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REVIEW 1 major objections 1 minor 5 references

Maximizing empowerment produces forward and backward state representations that ignore control-irrelevant features.

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2026-06-29 08:08 UTC pith:HVN3G56D

load-bearing objection The paper links empowerment to forward/backward invariant representations but leaves the required environment and optimization conditions unspecified. the 1 major comments →

arxiv 2605.30656 v1 pith:HVN3G56D submitted 2026-05-28 cs.LG

Learning to Perceive the World Through Control: Empowerment-Based Representation Learning

Mahsa Bastankhah , Sophie Broderick , Benjamin Eysenbach This is my paper
classification cs.LG
keywords empowermentrepresentation learningreinforcement learninginvariancecontrolunsupervised skill learningcausal learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether agents can learn representations that retain only features relevant to control by optimizing the empowerment objective. Empowerment measures how much an agent can influence its future observations through its actions. The analysis shows that this objective induces two complementary representations, one forward and one backward, both of which remain unchanged by dimensions of the observation that do not affect reachable states. Interaction aimed at control is required for these invariance properties to emerge, in contrast to learning from static datasets. A reader would care because the result suggests a mechanism for building control-centric models of the world without explicit labels or supervision.

Core claim

Empowerment agents induce two distinct representations—forward and backward—that capture complementary aspects of the state, and both of which are invariant to control-irrelevant features. Thus, empowerment maximization leads agents to learn an implicit, control-centric model of the world.

What carries the argument

The empowerment objective, which quantifies and maximizes mutual information between an agent's actions and future states.

Load-bearing premise

Optimizing the empowerment objective will automatically produce representations that stay unchanged by any observation features the agent cannot influence.

What would settle it

Train an empowerment agent in an environment containing an irrelevant observation dimension that does not affect reachable states, then check whether the learned representations still encode that dimension.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Representations will discard observation dimensions that cannot be affected by the agent's actions.
  • Both forward and backward views will focus exclusively on controllable state aspects.
  • Unsupervised skill learning will automatically acquire invariance properties useful for downstream control.
  • Learning through interaction will outperform passive observation for discovering control-relevant structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same invariance might appear in other control objectives such as mutual information between actions and rewards.
  • Environments with many irrelevant visual features could serve as testbeds to measure how cleanly the learned representations separate controllable from uncontrollable elements.
  • The two complementary representations could be combined explicitly to improve planning or exploration algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The manuscript argues that maximizing the empowerment objective in reinforcement learning induces two distinct representations (forward and backward) that capture complementary aspects of the state and are both invariant to control-irrelevant features, thereby yielding an implicit control-centric world model learned through interaction rather than passive observation.

Significance. If the invariance properties are shown to follow from the empowerment objective under stated conditions, the work would supply a theoretical account for why empowerment yields useful representations in unsupervised skill learning and would strengthen connections to causal representation learning by emphasizing active control.

major comments (1)
  1. [Theoretical analysis (main result on forward/backward representations)] The central claim that both forward and backward representations are invariant to control-irrelevant features lacks explicit conditions on the environment (e.g., Markovian dynamics, stationarity, or absence of non-stationary distractors) or on the optimization procedure under which the mutual-information objective automatically discards irrelevant dimensions. Without these, it is unclear whether the invariance holds generally or only in specially constructed cases.
minor comments (1)
  1. Notation for the forward and backward encoders should be introduced with explicit definitions before the invariance statements are made.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on the theoretical analysis. We address the major comment below and will revise the manuscript to improve clarity.

read point-by-point responses

  1. Referee: The central claim that both forward and backward representations are invariant to control-irrelevant features lacks explicit conditions on the environment (e.g., Markovian dynamics, stationarity, or absence of non-stationary distractors) or on the optimization procedure under which the mutual-information objective automatically discards irrelevant dimensions. Without these, it is unclear whether the invariance holds generally or only in specially constructed cases.

