The goal of machine learning is to find models that minimize prediction error on data that has not yet been seen. Its operational paradigm assumes access to a dataset S and articulates a scheme for evaluating how well a given model performs on an arbitrary sample. The sample can be S (in which case we speak of “in-sample” performance) or some entirely new S' (in which case we speak of “out-of-sample” performance). Traditional analysis of generalization assumes that both in- and out-of-sample data are i.i.d. draws from an infinite population. However, these probabilistic assumptions cannot be verified even in principle. This paper presents an alternative view of generalization through the lens of sensitivity analysis of solutions of optimization problems to perturbations in the problem data. Under this framework, generalization bounds are obtained by purely deterministic means and take the form of variational principles that relate in-sample and out-of-sample evaluations through an error term that quantifies how close out-of-sample data are to in-sample data. Statistical assumptions can then be used ex post to characterize the situations when this error term is small (either on average or with high probability).

ChatGPT was launched on November 30, 2022; the r/ChatGPT subreddit was created just one day later. Since then, chatbot-based AI products have gone from niche proofs-of-concept to widely-used household names. However, the ways in which adoption has developed, especially among non-experts, remains poorly understood. In this paper, we propose a principled framework for using social media as a data source for understanding the societal impact of widely-adopted consumer AI products, as well as a general approach to monitoring for societally-impactful trends in real time. We apply our framework to conduct what is, to the best of our knowledge, the first longitudinal study of r/ChatGPT. We find that, overall, r/ChatGPT posts over time illustrate the normalization of ChatGPT as an everyday consumer product rather than an exceptional, novel technology. However, our retrospective analysis also finds that posts about using ChatGPT for mental health support, and posts about developing emotional attachments to ChatGPT, both rise steadily in frequency immediately after the launch of GPT-4o in May 2024. We show that our real-time method can detect the increase in emotional engagement as early as October 2024—months before OpenAI made any (public) acknowledgment of this impact.

Paul Meehl's foundational work "Clinical versus Statistical Prediction," provided early theoretical justification and empirical evidence of the superiority of statistical methods over clinical judgment. Despite a century of empirical evidence supporting Meehl's central thesis, from early parole prediction studies in the 1920s to modern meta-analyses, confusion persists regarding when and why his troubling finding applies. This paper provides a contemporary theoretical justification for Meehl's result. Importantly, Meehl's prediction problems require a small set of possible outcomes and machine-readable data. Second, individual predictions and decisions are evaluated only on average. This formulation leads to a natural analysis from statistical decision theory, which shows that statistical rules are more accurate than clinical intuition almost by definition. Meehl's prediction paradox is an example of metrical determinism, where the rules of evaluation implicitly determine the best procedure. The decision-theoretic analysis of Meehl's problem elucidates the utility of algorithmic systems as decision-support tools, but also reveals their natural shortcomings, inducing expertise erosion, decision fatigue, and the usurpation of discretionary judgment.

N-of-1 experiments, where a unit serves as its own control and treatment in different time windows, have been used in certain medical contexts for decades. However, due to effects that accumulate over long time windows and interventions that have complex evolution, a lack of robust inference tools has limited the widespread applicability of such N-of-1 designs. This work combines techniques from experiment design in causal inference and system identification from control theory to provide such an inference framework. We derive a model of the dynamic interference effect that arises in linear time-invariant dynamical systems. We show that a family of causal estimands analogous to those studied in potential outcomes are estimable via a standard estimator derived from the method of moments. We derive formulae for higher moments of this estimator and describe conditions under which N-of-1 designs may provide faster ways to estimate the effects of interventions in dynamical systems. We also provide conditions under which our estimator is asymptotically normal and derive valid confidence intervals for this setting.

This commentary proposes a framework for understanding the role of statistics in policy-making, regulation, and bureaucratic systems. I introduce the concept of "ex ante policy," describing statistical rules and procedures designed before data collection to govern future actions. Through examining examples, particularly clinical trials, I explore how ex ante policy serves as a calculus of bureaucracy, providing numerical foundations for governance through clear, transparent rules. The ex ante frame obviates heated debates about inferential interpretations of probability and statistical tests, p-values, and rituals. I conclude by calling for a deeper appreciation of statistics' bureaucratic function and suggesting new directions for research in policy-oriented statistical methodology.

