Osbert Bastani

Marcus J. Min, Mike He, Zhaoyu Li et al.

Autoformalization translates informal natural language into formal, machine-verifiable languages. While most work focuses on individual statements, real formalization efforts are inherently theory-level: they require an entire web of axioms, definitions, and lemmas before target theorems can even be stated. In this position paper, we argue for theory-level autoformalization: formalizing complete theories, including all their inter-dependencies, as structured libraries. We examine the significance of this shift, address alternative views, identify open challenges, and propose three promising paths forward. Our survey of autoformalization is available at https://github.com/marcusm117/Awesome-Autoformalization.

Yecheng Jason Ma, William Liang, Guanzhi Wang et al.

Large Language Models (LLMs) have excelled as high-level semantic planners for sequential decision-making tasks. However, harnessing them to learn complex low-level manipulation tasks, such as dexterous pen spinning, remains an open problem. We bridge this fundamental gap and present Eureka, a human-level reward design algorithm powered by LLMs. Eureka exploits the remarkable zero-shot generation, code-writing, and in-context improvement capabilities of state-of-the-art LLMs, such as GPT-4, to perform evolutionary optimization over reward code. The resulting rewards can then be used to acquire complex skills via reinforcement learning. Without any task-specific prompting or pre-defined reward templates, Eureka generates reward functions that outperform expert human-engineered rewards. In a diverse suite of 29 open-source RL environments that include 10 distinct robot morphologies, Eureka outperforms human experts on 83% of the tasks, leading to an average normalized improvement of 52%. The generality of Eureka also enables a new gradient-free in-context learning approach to reinforcement learning from human feedback (RLHF), readily incorporating human inputs to improve the quality and the safety of the generated rewards without model updating. Finally, using Eureka rewards in a curriculum learning setting, we demonstrate for the first time, a simulated Shadow Hand capable of performing pen spinning tricks, adeptly manipulating a pen in circles at rapid speed.

Shuo Li, Sangdon Park, Insup Lee et al. · gatech

When applied to open-domain question answering, large language models (LLMs) frequently generate incorrect responses based on made-up facts, which are called $\textit{hallucinations}$. Retrieval augmented generation (RAG) is a promising strategy to avoid hallucinations, but it does not provide guarantees on its correctness. To address this challenge, we propose the Trustworthy Retrieval Augmented Question Answering, or $\textit{TRAQ}$, which provides the first end-to-end statistical correctness guarantee for RAG. TRAQ uses conformal prediction, a statistical technique for constructing prediction sets that are guaranteed to contain the semantically correct response with high probability. Additionally, TRAQ leverages Bayesian optimization to minimize the size of the constructed sets. In an extensive experimental evaluation, we demonstrate that TRAQ provides the desired correctness guarantee while reducing prediction set size by 16.2% on average compared to an ablation. The implementation is available at $\href{https://github.com/shuoli90/TRAQ.git}{TRAQ}$.

Sangdon Park, Edgar Dobriban, Insup Lee et al. · gatech

Uncertainty quantification is a key component of machine learning models targeted at safety-critical systems such as in healthcare or autonomous vehicles. We study this problem in the context of meta learning, where the goal is to quickly adapt a predictor to new tasks. In particular, we propose a novel algorithm to construct \emph{PAC prediction sets}, which capture uncertainty via sets of labels, that can be adapted to new tasks with only a few training examples. These prediction sets satisfy an extension of the typical PAC guarantee to the meta learning setting; in particular, the PAC guarantee holds with high probability over future tasks. We demonstrate the efficacy of our approach on four datasets across three application domains: mini-ImageNet and CIFAR10-C in the visual domain, FewRel in the language domain, and the CDC Heart Dataset in the medical domain. In particular, our prediction sets satisfy the PAC guarantee while having smaller size compared to other baselines that also satisfy this guarantee.

Shuo Li, Sangdon Park, Xiayan Ji et al. · gatech

Accurately detecting and tracking multi-objects is important for safety-critical applications such as autonomous navigation. However, it remains challenging to provide guarantees on the performance of state-of-the-art techniques based on deep learning. We consider a strategy known as conformal prediction, which predicts sets of labels instead of a single label; in the classification and regression settings, these algorithms can guarantee that the true label lies within the prediction set with high probability. Building on these ideas, we propose multi-object detection and tracking algorithms that come with probably approximately correct (PAC) guarantees. They do so by constructing both a prediction set around each object detection as well as around the set of edge transitions; given an object, the detection prediction set contains its true bounding box with high probability, and the edge prediction set contains its true transition across frames with high probability. We empirically demonstrate that our method can detect and track objects with PAC guarantees on the COCO and MOT-17 datasets.

Sangdon Park, Osbert Bastani, Taesoo Kim · gatech

Blockchains with smart contracts are distributed ledger systems that achieve block-state consistency among distributed nodes by only allowing deterministic operations of smart contracts. However, the power of smart contracts is enabled by interacting with stochastic off-chain data, which in turn opens the possibility to undermine the block-state consistency. To address this issue, an oracle smart contract is used to provide a single consistent source of external data; but, simultaneously, this introduces a single point of failure, which is called the oracle problem. To address the oracle problem, we propose an adaptive conformal consensus (ACon$^2$) algorithm that derives a consensus set of data from multiple oracle contracts via the recent advance in online uncertainty quantification learning. Interesting, the consensus set provides a desired correctness guarantee under distribution shift and Byzantine adversaries. We demonstrate the efficacy of the proposed algorithm on two price datasets and an Ethereum case study. In particular, the Solidity implementation of the proposed algorithm shows the potential practicality of the proposed algorithm, implying that online machine learning algorithms are applicable to address security issues in blockchains.

Yecheng Jason Ma, Shagun Sodhani, Dinesh Jayaraman et al.

