arxiv: 2604.17551 · v1 · submitted 2026-04-19 · 💻 cs.LG · cs.AI

Recognition: unknown

SVL: Goal-Conditioned Reinforcement Learning as Survival Learning

Fabian Schramm, Franki Nguimatsia Tiofack, Justin Carpentier, Th\'eotime Le Hellard

Pith reviewed 2026-05-10 06:41 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords goal-conditioned reinforcement learningsurvival analysishazard modelsvalue function estimationoffline reinforcement learninglong-horizon tasksdistributional reinforcement learning

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The pith

The goal-conditioned value function equals a discounted sum of survival probabilities over time.

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

Standard goal-conditioned reinforcement learning relies on temporal-difference methods whose bootstrapping can produce unstable estimates, especially over long horizons. This paper reframes the problem as survival analysis by treating the time to reach the goal as a random variable drawn from a probability distribution. It derives a closed-form identity showing that the value function is exactly the expected discounted sum of the probabilities that the goal has not yet been reached at each future step. These survival probabilities are learned by fitting a hazard model with maximum likelihood, using both trajectories that reach the goal and right-censored ones that do not. The resulting estimators are evaluated on offline benchmarks where they match or exceed strong baselines when paired with hierarchical actors.

Core claim

By modeling time-to-goal from each state as a probability distribution, the goal-conditioned value function admits an exact expression as a discounted sum of survival probabilities. This identity converts value estimation into maximum-likelihood training of a hazard model on observed times to goal, including right-censored trajectories. Three practical estimators—finite-horizon truncation and two binned infinite-horizon approximations—are introduced to implement the identity, and experiments show that policies derived from them perform competitively with temporal-difference and Monte Carlo baselines on long-horizon offline goal-conditioned tasks.

What carries the argument

the closed-form identity that rewrites the goal-conditioned value function as a discounted sum of survival probabilities obtained from a hazard model on time-to-goal

If this is right

Value estimation reduces to a supervised maximum-likelihood problem on time-to-goal data rather than a bootstrapped temporal-difference problem.
Both successful goal-reaching trajectories and right-censored trajectories contribute directly to training the value estimator.
Finite-horizon truncation together with binned infinite-horizon approximations allow the identity to be applied to tasks with long or unbounded horizons.
When the hazard model is combined with hierarchical actors, the resulting policies match or exceed those of temporal-difference and Monte Carlo baselines on offline goal-conditioned benchmarks.

Where Pith is reading between the lines

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

The survival framing may transfer to other sparse-reward or event-based reinforcement learning problems where the key signal is time until a terminal event.
Standard survival-analysis tools for handling censoring and competing risks could be imported to improve data efficiency in broader offline reinforcement learning settings.
Direct comparison of the binned approximations against exact long-horizon rollouts on continuous-control domains would quantify any bias introduced by discretization.

Load-bearing premise

Modeling time to goal as a probability distribution via a hazard function will produce stable value estimates that faithfully capture the objective without the instability of temporal-difference bootstrapping.

What would settle it

Compute exact Monte Carlo estimates of the goal-conditioned values on a test set of trajectories and compare them directly to the hazard-model estimates; systematic large discrepancies would show that the identity or the practical estimators do not hold.

Figures

Figures reproduced from arXiv: 2604.17551 by Fabian Schramm, Franki Nguimatsia Tiofack, Justin Carpentier, Th\'eotime Le Hellard.

**Figure 1.** Figure 1: Illustration of time-to-goal distribution models in GCRL. In the general GCRL setting, the time required to reach a goal is a random variable. As illustrated in the navigation task (left), an agent may reach the target (⋆) from the start (♦) via multiple distinct paths, resulting in a multi-modal distribution of arrival times (right) with modes at t ≈ 5, 9, 13. Traditional TD learning approaches estimate t… view at source ↗

**Figure 2.** Figure 2: Comparative analysis of the three estimators. Success rates (%) on AntMaze and HumanoidMaze navigation tasks at increasing scales (medium/large/giant), comparing finite-horizon (no bins), hazard-binned (PCH), and survival-binned (PCS). The three variants perform comparably across all scales. Full numerical results are provided Tab. 4. The visualization scheme is adapted from (Ahn et al., 2025). infinite-ho… view at source ↗

