We propose a novel estimator of the mutual information between two ordinal vectors $x$ and $y$. Our approach is inductive (as opposed to deductive) in that it depends on the data generating distribution solely through some nonparametric properties revealing associations in the data, and does not require having enough data to fully characterize the true joint distributions $P_{x, y}$. Specifically, our approach consists of (i) noting that $I\left(y; x\right) = I\left(u_y; u_x\right)$ where $u_y$ and $u_x$ are the copula-uniform dual representations of $y$ and $x$ (i.e. their images under the probability integral transform), and (ii) estimating the copula entropies $h\left(u_y\right)$, $h\left(u_x\right)$ and $h\left(u_y, u_x\right)$ by solving a maximum-entropy problem over the space of copula densities under a constraint of the type $\alpha_m = E\left[\phi_m(u_y, u_x)\right]$. We prove that, so long as the constraint is feasible, this problem admits a unique solution, it is in the exponential family, and it can be learned by solving a convex optimization problem. The resulting estimator, which we denote MIND, is marginal-invariant, always non-negative, unbounded for any sample size $n$, consistent, has MSE rate $O(1/n)$, and is more data-efficient than competing approaches. Beyond mutual information estimation, we illustrate that our approach may be used to mitigate mode collapse in GANs by maximizing the entropy of the copula of fake samples, a model we refer to as Copula Entropy Regularized GAN (CER-GAN).

Mixture models are an expressive hypothesis class that can approximate a rich set of policies. However, using mixture policies in the Maximum Entropy (MaxEnt) framework is not straightforward. The entropy of a mixture model is not equal to the sum of its components, nor does it have a closed-form expression in most cases. Using such policies in MaxEnt algorithms, therefore, requires constructing a tractable approximation of the mixture entropy. In this paper, we derive a simple, low-variance mixture-entropy estimator. We show that it is closely related to the sum of marginal entropies. Equipped with our entropy estimator, we derive an algorithmic variant of Soft Actor-Critic (SAC) to the mixture policy case and evaluate it on a series of continuous control tasks.

Nearly all real-world applications of reinforcement learning involve some degree of shift between the training environment and the testing environment. However, prior work has observed that even small shifts in the environment cause most RL algorithms to perform markedly worse. As we aim to scale reinforcement learning algorithms and apply them in the real world, it is increasingly important to learn policies that are robust to changes in the environment. Broadly, prior approaches to handling distribution shift in RL aim to maximize performance in either the average case or the worst case. While these methods have been successfully applied to a number of areas (e.g., self-driving cars, robot locomotion and manipulation), their success rests critically on the design of the distribution of environments.

Bounded rationality is an important consideration stemming from the fact that agents often have limits on their processing abilities, making the assumption of perfect rationality inapplicable to many real tasks. We propose an information-theoretic approach to the inference of agent decisions under Smithian competition. The model explicitly captures the boundedness of agents (limited in their information-processing capacity) as the cost of information acquisition for expanding their prior beliefs. The expansion is measured as the Kullblack-Leibler divergence between posterior decisions and prior beliefs. When information acquisition is free, the \textit{homo economicus} agent is recovered, while in cases when information acquisition becomes costly, agents instead revert to their prior beliefs. The maximum entropy principle is used to infer least-biased decisions, based upon the notion of Smithian competition formalised within the Quantal Response Statistical Equilibrium framework. The incorporation of prior beliefs into such a framework allowed us to systematically explore the effects of prior beliefs on decision-making, in the presence of market feedback. We verified the proposed model using Australian housing market data, showing how the incorporation of prior knowledge alters the resulting agent decisions. Specifically, it allowed for the separation (and analysis) of past beliefs and utility maximisation behaviour of the agent.

In this article, I will explain what the maximum entropy principle is, how to apply it and why it's useful in the context of Bayesian inference. The code to reproduce the results and figures can be found in this notebook. The maximum entropy principle is a method to create probability distributions that is most consistent with a given set of assumptions and nothing more. The rest of the article will explain what this means. First, we need to a way to measure the uncertainty in a probability distribution.