Reward and representation learning are two long-standing challenges for learning an expanding set of robot manipulation skills from sensory observations. Given the inherent cost and scarcity of in-domain, task-specific robot data, learning from large, diverse, offline human videos has emerged as a promising path towards acquiring a generally useful visual representation for control; however, how these human videos can be used for general-purpose reward learning remains an open question. We introduce $\textbf{V}$alue-$\textbf{I}$mplicit $\textbf{P}$re-training (VIP), a self-supervised pre-trained visual representation capable of generating dense and smooth reward functions for unseen robotic tasks. VIP casts representation learning from human videos as an offline goal-conditioned reinforcement learning problem and derives a self-supervised dual goal-conditioned value-function objective that does not depend on actions, enabling pre-training on unlabeled human videos. Theoretically, VIP can be understood as a novel implicit time contrastive objective that generates a temporally smooth embedding, enabling the value function to be implicitly defined via the embedding distance, which can then be used to construct the reward for any goal-image specified downstream task. Trained on large-scale Ego4D human videos and without any fine-tuning on in-domain, task-specific data, VIP's frozen representation can provide dense visual reward for an extensive set of simulated and $\textbf{real-robot}$ tasks, enabling diverse reward-based visual control methods and significantly outperforming all prior pre-trained representations. Notably, VIP can enable simple, $\textbf{few-shot}$ offline RL on a suite of real-world robot tasks with as few as 20 trajectories.

Zichen Zhang, Yunshuang Li, Osbert Bastani et al. · allen-ai

Real-world robotic tasks stretch over extended horizons and encompass multiple stages. Learning long-horizon manipulation tasks, however, is a long-standing challenge, and demands decomposing the overarching task into several manageable subtasks to facilitate policy learning and generalization to unseen tasks. Prior task decomposition methods require task-specific knowledge, are computationally intensive, and cannot readily be applied to new tasks. To address these shortcomings, we propose Universal Visual Decomposer (UVD), an off-the-shelf task decomposition method for visual long horizon manipulation using pre-trained visual representations designed for robotic control. At a high level, UVD discovers subgoals by detecting phase shifts in the embedding space of the pre-trained representation. Operating purely on visual demonstrations without auxiliary information, UVD can effectively extract visual subgoals embedded in the videos, while incurring zero additional training cost on top of standard visuomotor policy training. Goal-conditioned policies learned with UVD-discovered subgoals exhibit significantly improved compositional generalization at test time to unseen tasks. Furthermore, UVD-discovered subgoals can be used to construct goal-based reward shaping that jump-starts temporally extended exploration for reinforcement learning. We extensively evaluate UVD on both simulation and real-world tasks, and in all cases, UVD substantially outperforms baselines across imitation and reinforcement learning settings on in-domain and out-of-domain task sequences alike, validating the clear advantage of automated visual task decomposition within the simple, compact UVD framework.

Osbert Bastani, Varun Gupta, Christopher Jung et al.

We give a simple, generic conformal prediction method for sequential prediction that achieves target empirical coverage guarantees against adversarially chosen data. It is computationally lightweight -- comparable to split conformal prediction -- but does not require having a held-out validation set, and so all data can be used for training models from which to derive a conformal score. It gives stronger than marginal coverage guarantees in two ways. First, it gives threshold calibrated prediction sets that have correct empirical coverage even conditional on the threshold used to form the prediction set from the conformal score. Second, the user can specify an arbitrary collection of subsets of the feature space -- possibly intersecting -- and the coverage guarantees also hold conditional on membership in each of these subsets. We call our algorithm MVP, short for MultiValid Prediction. We give both theory and an extensive set of empirical evaluations.

Yecheng Jason Ma, William Liang, Vaidehi Som et al.

We present Language-Image Value learning (LIV), a unified objective for vision-language representation and reward learning from action-free videos with text annotations. Exploiting a novel connection between dual reinforcement learning and mutual information contrastive learning, the LIV objective trains a multi-modal representation that implicitly encodes a universal value function for tasks specified as language or image goals. We use LIV to pre-train the first control-centric vision-language representation from large human video datasets such as EpicKitchen. Given only a language or image goal, the pre-trained LIV model can assign dense rewards to each frame in videos of unseen robots or humans attempting that task in unseen environments. Further, when some target domain-specific data is available, the same objective can be used to fine-tune and improve LIV and even other pre-trained representations for robotic control and reward specification in that domain. In our experiments on several simulated and real-world robot environments, LIV models consistently outperform the best prior input state representations for imitation learning, as well as reward specification methods for policy synthesis. Our results validate the advantages of joint vision-language representation and reward learning within the unified, compact LIV framework.

Haydn Jones, Yimeng Zeng, Alden Rose et al.

Manually curated biomedical repositories -- spanning bioactivity, genomics, and chemistry -- are expensive to maintain, lag behind primary literature, and discard experimental context, obscuring nuances needed to assess data correctness and coverage. We show that PubMed itself can be autonomously and cost-effectively turned into structured datasets that are larger, more nuanced, and more accurate than the curated databases they replace. We present three coupled contributions: (1) an LLM-based entity-tagging pipeline, grounded in nine biomedical ontologies, that tags 4.5B entities across 19 categories in a 22.5M-paper, 2.5T-token PubMed corpus; (2) hybrid sparse-dense retrieval supporting entity-filtered semantic queries over the tagged corpus; and (3) Starling, a multi-agent deep research system that, given only a natural-language task description, designs precision- and recall-targeted retrieval filters, induces an extraction schema, and emits structured records with nuance-rich fields and supporting passages. Across six tasks -- blood-brain barrier permeability, oral bioavailability, acute toxicity (LD50), gene-disease associations, protein subcellular localization, and chemical reactions -- Starling produces ~6.3M records (91K-3M per task); several are, to our knowledge, the largest public datasets for their property. Frontier-model rejection of our extractions is 0.6-7.7% across tasks, far below error rates we measure on widely used curated counterparts (e.g., 16.5% on BBB_Martins, 7.3% on Bioavailability_Ma). Beyond scale and accuracy, the supporting passages carry nuance tabular databases discard -- e.g., oral bioavailability may depend on fed vs. fasted state. Together, the corpus, retrieval, and agent establish a foundation for AI-driven therapeutic design. Code and datasets: https://github.com/starling-labs/starling.

Kishor Jothimurugan, Suguman Bansal, Osbert Bastani et al.

Reinforcement learning has been shown to be an effective strategy for automatically training policies for challenging control problems. Focusing on non-cooperative multi-agent systems, we propose a novel reinforcement learning framework for training joint policies that form a Nash equilibrium. In our approach, rather than providing low-level reward functions, the user provides high-level specifications that encode the objective of each agent. Then, guided by the structure of the specifications, our algorithm searches over policies to identify one that provably forms an $ε$-Nash equilibrium (with high probability). Importantly, it prioritizes policies in a way that maximizes social welfare across all agents. Our empirical evaluation demonstrates that our algorithm computes equilibrium policies with high social welfare, whereas state-of-the-art baselines either fail to compute Nash equilibria or compute ones with comparatively lower social welfare.

