Daily Papers

Long-horizon agents continuously accumulate interaction history during task execution, yet the importance of past interactions changes as the agent state evolves. Existing context management methods largely compress history based on fixed windows, periodic schedules, or current relevance, overlooking a more fundamental question: when has a past interaction become safe to replace? Premature compression may remove information still needed for future actions, while overly conservative retention leads to substantial context overhead. To address this, we propose State Conditioned Compression (StateComp), a framework that determines when historical interactions can be safely compressed according to the current agent state. StateComp constructs KEEP and READY supervision through a two-stage annotation procedure and trains an imbalance-aware router on hidden representations from a frozen language model. A bounded state representation further reduces the cost of evaluating long histories, while adjacent READY interactions are grouped into continuous spans and replaced with compact summaries during execution. Experiments on WorkBuddyBench show that StateComp reduces total agent and summarization tokens by 52.27% while maintaining task performance, and achieves a 12.67-fold speedup in representation extraction.

The scalability of Large Language Models (LLMs) to long contexts is fundamentally constrained by the quadratic complexity of standard attention, motivating the adoption of linear attention mechanisms with sub-quadratic cost. To improve representation capacity under long contexts, recent approaches organize memory in a multi-state manner. However, existing multi-state linear attention methods rely on fixed state merging policies that cannot adapt to dynamically varying token importance, irreversibly obscuring critical tokens and causing severe error accumulation over long sequences. To address this limitation, we propose DLA, a dynamic memory modeling framework for multi-state linear attention. DLA introduces (i) Information-Aware Dynamic State Merging, which adaptively determines state boundaries based on token-level information variation, preserving high-resolution representations around semantic transitions while aggressively summarizing stable regions, and (ii) Capacity-Bounded Memory Modeling, which maintains a fixed-size, chronologically ordered state cache by selectively merging adjacent low-information states to control memory growth with minimal information loss. We pre-train DLA on two different linear attention models and evaluate on 16 datasets across three categories. Experimental results demonstrate the superiority of DLA over state-of-the-art.

Diffusion language models (DLMs) iteratively refine a sequence, allowing earlier predictions to be revised as context evolves. This rollback capability distinguishes them from irreversible autoregressive generation, but makes inference costly. Every denoising update alters the global context, forcing both prompt and response states to be recomputed even though only response tokens are revisable. Key-value (KV) caching could reduce this cost, yet conventional caching assumes immutable historical states and is therefore difficult to reconcile with rollback. In this paper, we introduce Adaptive Reuse of Cached Hidden States for Efficient Rollback (Archer), a training-free KV caching method for rollback-capable DLMs. Archer asymmetrically keeps the mutable response synchronized with the current hypothesis while reusing prompt K/V within a bounded state neighborhood. Although prompt representations also change under bidirectional attention, their token identities remain fixed; bounded reuse therefore amortizes repeated prompt computation without caching mutable response states. It also delays feedback from tentative tokens, reducing premature reinforcement of transient high-confidence errors and giving rollback more opportunity to correct them. Our analysis characterizes prompt reuse as a reversibility-aligned cache boundary, bounds its state-dependent approximation error, and gives a decoder-margin condition for preserving full-refresh decisions. Existing DLM acceleration often trades quality for speed. Archer shifts this frontier, attaining the best mean performance of 33.63% together with a 2.57x mean speedup on the main suite. Across evaluated settings, it improves Pass@1 by up to 3.05 points and reaches up to 2.95x speedup. Controlled analyses connect the quality gain to delayed prompt feedback and validate state-aware refresh. Our code is available at https://github.com/Hxnng/Archer.

State representation learning, or the ability to capture latent generative factors of an environment, is crucial for building intelligent agents that can perform a wide variety of tasks. Learning such representations without supervision from rewards is a challenging open problem. We introduce a method that learns state representations by maximizing mutual information across spatially and temporally distinct features of a neural encoder of the observations. We also introduce a new benchmark based on Atari 2600 games where we evaluate representations based on how well they capture the ground truth state variables. We believe this new framework for evaluating representation learning models will be crucial for future representation learning research. Finally, we compare our technique with other state-of-the-art generative and contrastive representation learning methods. The code associated with this work is available at https://github.com/mila-iqia/atari-representation-learning

Representation learning and exploration are among the key challenges for any deep reinforcement learning agent. In this work, we provide a singular value decomposition based method that can be used to obtain representations that preserve the underlying transition structure in the domain. Perhaps interestingly, we show that these representations also capture the relative frequency of state visitations, thereby providing an estimate for pseudo-counts for free. To scale this decomposition method to large-scale domains, we provide an algorithm that never requires building the transition matrix, can make use of deep networks, and also permits mini-batch training. Further, we draw inspiration from predictive state representations and extend our decomposition method to partially observable environments. With experiments on multi-task settings with partially observable domains, we show that the proposed method can not only learn useful representation on DM-Lab-30 environments (that have inputs involving language instructions, pixel images, and rewards, among others) but it can also be effective at hard exploration tasks in DM-Hard-8 environments.

Conventional state representations in reinforcement learning often omit critical task-related details, presenting a significant challenge for value networks in establishing accurate mappings from states to task rewards. Traditional methods typically depend on extensive sample learning to enrich state representations with task-specific information, which leads to low sample efficiency and high time costs. Recently, surging knowledgeable large language models (LLM) have provided promising substitutes for prior injection with minimal human intervention. Motivated by this, we propose LLM-Empowered State Representation (LESR), a novel approach that utilizes LLM to autonomously generate task-related state representation codes which help to enhance the continuity of network mappings and facilitate efficient training. Experimental results demonstrate LESR exhibits high sample efficiency and outperforms state-of-the-art baselines by an average of 29% in accumulated reward in Mujoco tasks and 30% in success rates in Gym-Robotics tasks.

Representations are at the core of all deep reinforcement learning (RL) methods for both Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs). Many representation learning methods and theoretical frameworks have been developed to understand what constitutes an effective representation. However, the relationships between these methods and the shared properties among them remain unclear. In this paper, we show that many of these seemingly distinct methods and frameworks for state and history abstractions are, in fact, based on a common idea of self-predictive abstraction. Furthermore, we provide theoretical insights into the widely adopted objectives and optimization, such as the stop-gradient technique, in learning self-predictive representations. These findings together yield a minimalist algorithm to learn self-predictive representations for states and histories. We validate our theories by applying our algorithm to standard MDPs, MDPs with distractors, and POMDPs with sparse rewards. These findings culminate in a set of preliminary guidelines for RL practitioners.

The ability to learn good representations of states is essential for solving large reinforcement learning problems, where exploration, generalization, and transfer are particularly challenging. The Laplacian representation is a promising approach to address these problems by inducing intrinsic rewards for temporally-extended action discovery and reward shaping, and informative state encoding. To obtain the Laplacian representation one needs to compute the eigensystem of the graph Laplacian, which is often approximated through optimization objectives compatible with deep learning approaches. These approximations, however, depend on hyperparameters that are impossible to tune efficiently, converge to arbitrary rotations of the desired eigenvectors, and are unable to accurately recover the corresponding eigenvalues. In this paper we introduce a theoretically sound objective and corresponding optimization algorithm for approximating the Laplacian representation. Our approach naturally recovers both the true eigenvectors and eigenvalues while eliminating the hyperparameter dependence of previous approximations. We provide theoretical guarantees for our method and we show that those results translate empirically into robust learning across multiple environments.

The Platonic Representation Hypothesis (PRH) holds that as models scale, representations of heterogeneous networks converge toward a shared model of reality. We propose its sequel and boundary, the Capability Convergence Hypothesis (CCH): under a fixed per-token inference budget, representational convergence does not entail capability convergence. Capability instead converges toward a class, the access-complete hybrid: any architecture holding both a compressive O(1)-state channel and a scalable verbatim-index channel. We anchor it on a witness task, the Newton's-apple problem in an infinite stream, and name three resource walls: a Shannon wall barring any o(Nb)-state architecture, a horizon wall barring any fixed window, and a circuit wall barring fixed-depth attention-only composition (conditional on TC0 != NC1). Under an explicit separability assumption a hybrid crosses all three by paying each wall's price, so capability is strictly super-additive under composition. We separate what we prove from what we conjecture: the access-completeness principle rests on information-theoretic lower bounds and pre-registered experiments, while the field-level convergence trend is an economics-motivated conjecture. We report the first pre-registered small-scale tests under criteria frozen before the data: the predicted scissors gap is measured (exact-retrieval error 0.994 vs. 0.000 once a 64-scalar state gains one global-attention layer), the state-tracking bifurcation lands at the registered boundary, and a conjunction witness shows an irreducibly two-channel solution; one prediction failed with its direction reversed and is reported as such. Representational convergence is given freely by scale; capability convergence must be purchased by access structure.

On-policy distillation (OPD) supervises the student only in output space by matching next-token probabilities. This output-only paradigm has two limits: (1) sampling variance from Monte Carlo KL estimates over large vocabularies (e.g., Qwen's ~150k tokens) persists throughout training, and (2) it treats the teacher as a black-box, discarding all intermediate hidden states after the LM head. We propose On-Policy Representation Distillation (OPRD), which lifts distillation into hidden-state space by aligning student and teacher representations across selected layers on the same rollouts, bypassing the LM head entirely. Theoretically, OPRD eliminates sampling variance and provides richer per-layer structural information. Empirically, OPRD closes the student-teacher gap on AIME 2024/2025 and AIMO, while output-space OPD baselines plateau below the teacher. OPRD also trains 1.44x faster and uses 54% less memory than top-k OPD. Code: https://github.com/ShenzhiYang2000/OPRD.

In this work, we introduce dual goal representations for goal-conditioned reinforcement learning (GCRL). A dual goal representation characterizes a state by "the set of temporal distances from all other states"; in other words, it encodes a state through its relations to every other state, measured by temporal distance. This representation provides several appealing theoretical properties. First, it depends only on the intrinsic dynamics of the environment and is invariant to the original state representation. Second, it contains provably sufficient information to recover an optimal goal-reaching policy, while being able to filter out exogenous noise. Based on this concept, we develop a practical goal representation learning method that can be combined with any existing GCRL algorithm. Through diverse experiments on the OGBench task suite, we empirically show that dual goal representations consistently improve offline goal-reaching performance across 20 state- and pixel-based tasks.

Building a world model requires a state representation useful for planning and decision-making---potentially over tasks unknown at training time. In the context of financial markets, planning and decision-making may require a model to reason about market-wide conditions, asset-specific expected returns, liquidity, volatility, and cross-asset relationships. Yet financial representation learning has largely been evaluated on individual predictive tasks, oftentimes on a single time period using comparatively narrow datasets. We address this through three primary contributions. First, we introduce Market-1T, a dataset containing nearly one trillion observations across U.S. equities from 2008 to 2025 at 1 Hz resolution. Second, we develop and implement a rigorous evaluation protocol. Third, we conduct a systematic large-scale study of financial representation learning, comparing 18 encoder-training strategies across nearly two decades of market regimes. We evaluate learned representations both by their predictive utility on common finance tasks and through probes of latent structure. We find that encoders with similar predictive performance can organize market state very differently. Collectively, we establish a foundation for training and evaluating financial market representations in support of world models such as DINO-WM, V-JEPA 2, and LeWM.

Masked diffusion language models (dLLMs) generate text by iteratively denoising masked tokens with bidirectional attention. Extending reasoning across generation chunks normally requires keeping earlier generated text in context. We ask whether a dLLM can instead continue reasoning after that text is cleared, using only a fixed-size carried state. We implement this state as a small number of register tokens: dedicated fixed-position tokens whose continuous hidden states are trained to carry reasoning progress across generation chunks. We post-train dLLMs to decode a chunk of text, clear it while preserving the register values, and continue decoding from the prompt and carried state. In our main comparisons on LLaDA and Dream, registers outperform discrete-text carry on every benchmark, with gains of up to 8.5 points on math and 19.5 points on code. Registers are especially effective for bounded code generation, where correct programs usually span several chunks. Finally, registers can be further refined with reinforcement learning on long-horizon reasoning tasks.

MDPs with low-rank transitions -- that is, the transition matrix can be factored into the product of two matrices, left and right -- is a highly representative structure that enables tractable learning. The left matrix enables expressive function approximation for value-based learning and has been studied extensively. In this work, we instead investigate sample-efficient learning with density features, i.e., the right matrix, which induce powerful models for state-occupancy distributions. This setting not only sheds light on leveraging unsupervised learning in RL, but also enables plug-in solutions for convex RL. In the offline setting, we propose an algorithm for off-policy estimation of occupancies that can handle non-exploratory data. Using this as a subroutine, we further devise an online algorithm that constructs exploratory data distributions in a level-by-level manner. As a central technical challenge, the additive error of occupancy estimation is incompatible with the multiplicative definition of data coverage. In the absence of strong assumptions like reachability, this incompatibility easily leads to exponential error blow-up, which we overcome via novel technical tools. Our results also readily extend to the representation learning setting, when the density features are unknown and must be learned from an exponentially large candidate set.

We present a representation-driven framework for reinforcement learning. By representing policies as estimates of their expected values, we leverage techniques from contextual bandits to guide exploration and exploitation. Particularly, embedding a policy network into a linear feature space allows us to reframe the exploration-exploitation problem as a representation-exploitation problem, where good policy representations enable optimal exploration. We demonstrate the effectiveness of this framework through its application to evolutionary and policy gradient-based approaches, leading to significantly improved performance compared to traditional methods. Our framework provides a new perspective on reinforcement learning, highlighting the importance of policy representation in determining optimal exploration-exploitation strategies.

