Recent advances in aligning large language models with human preferences have corroborated the growing importance of best-of-$N$ distillation (BOND). However, the iterative BOND algorithm is prohibitively expensive in practice due to the sample and computation inefficiency. This paper addresses the problem by revealing a unified game-theoretic connection between iterative BOND and self-play alignment, which unifies seemingly disparate algorithmic paradigms. Based on the connection, we establish a novel framework, \textbf{WIN} rate \textbf{D}ominance (WIND), with a series of efficient algorithms for regularized win rate dominance optimization that approximates iterative BOND in the parameter space. We provides provable sample efficiency guarantee for one of the WIND variant with the square loss objective. The experimental results confirm that our algorithm not only accelerates the computation, but also achieves superior sample efficiency compared to existing methods.

Reinforcement learning from human feedback (RLHF) has demonstrated great promise in aligning large language models (LLMs) with human preference. Depending on the availability of preference data, both online and offline RLHF are active areas of investigation. A key bottleneck is understanding how to incorporate uncertainty estimation in the reward function learned from the preference data for RLHF, regardless of how the preference data is collected. While the principles of optimism or pessimism under uncertainty are well-established in standard reinforcement learning (RL), a practically-implementable and theoretically-grounded form amenable to large language models is not yet available, as standard techniques for constructing confidence intervals become intractable under arbitrary policy parameterizations. In this paper, we introduce a unified approach to online and offline RLHF --- value-incentivized preference optimization (VPO) --- which regularizes the maximum-likelihood estimate of the reward function with the corresponding value function, modulated by a sign to indicate whether the optimism or pessimism is chosen. VPO also directly optimizes the policy with implicit reward modeling, and therefore shares a simpler RLHF pipeline similar to direct preference optimization. Theoretical guarantees of VPO are provided for both online and offline settings, matching the rates of their standard RL counterparts. Moreover, experiments on text summarization, dialogue, and standard benchmarks verify the practicality and effectiveness of VPO.

We prove that the classic REINFORCE stochastic policy gradient (SPG) method converges to globally optimal policies in finite-horizon Markov Decision Processes (MDPs) with $\textit{any}$ constant learning rate. To avoid the need for small or decaying learning rates, we introduce two key innovations in the stochastic bandit setting, which we then extend to MDPs. $\textbf{First}$, we identify a new exploration property of SPG: the online SPG method samples every action infinitely often (i.o.), improving on previous results that only guaranteed at least two actions would be sampled i.o. This means SPG inherently achieves asymptotic exploration without modification. $\textbf{Second}$, we eliminate the assumption of unique mean reward values, a condition that previous convergence analyses in the bandit setting relied on, but that does not translate to MDPs. Our results deepen the theoretical understanding of SPG in both bandit problems and MDPs, with a focus on how it handles the exploration-exploitation trade-off when standard optimization and stochastic approximation methods cannot be applied, as is the case with large constant learning rates.

We prove that the combination of a target network and over-parameterized linear function approximation establishes a weaker convergence condition for bootstrapped value estimation in certain cases, even with off-policy data. Our condition is naturally satisfied for expected updates over the entire state-action space or learning with a batch of complete trajectories from episodic Markov decision processes. Notably, using only a target network or an over-parameterized model does not provide such a convergence guarantee. Additionally, we extend our results to learning with truncated trajectories, showing that convergence is achievable for all tasks with minor modifications, akin to value truncation for the final states in trajectories. Our primary result focuses on temporal difference estimation for prediction, providing high-probability value estimation error bounds and empirical analysis on Baird's counterexample and a Four-room task. Furthermore, we explore the control setting, demonstrating that similar convergence conditions apply to Q-learning.

We provide a new understanding of the stochastic gradient bandit algorithm by showing that it converges to a globally optimal policy almost surely using \emph{any} constant learning rate. This result demonstrates that the stochastic gradient algorithm continues to balance exploration and exploitation appropriately even in scenarios where standard smoothness and noise control assumptions break down. The proofs are based on novel findings about action sampling rates and the relationship between cumulative progress and noise, and extend the current understanding of how simple stochastic gradient methods behave in bandit settings.

Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an $\varepsilon$-optimal policy with probability $1-\delta$ under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.

We show that the stochastic gradient bandit algorithm converges to a globally optimal policy at an $O(1/t)$ rate, even with a constant step size. Remarkably, global convergence of the stochastic gradient bandit algorithm has not been previously established, even though it is an old algorithm known to be applicable to bandits. The new result is achieved by establishing two novel technical findings: first, the noise of the stochastic updates in the gradient bandit algorithm satisfies a strong “growth condition” property, where the variance diminishes whenever progress becomes small, implying that additional noise control via diminishing step sizes is unnecessary; second, a form of “weak exploration” is automatically achieved through the stochastic gradient updates, since they prevent the action probabilities from decaying faster than $O(1/t)$, thus ensuring that every action is sampled infinitely often with probability $1$. These two findings can be used to show that the stochastic gradient update is already “sufficient” for bandits in the sense that exploration versus exploitation is automatically balanced in a manner that ensures almost sure convergence to a global optimum. These novel theoretical findings are further verified by experimental results.

We prove that, for finite-arm bandits with linear function approximation, the global convergence of policy gradient (PG) methods depends on inter-related properties between the policy update and the representation. textcolor{blue}{First}, we establish a few key observations that frame the study: \textbf{(i)} Global convergence can be achieved under linear function approximation without policy or reward realizability, both for the standard Softmax PG and natural policy gradient (NPG). \textbf{(ii)} Approximation error is not a key quantity for characterizing global convergence in either algorithm. \textbf{(iii)} The conditions on the representation that imply global convergence are different between these two algorithms. Overall, these observations call into question approximation error as an appropriate quantity for characterizing the global convergence of PG methods under linear function approximation. \textcolor{blue}{Second}, motivated by these observations, we establish new general results: \textbf{(i)} NPG with linear function approximation achieves global convergence \emph{if and only if} the projection of the reward onto the representable space preserves the optimal action's rank, a quantity that is not strongly related to approximation error. \textbf{(ii)} The global convergence of Softmax PG occurs if the representation satisfies a non-domination condition and can preserve the ranking of rewards, which goes well beyond policy or reward realizability. We provide experimental results to support these theoretical findings.

We study the effect of baselines in on-policy stochastic policy gradient optimization, and close the gap between the theory and practice of policy optimization methods. Our first contribution is to show that the \emph{state value} baseline allows on-policy stochastic \emph{natural} policy gradient (NPG) to converge to a globally optimal policy at an $O(1/t)$ rate, which was not previously known. The analysis relies on two novel findings: the expected progress of the NPG update satisfies a stochastic version of the non-uniform \L{}ojasiewicz (N\L{}) inequality, and with probability 1 the state value baseline prevents the optimal action's probability from vanishing, thus ensuring sufficient exploration. Importantly, these results provide a new understanding of the role of baselines in stochastic policy gradient: by showing that the variance of natural policy gradient estimates remains unbounded with or without a baseline, we find that variance reduction \emph{cannot} explain their utility in this setting. Instead, the analysis reveals that the primary effect of the value baseline is to \textbf{reduce the aggressiveness of the updates} rather than their variance. That is, we demonstrate that a finite variance is \emph{not necessary} for almost sure convergence of stochastic NPG, while controlling update aggressiveness is both necessary and sufficient. Additional experimental results verify these theoretical findings.

