Kullback-Leibler Regularization

While Reinforcement Learning from Human Feedback (RLHF) is the standard paradigm for aligning large language models with human preferences, its effectiveness in pluralistic settings has been called into question. Notably, recent work by G"olz et al. (2025) demonstrated that the \textit{distortion} -- defined as the multiplicative gap between the average user utility of the RLHF policy and the optimal average utility -- can scale exponentially with the Bradley-Terry temperature parameter $\beta$ when users have heterogeneous preferences. In this work, we present a fine-grained analysis of the distortion of RLHF with reward clipping and demonstrate that such exponential degradation is not a fundamental property of the algorithm but rather a consequence of distribution mismatch between the distribution generating preference data ($\mu$) and the KL reference policy ($\pi_{\mathrm{ref}}$). To this end, we establish tight upper and lower bounds on the distortion of RLHF across multiple regimes of the KL regularization strength. We show that in a representative regime, under the Bradley-Terry model, the distortion is $\tilde{\Theta}(\beta B + \beta)$, where $B$ is an upper bound on the log density ratio between $\mu$ and $\pi_{\mathrm{ref}}$. In particular, when there is no distribution mismatch (i.e., $\mu = \pi_{\mathrm{ref}}$), RLHF achieves the optimal distortion of $O(\beta)$ up to a constant. Our results suggest that, to reasonably maximize average utility with RLHF, it is preferable to use on-policy sampled preference data or to fine-tune before RLHF on data from a source close to $\mu$.

Regularization-based methods have become a standard approach for training Deep Reinforcement Learning policies against adversarial input perturbations. In this paper, we unify these methods by deriving new upper bounds on the performance gap between the nominal and worst-case policies. Each upper bound is expressed as an existing regularization objective plus a KL-divergence penalty between the nominal and worst-case policies, which further explains why adding a KL penalty improves robustness in practice. Building on these bounds, we formulate robust training as a constrained optimization problem, showing that existing methods correspond to the special case of a fixed Lagrange multiplier. We instead update the multiplier jointly with the policy to automatically tune the regularization weight. Finally, we conduct extensive adversarial evaluations across several continuous control tasks to validate our theoretical analysis.

We study KL-regularized contextual bandits under both reward and preference feedback. While existing regret guarantees typically depend on the eluder dimension, we show that simple greedy sampling can achieve polylogarithmic regret without explicit dependence on this complexity measure. For reward feedback, we analyze a greedy algorithm that samples directly from the Gibbs policy induced by the estimated reward. We extend the result to preference feedback under both general preference and Bradley--Terry models, while also sharpening existing dimension-dependent guarantees. Our analysis reveals a trade-off between greedy sampling and upper confidence bound-style exploration: greedy sampling enjoys stronger regret guarantees when KL regularization is sufficiently strong, whereas additional exploration yields sharper bounds as the regularization weakens.

We propose CVaR-penalized Generative Particle Algorithm (CVaR-GPA), a robust, tail-agnostic algorithm for fine-tuning generative models to learn heavy-tailed distributions and capture extreme events, requiring no prior knowledge or estimation of the target's tail characteristics. The method is the Wasserstein gradient flow of the Lipschitz-regularized Kullback-Leibler (KL) divergence penalized by a Conditional Value-at-Risk (CVaR) discrepancy term: the Lipschitz-regularized KL divergence enables robust learning under minimal assumptions on the target distribution, while the CVaR penalty restores the velocity that otherwise vanishes prematurely in the under-sampled tails. The penalized flow admits a bounded but non-Lipschitz velocity field. This departs from the Lipschitz transport maps of standard generators, which preserve the tail behavior of a light-tailed source, and enables transport toward heavier-tailed targets. To define this flow on empirical measures, we derive the first-variation subgradients of CVaR from its Rockafellar-Uryasev representation, valid precisely where the classical density-based formula fails. The particle algorithm CVaR-GPA fine-tunes the output samples of any pre-trained model, without access to its architecture, and runs on an adaptive time horizon set by a kinetic-energy stopping criterion rather than a preset depth. On synthetic isotropic and anisotropic Student-$t$ target distributions, Neal's funnel distribution, and the real-world high-dimensional Fama-French 25 portfolio dataset, CVaR-GPA dramatically improves global and tail accuracy on heavy-tailed targets over the pre-trained baseline.

