Stas Tiomkin

Stas Tiomkin, Naftali Tishby

The interaction between an artificial agent and its environment is bi-directional. The agent extracts relevant information from the environment, and affects the environment by its actions in return to accumulate high expected reward. Standard reinforcement learning (RL) deals with the expected reward maximization. However, there are always information-theoretic limitations that restrict the expected reward, which are not properly considered by the standard RL. In this work we consider RL objectives with information-theoretic limitations. For the first time we derive a Bellman-type recursive equa- tion for the causal information between the environment and the agent, which is combined plausibly with the Bellman recursion for the value function. The unified equitation serves to explore the typical behavior of artificial agents in an infinite time horizon.

Kyle Hollins Wray, Stas Tiomkin, Mykel J. Kochenderfer et al.

Multi-objective optimization models that encode ordered sequential constraints provide a solution to model various challenging problems including encoding preferences, modeling a curriculum, and enforcing measures of safety. A recently developed theory of topological Markov decision processes (TMDPs) captures this range of problems for the case of discrete states and actions. In this work, we extend TMDPs towards continuous spaces and unknown transition dynamics by formulating, proving, and implementing the policy gradient theorem for TMDPs. This theoretical result enables the creation of TMDP learning algorithms that use function approximators, and can generalize existing deep reinforcement learning (DRL) approaches. Specifically, we present a new algorithm for a policy gradient in TMDPs by a simple extension of the proximal policy optimization (PPO) algorithm. We demonstrate this on a real-world multiple-objective navigation problem with an arbitrary ordering of objectives both in simulation and on a real robot.

Stas Tiomkin, Ilya Nemenman, Daniel Polani et al.

Biological systems often choose actions without an explicit reward signal, a phenomenon known as intrinsic motivation. The computational principles underlying this behavior remain poorly understood. In this study, we investigate an information-theoretic approach to intrinsic motivation, based on maximizing an agent's empowerment (the mutual information between its past actions and future states). We show that this approach generalizes previous attempts to formalize intrinsic motivation, and we provide a computationally efficient algorithm for computing the necessary quantities. We test our approach on several benchmark control problems, and we explain its success in guiding intrinsically motivated behaviors by relating our information-theoretic control function to fundamental properties of the dynamical system representing the combined agent-environment system. This opens the door for designing practical artificial, intrinsically motivated controllers and for linking animal behaviors to their dynamical properties.

Tristan Shah, Ilya Nemenman, Daniel Polani et al.

Intrinsic motivations are receiving increasing attention, i.e. behavioral incentives that are not engineered, but emerge from the interaction of an agent with its surroundings. In this work we study the emergence of behaviors driven by one such incentive, empowerment, specifically in the context of more than one agent. We formulate a principled extension of empowerment to the multi-agent setting, and demonstrate its efficient calculation. We observe that this intrinsic motivation gives rise to characteristic modes of group-organization in two qualitatively distinct environments: a pair of agents coupled by a tendon, and a controllable Vicsek flock. This demonstrates the potential of intrinsic motivations such as empowerment to not just drive behavior for only individual agents but also higher levels of behavioral organization at scale.

Jacob Adamczyk, Argenis Arriojas, Stas Tiomkin et al.

In reinforcement learning (RL), the ability to utilize prior knowledge from previously solved tasks can allow agents to quickly solve new problems. In some cases, these new problems may be approximately solved by composing the solutions of previously solved primitive tasks (task composition). Otherwise, prior knowledge can be used to adjust the reward function for a new problem, in a way that leaves the optimal policy unchanged but enables quicker learning (reward shaping). In this work, we develop a general framework for reward shaping and task composition in entropy-regularized RL. To do so, we derive an exact relation connecting the optimal soft value functions for two entropy-regularized RL problems with different reward functions and dynamics. We show how the derived relation leads to a general result for reward shaping in entropy-regularized RL. We then generalize this approach to derive an exact relation connecting optimal value functions for the composition of multiple tasks in entropy-regularized RL. We validate these theoretical contributions with experiments showing that reward shaping and task composition lead to faster learning in various settings.

Derek Lilienthal, Paul Mello, Magdalini Eirinaki et al.