**Figure 3.** Figure 3: Network depth ablation study. Success rates (%) when varying actor depth and critic depth. This indicates that the distributional, survival-based critic alone provides a stronger learning signal than contrastive regression in an identical flat setup. Adding hierarchical extraction (HSVL) compounds this gain on the giant mazes, while not necessary for medium and large sizes. 5.3. Architectural ablation stud… view at source ↗

**Figure 4.** Figure 4: Hazard function architecture. From a given tuple of state s and goal g, the network predicts time-togoal behavior by combining a shared encoder with three heads: an immediate-hit predictor, coefficients describing temporal evolution, and weights that select among learned basis patterns. These components are combined to produce hazard predictions over time. The associated survival function is S π (t|s, a, … view at source ↗

read the original abstract

Standard approaches to goal-conditioned reinforcement learning (GCRL) that rely on temporal-difference learning can be unstable and sample-inefficient due to bootstrapping. While recent work has explored contrastive and supervised formulations to improve stability, we present a probabilistic alternative, called survival value learning (SVL), that reframes GCRL as a survival learning problem by modeling the time-to-goal from each state as a probability distribution. This structured distributional Monte Carlo perspective yields a closed-form identity that expresses the goal-conditioned value function as a discounted sum of survival probabilities, enabling value estimation via a hazard model trained via maximum likelihood on both event and right-censored trajectories. We introduce three practical value estimators, including finite-horizon truncation and two binned infinite-horizon approximations to capture long-horizon objectives. Experiments on offline GCRL benchmarks show that SVL combined with hierarchical actors matches or surpasses strong hierarchical TD and Monte Carlo baselines, excelling on complex, long-horizon tasks.

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.

Desk Editor's Note private letter to a colleague

SVL reframes GCRL as survival learning with a closed-form value identity, but the binned estimators risk bias on long horizons where the method claims its biggest edge.

read the letter

The main thing to know is that this paper treats goal reaching as a survival process and derives a closed-form identity linking the goal-conditioned value to a discounted sum of survival probabilities. That identity comes from a distributional Monte Carlo perspective and lets them estimate values with a hazard model trained by maximum likelihood on both successful and right-censored trajectories. If the identity is exact, it sidesteps the bootstrapping instability that plagues TD methods in long-horizon GCRL.

Referee Report

2 major / 2 minor

Summary. The paper proposes Survival Value Learning (SVL) for goal-conditioned reinforcement learning (GCRL). It reframes GCRL as a survival analysis problem by modeling time-to-goal from each state as a probability distribution via a hazard function. This yields a closed-form identity expressing the goal-conditioned value function as a discounted sum of survival probabilities, which is estimated via maximum likelihood training of the hazard model on both event and right-censored offline trajectories. Three practical estimators are introduced (finite-horizon truncation and two binned infinite-horizon approximations), and experiments on offline GCRL benchmarks show SVL with hierarchical actors matches or surpasses strong hierarchical TD and Monte Carlo baselines, with advantages on long-horizon tasks.

Significance. If the identity holds and the estimators accurately recover the value function, SVL offers a stable, bootstrapping-free alternative to temporal-difference methods in GCRL by drawing on survival analysis and distributional Monte Carlo ideas. The handling of right-censored trajectories via MLE is a concrete strength, and the approach could improve reliability in long-horizon goal-conditioned settings. Credit is due for the structured probabilistic reformulation and the explicit estimators that extend to infinite horizons.

major comments (2)

[§3 (Practical Value Estimators)] §3 (Practical Value Estimators) and the associated equations for the binned approximations: The manuscript claims the two binned infinite-horizon approximations faithfully capture long-horizon objectives. However, when the time-to-goal tail is heavy or the hazard varies within bins, replacing the infinite sum with a coarser discretization introduces non-negligible bias relative to the true E[γ^T]. This is load-bearing for the superiority claims on complex long-horizon tasks, and the paper provides neither an error bound on the discretization nor empirical checks that the bias remains smaller than the variance of TD baselines.
[§2 (Closed-form Identity)] §2 (Closed-form Identity): The central identity V(s) = 1 - (1-γ) ∑_{t=0}^∞ γ^t S(t|s) is presented as following directly from the distributional Monte Carlo perspective. While the identity itself may be parameter-free, the hazard model is fit by MLE to finite offline data, so extrapolation error in the tail (beyond observed censoring) is uncontrolled; the manuscript lacks a quantitative analysis of how this propagates to value estimates on the long-horizon benchmarks where SVL is claimed to excel.

minor comments (2)