Reinforcement learning (RL) has become a central paradigm for large language model (LLM) post-training, but optimization toward new objectives can degrade capabilities already present in the base model. KL regularization is widely used to mitigate such forgetting by constraining policy drift toward a reference model. However, standard full-policy KL regularization constrains the entire response distribution and may unnecessarily restrict exploration and target-task learning. This raises a natural question: can a more precise constraint preserve existing capabilities while minimizing interference with learning new tasks? To this end, we propose \underline{Co}rrectness-Conditioned \underline{KL} Regularization (CoKL), a conditional regularization framework that narrows the preservation constraint from the full output distribution to correctness-conditioned response distributions. We instantiate CoKL with forward KL divergence and derive a practical finite-group training objective for RL-based LLM post-training. At the population level, CoKL decouples the total probability assigned to correct responses from their correctness-conditioned distribution, thereby regularizing the relative probability allocation among reference-supported correct responses without directly anchoring incorrect outputs or total correctness mass. We further show that full-policy forward and reverse KL regularization induce a strict optimal correctness gap when the reference policy is imperfect, whereas CoKL avoids this limitation. Experiments in controlled multi-solution environments and continual post-training settings across multiple model scales demonstrate that CoKL achieves a more favorable balance between target-task improvement and prior-capability retention than existing regularization methods. Our code is available at https://github.com/Lumina04/CoKL.

Reinforcement learning (RL) fine-tuning is widely used in language model training to improve model performance on a target task while limiting drift from a reference policy. A standard way to balance this trade-off is via a KL-regularized RL objective, although this formulation does not by itself provide a principled way to set the regularization coefficient. In practice, the coefficient is typically chosen heuristically or via hyperparameter search, which can lead to unnecessary overhead in training cost or undesirable reward-retention trade-offs. We instead propose a game-theoretic framework that gives this trade-off an explicit statistical interpretation. Specifically, we study a sequential game in which an agent chooses a policy to maximize cumulative reward while a monitor observes policy outputs over time and tests for deviations from the reference policy. Although not originating from the same perspective, we show that the resulting equilibrium policy can nonetheless be expressed as the solution to a KL-regularized RL problem for an optimal regularization parameter that can be viewed as maximizing reward per unit of statistical distinguishability. Drawing on classical results from concave-convex fractional programming, we provide a principled method for learning this equilibrium coefficient via reduction to the KL-regularized RL objective, thus allowing for flexible integration into standard fine-tuning pipelines. In experiments with Qwen3-8B and Llama-3.2-1B, we demonstrate that our methods result in competitive reward-retention trade-offs in a continual learning setting, and illustrate how our framework may be used to audit API providers serving open-source models.

We introduce trajectory-regularized stochastic optimal control (TRSOC), which augments standard stochastic optimal control (SOC) with a Kullback--Leibler (KL) divergence between controlled and reference trajectory distributions. Using Girsanov's theorem, the trajectory KL reduces to a quadratic drift mismatch penalty, yielding a modified running cost that preserves the dynamic programming (DP) structure. We derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation and characterize the optimal policy. In the linear-quadratic (LQ) setting, the formulation admits a closed-form solution with an augmented control cost. Experiments show that the regularization parameter induces a trade-off between performance-driven and reference-preserving behavior, including cases with reference dynamics learned from offline data.

