arXiv subjects

Csaba Szepesvári

Publications and source records attributed to Csaba Szepesvári.

At least 37 records · Page 2Linked to original sources

arXiv2024

Optimistic Policy Optimization is Provably Efficient in Non-stationary MDPs

We study episodic reinforcement learning (RL) in non-stationary linear kernel Markov decision processes (MDPs). In this setting, both the reward function and the transition kernel are linear with respect to the given feature maps and are allowed to vary over time, as long as their respective parameter variations do not exceed certain variation budgets. We propose the \underline{p}eriodically \underline{r}estarted \underline{o}ptimistic \underline{p}olicy \underline{o}ptimization algorithm (PROPO), which is an optimistic policy optimization algorithm with linear function approximation. PROPO features two mechanisms: sliding-window-based policy evaluation and periodic-restart-based policy improvement, which are tailored for policy optimization in a non-stationary environment. In addition, only utilizing the technique of sliding window, we propose a value-iteration algorithm. We establish dynamic upper bounds for the proposed methods and a minimax lower bound which shows the (near-) optimality of the proposed methods. To our best knowledge, PROPO is the first provably efficient policy optimization algorithm that handles non-stationarity.

cs.LG↗

arXiv2024

Confident Natural Policy Gradient for Local Planning in $q_π$-realizable Constrained MDPs

The constrained Markov decision process (CMDP) framework emerges as an important reinforcement learning approach for imposing safety or other critical objectives while maximizing cumulative reward. However, the current understanding of how to learn efficiently in a CMDP environment with a potentially infinite number of states remains under investigation, particularly when function approximation is applied to the value functions. In this paper, we address the learning problem given linear function approximation with $q_π$-realizability, where the value functions of all policies are linearly representable with a known feature map, a setting known to be more general and challenging than other linear settings. Utilizing a local-access model, we propose a novel primal-dual algorithm that, after $\tilde{O}(\text{poly}(d) ε^{-3})$ queries, outputs with high probability a policy that strictly satisfies the constraints while nearly optimizing the value with respect to a reward function. Here, $d$ is the feature dimension and $ε> 0$ is a given error. The algorithm relies on a carefully crafted off-policy evaluation procedure to evaluate the policy using historical data, which informs policy updates through policy gradients and conserves samples. To our knowledge, this is the first result achieving polynomial sample complexity for CMDP in the $q_π$-realizable setting.

cs.LG↗

arXiv2024

Almost Free: Self-concordance in Natural Exponential Families and an Application to Bandits

We prove that single-parameter natural exponential families with subexponential tails are self-concordant with polynomial-sized parameters. For subgaussian natural exponential families we establish an exact characterization of the growth rate of the self-concordance parameter. Applying these findings to bandits allows us to fill gaps in the literature: We show that optimistic algorithms for generalized linear bandits enjoy regret bounds that are both second-order (scale with the variance of the optimal arm's reward distribution) and free of an exponential dependence on the bound of the problem parameter in the leading term. To the best of our knowledge, ours is the first regret bound for generalized linear bandits with subexponential tails, broadening the class of problems to include Poisson, exponential and gamma bandits.

cs.LG↗

arXiv2024

Switching the Loss Reduces the Cost in Batch (Offline) Reinforcement Learning

We propose training fitted Q-iteration with log-loss (FQI-log) for batch reinforcement learning (RL). We show that the number of samples needed to learn a near-optimal policy with FQI-log scales with the accumulated cost of the optimal policy, which is zero in problems where acting optimally achieves the goal and incurs no cost. In doing so, we provide a general framework for proving small-cost bounds, i.e. bounds that scale with the optimal achievable cost, in batch RL. Moreover, we empirically verify that FQI-log uses fewer samples than FQI trained with squared loss on problems where the optimal policy reliably achieves the goal.

cs.LG↗

arXiv2024

To Believe or Not to Believe Your LLM

We explore uncertainty quantification in large language models (LLMs), with the goal to identify when uncertainty in responses given a query is large. We simultaneously consider both epistemic and aleatoric uncertainties, where the former comes from the lack of knowledge about the ground truth (such as about facts or the language), and the latter comes from irreducible randomness (such as multiple possible answers). In particular, we derive an information-theoretic metric that allows to reliably detect when only epistemic uncertainty is large, in which case the output of the model is unreliable. This condition can be computed based solely on the output of the model obtained simply by some special iterative prompting based on the previous responses. Such quantification, for instance, allows to detect hallucinations (cases when epistemic uncertainty is high) in both single- and multi-answer responses. This is in contrast to many standard uncertainty quantification strategies (such as thresholding the log-likelihood of a response) where hallucinations in the multi-answer case cannot be detected. We conduct a series of experiments which demonstrate the advantage of our formulation. Further, our investigations shed some light on how the probabilities assigned to a given output by an LLM can be amplified by iterative prompting, which might be of independent interest.

