The cold posterior effect (CPE) (Wenzel et al., 2020) in Bayesian deep learning shows that, for posteriors with a temperature $T<1$, the resulting posterior predictive could have better performances than the Bayesian posterior ($T=1$). As the Bayesian posterior is known to be optimal under perfect model specification, many recent works have studied the presence of CPE as a model misspecification problem, arising from the prior and/or from the likelihood function. In this work, we provide a more nuanced understanding of the CPE as we show that misspecification leads to CPE only when the resulting Bayesian posterior underfits. In fact, we theoretically show that if there is no underfitting, there is no CPE.

PAC-Bayesian analysis is a frequentist framework for incorporating prior knowledge into learning. It was inspired by Bayesian learning, which allows sequential data processing and naturally turns posteriors from one processing step into priors for the next. However, despite two and a half decades of research, the ability to update priors sequentially without losing confidence information along the way remained elusive for PAC-Bayes. While PAC-Bayes allows construction of data-informed priors, the final confidence intervals depend only on the number of points that were not used for the construction of the prior, whereas confidence information in the prior, which is related to the number of points used to construct the prior, is lost. This limits the possibility and benefit of sequential prior updates, because the final bounds depend only on the size of the final batch. We present a novel and, in retrospect, surprisingly simple and powerful PAC-Bayesian procedure that allows sequential prior updates with no information loss. The procedure is based on a novel decomposition of the expected loss of randomized classifiers. The decomposition rewrites the loss of the posterior as an excess loss relative to a downscaled loss of the prior plus the downscaled loss of the prior, which is bounded recursively. As a side result, we also present a generalization of the split-kl and PAC-Bayes-split-kl inequalities to discrete random variables, which we use for bounding the excess losses, and which can be of independent interest. In empirical evaluation the new procedure significantly outperforms state-of-the-art.

Bayesian inference provides a principled probabilistic framework for quantifying uncertainty by updating beliefs based on prior knowledge and observed data through Bayes' theorem. In Bayesian deep learning, neural network weights are treated as random variables with prior distributions, allowing for a probabilistic interpretation and quantification of predictive uncertainty. However, Bayesian methods lack theoretical generalization guarantees for unseen data. PAC-Bayesian analysis addresses this limitation by offering a frequentist framework to derive generalization bounds for randomized predictors, thereby certifying the reliability of Bayesian methods in machine learning. Temperature T, or inverse-temperature λ = 1/T, originally from statistical mechanics in physics, naturally arises in various areas of statistical inference, including Bayesian inference and PAC-Bayesian analysis. In Bayesian inference, when T < 1 (“cold” posteriors), the likelihood is up-weighted, resulting in a sharper posterior distribution. Conversely, when T > 1 (“warm” posteriors), the likelihood is down-weighted, leading to a more diffuse posterior distribution. By balancing the influence of observed data and prior regularization, temperature adjustments can address issues of underfitting or overfitting in Bayesian models, bringing improved predictive performance.

The cold posterior effect (CPE) (Wenzel et al., 2020) in Bayesian deep learning shows that, for posteriors with a temperature $T<1$, the resulting posterior predictive could have better performance than the Bayesian posterior ($T=1$). As the Bayesian posterior is known to be optimal under perfect model specification, many recent works have studied the presence of CPE as a model misspecification problem, arising from the prior and/or from the likelihood. In this work, we provide a more nuanced understanding of CPE as we show that misspecification leads to CPE only when the resulting Bayesian posterior underfits. In fact, we theoretically show that if there is no underfitting, there is no CPE. Furthermore, we show that these tempered posteriors with $T < 1$ are indeed proper Bayesian posteriors with a different combination of likelihoods and priors parameterized by $T$. This observation validates the adjustment of the temperature hyperparameter $T$ as a straightforward approach to mitigate underfitting in the Bayesian posterior. In essence, we show that by fine-tuning the temperature $T$ we implicitly utilize alternative Bayesian posteriors, albeit with less misspecified likelihood and prior distributions. The code for replicating the experiments can be found at https://github.com/pyijiezhang/cpe-underfit.

Recent success stories in reinforcement learning have demonstrated that leveraging structural properties of the underlying environment is key in devising viable methods capable of solving complex tasks. We study off-policy learning in discounted reinforcement learning, where some equivalence relation in the environment exists. We introduce a new model-free algorithm, called QL-ES (Q-learning with equivalence structure), which is a variant of (asynchronous) Q-learning tailored to exploit the equivalence structure in the MDP. We report a non-asymptotic PAC-type sample complexity bound for QL-ES, thereby establishing its sample efficiency. This bound also allows us to quantify the superiority of QL-ES over Q-learning analytically, which shows that the theoretical gain in some domains can be massive. We report extensive numerical experiments demonstrating that QL-ES converges significantly faster than (structure-oblivious) Q-learning empirically. They imply that the empirical performance gain obtained by exploiting the equivalence structure could be massive, even in simple domains. To the best of our knowledge, QL-ES is the first provably efficient model-free algorithm to exploit the equivalence structure in finite MDPs.