Entropy regularization is widely used in continuous-time reinforcement learning (RL) to reduce sensitivity to environmental perturbations, yet its robustness benefits lack a rigorous theoretical foundation. This paper establishes the first robustness guarantees for entropy-regularized continuous-time Markov decision processes. We show that maximizing an entropy-regularized objective yields a lower bound on a worst-case robust RL problem with joint reward and transition perturbations. We analytically characterize the induced robust sets and prove that they expand monotonically with the regularization strength, justifying the empirical observation that stronger entropy improves robustness. In contrast to prior discrete-time analyses, our results remove the intractable state-distribution entropy term and provide guarantees invariant to action frequency. Experiments on queueing network control and market making confirm our theory, showing that entropy-regularized policies outperform greedy and $ε$-greedy baselines under dynamics perturbations.

Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data. When the data lies on a manifold with non-Euclidean topology, the standard Gaussian prior introduces a topological mismatch that degrades reconstruction quality and prevents faithful representation. We present a constructive mathematical framework that resolves this mismatch for all manifolds that admit a product covering space. These are manifolds expressible as products of elementary factors (circles, intervals, or lines) or as quotients of such products by a finite symmetry group. The class includes cylinders, tori, Möbius strips, Klein bottles, and real projective spaces. Factorized distributions over the elementary factors yield product topologies with closed-form, decoupled KL divergences, so that each latent factor can be shaped independently while keeping training tractable. We catalogue reparametrizable encoder-prior pairs for periodic, bounded, and unbounded supports, and provide coordinate transformations that allow standard neural networks to output non-Euclidean parameters with smooth gradients. For quotient manifolds, the decoder receives group-invariant features of the covering-space coordinates, so that identified points produce identical outputs. Anchor constraints fix the coordinate system relative to the data or create soft topological holes. Experiments on synthetic manifolds and real-image datasets (rotated and cyclically shifted MNIST) confirm that a topology-matched prior aligns KL regularization with the data manifold. The resulting topology-aware models outperform the Gaussian baseline at all practically relevant regularization strengths. The code is available at https://github.com/JvHulst/VAE-Topology.

We study KL-regularized contextual bandits and episodic reinforcement learning (RL) under general function approximation with model misspecification. Existing guarantees rely on realizability and therefore do not extend to misspecified models, where classical regret bounds may fail. This work introduces KL misspecification formulations for contextual bandits and episodic RL and analyzes regression-based algorithms with Gibbs policy updates. High-probability KL-regret guarantees with explicit misspecification terms are established, recovering the standard realizable KL-regularized setting as a special case.

On-policy self-distillation, where a language model conditions on privileged context to supervise its own generations, is a promising source of dense supervision for sparse-reward reinforcement learning. Actually, it can be instantiated as an auxiliary full-vocabulary student-to-teacher reverse Kullback-Leibler divergence loss. We therefore propose SDPG, a self-distilled policy-gradient framework that combines group-relative verifier advantages with normalized standard deviation, exact full-vocabulary on-policy self-distillation, as well as reference-policy KL regularization. Empirically, SDPG improves stability and performance over RLVR and self-distillation baselines. The code is available at https://github.com/lauyikfung/SDPG.

Low-Rank Adaptation (LoRA) has become one of the most widely used fine-tuning mechanisms for adapting large language models to new domains, tasks, and users. Yet adaptation performance alone can obscure an important failure mode: LoRA updates may improve performance on the target distribution while degrading prior capabilities learned during pretraining and alignment. We show that this forgetting becomes especially severe when the adaptation distribution differs substantially from the models original training or alignment distributions. The challenge is amplified in practical settings, where the original training and alignment data are typically unavailable. Motivated by this constraint, we study how LoRA based adaptation balances new learning against forgetting in a replay-free setting, and introduce a simple output space regularizer that can be added directly to existing training pipelines. Our method removes the ground-truth token from both the base and adapted model distributions, renormalizes the remaining probabilities, and applies KL regularization only over the non-target vocabulary. This preserves the base models relative preferences among alternative tokens without directly opposing the cross-entropy signal required for adaptation. As the regularizer acts only at the loss level, it requires no replay data, architectural changes, adapter redesign, or inference-time overhead, and can be applied directly to existing LoRA variants. Across all LoRA variants tested and across various backbones, our method improves the frontier between new learning and forgetting when the adaptation distribution differs substantially from the base models original training or alignment distributions, suggesting a broadly applicable route toward more reliable LLM updating.

