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We consider the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations. Under very mild assumptions on the process and measurement noises, we provide a finite time analysis for learning the unknown dynamics matrices (up to a similarity transform). Our analysis involves a regression problem with heavy-tailed and dependent data. Moreover, each row of our design matrix contains a Kronecker product of current input with a history of inputs, making it difficult to guarantee persistence of excitation. We overcome these challenges, first providing a data-dependent high probability error bound for arbitrary but fixed inputs. Then, we derive a data-independent error bound for inputs chosen according to a simple random design. Our main results provide an upper bound on the statistical error rates and sample complexity of learning the unknown dynamics matrices from a single finite trajectory of bilinear observations., Comment: 35 pages, 3 figures

We consider a recommender system that takes into account the interplay between recommendations, the evolution of user interests, and harmful content. We model the impact of recommendations on user behavior, particularly the tendency to consume harmful content. We seek recommendation policies that establish a tradeoff between maximizing click-through rate (CTR) and mitigating harm. We establish conditions under which the user profile dynamics have a stationary point, and propose algorithms for finding an optimal recommendation policy at stationarity. We experiment on a semi-synthetic movie recommendation setting initialized with real data and observe that our policies outperform baselines at simultaneously maximizing CTR and mitigating harm., Comment: Recommender Systems; Harm Mitigation; Amplification; User Preference Modeling

Modern data-driven control applications call for flexible nonlinear models that are amenable to principled controller synthesis and realtime feedback. Many nonlinear dynamical systems of interest are control affine. We propose two novel classes of nonlinear feature representations which capture control affine structure while allowing for arbitrary complexity in the state dependence. Our methods make use of random features (RF) approximations, inheriting the expressiveness of kernel methods at a lower computational cost. We formalize the representational capabilities of our methods by showing their relationship to the Affine Dot Product (ADP) kernel proposed by Casta\~neda et al. (2021) and a novel Affine Dense (AD) kernel that we introduce. We further illustrate the utility by presenting a case study of data-driven optimization-based control using control certificate functions (CCF). Simulation experiments on a double pendulum empirically demonstrate the advantages of our methods., Comment: 25 pages, 3 figures

In digital markets comprised of many competing services, each user chooses between multiple service providers according to their preferences, and the chosen service makes use of the user data to incrementally improve its model. The service providers' models influence which service the user will choose at the next time step, and the user's choice, in return, influences the model update, leading to a feedback loop. In this paper, we formalize the above dynamics and develop a simple and efficient decentralized algorithm to locally minimize the overall user loss. Theoretically, we show that our algorithm asymptotically converges to stationary points of of the overall loss almost surely. We also experimentally demonstrate the utility of our algorithm with real world data., Comment: Accepted by ICML24

Robot-assisted bite acquisition involves picking up food items that vary in their shape, compliance, size, and texture. A fully autonomous strategy for bite acquisition is unlikely to efficiently generalize to this wide variety of food items. We propose to leverage the presence of the care recipient to provide feedback when the system encounters novel food items. However, repeatedly asking for help imposes cognitive workload on the user. In this work, we formulate human-in-the-loop bite acquisition within a contextual bandit framework and propose a novel method, LinUCB-QG, that selectively asks for help. This method leverages a predictive model of cognitive workload in response to different types and timings of queries, learned using data from 89 participants collected in an online user study. We demonstrate that this method enhances the balance between task performance and cognitive workload compared to autonomous and querying baselines, through experiments in a food dataset-based simulator and a user study with 18 participants without mobility limitations., Comment: The second and third authors contributed equally. The last two authors advised equally

The Giant Magellan Telescope will use laser tomography adaptive optics to correct for atmospheric turbulence using artificial guide stars created in the sodium layer of the atmosphere (altitude ~95km). The sodium layer has appreciable thickness (~11km) and this results in the laser guide star being an elongated cylinder shape. Wavefront sensing with a Shack-Hartmann is challenging, as subapertures located further away from the laser launch position image an increasingly elongated perspective of the laser guide star. Large detectors can be used to adequately pack and sample the images on the detector, however, this increases readout noise and limits the design space available for the wavefront sensor. To tackle this challenge, we propose an original solution based on nano-engineered meta-optics tailored to produce a spatially varying anamorphic image scale compression. We present meta-lenslet array designs that can deliver ~100% of the full anamorphic image size reduction required for focal lengths down to 8mm, and greater than 50% image size reduction for focal lengths down to 2mm. This will allow greatly improved sampling of the available information across the whole wavefront sensor, while still being a viable design within the limits of current-generation fabrication facilities.