In policy search methods for reinforcement learning (RL), exploration is often performed by injecting noise either in action space at each step independently or in parameter space over each full trajectory. In prior work, it has been shown that with linear policies, a more balanced trade-off between these two exploration strategies is beneficial. However, that method did not scale to policies using deep neural networks. In this paper, we introduce Deep Coherent Exploration, a general and scalable exploration framework for deep RL algorithms on continuous control, that generalizes step-based and trajectory-based exploration. This framework models the last layer parameters of the policy network as latent variables and uses a recursive inference step within the policy update to handle these latent variables in a scalable manner. We find that Deep Coherent Exploration improves the speed and stability of learning of A2C, PPO, and SAC on several continuous control tasks.

Grünwald and Van Ommen (2017) show that Bayesian inference for linear regression can be inconsistent under model misspecification. In this paper, we extend their analysis to Bayesian neural networks (BNNs), investigating if they too can be inconsistent under misspecification. We find that BNNs exhibit the same inconsistency when Hamiltonian Monte Carlo is used for posterior inference. However, variational inference changes this behavior. Surprisingly, we find that variational Bayes leads to BNNs that are consistent in the setting studied by Grünwald and Van Ommen (2017). We conjecture that the success of variational Bayes is due to its optimization objective: the evidence lower bound (ELBO) implicitly encourages the posterior approximation to concentrate, mitigating the ill-effects of the misspecification.

Many real-world multi-agent interactions consider multiple distinct criteria, i.e. the payoffs are multi-objective in nature. However, the same multi-objective payoff vector may lead to different utilities for each participant. Therefore, it is essential for an agent to learn about the behaviour of other agents in the system. In this work, we present the first study of the effects of such opponent modelling on multi-objective multi-agent interactions with non-linear utilities. Specifically, we consider two-player multi-objective normal form games with non-linear utility functions under the scalarised expected returns optimisation criterion. We contribute novel actor-critic and policy gradient formulations to allow reinforcement learning of mixed strategies in this setting, along with extensions that incorporate opponent policy reconstruction and learning with opponent learning awareness (i.e., learning while considering the impact of one's policy when anticipating the opponent's learning step). Empirical results in five different MONFGs demonstrate that opponent learning awareness and modelling can drastically alter the learning dynamics in this setting. When equilibria are present, opponent modelling can confer significant benefits on agents that implement it. When there are no Nash equilibria, opponent learning awareness and modelling allows agents to still converge to meaningful solutions that approximate equilibria.

In multi-objective multi-agent systems (MOMASs), agents explicitly consider the possible trade-offs between conflicting objective functions. We argue that compromises between competing objectives in MOMAS should be analyzed on the basis of the utility that these compromises have for the users of a system, where an agent's utility function maps their payoff vectors to scalar utility values. This utility-based approach naturally leads to two different optimization criteria for agents in a MOMAS: expected scalarized returns (ESRs) and scalarized expected returns (SERs). In this article, we explore the differences between these two criteria using the framework of multi-objective normal-form games (MONFGs). We demonstrate that the choice of optimization criterion (ESR or SER) can radically alter the set of equilibria in a MONFG when nonlinear utility functions are used.

In this paper, we investigate the effects of opponent modelling on multi-objective multi-agent interactions with non-linear utilities. Specifically, we consider multi-objective normal form games (MONFGs) with non-linear utility functions under the scalarised expected returns optimisation criterion. We contribute a novel actor-critic formulation to allow reinforcement learning of mixed strategies in this setting, along with an extension that incorporates opponent policy reconstruction using conditional action frequencies. Our empirical results demonstrate that opponent modelling can drastically alter the learning dynamics in this setting.

In many multi-agent interactions in the real world, agents receive payoffs over multiple distinct criteria; i.e. the payoffs are multiobjective in nature. However, the same multi-objective payoff vector may lead to different utilities for each participant. Therefore, it is essential for an agent to learn about the behaviour of other agents in the system. In this work, we present the first study of the effects of such opponent modelling on multi-objective multiagent interactions with non-linear utilities. Specifically, we consider multi-objective normal form games with non-linear utility functions under the scalarised expected returns optimisation criterion. We contribute a novel actor-critic formulation to allow reinforcement learning of mixed strategies in this setting, along with an extension that incorporates opponent policy reconstruction using conditional action frequencies. Empirical results in five different MONFGs demonstrate that opponent modelling can drastically alter the learning dynamics in this setting. When equilibria are present opponent modelling can confer significant benefits on agents that implement it. However, when there are no Nash equilibria, opponent modelling can have adverse effects on utility, and has a neutral effect at best (after extensive hyper-parameter optimisation).