Online off-policy reinforcement learning (RL) is shaped by two coupled choices: the policy class and the update rule. Gaussian policies are fast and have tractable entropy, but struggle with multimodal action distributions. Generative policies are more expressive, but often require iterative sampling or lack tractable entropy estimates. On the optimisation side, SAC-style soft policy improvement and mirror descent (MD) can be viewed as minimising different KL divergences: the former moves the policy towards a value-induced Boltzmann distribution, while the latter regularises each update against the previous policy. Combining entropy regularisation with an MD constraint is therefore attractive, as it supports exploration while stabilising policy improvement; however, the resulting target can be multimodal and is poorly matched by unimodal Gaussian policies. We propose Stochastic MeanFlow Policies (SMFP), a one-step generative policy class that maps Gaussian noise to actions through a MeanFlow transformation. This stochastic reparameterisation yields a tractable entropy surrogate and allows MeanFlow policies to be trained within off-policy mirror descent under a unified objective for exploratory yet stable improvement. Across seven MuJoCo benchmarks, SMFP improves over Gaussian and generative baselines while retaining single-step inference efficiency.

Programmatic reinforcement learning (PRL) offers an interpretable alternative to deep reinforcement learning by representing policies as human-readable and -editable programs. While gradient-based methods have been developed to optimize continuous relaxations of programs, they face a significant performance drop when converting the continuous relaxations back into discrete programs. Post-hoc discretization can discard optimized branches and parameters in a program, which results in a collapse of policy expressivity and lowered task performance, leading in turn to a need for additional fine-tuning. To overcome these limitations, we propose Differentiable Discrete Programmatic Reinforcement Learning (DiPRL), a method that learns programmatic policies that become nearly discrete during training, avoiding a separate post-hoc fine-tuning stage. We first analyze the inherent risks of performance drop introduced by post-hoc discretization of gradient-based methods. Then, we introduce programmatic architecture entropy regularization, which enables smooth, differentiable training that encourages convergence toward a discrete program. DiPRL maintains the efficiency of gradient-based optimization while mitigating the risks of post-hoc discretization. Our experiments across multiple discrete and continuous RL tasks demonstrate that DiPRL can achieve strong performance via interpretable programmatic policies.

Latent diffusion models have emerged as the dominant paradigm for many generation tasks including audio generation such as text-to-audio, text-to-music and text-to-speech. A key component of latent diffusion is an autoencoder (VAE) that compresses high-dimensional signals into a low frame rate continuous representation that is conducive for downstream prediction. Regularizing these VAEs is challenging, as there is a trade-off between over-regularized (poor output quality) and under-regularized (difficult to predict) latent representations. We propose a framework for studying this trade-off through compression and train Audio VAEs at specific bitrates via target-KL regularization. This allows direct comparison to well-studied discrete neural audio codec models, and the construction of rate-distortion curves for audio VAEs. We evaluate the impact of target-KL regularization on text-to-sound generation and find that sweeping compression rates is helpful in identifying the optimal generation setting.

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast $\widetilde{\mathcal{O}}(1/n)$ convergence rate under \textit{unilateral concentrability}, improving over the standard $\widetilde{\mathcal{O}}(1/\sqrt{n})$ rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same $\widetilde{\mathcal{O}}(1/n)$ statistical rate up to a vanishing optimization error of order $\widetilde{\mathcal{O}}(1/\sqrt{T})$ in the number of self-play iterations $T$.