Zero-shot object-goal navigation (ZSON) requires a robot to find a named object category in a building it has never entered. The prevailing approach scores frontiers with a vision-language value map: every decision is another argmax over the map as it currently stands, and the evidence behind that score is discarded the moment it is taken. Systems that place a large vision-language model inside the perception-action loop typically query it on a fixed schedule from the current view alone; a room the robot walked through minutes earlier is never reconsidered, and a failed call has no defined fallback. We turn what the robot has already seen into the object of deliberation. Our hierarchical fast-slow agent leaves the value-map controller running at every step and writes a coordinate-anchored memory as it moves: a semantic grid of room types and confirmed object instances, together with a bounded store of pose-tagged keyframes. A VLM screens each candidate detection before it is written. A deliberative layer reads this memory in a bounded reason-retrieve-act loop. It wakes on structural events the reactive layer computes, reasons first over text, and recalls a first-person view only for candidates that text alone cannot separate. Per-invocation and per-run caps bound its calls, a call-free first tier resolves the most frequent stall, and any failure returns control to the reactive controller. Our system reaches 68.75% SR on HM3D v1 val and 47.29% on MP3D val, the highest success rate among the zero-shot methods compared here. Choosing among far frontiers by argmax instead of deliberating costs 3.40 SR points in a paired comparison over all 2000 HM3D episodes (95% CI [1.70, 5.05]); deliberating over every frontier does not recover them.

Communicating state machines provide a formal foundation for distributed computation. Unfortunately, they are Turing-complete and, thus, challenging to analyse. In this paper, we classify restrictions on channels which have been proposed to work around the undecidability of verification questions. We compare half-duplex communication, existential B-boundedness, and k-synchronisability. These restrictions do not prevent the communication channels from growing arbitrarily large but still restrict the power of the model. Each restriction gives rise to a set of languages so, for every pair of restrictions, we check whether one subsumes the other or if they are incomparable. We investigate their relationship in two different contexts: first, the one of communicating state machines, and, second, the one of communication protocol specifications using high-level message sequence charts. Surprisingly, these two contexts yield different conclusions. In addition, we integrate multiparty session types, another approach to specify communication protocols, into our classification. We show that multiparty session type languages are half-duplex, existentially 1-bounded, and 1-synchronisable. To~show this result, we provide the first formal embedding of multiparty session types into high-level message sequence charts.

State Space Models (SSMs) have recently emerged as efficient alternatives to Transformer-Based Models (TBMs) for long-sequence processing, offering linear scaling and lower memory use. Yet, how contextual information flows across layers and tokens in these architectures remains understudied. We present the first unified, token- and layer-level analysis of representation propagation in SSMs and TBMs. Using centered kernel alignment, stability metrics, and probing, we characterize how representations evolve within and across layers. We find a key divergence: TBMs rapidly homogenize token representations, with diversity reemerging only in later layers, while SSMs preserve token uniqueness early but converge to homogenization deeper. Theoretical analysis and parameter randomization further reveal that oversmoothing in TBMs stems from architectural design, whereas in SSMs it arises mainly from training dynamics. These insights clarify the inductive biases of both architectures and inform future model and training designs for long-context reasoning.

Recently, recurrent models based on linear state space models (SSMs) have shown promising performance in language modeling (LM), competititve with transformers. However, there is little understanding of the in-principle abilities of such models, which could provide useful guidance to the search for better LM architectures. We present a comprehensive theoretical study of the capacity of such SSMs as it compares to that of transformers and traditional RNNs. We find that SSMs and transformers have overlapping but distinct strengths. In star-free state tracking, SSMs implement straightforward and exact solutions to problems that transformers struggle to represent exactly. They can also model bounded hierarchical structure with optimal memory even without simulating a stack. On the other hand, we identify a design choice in current SSMs that limits their expressive power. We discuss implications for SSM and LM research, and verify results empirically on a recent SSM, Mamba.

TD-learning is a foundation reinforcement learning (RL) algorithm for value prediction. Critical to the accuracy of value predictions is the quality of state representations. In this work, we consider the question: how does end-to-end TD-learning impact the representation over time? Complementary to prior work, we provide a set of analysis that sheds further light on the representation dynamics under TD-learning. We first show that when the environments are reversible, end-to-end TD-learning strictly decreases the value approximation error over time. Under further assumptions on the environments, we can connect the representation dynamics with spectral decomposition over the transition matrix. This latter finding establishes fitting multiple value functions from randomly generated rewards as a useful auxiliary task for representation learning, as we empirically validate on both tabular and Atari game suites.

Language-model systems can separate contextual computation, persistent storage, and exact execution instead of updating all capabilities through one shared parameter system. We investigate FEM-ASM, a finite-element-method-inspired organization in which independently constructed document states and deterministic executable skills contribute typed proposals to a shared language-model state. An explicit residual operator reconciles proposals attached to common interface nodes. We evaluate this organization through controlled experiments and negative results rather than claiming a physical finite-element formulation of language. An attention-free Multi-Mesh prototype learns causal language modeling but does not establish competitive general capability. A versioned store contains 52,809 reconstructive memory elements near a 1.7-billion-floating-value budget; reconstruction is incomplete, with approximately 75\% token accuracy. Support-aware lexical indices make these elements addressable under provenance-controlled query construction. For executable arithmetic, positional result observations substantially improve neural rendering relative to a repeated global result vector, and output substitutions change the model's preferred answer. A bounded attachment demonstration further measures the effect of making selected evidence available, without establishing the utility of loading an entire multi-billion-value store. The results support a separation of storage, execution, and neural coordination, while identifying unresolved limitations in question-only retrieval, unrestricted answer generation, and end-to-end efficiency.

Representation rank is an important concept for understanding the role of Neural Networks (NNs) in Deep Reinforcement learning (DRL), which measures the expressive capacity of value networks. Existing studies focus on unboundedly maximizing this rank; nevertheless, that approach would introduce overly complex models in the learning, thus undermining performance. Hence, fine-tuning representation rank presents a challenging and crucial optimization problem. To address this issue, we find a guiding principle for adaptive control of the representation rank. We employ the Bellman equation as a theoretical foundation and derive an upper bound on the cosine similarity of consecutive state-action pairs representations of value networks. We then leverage this upper bound to propose a novel regularizer, namely BEllman Equation-based automatic rank Regularizer (BEER). This regularizer adaptively regularizes the representation rank, thus improving the DRL agent's performance. We first validate the effectiveness of automatic control of rank on illustrative experiments. Then, we scale up BEER to complex continuous control tasks by combining it with the deterministic policy gradient method. Among 12 challenging DeepMind control tasks, BEER outperforms the baselines by a large margin. Besides, BEER demonstrates significant advantages in Q-value approximation. Our code is available at https://github.com/sweetice/BEER-ICLR2024.

Modern Recurrent Neural Networks (RNNs), such as RWKV, are distinguished by their powerful short-range modeling capabilities and efficient fixed-size states, which constitute a core advantage over standard Transformers. However, there is a significant lack of research into their internal state as an editable knowledge representation. To fill this gap, we first explore the representational properties of the RWKV state by proposing the DREAMSTATE framework. This framework utilizes a conditional Diffusion Transformer (DiT) to directly model the probability manifold of the state, enabling its generation and editing. The structural nature of this representation is validated through t-SNE visualizations and controlled generation experiments. After successfully uncovering and modeling the state's representational potential, we further propose a novel hybrid architecture that combines the local advantages of RNNs with global context adaptability. This architecture features a parallel DiT that processes a variable-length global context to dynamically generate and adjust the core recurrent module's WKV parameters, transforming the fixed recurrence mechanism into a context-aware dynamic function. Experiments demonstrate that this hybrid model can be trained stably via a multi-objective loss, validating its design feasibility. Our work not only opens a new research direction for RNN state representation but also provides a concrete architectural reference for future model design. The code is publicly available at: https://huggingface.co/2dgx41s/DreamState.

A semantic decision engine such as Jev can return a valid answer and still miss a network deadline, select an infeasible action or leave the service unverified. We systematize 139 paper families by decision interface, execution path and check ownership. Fifty families claim that their engine fits a control loop or time budget, but only four support the claim with matched measurement. Across all 139, four report deadline attainment. The gap concentrates where the decision has no deterministic computation step. Those 72 families make 22 of the claims, none supported, and name a coverage owner in only two. Bounded tests under one event model show that each gap can reverse an admission verdict. A decision that meets a 10 s budget for every isolated request meets it for none once decisions queue ahead of replayed execution times. The same engine passes one coverage check and fails another. We derive a minimum reporting record, design rules and a research agenda for admitting decision engines to control loops.

While goal-conditioned behavior cloning (GCBC) methods can perform well on in-distribution training tasks, they do not necessarily generalize zero-shot to tasks that require conditioning on novel state-goal pairs, i.e. combinatorial generalization. In part, this limitation can be attributed to a lack of temporal consistency in the state representation learned by BC; if temporally correlated states are properly encoded to similar latent representations, then the out-of-distribution gap for novel state-goal pairs would be reduced. We formalize this notion by demonstrating how encouraging long-range temporal consistency via successor representations (SR) can facilitate generalization. We then propose a simple yet effective representation learning objective, BYOL-γ for GCBC, which theoretically approximates the successor representation in the finite MDP case through self-predictive representations, and achieves competitive empirical performance across a suite of challenging tasks requiring combinatorial generalization.

In real-world applications of reinforcement learning, it is often challenging to obtain a state representation that is parsimonious and satisfies the Markov property without prior knowledge. Consequently, it is common practice to construct a state larger than necessary, e.g., by concatenating measurements over contiguous time points. However, needlessly increasing the dimension of the state may slow learning and obfuscate the learned policy. We introduce the notion of a minimal sufficient state in a Markov decision process (MDP) as the subvector of the original state under which the process remains an MDP and shares the same reward function as the original process. We propose a novel SEquEntial Knockoffs (SEEK) algorithm that estimates the minimal sufficient state in a system with high-dimensional complex nonlinear dynamics. In large samples, the proposed method achieves selection consistency. As the method is agnostic to the reinforcement learning algorithm being applied, it benefits downstream tasks such as policy learning. Empirical experiments verify theoretical results and show the proposed approach outperforms several competing methods regarding variable selection accuracy and regret.

While deep reinforcement learning excels at solving tasks where large amounts of data can be collected through virtually unlimited interaction with the environment, learning from limited interaction remains a key challenge. We posit that an agent can learn more efficiently if we augment reward maximization with self-supervised objectives based on structure in its visual input and sequential interaction with the environment. Our method, Self-Predictive Representations(SPR), trains an agent to predict its own latent state representations multiple steps into the future. We compute target representations for future states using an encoder which is an exponential moving average of the agent's parameters and we make predictions using a learned transition model. On its own, this future prediction objective outperforms prior methods for sample-efficient deep RL from pixels. We further improve performance by adding data augmentation to the future prediction loss, which forces the agent's representations to be consistent across multiple views of an observation. Our full self-supervised objective, which combines future prediction and data augmentation, achieves a median human-normalized score of 0.415 on Atari in a setting limited to 100k steps of environment interaction, which represents a 55% relative improvement over the previous state-of-the-art. Notably, even in this limited data regime, SPR exceeds expert human scores on 7 out of 26 games. The code associated with this work is available at https://github.com/mila-iqia/spr

Classical simulation of quantum physics is a central approach to investigating physical phenomena. Quantum computers enhance computational capabilities beyond those of classical resources, but it remains unclear to what extent existing limited quantum computers can contribute to this enhancement. In this work, we explore a new hybrid, efficient quantum-classical representation of quantum states, the multiple-basis representation. This representation consists of a linear combination of states that are sparse in some given and different bases, specified by quantum circuits. Such representation is particularly appealing when considering depth-limited quantum circuits within reach of current hardware. We analyze the expressivity of multiple-basis representation states depending on the classical simulability of their quantum circuits. In particular, we show that multiple-basis representation states include, but are not restricted to, both matrix-product states and stabilizer states. Furthermore, we find cases in which this representation can be used, namely approximation of ground states, simulation of deeper computations by specifying bases with shallow circuits, and a tomographical protocol to describe states as multiple-basis representations. We envision this work to open the path of simultaneous use of several hardware-friendly bases, a natural description of hybrid computational methods accessible for near-term hardware.

A fundamental challenge in embodied intelligence is developing expressive and compact state representations for efficient world modeling and decision making. However, existing methods often fail to achieve this balance, yielding representations that are either overly redundant or lacking in task-critical information. We propose an unsupervised approach that learns a highly compressed two-token state representation using a lightweight encoder and a pre-trained Diffusion Transformer (DiT) decoder, capitalizing on its strong generative prior. Our representation is efficient, interpretable, and integrates seamlessly into existing VLA-based models, improving performance by 14.3% on LIBERO and 30% in real-world task success with minimal inference overhead. More importantly, we find that the difference between these tokens, obtained via latent interpolation, naturally serves as a highly effective latent action, which can be further decoded into executable robot actions. This emergent capability reveals that our representation captures structured dynamics without explicit supervision. We name our method StaMo for its ability to learn generalizable robotic Motion from compact State representation, which is encoded from static images, challenging the prevalent dependence to learning latent action on complex architectures and video data. The resulting latent actions also enhance policy co-training, outperforming prior methods by 10.4% with improved interpretability. Moreover, our approach scales effectively across diverse data sources, including real-world robot data, simulation, and human egocentric video.

While multitask representation learning has become a popular approach in reinforcement learning (RL) to boost the sample efficiency, the theoretical understanding of why and how it works is still limited. Most previous analytical works could only assume that the representation function is already known to the agent or from linear function class, since analyzing general function class representation encounters non-trivial technical obstacles such as generalization guarantee, formulation of confidence bound in abstract function space, etc. However, linear-case analysis heavily relies on the particularity of linear function class, while real-world practice usually adopts general non-linear representation functions like neural networks. This significantly reduces its applicability. In this work, we extend the analysis to general function class representations. Specifically, we consider an agent playing M contextual bandits (or MDPs) concurrently and extracting a shared representation function phi from a specific function class Phi using our proposed Generalized Functional Upper Confidence Bound algorithm (GFUCB). We theoretically validate the benefit of multitask representation learning within general function class for bandits and linear MDP for the first time. Lastly, we conduct experiments to demonstrate the effectiveness of our algorithm with neural net representation.

