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arxiv: 2605.06470 · v1 · submitted 2026-05-07 · 💻 cs.LG

Recognition: unknown

Hitting Time Isomorphism for Multi-Stage Planning with Foundation Policies

Magnus Victor Boock , Abdullah Akg\"ul , Mustafa Mert \c{C}elikok , Melih Kandemir
Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:41 UTC · model grok-4.3

classification 💻 cs.LG
keywords hitting timesoffline reinforcement learningrepresentation learningmulti-stage planningfoundation policiesMarkov decision processesoperator theory
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The pith

Expected hitting times form a directed Hilbert-space geometry for multi-stage planning in offline reinforcement learning.

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

The paper develops an operator-theoretic framework that recovers the directed temporal geometry of controlled Markov processes directly from hitting time observations. It proves that under latent linear closure this geometry exists as a Hilbert-space embedding where expected hitting times act as linear functionals of latent displacements, and that the embedding is uniquely identifiable up to bounded linear isomorphism. From this theory the authors derive the Isomorphic Embedding Learning algorithm, which trains goal-agnostic foundation policies by anchoring a consistency objective with explicit hitting-time regression. The resulting asymmetric, compositional structure supports graph-based multi-stage planning on long-horizon tasks and improves performance on offline maze locomotion data.

Core claim

Under the latent linear closure assumption, the directed temporal geometry of a controlled Markov process can be recovered from hitting time observations as a Hilbert-space displacement geometry in which expected hitting times are realized as linear functionals of latent displacements. This representation is uniquely identifiable up to a bounded linear isomorphism. In finite-dimensional implementations the global hitting-time error is bounded by one-step transition error scaled by the environment's transient spectral radius, and finite-sample guarantees are supplied that account for approximation, statistical complexity, and trajectory-label mismatch.

What carries the argument

The hitting time isomorphism, which realizes expected hitting times as linear functionals on latent state displacements inside a Hilbert space.

If this is right

  • The learned geometry supports robust graph-based multi-stage planning for long-horizon navigation.
  • Foundation policies can be trained in a goal-agnostic way from offline data by combining consistency objectives with hitting-time regression.
  • Finite-sample error bounds hold that incorporate approximation, statistical, and label-mismatch sources.
  • The asymmetric structure preserves triangle inequality and directedness, unlike many prior distance-based representations.

Where Pith is reading between the lines

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

  • The bounded linear isomorphism property may simplify transfer of learned policies across environments that share similar hitting-time statistics.
  • The spectral-radius error bound suggests a practical way to choose state representations that keep planning error controllable.
  • The same operator-theoretic approach could be applied to other temporal abstractions such as return times or visit frequencies.

Load-bearing premise

The latent linear closure assumption, under which hitting time observations are guaranteed to induce a Hilbert-space displacement geometry.

What would settle it

In a small finite-state MDP with known transient spectral radius, compute the learned embeddings from hitting time data and check whether the observed global hitting-time error exceeds the one-step transition error multiplied by that spectral radius.

read the original abstract

We present a new operator-theoretic representation learning framework for offline reinforcement learning that recovers the directed temporal geometry of a controlled Markov process from hitting time observations. While prior art often produces symmetric distances or fails to satisfy the triangle inequality, our framework learns a Hilbert-space displacement geometry where expected hitting times are realized as linear functionals of latent displacements. We prove that this representation exists under latent linear closure and is uniquely identifiable up to a bounded linear isomorphism. For finite-dimensional implementations, we show that global hitting-time error is bounded by one-step transition error amplified by the environment's transient spectral radius. Furthermore, we provide finite-sample guarantees accounting for approximation, statistical complexity, and trajectory-label mismatch. Derived from this theory, we curate Isomorphic Embedding Learning (IEL) as a new goal-agnostic foundation policy learning algorithm that anchors a HILP-style consistency objective with explicit hitting-time regression to ensure that the learned geometry reflects actual decision-time progress. This asymmetric and compositional structure enables robust graph-based multi-stage planning for long-horizon navigation. Our experiments demonstrate that IEL improves the state of the art of learning foundation policy policies from offline maze locomotion data. Our code can be found on https://github.com/MagnusBoock/IEL

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes an operator-theoretic representation learning framework for offline RL that recovers directed temporal geometry of controlled Markov processes from hitting time observations. It claims to learn a Hilbert-space displacement geometry in which expected hitting times act as linear functionals of latent displacements. Under the latent linear closure assumption, the paper proves existence and unique identifiability up to a bounded linear isomorphism, derives a global error bound in which hitting-time error is controlled by one-step transition error scaled by the transient spectral radius, and supplies finite-sample guarantees that incorporate approximation, statistical, and label-mismatch terms. From this theory the authors derive the Isomorphic Embedding Learning (IEL) algorithm, which augments a HILP-style consistency objective with explicit hitting-time regression to produce goal-agnostic foundation policies suitable for graph-based multi-stage planning. Experiments on offline maze locomotion data are reported to improve upon prior foundation-policy baselines.