Reinforcement Learning with Verifiable Rewards (RLVR) has become the standard paradigm for LLM mathematical reasoning, where Group Relative Policy Optimization (GRPO) serves as the mainstream algorithm. We point out two understudied inefficiencies existing in GRPO. First, the fixed KL penalty coefficient overly restricts policy exploration at stages where the model requires significant deviation from the reference policy. Second, uniform sampling of training questions ignores that moderately difficult problems provide the most informative gradient signals for optimization. We propose Exploration-Prioritized Policy Optimization (EXPO) with two lightweight plug-in modules. The Accuracy-Conditioned KL Scaling (AKL) dynamically adjusts KL regularization strength through a smooth nonlinear function of batch average accuracy, relaxing the penalty when the model underperforms and strengthening it when the model achieves good results. The Gaussian Curriculum Sampling (GCS) assigns sampling weights to questions following a Gaussian distribution centered at moderate accuracy around 0.5, focusing training on the model's learning frontier. We conduct extensive experiments on DeepSeek-R1-Distill-Qwen-1.5B and Qwen3-8B-Base over six mathematical reasoning benchmarks. The results show EXPO steadily surpasses vanilla GRPO. It obtains an absolute gain of 13.34 on AIME 2025 pass@32, rising from 63.33 percent to 76.67 percent, and achieves an average pass@32 improvement of 2.66 on the 8B model. The much larger performance gains on pass@32 compared with pass@1 demonstrate that EXPO effectively enlarges the model's exploration boundary under a fixed inference cost budget.

\emph{Kullback-Leibler} (KL) regularization is ubiquitous in reinforcement learning algorithms in the form of \emph{reverse} or \emph{forward} KL. Recent studies have demonstrated $ε^{-1}$-type fast rates for decision making under reverse KL regularization, in contrast to the standard $ε^{-2}$-type sample complexity. However, for forward-KL-regularized objectives, existing statistical analyses are either not applicable or result in $\tilde{O}(ε^{-2})$ slow rates. We take the first step towards addressing this problem via a streamlined analysis of forward-KL-regularized offline CBs. We give the first $\tilde{O}(ε^{-1})$ upper bounds in tabular and general function approximation settings, both under notions of \emph{single-policy concentrability}. In particular, our convex-analytical pipeline unifies these settings by exploiting the pessimism principle in a novel way and completely bypasses the proof routines in previous works based on the mean value theorem, which might be of independent interest. Moreover, we provide rate-optimal lower bounds, manifesting the tightness of our upper bounds in terms of statistical rates. Our lower bounds also demonstrate that the forward-KL-regularized sample complexity recovers the unregularized slow rate in the low-regularization regime, similarly to the reverse-KL regularization.

We introduce the Sinkhorn treatment effect, an entropic optimal transport measure of divergence between counterfactual distributions. Unlike classical quantities such as the average treatment effect, this measure captures differences across entire distributions. We analyze this divergence as a statistical functional and show it can be written as a smooth transformation of counterfactual mean embeddings with an appropriate kernel. This characterization allows us to establish first-order pathwise differentiability in general, and second-order pathwise differentiability under the null hypothesis of equal counterfactual distributions. Leveraging this smoothness, we construct debiased estimators and use them to obtain asymptotically valid tests for distributional treatment effects with a fixed entropic regularization parameter. Because the power of the test depends on this unknown parameter, we further propose an aggregated test that combines evidence across a grid of regularization choices. Experiments on simulated and image data demonstrate the practical advantages of our estimator and testing procedure.

Human Motion Segmentation (HMS), which aims to partition a video into non-overlapping segments corresponding to different human motions, has recently attracted increasing research attention. Existing HMS approaches are predominantly based on subspace clustering, which are grounded on the assumption that the distribution of high-dimensional temporal features well aligns with a Union-of-Subspaces (UoS). For videos in the real world, however, the raw frame-level features often violate the UoS assumption and yield unsatisfactory segmentation performance. To address this issue, we propose an efficient and effective approach for HMS, named Temporal Deep Self-expressive subspace Clustering (TDSC), which jointly learns temporally consistent structured representations and stabilized affinity for accurate and robust HMS. Specifically, in TDSC, we alternately learn structured representations of the input frame features and self-expressive coefficients via a properly regularized self-expressive model, in which a coding-rate maximization regularizer is incorporated to avoid representation collapse and conform the learned representations to span a desired UoS distribution, and meanwhile, temporal constraints are incorporated to promote temporally adjacent frames to be partitioned into the same groups. Moreover, we develop a temporal momentum averaging mechanism to stabilize affinity evolution and design a reparameterization strategy to enable efficient optimization. We conduct extensive experiments on five benchmark HMS datasets using both conventional (HoG) and up-to-date deep features (i.e., CLIP, DINOv2) to validate the effectiveness of our approach.