Yecheng Jason Ma, Jason Yan, Dinesh Jayaraman et al.

Offline goal-conditioned reinforcement learning (GCRL) promises general-purpose skill learning in the form of reaching diverse goals from purely offline datasets. We propose $\textbf{Go}$al-conditioned $f$-$\textbf{A}$dvantage $\textbf{R}$egression (GoFAR), a novel regression-based offline GCRL algorithm derived from a state-occupancy matching perspective; the key intuition is that the goal-reaching task can be formulated as a state-occupancy matching problem between a dynamics-abiding imitator agent and an expert agent that directly teleports to the goal. In contrast to prior approaches, GoFAR does not require any hindsight relabeling and enjoys uninterleaved optimization for its value and policy networks. These distinct features confer GoFAR with much better offline performance and stability as well as statistical performance guarantee that is unattainable for prior methods. Furthermore, we demonstrate that GoFAR's training objectives can be re-purposed to learn an agent-independent goal-conditioned planner from purely offline source-domain data, which enables zero-shot transfer to new target domains. Through extensive experiments, we validate GoFAR's effectiveness in various problem settings and tasks, significantly outperforming prior state-of-art. Notably, on a real robotic dexterous manipulation task, while no other method makes meaningful progress, GoFAR acquires complex manipulation behavior that successfully accomplishes diverse goals.

Alexander Shypula, Osbert Bastani, Edward Schwartz

Decompilers are useful tools used in reverse engineering to understand compiled source code. Reconstructing source code from compiled binaries is a challenging task, because high-level syntax, identifiers, and custom data types are generally lost as the compiler translates human-readable code to low-level machine code. Deterministic decompilers are useful tools for binary analysis, but can struggle to infer idiomatic syntax and identifier names. Generative AI models are a natural fit for reconstructing high-level syntax, identifiers, and types, but they can still suffer by hallucinating improper programming constructs and semantics. Instead of attempting to improve neural decompilers with more data and more training, we argue that compiler feedback can be used to dramatically improve the semantic correctness of neural decompiler outputs via search. Our system, Decaf (DECompilation with Automated Feedback), raises the neural decompilation rate from 26.0% on ExeBench to 83.9% on the Real -O2 split without sacrificing similarity to the original source code. We also find our automatic feedback methodology is highly effective for improving weaker neural decompilation models.

Kishor Jothimurugan, Steve Hsu, Osbert Bastani et al.

Compositional reinforcement learning is a promising approach for training policies to perform complex long-horizon tasks. Typically, a high-level task is decomposed into a sequence of subtasks and a separate policy is trained to perform each subtask. In this paper, we focus on the problem of training subtask policies in a way that they can be used to perform any task; here, a task is given by a sequence of subtasks. We aim to maximize the worst-case performance over all tasks as opposed to the average-case performance. We formulate the problem as a two agent zero-sum game in which the adversary picks the sequence of subtasks. We propose two RL algorithms to solve this game: one is an adaptation of existing multi-agent RL algorithms to our setting and the other is an asynchronous version which enables parallel training of subtask policies. We evaluate our approach on two multi-task environments with continuous states and actions and demonstrate that our algorithms outperform state-of-the-art baselines.

Tsai-Hsuan Chung, Vahid Rostami, Hamsa Bastani et al.

We study the problem of allocating limited supply of medical resources in developing countries, in particular, Sierra Leone. We address this problem by combining machine learning (to predict demand) with optimization (to optimize allocations). A key challenge is the need to align the loss function used to train the machine learning model with the decision loss associated with the downstream optimization problem. Traditional solutions have limited flexibility in the model architecture and scale poorly to large datasets. We propose a decision-aware learning algorithm that uses a novel Taylor expansion of the optimal decision loss to derive the machine learning loss. Importantly, our approach only requires a simple re-weighting of the training data, ensuring it is both flexible and scalable, e.g., we incorporate it into a random forest trained using a multitask learning framework. We apply our framework to optimize the distribution of essential medicines in collaboration with policymakers in Sierra Leone; highly uncertain demand and limited budgets currently result in excessive unmet demand. Out-of-sample results demonstrate that our end-to-end approach can significantly reduce unmet demand across 1040 health facilities throughout Sierra Leone.

Wenwen Si, Sangdon Park, Insup Lee et al.

Prediction sets capture uncertainty by predicting sets of labels rather than individual labels, enabling downstream decisions to conservatively account for all plausible outcomes. Conformal inference algorithms construct prediction sets guaranteed to contain the true label with high probability. These guarantees fail to hold in the face of distribution shift, which is precisely when reliable uncertainty quantification can be most useful. We propose a novel algorithm for constructing prediction sets with PAC guarantees in the label shift setting. This method estimates the predicted probabilities of the classes in a target domain, as well as the confusion matrix, then propagates uncertainty in these estimates through a Gaussian elimination algorithm to compute confidence intervals for importance weights. Finally, it uses these intervals to construct prediction sets. We evaluate our approach on five datasets: the CIFAR-10, ChestX-Ray and Entity-13 image datasets, the tabular CDC Heart dataset, and the AGNews text dataset. Our algorithm satisfies the PAC guarantee while producing smaller, more informative, prediction sets compared to several baselines.

Adam Khakhar, Stephen Mell, Osbert Bastani

Prediction sets have recently been shown to be a promising strategy for quantifying the uncertainty of deep neural networks in a way that provides theoretical guarantees. However, existing techniques have largely targeted settings where the space of labels is simple, so prediction sets can be arbitrary subsets of labels. For structured prediction problems where the space of labels is exponential in size, even prediction sets containing a small fraction of all labels can be exponentially large. In the context of code generation, we propose a solution that considers a restricted set of prediction sets that can compactly be represented as partial programs, which are programs with portions replaced with holes. Given a trained code generation model, our algorithm leverages a programming language's abstract syntax tree to generate a set of programs such that the correct program is in the set with high-confidence. Valuable applications of our algorithm include a Codex-style code generator with holes in uncertain parts of the generated code, which provides a partial program with theoretical guarantees. We evaluate our approach on PICARD (a T5 model for SQL semantic parsing) and Codex (a GPT model for over a dozen programming languages, including Python), demonstrating that our approach generates compact PAC prediction sets. This is the first research contribution that generates PAC prediction sets for generative code models.

George Tolkachev, Stephen Mell, Steve Zdancewic et al.