While recommender systems have become an integral component of the Web experience, their heavy reliance on user data raises privacy and security concerns. Substituting user data with synthetic data can address these concerns, but accurately replicating these real-world datasets has been a notoriously challenging problem. Recent advancements in generative AI have demonstrated the impressive capabilities of diffusion models in generating realistic data across various domains. In this work we introduce a Score-based Diffusion Recommendation Module (SDRM), which captures the intricate patterns of real-world datasets required for training highly accurate recommender systems. SDRM allows for the generation of synthetic data that can replace existing datasets to preserve user privacy, or augment existing datasets to address excessive data sparsity. Our method outperforms competing baselines such as generative adversarial networks, variational autoencoders, and recently proposed diffusion models in synthesizing various datasets to replace or augment the original data by an average improvement of 4.30% in Recall@k and 4.65% in NDCG@k.

Jacob Adamczyk, Volodymyr Makarenko, Argenis Arriojas et al.

In the field of reinforcement learning (RL), agents are often tasked with solving a variety of problems differing only in their reward functions. In order to quickly obtain solutions to unseen problems with new reward functions, a popular approach involves functional composition of previously solved tasks. However, previous work using such functional composition has primarily focused on specific instances of composition functions whose limiting assumptions allow for exact zero-shot composition. Our work unifies these examples and provides a more general framework for compositionality in both standard and entropy-regularized RL. We find that, for a broad class of functions, the optimal solution for the composite task of interest can be related to the known primitive task solutions. Specifically, we present double-sided inequalities relating the optimal composite value function to the value functions for the primitive tasks. We also show that the regret of using a zero-shot policy can be bounded for this class of functions. The derived bounds can be used to develop clipping approaches for reducing uncertainty during training, allowing agents to quickly adapt to new tasks.

Jacob Adamczyk, Stas Tiomkin, Rahul Kulkarni

An agent's ability to leverage past experience is critical for efficiently solving new tasks. Approximate solutions for new tasks can be obtained from previously derived value functions, as demonstrated by research on transfer learning, curriculum learning, and compositionality. However, prior work has primarily focused on using value functions to obtain zero-shot approximations for solutions to a new task. In this work, we show how an arbitrary approximation for the value function can be used to derive double-sided bounds on the optimal value function of interest. We further extend the framework with error analysis for continuous state and action spaces. The derived results lead to new approaches for clipping during training which we validate numerically in simple domains.

Tristan Shah, Stas Tiomkin

Intrinsic Motivation (IM) is a paradigm for generating intelligent behavior without external utilities. The existing information-theoretic methods for IM are predominantly based on information transmission, which explicitly depends on the designer's choice of which random variables engage in transmission. In this work, we introduce a novel IM principle, Controllable Information Production (CIP), that avoids both external utilities and designer-specified variables. We derive the CIP objective from Optimal Control, showing a connection between extrinsic and intrinsic behaviors. CIP appears as the gap between open-loop and closed-loop Kolmogorov-Sinai entropies, which simultaneously rewards the pursuit and regulation of chaos. We establish key theoretical properties of CIP and demonstrate its effectiveness on standard IM benchmarks.

Tristan Shah, Feruza Amirkulova, Stas Tiomkin

Controlling systems governed by partial differential equations is an inherently hard problem. Specifically, control of wave dynamics is challenging due to additional physical constraints and intrinsic properties of wave phenomena such as dissipation, attenuation, reflection, and scattering. In this work, we introduce an environment designed for the study of the control of acoustic waves by actuated metamaterial designs. We utilize this environment for the development of a novel machine-learning method, based on deep neural networks, for efficiently learning the dynamics of an acoustic PDE from samples. Our model is fully interpretable and maps physical constraints and intrinsic properties of the real acoustic environment into its latent representation of information. Within our model we use a trainable perfectly matched layer to explicitly learn the property of acoustic energy dissipation. Our model can be used to predict and control scattered wave energy. The capabilities of our model are demonstrated on an important problem in acoustics, which is the minimization of total scattered energy. Furthermore, we show that the prediction of scattered energy by our model generalizes in time and can be extended to long time horizons. We make our code repository publicly available.

Suruchi Sharma, Volodymyr Makarenko, Gautam Kumar et al.