Kullback-Leibler (KL) regularization is widely used in offline decision-making and offers several benefits, motivating recent work on the sample complexity of offline learning with respect to KL-regularized performance metrics. Nevertheless, the exact sample complexity of KL-regularized offline learning remains largely from fully characterized. In this paper, we study this question in the setting of multi-armed bandits (MABs). We provide a sharp analysis of KL-PCB (Zhao et al., 2026), showing that it achieves a sample complexity of $\tilde{O}(ηSAC^{π^*}/ε)$ under large regularization $η= \tilde{O}(ε^{-1})$, and a sample complexity of $\tildeΩ(SAC^{π^*}/ε^2)$ under small regularization $η= \tildeΩ(ε^{-1})$, where $η$ is the regularization parameter, $S$ is the number of contexts, $A$ is the number of arms, $C^{π^*}$ policy coverage coefficient at the optimal policy $π^*$, $ε$ is the desired sub-optimality, and $\tilde{O}$ and $\tildeΩ$ hide all poly-logarithmic factors. We further provide a pair of sharper sample complexity lower bounds, which matches the upper bounds over the entire range of regularization strengths. Overall, our results provide a nearly complete characterization of offline multi-armed bandits with KL regularization.

Clinical reasoning demands multi-step interactions -- gathering patient history, ordering tests, interpreting results, and making safe treatment decisions -- yet a unified training environment provides the breadth of clinical domains and specialized tools to train generalizable medical AI agents through reinforcement learning remains elusive. We present a comprehensive empirical study of multi-turn agentic RL for medical AI, built on \gym{}, a gymnasium-compatible environment spanning 10 clinical domains with 3.6K+ tasks, 135 domain-specific tools, and a knowledge base of 828K medical passages. Our analysis reveals that agentic multi-turn structure degrades into verbose single-turn monologues, characterized by monotonic length explosion and a simultaneous erosion of tool-use frequency. We characterize how this collapse, alongside distillation instability, stems from the misalignment of sparse terminal rewards with sequential clinical trajectories. We find that vanilla GRPO achieves strong final accuracy on some benchmarks but suffers from training instability, evidenced by significant oscillations in response length and prolonged convergence periods. To improve training efficiency and stability, we propose Turn-level Truncated On-Policy Distillation (TT-OPD), a self-distillation framework where a gradient-free EMA teacher leverages outcome-privileged information to provide dense, outcome-aware KL regularization at every conversation turn. TT-OPD achieves the best performance on 10 of 18 benchmarks with an average +3.9~pp improvement over the non-RL baseline with faster early convergence, controlled response length, and sustained multi-turn tool use.

Offline multi-agent reinforcement learning in general-sum settings is challenged by the distribution shift between logged datasets and target equilibrium policies. While standard methods rely on manual pessimistic penalties, we demonstrate that KL regularization suffices to stabilize learning and achieve equilibrium recovery. We propose General-sum Anchored Nash Equilibrium (GANE), which recovers regularized Nash equilibria at an accelerated statistical rate of $\widetilde{O}(1/n)$. For computational tractability, we develop General-sum Anchored Mirror Descent (GAMD), an iterative algorithm converging to a Coarse Correlated Equilibrium at the standard rate of $\widetilde{O}(1/\sqrt{n}+1/T)$. These results establish KL regularization as a standalone mechanism for pessimism-free offline learning that achieves equivalent or accelerated rates in multi-player general-sum games.

