Plasticity as the Mirror of Empowerment (2505.10361v1)

Abstract: Agents are minimally entities that are influenced by their past observations and act to influence future observations. This latter capacity is captured by empowerment, which has served as a vital framing concept across artificial intelligence and cognitive science. This former capacity, however, is equally foundational: In what ways, and to what extent, can an agent be influenced by what it observes? In this paper, we ground this concept in a universal agent-centric measure that we refer to as plasticity, and reveal a fundamental connection to empowerment. Following a set of desiderata on a suitable definition, we define plasticity using a new information-theoretic quantity we call the generalized directed information. We show that this new quantity strictly generalizes the directed information introduced by Massey (1990) while preserving all of its desirable properties. Our first finding is that plasticity is the mirror of empowerment: The agent's plasticity is identical to the empowerment of the environment, and vice versa. Our second finding establishes a tension between the plasticity and empowerment of an agent, suggesting that agent design needs to be mindful of both characteristics. We explore the implications of these findings, and suggest that plasticity, empowerment, and their relationship are essential to understanding agency.

• The paper introduces the Generalized Directed Information (GDI) as a universal measure to capture the bidirectional influence between agent actions and environmental observations.
• The paper demonstrates that plasticity and empowerment are mirror properties, establishing that an agent’s capacity to be influenced is equivalent to its ability to influence.
• The paper reveals a trade-off where maximizing empowerment reduces plasticity, suggesting agents alternate between phases of adaptation and control.

This paper, "Plasticity as the Mirror of Empowerment" (2505.10361), explores two fundamental capacities of artificial agents: empowerment and plasticity. Empowerment, which has been extensively studied, relates to an agent's ability to influence its future observations. Plasticity, often discussed in neuroscience and machine learning in terms of adaptability or learning, is framed here as the agent's capacity to be influenced by its past observations. The paper proposes a universal, agent-centric measure for plasticity and reveals a deep, reciprocal relationship between these two concepts, suggesting they are critical for understanding agency itself.

The authors begin by setting up a general agent-environment interaction framework based on a discrete-time exchange of actions ($A$) and observations ($O$). This setup avoids strong assumptions like the Markov property, common in standard reinforcement learning. They build upon the concept of directed information ($I(X_{1:n} \rightarrow Y_{1:n})$), which quantifies the information flow from a sequence $X$ to a sequence $Y$ over time, accounting for feedback. Existing notions of empowerment, such as those by Klyubin et al. (open-loop) and Capdepuy (closed-loop, using directed information), are reviewed.

A key contribution is the introduction of the Generalized Directed Information (GDI), denoted $I(X_{a:b} \rightarrow Y_{c:d})$. Standard directed information requires sequences of the same length starting from time 1. GDI generalizes this to arbitrary intervals of time, $[a:b]$ and $[c:d]$. The GDI is defined as:

$$I(X_{a:b} \rightarrow Y_{c:d}) \stackrel{def}{=} \sum_{i=\max(a,c)}^d I(X_{a:\min(b,i)}; Y_i | X_{1:a-1}, Y_{1:i-1})$$

This formulation captures the influence of past $X$ variables within the interval $[a:b]$ on future $Y_i$ variables within the interval $[c:d]$, conditioned on everything observed before the intervals start.

The paper establishes several properties for GDI, including:

• It strictly generalizes standard directed information when intervals start at 1 and have the same length (Proposition 3.2).
• It is temporally consistent, being zero if the influencing interval is strictly after the influenced interval (Proposition 3.3).
• It satisfies a summation property allowing intervals to be broken down (Proposition 3.4).
• Crucially, it satisfies a Conservation Law of GDI (Theorem 3.5):

  $$I(X_{a:b}; Y_{c:d} | X_{1:a-1}, Y_{1:c-1}) = I(X_{a:b} \rightarrow Y_{c:d}) + I(Y_{c:d} \rightarrow X_{a:b})$$

  This law states that the total conditional mutual information between the two sequences, given the history before their intervals, can be decomposed into the directed information from X to Y and from Y to X. This is a generalization of Massey's conservation law for standard directed information.

Using GDI, the paper provides a universal definition for plasticity. Guided by desiderata (precision, simplicity, minimal assumptions, flexibility), agent plasticity relative to a set of environments $\mathcal{E}$ is defined as:

$$\Psi_{a:b}^{c:d}(\Lambda, \mathcal{E}) \stackrel{def}{=} \max_{e \in \mathcal{E}} I(O_{a:b} \rightarrow A_{c:d})$$

This measures the maximum directed information from a sequence of observations $O_{a:b}$ to a sequence of agent actions $A_{c:d}$ over a set of possible environments. For a specific environment $e$, it's $\Psi_{a:b}^{c:d}(\Lambda, e)$. The authors show this definition satisfies the desiderata. Notably, an agent has positive plasticity relative to an environment if and only if its actions are influenced by observations at some timestep within the relevant intervals (Lemma 4.2). Trivial agents like constant agents or those whose actions only depend on history length have zero plasticity (Theorem 4.3).

The GDI framework also enriches the definition of empowerment. Agent empowerment in environment $e$ is defined as:

$$\mathcal{E}_{a:b}^{c:d}(\Lambda, e) \stackrel{def}{=} \max_{\lambda \in \Lambda} I(A_{a:b} \rightarrow O_{c:d})$$

This measures the maximum directed information from a sequence of agent actions $A_{a:b}$ to a sequence of observations $O_{c:d}$ over a set of possible agents $\Lambda$. This definition generalizes previous notions of empowerment by allowing arbitrary time intervals (Proposition 4.5).