The need for developing model evaluations beyond static benchmarking, especially in the post-deployment phase, is now well-understood. At the same time, concerns about the concentration of power in deployed AI systems have sparked a keen interest in 'democratic' or 'public' AI. In this work, we bring these two ideas together by proposing mechanisms for aggregated individual reporting (AIR), a framework for post-deployment evaluation that relies on individual reports from the public. An AIR mechanism allows those who interact with a specific, deployed (AI) system to report when they feel that they may have experienced something problematic; these reports are then aggregated over time, with the goal of evaluating the relevant system in a fine-grained manner. This position paper argues that individual experiences should be understood as an integral part of post-deployment evaluation, and that the scope of our proposed aggregated individual reporting mechanism is a practical path to that end. On the one hand, individual reporting can identify substantively novel insights about safety and performance; on the other, aggregation can be uniquely useful for informing action. From a normative perspective, the post-deployment phase completes a missing piece in the conversation about 'democratic' AI. As a pathway to implementation, we provide a workflow of concrete design decisions and pointers to areas requiring further research and methodological development.

We show that several popular algorithms for reinforcement learning in large language models with binary rewards can be viewed as stochastic gradient ascent on a monotone transform of the probability of a correct answer given a prompt. In particular, the transformation associated with rejection sampling algorithms is the logarithm and that associated with the GRPO algorithm is the arcsine of the square root.

In computed tomography (CT), the forward model consists of a linear Radon transform followed by an exponential nonlinearity based on the attenuation of light according to the Beer–Lambert Law. Conventional reconstruction often involves inverting this nonlinearity as a preprocessing step and then solving a convex inverse problem. However, this nonlinear measurement preprocessing required to use the Radon transform is poorly conditioned in the vicinity of high-density materials, such as metal. This preprocessing makes CT reconstruction methods numerically sensitive and susceptible to artifacts near high-density regions. In this paper, we study a technique where the signal is directly reconstructed from raw measurements through the nonlinear forward model. Though this optimization is nonconvex, we show that gradient descent provably converges to the global optimum at a geometric rate, perfectly reconstructing the underlying signal with a near minimal number of random measurements. We also prove similar results in the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal. This is achieved by enforcing prior structural information about the signal through constraints on the optimization variables. We illustrate the benefits of direct nonlinear CT reconstruction with cone-beam CT experiments on synthetic and real 3D volumes. We show that this approach reduces metal artifacts compared to a commercial reconstruction of a human skull with metal dental crowns.

This paper provides elementary analyses of the regret and generalization of minimum-norm interpolating classifiers (MNIC). The MNIC is the function of smallest Reproducing Kernel Hilbert Space norm that perfectly interpolates a label pattern on a finite data set. We derive a mistake bound for MNIC and a regularized variant that holds for all data sets. This bound follows from elementary properties of matrix inverses. Under the assumption that the data is independently and identically distributed, the mistake bound implies that MNIC generalizes at a rate proportional to the norm of the interpolating solution and inversely proportional to the number of data points. This rate matches similar rates derived for margin classifiers and perceptrons. We derive several plausible generative models where the norm of the interpolating classifier is bounded or grows at a rate sublinear in $n$. We also show that as long as the population class conditional distributions are sufficiently separable in total variation, then MNIC generalizes with a fast rate.

We investigate adaptive protocols for the elimination or reduction of the use of medications or addictive substances. We formalize this problem as online optimization, minimizing the cumulative dose subject to constraints on individual well-being. We adapt a model of addiction from the psychology literature and show how it can be described by a class of linear time-invariant systems. For such systems, the optimal policy amounts to taking the smallest dose that maintains well-being. We derive a simple protocol based on integral control that requires no system identification, only needing approximate knowledge of the instantaneous dose response. This protocol is robust to model misspecification and is able to maintain an individual's well-being during the tapering process. Numerical experiments demonstrate that the adaptive protocol outperforms non-adaptive methods in terms of both maintenance of well-being and rate of dose reduction.

We introduce k-planes, a white-box model for radiance fields in arbitrary dimensions. Our model uses planes to represent a d-dimensional scene, providing a seamless way to go from static (d = 3) to dynamic (d= 4) scenes. This planar factorization makes adding dimension-specific priors easy, e.g. temporal smoothness and multi-resolution spatial structure, and induces a natural decomposition of static and dynamic components of a scene. We use a linear feature decoder with a learned color basis that yields similar performance as a nonlinear black-box MLP decoder. Across a range of synthetic and real, static and dynamic, fixed and varying appearance scenes, k-planes yields competitive and often state-of-the-art recon- struction fidelity with low memory usage, achieving 1000x compression over a full 4D grid, and fast optimization with a pure PyTorch implementation. For video results and code, please see sarafridov.github.io/K-Planes.

We introduce Plenoxels (plenoptic voxels), a systemfor photorealistic view synthesis. Plenoxels represent a scene as a sparse 3D grid with spherical harmonics. This representation can be optimized from calibrated images via gradient methods and regularization without any neural components. On standard, benchmark tasks, Plenoxels are optimized two orders of magnitude faster than Neural Radiance Fields with no loss in visual quality. For video and code, please see https://alexyu.net/plenoxels.