As AI systems enter into a growing number of societal domains, these systems increasingly shape and are shaped by user preferences, opinions, and behaviors. However, the design of AI systems rarely accounts for how AI and users shape one another. In this position paper, we argue for the development of formal interaction models which mathematically specify how AI and users shape one another. Formal interaction models can be leveraged to (1) specify interactions for implementation, (2) monitor interactions through empirical analysis, (3) anticipate societal impacts via counterfactual analysis, and (4) control societal impacts via interventions. The design space of formal interaction models is vast, and model design requires careful consideration of factors such as style, granularity, mathematical complexity, and measurability. Using content recommender systems as a case study, we critically examine the nascent literature of formal interaction models with respect to these use-cases and design axes. More broadly, we call for the community to leverage formal interaction models when designing, evaluating, or auditing any AI system which interacts with users.

Real-world systems often involve some pool of users choosing between a set of services. With the increase in popularity of online learning algorithms, these services can now self-optimize, leveraging data collected on users to maximize some reward such as service quality. On the flipside, users may strategically choose which services to use in order to pursue their own reward functions, in the process wielding power over which services can see and use their data. Extensive prior research has been conducted on the effects of strategic users in single-service settings, with strategic behavior manifesting in the manipulation of observable features to achieve a desired classification; however, this can often be costly or unattainable for users and fails to capture the full behavior of multi-service dynamic systems. As such, we analyze a setting in which strategic users choose among several available services in order to pursue positive classifications, while services seek to minimize loss functions on their observations. We focus our analysis on realizable settings, and show that naive retraining can still lead to oscillation even if all users are observed at different times; however, if this retraining uses memory of past observations, convergent behavior can be guaranteed for certain loss function classes. We provide results obtained from synthetic and real-world data to empirically validate our theoretical findings., Comment: 18 pages, 9 figures

This paper studies ML systems that interactively learn from users across multiple subpopulations with heterogeneous data distributions. The primary objective is to provide specialized services for different user groups while also predicting user preferences. Once the users select a service based on how well the service anticipated their preference, the services subsequently adapt and refine themselves based on the user data they accumulate, resulting in an iterative, alternating minimization process between users and services (learning dynamics). Employing such tailored approaches has two main challenges: (i) Unknown user preferences: Typically, data on user preferences are unavailable without interaction, and uniform data collection across a large and diverse user base can be prohibitively expensive. (ii) Suboptimal Local Solutions: The total loss (sum of loss functions across all users and all services) landscape is not convex even if the individual losses on a single service are convex, making it likely for the learning dynamics to get stuck in local minima. The final outcome of the aforementioned learning dynamics is thus strongly influenced by the initial set of services offered to users, and is not guaranteed to be close to the globally optimal outcome. In this work, we propose a randomized algorithm to adaptively select very few users to collect preference data from, while simultaneously initializing a set of services. We prove that under mild assumptions on the loss functions, the expected total loss achieved by the algorithm right after initialization is within a factor of the globally optimal total loss with complete user preference data, and this factor scales only logarithmically in the number of services. Our theory is complemented by experiments on real as well as semi-synthetic datasets.

The feedback that users provide through their choices (e.g., clicks, purchases) is one of the most common types of data readily available for training search and recommendation algorithms. However, myopically training systems based on choice data may only improve short-term engagement, but not the long-term sustainability of the platform and the long-term benefits to its users, content providers, and other stakeholders. In this paper, we thus develop a new framework in which decision makers (e.g., platform operators, regulators, users) can express long-term goals for the behavior of the platform (e.g., fairness, revenue distribution, legal requirements). These goals take the form of exposure or impact targets that go well beyond individual sessions, and we provide new control-based algorithms to achieve these goals. In particular, the controllers are designed to achieve the stated long-term goals with minimum impact on short-term engagement. Beyond the principled theoretical derivation of the controllers, we evaluate the algorithms on both synthetic and real-world data. While all controllers perform well, we find that they provide interesting trade-offs in efficiency, robustness, and the ability to plan ahead.

