Siyuan He, Bokai Yang, Jie Hu +2stat.ML cs.LG math.ST
Mixture-of-experts (MoE) architectures increase model capacity by combining a collection of expert predictors through input-dependent routing, while often activating only a small subset of experts for each input. Despite their growing importance in modern large-scale models, the statistical roles of their design choices, especially routing, sparse activation, and shared experts, remain only partially understood, as existing theory has largely focused on parametric or correctly specified MoE models. In this paper, we view MoE as a form of localized aggregation and show how this localization reshapes the approximation-estimation-computation tradeoff. We derive oracle risk bounds for learning dense and sparse routing with evolving experts, separating approximation, expert-learning, and router-estimation errors, and characterize how sparse Top-K routing can retain the benefits of localized aggregation while controlling per-input computation. We also interpret gating through the geometry of input space, relating routing performance to regions of local expert advantage, and show how shared experts, as adopted in architectures such as DeepSeekMoE, can extract common predictive structure so that routed experts focus on residual local variation. Together, these results provide a unified statistical framework for understanding MoE through input-dependent expert aggregation, in which expert specialization and computational tradeoffs are governed by local predictive structure.
Generative models have been studied experimentally and theoretically as priors for inverse problems such as compressed sensing. Recent work by Gunn et al. studied the use of generative priors with tunable complexity, where a family of generative priors with varying complexity is maintained and a specific complexity can be selected at inversion time. They demonstrated that lower reconstruction errors can be experimentally attained for a variety of inverse problems by appropriately tuning the complexity of the generative prior. In the present paper, we establish theory for compressed sensing in the setting of a tunable family of linear generative priors naturally related through their singular value decompositions. We prove that in noiseless Gaussian compressed sensing, the full-dimensional linear prior attains the minimum expected reconstruction error over the entire family of linear priors. Thus, in this idealized linear noiseless setting, tuning to a lower-complexity prior does not improve the expected reconstruction error. This result is in contract to the behavior of denoising, where lower complexity priors attain lower reconstruction errors due to a standard bias-variance tradeoff. This result indicates that the experimental benefits of tunability in compressed sensing with neural network priors arises due to nonlinearities in the generative models.
We extend recent work establishing an equivalence between one-layer transformers and nearest-neighbor classifiers in the binary setting to the multiclass case. By leveraging the simplex encoding, we show that one-layer transformers with an argmax classification head behave identically to a one-nearest-neighbor classifier in the multiclass setting. This closes a gap left by prior work, whose multiclass result relied on a non-standard rounding-based approach rather than the typical argmax head used in practice.
Token prediction is a central pre-training objective for modern language models. Despite its empirical success, why token prediction learns broadly useful representations remains incompletely understood. We develop a statistical framework connecting token prediction with representation geometry, encoder approximation, and downstream performance. Under a softmax prediction head, we show that accurate token prediction organizes token embeddings according to similarities between the distributions of contexts in which different token types appear, as measured by Hellinger distance, with explicit errors governed by prediction accuracy and token frequency. Meanwhile, the contextual representation provides a low-dimensional coordinate for the conditional distribution of the target token relative to these embeddings. We further introduce a self-consistency principle showing that repeated applications of a shared representation block can progressively refine the contextual representation without introducing additional block parameters. Among representations with the same prediction accuracy, this recurrent construction favors those that can be stably reconstructed from their contexts. Finally, we establish downstream guarantees for token generation, token community recovery, and classification by a linear probe, showing how prediction accuracy and recovered geometry translate into performance beyond the pre-training objective. Together, these results explain how the simple objective of predicting tokens can recover semantic geometry and produce broadly useful representations. A controlled simulation illustrates the theoretical mechanisms.
