Exploration-Exploitation

We study the problem of fixed budget pure exploration in reinforcement learning. The goal is to identify a near-optimal policy, given a fixed budget on the number of interactions with the environment. Unlike the standard PAC setting, we do not require the target error level $\epsilon$ and failure rate $\delta$ as input. We propose novel algorithms and provide, to the best of our knowledge, the first instance-dependent $\epsilon$-uniform guarantee, meaning that the probability that $\epsilon$-correctness is ensured can be obtained simultaneously for all $\epsilon$ above a budget-dependent threshold. It characterizes the budget requirements in terms of the problem-specific hardness of exploration. As a core component of our analysis, we derive a $\epsilon$-uniform guarantee for the multiple bandit problem—solving multiple multi-armed bandit instances simultaneously—which may be of independent interest. To enable our analysis, we also develop tools for reward-free exploration under the fixed-budget setting, which we believe will be useful for future work.

When there exists an unlearnable source of randomness (noisy-TV) in the environment, a naively intrinsic reward driven exploring agent gets stuck at that source of randomness and fails at exploration. Intrinsic reward based on uncertainty estimation or distribution similarity, while eventually escapes noisy-TVs as time unfolds, suffers from poor sample efficiency and high computational cost. Inspired by recent findings from neuroscience that humans monitor their improvements during exploration, we propose a novel method for intrinsically-motivated exploration, named Learning Progress Monitoring (LPM). During exploration, LPM rewards model improvements instead of prediction error or novelty, effectively rewards the agent for observing learnable transitions rather than the unlearnable transitions. We introduce a dual-network design that uses an error model to predict the expected prediction error of the dynamics model in its previous iteration, and use the difference between the model errors of the current iteration and previous iteration to guide exploration. We theoretically show that the intrinsic reward of LPM is zero-equivariant and a monotone indicator of Information Gain (IG), and that the error model is necessary to achieve monotonicity correspondence with IG. We empirically compared LPM against state-of-the-art baselines in noisy environments based on MNIST, 3D maze with 160x120 RGB inputs, and Atari. Results show that LPM's intrinsic reward converges faster, explores more states in the maze experiment, and achieves higher extrinsic reward in Atari. This conceptually simple approach marks a shift-of-paradigm of noise-robust exploration.

In pure exploration problems, a statistician sequentially collects information to answer a question about some stochastic and unknown environment. The probability of returning a wrong answer should not exceed a maximum risk parameter $\delta$ and good algorithms make as few queries to the environment as possible. The Track-and-Stop algorithm is a pioneering method to solve these problems. Specifically, it is well-known that it enjoys asymptotic optimality sample complexity guarantees for $\delta \to 0$ whenever the map from the environment to its correct answers is single-valued (e.g., best-arm identification with a unique optimal arm). The Sticky Track-and-Stop algorithm extends these results to settings where, for each environment, there might exist multiple correct answers (e.g., $\epsilon$-optimal arm identification). Although both methods are optimal in the asymptotic regime, their non-asymptotic guarantees remain unknown. In this work, we fill this gap and provide non-asymptotic guarantees for both algorithms.

This paper examines the exploration–exploitation trade-off in reinforcement learning with verifiable rewards (RLVR), a framework for improving the reasoning of Large Language Models (LLMs). Recent studies suggest that RLVR can elicit strong mathematical reasoning in LLMs through two seemingly paradoxical mechanisms: \textit{spurious rewards}, which suppress exploitation by rewarding outcomes unrelated to the ground truth, and \textit{entropy minimization}, which suppresses exploration by pushing the model toward more confident and deterministic outputs, highlighting a puzzling dynamic: both discouraging exploitation and discouraging exploration improve reasoning performance, yet the underlying principles that reconcile these effects remain poorly understood. We focus on two fundamental questions: (i) how policy entropy relates to performance, and (ii) whether spurious rewards yield gains, potentially through the interplay of clipping bias and model contamination. Our results show that clipping bias under spurious rewards reduces policy entropy, leading to more confident and deterministic outputs, while entropy minimization alone is insufficient for improvement. We further propose a reward-misalignment model explaining why spurious rewards can enhance performance beyond contaminated settings. Our findings clarify the mechanisms behind spurious-reward benefits and provide principles for more effective RLVR training.