cs.LG↗

arXiv2024

Trajectory Data Suffices for Statistically Efficient Learning in Offline RL with Linear $q^π$-Realizability and Concentrability

We consider offline reinforcement learning (RL) in $H$-horizon Markov decision processes (MDPs) under the linear $q^π$-realizability assumption, where the action-value function of every policy is linear with respect to a given $d$-dimensional feature function. The hope in this setting is that learning a good policy will be possible without requiring a sample size that scales with the number of states in the MDP. Foster et al. [2021] have shown this to be impossible even under $\textit{concentrability}$, a data coverage assumption where a coefficient $C_\text{conc}$ bounds the extent to which the state-action distribution of any policy can veer off the data distribution. However, the data in this previous work was in the form of a sequence of individual transitions. This leaves open the question of whether the negative result mentioned could be overcome if the data was composed of sequences of full trajectories. In this work we answer this question positively by proving that with trajectory data, a dataset of size $\text{poly}(d,H,C_\text{conc})/ε^2$ is sufficient for deriving an $ε$-optimal policy, regardless of the size of the state space. The main tool that makes this result possible is due to Weisz et al. [2023], who demonstrate that linear MDPs can be used to approximate linearly $q^π$-realizable MDPs. The connection to trajectory data is that the linear MDP approximation relies on "skipping" over certain states. The associated estimation problems are thus easy when working with trajectory data, while they remain nontrivial when working with individual transitions. The question of computational efficiency under our assumptions remains open.

cs.LG↗

arXiv2024

Stochastic Gradient Descent for Gaussian Processes Done Right

As is well known, both sampling from the posterior and computing the mean of the posterior in Gaussian process regression reduces to solving a large linear system of equations. We study the use of stochastic gradient descent for solving this linear system, and show that when \emph{done right} -- by which we mean using specific insights from the optimisation and kernel communities -- stochastic gradient descent is highly effective. To that end, we introduce a particularly simple \emph{stochastic dual descent} algorithm, explain its design in an intuitive manner and illustrate the design choices through a series of ablation studies. Further experiments demonstrate that our new method is highly competitive. In particular, our evaluations on the UCI regression tasks and on Bayesian optimisation set our approach apart from preconditioned conjugate gradients and variational Gaussian process approximations. Moreover, our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.

cs.LG↗

arXiv2024

Mitigating LLM Hallucinations via Conformal Abstention

We develop a principled procedure for determining when a large language model (LLM) should abstain from responding (e.g., by saying "I don't know") in a general domain, instead of resorting to possibly "hallucinating" a non-sensical or incorrect answer. Building on earlier approaches that use self-consistency as a more reliable measure of model confidence, we propose using the LLM itself to self-evaluate the similarity between each of its sampled responses for a given query. We then further leverage conformal prediction techniques to develop an abstention procedure that benefits from rigorous theoretical guarantees on the hallucination rate (error rate). Experimentally, our resulting conformal abstention method reliably bounds the hallucination rate on various closed-book, open-domain generative question answering datasets, while also maintaining a significantly less conservative abstention rate on a dataset with long responses (Temporal Sequences) compared to baselines using log-probability scores to quantify uncertainty, while achieveing comparable performance on a dataset with short answers (TriviaQA). To evaluate the experiments automatically, one needs to determine if two responses are equivalent given a question. Following standard practice, we use a thresholded similarity function to determine if two responses match, but also provide a method for calibrating the threshold based on conformal prediction, with theoretical guarantees on the accuracy of the match prediction, which might be of independent interest.