Flow matching enables likelihood-free training, yet alignment methods increasingly reuse conditional flow matching (CFM) losses as endpoint negative log-likelihoods (NLLs) and their old/new differences as log-likelihood ratios. We characterize when these substitutions are valid. For linear Gaussian paths, we exactly decompose endpoint NLL into entropy, a weighted CFM objective, an interior velocity--score residual, and a boundary residual. Thus CFM-only estimates and differences are exact only when the corresponding residuals cancel. At the off-policy population optimum, ordinary CFM is not generally a pointwise NLL estimator, whereas \(w_{\mathrm{sc}}(t)=(1-t)/t\) removes the interior residual; this positive result does not extend generally to training or on-policy alignment. On-policy log-ratios can remain biased even for identical endpoint laws or after surrogate optimization. Experiments across dimensions, distributions, and geometries support these conclusions and the mechanisms that make inexact ratios useful. **More broadly, the decomposition provides a theoretical basis for adapting likelihood-based LLM methods to flow matching, while distinguishing exact substitutions from controlled surrogates.**
One implicit DDIM inversion step is the cheapest probe of whether a pretrained diffusion model encodes local manifold geometry. It is the stationarity condition of an explicit potential, $x-G(x)=\nablaΨ_t(x)$, strongly convex at the Bayes limit with modulus exactly $e^{-h_t}$ for the step's log-SNR gap $h_t$ $-$ for every data law, schedule and point, with no manifold, reach or unimodality hypothesis. Three consequences must be kept apart. (i) The solution is unique at the Bayes limit; a second one requires the trained score to violate the posterior-covariance bound by $1/(1-e^{-h_t})$, a hypothesis-free certificate of model error; the same bound makes contraction a schedule constant, $ρ_g^{\star}=1-e^{-h_t}<0.326$ throughout the standard DDPM schedule. (ii) The solver can still fail: Picard iteration is unit-step gradient descent on $Ψ_t$, unstable wherever $λ_{\max}(\nabla^2Ψ_t)>2$, so oscillation certifies nothing; damping below $2/λ_{\max}$ cures it. (iii) The geometry lives in the convergence domain: on the scale-free depth $w=rκ_{\max}$ the oscillation shell sits at $w=\tfrac12$, schedule-free, and the divergence shell at $w=1/(1+ρ_g^{\star})$, with a measured finite-noise correction in $\|\mathrm{II}\|^2$. Exact scores reproduce both to within $0.54\%$ on three classes; no trained score we probe shows a shell $-$ a derived limitation, not a null result: the Fermi window conflicts with the model's own training support by $3.6$-$5.6\times$, and the trained Hessian-Lipschitz constant is $2$-$12\%$ of the curvature the law reads, $0$ on a ReLU net. Finally the unconditional ceiling $σ_tλ_{\max}(\mathrm{sym}\,J)\le1$, from $\mathrm{Cov}(x_0\mid x_t)\succeq0$ alone, holds for the exact score to $3\times10^{-7}$ but is violated in all DDPM CIFAR-10/CelebA-HQ-256 settings, by $1.26$-$4.66\times$.
World-action models predict a future outcome and then infer an associated action. Although this factorization can improve representation learning and data efficiency, it is unclear whether it provides stronger control capability than direct behavior cloning when both are trained from the same observational demonstrations. We compare a direct behavior-cloning policy, an imitation-trained world-action policy, and a policy optimized with an action-conditioned world model. At the controller-class level, every world-action policy can be flattened into a direct stochastic policy with the same closed-loop trajectory distribution. At the population level, under realizability, exact optimization, common deployment information, and distribution-preserving deployment, direct behavior cloning and world-action imitation both recover the observational behavior policy. Thus, future prediction changes the learning factorization but not the unrestricted external policy class or ideal imitation target. Action-conditioned world-model learning differs by predicting outcomes under specified actions and comparing them through a control objective. We characterize the irreducible action-specific prediction error of future models that do not condition on the candidate action, identify conditions under which a world-action joint can recover an interventional forward model, and show that observational demonstrations do not identify action effects in general. Finally, we construct an environment family in which every observational learner has positive worst-case regret, whereas one informative intervention permits zero regret. The key distinction is therefore between predicting futures associated with observed behavior and predicting consequences of specified actions for policy optimization.
The empirical success of diffusion models in generative modelling has motivated theoretical work, including quantitative error bounds and qualitative analyses that characterise the different phases of denoising. We bring these two areas together by studying the adaptivity of diffusion models to the structured geometry of multimodal high-dimensional data that consists of multiple clusters in $\mathbb{R}^D$, each with its own low-dimensional structure, and inter-cluster separation depending on $D$. We employ $K$-mixture Gaussian distributions as a canonical framework to capture this geometry and establish two theoretical results. First, we interpret denoising as a dynamical Bayesian classifier: the mixture score is a posterior-weighted average of cluster-wise scores, and we show that, with high probability, the posterior class probabilities concentrate on a single cluster once the signal-to-noise ratio reaches the scale $Θ(\log (KD)/D)$. Second, by separately analysing the denoising process in its mixing and cluster-commitment phases, we prove that the KL error bound depends linearly on the maximum intrinsic dimension of a cluster, up to a logarithmic factor, even when $K$ grows polynomially with $D$. This improves on ambient-dimensional bounds and extends existing low-dimensional adaptivity analyses to multimodal distributions with heterogeneous, approximately low-rank covariances.