Softmax policy gradient is a popular algorithm for policy optimization in single-agent reinforcement learning, particularly since projection is not needed for each gradient update. However, in multi-agent systems, the lack of central coordination introduces significant additional difficulties in the convergence analysis. Even for a stochastic game with identical interest, there can be multiple Nash Equilibria (NEs), which disables proof techniques that rely on the existence of a unique global optimum. Moreover, the softmax parameterization introduces non-NE policies with zero gradient, making it difficult for gradient-based algorithms in seeking NEs. In this paper, we study the finite time convergence of decentralized softmax gradient play in a special form of game, Markov Potential Games (MPGs), which includes the identical interest game as a special case. We investigate both gradient play and natural gradient play, with and without $\log$-barrier regularization. The established convergence rates for the unregularized cases contain a trajectory dependent constant that can be \emph{arbitrarily large}, whereas the $\log$-barrier regularization overcomes this drawback, with the cost of slightly worse dependence on other factors such as the action set size. An empirical study on an identical interest matrix game confirms the theoretical findings.

Prioritized Experience Replay (ER) has been empirically shown to improve sample efficiency across many domains and attracted great attention; however, there is little theoretical understanding of why such prioritized sampling helps and its limitations. In this work, we take a deep look at the prioritized ER. In a supervised learning setting, we show the equivalence between the error-based prioritized sampling method for minimizing mean squared error and the uniform sampling for cubic power loss. We then provide theoretical insight into why error-based prioritized sampling improves convergence rate upon uniform sampling when minimizing mean squared error during early learning. Based on the insight, we further point out two limitations of the prioritized ER method: 1) outdated priorities and 2) insufficient coverage of the sample space. To mitigate the limitations, we propose our model-based stochastic gradient Langevin dynamics sampling method. We show that our method does provide states distributed close to an ideal prioritized sampling distribution estimated by the brute-force method, which does not suffer from the two limitations. We conduct experiments on both discrete and continuous control problems to show our approach’s efficacy and examine the practical implication of our method in an autonomous driving application.

Classical global convergence results for first-order methods rely on uniform smoothness and the Ł{}ojasiewicz inequality. Motivated by properties of objective functions that arise in machine learning, we propose a non-uniform refinement of these notions, leading to \emph{Non-uniform Smoothness} (NS) and \emph{Non-uniform Ł{}ojasiewicz inequality} (NŁ{}). The new definitions inspire new geometry-aware first-order methods that are able to converge to global optimality faster than the classical $\Omega(1/t^2)$ lower bounds. To illustrate the power of these geometry-aware methods and their corresponding non-uniform analysis, we consider two important problems in machine learning: policy gradient optimization in reinforcement learning (PG), and generalized linear model training in supervised learning (GLM). For PG, we find that normalizing the gradient ascent method can accelerate convergence to $O(e^{- c \cdot t})$ (where $c > 0$) while incurring less overhead than existing algorithms. For GLM, we show that geometry-aware normalized gradient descent can also achieve a linear convergence rate, which significantly improves the best known results. We additionally show that the proposed geometry-aware gradient descent methods escape landscape plateaus faster than standard gradient descent. Experimental results are used to illustrate and complement the theoretical findings.

Batch policy optimization considers leveraging existing data for policy construction before interacting with an environment. Although interest in this problem has grown significantly in recent years, its theoretical foundations remain under-developed. To advance the understanding of this problem, we provide three results that characterize the limits and possibilities of batch policy optimization in the finite-armed stochastic bandit setting. First, we introduce a class of confidence-adjusted index algorithms that unifies optimistic and pessimistic principles in a common framework, which enables a general analysis. For this family, we show that any confidence-adjusted index algorithm is minimax optimal, whether it be optimistic, pessimistic or neutral. Our analysis reveals that instance-dependent optimality, commonly used to establish optimality of on-line stochastic bandit algorithms, cannot be achieved by any algorithm in the batch setting. In particular, for any algorithm that performs optimally in some environment, there exists another environment where the same algorithm suffers arbitrarily larger regret. Therefore, to establish a framework for distinguishing algorithms, we introduce a new weighted-minimax criterion that considers the inherent difficulty of optimal value prediction. We demonstrate how this criterion can be used to justify commonly used pessimistic principles for batch policy optimization.