We examine a multi-armed bandit problem with contextual information, where the objective is to ensure that each arm receives a minimum aggregated reward across contexts while simultaneously maximizing the total cumulative reward. This framework captures a broad class of real-world applications where fair revenue allocation is critical and contextual variation is inherent. The cross-context aggregation of minimum reward constraints, while enabling better performance and easier feasibility, introduces significant technical challenges—particularly the absence of closed-form optimal allocations typically available in standard MAB settings. We design and analyze algorithms that either optimistically prioritize performance or pessimistically enforce constraint satisfaction. For each algorithm, we derive problem-dependent upper bounds on both regret and constraint violations. Furthermore, we establish a lower bound demonstrating that the dependence on the time horizon in our results is optimal in general and revealing fundamental limitations of the free exploration principle leveraged in prior work.

In this work, we take a first step toward elucidating the mechanisms behind emergent exploration in unsupervised reinforcement learning. We study Single-Goal Contrastive Reinforcement Learning (SGCRL) (Liu et al., 2025), a self-supervised algorithm capable of solving challenging long-horizon goal-reaching tasks without external rewards or curricula. We combine theoretical analysis of the algorithm’s objective function with controlled experiments to understand what drives its exploration. We show that SGCRL maximizes implicit rewards shaped by its learned representations. These representations automatically modify the reward landscape to promote exploration before reaching the goal and exploitation thereafter. Our experiments also demonstrate that these exploration dynamics arise from learning low-rank representations of the state space rather than from neural network function approximation. Our improved understanding enables us to adapt SGCRL to perform safety-aware exploration.

Exogenous MDPs (Exo-MDPs) capture sequential decision-making where uncertainty comes solely from exogenous inputs that evolve independently of the learner’s actions. This structure is especially common in operations research applications such as inventory control, energy storage, and resource allocation, where exogenous randomness (e.g., demand, arrivals, or prices) drives system behavior. Despite decades of empirical evidence that greedy, exploitation-only methods work remarkably well in these settings, theory has lagged behind: all existing regret guarantees for Exo-MDPs rely on explicit exploration or tabular assumptions. We show that exploration is unnecessary. We propose Pure Exploitation Learning ($\texttt{PEL}$) and prove the first general finite-sample regret bounds for exploitation-only algorithms in Exo-MDPs. In the tabular case, PEL achieves $\widetilde{O}(H^2|\Xi|\sqrt{K})$. For large, continuous endogenous state spaces, we introduce $\texttt{LSVI-PE}$, a simple linear-approximation method whose regret is polynomial in the feature dimension, exogenous state space, and horizon, independent of the endogenous state and action spaces. Our analysis introduces two new tools: counterfactual trajectories and Bellman-closed feature transport, which together allow greedy policies to have accurate value estimates without optimism. Experiments on synthetic and resource-management tasks show $\texttt{PEL}$ consistently outperforming baselines. Overall, our results overturn the conventional wisdom that exploration is required, demonstrating that in Exo-MDPs, pure exploitation is enough.

We propose DIVERSE, a framework for systematically exploring the Rashomon set of deep neural networks, the collection of models that match a reference model’s accuracy while differing in their predictive behavior. DIVERSE augments a pretrained model with Feature-wise Linear Modulation (FiLM) layers and uses Covariance Matrix Adaptation Evolution Strategy (CMA-ES) to search a latent modulation space, generating diverse model variants without retraining or gradient access. Across MNIST, PneumoniaMNIST, and CIFAR-10, DIVERSE uncovers multiple high-performing yet functionally distinct models. Our experiments show that DIVERSE offers a competitive and efficient exploration of the Rashomon set, making it feasible to construct diverse sets that maintain robustness and performance while supporting well-balanced model multiplicity. While retraining remains the baseline to generate Rashomon sets, DIVERSE achieves comparable diversity at reduced computational cost.

Reinforcement learning with verifiable rewards has significantly advanced the reasoning capabilities of large language models, yet how to explicitly steer training toward exploration or exploitation remains an open problem. We introduce Token Hidden Reward (THR), a token-level metric that quantifies each token’s influence on the likelihood of correct responses under Group Relative Policy Optimization (GRPO). We find that training dynamics are dominated by a small subset of tokens with high absolute THR values. Most interestingly, tokens with positive THR strengthen confidence in correct outputs, thus favoring exploitation, while tokens with negative THR preserve probability mass for alternative outputs, enabling exploration. This insight suggests a natural intervention: a THR-guided reweighting algorithm that modulates GRPO’s learning signals to explicitly bias training toward exploitation or exploration. We validate the efficacy of this algorithm on diverse math reasoning benchmarks. By amplifying tokens with positive THR value and weakening negative ones, our algorithm improves greedy-decoding accuracy, favoring exploitation. The reverse strategy yields consistent gains in Pass@K accuracy, favoring exploration. We further demonstrate that our algorithm integrates seamlessly with other RL objectives such as GSPO and generalizes across architectures including Llama. These findings establish THR as a principled and fine-grained mechanism for dynamically controlling exploration and exploitation in RL-tuned LLMs, providing new tools for targeted fine-tuning in reasoning-intensive applications.

