Successor Representations

Time is of the essence when dealing with multiple reward signals and non-linear utility. In this paper we argue that the current main approaches in multi-objective RL (SER and ESR), and successor features, are insufficient. While each approach deals with non-linear effects on user utility on different timescales, none of them take into account that different effects happening on different timescales can happen within the same decision problem. We motivate that this can indeed be the case by an example, both intuitively and numerically, leading to a new perspective, and a significant and non-trivial gap in the literature.

Many reinforcement learning systems, from fleet management to traffic signal control, must serve an objective that changes dynamically after deployment, and retraining a policy for each new objective is prohibitively expensive. For a single agent, this problem is well understood: successor features with generalized policy improvement, together with their universal extension, recombine a library of learned policies into a policy for any new objective, with a guarantee that the result is never worse than any policy in the library. However, multi-agent transfer has received far less attention, and the common practice of letting each agent recombine its own library independently inherits the recipe but not the guarantee. We prove that this independent composition can produce joint behavior strictly worse than every policy in the library, because recombining teammates changes the environment each agent faces and invalidates the values it relies on, a failure with no single-agent counterpart. We further show that the only unconditionally safe fixed rule is synchronized composition, which moves the whole team to one jointly trained policy but cannot serve objectives that assign different goals to different agents. To attain safety and flexibility at once, we propose MA-USFA, a hierarchical method with two layers: a lower layer of universal successor feature approximators that predicts each agent's successor features while conditioned on its teammates' objectives, and an upper composer that selects, across agents, which library entry each agent should follow and supplies the cross-agent correction a per-agent value cannot represent. Trained once over the distribution of objectives, it is applied at deployment with no per-task adaptation.

Contemporary model-free reinforcement learning algorithms can achieve very high performance, but have low sample efficiency and are not robust to changes in the environment. Model-based algorithms have much higher sample efficiency, but still fail when the environment shifts. This paper introduces Adaptive Topological Learning with Abstract Successors (ATLAS) to combat these challenges. ATLAS uses a Grow When Required network with Successor Features in order to achieve high sample efficiency while also robustly tackling catastrophic forgetting. We evaluate ATLAS in spatial navigation tasks, benchmarking its performance against common on-policy and off-policy algorithms. Our empirical results demonstrate that by structurally decoupling transition dynamics from the reward signal, ATLAS achieves near-instantaneous adaptation to new goals and can exhibit positive backward transfer, significantly outperforming baseline methods in non-stationary environments.

Real-world Reinforcement Learning depends on the ability to formulate safety constraints into a policy. A common way to model such constraints is to introduce an additional cost signal in the Markov Decision Process, which notifies the agent of unwanted behavior independently of the reward signal. Unfortunately, current methods are hard to adapt to changes in the cost function introduced by, e.g., domain shift or obstacles moving over time. The lack of adaptability means that policies are too unflexible to deal with complex real-world conditions. We propose the Safe Deep Successor Representation (SafeDSR), a novel method that allows quick retraining of policies towards new cost structures. SafeDSR extends the Deep Successor Representation (Kulkarni et al., 2016) to Constrained Reinforcement Learning by introducing a single learnable weight matrix to decouple the learned value function across dynamics, rewards, and costs. This matrix can be updated in a supervised manner instead of having to adapt the whole network if the cost structure of the environment changes. We demonstrate this ability in a freely configurable two-dimensional navigation environment and show that our method is competitive on a simple navigation task while being considerably more flexible

Safety alignment in LLMs does not improve monotonically across model generations. Studying four generations of Google's Gemma family (7B-31B) with quality-diversity evolution (MAP-Elites) as an automated red-teaming probe, we find that Gemma 3 (12B) exhibits 68.7% +/- 5.7% attack success rate (ASR; mean +/- std, 3 seeds), significantly higher than its predecessor Gemma 2 (45.5% +/- 7.2%; p = 0.030, paired bootstrap) and its successor Gemma 4 (33.9% +/- 1.8%). Replaying evolved attack archives across generations reveals that attacks from other generations transfer to Gemma 3 at 44-46% but only 14-18% to Gemma 4, indicating that Gemma 4's safety gains generalize beyond the attack distributions evolved against earlier generations. Under our 8B judge, copyright and cybercrime vulnerabilities register at near-100% across all generations, though a second-judge audit (Section 6) suggests the copyright result is sensitive to judge choice. Misinformation ASR jumps from 29% to 99% between Gemma 2 and Gemma 3 and remains elevated at 77% in Gemma 4, indicating the regression was not fully addressed. These patterns are invisible to static benchmarks and emerge only through adaptive, longitudinal probing. All experiments use 3 random seeds with a unified self-hosted judge; code and artifacts are available at https://github.com/bassrehab/red-queen.