We study the effect of stochasticity in on-policy policy optimization, and make the following four contributions. \emph{First}, we show that the preferability of optimization methods depends critically on whether stochastic versus exact gradients are used. In particular, unlike the true gradient setting, geometric information cannot be easily exploited in the stochastic case for accelerating policy optimization without detrimental consequences or impractical assumptions. \emph{Second}, to explain these findings we introduce the concept of committal rate for stochastic policy optimization, and show that this can serve as a criterion for determining almost sure convergence to global optimality. \emph{Third}, we show that in the absence of external oracle information, which allows an algorithm to determine the difference between optimal and sub-optimal actions given only on-policy samples, there is an inherent trade-off between exploiting geometry to accelerate convergence versus achieving optimality almost surely. That is, an uninformed algorithm either converges to a globally optimal policy with probability $1$ but at a rate no better than $O(1/t)$, or it achieves faster than $O(1/t)$ convergence but then must fail to converge to the globally optimal policy with some positive probability. \emph{Finally}, we use the committal rate theory to explain why practical policy optimization methods are sensitive to random initialization, then develop an ensemble method that can be guaranteed to achieve near-optimal solutions with high probability.

We make three contributions toward better understanding policy gradient methods in the tabular setting. First, we show that with the true gradient, policy gradient with a softmax parametrization converges at a $O(1/t)$ rate, with constants depending on the problem and initialization. This result significantly expands the recent asymptotic convergence results. The analysis relies on two findings: that the softmax policy gradient satisfies a Ł{}ojasiewicz inequality, and the minimum probability of an optimal action during optimization can be bounded in terms of its initial value. Second, we analyze entropy regularized policy gradient and show that it enjoys a significantly faster linear convergence rate $O(e^{-t})$ toward softmax optimal policy. This result resolves an open question in the recent literature. Finally, combining the above two results and additional new $\Omega(1/t)$ lower bound results, we explain how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate. The separation of rates is further explained using the notion of non-uniform Ł{}ojasiewicz degree. These results provide a theoretical understanding of the impact of entropy and corroborate existing empirical studies.

The softmax is the standard transformation used in machine learning to map real-valued vectors to categorical distributions. Unfortunately, this transform poses serious drawbacks for gradient descent (ascent) optimization. We reveal this difficulty by establishing two negative results: (1) optimizing any expectation with respect to the softmax must exhibit sensitivity to parameter initialization (softmax gravity well''), and (2) optimizing log-probabilities under the softmax must exhibit slow convergence (softmax damping''). Both findings are based on an analysis of convergence rates using the Non-uniform \L{}ojasiewicz (N\L{}) inequalities. To circumvent these shortcomings we investigate an alternative transformation, the \emph{escort} mapping, that demonstrates better optimization properties. The disadvantages of the softmax and the effectiveness of the escort transformation are further explained using the concept of N\L{} coefficient. In addition to proving bounds on convergence rates to firmly establish these results, we also provide experimental evidence for the superiority of the escort transformation.

In this paper, we investigate Exploratory Conservative Policy Optimization (ECPO), a policy optimization strategy that improves exploration behavior while assuring monotonic progress in a principled objective. ECPO conducts maximum entropy exploration within a mirror descent framework, but updates policies using reversed KL projection. This formulation bypasses undesirable mode seeking behavior and avoids premature convergence to sub-optimal policies, while still supporting strong theoretical properties such as guaranteed policy improvement. Experimental evaluations demonstrate that the proposed method significantly improves practical exploration and surpasses the empirical performance of state-of-the art policy optimization methods in a set of benchmark tasks.

We develop a new algorithm for online planning in large scale sequential decision problems that improves upon the worst case efficiency of UCT. The idea is to augment Monte-Carlo Tree Search (MCTS) with maximum entropy policy optimization, evaluating each search node by softmax values back-propagated from simulation. To establish the effectiveness of this approach, we first investigate the single-step decision problem, stochastic softmax bandits, and show that softmax values can be estimated at an optimal convergence rate in terms of mean squared error. We then extend this approach to general sequential decision making by developing a general MCTS algorithm, Maximum Entropy for Tree Search (MENTS). We prove that the probability of MENTS failing to identify the best decision at the root decays exponentially, which fundamentally improves the polynomial convergence rate of UCT. Our experimental results also demonstrate that MENTS is more sample efficient than UCT in both synthetic problems and Atari 2600 games.