A recent randomized trial evaluated the impact of mask promotion on COVID-19-related outcomes. We find that staff behavior in both unblinded and supposedly blinded steps caused large and statistically significant imbalances in population sizes. These denominator differences constitute the rate differences observed in the trial, complicating inferences of causality.

Optimization techniques are at the core of data science, including data analysis and machine learning. An understanding of basic optimization techniques and their fundamental properties provides important grounding for students, researchers, and practitioners in these areas. This text covers the fundamentals of optimization algorithms in a compact, self-contained way, focusing on the techniques most relevant to data science. An introductory chapter demonstrates that many standard problems in data science can be formulated as optimization problems. Next, many fundamental methods in optimization are described and analyzed, including: gradient and accelerated gradient methods for unconstrained optimization of smooth (especially convex) functions; the stochastic gradient method, a workhorse algorithm in machine learning; the coordinate descent approach; several key algorithms for constrained optimization problems; algorithms for minimizing nonsmooth functions arising in data science; foundations of the analysis of nonsmooth functions and optimization duality; and the back-propagation approach, relevant to neural networks.

Designing recommendation systems that serve content aligned with time varying preferences requires proper accounting of the feedback effects of recommendations on human behavior and psychological condition. We argue that modeling the influence of recommendations on people's preferences must be grounded in psychologically plausible models. We contribute a methodology for developing grounded dynamic preference models. We demonstrate this method with models that capture three classic effects from the psychology literature: Mere-Exposure, Operant Conditioning, and Hedonic Adaptation. We conduct simulation-based studies to show that the psychological models manifest distinct behaviors that can inform system design. Our study has two direct implications for dynamic user modeling in recommendation systems. First, the methodology we outline is broadly applicable for psychologically grounding dynamic preference models. It allows us to critique recent contributions based on their limited discussion of psychological foundation and their implausible predictions. Second, we discuss implications of dynamic preference models for recommendation systems evaluation and design. In an example, we show that engagement and diversity metrics may be unable to capture desirable recommendation system performance.

While the identification of nonlinear dynamical systems is a fundamental building block of model-based reinforcement learning and feedback control, its sample complexity is only understood for systems that either have discrete states and actions or for systems that can be identified from data generated by i.i.d. random inputs. Nonetheless, many interesting dynamical systems have continuous states and actions and can only be identified through a judicious choice of inputs. Motivated by practical settings, we study a class of nonlinear dynamical systems whose state transitions depend linearly on a known feature embedding of state-action pairs. To estimate such systems in finite time identification methods must explore all directions in feature space. We propose an active learning approach that achieves this by repeating three steps: trajectory planning, trajectory tracking, and re-estimation of the system from all available data. We show that our method estimates nonlinear dynamical systems at a parametric rate, similar to the statistical rate of standard linear regression.

Modern nonlinear control theory seeks to endow systems with properties such as stability and safety, and has been deployed successfully across various domains. Despite this success, model uncertainty remains a significant challenge in ensuring that model-based controllers transfer to real world systems. This paper develops a data-driven approach to robust control synthesis in the presence of model uncertainty using Control Certificate Functions (CCFs), resulting in a convex optimization based controller for achieving properties like stability and safety. An important benefit of our framework is nuanced data-dependent guarantees, which in principle can yield sample-efficient data collection approaches that need not fully determine the input-to-state relationship. This work serves as a starting point for addressing important questions at the intersection of nonlinear control theory and non-parametric learning, both theoretical and in application. We demonstrate the efficiency of the proposed method with respect to input data in simulation with an inverted pendulum in multiple experimental settings.

Collecting more diverse and representative training data is often touted as a remedy for the disparate performance of machine learning predictors across subpopulations. However, a precise framework for understanding how dataset properties like diversity affect learning outcomes is largely lacking. By casting data collection as part of the learning process, we demonstrate that diverse representation in training data is key not only to increasing subgroup performances, but also to achieving population level objectives. Our analysis and experiments describe how dataset compositions influence performance and provide constructive results for using trends in existing data, alongside domain knowledge, to help guide intentional, objective-aware dataset design.

This graduate textbook on machine learning tells a story of how patterns in data support predictions and consequential actions. Starting with the foundations of decision making, we cover representation, optimization, and generalization as the constituents of supervised learning. A chapter on datasets as benchmarks examines their histories and scientific bases. Self-contained introductions to causality, the practice of causal inference, sequential decision making, and reinforcement learning equip the reader with concepts and tools to reason about actions and their consequences. Throughout, the text discusses historical context and societal impact. We invite readers from all backgrounds; some experience with probability, calculus, and linear algebra suffices.