Developing robust representations of chemical structures that enable models to learn topological inductive biases is challenging. In this manuscript, we present a representation of atomistic systems. We begin by proving that our representation satisfies all structural, geometric, efficiency, and generalizability constraints. Afterward, we provide a general algorithm to encode any atomistic system. Finally, we report performance comparable to state-of-the-art methods on numerous tasks. We open-source all code and datasets. The code and data are available at https://github.com/rahulkhorana/PolyatomicComplexes.

Data-efficiency and generalization are key challenges in deep learning and deep reinforcement learning as many models are trained on large-scale, domain-specific, and expensive-to-label datasets. Self-supervised models trained on large-scale uncurated datasets have shown successful transfer to diverse settings. We investigate using pretrained image representations and spatio-temporal attention for state representation learning in Atari. We also explore fine-tuning pretrained representations with self-supervised techniques, i.e., contrastive predictive coding, spatio-temporal contrastive learning, and augmentations. Our results show that pretrained representations are at par with state-of-the-art self-supervised methods trained on domain-specific data. Pretrained representations, thus, yield data and compute-efficient state representations. https://github.com/PAL-ML/PEARL_v1

Most contemporary neural learning systems rely on epoch-based optimization and repeated access to historical data, implicitly assuming reversible computation. In contrast, real-world environments often present information as irreversible streams, where inputs cannot be replayed or revisited. Under such conditions, conventional architectures degrade into reactive filters lacking long-horizon coherence. This paper introduces Stream Neural Networks (StNN), an execution paradigm designed for irreversible input streams. StNN operates through a stream-native execution algorithm, the Stream Network Algorithm (SNA), whose fundamental unit is the stream neuron. Each stream neuron maintains a persistent temporal state that evolves continuously across inputs. We formally establish three structural guarantees: (1) stateless mappings collapse under irreversibility and cannot encode temporal dependencies; (2) persistent state dynamics remain bounded under mild activation constraints; and (3) the state transition operator is contractive for λ < 1, ensuring stable long-horizon execution. Empirical phase-space analysis and continuous tracking experiments validate these theoretical results. The execution principles introduced in this work define a minimal substrate for neural computation under irreversible streaming constraints.

Safety heads are lightweight classifiers attached to pretrained language models for flagging harmful inputs before generation. Their empirical detection performance has been studied, but their formal robustness properties remain largely unexplored. We ask when a State Space Model (SSM)-based safety head can be certified to produce the same prediction for all inputs within a bounded embedding-space perturbation. We prove that the answer turns on a single condition: the l_infty norm of the state transition matrix must satisfy A_infty<1 (the contraction condition), which enables exact interval bound propagation (IBP) certification for linear time-invariant classifiers. When the contraction condition holds, the reachable output interval has bounded steady-state width and examples can be certified as robustly classified. When it fails, the interval grows exponentially with sequence length and certification is impossible at any practical perturbation radius. We enforce contraction with a hinge penalty and show on toxic comment data that certified fraction improves from 41\% to 59\%, with a sharp empirical phase transition at A_infty=1 matching the theory. Applying a contraction-regularized S4 head to jailbreak detection on JailbreakBench, we achieve a zero-shot transfer to AdvBench (DR=0.994) and HarmBench (DR=0.988). A logistic regression on mean-pooled Mamba-130M embeddings matches or exceeds the S4 head on every detection metric, confirming that harmful intent is already linearly separable in the embedding space. The S4 safety head's contribution is not superior discrimination but the formal certification that no probe-based approach provides.

State Space Models (SSMs) offer an efficient alternative to Transformers for sequence modeling, yet conditioning pre-trained SSMs for iterative generation typically operates outside the recurrent operator, through input injection or activation modulation. While such mechanisms expose the model to conditioning information, they leave the underlying temporal dynamics fixed. We introduce MaRK (Markov-adapted Recurrent Kernels), a dynamic operator-conditioning framework that maps context vectors directly into bounded modulations of a frozen SSM's recurrence (A), read-in (B), read-out (C), skip (D), and discretization (Δ) parameters. Viewed through the lens of LPV-SSM systems, MaRK induces a context-indexed family of Markov parameter sequences, allowing each diffusion timestep to reshape the model's input-output memory kernel. We instantiate MaRK on a frozen 111M-parameter Hydra SSM backbone and study three adapter geometries: Hypernet, Chebyshev polynomial, and Discrete Cosine Transform kernels. Since these adapters modify the Markov parameter sequence through low-rank auxiliary maps on the frozen backbone, parameter-efficient fine-tuning arises as a structural consequence of the adaptation mechanism itself, requiring only 6.3--11M trainable auxiliary parameters to transition from a bidirectional objective to an iterative diffusion regime. The bounded recurrence parameterization further yields an analytic Affine Quadratic Stability certificate for the modulated recurrence. Through synthetic LPV recovery experiments and Markov-operator diagnostics, we show that MaRK recovers coordinate-invariant temporal operators under matched assumptions and produces distinct, stable timestep-conditioned memory profiles. Empirically, the Chebyshev variant yields the strongest performance, achieving an average validation loss of 2.55, followed by the DCT (2.59) and Hypernet (3.77) geometries.

This paper investigates robust representation learning in offline goal-conditioned reinforcement learning (GCRL). Particularly in sparse reward scenarios, learning representations that align state and goal latents is a challenge that frequently culminates in representation divergence where the encoder drifts toward a low-dimensional, goal-agnostic subspace that destabilizes policy learning. We address this issue by showing that an agent must acquire a fundamental understanding of its environment across multiple scales, from local physical dynamics to long-horizon goal-directed structure. Building on this insight, we propose Ms.PR, a framework that leverages multi-scale predictive supervision to enforce goal-directed alignment within the latent space. We demonstrate that Ms.PR leads to improved representation quality and strong performance on both vision and state-based tasks. Furthermore, we show that our approach is exceptionally resilient under realistic, challenging data regimes, maintaining state-of-the-art performance across a wide variety of tasks, trajectory stitching scenarios, and extreme noise conditions.

Large language model (LLM) agents coordinate complex tasks through multi-role and multi-stage workflows. Upstream state is repeatedly transformed into intermediate language artifacts, such as summaries, plans, tickets, memories, and handoff notes, from which downstream components act. For action-constraining state, topical retention is insufficient: an artifact may mention an unresolved condition while changing it from a requirement that must be resolved before execution into information that may merely inform the next action. We study this action-binding role as operational state preservation. Safety blockers provide a controlled instance because each source state has an explicit prerequisite, authority, fallback, and execution consequence. We condition on correct upstream identification, vary the handoff transformation, and evaluate an executor restricted to the resulting artifact. Across 1,296 controlled synthetic episodes, direct-handoff controls preserve every blocker, whereas compression, plan assimilation, convergence, ownership deferral, and precedent substitution repeatedly turn binding state into caveats or non-binding considerations. Normal handoff compression produces 100.0% deactivation and 54.2% forbidden action. Restoring all four state fields raises preservation to 100.0% and reduces forbidden action to 0.0%. Fixed-artifact interventions further separate preservation from containment: downstream verification eliminates forbidden action while artifact deactivation remains 95.3%. These results identify a state-transmission failure between information extraction and action. Handoff transformations can retain state content while weakening its constraints on downstream action. Semantic availability does not guarantee operational preservation.

Large language models are increasingly used as agents in social, economic, and policy simulations. A common assumption is that stronger reasoning should improve simulation fidelity. We argue that this assumption can fail when the objective is not to solve a strategic problem, but to sample plausible boundedly rational behavior. In such settings, reasoning-enhanced models can become better solvers and worse simulators: they can over-optimize for strategically dominant actions, collapse compromise-oriented terminal behavior, and sometimes exhibit a diversity-without-fidelity pattern in which local variation survives without outcome-level fidelity. We study this solver-sampler mismatch in three multi-agent negotiation environments adapted from earlier simulation work: an ambiguous fragmented-authority trading-limits scenario, an ambiguous unified-opposition trading-limits scenario, and a new-domain grid-curtailment case in emergency electricity management. We compare three reflection conditions, no reflection, bounded reflection, and native reasoning, across two primary model families and then extend the same protocol to direct OpenAI runs with GPT-4.1 and GPT-5.2. Across all three experiments, bounded reflection produces substantially more diverse and compromise-oriented trajectories than either no reflection or native reasoning. In the direct OpenAI extension, GPT-5.2 native ends in authority decisions in 45 of 45 runs across the three experiments, while GPT-5.2 bounded recovers compromise outcomes in every environment. The contribution is not a claim that reasoning is generally harmful. It is a methodological warning: model capability and simulation fidelity are different objectives, and behavioral simulation should qualify models as samplers, not only as solvers.

We study the learning dynamics of self-predictive learning for reinforcement learning, a family of algorithms that learn representations by minimizing the prediction error of their own future latent representations. Despite its recent empirical success, such algorithms have an apparent defect: trivial representations (such as constants) minimize the prediction error, yet it is obviously undesirable to converge to such solutions. Our central insight is that careful designs of the optimization dynamics are critical to learning meaningful representations. We identify that a faster paced optimization of the predictor and semi-gradient updates on the representation, are crucial to preventing the representation collapse. Then in an idealized setup, we show self-predictive learning dynamics carries out spectral decomposition on the state transition matrix, effectively capturing information of the transition dynamics. Building on the theoretical insights, we propose bidirectional self-predictive learning, a novel self-predictive algorithm that learns two representations simultaneously. We examine the robustness of our theoretical insights with a number of small-scale experiments and showcase the promise of the novel representation learning algorithm with large-scale experiments.

Goal-Conditioned Reinforcement Learning (GCRL) enables agents to autonomously acquire diverse behaviors, but faces major challenges in visual environments due to high-dimensional, semantically sparse observations. In the online setting, where agents learn representations while exploring, the latent space evolves with the agent's policy, to capture newly discovered areas of the environment. However, without incentivization to maximize state coverage in the representation, classical approaches based on auto-encoders may converge to latent spaces that over-represent a restricted set of states frequently visited by the agent. This is exacerbated in an intrinsic motivation setting, where the agent uses the distribution encoded in the latent space to sample the goals it learns to master. To address this issue, we propose to progressively enforce distributional shifts towards a uniform distribution over the full state space, to ensure a full coverage of skills that can be learned in the environment. We introduce DRAG (Distributionally Robust Auto-Encoding for GCRL), a method that combines the β-VAE framework with Distributionally Robust Optimization. DRAG leverages an adversarial neural weighter of training states of the VAE, to account for the mismatch between the current data distribution and unseen parts of the environment. This allows the agent to construct semantically meaningful latent spaces beyond its immediate experience. Our approach improves state space coverage and downstream control performance on hard exploration environments such as mazes and robotic control involving walls to bypass, without pre-training nor prior environment knowledge.

State space models (SSM) have recently been shown to be very effective as a deep learning layer as a promising alternative to sequence models such as RNNs, CNNs, or Transformers. The first version to show this potential was the S4 model, which is particularly effective on tasks involving long-range dependencies by using a prescribed state matrix called the HiPPO matrix. While this has an interpretable mathematical mechanism for modeling long dependencies, it introduces a custom representation and algorithm that can be difficult to implement. On the other hand, a recent variant of S4 called DSS showed that restricting the state matrix to be fully diagonal can still preserve the performance of the original model when using a specific initialization based on approximating S4's matrix. This work seeks to systematically understand how to parameterize and initialize such diagonal state space models. While it follows from classical results that almost all SSMs have an equivalent diagonal form, we show that the initialization is critical for performance. We explain why DSS works mathematically, by showing that the diagonal restriction of S4's matrix surprisingly recovers the same kernel in the limit of infinite state dimension. We also systematically describe various design choices in parameterizing and computing diagonal SSMs, and perform a controlled empirical study ablating the effects of these choices. Our final model S4D is a simple diagonal version of S4 whose kernel computation requires just 2 lines of code and performs comparably to S4 in almost all settings, with state-of-the-art results for image, audio, and medical time-series domains, and averaging 85\% on the Long Range Arena benchmark.

In the classical transformer attention scheme, we are given three n times d size matrices Q, K, V (the query, key, and value tokens), and the goal is to compute a new n times d size matrix D^{-1} exp(QK^top) V where D = diag( exp(QK^top) {bf 1}_n ). In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in n. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations: bullet On the positive side, if all entries of the input matrices are bounded above by o(sqrt[3]{log n}) then we show how to approximate the ``tensor-type'' attention matrix in n^{1+o(1)} time. bullet On the negative side, we show that if the entries of the input matrices may be as large as Omega(sqrt[3]{log n}), then there is no algorithm that runs faster than n^{3-o(1)} (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory). We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.

We formalize a transfinite Phi process that treats all possibility embeddings as operators on structured state spaces including complete lattices, Banach and Hilbert spaces, and orthomodular lattices. We prove a determinization lemma showing that lifting to sets or distributions yields a deterministic global dynamic, an ordinal stabilization theorem sending operator transforms to the fixed subspace by stage omega under normal spectral contraction, and a product of Riesz projections theorem for commuting layers. We establish a compositionality law for lifted maps, show closure of Phi packings, and present a quantitative application to sensory depletion that models tissue removal as a projection and derives strict decreases in the attainable fixed point under minimal monotonicity and positivity assumptions. We also state measurable conditions for probabilistic lifts, give explicit non normal and non commuting counterexamples, and provide finite dimensional and stochastic witnesses together with per theorem scope tables and a small reproducible code appendix.