A key challenge facing natural language interfaces is enabling users to understand the capabilities of the underlying system. We propose a novel approach for generating explanations of a natural language interface based on semantic parsing. We focus on counterfactual explanations, which are post-hoc explanations that describe to the user how they could have minimally modified their utterance to achieve their desired goal. In particular, the user provides an utterance along with a demonstration of their desired goal; then, our algorithm synthesizes a paraphrase of their utterance that is guaranteed to achieve their goal. In two user studies, we demonstrate that our approach substantially improves user performance, and that it generates explanations that more closely match the user's intent compared to two ablations.

Vashist Avadhanula, Omar Abdul Baki, Hamsa Bastani et al.

We describe the current content moderation strategy employed by Meta to remove policy-violating content from its platforms. Meta relies on both handcrafted and learned risk models to flag potentially violating content for human review. Our approach aggregates these risk models into a single ranking score, calibrating them to prioritize more reliable risk models. A key challenge is that violation trends change over time, affecting which risk models are most reliable. Our system additionally handles production challenges such as changing risk models and novel risk models. We use a contextual bandit to update the calibration in response to such trends. Our approach increases Meta's top-line metric for measuring the effectiveness of its content moderation strategy by 13%.

Stephen Mell, David Mell, Konstantinos Kallas et al.

Compound AI applications, which compose calls to ML models using a general-purpose programming language like Python, are widely used for a variety of user-facing tasks, from software engineering to enterprise automation, making their end-to-end latency a critical bottleneck. In contrast to traditional applications, execution time is dominated by the external components, which cannot be handled by traditional language optimization systems, like optimizing compilers. To address this problem, we develop PopPy, a system that can uncover parallelization opportunities in Python applications that invoke these heavy external components, including those used in compound AI applications. PopPy supports a very expressive fragment of Python and requires minimal developer input to uncover parallelism. It combines an ahead-of-time compiler with a runtime, addressing three key challenges in extracting parallelism from Python applications: language complexity, dynamic dispatch, and variable mutation. On a set of real-world compound AI applications, PopPy achieves up to $6.4\times$ speedups in end-to-end execution time compared to standard Python execution while preserving the sequential program semantics.

Halley Young, Yimeng Zeng, Jacob Gardner et al.

The capability to generate diverse text is a key challenge facing large language models (LLMs). Thus far, diversity has been studied via metrics such as $n$-gram diversity or diversity of BERT embeddings. However, for these kinds of diversity, the user has little control over the dimensions along which diversity is considered. For example, in the poetry domain, one might desire diversity in terms of rhyme and meter, whereas in the code domain, one might desire diversity in terms of the kinds of expressions used to solve a problem. We propose a diversity metric called structural diversity, where the user provides a mapping from generated text to features capturing the kinds of diversity that they care about. In addition, we propose a novel strategy called chain-of-specification (CoS) prompting for improving diversity by first having the LLM generate a specification encoding one instance of structural features, and then prompting the LLM to generate text that satisfies these features; notably, our strategy works with blackbox LLMs. In our experiments, we show that for structural diversity in the poetry and code domains, CoS significantly improves diversity compared to several baselines.

Haosen Ge, Hamsa Bastani, Osbert Bastani

Existing approaches to algorithmic fairness aim to ensure equitable outcomes if human decision-makers comply perfectly with algorithmic decisions. However, perfect compliance with the algorithm is rarely a reality or even a desirable outcome in human-AI collaboration. Yet, recent studies have shown that selective compliance with fair algorithms can amplify discrimination relative to the prior human policy. As a consequence, ensuring equitable outcomes requires fundamentally different algorithmic design principles that ensure robustness to the decision-maker's (a priori unknown) compliance pattern. We define the notion of compliance-robustly fair algorithmic recommendations that are guaranteed to (weakly) improve fairness in decisions, regardless of the human's compliance pattern. We propose a simple optimization strategy to identify the best performance-improving compliance-robustly fair policy. However, we show that it may be infeasible to design algorithmic recommendations that are simultaneously fair in isolation, compliance-robustly fair, and more accurate than the human policy; thus, if our goal is to improve the equity and accuracy of human-AI collaboration, it may not be desirable to enforce traditional algorithmic fairness constraints. We illustrate the value of our approach on criminal sentencing data before and after the introduction of an algorithmic risk assessment tool in Virginia.

Kavi Gupta, Osbert Bastani, Armando Solar-Lezama

Real-world processes often contain intermediate state that can be modeled as an extremely sparse activation tensor. In this work, we analyze the identifiability of such sparse and local latent intermediate variables, which we call motifs. We prove our Motif Identifiability Theorem, stating that under certain assumptions it is possible to precisely identify these motifs exclusively by reducing end-to-end error. Notably, we do not assume identifiability of parameters, but rather of a latent intermediate representation output by a local model, thus allowing these representations to be arbitrarily complex functions of the input. Additionally, we provide the Sparling algorithm, which uses a new kind of informational bottleneck that enforces levels of activation sparsity unachievable using other techniques. We confirm empirically that extreme sparsity is necessary to achieve good intermediate state modeling. On synthetic domains, we are able to precisely localize the intermediate states up to feature permutation with > 90% accuracy, even though we only train end-to-end.

Wenwen Si, Sooyong Jang, Insup Lee et al.

While large language models (LLMs) have recently made tremendous progress towards solving challenging AI problems, they have done so at increasingly steep computational and API costs. We propose a novel strategy where we combine multiple LLM models with varying cost/accuracy tradeoffs in an agentic manner, where models and tools are run in sequence as determined by an orchestration model to minimize cost subject to a user-specified level of reliability; this constraint is formalized using conformal prediction to provide guarantees. To solve this problem, we propose Conformal Constrained Policy Optimization (CCPO), a training paradigm that integrates constrained policy optimization with off-policy reinforcement learning and recent advances in online conformal prediction. CCPO jointly optimizes a cost-aware policy (score function) and an adaptive threshold. Across two multi-hop question answering benchmarks, CCPO achieves up to a 30% cost reduction compared to other cost-aware baselines and LLM-guided methods without compromising reliability. Our approach provides a principled and practical framework for deploying LLM agents that are significantly more cost-effective while maintaining reliability.

