A general Markov decision process formalism for action-state entropy-regularized reward maximization
Previous work has separately addressed different forms of action, state and action-state entropy regularization, pure exploration and space occupation. These problems have become extremely relevant for regularization, generalization, speeding up learning and providing robust solutions at unprecedented levels. However, solutions of those problems are hectic, ranging from convex and non-convex optimization, and unconstrained optimization to constrained optimization. Here we provide a general dual function formalism that transforms the constrained optimization problem into an unconstrained convex one for any mixture of action and state entropies. The cases with pure action entropy and pure state entropy are understood as limits of the mixture.
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We formulate a stochastic process, FiLex, as a mathematical model of lexicon entropy in deep learning-based emergent language systems. Defining a model mathematically allows it to generate clear predictions which can be directly and decisively tested. We empirically verify across four different environments that FiLex predicts the correct correlation between hyperparameters (training steps, lexicon size, learning rate, rollout buffer size, and Gumbel-Softmax temperature) and the emergent language's entropy in 20 out of 20 environment-hyperparameter combinations. Furthermore, our experiments reveal that different environments show diverse relationships between their hyperparameters and entropy which demonstrates the need for a model which can make well-defined predictions at a precise level of granularity.
The aim of Inverse Reinforcement Learning (IRL) is to infer a reward function $R$ from a policy $\pi$. To do this, we need a model of how $\pi$ relates to $R$. In the current literature, the most common models are optimality, Boltzmann rationality, and causal entropy maximisation. One of the primary motivations behind IRL is to infer human preferences from human behaviour. However, the true relationship between human preferences and human behaviour is much more complex than any of the models currently used in IRL. This means that they are misspecified, which raises the worry that they might lead to unsound inferences if applied to real-world data. In this paper, we provide a mathematical analysis of how robust different IRL models are to misspecification, and answer precisely how the demonstrator policy may differ from each of the standard models before that model leads to faulty inferences about the reward function $R$. We also introduce a framework for reasoning about misspecification in IRL, together with formal tools that can be used to easily derive the misspecification robustness of new IRL models.
Empirical studies of the loss landscape of deep networks have revealed that many local minima are connected through low-loss valleys. Yet, little is known about the theoretical origin of such valleys. We present a general framework for finding continuous symmetries in the parameter space, which carve out low-loss valleys. Our framework uses equivariances of the activation functions and can be applied to different layer architectures. To generalize this framework to nonlinear neural networks, we introduce a novel set of nonlinear, data-dependent symmetries. These symmetries can transform a trained model such that it performs similarly on new samples, which allows ensemble building that improves robustness under certain adversarial attacks. We then show that conserved quantities associated with linear symmetries can be used to define coordinates along low-loss valleys. The conserved quantities help reveal that using common initialization methods, gradient flow only explores a small part of the global minimum. By relating conserved quantities to convergence rate and sharpness of the minimum, we provide insights on how initialization impacts convergence and generalizability.
Most of the literature on learning in games has focused on the restrictive setting where the underlying repeated game does not change over time. Much less is known about the convergence of no-regret learning algorithms in dynamic multiagent settings. In this paper, we characterize the convergence of optimistic gradient descent (OGD) in time-varying games. Our framework yields sharp convergence bounds for the equilibrium gap of OGD in zero-sum games parameterized on natural variation measures of the sequence of games, subsuming known results for static games. Furthermore, we establish improved second-order variation bounds under strong convexity-concavity, as long as each game is repeated multiple times. Our results also apply to time-varying general-sum multi-player games via a bilinear formulation of correlated equilibria, which has novel implications for meta-learning and for obtaining refined variation-dependent regret bounds, addressing questions left open in prior papers. Finally, we leverage our framework to also provide new insights on dynamic regret guarantees in static games.
A novel Policy Gradient (PG) algorithm, called Matryoshka Policy Gradient (MPG), is introduced and studied, in the context of max-entropy reinforcement learning, where an agent aims at maximising entropy bonuses additional to its cumulative rewards. MPG differs from standard PG in that it trains a sequence of policies to learn finite horizon tasks simultaneously, instead of a single policy for the single standard objective. For softmax policies, we prove convergence of MPG and global optimality of the limit by showing that the only critical point of the MPG objective is the optimal policy; these results hold true even in the case of continuous compact state space. MPG is intuitive, theoretically sound and we furthermore show that the optimal policy of the standard max-entropy objective can be approximated arbitrarily well by the optimal policy of the MPG framework. Finally, we justify that MPG is well suited when the policies are parametrized with neural networks and we provide an simple criterion to verify the global optimality of the policy at convergence. As a proof of concept, we evaluate numerically MPG on standard test benchmarks.