While LLMs are powerful embedding backbones, their application in training-free settings faces two structural challenges: causal attention restricts early tokens from accessing subsequent context, and the next-token prediction objective biases representations toward generation rather than semantic compression. To address these limitations, we propose KV-Embedding, a framework that activates the latent representation power of frozen LLMs. Our method leverages the observation that the key-value (KV) states of the final token at each layer encode a compressed view of the sequence. By re-routing these states as a prepended prefix, we enable all tokens to access sequence-level context within a single forward pass. To ensure model-agnostic applicability, we introduce an automated layer selection strategy based on intrinsic dimensionality. Evaluations on MTEB across Qwen, Mistral, and Llama backbones show that KV-Embedding outperforms existing training-free baselines by up to 10%, while maintaining robust performance on sequences up to 4,096 tokens. These results demonstrate that internal state manipulation offers an efficient alternative to input modification, and we hope this work encourages further exploration of LLM internals for representation learning.

The Transformer architecture, despite its widespread success, struggles with long-context scenarios due to quadratic computation and linear memory growth. While various linear attention variants mitigate these efficiency constraints by compressing context into fixed-size states, they often degrade performance in tasks such as in-context retrieval and reasoning. To address this limitation and achieve more effective context compression, we propose two key innovations. First, we introduce a row-sparse update formulation for linear attention by conceptualizing state updating as information classification. This enables sparse state updates via softmax-based top-k hard classification, thereby extending receptive fields and reducing inter-class interference. Second, we present Sparse State Expansion (SSE) within the sparse framework, which expands the contextual state into multiple partitions, effectively decoupling parameter size from state capacity while maintaining the sparse classification paradigm. Our design, supported by efficient parallelized implementations, yields effective classification and discriminative state representations. We extensively validate SSE in both pure linear and hybrid (SSE-H) architectures across language modeling, in-context retrieval, and mathematical reasoning benchmarks. SSE demonstrates strong retrieval performance and scales favorably with state size. Moreover, after reinforcement learning (RL) training, our 2B SSE-H model achieves state-of-the-art mathematical reasoning performance among small reasoning models, scoring 64.7 on AIME24 and 51.3 on AIME25, significantly outperforming similarly sized open-source Transformers. These results highlight SSE as a promising and efficient architecture for long-context modeling.

A user's movie, news, and dialogue histories differ in their native actions and outputs, yet each interaction supplies evidence that can update user memory. We study whether these histories can train one reusable update mechanism. An action-on-item schema pairs a mapped interaction role with a content embedding, allowing shared update parameters to operate on separate user states. We establish invariance to native relabeling, bounded state changes under item-embedding perturbations, and a pooled-training bound under explicit compatibility conditions. The Multi-Timescale State Hypothesis (MTSH) specifies how this evidence enters, persists, and is consumed; PerTIDE implements it with action gating, three state-space traces, fusion, and command-conditioned readout. On PENS, the same history encoder supports both next-news prediction and personalized headline generation. In a controlled PENS-to-MovieLens experiment, a frozen source-trained core exceeds an identically structured random core by 15.23 MRR points after fitting the same target consumer. On MIND, PerTIDE retains a 4.12-point MRR advantage over a same-input three-branch state-space control. Action, readout, and trace interventions identify complementary contributions to these gains. Together, the theory and experiments support learning history updates across compatible sources and reusing them through predictive and generative consumers.

We study how representation learning can accelerate reinforcement learning from rich observations, such as images, without relying either on domain knowledge or pixel-reconstruction. Our goal is to learn representations that both provide for effective downstream control and invariance to task-irrelevant details. Bisimulation metrics quantify behavioral similarity between states in continuous MDPs, which we propose using to learn robust latent representations which encode only the task-relevant information from observations. Our method trains encoders such that distances in latent space equal bisimulation distances in state space. We demonstrate the effectiveness of our method at disregarding task-irrelevant information using modified visual MuJoCo tasks, where the background is replaced with moving distractors and natural videos, while achieving SOTA performance. We also test a first-person highway driving task where our method learns invariance to clouds, weather, and time of day. Finally, we provide generalization results drawn from properties of bisimulation metrics, and links to causal inference.

Perception for action suggests that representations of the world should be shaped not by visual fidelity alone, but by their relevance for actions. At the same time, latent JEPA-style world models advocate learning compact predictive states from high-dimensional observations to facilitate the prediction of future states, but end-to-end training of these models is nontrivial because representations may collapse if our only goal is to construct a latent state that is easy to predict. We introduce a sensorimotor world model (SMWM): a latent world model trained end-to-end with inverse dynamics regularization. This single regularizer addresses both issues: it prevents representation collapse and induces action-aligned representations. By forcing latent states to preserve information about the action underlying a transition, it biases the model toward the controllable degrees of freedom of the environment while discarding uncontrollable distractors. This yields stable latent world models trained from offline, reward-free trajectories, without frozen encoders, exponential moving averages, or complex latent regularizers. Empirically, SMWM learns compact, interpretable latent spaces and enables competitive planning performance across simple 2D and 3D control tasks.

Learning a good history representation is one of the core challenges of reinforcement learning (RL) in partially observable environments. Recent works have shown the advantages of various auxiliary tasks for facilitating representation learning. However, the effectiveness of such auxiliary tasks has not been fully convincing, especially in partially observable environments that require long-term memorization and inference. In this empirical study, we investigate the effectiveness of future prediction for learning the representations of histories, possibly of extensive length, in partially observable environments. We first introduce an approach that decouples the task of learning history representations from policy optimization via future prediction. Then, our main contributions are two-fold: (a) we demonstrate that the performance of reinforcement learning is strongly correlated with the prediction accuracy of future observations in partially observable environments, and (b) our approach can significantly improve the overall end-to-end approach by preventing high-variance noisy signals from reinforcement learning objectives to influence the representation learning. We illustrate our claims on three types of benchmarks that necessitate the ability to process long histories for high returns.

Voting over multiple LLM responses is a common primitive in test-time scaling and ensemble inference. Collecting more responses can expand the candidate pool and increase the chance that a correct answer is discovered. Under a fixed call budget, a discovered answer still needs to accumulate enough support within the remaining calls to become the final plurality winner, creating a discovery-to-decision gap. In this work, we characterize this gap through the realized vote state and remaining call budget. We derive a sharp recoverability threshold and show that, as sampling proceeds, the observed candidate set can only expand while the set of reachable endpoint winners can only contract, inducing a candidate-level conversion window. Under a specified iid response law, the same state yields exact finite-horizon endpoint probabilities. We further show that merging wrong-answer identities preserves single-call correctness and cannot improve plurality accuracy, and that the effect of redistributing wrong-answer probability depends on the realized vote state. Singleton reachability yields a gold-free exact locking certificate. For a known answer universe, its first trigger is the earliest prefix at which all admissible continuations yield the same fixed-budget output. Empirically, most discovered-but-unselected correct answers lose reachability only after discovery. In a controlled Word16 study, input permutation improves raw-plurality accuracy by 21.1 points with essentially unchanged single-call correctness. Exact locking saves 28-30% of calls at a 16-call budget while preserving every fixed-budget output.

In Reinforcement Learning (RL), Laplacian Representation (LapRep) is a task-agnostic state representation that encodes the geometry of the environment. A desirable property of LapRep stated in prior works is that the Euclidean distance in the LapRep space roughly reflects the reachability between states, which motivates the usage of this distance for reward shaping. However, we find that LapRep does not necessarily have this property in general: two states having small distance under LapRep can actually be far away in the environment. Such mismatch would impede the learning process in reward shaping. To fix this issue, we introduce a Reachability-Aware Laplacian Representation (RA-LapRep), by properly scaling each dimension of LapRep. Despite the simplicity, we demonstrate that RA-LapRep can better capture the inter-state reachability as compared to LapRep, through both theoretical explanations and experimental results. Additionally, we show that this improvement yields a significant boost in reward shaping performance and also benefits bottleneck state discovery.

Sequential decision-making agents struggle with long horizon tasks, since solving them requires multi-step reasoning. Most reinforcement learning (RL) algorithms address this challenge by improved credit assignment, introducing memory capability, altering the agent's intrinsic motivation (i.e. exploration) or its worldview (i.e. knowledge representation). Many of these components could be learned from offline data. In this work, we follow the hypothesis that exploration and representation learning can be improved by separately learning two different models from a single offline dataset. We show that learning a state representation using noise-contrastive estimation and a model of auxiliary reward separately from a single collection of human demonstrations can significantly improve the sample efficiency on the challenging NetHack benchmark. We also ablate various components of our experimental setting and highlight crucial insights.

This paper describes pi2vec, a method for representing behaviors of black box policies as feature vectors. The policy representations capture how the statistics of foundation model features change in response to the policy behavior in a task agnostic way, and can be trained from offline data, allowing them to be used in offline policy selection. This work provides a key piece of a recipe for fusing together three modern lines of research: Offline policy evaluation as a counterpart to offline RL, foundation models as generic and powerful state representations, and efficient policy selection in resource constrained environments.

Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership i in S and estimating set intersection sizes |X cap Y| for two sets of symbols X and Y, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

Recently, there have been numerous attempts to enhance the sample efficiency of off-policy reinforcement learning (RL) agents when interacting with the environment, including architecture improvements and new algorithms. Despite these advances, they overlook the potential of directly constraining the initial representations of the input data, which can intuitively alleviate the distribution shift issue and stabilize training. In this paper, we introduce the Tanh function into the initial layer to fulfill such a constraint. We theoretically unpack the convergence property of the temporal difference learning with the Tanh function under linear function approximation. Motivated by theoretical insights, we present our Constrained Initial Representations framework, tagged CIR, which is made up of three components: (i) the Tanh activation along with normalization methods to stabilize representations; (ii) the skip connection module to provide a linear pathway from the shallow layer to the deep layer; (iii) the convex Q-learning that allows a more flexible value estimate and mitigates potential conservatism. Empirical results show that CIR exhibits strong performance on numerous continuous control tasks, even being competitive or surpassing existing strong baseline methods.

Black-box LLM distillation is usually evaluated as an output-matching problem: a student is considered successful when its responses are semantically similar to, or task-consistent with, those of a teacher. However, output similarity does not imply that the student is behaviorally indistinguishable from the model it imitates. We introduce bounded behavioral indistinguishability, formalized as (ε,q,t,A)-behavioral indistinguishability over an explicit prompt distribution, where ε bounds distinguishing advantage, q bounds oracle queries, t bounds computation, and A denotes the adversary class. We instantiate this notion on Qwen and Llama teacher-student pairs using a controlled 5,000-prompt behavioral probe suite. For each family, we compare the teacher with both the base student and the LoRA-distilled student, measuring whether distillation reduces distinguishability rather than merely improving similarity. LoRA raises semantic similarity from 0.788 to 0.862 for Qwen and from 0.814 to 0.874 for Llama. Yet adversarial evaluation reveals remaining behavioral differences: learned discriminators retain nonzero advantage, and pairwise category analysis shows artifacts concentrated in style/format, robustness, and domain-technical prompts. A pairwise teacher-identification adversary confirms this trend. With a different-family Llama judge and A/B-swap consistency filtering, Qwen distinguishing advantage drops from 0.158 for the base student to 0.081 after LoRA distillation. Query-budget experiments show that disagreement-guided acquisition does not consistently outperform stratified random sampling, indicating that coverage and diversity remain strong baselines. Our results show that semantic fidelity is useful but insufficient: black-box LLM distillation requires bounded, adversarial, and category-aware evaluation.

The general sequential decision-making problem, which includes Markov decision processes (MDPs) and partially observable MDPs (POMDPs) as special cases, aims at maximizing a cumulative reward by making a sequence of decisions based on a history of observations and actions over time. Recent studies have shown that the sequential decision-making problem is statistically learnable if it admits a low-rank structure modeled by predictive state representations (PSRs). Despite these advancements, existing approaches typically involve oracles or steps that are computationally intractable. On the other hand, the upper confidence bound (UCB) based approaches, which have served successfully as computationally efficient methods in bandits and MDPs, have not been investigated for more general PSRs, due to the difficulty of optimistic bonus design in these more challenging settings. This paper proposes the first known UCB-type approach for PSRs, featuring a novel bonus term that upper bounds the total variation distance between the estimated and true models. We further characterize the sample complexity bounds for our designed UCB-type algorithms for both online and offline PSRs. In contrast to existing approaches for PSRs, our UCB-type algorithms enjoy computational tractability, last-iterate guaranteed near-optimal policy, and guaranteed model accuracy.

To make claims about representations that generalize beyond a particular trained model, we need to specify when two representations should count as equivalent. The Linear Representation Hypothesis is often discussed without making this equivalence explicit. Different notions of equivalence preserve different structures, so metrics, probes, and interventions that appear to study the same representation may in fact correspond to different hypotheses. We therefore argue that the Linear Representation Hypothesis is not one hypothesis but a family of claims distinguished by representation equivalence. We formalize this idea using group actions, specifying the representation object, the procedure that produces it, and the property ultimately asserted, while accounting for equivalences imposed by the model architecture. This framework clarifies how assumptions can change across metrics, reading points, and analysis stages, and we use it to audit common representation quantities and recent interpretability analyses.

Continual learning systems are increasingly deployed in environments where retraining or reset is infeasible, yet many approaches emphasize task performance rather than the evolution of internal representations over time. In this work, we study a minimal continual learning agent designed to isolate representational dynamics from architectural complexity and optimization objectives. The agent maintains a persistent state vector across executions and incrementally updates it as new textual data is introduced. We quantify representational change using cosine similarity between successive normalized state vectors and define a stability metric over time intervals. Longitudinal experiments across eight executions reveal a transition from an initial plastic regime to a stable representational regime under consistent input. A deliberately introduced semantic perturbation produces a bounded decrease in similarity, followed by recovery and restabilization under subsequent coherent input. These results demonstrate that meaningful stability plasticity tradeoffs can emerge in a minimal, stateful learning system without explicit regularization, replay, or architectural complexity. The work establishes a transparent empirical baseline for studying representational accumulation and adaptation in continual learning systems.