    Authors: We agree that the conditions should be stated explicitly. The analysis in the manuscript is developed under the standard assumptions of a Markov decision process with stationary transition dynamics; the empowerment objective is the mutual information between a sequence of actions and the resulting future states, optimized over policies. Under these conditions, dimensions that do not affect the controllable future states contribute zero to the mutual information and are therefore discarded at optimality. In the revision we will add a dedicated assumptions paragraph immediately preceding the main theorem that lists these conditions (Markovian stationary dynamics, global optimization of the mutual-information objective) and briefly notes that the invariance result does not extend to non-stationary distractors that alter the controllable dynamics. This makes the scope of the claim precise while preserving the original proof strategy. revision: yes

Circularity Check

0 steps flagged

No circularity; abstract-level claims lack equations or derivations to inspect

full rationale

The provided manuscript text consists solely of an abstract and high-level description with no equations, theorems, or explicit derivation steps. The central claim that empowerment induces forward/backward representations invariant to control-irrelevant features is stated conceptually without any mathematical reduction, fitted parameters, or self-citation chains that could be checked for equivalence to inputs by construction. No load-bearing steps exist to evaluate under the enumerated circularity patterns, so the result is self-contained at the level of stated motivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No full manuscript text is available, so specific free parameters, axioms, or invented entities cannot be identified from the abstract alone.

pith-pipeline@v0.9.1-grok · 5666 in / 963 out tokens · 26658 ms · 2026-06-29T08:08:32.730977+00:00 · methodology

0 comments
read the original abstract

In many practical reinforcement learning environments, observations are far higher-dimensional than the variables that matter for control. In this work, we ask: can we learn representations that capture only control-relevant features of the environment? We study this question through the empowerment objective, which maximizes an agent's influence over the environment and is widely used for unsupervised skill learning. We show that empowerment agents induce two distinct representations -- forward and backward -- that capture complementary aspects of the state, and both of which are invariant to control-irrelevant features. Thus, empowerment maximization leads agents to learn an implicit, control-centric model of the world. Our analysis highlights the importance of learning representations through interaction rather than from passive datasets: interaction aimed at maximizing control is essential for learning useful invariance properties, a perspective that aligns closely with the causal learning literature.

Figures

Figures reproduced from arXiv: 2605.30656 by Benjamin Eysenbach, Mahsa Bastankhah, Sophie Broderick.