Control of a dynamical system without the knowledge of dynamics is an important and challenging task. Modern machine learning approaches, such as deep neural networks (DNNs), allow for the estimation of a dynamics model from control inputs and corresponding state observation outputs. Such data-driven models are often utilized for the derivation of model-based controllers. However, in general, there are no guarantees that a model represented by DNNs will be controllable according to the formal control-theoretical meaning of controllability, which is crucial for the design of effective controllers. This often precludes the use of DNN-estimated models in applications, where formal controllability guarantees are required. In this proof-of-the-concept work, we propose a control-theoretical method that explicitly enhances models estimated from data with controllability. That is achieved by augmenting the model estimation objective with a controllability constraint, which penalizes models with a low degree of controllability. As a result, the models estimated with the proposed controllability constraint allow for the derivation of more efficient controllers, they are interpretable by the control-theoretical quantities and have a lower long-term prediction error. The proposed method provides new insights on the connection between the DNN-based estimation of unknown dynamics and the control-theoretical guarantees of the solution properties. We demonstrate the superiority of the proposed method in two standard classical control systems with state observation given by low resolution high-dimensional images.

Phu Nguyen, Daniel Polani, Stas Tiomkin

The reward function is an essential component in robot learning. Reward directly affects the sample and computational complexity of learning, and the quality of a solution. The design of informative rewards requires domain knowledge, which is not always available. We use the properties of the dynamics to produce system-appropriate reward without adding external assumptions. Specifically, we explore an approach to utilize the Lyapunov exponents of the system dynamics to generate a system-immanent reward. We demonstrate that the `Sum of the Positive Lyapunov Exponents' (SuPLE) is a strong candidate for the design of such a reward. We develop a computational framework for the derivation of this reward, and demonstrate its effectiveness on classical benchmarks for sample-based stabilization of various dynamical systems. It eliminates the need to start the training trajectories at arbitrary states, also known as auxiliary exploration. While the latter is a common practice in simulated robot learning, it is unpractical to consider to use it in real robotic systems, since they typically start from natural rest states such as a pendulum at the bottom, a robot on the ground, etc. and can not be easily initialized at arbitrary states. Comparing the performance of SuPLE to commonly-used reward functions, we observe that the latter fail to find a solution without auxiliary exploration, even for the task of swinging up the double pendulum and keeping it stable at the upright position, a prototypical scenario for multi-linked robots. SuPLE-induced rewards for robot learning offer a novel route for effective robot learning in typical as opposed to highly specialized or fine-tuned scenarios. Our code is publicly available for reproducibility and further research.

Jacob Adamczyk, Volodymyr Makarenko, Stas Tiomkin et al.

The average-reward formulation of reinforcement learning (RL) has drawn increased interest in recent years for its ability to solve temporally-extended problems without relying on discounting. Meanwhile, in the discounted setting, algorithms with entropy regularization have been developed, leading to improvements over deterministic methods. Despite the distinct benefits of these approaches, deep RL algorithms for the entropy-regularized average-reward objective have not been developed. While policy-gradient based approaches have recently been presented for the average-reward literature, the corresponding actor-critic framework remains less explored. In this paper, we introduce an average-reward soft actor-critic algorithm to address these gaps in the field. We validate our method by comparing with existing average-reward algorithms on standard RL benchmarks, achieving superior performance for the average-reward criterion.

Himaja Papala, Daniel Polani, Stas Tiomkin

Traffic congestion has long been an ubiquitous problem that is exacerbating with the rapid growth of megacities. In this proof-of-concept work we study intrinsic motivation, implemented via the empowerment principle, to control autonomous car behavior to improve traffic flow. In standard models of traffic dynamics, self-organized traffic jams emerge spontaneously from the individual behavior of cars, affecting traffic over long distances. Our novel car behavior strategy improves traffic flow while still being decentralized and using only locally available information without explicit coordination. Decentralization is essential for various reasons, not least to be able to absorb robustly substantial levels of uncertainty. Our scenario is based on the well-established traffic dynamics model, the Nagel-Schreckenberg cellular automaton. In a fraction of the cars in this model, we substitute the default behavior by empowerment, our intrinsic motivation-based method. This proposed model significantly improves overall traffic flow, mitigates congestion, and reduces the average traffic jam time.