This paper proposes and evaluates Memory-Augmented Monte Carlo Tree Search (M-MCTS), which provides a new approach to exploit generalization in online real-time search. The key idea of M-MCTS is to incorporate MCTS with a memory structure, where each entry contains information of a particular state. This memory is used to generate an approximate value estimation by combining the estimations of similar states. We show that the memory based value approximation is better than the vanilla Monte Carlo estimation with high probability under mild conditions. We evaluate M-MCTS in the game of Go. Experimental results show that M-MCTS outperforms the original MCTS with the same number of simulations.

We study the problem of identifying the topics and sentiments and tracking their shifts from social media texts in different geographical regions during emergencies and disasters.We propose a location-based dynamic sentiment-topic model (LDST) which can jointly model topic, sentiment, time and Geolocation information.The experimental results demonstrate that LDST performs very well at discovering topics and sentiments from social media and tracking their shifts in different geographical regions during emergencies and disasters 1 .

The scalability of submodular optimization methods is critical for their usability in practice. In this paper, we study the reducibility of submodular functions, a property that enables us to reduce the solution space of submodular optimization problems without performance loss. We introduce the concept of reducibility using marginal gains. Then we show that by adding perturbation, we can endow irreducible functions with reducibility, based on which we propose the perturbation-reduction optimization framework. Our theoretical analysis proves that given the perturbation scales, the reducibility gain could be computed, and the performance loss has additive upper bounds. We further conduct empirical studies and the results demonstrate that our proposed framework significantly accelerates existing optimization methods for irreducible submodular functions with a cost of only small performance losses.

Discovering the author's interest over time from documents has important applications in recommendation systems, authorship identification and opinion extraction. In this paper, we propose an interest drift model (IDM), which monitors the evolution of author interests in time-stamped documents. The model further uses the discovered author interest information to help finding better topics. Unlike traditional topic models, our model is sensitive to the ordering of words, thus it extracts more information from the semantic meaning of the context. The experiment results show that the IDM model learns better topics than state-of-the-art topic models.

With the extensive application of submodularity, its generalizations are constantly being proposed. However, most of them are tailored for special problems. In this paper, we focus on quasi-submodularity, a universal generalization, which satisfies weaker properties than submodularity but still enjoys favorable performance in optimization. Similar to the diminishing return property of submodularity, we first define a corresponding property called the single sub-crossing, then we propose two algorithms for unconstrained quasi-submodular function minimization and maximization, respectively. The proposed algorithms return the reduced lattices in O(n) iterations, and guarantee the objective function values are strictly monotonically increased or decreased after each iteration. Moreover, any local and global optima are definitely contained in the reduced lattices. Experimental results verify the effectiveness and efficiency of the proposed algorithms on lattice reduction.

Hashing has recently attracted considerable attention for large scale similarity search. However, learning compact codes with good performance is still a challenge. In many cases, the real-world data lies on a low-dimensional manifold embedded in high-dimensional ambient space. To capture meaningful neighbors, a compact hashing representation should be able to uncover the intrinsic geometric structure of the manifold, e.g., the neighborhood relationships between subregions. Most existing hashing methods only consider this issue during mapping data points into certain projected dimensions. When getting the binary codes, they either directly quantize the projected values with a threshold, or use an orthogonal matrix to refine the initial projection matrix, which both consider projection and quantization separately, and will not well preserve the locality structure in the whole learning process. In this paper, we propose a novel hashing algorithm called Locality Preserving Hashing to effectively solve the above problems. Specifically, we learn a set of locality preserving projections with a joint optimization framework, which minimizes the average projection distance and quantization loss simultaneously. Experimental comparisons with other state-of-the-art methods on two large scale datasets demonstrate the effectiveness and efficiency of our method.