Uncertainty quantification is a critical aspect of reinforcement learning and deep learning, with numerous applications ranging from efficient exploration and stable offline reinforcement learning to outlier detection in medical diagnostics. The scale of modern neural networks, however, complicates the use of many theoretically well-motivated approaches such as full Bayesian inference. Approximate methods like deep ensembles can provide reliable uncertainty estimates but still remain computationally expensive. In this work, we propose contextual similarity distillation, a novel approach that explicitly estimates the variance of an ensemble of deep neural networks with a single model, without ever learning or evaluating such an ensemble in the first place. Our method builds on the predictable learning dynamics of wide neural networks, governed by the neural tangent kernel, to derive an efficient approximation of the predictive variance of an infinite ensemble. Specifically, we reinterpret the computation of ensemble variance as a supervised regression problem with kernel similarities as regression targets. The resulting model can estimate predictive variance at inference time with a single forward pass, and can make use of unlabeled target-domain data or data augmentations to refine its uncertainty estimates. We empirically validate our method across a variety of out-of-distribution detection benchmarks and sparse-reward reinforcement learning environments. We find that our single-model method performs competitively and sometimes superior to ensemble-based baselines and serves as a reliable signal for efficient exploration. These results, we believe, position contextual similarity distillation as a principled and scalable alternative for uncertainty quantification in reinforcement learning and general deep learning.

While Large Language Models (LLMs) hold promise to become autonomous agents, they often explore suboptimally in sequential decision-making. Recent work has sought to enhance this capability via supervised fine-tuning (SFT) or reinforcement learning (RL), improving regret on the classic multi-armed bandit task. However, it remains unclear how these learning methods shape exploration strategies and how well they generalize. We investigate both paradigms by training LLMs with SFT on expert trajectories and RL with a range of tailored reward signals including a strategic, regret-shaped reward to reduce variance, and an algorithmic reward that enables oracle imitation. The resulting agents outperform pre-trained models and achieve performance comparable to Upper Confidence Bound (UCB) and Thompson Sampling, with robust generalization to 6$\times$ longer horizons and across bandit families. Behavioral analysis reveals that gains often stem from more sophisticated but greedier exploitation: RL/SFT agents are more prone to early catastrophic failure than pre-trained models, prematurely abandoning exploration. Furthermore, agents trained to imitate UCB learn to outperform their teacher by adopting more exploitative variants. Our findings clarify when each training paradigm is preferable and advocate tailored reward design and evaluation beyond average regret to promote robust exploratory behavior.

Auto-regressive large language models (LLMs) exhibit a non-trivial capacity to "anticipate'' long-range future tokens despite being trained to predict only one token at a time. Nevertheless, how to systematically profile, enhance and leverage such capacity to practically improve LLM reasoning performance remains unclear. In this paper, we propose **Next Token-Bag Exploitation (Next-ToBE)** to tackle this challenge. Next-ToBE quantifies LLM’s anticipatory capacity by measuring how well tokens in the future window are pre-captured by the model’s current softmax probabilities. This capacity is strongly correlated with LLM generative quality but often suppressed by the rigid one-hot objective in next-token prediction. To address this, we replace the {one-hot target vector} in next-token prediction with a soft target distribution spanning additional future tokens. Specifically, the immediate next token retains the highest importance, while more distant ``look-ahead tokens'' are also included to enrich supervision, with their importance dynamically determined by temporal and semantic relevance patterns to inject forward-looking pressure. Besides, the fitting process emphasizes the model’s intrinsic anticipatory tendency, thus preserving the confidence and fidelity of the pre-trained model to improve training stability. Overall, Next-ToBE not only effectively activates LLM anticipatory capacity through fine-tuning, yielding notable gains in reasoning performance with higher memory and computational efficiency against the MTP baselines, but also shows great potential in pretraining setting by successfully cultivating this capacity from scratch. These highlight its value as an effective strategy to extend the prediction horizon of LLMs, enabling them to see further, and reason better.