cs.LG↗

arXiv2024

Regret Minimization via Saddle Point Optimization

A long line of works characterizes the sample complexity of regret minimization in sequential decision-making by min-max programs. In the corresponding saddle-point game, the min-player optimizes the sampling distribution against an adversarial max-player that chooses confusing models leading to large regret. The most recent instantiation of this idea is the decision-estimation coefficient (DEC), which was shown to provide nearly tight lower and upper bounds on the worst-case expected regret in structured bandits and reinforcement learning. By re-parametrizing the offset DEC with the confidence radius and solving the corresponding min-max program, we derive an anytime variant of the Estimation-To-Decisions (E2D) algorithm. Importantly, the algorithm optimizes the exploration-exploitation trade-off online instead of via the analysis. Our formulation leads to a practical algorithm for finite model classes and linear feedback models. We further point out connections to the information ratio, decoupling coefficient and PAC-DEC, and numerically evaluate the performance of E2D on simple examples.

cs.LG↗

arXiv2023

Online RL in Linearly $q^π$-Realizable MDPs Is as Easy as in Linear MDPs If You Learn What to Ignore

We consider online reinforcement learning (RL) in episodic Markov decision processes (MDPs) under the linear $q^π$-realizability assumption, where it is assumed that the action-values of all policies can be expressed as linear functions of state-action features. This class is known to be more general than linear MDPs, where the transition kernel and the reward function are assumed to be linear functions of the feature vectors. As our first contribution, we show that the difference between the two classes is the presence of states in linearly $q^π$-realizable MDPs where for any policy, all the actions have approximately equal values, and skipping over these states by following an arbitrarily fixed policy in those states transforms the problem to a linear MDP. Based on this observation, we derive a novel (computationally inefficient) learning algorithm for linearly $q^π$-realizable MDPs that simultaneously learns what states should be skipped over and runs another learning algorithm on the linear MDP hidden in the problem. The method returns an $ε$-optimal policy after $\text{polylog}(H, d)/ε^2$ interactions with the MDP, where $H$ is the time horizon and $d$ is the dimension of the feature vectors, giving the first polynomial-sample-complexity online RL algorithm for this setting. The results are proved for the misspecified case, where the sample complexity is shown to degrade gracefully with the misspecification error.

cs.LG↗

arXiv2023

The Optimal Approximation Factors in Misspecified Off-Policy Value Function Estimation

Theoretical guarantees in reinforcement learning (RL) are known to suffer multiplicative blow-up factors with respect to the misspecification error of function approximation. Yet, the nature of such \emph{approximation factors} -- especially their optimal form in a given learning problem -- is poorly understood. In this paper we study this question in linear off-policy value function estimation, where many open questions remain. We study the approximation factor in a broad spectrum of settings, such as with the weighted $L_2$-norm (where the weighting is the offline state distribution), the $L_\infty$ norm, the presence vs. absence of state aliasing, and full vs. partial coverage of the state space. We establish the optimal asymptotic approximation factors (up to constants) for all of these settings. In particular, our bounds identify two instance-dependent factors for the $L_2(μ)$ norm and only one for the $L_\infty$ norm, which are shown to dictate the hardness of off-policy evaluation under misspecification.

cs.LG↗

arXiv2023

Optimistic Natural Policy Gradient: a Simple Efficient Policy Optimization Framework for Online RL

While policy optimization algorithms have played an important role in recent empirical success of Reinforcement Learning (RL), the existing theoretical understanding of policy optimization remains rather limited -- they are either restricted to tabular MDPs or suffer from highly suboptimal sample complexity, especial in online RL where exploration is necessary. This paper proposes a simple efficient policy optimization framework -- Optimistic NPG for online RL. Optimistic NPG can be viewed as a simple combination of the classic natural policy gradient (NPG) algorithm [Kakade, 2001] with optimistic policy evaluation subroutines to encourage exploration. For $d$-dimensional linear MDPs, Optimistic NPG is computationally efficient, and learns an $\varepsilon$-optimal policy within $\tilde{O}(d^2/\varepsilon^3)$ samples, which is the first computationally efficient algorithm whose sample complexity has the optimal dimension dependence $\tildeΘ(d^2)$. It also improves over state-of-the-art results of policy optimization algorithms [Zanette et al., 2021] by a factor of $d$. In the realm of general function approximation, which subsumes linear MDPs, Optimistic NPG, to our best knowledge, stands as the first policy optimization algorithm that achieves polynomial sample complexity for learning near-optimal policies.

cs.LG↗

arXiv2023

Context-lumpable stochastic bandits

We consider a contextual bandit problem with $S$ contexts and $K$ actions. In each round $t=1,2,\dots$, the learner observes a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into $r\le \min\{S,K\}$ groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an $ε$-optimal policy after using at most $\widetilde O(r (S +K )/ε^2)$ samples with high probability and provide a matching $Ω(r(S+K)/ε^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r^3(S+K)T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O(\sqrt{{poly}(r)(S+K)T})$ minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.