Osbert Bastani

We prove an asymptotic normality guarantee for generalized low-rank matrix sensing -- i.e., matrix sensing under a general convex loss $\bar\ell(\langle X,M\rangle,y^*)$, where $M\in\mathbb{R}^{d\times d}$ is the unknown rank-$k$ matrix, $X$ is a measurement matrix, and $y^*$ is the corresponding measurement. Our analysis relies on tools from Riemannian geometry to handle degeneracy of the Hessian of the loss due to rotational symmetry in the parameter space. In particular, we parameterize the manifold of low-rank matrices by $\barθ\barθ^\top$, where $\barθ\in\mathbb{R}^{d\times k}$. Then, assuming the minimizer of the empirical loss $\barθ^0\in\mathbb{R}^{d\times k}$ is in a constant size ball around the true parameters $\barθ^*$, we prove $\sqrt{n}(φ^0-φ^*)\xrightarrow{D}N(0,(H^*)^{-1})$ as $n\to\infty$, where $φ^0$ and $φ^*$ are representations of $\barθ^*$ and $\barθ^0$ in the horizontal space of the Riemannian quotient manifold $\mathbb{R}^{d\times k}/\text{O}(k)$, and $H^*$ is the Hessian of the true loss in the same representation.

Celeste Barnaby, Danny Ding, Osbert Bastani et al.

High-level specifications of code are inherently ambiguous, and prior systems have explored interactive techniques to help users clarify their intent and resolve such ambiguities. However, most existing approaches elicit supervision through labeled examples, which are often error-prone and may fail to capture user intent. This paper introduces a new active learning paradigm for program disambiguation based on multiple-choice queries. In this paradigm, the system presents a small set of high-level behaviors as multiple-choice options, and the user simply selects the intended one. Technically, each answer option corresponds to a Hoare triple that characterizes a cluster of semantically similar candidate programs. This formulation enables formal reasoning about the informativeness and interpretability of queries, and supports systematic construction of optimal queries. Building on this insight, we develop a new active learning algorithm and implement it in a tool called Socrates, which automatically synthesizes informative multiple-choice queries for program disambiguation. We evaluate Socrates across four domains spanning both symbolic and neurosymbolic settings and show that it produces intuitive, easy-to-answer queries and achieves efficient convergence. Most importantly, Socrates identifies the intended program more reliably than existing methods, while maintaining competitive runtime performance.

Angel Tsai-Hsuan Chung, Jatu Abdulai, Patrick Bayoh et al.

A critical challenge in healthcare systems in low- and middle-income countries (LMICs) is the efficient and equitable allocation of scarce resources, particularly essential medicines. This problem is complicated by limited high-quality data, which restricts the applicability of traditional data-driven techniques. We propose a novel decision-aware machine learning framework for essential medicines allocation, which additionally leverages multi-task learning to ensure sample efficiency and catalytic priors to ensure equitable allocation. In collaboration with the Sierra Leone national government, we performed a staggered, nationwide deployment of our system as a decision support tool. Our econometric evaluation finds an estimated 19% increase in consumption of allocated products in treated districts, demonstrating its efficacy at improving access to essential medicines. Our tool was subsequently scaled nationwide, covering an estimated 2 million women and children under five. Our work demonstrates how machine learning methods can improve efficiency at very low cost in resource-constrained global health settings.

Michael S. Yao, Yimeng Zeng, Hamsa Bastani et al.

Offline model-based optimization seeks to optimize against a learned surrogate model without querying the true oracle objective function during optimization. Such tasks are commonly encountered in protein design, robotics, and clinical medicine where evaluating the oracle function is prohibitively expensive. However, inaccurate surrogate model predictions are frequently encountered along offline optimization trajectories. To address this limitation, we propose generative adversarial model-based optimization using adaptive source critic regularization (aSCR) -- a task- and optimizer- agnostic framework for constraining the optimization trajectory to regions of the design space where the surrogate function is reliable. We propose a computationally tractable algorithm to dynamically adjust the strength of this constraint, and show how leveraging aSCR with standard Bayesian optimization outperforms existing methods on a suite of offline generative design tasks. Our code is available at https://github.com/michael-s-yao/gabo

Yecheng Jason Ma, Andrew Shen, Osbert Bastani et al.

Reinforcement Learning (RL) agents in the real world must satisfy safety constraints in addition to maximizing a reward objective. Model-based RL algorithms hold promise for reducing unsafe real-world actions: they may synthesize policies that obey all constraints using simulated samples from a learned model. However, imperfect models can result in real-world constraint violations even for actions that are predicted to satisfy all constraints. We propose Conservative and Adaptive Penalty (CAP), a model-based safe RL framework that accounts for potential modeling errors by capturing model uncertainty and adaptively exploiting it to balance the reward and the cost objectives. First, CAP inflates predicted costs using an uncertainty-based penalty. Theoretically, we show that policies that satisfy this conservative cost constraint are guaranteed to also be feasible in the true environment. We further show that this guarantees the safety of all intermediate solutions during RL training. Further, CAP adaptively tunes this penalty during training using true cost feedback from the environment. We evaluate this conservative and adaptive penalty-based approach for model-based safe RL extensively on state and image-based environments. Our results demonstrate substantial gains in sample-efficiency while incurring fewer violations than prior safe RL algorithms. Code is available at: https://github.com/Redrew/CAP

Yecheng Jason Ma, Jeevana Priya Inala, Dinesh Jayaraman et al.

Forecasting complex vehicle and pedestrian multi-modal distributions requires powerful probabilistic approaches. Normalizing flows (NF) have recently emerged as an attractive tool to model such distributions. However, a key drawback is that independent samples drawn from a flow model often do not adequately capture all the modes in the underlying distribution. We propose Likelihood-Based Diverse Sampling (LDS), a method for improving the quality and the diversity of trajectory samples from a pre-trained flow model. Rather than producing individual samples, LDS produces a set of trajectories in one shot. Given a pre-trained forecasting flow model, we train LDS using gradients from the model, to optimize an objective function that rewards high likelihood for individual trajectories in the predicted set, together with high spatial separation among trajectories. LDS outperforms state-of-art post-hoc neural diverse forecasting methods for various pre-trained flow models as well as conditional variational autoencoder (CVAE) models. Crucially, it can also be used for transductive trajectory forecasting, where the diverse forecasts are trained on-the-fly on unlabeled test examples. LDS is easy to implement, and we show that it offers a simple plug-in improvement over baselines on two challenging benchmarks. Code is at: https://github.com/JasonMa2016/LDS

Brian Heath, Neelay Velingker, Osbert Bastani et al.