Reinforcement learning algorithms are typically designed for discrete-time dynamics, even though the underlying real-world control systems are often continuous in time. In this paper, we study the problem of continuous-time reinforcement learning, where the unknown system dynamics are represented using nonlinear ordinary differential equations (ODEs). We leverage probabilistic models, such as Gaussian processes and Bayesian neural networks, to learn an uncertainty-aware model of the underlying ODE. Our algorithm, COMBRL, greedily maximizes a weighted sum of the extrinsic reward and model epistemic uncertainty. This yields a scalable and sample-efficient approach to continuous-time model-based RL. We show that COMBRL achieves sublinear regret in the reward-driven setting, and in the unsupervised RL setting (i.e., without extrinsic rewards), we provide a sample complexity bound. In our experiments, we evaluate COMBRL in both standard and unsupervised RL settings and demonstrate that it scales better, is more sample-efficient than prior methods, and outperforms baselines across several deep RL tasks.

Balancing exploration and exploitation remains a central challenge in reinforcement learning with verifiable rewards (RLVR) for large language models (LLMs). Current RLVR methods often overemphasize exploitation, leading to entropy collapse, reduced exploratory capacity, and ultimately limited performance gains. Although techniques that add randomness increase policy stochasticity, they frequently fail to escape dominant behavioral modes. The resulting sample-and-reward dynamics amplify these modes, eroding exploration and leading to entropy collapse. We introduce Exploration-Enhanced Policy Optimization (EEPO), a novel framework that promotes exploration through two-stage rollouts with adaptive unlearning. In the first stage, the model generates half of the trajectories; it then undergoes a lightweight, temporary unlearning step to suppress these sampled responses, forcing the second stage to explore different regions of the output space. This sample-then-forget mechanism actively steers the policy away from dominant modes and encourages mode-seeking exploration. Across five reasoning benchmarks, EEPO consistently outperforms baselines, achieving average gains of 24.3% on Qwen2.5-3B, 33.0% on Llama3.2-3B-Instruct, and 10.4% on Qwen3-8B-Base.

Estimating epistemic uncertainty in value functions is a crucial challenge for many aspects of reinforcement learning (RL), including efficient exploration, safe decision-making, and offline RL. While deep ensembles provide a robust method for quantifying value uncertainty, they come with significant computational overhead. Single-model methods, while computationally favorable, often rely on heuristics and typically require additional propagation mechanisms for myopic uncertainty estimates. In this work we introduce universal value-function uncertainties (UVU), which, similar in spirit to random network distillation (RND), quantify uncertainty as squared prediction errors between an online learner and a fixed, randomly initialized target network. Unlike RND, UVU errors reflect policy-conditional $\textit{value uncertainty}$, incorporating the future uncertainties $\textit{any policy}$ may encounter. This is due to the training procedure employed in UVU: the online network is trained using temporal difference learning with a synthetic reward derived from the fixed, randomly initialized target network. We provide an extensive theoretical analysis of our approach using neural tangent kernel (NTK) theory and show that in the limit of infinite network width, UVU errors are exactly equivalent to the variance of an ensemble of independent universal value functions. Empirically, we show that UVU achieves equal performance to large ensembles on challenging multi-task offline RL settings, while offering simplicity and substantial computational savings.

We study the _active sequential hypothesis testing_ problem, also known as _pure exploration_: given a new task, the learner _adaptively collects data_ from the environment to efficiently determine an underlying correct hypothesis. A classical instance of this problem is the task of identifying the best arm in a multi-armed bandit problem (a.k.a. BAI, Best-Arm Identification), where actions index hypotheses. Another important case is generalized search, a problem of determining the correct label through a sequence of strategically selected queries that indirectly reveal information about the label. In this work, we introduce _In-Context Pure Exploration_ (ICPE), which meta-trains Transformers to map _observation histories_ to _query actions_ and a _predicted hypothesis_, yielding a model that transfers in-context. At inference time, ICPE actively gathers evidence on new tasks and infers the true hypothesis without parameter updates. Across deterministic, stochastic, and structured benchmarks, including BAI and generalized search, ICPE is competitive with adaptive baselines while requiring no explicit modeling of information structure. Our results support Transformers as practical architectures for _general sequential testing_.