cs.LG↗

arXiv2023

Regularization and Variance-Weighted Regression Achieves Minimax Optimality in Linear MDPs: Theory and Practice

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 $1-δ$ 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.

cs.LG↗

arXiv2023

Learning Lipschitz Functions by GD-trained Shallow Overparameterized ReLU Neural Networks

We explore the ability of overparameterized shallow ReLU neural networks to learn Lipschitz, nondifferentiable, bounded functions with additive noise when trained by Gradient Descent (GD). To avoid the problem that in the presence of noise, neural networks trained to nearly zero training error are inconsistent in this class, we focus on the early-stopped GD which allows us to show consistency and optimal rates. In particular, we explore this problem from the viewpoint of the Neural Tangent Kernel (NTK) approximation of a GD-trained finite-width neural network. We show that whenever some early stopping rule is guaranteed to give an optimal rate (of excess risk) on the Hilbert space of the kernel induced by the ReLU activation function, the same rule can be used to achieve minimax optimal rate for learning on the class of considered Lipschitz functions by neural networks. We discuss several data-free and data-dependent practically appealing stopping rules that yield optimal rates.

cs.LG↗

arXiv2023

A Free Lunch from the Noise: Provable and Practical Exploration for Representation Learning

Representation learning lies at the heart of the empirical success of deep learning for dealing with the curse of dimensionality. However, the power of representation learning has not been fully exploited yet in reinforcement learning (RL), due to i), the trade-off between expressiveness and tractability; and ii), the coupling between exploration and representation learning. In this paper, we first reveal the fact that under some noise assumption in the stochastic control model, we can obtain the linear spectral feature of its corresponding Markov transition operator in closed-form for free. Based on this observation, we propose Spectral Dynamics Embedding (SPEDE), which breaks the trade-off and completes optimistic exploration for representation learning by exploiting the structure of the noise. We provide rigorous theoretical analysis of SPEDE, and demonstrate the practical superior performance over the existing state-of-the-art empirical algorithms on several benchmarks.

stat.ML↗

arXiv2023

Exponential Hardness of Reinforcement Learning with Linear Function Approximation

A fundamental question in reinforcement learning theory is: suppose the optimal value functions are linear in given features, can we learn them efficiently? This problem's counterpart in supervised learning, linear regression, can be solved both statistically and computationally efficiently. Therefore, it was quite surprising when a recent work \cite{kane2022computational} showed a computational-statistical gap for linear reinforcement learning: even though there are polynomial sample-complexity algorithms, unless NP = RP, there are no polynomial time algorithms for this setting. In this work, we build on their result to show a computational lower bound, which is exponential in feature dimension and horizon, for linear reinforcement learning under the Randomized Exponential Time Hypothesis. To prove this we build a round-based game where in each round the learner is searching for an unknown vector in a unit hypercube. The rewards in this game are chosen such that if the learner achieves large reward, then the learner's actions can be used to simulate solving a variant of 3-SAT, where (a) each variable shows up in a bounded number of clauses (b) if an instance has no solutions then it also has no solutions that satisfy more than (1-$ε$)-fraction of clauses. We use standard reductions to show this 3-SAT variant is approximately as hard as 3-SAT. Finally, we also show a lower bound optimized for horizon dependence that almost matches the best known upper bound of $\exp(\sqrt{H})$.

cs.LG↗

arXiv2023

Efficient Planning in Combinatorial Action Spaces with Applications to Cooperative Multi-Agent Reinforcement Learning

A practical challenge in reinforcement learning are combinatorial action spaces that make planning computationally demanding. For example, in cooperative multi-agent reinforcement learning, a potentially large number of agents jointly optimize a global reward function, which leads to a combinatorial blow-up in the action space by the number of agents. As a minimal requirement, we assume access to an argmax oracle that allows to efficiently compute the greedy policy for any Q-function in the model class. Building on recent work in planning with local access to a simulator and linear function approximation, we propose efficient algorithms for this setting that lead to polynomial compute and query complexity in all relevant problem parameters. For the special case where the feature decomposition is additive, we further improve the bounds and extend the results to the kernelized setting with an efficient algorithm.

cs.LG↗