Algorithms are used to aid human decision makers by making predictions and recommending decisions. Currently, these algorithms are trained to optimize prediction accuracy. What if they were optimized to control final decisions? In this paper, we study a decision-aid algorithm that learns about the human decision maker and provides ''personalized recommendations'' to influence final decisions. We first consider fixed human decision functions which map observable features and the algorithm's recommendations to final decisions. We characterize the conditions under which perfect control over final decisions is attainable. Under fairly general assumptions, the parameters of the human decision function can be identified from past interactions between the algorithm and the human decision maker, even when the algorithm was constrained to make truthful recommendations. We then consider a decision maker who is aware of the algorithm's manipulation and responds strategically. By posing the setting as a variation of the cheap talk game [Crawford and Sobel, 1982], we show that all equilibria are partition equilibria where only coarse information is shared: the algorithm recommends an interval containing the ideal decision. We discuss the potential applications of such algorithms and their social implications.

Datasets for training recommender systems are often subject to distribution shift induced by users' and recommenders' selection biases. In this paper, we study the impact of selection bias on datasets with different quantization. We then leverage two differently quantized datasets from different source distributions to mitigate distribution shift by applying the inverse probability scoring method from causal inference. Empirically, our approach gains significant performance improvement over single-dataset methods and alternative ways of combining two datasets., Comment: In Workshop on Distribution Shifts, 36th Conference on Neural Information Processing Systems (NeurIPS 2022)

Motivated by perception-based control problems in autonomous systems, this paper addresses the problem of developing feedback controllers to regulate the inputs and the states of a dynamical system to optimal solutions of an optimization problem when one has no access to exact measurements of the system states. In particular, we consider the case where the states need to be estimated from high-dimensional sensory data received only at discrete time intervals. We develop a sampled-data feedback controller that is based on adaptations of a projected gradient descent method, and that includes neural networks as integral components to estimate the state of the system from perceptual information. We derive sufficient conditions to guarantee (local) input-to-state stability of the control loop. Moreover, we show that the interconnected system tracks the solution trajectory of the underlying optimization problem up to an error that depends on the approximation errors of the neural network and on the time-variability of the optimization problem; the latter originates from time-varying safety and performance objectives, input constraints, and unknown disturbances. As a representative application, we illustrate our results with numerical simulations for vision-based autonomous driving., Comment: This is an extended version of the paper accepted to IFAC World Congress 2023 for publication, containing proofs, and recently updated to address a typo in Assumption 3

Online convex optimization (OCO) is a widely used framework in online learning. In each round, the learner chooses a decision in a convex set and an adversary chooses a convex loss function, and then the learner suffers the loss associated with their current decision. However, in many applications the learner's loss depends not only on the current decision but on the entire history of decisions until that point. The OCO framework and its existing generalizations do not capture this, and they can only be applied to many settings of interest after a long series of approximation arguments. They also leave open the question of whether the dependence on memory is tight because there are no non-trivial lower bounds. In this work we introduce a generalization of the OCO framework, "Online Convex Optimization with Unbounded Memory", that captures long-term dependence on past decisions. We introduce the notion of $p$-effective memory capacity, $H_p$, that quantifies the maximum influence of past decisions on present losses. We prove an $O(\sqrt{H_p T})$ upper bound on the policy regret and a matching (worst-case) lower bound. As a special case, we prove the first non-trivial lower bound for OCO with finite memory \citep{anavaHM2015online}, which could be of independent interest, and also improve existing upper bounds. We demonstrate the broad applicability of our framework by using it to derive regret bounds, and to improve and simplify existing regret bound derivations, for a variety of online learning problems including online linear control and an online variant of performative prediction., Comment: Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS 2023)