Nadav Amir, Stas Tiomkin, Angela Langdon

Purposeful behavior is a hallmark of natural and artificial intelligence. Its acquisition is often believed to rely on world models, comprising both descriptive (what is) and prescriptive (what is desirable) aspects that identify and evaluate state of affairs in the world, respectively. Canonical computational accounts of purposeful behavior, such as reinforcement learning, posit distinct components of a world model comprising a state representation (descriptive aspect) and a reward function (prescriptive aspect). However, an alternative possibility, which has not yet been computationally formulated, is that these two aspects instead co-emerge interdependently from an agent's goal. Here, we describe a computational framework of goal-directed state representation in cognitive agents, in which the descriptive and prescriptive aspects of a world model co-emerge from agent-environment interaction sequences, or experiences. Drawing on Buddhist epistemology, we introduce a construct of goal-directed, or telic, states, defined as classes of goal-equivalent experience distributions. Telic states provide a parsimonious account of goal-directed learning in terms of the statistical divergence between behavioral policies and desirable experience features. We review empirical and theoretical literature supporting this novel perspective and discuss its potential to provide a unified account of behavioral, phenomenological and neural dimensions of purposeful behaviors across diverse substrates.

Tristan Shah, Noam Smilovich, Feruza Amirkulova et al.

Recent advancements in robotics, control, and machine learning have facilitated progress in the challenging area of object manipulation. These advancements include, among others, the use of deep neural networks to represent dynamics that are partially observed by robot sensors, as well as effective control using sparse control signals. In this work, we explore a more general problem: the manipulation of acoustic waves, which are partially observed by a robot capable of influencing the waves through spatially sparse actuators. This problem holds great potential for the design of new artificial materials, ultrasonic cutting tools, energy harvesting, and other applications. We develop an efficient data-driven method for robot learning that is applicable to either focusing scattered acoustic energy in a designated region or suppressing it, depending on the desired task. The proposed method is better in terms of a solution quality and computational complexity as compared to a state-of-the-art learning based method for manipulation of dynamical systems governed by partial differential equations. Furthermore our proposed method is competitive with a classical semi-analytical method in acoustics research on the demonstrated tasks. We have made the project code publicly available, along with a web page featuring video demonstrations: https://gladisor.github.io/waves/.

Jacob Adamczyk, Volodymyr Makarenko, Stas Tiomkin et al.

In reinforcement learning, two objective functions have been developed extensively in the literature: discounted and averaged rewards. The generalization to an entropy-regularized setting has led to improved robustness and exploration for both of these objectives. Recently, the entropy-regularized average-reward problem was addressed using tools from large deviation theory in the tabular setting. This method has the advantage of linearity, providing access to both the optimal policy and average reward-rate through properties of a single matrix. In this paper, we extend that framework to more general settings by developing approaches based on function approximation by neural networks. This formulation reveals new theoretical insights into the relationship between different objectives used in RL. Additionally, we combine our algorithm with a posterior policy iteration scheme, showing how our approach can also solve the average-reward RL problem without entropy-regularization. Using classic control benchmarks, we experimentally find that our method compares favorably with other algorithms in terms of stability and rate of convergence.

Jacob Adamczyk, Volodymyr Makarenko, Stas Tiomkin et al.

In reinforcement learning, especially in sparse-reward domains, many environment steps are required to observe reward information. In order to increase the frequency of such observations, "potential-based reward shaping" (PBRS) has been proposed as a method of providing a more dense reward signal while leaving the optimal policy invariant. However, the required "potential function" must be carefully designed with task-dependent knowledge to not deter training performance. In this work, we propose a "bootstrapped" method of reward shaping, termed BSRS, in which the agent's current estimate of the state-value function acts as the potential function for PBRS. We provide convergence proofs for the tabular setting, give insights into training dynamics for deep RL, and show that the proposed method improves training speed in the Atari suite.

Jacob Adamczyk, Volodymyr Makarenko, Stas Tiomkin et al.

An agent's ability to leverage past experience is critical for efficiently solving new tasks. Prior work has focused on using value function estimates to obtain zero-shot approximations for solutions to a new task. In soft Q-learning, we show how any value function estimate can also be used to derive double-sided bounds on the optimal value function. The derived bounds lead to new approaches for boosting training performance which we validate experimentally. Notably, we find that the proposed framework suggests an alternative method for updating the Q-function, leading to boosted performance.