Generative approaches to clustering provide information on geometric properties of clusters, whereas discriminative approaches provide boundaries between clusters. Ideas from both approaches are incorporated to present a fully unsupervised, probabilistic, and discriminative clustering method via a regularized mutual information objective function, wherein a mixture of mixtures of Gaussian and uniform distributions is used for formulation of the conditional model. Automatic selection of the number of components is established with the introduction of the regularizing term and a merge step, similar to those applied in reversible jump Markov chain Monte Carlo methods used in Bayesian clustering. Consequently, the turtle shell method -- a fully unsupervised clustering method capable of estimating non-linear boundary lines, automatically selecting the number of components, and capturing intuitive clusters in the presence of data abnormalities such as noise and/or irregular cluster shapes -- is introduced. We test this method on various simulated and real datasets commonly explored in clustering research, and extend the analysis to datasets arising from flow cytometry experiments.

Iterative alignment methods based on purely greedy updates are remarkably effective in practice, yet existing theoretical guarantees of $O(\log T)$ KL-regularized regret can seem pessimistic relative to their empirical performance. In this paper, we argue that this mismatch arises from the regret criterion itself: KL-regularized regret conflates the statistical cost of learning with the exploratory randomization induced by the softened training policy. To separate these effects, we study the traditional temperature-zero regret criterion, which evaluates only the top-ranked response at inference time. Under this decision-centric notion of performance, we prove that standard greedy online alignment methods, including online RLHF and online DPO, achieve constant $(O(1))$ cumulative regret. By isolating the cost of identifying the best response from the stochasticity induced by regularization, our results provide a sharper theoretical explanation for the practical superb efficiency of greedy alignment.

A major bottleneck in scientific discovery consists of narrowing an exponentially large set of objects, such as proteins or molecules, to a small set of promising candidates with desirable properties. While this process can rely on expert knowledge, recent methods leverage reinforcement learning (RL) guided by a proxy reward function to enable this filtering. By employing various forms of entropy regularization, these methods aim to learn samplers that generate diverse candidates that are highly rated by the proxy function. In this work, we make two main contributions. First, we show that these methods are liable to generate overly diverse, suboptimal candidates in large search spaces. To address this issue, we introduce a novel unified operator that combines several regularized RL operators into a general framework that better targets peakier sampling distributions. Secondly, we offer a novel, robust RL perspective of this filtering process. The regularization can be interpreted as robustness to a compositional form of uncertainty in the proxy function (i.e., the true evaluation of a candidate differs from the proxy's evaluation). Our analysis leads us to a novel, easy-to-use algorithm we name trajectory general mellowmax (TGM): we show it identifies higher quality, diverse candidates than baselines in both synthetic and real-world tasks. Code: https://github.com/marcojira/tgm.

How can we effectively remove or ``unlearn'' undesirable information, such as specific features or the influence of individual data points, from a learning outcome while minimizing utility loss and ensuring rigorous guarantees? We introduce a unified mathematical framework based on information-theoretic regularization to address both data-point unlearning and feature unlearning. For data-point unlearning, we introduce the \emph{Marginal Unlearning Principle}, an auditable and provable framework. Moreover, we provide an information-theoretic unlearning definition based on the proposed principle and provable guarantees on sufficiency and necessity of marginal unlearning. We then show that the proposed framework provides a natural solution to the marginal unlearning problem and yields auditable high-probability marginal-unlearning guarantees. For feature unlearning, the framework applies to deep learning with flexible training objectives. By combining flexibility in learning objectives with simplicity in regularization design, our approach is highly adaptable and practical for a wide range of machine learning and AI applications. From a mathematical perspective, we provide a unified analytic solution to the optimal feature unlearning problem with a variety of information-theoretic training objectives. Our theoretical analysis reveals intriguing connections between machine unlearning, information theory, optimal transport, and extremal sigma algebras. Numerical simulations support our theoretical findings.