Content creators compete for user attention. Their reach crucially depends on algorithmic choices made by developers on online platforms. To maximize exposure, many creators adapt strategically, as evidenced by examples like the sprawling search engine optimization industry. This begets competition for the finite user attention pool. We formalize these dynamics in what we call an exposure game, a model of incentives induced by algorithms, including modern factorization and (deep) two-tower architectures. We prove that seemingly innocuous algorithmic choices, e.g., non-negative vs. unconstrained factorization, significantly affect the existence and character of (Nash) equilibria in exposure games. We proffer use of creator behavior models, like exposure games, for an (ex-ante) pre-deployment audit. Such an audit can identify misalignment between desirable and incentivized content, and thus complement post-hoc measures like content filtering and moderation. To this end, we propose tools for numerically finding equilibria in exposure games, and illustrate results of an audit on the MovieLens and LastFM datasets. Among else, we find that the strategically produced content exhibits strong dependence between algorithmic exploration and content diversity, and between model expressivity and bias towards gender-based user and creator groups., Comment: presented at ICLR 2023 (top 5%)

Numerous online services are data-driven: the behavior of users affects the system's parameters, and the system's parameters affect the users' experience of the service, which in turn affects the way users may interact with the system. For example, people may choose to use a service only for tasks that already works well, or they may choose to switch to a different service. These adaptations influence the ability of a system to learn about a population of users and tasks in order to improve its performance broadly. In this work, we analyze a class of such dynamics -- where users allocate their participation amongst services to reduce the individual risk they experience, and services update their model parameters to reduce the service's risk on their current user population. We refer to these dynamics as \emph{risk-reducing}, which cover a broad class of common model updates including gradient descent and multiplicative weights. For this general class of dynamics, we show that asymptotically stable equilibria are always segmented, with sub-populations allocated to a single learner. Under mild assumptions, the utilitarian social optimum is a stable equilibrium. In contrast to previous work, which shows that repeated risk minimization can result in (Hashimoto et al., 2018; Miller et al., 2021), we find that repeated myopic updates with multiple learners lead to better outcomes. We illustrate the phenomena via a simulated example initialized from real data., Comment: AISTATS 2024

Many projects (both practical and academic) have designed algorithms to match users to content they will enjoy under the assumption that user's preferences and opinions do not change with the content they see. Evidence suggests that individuals' preferences are directly shaped by what content they see -- radicalization, rabbit holes, polarization, and boredom are all example phenomena of preferences affected by content. Polarization in particular can occur even in ecosystems with "mass media," where no personalization takes place, as recently explored in a natural model of preference dynamics by~\citet{hkazla2019geometric} and~\citet{gaitonde2021polarization}. If all users' preferences are drawn towards content they already like, or are repelled from content they already dislike, uniform consumption of media leads to a population of heterogeneous preferences converging towards only two poles. In this work, we explore whether some phenomenon akin to polarization occurs when users receive \emph{personalized} content recommendations. We use a similar model of preference dynamics, where an individual's preferences move towards content the consume and enjoy, and away from content they consume and dislike. We show that standard user reward maximization is an almost trivial goal in such an environment (a large class of simple algorithms will achieve only constant regret). A more interesting objective, then, is to understand under what conditions a recommendation algorithm can ensure stationarity of user's preferences. We show how to design a content recommendations which can achieve approximate stationarity, under mild conditions on the set of available content, when a user's preferences are known, and how one can learn enough about a user's preferences to implement such a strategy even when user preferences are initially unknown., Comment: EC 2022

Building systems that are good for society in the face of complex societal effects requires a dynamic approach. Recent approaches to machine learning (ML) documentation have demonstrated the promise of discursive frameworks for deliberation about these complexities. However, these developments have been grounded in a static ML paradigm, leaving the role of feedback and post-deployment performance unexamined. Meanwhile, recent work in reinforcement learning has shown that the effects of feedback and optimization objectives on system behavior can be wide-ranging and unpredictable. In this paper we sketch a framework for documenting deployed and iteratively updated learning systems, which we call Reward Reports. Taking inspiration from various contributions to the technical literature on reinforcement learning, we outline Reward Reports as living documents that track updates to design choices and assumptions behind what a particular automated system is optimizing for. They are intended to track dynamic phenomena arising from system deployment, rather than merely static properties of models or data. After presenting the elements of a Reward Report, we discuss a concrete example: Meta's BlenderBot 3 chatbot. Several others for game-playing (DeepMind's MuZero), content recommendation (MovieLens), and traffic control (Project Flow) are included in the appendix.