Optimizer-state quantization is commonly designed for Adam's dense, parameter-aligned first- and second-moment arrays. This abstraction breaks for memory-efficient optimizers, whose states may be factored, confidence-modulated, or maintained in a projected space, so similar reconstruction error can produce different update error. We formulate optimizer-state quantization as a joint problem over representation, topology, and update semantics. We then introduce Adaptive Log-Space (AL) quantization for non-negative states. AL fits each block's observed nonzero logarithmic interval and reserves a separate code for exact zero, enforcing q = 0 Leftrightarrow x = 0; signed momentum and state precision remain independently selectable. Controlled probes show that adaptive ranges reduce update error and temporal drift, exact-zero reservation preserves dormant states, and state topology constrains useful block granularity. End-to-end language-model training evaluates the resulting policy across dense, factored, confidence, and projected optimizer states. On TinyLlama-1.1B, AL8 with uniform 8-bit momentum reaches 72.90 perplexity versus 73.54 for bitsandbytes 8-bit AdamW, with comparable optimizer-state storage and higher throughput. CAME matches reference-level final perplexity across three seeds when its non-negative states use AL16, while a semantic grouping-and-protection policy closes most of quantized Adafactor's 100K-step late-loss gap. These results make state topology and update semantics first-class design constraints for optimizer quantization.

In a class of quantum circuits known as peaked circuits, the goal is to predict the most probable bit string at the output of the circuit. Since these circuits are designed to have a sharp peak in their output distribution, in principle it should be possible to simulate them using a truncated state vector with a limited number of terms, or a fraction of the total probability mass. This approximate simulation can be carried out on a classical computer with a sparse representation that stores only the nonzero amplitudes of the state vector, in contrast to the dense representations that are common in most quantum simulators. For efficiency, all operations on the state vector should be vectorized to the furthest possible extent and, if available, hardware acceleration can also be used. This work describes how these requirements were met in an open-source implementation, and discusses its performance and limitations.

Deep learning models are increasingly employed for perception, prediction, and control in complex systems. Embedding physical knowledge into these models is crucial for achieving realistic and consistent outputs, a challenge often addressed by physics-informed machine learning. However, integrating physical knowledge with representation learning becomes difficult when dealing with high-dimensional observation data, such as images, particularly under conditions of incomplete or imprecise state information. To address this, we propose Physically Interpretable World Models, a novel architecture that aligns learned latent representations with real-world physical quantities. Our method combines a variational autoencoder with a dynamical model that incorporates unknown system parameters, enabling the discovery of physically meaningful representations. By employing weak supervision with interval-based constraints, our approach eliminates the reliance on ground-truth physical annotations. Experimental results demonstrate that our method improves the quality of learned representations while achieving accurate predictions of future states, advancing the field of representation learning in dynamic systems.

Embodied AI agents require a fine-grained understanding of the physical world mediated through visual and language inputs. Such capabilities are difficult to learn solely from task-specific data. This has led to the emergence of pre-trained vision-language models as a tool for transferring representations learned from internet-scale data to downstream tasks and new domains. However, commonly used contrastively trained representations such as in CLIP have been shown to fail at enabling embodied agents to gain a sufficiently fine-grained scene understanding -- a capability vital for control. To address this shortcoming, we consider representations from pre-trained text-to-image diffusion models, which are explicitly optimized to generate images from text prompts and as such, contain text-conditioned representations that reflect highly fine-grained visuo-spatial information. Using pre-trained text-to-image diffusion models, we construct Stable Control Representations which allow learning downstream control policies that generalize to complex, open-ended environments. We show that policies learned using Stable Control Representations are competitive with state-of-the-art representation learning approaches across a broad range of simulated control settings, encompassing challenging manipulation and navigation tasks. Most notably, we show that Stable Control Representations enable learning policies that exhibit state-of-the-art performance on OVMM, a difficult open-vocabulary navigation benchmark.

This letter summarizes and proves the concept of bounded-input bounded-state (BIBS) stability for weight convergence of a broad family of in-parameter-linear nonlinear neural architectures as it generally applies to a broad family of incremental gradient learning algorithms. A practical BIBS convergence condition results from the derived proofs for every individual learning point or batches for real-time applications.

This is an extended version of our publication Learning state machines from data streams: A generic strategy and an improved heuristic, International Conference on Grammatical Inference (ICGI) 2023, Rabat, Morocco. It has been extended with a formal proof on PAC-bounds, and the discussion and analysis of a similar approach has been moved from the appendix and now has a full dedicated section. State machine models are models that simulate the behavior of discrete event systems, capable of representing systems such as software systems, network interactions, and control systems, and have been researched extensively. The nature of most learning algorithms however is the assumption that all data be available at the beginning of the algorithm, and little research has been done in learning state machines from streaming data. In this paper, we want to close this gap further by presenting a generic method for learning state machines from data streams, as well as a merge heuristic that uses sketches to account for incomplete prefix trees. We implement our approach in an open-source state merging library and compare it with existing methods. We show the effectiveness of our approach with respect to run-time, memory consumption, and quality of results on a well known open dataset. Additionally, we provide a formal analysis of our algorithm, showing that it is capable of learning within the PAC framework, and show a theoretical improvement to increase run-time, without sacrificing correctness of the algorithm in larger sample sizes.

We propose the Representation World Model (RWM), which learns states, transitions, and executable plans directly in representation space. Unlike existing world models that typically learn latent representations together with explicit dynamics models and perform planning through search, optimization, or policy-based prediction, RWM directly incorporates planning into the learned representation geometry. RWM learns the representation geometry by applying inverse-dynamics supervision locally along latent paths constructed from endpoint representations, requiring these paths to preserve task-relevant state and transition information. At inference, planning is performed by directly constructing a latent path between the current and goal representations, with inverse dynamics used to recover the corresponding actions, without recursive rollouts or action-space search. Experiments on continuous-control benchmarks demonstrate the effectiveness of RWM for direct planning, while results on robotic manipulation further show its potential to extend to more complex embodied control tasks. These results suggest that planning directly in representation space provides a promising alternative to conventional world-model planning.

One method for obtaining generalizable solutions to machine learning tasks when presented with diverse training environments is to find invariant representations of the data. These are representations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a {\em finite sample} setting, we consider the notion of epsilon-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen SEMs? This larger collection of SEMs is generated through a parameterized family of interventions. Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds probabilistically over a family of linear SEMs without faithfulness assumptions. Our results show bounds that do not scale in ambient dimension when intervention sites are restricted to lie in a constant size subset of in-degree bounded nodes. We also show how to extend our results to a linear indirect observation model that incorporates latent variables.

How do sequence models represent their decision-making process? Prior work suggests that Othello-playing neural network learned nonlinear models of the board state (Li et al., 2023). In this work, we provide evidence of a closely related linear representation of the board. In particular, we show that probing for "my colour" vs. "opponent's colour" may be a simple yet powerful way to interpret the model's internal state. This precise understanding of the internal representations allows us to control the model's behaviour with simple vector arithmetic. Linear representations enable significant interpretability progress, which we demonstrate with further exploration of how the world model is computed.

We show that the MinMax algebra provides a form of recurrence that is expressively powerful, efficiently implementable, and most importantly it is not affected by vanishing or exploding gradient. We call MinMax Recurrent Neural Cascades (RNCs) the models obtained by cascading several layers of neurons that employ such recurrence. We show that MinMax RNCs enjoy many favourable theoretical properties. First, their formal expressivity includes all regular languages, arguably the maximal expressivity for a finite-memory system. Second, they can be evaluated in parallel with a runtime that is logarithmic in the input length given enough processors; and they can also be evaluated sequentially. Third, their state and activations are bounded uniformly for all input lengths. Fourth, at almost all points, their loss gradient exists and it is bounded. Fifth, they do not exhibit a vanishing state gradient: the gradient of a state w.r.t. a past state can have constant value one regardless of the time distance between the two states. Finally, we find empirical evidence that the favourable theoretical properties of MinMax RNCs are matched by their practical capabilities: they are able to perfectly solve a number of synthetic tasks, showing superior performance compared to the considered state-of-the-art recurrent neural networks; also, we train a MinMax RNC of 127M parameters on next-token prediction, and the obtained model shows competitive performance for its size, providing evidence of the potential of MinMax RNCs on real-world tasks.

A recurring challenge in self-supervised learning is preventing representation collapse. Existing solutions typically rely on global regularization, such as maximizing distances, decorrelating dimensions or enforcing certain distributions. We instead reinterpret representation learning as a discrete packing problem, where preserving information simplifies to maintaining injectivity. We operationalize this in Hypersolid, a method using short-range hard-ball repulsion to prevent local collisions. This constraint results in a high-separation geometric regime that preserves augmentation diversity, excelling on fine-grained and low-resolution classification tasks.

While researchers are finding concepts represented as linear directions in language models, a bag of linear directions fails to capture relational structure. To better understand this dichotomy, we study a model with known linear representations, but trained in a highly structured domain -- the board game Othello. While the model's internal board-state representation is linearly decodable, we find additional structure in the form of tensor product representations (TPRs). We train TPR probes to recover shared structure amongst the linear probes, yielding a factorization into square-embeddings, color-embeddings, and a binding matrix that composes them to construct the model's board-state representation. We find geometric signatures within the weights of our TPR probe that align with the structure of the board, but perhaps more importantly, that the linear probes can be recovered directly from the parameters of our TPR probe. Our findings suggest that directional representations may be projections of more structured underlying representations.

Learned representations are a central component in modern ML systems, serving a multitude of downstream tasks. When training such representations, it is often the case that computational and statistical constraints for each downstream task are unknown. In this context rigid, fixed capacity representations can be either over or under-accommodating to the task at hand. This leads us to ask: can we design a flexible representation that can adapt to multiple downstream tasks with varying computational resources? Our main contribution is Matryoshka Representation Learning (MRL) which encodes information at different granularities and allows a single embedding to adapt to the computational constraints of downstream tasks. MRL minimally modifies existing representation learning pipelines and imposes no additional cost during inference and deployment. MRL learns coarse-to-fine representations that are at least as accurate and rich as independently trained low-dimensional representations. The flexibility within the learned Matryoshka Representations offer: (a) up to 14x smaller embedding size for ImageNet-1K classification at the same level of accuracy; (b) up to 14x real-world speed-ups for large-scale retrieval on ImageNet-1K and 4K; and (c) up to 2% accuracy improvements for long-tail few-shot classification, all while being as robust as the original representations. Finally, we show that MRL extends seamlessly to web-scale datasets (ImageNet, JFT) across various modalities -- vision (ViT, ResNet), vision + language (ALIGN) and language (BERT). MRL code and pretrained models are open-sourced at https://github.com/RAIVNLab/MRL.

Offline reinforcement learning (RL) can in principle synthesize more optimal behavior from a dataset consisting only of suboptimal trials. One way that this can happen is by "stitching" together the best parts of otherwise suboptimal trajectories that overlap on similar states, to create new behaviors where each individual state is in-distribution, but the overall returns are higher. However, in many interesting and complex applications, such as autonomous navigation and dialogue systems, the state is partially observed. Even worse, the state representation is unknown or not easy to define. In such cases, policies and value functions are often conditioned on observation histories instead of states. In these cases, it is not clear if the same kind of "stitching" is feasible at the level of observation histories, since two different trajectories would always have different histories, and thus "similar states" that might lead to effective stitching cannot be leveraged. Theoretically, we show that standard offline RL algorithms conditioned on observation histories suffer from poor sample complexity, in accordance with the above intuition. We then identify sufficient conditions under which offline RL can still be efficient -- intuitively, it needs to learn a compact representation of history comprising only features relevant for action selection. We introduce a bisimulation loss that captures the extent to which this happens, and propose that offline RL can explicitly optimize this loss to aid worst-case sample complexity. Empirically, we show that across a variety of tasks either our proposed loss improves performance, or the value of this loss is already minimized as a consequence of standard offline RL, indicating that it correlates well with good performance.

World generative models are typically used through what they produce: a rendered future, a video-conditioned action, or latent context computed by a costly generative branch. We argue that their more reusable asset is the computation that constructs a future. As a generator transforms a corrupted future into a coherent trajectory, its intermediate states organize appearance, spatial layout, and interaction across levels of abstraction. Can this future-generative computation be internalized in a representation inferred from the present alone? We present Enfold, which transfers this computation into a representation predicted from the current visual context and language instruction. During training, multi-level states exposed as the generator processes the observed future supervise a current-only encoder. The learned representation is fed back to condition future generation and is read by task heads without allowing task gradients to reshape the encoder. At deployment, action prediction no longer executes the generator. Across LIBERO, RoboTwin2.0, and real-robot tasks, Enfold supports strong control while reducing action latency by 3.7times relative to Fast--WAM, Enfold-Flash reaches 10.1times. Representation analyses show that it suppresses nuisance variation and preferentially captures changes that emerge over longer horizons. When the current scene is altered by human intervention, both the generated continuation and the executed actions adapt, which is inconsistent with fixed trajectory replay. These results recast a world generator as a source of predictive control representations: its future need not be materialized at every step if its internal structure can be enfolded into the present.