Figure 1
Figure 1. Figure 1: Forward and backward representations are asymmetric s + and sˆ + satisfy the following condition: for every initial state s0, either (i) Both s + and sˆ + are unreachable from s0, or (ii) There exists a constant α(s0) > 0 such that d π γ (s + | s0) = α(s0) d π γ (ˆs + | s0) ∀ π ∈ Π. Then, under the minimal representation assump￾tion, the MISL backward representations alias these states: ψ(s +) = ψ(ˆs +). I… view at source ↗
Figure 2
Figure 2. Figure 2: y is controllable; w is uncontrollable but control-relevant; e is uncontrollable and control-irrelevant. If the dashed line exists, y and e share a confounding parent. Minimal empowerment-based representations and policies ignore e. composed of policies that are invariant to the control￾irrelevant feature e: p ∗ (z | s0) > 0 =⇒ π(a | (x, e), z) = π(a | (x, eˆ), z) ∀ x, e, e, a, a. ˆ Proof in Appendix E. Pr… view at source ↗
Figure 4
Figure 4. Figure 4: MISL representations are noise-invariant and useful for downstream task across environments. noise, the learned representations are not noise-invariant. In contrast, MISL requires no pre-collected data. Furthermore, the appendix includes additional experiments on invariance under action relabeling H.2. Experimental setup. We evaluate invariance to control￾irrelevant features in the point-Maze environment b… view at source ↗
Figure 6
Figure 6. Figure 6: MISL representations are invariant to noise distributions unseen during training. RQ4. Forward and backward representation asymmetry To illustrate the asymmetry between forward and backward representations, we consider the simple MDP shown in Fig￾environments and PPO for the Ant environment. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Variance of the representations to the noise relative to the state dimensions (lower is better). Even as the network sizes increase, the representations remain noise-invariant without requiring explicit regularization. emerges empirically. To investigate whether this effect is due to limited network size acting as an implicit regularizer, we scale each component independently: increasing the pol￾icy to a 5… view at source ↗
Figure 7
Figure 7. Figure 7: Linear probes on MISL representations can accurately recover the control-relevant features (x, y) in PointMaze. RQ5. Do the networks require regularization for noise￾invariance? Theorem 5.4 requires information bottleneck regularization to guarantee noise-invariant policies and rep￾resentations. However, in practice, we use the implementa￾tion of (Park et al., 2024), which applies no explicit regu￾larizati… view at source ↗
Figure 9
Figure 9. Figure 9: Graphical models illustrating the effect of policy invariance on conditional independence. T is sampled from Geom(1 − γ) Therefore, we can simplify the mutual information using the chain rule: I(S +;Z | s0) = I(X+, E+;Z | x0, e0) =✘✘✘✘✘✘✘✘✿0 I(E +;Z | x0, e0) + I(X+;Z | E +, x0, e0) (10) In general, from writing both sides of the chain rule we have: I(X+, E+;Z | x0, e0) =✘✘✘✘✘✘✘✘✿0 I(E +;Z | x0, e0) + I(X+… view at source ↗
Figure 10
Figure 10. Figure 10: Graphical model in the presence of a w that is a confounding factor for both e and y. Purple and gray shades are, respectively, primary and secondary shades in the Bayesian ball algorithm for determining d-separation. G.1. Extending the proof of Theorem 5.4 We start by using the chain rule to rewrite the MI objective. I(S +;Z | s0) = I(Y +, W+, E+;Z | y0, w0, e0) = ✘✘✘✘✘✘✘✘✘✘✿0 I(E +;Z | y0, w0, e0) + ✘✘✘… view at source ↗
Figure 11
Figure 11. Figure 11: States with similar future state distribution but different action scaling have the same representation. follows: RelNMSE = Ex,y∥ϕ(x, y, 0) − ϕ(x, y, 1)∥ 2 Ex,y∥ϕ(x, y, 0)∥ 2 + Ex,yϕ(x, y, 1)∥ 2 Lower values of RelNMSE shows that the corresponding states have a similar representations. As indicated by [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: AC-state representations noise invariance (Lower better). This method are unable of learning noise-invariant representations if the data collection is not noise invariant. environment augmented with one Gaussian noise dimension. We collect datasets using policies with different levels of noise dependence, and then measure the ratio of representation variance along the noise dimension to the variance along… view at source ↗
Figure 13
Figure 13. Figure 13: Bisimulation representation variance to noise and state dims. (lower better , higher better , higher better). This result is surprising, since maximal bisimulation is often interpreted as enforcing noise invariance. However, in practice, methods such as Zhang et al. (2020b) do not guarantee the maximal bisimulation relation. Instead, capturing temporally correlated noise arises as a byproduct of using a p… view at source ↗
Figure 14
Figure 14. Figure 14: The variance of the mean of the transition model to noise and state dims. (lower better , higher better , higher better). J. Additional figures We evaluate whether MISL representations trained with 5-dimensional i.i.d. Gaussian noise remain invariant when tested on Gaussian noise with cross-dimensional correlations in the Point-Maze environment [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: MISL representations trained with i.i.d. noise dimensions are also invariant to correlated noise dimensions. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗

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Reference graph

Works this paper leans on

5 extracted references · 2 canonical work pages · 1 internal anchor

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    Step 1We first show that for any distribution over skills p(z|s 0)∈∆ |Z|−1 that assigns nonzero probability to policies that take actions based on e, we can construct a new distribution pinv(z|s 0)∈∆ |Zinv|−1 whose support consists only ofe-invariant policies (which may be nonstationary inx), without decreasing the mutual information. Formally, Ip(X+;Z|E ...

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    Step 2In the second step of the proof, we show that restricting the skills further to non-stationary policies in x does not further increase the mutual information. In particular, Ipinv(X+;Z|x 0)≤max p(z|s0)∈∆|Zinv|−1 I(X +;Z|x 0) = max p(z|s0)∈∆|Zstat |−1 I(X +;Z|x 0),(13) where Zstat denotes the set of skills corresponding to e-invariant, Markovian, and...

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