In the long term, reinforcement learning (RL) is considered by many AI theorists to be the most promising path to artificial general intelligence. This places RL practitioners in a position to design systems that have never existed before and lack prior documentation in law and policy. Public agencies could intervene on complex dynamics that were previously too opaque to deliberate about, and long-held policy ambitions would finally be made tractable. In this whitepaper we illustrate this potential and how it might be technically enacted in the domains of energy infrastructure, social media recommender systems, and transportation. Alongside these unprecedented interventions come new forms of risk that exacerbate the harms already generated by standard machine learning tools. We correspondingly present a new typology of risks arising from RL design choices, falling under four categories: scoping the horizon, defining rewards, pruning information, and training multiple agents. Rather than allowing RL systems to unilaterally reshape human domains, policymakers need new mechanisms for the rule of reason, foreseeability, and interoperability that match the risks these systems pose. We argue that criteria for these choices may be drawn from emerging subfields within antitrust, tort, and administrative law. It will then be possible for courts, federal and state agencies, and non-governmental organizations to play more active roles in RL specification and evaluation. Building on the "model cards" and "datasheets" frameworks proposed by Mitchell et al. and Gebru et al., we argue the need for Reward Reports for AI systems. Reward Reports are living documents for proposed RL deployments that demarcate design choices., Comment: 60 pages

In this work, we consider how preference models in interactive recommendation systems determine the availability of content and users' opportunities for discovery. We propose an evaluation procedure based on stochastic reachability to quantify the maximum probability of recommending a target piece of content to an user for a set of allowable strategic modifications. This framework allows us to compute an upper bound on the likelihood of recommendation with minimal assumptions about user behavior. Stochastic reachability can be used to detect biases in the availability of content and diagnose limitations in the opportunities for discovery granted to users. We show that this metric can be computed efficiently as a convex program for a variety of practical settings, and further argue that reachability is not inherently at odds with accuracy. We demonstrate evaluations of recommendation algorithms trained on large datasets of explicit and implicit ratings. Our results illustrate how preference models, selection rules, and user interventions impact reachability and how these effects can be distributed unevenly., Comment: to appear ICML 2021

The development of artificial intelligence (AI) technologies has far exceeded the investigation of their relationship with society. Sociotechnical inquiry is needed to mitigate the harms of new technologies whose potential impacts remain poorly understood. To date, subfields of AI research develop primarily individual views on their relationship with sociotechnics, while tools for external investigation, comparison, and cross-pollination are lacking. In this paper, we propose four directions for inquiry into new and evolving areas of technological development: value--what progress and direction does a field promote, optimization--how the defined system within a problem formulation relates to broader dynamics, consensus--how agreement is achieved and who is included in building it, and failure--what methods are pursued when the problem specification is found wanting. The paper provides a lexicon for sociotechnical inquiry and illustrates it through the example of consumer drone technology., Comment: 9 pages, 1 figure

Despite interest in communicating ethical problems and social contexts within the undergraduate curriculum to advance Public Interest Technology (PIT) goals, interventions at the graduate level remain largely unexplored. This may be due to the conflicting ways through which distinct Artificial Intelligence (AI) research tracks conceive of their interface with social contexts. In this paper we track the historical emergence of sociotechnical inquiry in three distinct subfields of AI research: AI Safety, Fair Machine Learning (Fair ML) and Human-in-the-Loop (HIL) Autonomy. We show that for each subfield, perceptions of PIT stem from the particular dangers faced by past integration of technical systems within a normative social order. We further interrogate how these histories dictate the response of each subfield to conceptual traps, as defined in the Science and Technology Studies literature. Finally, through a comparative analysis of these currently siloed fields, we present a roadmap for a unified approach to sociotechnical graduate pedagogy in AI., Comment: 8 Pages

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 validate the proposed method in simulation with an inverted pendulum in multiple experimental configurations., Comment: 8 pages, 2 figures, submitted to Conference on Decision & Control (CDC) 2021

Recommender systems operate in an inherently dynamical setting. Past recommendations influence future behavior, including which data points are observed and how user preferences change. However, experimenting in production systems with real user dynamics is often infeasible, and existing simulation-based approaches have limited scale. As a result, many state-of-the-art algorithms are designed to solve supervised learning problems, and progress is judged only by offline metrics. In this work we investigate the extent to which offline metrics predict online performance by evaluating eleven recommenders across six controlled simulated environments. We observe that offline metrics are correlated with online performance over a range of environments. However, improvements in offline metrics lead to diminishing returns in online performance. Furthermore, we observe that the ranking of recommenders varies depending on the amount of initial offline data available. We study the impact of adding exploration strategies, and observe that their effectiveness, when compared to greedy recommendation, is highly dependent on the recommendation algorithm. We provide the environments and recommenders described in this paper as Reclab: an extensible ready-to-use simulation framework at https://github.com/berkeley-reclab/RecLab.