Significance. If the central claims hold, the work supplies a theoretically grounded route to asymmetric, compositional latent geometries that respect the directed temporal structure of decision processes, addressing a recognized limitation of symmetric distance embeddings in prior RL representation learning. The explicit use of hitting-time regression to anchor the objective, the identifiability result, the spectral-radius error bound, and the release of reproducible code constitute concrete strengths that would advance foundation-policy methods for long-horizon navigation.

major comments (2)
  1. [theoretical development and experimental section] The existence, uniqueness, and error-bound results are stated to hold under the latent linear closure assumption (abstract and the theoretical development). No diagnostic is supplied in the experimental section to verify that the embeddings learned by IEL on the maze data satisfy closure under the linear operations induced by the transition operator; without such a check the applicability of the identifiability and finite-sample guarantees to the reported results remains unconfirmed.
  2. [finite-dimensional error bound] The finite-dimensional error bound asserts that global hitting-time error is controlled by one-step transition error amplified by the transient spectral radius. The manuscript does not indicate how this radius is computed or bounded for the maze environments, nor does it report empirical values, making it impossible to assess the practical tightness of the bound or its relation to the observed performance gains.
minor comments (2)
  1. The notation distinguishing the hitting-time functional from the one-step transition operator could be introduced more explicitly to aid readers outside operator theory.
  2. [experiments] Table or figure captions for the maze experiments should state the number of independent runs and the precise statistical test used to claim improvement over baselines.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and outline the revisions we will make to strengthen the connection between the theoretical results and the experimental validation.

read point-by-point responses

  1. Referee: [theoretical development and experimental section] The existence, uniqueness, and error-bound results are stated to hold under the latent linear closure assumption (abstract and the theoretical development). No diagnostic is supplied in the experimental section to verify that the embeddings learned by IEL on the maze data satisfy closure under the linear operations induced by the transition operator; without such a check the applicability of the identifiability and finite-sample guarantees to the reported results remains unconfirmed.

    Authors: We agree that an explicit diagnostic would better substantiate the applicability of the identifiability and error-bound results to the reported experiments. The IEL objective combines a HILP-style consistency loss with hitting-time regression precisely to encourage the learned embeddings to respect latent linear closure under the transition operator. However, the original manuscript did not include a post-training verification step on the maze embeddings. In the revision we will add a diagnostic that quantifies the average deviation from linearity (e.g., the norm of the residual when applying the empirical transition operator in latent space) on held-out transitions, and we will report these values for each maze environment. revision: yes

  2. Referee: [finite-dimensional error bound] The finite-dimensional error bound asserts that global hitting-time error is controlled by one-step transition error amplified by the transient spectral radius. The manuscript does not indicate how this radius is computed or bounded for the maze environments, nor does it report empirical values, making it impossible to assess the practical tightness of the bound or its relation to the observed performance gains.

    Authors: The transient spectral radius is defined with respect to the sub-stochastic transition matrix on the transient (non-absorbing) states of each finite-state maze. For the offline datasets this quantity can be obtained by constructing the empirical transition matrix, removing the absorbing goal states, and computing the spectral radius of the resulting matrix via the power iteration method or direct eigendecomposition. We will revise the manuscript to describe this procedure explicitly and to report the numerical values obtained for each maze environment, thereby allowing readers to gauge the practical scale of the amplification factor relative to the observed performance improvements. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained under explicit assumptions with no reduction to inputs by construction.

full rationale

The paper states its core representation exists and is identifiable under the latent linear closure assumption, which is presented as a precondition rather than derived from the representation itself. The error bounds follow from standard operator theory applied to the transient spectral radius. The IEL algorithm incorporates hitting-time regression as an explicit objective term to enforce the geometry, without the regression being a renamed fit of the same quantity. No self-citations or ansatzes are invoked in a load-bearing way in the provided text. Thus, the derivation does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims depend on the latent linear closure assumption for existence and uniqueness; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption latent linear closure
    Invoked to guarantee that the representation exists and is uniquely identifiable up to bounded linear isomorphism.

pith-pipeline@v0.9.0 · 5528 in / 1179 out tokens · 47069 ms · 2026-05-08T12:41:30.225895+00:00 · methodology

discussion (0)

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