Predicting and reasoning about the future lie at the heart of many time-series questions. For example, goal-conditioned reinforcement learning can be viewed as learning representations to predict which states are likely to be visited in the future. While prior methods have used contrastive predictive coding to model time series data, learning representations that encode long-term dependencies usually requires large amounts of data. In this paper, we introduce a temporal difference version of contrastive predictive coding that stitches together pieces of different time series data to decrease the amount of data required to learn predictions of future events. We apply this representation learning method to derive an off-policy algorithm for goal-conditioned RL. Experiments demonstrate that, compared with prior RL methods, ours achieves 2 times median improvement in success rates and can better cope with stochastic environments. In tabular settings, we show that our method is about 20 times more sample efficient than the successor representation and 1500 times more sample efficient than the standard (Monte Carlo) version of contrastive predictive coding.

Learning compatible representations enables the interchangeable use of semantic features as models are updated over time. This is particularly relevant in search and retrieval systems where it is crucial to avoid reprocessing of the gallery images with the updated model. While recent research has shown promising empirical evidence, there is still a lack of comprehensive theoretical understanding about learning compatible representations. In this paper, we demonstrate that the stationary representations learned by the d-Simplex fixed classifier optimally approximate compatibility representation according to the two inequality constraints of its formal definition. This not only establishes a solid foundation for future works in this line of research but also presents implications that can be exploited in practical learning scenarios. An exemplary application is the now-standard practice of downloading and fine-tuning new pre-trained models. Specifically, we show the strengths and critical issues of stationary representations in the case in which a model undergoing sequential fine-tuning is asynchronously replaced by downloading a better-performing model pre-trained elsewhere. Such a representation enables seamless delivery of retrieval service (i.e., no reprocessing of gallery images) and offers improved performance without operational disruptions during model replacement. Code available at: https://github.com/miccunifi/iamcl2r.

Reinforcement learning (RL) has become a standard paradigm for post-training and aligning Large Language Models (LLMs), yet recent evidence suggests it faces a persistent "capability ceiling": unlike classical RL systems that discover novel strategies, RL for LLMs often acts as a mere refiner of patterns already latent in pre-trained weights. In this work, we identify a fundamental structural bottleneck: while classical RL relies on compact, informative Markov states, current LLM post-training formulations are tethered to an ever-expanding history of actions. We revisit a classical principle long central to RL yet absent from LLM post-training: explicit Markov states. Theoretically, we provide rigorous guarantees demonstrating that leveraging estimated Markov states can significantly reduce sample complexity. Empirically, we show that introducing Markov states consistently breaks the performance boundaries of standard RL post-training across a suite of complex logic puzzles. Our findings suggest that moving beyond "history-as-state" modeling in favor of structured Markovian representations is essential for unlocking open-ended discovery and genuinely new reasoning capabilities in Generative AI.

Transformer-based architectures have dominated sequence modeling, largely due to the expressive power of attention mechanisms. However, for a class of deterministic state tracking tasks---such as parity checking, modular counting, and parenthesis matching---attention may be overkill. In this paper, we show that state propagation alone is sufficient. We propose the Complex State Propagator (CSP), a minimalistic recurrent architecture that only propagates hidden states across layers without output projections at intermediate steps. The state is represented as a complex-valued vector, updated via input-dependent rotations in the complex domain. To enable deep propagation without gradient vanishing or degradation, we introduce a block-level skip connection alongside element-wise complex normalization and SiLU activation at sequence boundaries. Applied with Focal Loss, CSP achieves 100\% accuracy with perfect F1 scores across canonical tasks.

Recently, unsupervised representation learning (URL) has improved the sample efficiency of Reinforcement Learning (RL) by pretraining a model from a large unlabeled dataset. The underlying principle of these methods is to learn temporally predictive representations by predicting future states in the latent space. However, an important challenge of this approach is the representational collapse, where the subspace of the latent representations collapses into a low-dimensional manifold. To address this issue, we propose a novel URL framework that causally predicts future states while increasing the dimension of the latent manifold by decorrelating the features in the latent space. Through extensive empirical studies, we demonstrate that our framework effectively learns predictive representations without collapse, which significantly improves the sample efficiency of state-of-the-art URL methods on the Atari 100k benchmark. The code is available at https://github.com/dojeon-ai/SimTPR.

As systems trend toward superintelligence, a natural modeling premise is that agents can self-improve along every facet of their own design. We formalize this with a five-axis decomposition and a decision layer, separating incentives from learning behavior and analyzing axes in isolation. Our central result identifies and introduces a sharp utility--learning tension, the structural conflict in self-modifying systems whereby utility-driven changes that improve immediate or expected performance can also erode the statistical preconditions for reliable learning and generalization. Our findings show that distribution-free guarantees are preserved iff the policy-reachable model family is uniformly capacity-bounded; when capacity can grow without limit, utility-rational self-changes can render learnable tasks unlearnable. Under standard assumptions common in practice, these axes reduce to the same capacity criterion, yielding a single boundary for safe self-modification. Numerical experiments across several axes validate the theory by comparing destructive utility policies against our proposed two-gate policies that preserve learnability.

We develop an econometric framework integrating heavy-tailed Student's t distributions with behavioral probability weighting while preserving infinite divisibility. Using 432{,}752 observations across 86 assets (2004--2024), we demonstrate Student's t specifications outperform Gaussian models in 88.4\% of cases. Bounded probability-weighting transformations preserve mathematical properties required for dynamic pricing. Gaussian models underestimate 99\% Value-at-Risk by 19.7\% versus 3.2\% for our specification. Joint estimation procedures identify tail and behavioral parameters with established asymptotic properties. Results provide robust inference for asset-pricing applications where heavy tails and behavioral distortions coexist.

A common assumption holds that enough observational and interventional data, given to a strong enough predictor, suffices. We report a failure mode that contradicts it. Across hundreds of structural causal models, on identified quantities a strong predictor and a Bayesian baseline both succeed, but on unidentified quantities (the couplings between counterfactual worlds) the predictor collapses to a point, on 28% of models to one no valid model can produce, while the truth is an admissible interval more data never narrows. The gap is structural: prediction cannot represent uncertainty over counterfactual couplings. We cast a world model as a single positive semidefinite coupling kernel K(T,T') over admissible worlds, whose diagonal is the ordinary posterior (what a predictor recovers) and whose off-diagonal is the cross-world coupling it cannot, which every counterfactual reads. The paper is the theory of that off-diagonal. It is real: two states with identical posteriors differ on a cross-world query, and the off-diagonal is the coupling that fixes counterfactuals. It can be bounded: positive semidefiniteness is partial-identifying information the marginals lack, and enforcing it bounds counterfactuals in polynomial time where the exact response-type program is intractable. Logical structure sharpens it: ontology axioms tighten the bound by up to a third, propagating to couplings they never touch. It can be acquired: targeted scars, constraints learned from encountered infeasibilities, close the gap several times faster than untargeted ones. Its full reconstruction is approximate counting of the admissible worlds, tractable below the Sly-Sun threshold and inapproximable above; we do not claim to beat the worst case.

We consider the question of how to employ next-token prediction algorithms in adversarial online decision-making environments. Specifically, if we train a next-token prediction model on a distribution D over sequences of opponent actions, when is it the case that the induced online decision-making algorithm (by approximately best responding to the model's predictions) has low adversarial regret (i.e., when is D a low-regret distribution)? For unbounded context windows (where the prediction made by the model can depend on all the actions taken by the adversary thus far), we show that although not every distribution D is a low-regret distribution, every distribution D is exponentially close (in TV distance) to one low-regret distribution, and hence sublinear regret can always be achieved at negligible cost to the accuracy of the original next-token prediction model. In contrast to this, for bounded context windows (where the prediction made by the model can depend only on the past w actions taken by the adversary, as may be the case in modern transformer architectures), we show that there are some distributions D of opponent play that are Θ(1)-far from any low-regret distribution D' (even when w = Ω(T) and such distributions exist). Finally, we complement these results by showing that the unbounded context robustification procedure can be implemented by layers of a standard transformer architecture, and provide empirical evidence that transformer models can be efficiently trained to represent these new low-regret distributions.

As machine learning has moved towards leveraging large models as priors for downstream tasks, the community has debated the right form of prior for solving reinforcement learning (RL) problems. If one were to try to prefetch as much computation as possible, they would attempt to learn a prior over the policies for some yet-to-be-determined reward function. Recent work (forward-backward (FB) representation learning) has tried this, arguing that an unsupervised representation learning procedure can enable optimal control over arbitrary rewards without further fine-tuning. However, FB's training objective and learning behavior remain mysterious. In this paper, we demystify FB by clarifying when such representations can exist, what its objective optimizes, and how it converges in practice. We draw connections with rank matching, fitted Q-evaluation, and contraction mapping. Our analysis suggests a simplified unsupervised pre-training method for RL that, instead of enabling optimal control, performs one step of policy improvement. We call our proposed method one-step forward-backward representation learning (one-step FB). Experiments in didactic settings, as well as in 10 state-based and image-based continuous control domains, demonstrate that one-step FB converges to errors 10^5 smaller and improves zero-shot performance by +24% on average. Our project website is available at https://chongyi-zheng.github.io/onestep-fb.

Representational similarity is foundational to analyses of deep networks, yet distances between point-valued representations are not intrinsically tied to downstream function: nearby states may produce different behaviors, while distant states may behave similarly. We instead give representations volume, turning similarity into statistical distinguishability. Overlapping stochastic representations necessarily induce overlapping downstream distributions, grounding latent comparison in model function and bringing it under information-theoretic tools such as the data-processing inequality. We realize this idea in pretrained transformers through a light-touch modification to LayerNorm: at each residual-stream read, we normalize the state, add isotropic Gaussian noise, and renormalize. During distillation fine-tuning, one learned allocation parameter per residual-stream read distributes a fixed global rate budget across the processing stack. The resulting model can be viewed as transformer blocks reading the residual stream with learned finite precision under a shared global rate budget. Using the Bhattacharyya coefficient, we trace which counterfactual distinctions are preserved through MLP blocks or selectively exposed to the query, key, and value computations of individual attention heads. Experiments on ViT-S and GPT-2 small reveal the depthwise propagation of continuous visual perturbations and head-specific sensitivity to token distinctions aligned with known attention motifs. These results establish distinguishability as a functionally grounded lens on transformer computation that complements existing interpretability approaches.

The successes of modern deep machine learning methods are founded on their ability to transform inputs across multiple layers to build good high-level representations. It is therefore critical to understand this process of representation learning. However, standard theoretical approaches (formally NNGPs) involving infinite width limits eliminate representation learning. We therefore develop a new infinite width limit, the Bayesian representation learning limit, that exhibits representation learning mirroring that in finite-width models, yet at the same time, retains some of the simplicity of standard infinite-width limits. In particular, we show that Deep Gaussian processes (DGPs) in the Bayesian representation learning limit have exactly multivariate Gaussian posteriors, and the posterior covariances can be obtained by optimizing an interpretable objective combining a log-likelihood to improve performance with a series of KL-divergences which keep the posteriors close to the prior. We confirm these results experimentally in wide but finite DGPs. Next, we introduce the possibility of using this limit and objective as a flexible, deep generalisation of kernel methods, that we call deep kernel machines (DKMs). Like most naive kernel methods, DKMs scale cubically in the number of datapoints. We therefore use methods from the Gaussian process inducing point literature to develop a sparse DKM that scales linearly in the number of datapoints. Finally, we extend these approaches to NNs (which have non-Gaussian posteriors) in the Appendices.

We introduce a differentiable estimator of range-partition entropy, a recent concept from computational geometry that enables algorithms to adapt to the "sortedness" of their input. While range-partition entropy provides strong guarantees in algorithm design, it has not yet been made accessible to deep learning. In this work, we (i) propose the first differentiable approximation of range-partition entropy, enabling its use as a trainable loss or regularizer; (ii) design EntropyNet, a neural module that restructures data into low-entropy forms to accelerate downstream instance-optimal algorithms; and (iii) extend this principle beyond geometry by applying entropy regularization directly to Transformer attention. Across tasks, we demonstrate that differentiable entropy improves efficiency without degrading correctness: in geometry, our method achieves up to 4.1times runtime speedups with negligible error (<0.2%); in deep learning, it induces structured attention patterns that yield 6% higher accuracy at 80% sparsity compared to L1 baselines. Our theoretical analysis provides approximation bounds for the estimator, and extensive ablations validate design choices. These results suggest that entropy-bounded computation is not only theoretically elegant but also a practical mechanism for adaptive learning, efficiency, and structured representation.

In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic representation, then verify it satisfies the Cauchy problem. The corresponding SDE, on the other hand, allows the drift and diffusion coefficients to respond nonlinearly to the current state through the state-dependent variable exponents, and thus, extends the expressive power of classical SDEs to better capture complex dynamics. To validate our theoretical framework, we conduct comprehensive numerical experiments comparing finite difference solutions (Crank-Nicolson on logarithmic grids) with Monte Carlo simulations of the SDE.

We study pre-training representations for decision-making using video data, which is abundantly available for tasks such as game agents and software testing. Even though significant empirical advances have been made on this problem, a theoretical understanding remains absent. We initiate the theoretical investigation into principled approaches for representation learning and focus on learning the latent state representations of the underlying MDP using video data. We study two types of settings: one where there is iid noise in the observation, and a more challenging setting where there is also the presence of exogenous noise, which is non-iid noise that is temporally correlated, such as the motion of people or cars in the background. We study three commonly used approaches: autoencoding, temporal contrastive learning, and forward modeling. We prove upper bounds for temporal contrastive learning and forward modeling in the presence of only iid noise. We show that these approaches can learn the latent state and use it to do efficient downstream RL with polynomial sample complexity. When exogenous noise is also present, we establish a lower bound result showing that the sample complexity of learning from video data can be exponentially worse than learning from action-labeled trajectory data. This partially explains why reinforcement learning with video pre-training is hard. We evaluate these representational learning methods in two visual domains, yielding results that are consistent with our theoretical findings.