Representation learning is a powerful tool for spatio-temporal abstraction within reinforcement learning (RL). Two well established approaches are through the successor representation (SR) and the default representation (DR). The SR encodes states by the future trajectories they induce, capturing information flow decoupled from reward. The DR builds on this by weighting trajectories with reward, integrating credit-assignment structure into the representation. Eigenvectors of both representations have been used to support a range of downstream tasks -- including option discovery, reward shaping, transfer learning, and exploration. We introduce a structurally distinct formulation: the terminal representation (TR). The TR encodes reward-weighted trajectories similarly to the DR, but can be learned as a lower-dimensionality object, and can be used directly for the mentioned applications without eigenvector computations. Eigendecomposition also imposes the assumption of symmetric transition dynamics, which the TR can bypass. In this work we develop the theoretical foundations of the TR: its derivation, convergence of two learning algorithms, its use for zero-shot compositionality, and equivalences between alternative reward formulations. We further show the TR is embedded in the top DR eigenvector, allowing it to capture the same underlying knowledge without eigendecomposition. Additionally, we provide empirical evidence of the TR as a viable alternative to existing representations in subsidiary applications, while requiring less computational overhead to learn, store, and use.

A hallmark of intelligence is the ability to adapt in non-stationary environments, yet deep Reinforcement Learning (RL) agents often struggle in such settings. Prior studies introduce non-stationarity through abrupt shifts in features or dynamics, whereas real-world environments often evolve gradually through continual drift. This distinction has important implications for the "stability-plasticity dilemma" in RL, as abrupt task changes may demand more plasticity than naturalistic settings. To address this, we modify existing 3D Miniworld and MuJoCo environments to incorporate naturalistic, continual non-stationarity, and use them to examine how stability and adaptation affect performance under continuous environmental change. We find that methods favoring stability, such as synaptic consolidation, outperform approaches focused on plasticity, such as parameters resetting. Motivated by this result, and prior evidence that Successor Features (SFs) reduce interference, we investigate whether SFs are better consolidation targets than Q-values. Across both environments, applying neuro-inspired synaptic consolidation to SFs yields superior performance on continually changing settings. Moreover, consolidation is most effective when SFs are stabilized across multiple timescales, which capture complementary aspects of gradual environmental change. Together, these results suggest that stability is more critical in continual learning when changes are gradual, and that multi-timescale consolidation of predictive representations is an effective approach.

Language models are typically trained to predict the next token in a sequence. Here, we explore an alternative predictive principle from reinforcement learning: Successor Representations (SRs), which model the expected discounted distribution of future states rather than the immediate next state. We transfer this framework to natural language and train neural networks to predict future word distributions across multiple temporal horizons, thereby learning representations of long-range transition structure. We train a deep residual neural network on WikiText-103 (103 million tokens; 20,000-word vocabulary) and optimize successor representations as probability distributions using KL divergence. Without explicit linguistic supervision, structured language representations emerge spontaneously. After training, the learned space develops a clear geometric organization with respect to part-of-speech (POS) categories: nouns, verbs, and adjectives become separable and recoverable through unsupervised clustering. This organization depends systematically on predictive horizon, with short horizons producing the strongest syntactic structure and longer horizons increasingly integrating broader contextual and semantic information. At finer resolutions, additional interpretable lexical substructure emerges, revealing coherent subclasses within major word categories. These findings suggest that syntactic categories need not be explicitly encoded but may arise as a consequence of predictive sequence learning. To our knowledge, this work provides the first systematic application of successor representations to natural language and establishes a conceptual bridge between reinforcement learning, linguistics, and cognitive neuroscience.

Hierarchical reinforcement learning can improve generalization by decomposing long-horizon decision-making into simpler subproblems. However, existing approaches often rely on restrictive design choices, such as fixed temporal abstractions or goal-conditioned objectives, which largely confine them to goal-reaching tasks and limit their applicability to general reward functions. In this paper, we introduce switching successor measures, an extension of successor measures that enables hierarchical control in zero-shot reinforcement learning without additional supervision, fixed horizons, or manually designed subgoals. We show that switching successor measures arise naturally from classical successor measures while preserving their underlying structure. Building on this result, we propose FB $π$-Switch, an algorithm that extracts both a high-level subgoal-selection policy and a low-level control policy directly from forward-backward (FB) representations, allowing hierarchical behavior to emerge from a single learned representation. Experiments on both goal-conditioned and general reward-based tasks show that FB $π$-Switch improves over non-hierarchical baselines and matches state-of-the-art hierarchical methods in goal-conditioned settings. These results demonstrate that structured successor representations provide a flexible foundation for hierarchical zero-shot reinforcement learning beyond goal-reaching tasks. Our project website is available at: https://stestokth.github.io/switching-successors/.