The notation for the survival function S(t|s) and its relation to the learned hazard should be stated explicitly once, including how right-censoring is incorporated in the likelihood.
[Experiments] Table or figure reporting the benchmark results should include standard deviations across seeds and a clear statement of which tasks are long-horizon to allow direct assessment of the claimed advantages.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the strengths of the probabilistic reformulation and censored-data handling in SVL. We address the two major comments point by point below, acknowledging where additional analysis would strengthen the claims, and describe the revisions we will incorporate.

read point-by-point responses

Referee: [§3 (Practical Value Estimators)] §3 (Practical Value Estimators) and the associated equations for the binned approximations: The manuscript claims the two binned infinite-horizon approximations faithfully capture long-horizon objectives. However, when the time-to-goal tail is heavy or the hazard varies within bins, replacing the infinite sum with a coarser discretization introduces non-negligible bias relative to the true E[γ^T]. This is load-bearing for the superiority claims on complex long-horizon tasks, and the paper provides neither an error bound on the discretization nor empirical checks that the bias remains smaller than the variance of TD baselines.

Authors: We agree that binning can introduce discretization bias when tails are heavy or hazards vary within bins. The manuscript selects bin widths from the empirical time-to-goal distribution in the offline dataset to keep this bias small relative to the stability gains, and the reported outperformance on long-horizon tasks provides indirect support. To directly address the concern, we will add an appendix containing an empirical comparison of the binned estimators against a fine-grained Monte Carlo approximation of the infinite-horizon sum on the same benchmarks; this will quantify the discretization error and confirm it remains smaller than the performance margin over TD baselines. We will also expand the main text to discuss the bias-variance tradeoff guiding bin selection. revision: partial
Referee: [§2 (Closed-form Identity)] §2 (Closed-form Identity): The central identity V(s) = 1 - (1-γ) ∑_{t=0}^∞ γ^t S(t|s) is presented as following directly from the distributional Monte Carlo perspective. While the identity itself may be parameter-free, the hazard model is fit by MLE to finite offline data, so extrapolation error in the tail (beyond observed censoring) is uncontrolled; the manuscript lacks a quantitative analysis of how this propagates to value estimates on the long-horizon benchmarks where SVL is claimed to excel.

Authors: The identity is exact and parameter-free; it follows directly from the definition of the survival function and the geometric discounting without reference to any particular hazard model. Maximum-likelihood estimation on right-censored trajectories is a standard, consistent procedure in survival analysis that permits tail extrapolation under the model's functional assumptions. While the current manuscript does not contain a dedicated propagation analysis, the empirical results on long-horizon tasks (where tail behavior is critical) indicate that any extrapolation error does not negate the observed advantages. In revision we will add a quantitative sensitivity study that varies the degree of censoring in the training data and measures the resulting change in value estimates, thereby bounding the practical impact of tail extrapolation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; core identity is a standard mathematical re-expression independent of fitted parameters

full rationale

The paper derives a closed-form identity V(s) = 1 - (1-γ) ∑ γ^t S(t|s) from the distributional Monte Carlo view of goal-reaching returns. This is exactly the known relation E[γ^T] = 1 - (1-γ) ∑ γ^t P(T > t) for hitting time T, which follows directly from the definition of the discounted return under sparse goal rewards and holds without reference to any model, fit, or self-citation. The subsequent hazard-model MLE step is ordinary statistical estimation on censored data; the resulting value estimator is an approximation to this identity rather than a quantity forced to equal its own training objective. Finite-horizon truncation and binned approximations are explicitly introduced as practical compromises, not claimed to be exact by construction. No load-bearing self-citation or ansatz smuggling appears in the provided derivation chain. The approach therefore remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard MDP assumptions plus the novel modeling choice of time-to-goal as a survival random variable; the practical estimators introduce truncation and binning choices that function as free parameters.

free parameters (1)

horizon truncation length or bin sizes
The three value estimators rely on finite-horizon truncation and binned infinite-horizon approximations whose specific lengths or bin widths must be chosen.

axioms (1)

domain assumption The environment is a goal-conditioned MDP in which time-to-goal is a well-defined random variable with right-censoring possible.
Invoked when modeling trajectories as event or censored data for maximum-likelihood hazard estimation.

pith-pipeline@v0.9.0 · 5483 in / 1420 out tokens · 77387 ms · 2026-05-10T06:41:08.833872+00:00 · methodology

discussion (0)

Reference graph

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