The core analysis reveals two main findings about the relationship between plasticity and empowerment:

1. Plasticity is the Mirror of Empowerment (Proposition 4.6): Due to the inherent symmetry between the agent and the environment (both are stochastic functions mapping histories to symbols), the agent's empowerment is mathematically equivalent to the environment's plasticity, and the agent's plasticity is equivalent to the environment's empowerment. $\mathcal{E}(\lambda) = \Psi(e)$ and $\Psi(\lambda) = \mathcal{E}(e)$, where the intervals are implicitly the same for both quantities. This highlights that understanding how an agent influences its environment is symmetric to understanding how the environment influences the agent. A corollary is that if an environment has zero empowerment (i.e., cannot be influenced by any agent), then no agent interacting with it can have non-zero plasticity (i.e., cannot be influenced by observations from that environment), and vice versa (Corollary 4.7).
2. Plasticity-Empowerment Tension (Theorem 4.8): For any agent-environment pair and fixed intervals $[a:b]$ and $[c:d]$, there is a tight upper bound on the sum of the agent's plasticity and empowerment over these same intervals: $$\Psi_{c:d}^{a:b}(\lambda, e) + \mathcal{E}_{a:b}^{c:d}(\lambda, e) \leq m$$, where $m$ depends on the interval lengths and the sizes of the action and observation spaces. This implies that an agent cannot simultaneously maximize both its plasticity (its capacity to be shaped by observations) and its empowerment (its capacity to shape observations) over the same periods. Maximizing one quantity pushes the other towards zero. The authors suggest agents might oscillate between phases of high plasticity (learning, adapting) and high empowerment (acting, controlling).

Practical Implications and Implementation:

While the paper is theoretical, the concepts have significant implications for designing and analyzing AI agents:

• Agent Design: The revealed tension suggests that agents must navigate a trade-off between being receptive to environmental signals (plasticity) and being able to influence the environment (empowerment). Current agent designs often implicitly favor empowerment (e.g., maximizing reward), but the paper suggests explicit consideration and balancing of plasticity might be crucial for developing more general, adaptive, and safe agents. An agent with low plasticity cannot adapt to changing environments, while one with low empowerment cannot achieve goals.
• Understanding Agency: The paper proposes that non-zero plasticity and non-zero empowerment are necessary and potentially sufficient conditions for an input-output system to be considered an agent. Analyzing the dynamics of these quantities in existing agents could provide insight into their behavior and capabilities.
• Continual Learning: The concept of plasticity aligns closely with the capacity for adaptation in continual learning settings, where the environment (data distribution) shifts. Measuring plasticity (e.g., $I(O_{new} \rightarrow A_{new})$) could provide a quantitative metric for an agent's ability to adapt to these shifts.
• Intrinsic Motivation: Empowerment has been used as an intrinsic motivation signal. This work suggests that plasticity could also be a source of intrinsic motivation, driving agents to seek informative environments where their actions can be influenced by observations, thereby increasing their learning potential.
• Estimation: The paper points to the need for efficient algorithms to estimate GDI from observed interaction data. This is critical for applying these measures in practice. Existing methods for estimating directed information from time series data [Jiao et al., 2013] could potentially be extended or adapted. Implementing this would involve collecting sequences of actions and observations from an agent interacting with an environment and applying information-theoretic estimators. This would require careful consideration of sample efficiency, especially for complex or high-dimensional observation/action spaces.
• Alternative Plasticity Definitions: The paper mentions extending the plasticity definition to measure the influence of observations on internal agent components like policy parameters or agent state $S$ ($I(O_{a:b} \rightarrow L_{c:d})$ or $I(O_{a:b} \rightarrow S_{c:d})$). Implementing this would involve exposing or tracking these internal variables during agent operation and applying GDI estimation. This could be particularly relevant for analyzing model-based or stateful agents.

Implementation Considerations:

• Data Requirements: Estimating information-theoretic quantities like GDI requires sufficient samples of the random variables involved (sequences of actions and observations). The amount of data needed grows with the length of the sequences and the complexity of the dependencies.
• Computational Complexity: Direct estimation of mutual information and its variants can be computationally expensive, particularly in high-dimensional or continuous state/action spaces. Non-parametric methods might be necessary but can have their own sample efficiency issues.
• Discretization: For agents operating in continuous spaces, discretization of observations and actions might be necessary to apply the information-theoretic definitions based on discrete random variables. The choice of discretization can significantly impact the results.
• Max over Sets: The definitions of plasticity ($\max_{e \in \mathcal{E}}$) and empowerment ($\max_{\lambda \in \Lambda}$) involve optimization over sets of environments or agents. In practice, this maximum might be hard to compute or approximate. For a single agent interacting with a single environment, these reduce to direct GDI calculations, which are more tractable. Analyzing performance over a set would require simulating or analyzing agent behavior across multiple environment instances or agent variations.

In summary, the paper provides a fundamental theoretical framework linking plasticity and empowerment through the Generalized Directed Information. It reveals a crucial symmetry and a tension between these capacities, offering new perspectives for the analysis and design of agents beyond traditional reward-centric views. The practical application hinges on developing efficient methods for estimating GDI in complex interactive systems.