Practitioners deploying multi-agent large language model (LLM) systems must currently choose between communication topologies such as chain, star, mesh, and richer variants without any pre-inference diagnostic for which topology will amplify drift, converge to consensus, or remain robust under perturbation. Existing evaluation answers these questions only post hoc and only for the task measured. We introduce a structural diagnostic for multi-agent LLM communication graphs based on the successor representation $M = (I - γP)^{-1}$ of the row-stochastic communication operator, and we connect three of its spectral quantities, the spectral radius $ρ(M)$, the spectral gap $Δ(M)$, and the condition number $κ(M)$, to three distinct failure modes. We derive closed-form spectra for the chain, star, and mesh under row-stochastic normalization, and validate the predictions on a 12-step structured state-tracking task with Qwen2.5-7B-Instruct over 100 independent trials. The condition number is a perfect rank-order predictor of empirical perturbation robustness ($r_s = 1.0$); the spectral gap partially predicts consensus dynamics ($r_s = 0.5$); and the spectral radius is perfectly \emph{inverted} with respect to cumulative error ($r_s = -1.0$). We trace this inversion to a regime in which linear spectra are blind to non-contracting bias drift, and we propose an affine-noise extension of the predictive map that recovers the empirical ordering. We read this as a first step toward representational, drift-aware structural diagnostics for multi-agent LLM systems, sitting alongside classical spectral and consensus theory.

The challenge of engineering autonomous agents capable of navigating the stochastic and adversarial nature of the physical world has historically resided at the intersection of symbolic logic and control theory. Traditional multi-agent system (MAS) frameworks have relied heavily on monolithic logical models -- primarily variations of the event calculus and situation calculus -- to represent action, change, and temporal persistence. While these classical systems provide robust solutions to the frame problem through mechanisms like circumscription and successor state axioms, they are inherently limited by a closed-world assumption that fails in the face of unobserved agent interventions, plan interruptions, and divergent belief-reality states. The paradigm of Sheaf-Theoretic Planning (STP) emerges as a transformative alternative, grounding the problem of multi-agent coordination under the mathematical structures of topos theory and sheaf semantics. This report provides an exhaustive analysis, justification, and extension of the STP framework, exploring its categorical foundations, implementation feasibility, and role in the future of resilient autonomous systems.

Recent advances in large language models (LLMs) have shown promise in formal theorem proving, yet evaluating semantic correctness remains challenging. Existing evaluations rely on indirect proxies such as lexical overlap with human-annotated proof, or expensive manual inspection. Inspired by the shift from lexical comparison to test-based evaluation in code generation, we propose T , a framework that evaluates the semantic correctness of formal theorems: a generated theorem is considered correct only if all dependent successor theorems compile successfully, analogous to integration testing. We construct a benchmark from 5 real-world Lean 4 repositories, comprising 2,206 problems paired with 41 successor theorems on average, automatically extracted without human effort. Experiments demonstrate that while state-of-the-art models achieve high compilation success, they perform significantly worse under our semantic metric. The best model, Claude-Sonnet-4.5, achieves only 38.9% Testing Accuracy on the full set, given both natural language proof and successor theorems as context, revealing a critical gap in current theorem generation capabilities.

Active inference, a neurally-inspired model for inferring actions based on the free energy principle (FEP), has been proposed as a unifying framework for understanding perception, action, and learning in the brain. Active inference has previously been used to model ecologically important tasks such as navigation and planning, but scaling it to solve complex large-scale problems in real-world environments has remained a challenge. Inspired by the existence of multi-scale hierarchical representations in the brain, we propose a model for planning of actions based on hierarchical active inference. Our approach combines a hierarchical model of the environment with successor representations for efficient planning. We present results demonstrating (1) how lower-level successor representations can be used to learn higher-level abstract states, (2) how planning based on active inference at the lower-level can be used to bootstrap and learn higher-level abstract actions, and (3) how these learned higher-level abstract states and actions can facilitate efficient planning. We illustrate the performance of the approach on several planning and reinforcement learning (RL) problems including a variant of the well-known four rooms task, a key-based navigation task, a partially observable planning problem, the Mountain Car problem, and PointMaze, a family of navigation tasks with continuous state and action spaces. Our results represent, to our knowledge, the first application of learned hierarchical state and action abstractions to active inference in FEP-based theories of brain function.

Heuristics are a central component of deterministic planning, particularly in domain-independent settings where general applicability is prioritized over task-specific tuning. This work revisits that paradigm in light of recent advances in large language models (LLMs), which enable the automatic synthesis of heuristics directly from problem definitions -- bypassing the need for handcrafted domain knowledge. We present a method that employs LLMs to generate problem-specific heuristic functions from planning tasks specified through successor generators, goal tests, and initial states written in a general-purpose programming language. These heuristics are compiled and integrated into standard heuristic search algorithms, such as greedy best-first search. Our approach achieves competitive, and in many cases state-of-the-art, performance across a broad range of established planning benchmarks. Moreover, it enables the solution of problems that are difficult to express in traditional formalisms, including those with complex numeric constraints or custom transition dynamics. We provide an extensive empirical evaluation that characterizes the strengths and limitations of the approach across diverse planning settings, demonstrating its effectiveness.