A central task in control theory, artificial intelligence, and formal methods is to synthesize reward-maximizing strategies for agents that operate in partially unknown environments. In environments modeled by gray-box Markov decision processes (MDPs), the impact of the agents' actions are known in terms of successor states but not the stochastics involved. In this paper, we devise a strategy synthesis algorithm for gray-box MDPs via reinforcement learning that utilizes interval MDPs as internal model. To compete with limited sampling access in reinforcement learning, we incorporate two novel concepts into our algorithm, focusing on rapid and successful learning rather than on stochastic guarantees and optimality: lower confidence bound exploration reinforces variants of already learned practical strategies and action scoping reduces the learning action space to promising actions. We illustrate benefits of our algorithms by means of a prototypical implementation applied on examples from the AI and formal methods communities.
Efficient and accurate estimation of multivariate empirical probability distributions is fundamental to the calculation of information-theoretic measures such as mutual information and transfer entropy. Common techniques include variations on histogram estimation which, whilst computationally efficient, are often unable to precisely capture the probability density of samples with high correlation, kurtosis or fine substructure, especially when sample sizes are small. Adaptive partitions, which adjust heuristically to the sample, can reduce the bias imparted from the geometry of the histogram itself, but these have commonly focused on the location, scale and granularity of the partition, the effects of which are limited for highly correlated distributions. In this paper, I reformulate the differential entropy estimator for the special case of an equiprobable histogram, using a k-d tree to partition the sample space into bins of equal probability mass. By doing so, I expose an implicit rotational orientation parameter, which is conjectured to be suboptimally specified in the typical marginal alignment. I propose that the optimal orientation minimises the variance of the bin volumes, and demonstrate that improved entropy estimates can be obtained by rotationally aligning the partition to the sample distribution accordingly. Such optimal partitions are observed to be more accurate than existing techniques in estimating entropies of correlated bivariate Gaussian distributions with known theoretical values, across varying sample sizes (99% CI).
High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted $\ell_1$-penalization which reduces the estimation bias from $\ell_1$-penalization and leads to oracle properties. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which $d \ll n$; (ii) high-dimensional regime in which $s\ll n\ll d$ with $s$ denoting the number of significant predictors. In the high-dimensional setting, we carefully characterize the solution path of the iteratively reweighted $\ell_1$-penalized retire estimation, adapted from the local linear approximation algorithm for folded-concave regularization. Under a mild minimum signal strength condition, we show that after as many as $\log(\log d)$ iterations the final iterate enjoys the oracle convergence rate. At each iteration, the weighted $\ell_1$-penalized convex program can be efficiently solved by a semismooth Newton coordinate descent algorithm. Numerical studies demonstrate the competitive performance of the proposed procedure compared with either non-robust or quantile regression based alternatives.
Conventionally, since the natural language action space is astronomical, approximate dynamic programming applied to dialogue generation involves policy improvement with action sampling. However, such a practice is inefficient for reinforcement learning (RL) because the eligible (high action value) responses are very sparse, and the greedy policy sustained by the random sampling is flabby. This paper shows that the performance of dialogue policy positively correlated with sampling size by theoretical and experimental. We introduce a novel dual-granularity Q-function to alleviate this limitation by exploring the most promising response category to intervene in the sampling. It extracts the actions following the grained hierarchy, which can achieve the optimum with fewer policy iterations. Our approach learns in the way of offline RL from multiple reward functions designed to recognize human emotional details. Empirical studies demonstrate that our algorithm outperforms the baseline methods. Further verification presents that ours can generate responses with higher expected rewards and controllability.
We inspected 45 actively deployed Operational Technology (OT) product families from ten major vendors and found that every system suffers from at least one trivial vulnerability. We reported a total of 53 weaknesses, stemming from insecure by design practices or basic security design failures. They enable attackers to take a device offline, manipulate its operational parameters, and execute arbitrary code without any constraint. We discuss why vulnerable products are often security certified and appear to be more secure than they actually are, and we explain complicating factors of OT risk management.
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