The increasing prevalence of mobile apps has led to a proliferation of resource usage scenarios in which they are deployed. This motivates the need to specialize mobile apps based on diverse and varying preferences of users. We propose a system, called PolyDroid, for automatically specializing mobile apps based on user preferences. The app developer provides a number of candidate configurations, called reductions, that limit the resource usage of the original app. The key challenge underlying PolyDroid concerns learning the quality of user experience under different reductions. We propose an active learning technique that requires few user experiments to determine the optimal reduction for a given resource usage specification. On a benchmark suite comprising 20 diverse, open-source Android apps, we demonstrate that on average, PolyDroid obtains more than 85% of the optimal performance using just two user experiments.

Yecheng Jason Ma, Joey Hejna, Ayzaan Wahid et al.

Predicting temporal progress from visual trajectories is important for intelligent robots that can learn, adapt, and improve. However, learning such progress estimator, or temporal value function, across different tasks and domains requires both a large amount of diverse data and methods which can scale and generalize. To address these challenges, we present Generative Value Learning (\GVL), a universal value function estimator that leverages the world knowledge embedded in vision-language models (VLMs) to predict task progress. Naively asking a VLM to predict values for a video sequence performs poorly due to the strong temporal correlation between successive frames. Instead, GVL poses value estimation as a temporal ordering problem over shuffled video frames; this seemingly more challenging task encourages VLMs to more fully exploit their underlying semantic and temporal grounding capabilities to differentiate frames based on their perceived task progress, consequently producing significantly better value predictions. Without any robot or task specific training, GVL can in-context zero-shot and few-shot predict effective values for more than 300 distinct real-world tasks across diverse robot platforms, including challenging bimanual manipulation tasks. Furthermore, we demonstrate that GVL permits flexible multi-modal in-context learning via examples from heterogeneous tasks and embodiments, such as human videos. The generality of GVL enables various downstream applications pertinent to visuomotor policy learning, including dataset filtering, success detection, and advantage-weighted regression -- all without any model training or finetuning.

Xinmeng Huang, Shuo Li, Mengxin Yu et al.

Language Models (LMs) have shown promising performance in natural language generation. However, as LMs often generate incorrect or hallucinated responses, it is crucial to correctly quantify their uncertainty in responding to given inputs. In addition to verbalized confidence elicited via prompting, many uncertainty measures ($e.g.$, semantic entropy and affinity-graph-based measures) have been proposed. However, these measures can differ greatly, and it is unclear how to compare them, partly because they take values over different ranges ($e.g.$, $[0,\infty)$ or $[0,1]$). In this work, we address this issue by developing a novel and practical framework, termed $Rank$-$Calibration$, to assess uncertainty and confidence measures for LMs. Our key tenet is that higher uncertainty (or lower confidence) should imply lower generation quality, on average. Rank-calibration quantifies deviations from this ideal relationship in a principled manner, without requiring ad hoc binary thresholding of the correctness score ($e.g.$, ROUGE or METEOR). The broad applicability and the granular interpretability of our methods are demonstrated empirically.

Alexander Shypula, Shuo Li, Botong Zhang et al.

Recent work suggests that preference-tuning techniques--including Reinforcement Learning from Human Preferences (RLHF) methods like PPO and GRPO, as well as alternatives like DPO--reduce diversity, creating a dilemma given that such models are widely deployed in applications requiring diverse outputs. To address this, we introduce a framework for measuring effective semantic diversity--diversity among outputs that meet quality thresholds--which better reflects the practical utility of large language models (LLMs). Using open-ended tasks that require no human intervention, we find counterintuitive results: although preference-tuned models--especially those trained via RL--exhibit reduced lexical and syntactic diversity, they produce greater effective semantic diversity than SFT or base models, not from increasing diversity among high-quality outputs, but from generating more high-quality outputs overall. We discover that preference tuning reduces syntactic diversity while preserving semantic diversity--revealing a distinction between diversity in form and diversity in content that traditional metrics often overlook. Our analysis further shows that smaller models are consistently more parameter-efficient at generating unique content within a fixed sampling budget, offering insights into the relationship between model scaling and diversity. These findings have important implications for applications that require diverse yet high-quality outputs, from creative assistance to synthetic data generation.

William Liang, Sam Wang, Hung-Ju Wang et al.

Recent work has demonstrated that a promising strategy for teaching robots a wide range of complex skills is by training them on a curriculum of progressively more challenging environments. However, developing an effective curriculum of environment distributions currently requires significant expertise, which must be repeated for every new domain. Our key insight is that environments are often naturally represented as code. Thus, we probe whether effective environment curriculum design can be achieved and automated via code generation by large language models (LLM). In this paper, we introduce Eurekaverse, an unsupervised environment design algorithm that uses LLMs to sample progressively more challenging, diverse, and learnable environments for skill training. We validate Eurekaverse's effectiveness in the domain of quadrupedal parkour learning, in which a quadruped robot must traverse through a variety of obstacle courses. The automatic curriculum designed by Eurekaverse enables gradual learning of complex parkour skills in simulation and can successfully transfer to the real-world, outperforming manual training courses designed by humans.

Ramya Ramalingam, Sangdon Park, Osbert Bastani

Machine learning has become an effective tool for automatically annotating unstructured data (e.g., images) with structured labels (e.g., object detections). As a result, a new programming paradigm called neurosymbolic programming has emerged where users write queries against these predicted annotations. However, due to the intrinsic fallibility of machine learning models, these programs currently lack any notion of correctness. In many domains, users may want some kind of conservative guarantee that the results of their queries contain all possibly relevant instances. Conformal prediction has emerged as a promising strategy for quantifying uncertainty in machine learning by modifying models to predict sets of labels instead of individual labels; it provides a probabilistic guarantee that the prediction set contains the true label with high probability. We propose a novel framework for adapting conformal prediction to neurosymbolic programs; our strategy is to represent prediction sets as abstract values in some abstract domain, and then to use abstract interpretation to propagate prediction sets through the program. Our strategy satisfies three key desiderata: (i) correctness (i.e., the program outputs a prediction set that contains the true output with high probability), (ii) compositionality (i.e., we can quantify uncertainty separately for different modules and then compose them together), and (iii) structured values (i.e., we can provide uncertainty quantification for structured values such as lists). When the full program is available ahead-of-time, we propose an optimization that incorporates conformal prediction at intermediate program points to reduce imprecision in abstract interpretation. We evaluate our approach on programs that take MNIST and MS-COCO images as input, demonstrating that it produces reasonably sized prediction sets while satisfying a coverage guarantee.