Nadav Amir, Stas Tiomkin

Computational models of purposeful behavior comprise both descriptive and prescriptive aspects, used respectively to ascertain and evaluate situations in the world. In reinforcement learning, prescriptive reward functions are assumed to depend on predefined and fixed descriptive state representations. Alternatively, these two aspects may emerge interdependently: goals can shape the acquired state representations and vice versa. Here, we present a computational framework for state representation learning in bounded agents, where descriptive and prescriptive aspects are coupled through the notion of goal-directed, or telic, states. We introduce the concept of telic-controllability to characterize the tradeoff between the granularity of a telic state representation and the policy complexity required to reach all telic states. We propose an algorithm for learning telic-controllable state representations, illustrating it using a simulated navigation task. Our framework highlights the role of deliberate ignorance -- knowing what to ignore -- for learning state representations that balance goal flexibility and cognitive complexity.

Argenis Arriojas, Jacob Adamczyk, Stas Tiomkin et al.

Reinforcement learning (RL) is an important field of research in machine learning that is increasingly being applied to complex optimization problems in physics. In parallel, concepts from physics have contributed to important advances in RL with developments such as entropy-regularized RL. While these developments have led to advances in both fields, obtaining analytical solutions for optimization in entropy-regularized RL is currently an open problem. In this paper, we establish a mapping between entropy-regularized RL and research in non-equilibrium statistical mechanics focusing on Markovian processes conditioned on rare events. In the long-time limit, we apply approaches from large deviation theory to derive exact analytical results for the optimal policy and optimal dynamics in Markov Decision Process (MDP) models of reinforcement learning. The results obtained lead to a novel analytical and computational framework for entropy-regularized RL which is validated by simulations. The mapping established in this work connects current research in reinforcement learning and non-equilibrium statistical mechanics, thereby opening new avenues for the application of analytical and computational approaches from one field to cutting-edge problems in the other.

Philippe Hansen-Estruch, Wenling Shang, Lerrel Pinto et al.

Learning the dynamics of a physical system wherein an autonomous agent operates is an important task. Often these systems present apparent geometric structures. For instance, the trajectories of a robotic manipulator can be broken down into a collection of its transitional and rotational motions, fully characterized by the corresponding Lie groups and Lie algebras. In this work, we take advantage of these structures to build effective dynamical models that are amenable to sample-based learning. We hypothesize that learning the dynamics on a Lie algebra vector space is more effective than learning a direct state transition model. To verify this hypothesis, we introduce the Group Enhanced Model (GEM). GEMs significantly outperform conventional transition models on tasks of long-term prediction, planning, and model-based reinforcement learning across a diverse suite of standard continuous-control environments, including Walker, Hopper, Reacher, Half-Cheetah, Inverted Pendulums, Ant, and Humanoid. Furthermore, plugging GEM into existing state of the art systems enhances their performance, which we demonstrate on the PETS system. This work sheds light on a connection between learning of dynamics and Lie group properties, which opens doors for new research directions and practical applications along this direction. Our code is publicly available at: https://tinyurl.com/GEMMBRL.

Xingyu Lu, Kimin Lee, Pieter Abbeel et al.

Despite the significant progress of deep reinforcement learning (RL) in solving sequential decision making problems, RL agents often overfit to training environments and struggle to adapt to new, unseen environments. This prevents robust applications of RL in real world situations, where system dynamics may deviate wildly from the training settings. In this work, our primary contribution is to propose an information theoretic regularization objective and an annealing-based optimization method to achieve better generalization ability in RL agents. We demonstrate the extreme generalization benefits of our approach in different domains ranging from maze navigation to robotic tasks; for the first time, we show that agents can generalize to test parameters more than 10 standard deviations away from the training parameter distribution. This work provides a principled way to improve generalization in RL by gradually removing information that is redundant for task-solving; it opens doors for the systematic study of generalization from training to extremely different testing settings, focusing on the established connections between information theory and machine learning.

Ruihan Zhao, Kevin Lu, Pieter Abbeel et al.

Intrinsically motivated artificial agents learn advantageous behavior without externally-provided rewards. Previously, it was shown that maximizing mutual information between agent actuators and future states, known as the empowerment principle, enables unsupervised stabilization of dynamical systems at upright positions, which is a prototypical intrinsically motivated behavior for upright standing and walking. This follows from the coincidence between the objective of stabilization and the objective of empowerment. Unfortunately, sample-based estimation of this kind of mutual information is challenging. Recently, various variational lower bounds (VLBs) on empowerment have been proposed as solutions; however, they are often biased, unstable in training, and have high sample complexity. In this work, we propose an alternative solution based on a trainable representation of a dynamical system as a Gaussian channel, which allows us to efficiently calculate an unbiased estimator of empowerment by convex optimization. We demonstrate our solution for sample-based unsupervised stabilization on different dynamical control systems and show the advantages of our method by comparing it to the existing VLB approaches. Specifically, we show that our method has a lower sample complexity, is more stable in training, possesses the essential properties of the empowerment function, and allows estimation of empowerment from images. Consequently, our method opens a path to wider and easier adoption of empowerment for various applications.