Modern nonlinear control theory seeks to develop feedback controllers that endow systems with properties such as safety and stability. The guarantees ensured by these controllers often rely on accurate estimates of the system state for determining control actions. In practice, measurement model uncertainty can lead to error in state estimates that degrades these guarantees. In this paper, we seek to unify techniques from control theory and machine learning to synthesize controllers that achieve safety in the presence of measurement model uncertainty. We define the notion of a Measurement-Robust Control Barrier Function (MR-CBF) as a tool for determining safe control inputs when facing measurement model uncertainty. Furthermore, MR-CBFs are used to inform sampling methodologies for learning-based perception systems and quantify tolerable error in the resulting learned models. We demonstrate the efficacy of MR-CBFs in achieving safety with measurement model uncertainty on a simulated Segway system.

In order to certify performance and safety, feedback control requires precise characterization of sensor errors. In this paper, we provide guarantees on such feedback systems when sensors are characterized by solving a supervised learning problem. We show a uniform error bound on nonparametric kernel regression under a dynamically-achievable dense sampling scheme. This allows for a finite-time convergence rate on the sub-optimality of using the regressor in closed-loop for waypoint tracking. We demonstrate our results in simulation with simplified unmanned aerial vehicle and autonomous driving examples., Comment: to appear at L4DC 2021

While real-world decisions involve many competing objectives, algorithmic decisions are often evaluated with a single objective function. In this paper, we study algorithmic policies which explicitly trade off between a private objective (such as profit) and a public objective (such as social welfare). We analyze a natural class of policies which trace an empirical Pareto frontier based on learned scores, and focus on how such decisions can be made in noisy or data-limited regimes. Our theoretical results characterize the optimal strategies in this class, bound the Pareto errors due to inaccuracies in the scores, and show an equivalence between optimal strategies and a rich class of fairness-constrained profit-maximizing policies. We then present empirical results in two different contexts -- online content recommendation and sustainable abalone fisheries -- to underscore the applicability of our approach to a wide range of practical decisions. Taken together, these results shed light on inherent trade-offs in using machine learning for decisions that impact social welfare.

Recommender systems often rely on models which are trained to maximize accuracy in predicting user preferences. When the systems are deployed, these models determine the availability of content and information to different users. The gap between these objectives gives rise to a potential for unintended consequences, contributing to phenomena such as filter bubbles and polarization. In this work, we consider directly the information availability problem through the lens of user recourse. Using ideas of reachability, we propose a computationally efficient audit for top-$N$ linear recommender models. Furthermore, we describe the relationship between model complexity and the effort necessary for users to exert control over their recommendations. We use this insight to provide a novel perspective on the user cold-start problem. Finally, we demonstrate these concepts with an empirical investigation of a state-of-the-art model trained on a widely used movie ratings dataset., Comment: appeared at FAccT '20

Motivated by vision-based control of autonomous vehicles, we consider the problem of controlling a known linear dynamical system for which partial state information, such as vehicle position, is extracted from complex and nonlinear data, such as a camera image. Our approach is to use a learned perception map that predicts some linear function of the state and to design a corresponding safe set and robust controller for the closed loop system with this sensing scheme. We show that under suitable smoothness assumptions on both the perception map and the generative model relating state to complex and nonlinear data, parameters of the safe set can be learned via appropriately dense sampling of the state space. We then prove that the resulting perception-control loop has favorable generalization properties. We illustrate the usefulness of our approach on a synthetic example and on the self-driving car simulation platform CARLA., Comment: This revision includes reframing the local generalization problem, with relaxed the assumptions so that the robust problem depends on a local slope bound rather than a Lipschitz constant, and provide a method for learning the slope bound from data. We also include additional experiments with a CNN perception module