Sagnik Anupam, Alexander Shypula, Osbert Bastani

With the advent of large language models (LLMs), there has been a great deal of interest in applying them to solve difficult programming tasks. Recent work has demonstrated their potential at program optimization, a key challenge in programming languages research. We propose a blackbox adaptation method called Retrieval Augmented Search (RAS) that performs beam search over candidate optimizations; at each step, it retrieves in-context examples from a given training dataset of slow-fast program pairs to guide the LLM. Critically, we find that performing contextual retrieval based on an LLM-generated natural language description significantly outperforms retrieval based on the source code. In addition, we propose a method called AEGIS for improving interpretability by decomposing training examples into "atomic edits" that are significantly more incremental in nature. We show that RAS performs 1.8$\times$ better than prior state-of-the-art blackbox adaptation strategies, and that AEGIS performs 1.37$\times$ better while performing significantly smaller edits.

Haosen Ge, Hamsa Bastani, Osbert Bastani

Conformal prediction has emerged as an effective strategy for uncertainty quantification by modifying a model to output sets of labels instead of a single label. These prediction sets come with the guarantee that they contain the true label with high probability. However, conformal prediction typically requires a large calibration dataset of i.i.d. examples. We consider the online learning setting, where examples arrive over time, and the goal is to construct prediction sets dynamically. Departing from existing work, we assume semi-bandit feedback, where we only observe the true label if it is contained in the prediction set. For instance, consider calibrating a document retrieval model to a new domain; in this setting, a user would only be able to provide the true label if the target document is in the prediction set of retrieved documents. We propose a novel conformal prediction algorithm targeted at this setting, and prove that it obtains sublinear regret compared to the optimal conformal predictor. We evaluate our algorithm on a retrieval task, an image classification task, and an auction price-setting task, and demonstrate that it empirically achieves good performance compared to several baselines.

Lianghuan Huang, Sagnik Anupam, Insup Lee et al.

Reinforcement learning (RL) has emerged as a promising strategy for finetuning small language models (SLMs) to solve targeted tasks such as math and coding. However, RL algorithms tend to be resource-intensive, taking a significant amount of time to train. We propose RAPID, a novel RL algorithm that can substantially reduce the running time of RL. Our key insight is that RL tends to be costly due to the need to perform both inference and backpropagation during training. To maximize use of computational resources, our algorithm performs inference in large batches, and then performs off-policy policy gradient updates in mini-batches. For off-policy updates, we incorporate group advantage estimation into the policy gradient algorithm, and derive an importance weighted estimator to correct for the bias arising from off-policy learning. Our experiments demonstrate that our algorithm can reduce running time by 11%-34% on three benchmarks compared to state-of-the-art RL algorithms while maintaining similar or better accuracy.

Michael S. Yao, James C. Gee, Osbert Bastani

The goal of offline model-based optimization (MBO) is to propose new designs that maximize a reward function given only an offline dataset. However, an important desiderata is to also propose a diverse set of final candidates that capture many optimal and near-optimal design configurations. We propose Diversity in Adversarial Model-based Optimization (DynAMO) as a novel method to introduce design diversity as an explicit objective into any MBO problem. Our key insight is to formulate diversity as a distribution matching problem where the distribution of generated designs captures the inherent diversity contained within the offline dataset. Extensive experiments spanning multiple scientific domains show that DynAMO can be used with common optimization methods to significantly improve the diversity of proposed designs while still discovering high-quality candidates.

Hamsa Bastani, Osbert Bastani, Bryce McLaughlin

There has been a surge of recent interest in automatically learning policies to target treatment decisions based on rich individual covariates. A common approach is to train a machine learning model to predict counterfactual outcomes, and then select the policy that optimizes the predicted objective value. In addition, practitioners also want confidence that the learned policy has better performance than the incumbent policy according to downstream policy evaluation. However, due to the winner's curse-an issue where the policy optimization procedure exploits prediction errors rather than finding actual improvements-predicted performance improvements are often not substantiated by downstream policy optimization. To address this challenge, we propose a novel strategy called inference-aware policy optimization, which modifies policy optimization to account for how the policy will be evaluated downstream. Specifically, it optimizes not only for the estimated objective value, but also for the chances that the policy will be statistically significantly better than the observational policy used to collect data. We mathematically characterize the Pareto frontier of policies according to the tradeoff of these two goals. Based on our characterization, we design a policy optimization algorithm that uses machine learning to predict counterfactual outcomes, and then plugs in these predictions to estimate the Pareto frontier; then, the decision-maker can select the policy that optimizes their desired tradeoff, after which policy evaluation can be performed on the test set as usual. Finally, we perform simulations to illustrate the effectiveness of our methodology.

Sagnik Anupam, Davis Brown, Shuo Li et al.

LLM web agents now browse and take actions on the open web, yet current agent evaluations are constrained to sandboxed environments or artificial tasks. We introduce BrowserArena, a live open-web agent evaluation platform that collects user-submitted tasks, runs Arena-style head-to-head comparisons, and uses step-level human feedback to surface failure modes. Collecting and analyzing step-level annotations on the agent traces, we identify three consistent failure modes: captcha resolution, pop-up banner removal, and direct navigation to URLs. By constructing targeted datasets to further study these tasks, we discover variations in how different language models navigate these failure modes. We find, for example, that o4-mini deploys a wider variety of strategies to circumvent captcha resolution than other models and DeepSeek-R1 consistently misleads users about pop-up banner closure. Our findings surface both the diversity and brittleness of current web agents. More broadly, our benchmarking methodology provides an approach to evaluating and understanding web agent failure modes at scale.

Michael S. Yao, Osbert Bastani, Alma Andersson et al.

The goal of personalized medicine is to discover a treatment regimen that optimizes a patient's clinical outcome based on their personal genetic and environmental factors. However, candidate treatments cannot be arbitrarily administered to the patient to assess their efficacy; we often instead have access to an in silico surrogate model that approximates the true fitness of a proposed treatment. Unfortunately, such surrogate models have been shown to fail to generalize to previously unseen patient-treatment combinations. We hypothesize that domain-specific prior knowledge - such as medical textbooks and biomedical knowledge graphs - can provide a meaningful alternative signal of the fitness of proposed treatments. To this end, we introduce LLM-based Entropy-guided Optimization with kNowledgeable priors (LEON), a mathematically principled approach to leverage large language models (LLMs) as black-box optimizers without any task-specific fine-tuning, taking advantage of their ability to contextualize unstructured domain knowledge to propose personalized treatment plans in natural language. In practice, we implement LEON via 'optimization by prompting,' which uses LLMs as stochastic engines for proposing treatment designs. Experiments on real-world optimization tasks show LEON outperforms both traditional and LLM-based methods in proposing individualized treatments for patients.