Yuqing Du, Stas Tiomkin, Emre Kiciman et al.

One difficulty in using artificial agents for human-assistive applications lies in the challenge of accurately assisting with a person's goal(s). Existing methods tend to rely on inferring the human's goal, which is challenging when there are many potential goals or when the set of candidate goals is difficult to identify. We propose a new paradigm for assistance by instead increasing the human's ability to control their environment, and formalize this approach by augmenting reinforcement learning with human empowerment. This task-agnostic objective preserves the person's autonomy and ability to achieve any eventual state. We test our approach against assistance based on goal inference, highlighting scenarios where our method overcomes failure modes stemming from goal ambiguity or misspecification. As existing methods for estimating empowerment in continuous domains are computationally hard, precluding its use in real time learned assistance, we also propose an efficient empowerment-inspired proxy metric. Using this, we are able to successfully demonstrate our method in a shared autonomy user study for a challenging simulated teleoperation task with human-in-the-loop training.

Albert Zhan, Stas Tiomkin, Pieter Abbeel

Imitation learning can reproduce policies by observing experts, which poses a problem regarding policy privacy. Policies, such as human, or policies on deployed robots, can all be cloned without consent from the owners. How can we protect against external observers cloning our proprietary policies? To answer this question we introduce a new reinforcement learning framework, where we train an ensemble of near-optimal policies, whose demonstrations are guaranteed to be useless for an external observer. We formulate this idea by a constrained optimization problem, where the objective is to improve proprietary policies, and at the same time deteriorate the virtual policy of an eventual external observer. We design a tractable algorithm to solve this new optimization problem by modifying the standard policy gradient algorithm. Our formulation can be interpreted in lenses of confidentiality and adversarial behaviour, which enables a broader perspective of this work. We demonstrate the existence of "non-clonable" ensembles, providing a solution to the above optimization problem, which is calculated by our modified policy gradient algorithm. To our knowledge, this is the first work regarding the protection of policies in Reinforcement Learning.

Xingyu Lu, Stas Tiomkin, Pieter Abbeel

While recent progress in deep reinforcement learning has enabled robots to learn complex behaviors, tasks with long horizons and sparse rewards remain an ongoing challenge. In this work, we propose an effective reward shaping method through predictive coding to tackle sparse reward problems. By learning predictive representations offline and using these representations for reward shaping, we gain access to reward signals that understand the structure and dynamics of the environment. In particular, our method achieves better learning by providing reward signals that 1) understand environment dynamics 2) emphasize on features most useful for learning 3) resist noise in learned representations through reward accumulation. We demonstrate the usefulness of this approach in different domains ranging from robotic manipulation to navigation, and we show that reward signals produced through predictive coding are as effective for learning as hand-crafted rewards.

Ruihan Zhao, Stas Tiomkin, Pieter Abbeel

Mutual Information between agent Actions and environment States (MIAS) quantifies the influence of agent on its environment. Recently, it was found that the maximization of MIAS can be used as an intrinsic motivation for artificial agents. In literature, the term empowerment is used to represent the maximum of MIAS at a certain state. While empowerment has been shown to solve a broad range of reinforcement learning problems, its calculation in arbitrary dynamics is a challenging problem because it relies on the estimation of mutual information. Existing approaches, which rely on sampling, are limited to low dimensional spaces, because high-confidence distribution-free lower bounds for mutual information require exponential number of samples. In this work, we develop a novel approach for the estimation of empowerment in unknown dynamics from visual observation only, without the need to sample for MIAS. The core idea is to represent the relation between action sequences and future states using a stochastic dynamic model in latent space with a specific form. This allows us to efficiently compute empowerment with the "Water-Filling" algorithm from information theory. We construct this embedding with deep neural networks trained on a sophisticated objective function. Our experimental results show that the designed embedding preserves information-theoretic properties of the original dynamics.