We study the problem of counting the number of distinct elements in a dataset subject to the constraint of differential privacy. We consider the challenging setting of person-level DP (a.k.a. user-level DP) where each person may contribute an unbounded number of items and hence the sensitivity is unbounded. Our approach is to compute a bounded-sensitivity version of this query, which reduces to solving a max-flow problem. The sensitivity bound is optimized to balance the noise we must add to privatize the answer against the error of the approximation of the bounded-sensitivity query to the true number of unique elements.

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.

The theory of state tracking in recurrent architectures has predominantly focused on expressive capacity: whether a fixed architecture can theoretically realize a set of symbolic transition rules. We argue that equally important is error control, the dynamics governing hidden-state drift along the directions that distinguish symbolic states. We prove that affine recurrent networks, a class of models encompassing State-Space Models and Linear Attention, cannot correct errors along state-separating subspaces once they preserve state representations. Consequently, practical affine trackers do not learn robust state tracking; rather, they learn finite horizon solutions governed by accumulated state-relevant error. We characterize the mechanics of this failure, showing that tracking remains readable only while the accumulating within-class spread remains small relative to the initial between-class separation. We demonstrate empirically on group state-tracking tasks that this breakdown is predictable: tracking collapses when the distinguishability ratio crosses the readability threshold of the trained decoder. Across trained models, the point of this crossing predicts the horizon at which downstream accuracy fails. These results establish that robust state tracking is determined not only by an architecture's theoretical expressivity but crucially by its error control.

We describe a framework for using natural language to design state abstractions for imitation learning. Generalizable policy learning in high-dimensional observation spaces is facilitated by well-designed state representations, which can surface important features of an environment and hide irrelevant ones. These state representations are typically manually specified, or derived from other labor-intensive labeling procedures. Our method, LGA (language-guided abstraction), uses a combination of natural language supervision and background knowledge from language models (LMs) to automatically build state representations tailored to unseen tasks. In LGA, a user first provides a (possibly incomplete) description of a target task in natural language; next, a pre-trained LM translates this task description into a state abstraction function that masks out irrelevant features; finally, an imitation policy is trained using a small number of demonstrations and LGA-generated abstract states. Experiments on simulated robotic tasks show that LGA yields state abstractions similar to those designed by humans, but in a fraction of the time, and that these abstractions improve generalization and robustness in the presence of spurious correlations and ambiguous specifications. We illustrate the utility of the learned abstractions on mobile manipulation tasks with a Spot robot.

Recently, recurrent models such as state space models and linear attention have become popular due to their linear complexity in the sequence length. Thanks to their recurrent nature, in principle they can process arbitrarily long sequences, but their performance sometimes drops considerably beyond their training context lengths-i.e. they fail to length generalize. In this work, we provide comprehensive empirical and theoretical analysis to support the unexplored states hypothesis, which posits that models fail to length generalize when during training they are only exposed to a limited subset of the distribution of all attainable states (i.e. states that would be attained if the recurrence was applied to long sequences). Furthermore, we investigate simple training interventions that aim to increase the coverage of the states that the model is trained on, e.g. by initializing the state with Gaussian noise or with the final state of a different input sequence. With only 500 post-training steps (sim 0.1% of the pre-training budget), these interventions enable length generalization for sequences that are orders of magnitude longer than the training context (e.g. 2klongrightarrow 128k) and show improved performance in long context tasks, thus presenting a simple and efficient way to enable robust length generalization in general recurrent models.

We study the autonomous exploration (AX) problem proposed by Lim & Auer (2012). In this setting, the objective is to discover a set of epsilon-optimal policies reaching a set S_L^{rightarrow} of incrementally L-controllable states. We introduce a novel layered decomposition of the set of incrementally L-controllable states that is based on the iterative application of a state-expansion operator. We leverage these results to design Layered Autonomous Exploration (LAE), a novel algorithm for AX that attains a sample complexity of mathcal{O}(LS^{rightarrow}_{L(1+epsilon)}Gamma_{L(1+epsilon)} A ln^{12}(S^{rightarrow}_{L(1+epsilon)})/epsilon^2), where S^{rightarrow}_{L(1+epsilon)} is the number of states that are incrementally L(1+epsilon)-controllable, A is the number of actions, and Gamma_{L(1+epsilon)} is the branching factor of the transitions over such states. LAE improves over the algorithm of Tarbouriech et al. (2020a) by a factor of L^2 and it is the first algorithm for AX that works in a countably-infinite state space. Moreover, we show that, under a certain identifiability assumption, LAE achieves minimax-optimal sample complexity of mathcal{O}(LS^{rightarrow}_{L}Aln^{12}(S^{rightarrow}_{L})/epsilon^2), outperforming existing algorithms and matching for the first time the lower bound proved by Cai et al. (2022) up to logarithmic factors.

Evaluations of LLM planning agents largely ask whether a task succeeds or a declared plan is followed. In strategic cyber-physical systems, a stronger question is whether the planning architecture remains appropriate after autonomous participants respond and physics constrains the outcome. We introduce a controlled, physics-grounded benchmark built around planning-induced control trajectories: the ordered planning operations and directives through which an execution architecture acts on other agents and the physical process. It implements predefined, sequential, hierarchical, and search executors in a smart-grid demand-response system with 40 heterogeneous prosumers and an independently simulated radial feeder. The LLM is bounded to typed policy declaration and short operator messages, while schedule construction, prosumer dynamics, and power flow remain explicit code. The protocol uses paired forced-mode counterfactuals, common random response draws, and event-level deadline feasibility. Three properties follow. Architecture materially changes outcomes: forced search is the oracle in all five baseline seeds. Execution fidelity needs more than mode agreement: objective substitution holds agreement at 1.0 while increasing voltage shortfall by 2.68x. A 144-scenario, 576-episode bank has feasible oracles from three of the four architectures. A prespecified stress-held-out ridge has mean regret 90.7 (95% interval [73.8, 108.6]) and no detectable value over fixed sequential; applying known deadline feasibility before quality prediction cuts regret to 29.0 and improves over fixed sequential by 61.1. An all-feasible ablation does not beat fixed search, localising the remaining challenge to within-feasible quality selection. A five-model extension separates stress-conditioned, state-blind, and invariant declarers; latency tails show that live feasibility should be treated probabilistically.

Recent years have seen the emergence of pre-trained representations as a powerful abstraction for AI applications in computer vision, natural language, and speech. However, policy learning for control is still dominated by a tabula-rasa learning paradigm, with visuo-motor policies often trained from scratch using data from deployment environments. In this context, we revisit and study the role of pre-trained visual representations for control, and in particular representations trained on large-scale computer vision datasets. Through extensive empirical evaluation in diverse control domains (Habitat, DeepMind Control, Adroit, Franka Kitchen), we isolate and study the importance of different representation training methods, data augmentations, and feature hierarchies. Overall, we find that pre-trained visual representations can be competitive or even better than ground-truth state representations to train control policies. This is in spite of using only out-of-domain data from standard vision datasets, without any in-domain data from the deployment environments. Source code and more at https://sites.google.com/view/pvr-control.

We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator P: L^2(M,μ) to L^2(M,μ) encodes the propagation of semantic mass. Under mild regularity assumptions (compactness, ergodicity, bounded Jacobian), P is compact with discrete spectrum. Within this setting, we prove the Semantic Characterization Theorem (SCT): the leading eigenfunctions of P induce finitely many spectral basins of invariant meaning, each definable in an o-minimal structure over R. Thus spectral lumpability and logical tameness coincide. This explains how discrete symbolic semantics can emerge from continuous computation: the continuous activation manifold collapses into a finite, logically interpretable ontology. We further extend the SCT to stochastic and adiabatic (time-inhomogeneous) settings, showing that slowly drifting kernels preserve compactness, spectral coherence, and basin structure.

Large Language Models (LLMs) have benefited enormously from scaling, yet these gains are bounded by five fundamental limitations: (1) hallucination, (2) context compression, (3) reasoning degradation, (4) retrieval fragility, and (5) multimodal misalignment. While existing surveys describe these phenomena empirically, they lack a rigorous theoretical synthesis connecting them to the foundational limits of computation, information, and learning. This work closes that gap by presenting a unified, proof-informed framework that formalizes the innate theoretical ceilings of LLM scaling. First, computability and uncomputability imply an irreducible residue of error: for any computably enumerable model family, diagonalization guarantees inputs on which some model must fail, and undecidable queries (e.g., halting-style tasks) induce infinite failure sets for all computable predictors. Second, information-theoretic and statistical constraints bound attainable accuracy even on decidable tasks, finite description length enforces compression error, and long-tail factual knowledge requires prohibitive sample complexity. Third, geometric and computational effects compress long contexts far below their nominal size due to positional under-training, encoding attenuation, and softmax crowding. We further show how likelihood-based training favors pattern completion over inference, how retrieval under token limits suffers from semantic drift and coupling noise, and how multimodal scaling inherits shallow cross-modal alignment. Across sections, we pair theorems and empirical evidence to outline where scaling helps, where it saturates, and where it cannot progress, providing both theoretical foundations and practical mitigation paths like bounded-oracle retrieval, positional curricula, and sparse or hierarchical attention.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict g_1 star g_2 for elements of a finite group G. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

How should hidden states generated autoregressively be collapsed into a representation that reflects a language model's internal state? Despite tokens being generated under causal masking, we find that mean pooling across their hidden states yields more semantic representations than any individual token alone. We quantify this through kernel alignment to reference spaces in language, vision, and protein domains. The improvement through mean pooling is consistent with information being distributed across generated tokens rather than localized to a single position. Furthermore, representations derived from generated tokens outperform those from prompt tokens, and alignment across generation reveals interpretable dynamics in model behavior.

LLM agents excel when environments are mostly static and the needed information fits in a model's context window, but they often fail in open-ended investigations where explanations must be constructed by iteratively mining evidence from massive, heterogeneous operational data. These investigations exhibit hidden dependency structure: entities interact, signals co-vary, and the importance of a fact may only become clear after other evidence is discovered. Because the context window is bounded, agents must summarize intermediate findings before their significance is known, increasing the risk of discarding key evidence. ReAct-style agents are especially brittle in this regime. Their retrieve-summarize-reason loop makes conclusions sensitive to exploration order and introduces run-to-run non-determinism, producing a reliability gap where Pass-at-k may be high but Majority-at-k remains low. Simply sampling more rollouts or generating longer reasoning traces does not reliably stabilize results, since hypotheses cannot be autonomously checked as new evidence arrives and there is no explicit mechanism for belief bookkeeping and revision. In addition, ReAct entangles semantic reasoning with controller duties such as tool orchestration and state tracking, so execution errors and plan drift degrade reasoning while consuming scarce context. We address these issues by formulating investigation as abductive reasoning over a dependency graph and proposing EoG (Explanations over Graphs), a disaggregated framework in which an LLM performs bounded local evidence mining and labeling (cause vs symptom) while a deterministic controller manages traversal, state, and belief propagation to compute a minimal explanatory frontier. On a representative ITBench diagnostics task, EoG improves both accuracy and run-to-run consistency over ReAct baselines, including a 7x average gain in Majority-at-k entity F1.

Policy-adherent tool-calling agents in customer-service domains must maintain task states across turns while calling tools and obeying domain policies. Task states consist of relevant facts, identifiers, constraints, and conditions observed through user interaction and tool calls. In standard agents, task states are not represented separately. Observations, tool returns, and policy instructions are placed in the prompt, leaving agents to reconstruct the relevant states from the prompt each time they decide what to do next. This design makes state management implicit, creating two common failure modes. An agent may retrieve the right facts but later ground its decision in stale, missing, or incorrect information; and a syntactically valid tool call may still violate a domain policy that depends on the current task state. We introduce LedgerAgent, an inference-time method for tool-calling agents that maintains observed task states in a separate ledger and renders the states into the prompt. The ledger is also used to check state-dependent policy constraints before environment-changing tool calls are executed, blocking policy violations. Across four customer-service domains and a mixed panel of open- and closed-weight models, LedgerAgent improves average passk over a standard prompt-based tool-calling approach, with the largest gains under stricter multi-trial consistency metrics.

Set representation has become ubiquitous in deep learning for modeling the inductive bias of neural networks that are insensitive to the input order. DeepSets is the most widely used neural network architecture for set representation. It involves embedding each set element into a latent space with dimension L, followed by a sum pooling to obtain a whole-set embedding, and finally mapping the whole-set embedding to the output. In this work, we investigate the impact of the dimension L on the expressive power of DeepSets. Previous analyses either oversimplified high-dimensional features to be one-dimensional features or were limited to analytic activations, thereby diverging from practical use or resulting in L that grows exponentially with the set size N and feature dimension D. To investigate the minimal value of L that achieves sufficient expressive power, we present two set-element embedding layers: (a) linear + power activation (LP) and (b) linear + exponential activations (LE). We demonstrate that L being poly(N, D) is sufficient for set representation using both embedding layers. We also provide a lower bound of L for the LP embedding layer. Furthermore, we extend our results to permutation-equivariant set functions and the complex field.

State-space models (SSMs) are a new class of foundation models that have emerged as a compelling alternative to Transformers and their attention mechanisms for sequence processing tasks. This paper provides a detailed theoretical analysis of selective SSMs, the core components of the Mamba and Mamba-2 architectures. We leverage the connection between selective SSMs and the self-attention mechanism to highlight the fundamental similarities between these models. Building on this connection, we establish a length independent covering number-based generalization bound for selective SSMs, providing a deeper understanding of their theoretical performance guarantees. We analyze the effects of state matrix stability and input-dependent discretization, shedding light on the critical role played by these factors in the generalization capabilities of selective SSMs. Finally, we empirically demonstrate the sequence length independence of the derived bounds on two tasks.

