-policy {\delta}-PI algorithms is shown. The suitability of the model-free {\delta}-PI algorithm is illustrated with a nonlinear system simulation.

Datasets encountered in scientific and engineering applications appear in complex formats (e.g., images, multivariate time series, molecules, video, text strings, networks). Graph theory provides a unifying framework to model such datasets and enables the use of powerful tools that can help analyze, visualize, and extract value from data. In this work, we present PlasmoData.jl, an open-source, Julia framework that uses concepts of graph theory to facilitate the modeling and analysis of complex datasets. The core of our framework is a general data modeling abstraction, which we call a DataGraph. We show how the abstraction and software implementation can be used to represent diverse data objects as graphs and to enable the use of tools from topology, graph theory, and machine learning (e.g., graph neural networks) to conduct a variety of tasks. We illustrate the versatility of the framework by using real datasets: i) an image classification problem using topological data analysis to extract features from the graph model to train machine learning models; ii) a disease outbreak problem where we model multivariate time series as graphs to detect abnormal events; and iii) a technology pathway analysis problem where we highlight how we can use graphs to navigate connectivity. Our discussion also highlights how PlasmoData.jl leverages native Julia capabilities to enable compact syntax, scalable computations, and interfaces with diverse packages.

Lambert's problem concerns with transferring a spacecraft from a given initial to a given terminal position within prescribed flight time via velocity control subject to a gravitational force field. We consider a probabilistic variant of the Lambert problem where the knowledge of the endpoint constraints in position vectors are replaced by the knowledge of their respective joint probability density functions. We show that the Lambert problem with endpoint joint probability density constraints is a generalized optimal mass transport (OMT) problem, thereby connecting this classical astrodynamics problem with a burgeoning area of research in modern stochastic control and stochastic machine learning. This newfound connection allows us to rigorously establish the existence and uniqueness of solution for the probabilistic Lambert problem. The same connection also helps to numerically solve the probabilistic Lambert problem via diffusion regularization, i.e., by leveraging further connection of the OMT with the Schr\"odinger bridge problem (SBP). This also shows that the probabilistic Lambert problem with additive dynamic process noise is in fact a generalized SBP, and can be solved numerically using the so-called Schr\"odinger factors, as we do in this work. We explain how the resulting analysis leads to solving a boundary-coupled system of reaction-diffusion PDEs where the nonlinear gravitational potential appears as the reaction rate. We propose novel algorithms for the same, and present illustrative numerical results. Our analysis and the algorithmic framework are nonparametric, i.e., we make neither statistical (e.g., Gaussian, first few moments, mixture or exponential family, finite dimensionality of the sufficient statistic) nor dynamical (e.g., Taylor series) approximations.