The maximum entropy principle advocates to evaluate events' probabilities using a distribution that maximizes entropy among those that satisfy certain expectations' constraints. Such principle can be generalized for arbitrary decision problems where it corresponds to minimax approaches. This paper establishes a framework for supervised classification based on the generalized maximum entropy principle that leads to minimax risk classifiers (MRCs). We develop learning techniques that determine MRCs for general entropy functions and provide performance guarantees by means of convex optimization. In addition, we describe the relationship of the presented techniques with existing classification methods, and quantify MRCs performance in comparison with the proposed bounds and conventional methods.

What goals should a multi-goal reinforcement learning agent pursue during training in long-horizon tasks? When the desired (test time) goal distribution is too distant to offer a useful learning signal, we argue that the agent should not pursue unobtainable goals. Instead, it should set its own intrinsic goals that maximize the entropy of the historical achieved goal distribution. We propose to optimize this objective by having the agent pursue past achieved goals in sparsely explored areas of the goal space, which focuses exploration on the frontier of the achievable goal set. We show that our strategy achieves an order of magnitude better sample efficiency than the prior state of the art on long-horizon multi-goal tasks including maze navigation and block stacking.

Model usage is the central challenge of model-based reinforcement learning. Although dynamics model based on deep neural networks provide good generalization for single step prediction, such ability is over exploited when it is used to predict long horizon trajectories due to compounding errors. In this work, we propose a Dyna-style model-based reinforcement learning algorithm, which we called Maximum Entropy Model Rollouts (MEMR). To eliminate the compounding errors, we only use our model to generate single-step rollouts. Furthermore, we propose to generate \emph{diverse} model rollouts by non-uniform sampling of the environment states such that the entropy of the model rollouts is maximized. We mathematically derived the maximum entropy sampling criteria for one data case under Gaussian prior. To accomplish this criteria, we propose to utilize a prioritized experience replay. Our preliminary experiments in challenging locomotion benchmarks show that our approach achieves the same sample efficiency of the best model-based algorithms, matches the asymptotic performance of the best model-free algorithms, and significantly reduces the computation requirements of other model-based methods.

In the past decades, we have witnessed significant progress in the domain of autonomous driving. Advanced techniques based on optimization and reinforcement learning (RL) become increasingly powerful at solving the forward problem: given designed reward/cost functions, how should we optimize them and obtain driving policies that interact with the environment safely and efficiently. Such progress has raised another equally important question: \emph{what should we optimize}? Instead of manually specifying the reward functions, it is desired that we can extract what human drivers try to optimize from real traffic data and assign that to autonomous vehicles to enable more naturalistic and transparent interaction between humans and intelligent agents. To address this issue, we present an efficient sampling-based maximum-entropy inverse reinforcement learning (IRL) algorithm in this paper. Different from existing IRL algorithms, by introducing an efficient continuous-domain trajectory sampler, the proposed algorithm can directly learn the reward functions in the continuous domain while considering the uncertainties in demonstrated trajectories from human drivers. We evaluate the proposed algorithm on real driving data, including both non-interactive and interactive scenarios. The experimental results show that the proposed algorithm achieves more accurate prediction performance with faster convergence speed and better generalization compared to other baseline IRL algorithms.

This work focuses on off-policy evaluation (OPE) with function approximation in infinite-horizon undiscounted Markov decision processes (MDPs). For MDPs that are ergodic and linear (i.e. where rewards and dynamics are linear in some known features), we provide the first finite-sample OPE error bound, extending existing results beyond the episodic and discounted cases. In a more general setting, when the feature dynamics are approximately linear and for arbitrary rewards, we propose a new approach for estimating stationary distributions with function approximation. We formulate this problem as finding the maximum-entropy distribution subject to matching feature expectations under empirical dynamics. We show that this results in an exponential-family distribution whose sufficient statistics are the features, paralleling maximum-entropy approaches in supervised learning. We demonstrate the effectiveness of the proposed OPE approaches in multiple environments.