Stephen Mell, Botong Zhang, David Mell et al.

Modern large language models (LLMs) are often deployed as agents, calling external tools adaptively to solve tasks. Rather than directly calling tools, it can be more effective for LLMs to write code to perform the tool calls, enabling them to automatically generate complex control flow such as conditionals and loops. Such code actions are typically provided as Python code, since LLMs are quite proficient at it; however, Python may not be the ideal language due to limited built-in support for performance, security, and reliability. We propose a novel programming language for code actions, called Quasar, which has several benefits: (1) automated parallelization to improve performance, (2) uncertainty quantification to improve reliability and mitigate hallucinations, and (3) security features enabling the user to validate actions. LLMs can write code in a subset of Python, which is automatically transpiled to Quasar. We evaluate our approach on the ViperGPT visual question answering agent, applied to the GQA dataset, demonstrating that LLMs with Quasar actions instead of Python actions retain strong performance, while reducing execution time when possible by 42%, improving security by reducing user approval interactions when possible by 52%, and improving reliability by applying conformal prediction to achieve a desired target coverage level.

Botong Zhang, Shuo Li, Ignacio Hounie et al.

We study the problem of computing an optimal large language model (LLM) policy for a constrained alignment problem, where the goal is to maximize a primary reward objective while satisfying constraints on secondary utilities. Despite the popularity of Lagrangian-based LLM policy search in constrained alignment, iterative primal-dual methods often fail to converge, and non-iterative dual-based methods do not achieve optimality in the LLM parameter space. To address these challenges, we employ Lagrangian duality to develop an iterative dual-based alignment method that alternates between updating the LLM policy via Lagrangian maximization and updating the dual variable via dual descent. In theory, we characterize the primal-dual gap between the primal value in the distribution space and the dual value in the LLM parameter space. We further quantify the optimality gap of the learned LLM policies at near-optimal dual variables with respect to both the objective and the constraint functions. These results prove that dual-based alignment methods can find an optimal constrained LLM policy, up to an LLM parametrization gap. We demonstrate the effectiveness and merits of our approach through extensive experiments conducted on the PKU-SafeRLHF dataset.

Yimeng Zeng, Natalie Maus, Haydn Thomas Jones et al.

In multi-task Bayesian optimization, the goal is to leverage experience from optimizing existing tasks to improve the efficiency of optimizing new ones. While approaches using multi-task Gaussian processes or deep kernel transfer exist, the performance improvement is marginal when scaling beyond a moderate number of tasks. We introduce a novel approach leveraging large language models (LLMs) to learn from, and improve upon, previous optimization trajectories, scaling to approximately 1500 distinct tasks. Specifically, we propose a feedback loop in which an LLM is fine-tuned on the high quality solutions to specific tasks found by Bayesian optimization (BO). This LLM is then used to generate initialization points for future BO searches for new tasks. The trajectories of these new searches provide additional training data for fine-tuning the LLM, completing the loop. We evaluate our method on two distinct domains: database query optimization and antimicrobial peptide design. Results demonstrate that our approach creates a positive feedback loop, where the LLM's generated initializations gradually improve, leading to better optimization performance. As this feedback loop continues, we find that the LLM is eventually able to generate solutions to new tasks in just a few shots that are better than the solutions produced by "from scratch" by Bayesian optimization while simultaneously requiring significantly fewer oracle calls.

Wentao Guo, Jikai Long, Yimeng Zeng et al.

Zeroth-order optimization (ZO) is a memory-efficient strategy for fine-tuning Large Language Models using only forward passes. However, the application of ZO fine-tuning in memory-constrained settings such as mobile phones and laptops is still challenging since full precision forward passes are infeasible. In this study, we address this limitation by integrating sparsity and quantization into ZO fine-tuning of LLMs. Specifically, we investigate the feasibility of fine-tuning an extremely small subset of LLM parameters using ZO. This approach allows the majority of un-tuned parameters to be quantized to accommodate the constraint of limited device memory. Our findings reveal that the pre-training process can identify a set of "sensitive parameters" that can guide the ZO fine-tuning of LLMs on downstream tasks. Our results demonstrate that fine-tuning 0.1% sensitive parameters in the LLM with ZO can outperform the full ZO fine-tuning performance, while offering wall-clock time speedup. Additionally, we show that ZO fine-tuning targeting these 0.1% sensitive parameters, combined with 4 bit quantization, enables efficient ZO fine-tuning of an Llama2-7B model on a GPU device with less than 8 GiB of memory and notably reduced latency.

Yecheng Jason Ma, William Liang, Hung-Ju Wang et al.

Transferring policies learned in simulation to the real world is a promising strategy for acquiring robot skills at scale. However, sim-to-real approaches typically rely on manual design and tuning of the task reward function as well as the simulation physics parameters, rendering the process slow and human-labor intensive. In this paper, we investigate using Large Language Models (LLMs) to automate and accelerate sim-to-real design. Our LLM-guided sim-to-real approach, DrEureka, requires only the physics simulation for the target task and automatically constructs suitable reward functions and domain randomization distributions to support real-world transfer. We first demonstrate that our approach can discover sim-to-real configurations that are competitive with existing human-designed ones on quadruped locomotion and dexterous manipulation tasks. Then, we showcase that our approach is capable of solving novel robot tasks, such as quadruped balancing and walking atop a yoga ball, without iterative manual design.

Kan Xu, Hamsa Bastani, Surbhi Goel et al.

We study the stochastic bandit problem with ReLU neural network structure. We show that a $\tilde{O}(\sqrt{T})$ regret guarantee is achievable by considering bandits with one-layer ReLU neural networks; to the best of our knowledge, our work is the first to achieve such a guarantee. In this specific setting, we propose an OFU-ReLU algorithm that can achieve this upper bound. The algorithm first explores randomly until it reaches a linear regime, and then implements a UCB-type linear bandit algorithm to balance exploration and exploitation. Our key insight is that we can exploit the piecewise linear structure of ReLU activations and convert the problem into a linear bandit in a transformed feature space, once we learn the parameters of ReLU relatively accurately during the exploration stage. To remove dependence on model parameters, we design an OFU-ReLU+ algorithm based on a batching strategy, which can provide the same theoretical guarantee.