Reinforcement learning (RL) is the dominant paradigm for sharpening strategic tool use capabilities of LLMs on long-horizon, sparsely-rewarded agent tasks, yet it faces a fundamental challenge of exploration-exploitation trade-off. Existing studies stimulate exploration through the lens of policy entropy, but such mechanical entropy maximization is prone to RL instability due to the multi-turn distribution shifting. In this paper, we target the progressive exploration-exploitation balance under the guidance of the agent's own experiences without succumbing to either entropy collapsing or runaway divergence. We propose SPEAR, a self-imitation learning (SIL) recipe for training agentic LLMs. It extends the vanilla SIL, where a replay buffer stores good experience for off-policy update, by gradually steering the policy entropy across stages. Specifically, the proposed curriculum scheduling harmonizes intrinsic reward shaping and self-imitation to 1) expedite exploration via frequent tool interactions at the beginning, and 2) strengthen exploitation of successful tactics upon convergence towards familiarity with the environment. We also combine bag-of-tricks of industrial RL optimizations for a strong baseline Dr.BoT to demonstrate our effectiveness. In ALFWorld and WebShop, SPEAR increases the success rates of GRPO/GiGPO/Dr.BoT by up to 16.1\%/5.1\%/8.6\% and 20.7\%/11.8\%/13.9\%, respectively. In AIME24 and AIME25, SPEAR boosts Dr.BoT by up to 3.8\% and 6.1\%, respectively. Such gains incur only 10\%–25\% extra theoretical complexity and negligible runtime overhead in practice, demonstrating the plug-and-play scalability of SPEAR.

Effective exploration in reinforcement learning requires not only tracking where an agent has been, but also understanding how the agent perceives and represents the world. To learn powerful representations, an agent should actively explore states that contribute to its knowledge of the environment. Temporal representations can capture the information necessary to solve a wide range of potential tasks while avoiding the computational cost associated with full state reconstruction. In this paper, we propose an exploration method that leverages temporal contrastive representations to guide exploration, prioritizing states with unpredictable future outcomes. We demonstrate that such representations can enable the learning of complex exploratory behaviors in locomotion, manipulation, and embodied-AI tasks, revealing capabilities and behaviors that traditionally require extrinsic rewards. Unlike approaches that rely on explicit distance learning or episodic memory mechanisms (e.g., quasimetric-based methods), our method builds directly on temporal similarities, yielding a simpler yet effective strategy for exploration.

Flow and diffusion models are typically pre-trained on limited available data (e.g., molecular samples), covering only a fraction of the valid design space (e.g., the full molecular space). As a consequence, they tend to generate samples from only a narrow portion of the feasible domain. This is a fundamental limitation for scientific discovery applications, where one typically aims to sample valid designs beyond the available data distribution. To this end, we address the challenge of leveraging access to a verifier (e.g., an atomic bonds checker), to adapt a pre-trained flow model so that its induced density expands beyond regions of high data availability, while preserving samples validity. We introduce formal notions of strong and weak verifiers and propose algorithmic frameworks for global and local flow expansion via probability-space optimization. Then, we present Flow Expander (FE), a scalable mirror descent scheme that provably tackles both problems by verifier-constrained entropy maximization over the flow process noised state space. Next, we provide a thorough theoretical analysis of the proposed method, and state convergence guarantees under both idealized and general assumptions. Ultimately, we empirically evaluate our method on both illustrative, yet visually interpretable settings, and on a molecular design task showcasing the ability of FE to expand a pre-trained flow model increasing conformer diversity while preserving validity.

Reinforcement Learning with Verifiable Rewards (RLVR) has proven effective for enhancing Large Language Models (LLMs) on complex reasoning tasks. Yet current methods face an exploration dilemma: standard RL struggles to escape the local optima of pre-trained LLMs’ sharply peaked initial policies, boosting single-solution accuracy (pass@1) but suppressing solution diversity and multi-solution performance (pass@k). As a result, RLVR often distills existing capabilities rather than discovering new reasoning strategies. We address this with a Risk-Sensitive Reinforcement Learning framework. By adopting a risk-seeking objective that interpolates between mean and maximum rewards, we derive a novel Risk-Sensitive GRPO (RS-GRPO) algorithm that emphasizes hard prompts to drive exploration. Across six mathematical reasoning benchmarks and five LLMs, RS-GRPO consistently improves pass@k performance while enhancing or maintaing pass@1.