We study regret minimization for reinforcement learning (RL) in Latent Markov Decision Processes (LMDPs) with context in hindsight. We design a novel model-based algorithmic framework which can be instantiated with both a model-optimistic and a value-optimistic solver. We prove an O(mathsf{Var^star M Gamma S A K}) regret bound where O hides logarithm factors, M is the number of contexts, S is the number of states, A is the number of actions, K is the number of episodes, Gamma le S is the maximum transition degree of any state-action pair, and Var^star is a variance quantity describing the determinism of the LMDP. The regret bound only scales logarithmically with the planning horizon, thus yielding the first (nearly) horizon-free regret bound for LMDP. This is also the first problem-dependent regret bound for LMDP. Key in our proof is an analysis of the total variance of alpha vectors (a generalization of value functions), which is handled with a truncation method. We complement our positive result with a novel Omega(mathsf{Var^star M S A K}) regret lower bound with Gamma = 2, which shows our upper bound minimax optimal when Gamma is a constant for the class of variance-bounded LMDPs. Our lower bound relies on new constructions of hard instances and an argument inspired by the symmetrization technique from theoretical computer science, both of which are technically different from existing lower bound proof for MDPs, and thus can be of independent interest.

Diffusion models have become the go-to method for large-scale generative models in real-world applications. These applications often involve data distributions confined within bounded domains, typically requiring ad-hoc thresholding techniques for boundary enforcement. Reflected diffusion models (Lou23) aim to enhance generalizability by generating the data distribution through a backward process governed by reflected Brownian motion. However, reflected diffusion models may not easily adapt to diverse domains without the derivation of proper diffeomorphic mappings and do not guarantee optimal transport properties. To overcome these limitations, we introduce the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive elegant reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and explore natural connections to entropic optimal transport for the study of approximate linear convergence - a valuable insight for practical training. Our algorithm yields robust generative modeling in diverse domains, and its scalability is demonstrated in real-world constrained generative modeling through standard image benchmarks.

The field of image generation is currently bifurcated into autoregressive (AR) models operating on discrete tokens and diffusion models utilizing continuous latents. This divide, rooted in the distinction between VQ-VAEs and VAEs, hinders unified modeling and fair benchmarking. Finite Scalar Quantization (FSQ) offers a theoretical bridge, yet vanilla FSQ suffers from a critical flaw: its equal-interval quantization can cause activation collapse. This mismatch forces a trade-off between reconstruction fidelity and information efficiency. In this work, we resolve this dilemma by simply replacing the activation function in original FSQ with a distribution-matching mapping to enforce a uniform prior. Termed iFSQ, this simple strategy requires just one line of code yet mathematically guarantees both optimal bin utilization and reconstruction precision. Leveraging iFSQ as a controlled benchmark, we uncover two key insights: (1) The optimal equilibrium between discrete and continuous representations lies at approximately 4 bits per dimension. (2) Under identical reconstruction constraints, AR models exhibit rapid initial convergence, whereas diffusion models achieve a superior performance ceiling, suggesting that strict sequential ordering may limit the upper bounds of generation quality. Finally, we extend our analysis by adapting Representation Alignment (REPA) to AR models, yielding LlamaGen-REPA. Codes is available at https://github.com/Tencent-Hunyuan/iFSQ

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

State-of-the-art methods for conditional average treatment effect (CATE) estimation make widespread use of representation learning. Here, the idea is to reduce the variance of the low-sample CATE estimation by a (potentially constrained) low-dimensional representation. However, low-dimensional representations can lose information about the observed confounders and thus lead to bias, because of which the validity of representation learning for CATE estimation is typically violated. In this paper, we propose a new, representation-agnostic framework for estimating bounds on the representation-induced confounding bias that comes from dimensionality reduction (or other constraints on the representations) in CATE estimation. First, we establish theoretically under which conditions CATEs are non-identifiable given low-dimensional (constrained) representations. Second, as our remedy, we propose to perform partial identification of CATEs or, equivalently, aim at estimating of lower and upper bounds of the representation-induced confounding bias. We demonstrate the effectiveness of our bounds in a series of experiments. In sum, our framework is of direct relevance in practice where the validity of CATE estimation is of importance.

State-Space Models (SSMs) have recently emerged as a powerful and efficient alternative to the long-standing transformer architecture. However, existing SSM conceptualizations retain deeply rooted biases from their roots in natural language processing. This constrains their ability to appropriately model the spatially-dependent characteristics of visual inputs. In this paper, we address these limitations by re-deriving modern selective state-space techniques, starting from a natively multidimensional formulation. Currently, prior works attempt to apply natively 1D SSMs to 2D data (i.e. images) by relying on arbitrary combinations of 1D scan directions to capture spatial dependencies. In contrast, Mamba2D improves upon this with a single 2D scan direction that factors in both dimensions of the input natively, effectively modelling spatial dependencies when constructing hidden states. Mamba2D shows comparable performance to prior adaptations of SSMs for vision tasks, on standard image classification evaluations with the ImageNet-1K dataset. Source code is available at https://github.com/cocoalex00/Mamba2D.

State space models (SSMs) have shown impressive results on tasks that require modeling long-range dependencies and efficiently scale to long sequences owing to their subquadratic runtime complexity. Originally designed for continuous signals, SSMs have shown superior performance on a plethora of tasks, in vision and audio; however, SSMs still lag Transformer performance in Language Modeling tasks. In this work, we propose a hybrid layer named Block-State Transformer (BST), that internally combines an SSM sublayer for long-range contextualization, and a Block Transformer sublayer for short-term representation of sequences. We study three different, and completely parallelizable, variants that integrate SSMs and block-wise attention. We show that our model outperforms similar Transformer-based architectures on language modeling perplexity and generalizes to longer sequences. In addition, the Block-State Transformer demonstrates more than tenfold increase in speed at the layer level compared to the Block-Recurrent Transformer when model parallelization is employed.

Selective state-space models (SSMs) are an emerging alternative to the Transformer, offering the unique advantage of parallel training and sequential inference. Although these models have shown promising performance on a variety of tasks, their formal expressiveness and length generalization properties remain underexplored. In this work, we provide insight into the workings of selective SSMs by analyzing their expressiveness and length generalization performance on regular language tasks, i.e., finite-state automaton (FSA) emulation. We address certain limitations of modern SSM-based architectures by introducing the Selective Dense State-Space Model (SD-SSM), the first selective SSM that exhibits perfect length generalization on a set of various regular language tasks using a single layer. It utilizes a dictionary of dense transition matrices, a softmax selection mechanism that creates a convex combination of dictionary matrices at each time step, and a readout consisting of layer normalization followed by a linear map. We then proceed to evaluate variants of diagonal selective SSMs by considering their empirical performance on commutative and non-commutative automata. We explain the experimental results with theoretical considerations. Our code is available at https://github.com/IBM/selective-dense-state-space-model.

Reinforcement Learning (RL) has demonstrated tremendous empirical success across numerous challenging domains. However, we lack a strong theoretical understanding of the statistical complexity of RL in environments with large state spaces, where function approximation is required for sample-efficient learning. This thesis addresses this gap by rigorously examining the statistical complexity of RL with function approximation from a learning theoretic perspective. Departing from a long history of prior work, we consider the weakest form of function approximation, called agnostic policy learning, in which the learner seeks to find the best policy in a given class Pi, with no guarantee that Pi contains an optimal policy for the underlying task. We systematically explore agnostic policy learning along three key axes: environment access -- how a learner collects data from the environment; coverage conditions -- intrinsic properties of the underlying MDP measuring the expansiveness of state-occupancy measures for policies in the class Pi, and representational conditions -- structural assumptions on the class Pi itself. Within this comprehensive framework, we (1) design new learning algorithms with theoretical guarantees and (2) characterize fundamental performance bounds of any algorithm. Our results reveal significant statistical separations that highlight the power and limitations of agnostic policy learning.

Memory for a long-horizon LLM agent is a contract about what each future decision is allowed to see. The simplest contract appends past observations, tool calls, and reflections to every prompt, which makes prior context easy to access but also turns it into a jumbled mixture in which the effect of any single memory component is hard to isolate. We introduce and instrument an alternative bounded contract: every decision is made from a fresh user message assembled by typed retrieval, with no raw cross-decision transcript appended. The prompt thus stays bounded across runs of any length, and any single layer can be ablated in isolation. We instantiate the contract in Slay the Spire 2, a closed-rule stochastic deck-building game whose runs require hundreds of tactical and strategic decisions. A public online benchmark of frontier LLMs on the same game reports zero wins at the lowest difficulty across five configurations, and the developer-reported human win rate at the same difficulty is 16%; the task is hard but not saturated. Within our harness, a fixed-A0 ablation shows the largest observed difference when triggered strategic skills are enabled: the no-store baseline wins 3/10 games and adding the skill layer 6/10. At this sample size the comparison is directional rather than statistically decisive (Fisher exact p\approx0.37); a cross-backbone probe and public accumulating-context baselines are reported as operational comparisons rather than controlled tests of the contract variable itself. We release a reproducible testbed: 298 completed trajectories with condition tags, frozen memory/skill snapshots, prompt records, and analysis scripts -- an agent design and a validated, reusable methodology for studying how explicit memory layers shape long-horizon LLM-agent decisions.

User modeling is critical for developing personalized services in industry. A common way for user modeling is to learn user representations that can be distinguished by their interests or preferences. In this work, we focus on developing universal user representation model. The obtained universal representations are expected to contain rich information, and be applicable to various downstream applications without further modifications (e.g., user preference prediction and user profiling). Accordingly, we can be free from the heavy work of training task-specific models for every downstream task as in previous works. In specific, we propose Self-supervised User Modeling Network (SUMN) to encode behavior data into the universal representation. It includes two key components. The first one is a new learning objective, which guides the model to fully identify and preserve valuable user information under a self-supervised learning framework. The other one is a multi-hop aggregation layer, which benefits the model capacity in aggregating diverse behaviors. Extensive experiments on benchmark datasets show that our approach can outperform state-of-the-art unsupervised representation methods, and even compete with supervised ones.

State Space Models (SSMs) have emerged as a promising alternative to the popular transformer-based models and have been increasingly gaining attention. Compared to transformers, SSMs excel at tasks with sequential data or longer contexts, demonstrating comparable performances with significant efficiency gains. In this survey, we provide a coherent and systematic overview for SSMs, including their theoretical motivations, mathematical formulations, comparison with existing model classes, and various applications. We divide the SSM series into three main sections, providing a detailed introduction to the original SSM, the structured SSM represented by S4, and the selective SSM typified by Mamba. We put an emphasis on technicality, and highlight the various key techniques introduced to address the effectiveness and efficiency of SSMs. We hope this manuscript serves as an introduction for researchers to explore the theoretical foundations of SSMs.

In this paper, we propose a method to partially mimic natural intelligence for the problem of lifelong learning representations that are compatible. We take the perspective of a learning agent that is interested in recognizing object instances in an open dynamic universe in a way in which any update to its internal feature representation does not render the features in the gallery unusable for visual search. We refer to this learning problem as Compatible Lifelong Learning Representations (CL2R) as it considers compatible representation learning within the lifelong learning paradigm. We identify stationarity as the property that the feature representation is required to hold to achieve compatibility and propose a novel training procedure that encourages local and global stationarity on the learned representation. Due to stationarity, the statistical properties of the learned features do not change over time, making them interoperable with previously learned features. Extensive experiments on standard benchmark datasets show that our CL2R training procedure outperforms alternative baselines and state-of-the-art methods. We also provide novel metrics to specifically evaluate compatible representation learning under catastrophic forgetting in various sequential learning tasks. Code at https://github.com/NiccoBiondi/CompatibleLifelongRepresentation.

We introduce Psi-Turing Machines (Psi-TM): classical Turing machines equipped with a constant-depth introspection interface iota and an explicit per-step information budget B(d,n)=c,dlog_2 n . With the interface frozen, we develop an information-theoretic lower-bound toolkit: Budget counting, Psi -Fooling, and Psi -Fano, with worked examples L_k and L_k^{phase} . We prove an oracle-relative separation P^{Psi} neq NP^{Psi} and a strict depth hierarchy, reinforced by an Anti-Simulation Hook that rules out polynomial emulation of iota_k using many calls to iota_{k-1} under the budget regime. We also present two independent platforms (Psi-decision trees and interface-constrained circuits IC-AC^{0}/IC-NC^{1}) and bridges that transfer bounds among machine, tree, and circuit with explicit poly/log losses. The model preserves classical computational power outside iota yet enables precise oracle-aware statements about barriers (relativization; partial/conditional progress on natural proofs and proof complexity). The aim is a standardized minimal introspection interface with clearly accounted information budgets.

For robotic agents operating in dynamic environments, learning visual state representations from streaming video observations is essential for sequential decision making. Recent self-supervised learning methods have shown strong transferability across vision tasks, but they do not explicitly address what a good visual state should encode. We argue that effective visual states must capture what-is-where by jointly encoding the semantic identities of scene elements and their spatial locations, enabling reliable detection of subtle dynamics across observations. To this end, we propose CroBo, a visual state representation learning framework based on a global-to-local reconstruction objective. Given a reference observation compressed into a compact bottleneck token, CroBo learns to reconstruct heavily masked patches in a local target crop from sparse visible cues, using the global bottleneck token as context. This learning objective encourages the bottleneck token to encode a fine-grained representation of scene-wide semantic entities, including their identities, spatial locations, and configurations. As a result, the learned visual states reveal how scene elements move and interact over time, supporting sequential decision making. We evaluate CroBo on diverse vision-based robot policy learning benchmarks, where it achieves state-of-the-art performance. Reconstruction analyses and perceptual straightness experiments further show that the learned representations preserve pixel-level scene composition and encode what-moves-where across observations. Project page available at: https://seokminlee-chris.github.io/CroBo-ProjectPage.