Contrastive learning has become a cornerstone of modern representation learning, powering CLIP-style models that underpin text-to-image generation, vision-language models, and retrieval across a rapidly growing range of modalities. Despite this empirical success, the expressive power of these architectures remains poorly understood. To gain insight, we study expressivity by adopting a population-level, density-estimation viewpoint: each architecture comprises a parameterized set of densities whose parameters may be chosen to approximate the joint distribution of the modalities. This isolates a question of pure representational capacity: which joint distributions can a given contrastive family of parameterizations approximate to arbitrary accuracy? We show that expressivity is sharply architecture-dependent. For two modalities, the simple two-tower CLIP architecture is a universal approximator. A natural generalization of CLIP, widely used in practice when three or more modalities are present, is based on a loss found by summing over all pairwise similarities. This provably cannot represent arbitrary joint distributions, although we prove that it remains expressive enough to match all pairwise conditionals. Motivated by this gap, we propose Hadamard-CLIP, which adds a single learned weight vector on top of the existing encoders and restores universal approximation of the joint for any number of modalities while preserving CLIP's fast, precomputable-embedding retrieval.
Multi-layer transformers form the critical component of essentially all large language models (LLMs) in use today. Because of their ubiquity and computational capability, there is a rapidly growing body of work that aims to precisely calibrate the expressive power of transformers as language recognizers by comparing them against standard models of computation studied for decades by the theoretical computer science community. In this endeavor, circuit complexity has by and large emerged as the "correct" branch of computational complexity to analyze the expressive power of transformers; the reason is that parameterizing transformers by the various resources they use, such as attention and precision, leads to direct comparisons with different classes of circuits parameterized by resources such as type of gates, size, and depth. Here, we present an overview of selected results that delineate the expressive power of transformers using concepts and methods from circuit complexity.
Two accounts recur in explanations of the success of rotary position embeddings (RoPE). Expressivity studies associate periodic position information with modular predicates, whereas mechanistic and long-context studies emphasize positional anchors and local offsets. We formalize both accounts for fully uniform, finite-precision soft-attention transformers. We find that, if every rotary component is periodic, RoPE transformers recognize exactly the languages definable in past temporal logic with modular predicates. Conventional RoPE is different: The rotations it computes never repeat. This yields a precision-dependent bounded simulation of fixed-offset look-back operators, rather than an all-length modular characterization. Controlled experiments match this separation: Constructed periodic schedules length-generalize on modular languages, while conventional RoPE behaves more like a bounded locality bias and can impair tasks requiring position-invariant access to distant context. Altogether, our findings shed light on RoPE transformers, bringing theoretical expressivity characterizations closer to models used in practice.
Karim Zaghw, Andrew Pashea, Marc Pritsch +3cs.AI cs.CV
Active inference offers a unified framework for perception, learning, and action, but scaling discrete active-inference models to rich spatial and temporal domains remains difficult. Renormalising generative models (RGMs) address this challenge by composing discrete generative models across spatial and temporal scales, coarse-graining lower-level states and paths into higher-level causes for objects, events, and action. However, fully reproducing and adapting the framework remains difficult: the mathematical exposition is compact, and the reference implementations are deeply integrated within specialized software environments, leaving many algorithmic details implicit. This paper addresses these challenges by providing a self-contained, derivation-oriented account of RGMs together with an open, verified implementation. We explain how the hierarchy is built, how beliefs and actions are updated within it, and how information is passed between levels. Where the published equations and implementation differ in emphasis, we make those choices explicit and explain their modelling consequences. By clarifying the theory and separating it from its original implementation context, this work lowers practical barriers to entry and makes RGMs more transparent, auditable, and reproducible, providing a foundation for future quantitative evaluation and development on machine-learning benchmarks.
We argue that learning visual representations without labels requires a training signal jointly complete across three non-overlapping objectives: semantic invariance across augmented views, patch-level spatial prediction, and representational non-degeneracy. We formalize these as the observation, prediction, and regularization principles and prove (i) that combining observation and prediction without regularization admits the constant encoder as a global minimizer under negative-free alignment; (ii) that the two objectives are gradient-complementary and structurally non-conflicting at the encoder output; and (iii) that the momentum encoder converges to the same fixed point as the online encoder and provides no collapse guarantee at convergence. Contrastive alignment provides only self-limiting collapse resistance, formalized via an explicit gradient-decay argument. Dropping prediction withholds the spatial training signal by construction; dropping observation forfeits cross-view semantic invariance by construction; at the scale we study, no pair substitutes for the third. Every major self-supervised method is a special case of a single unified energy decomposition. We pair every theoretical claim with a controlled experiment, including a patch-retrieval evaluation for the spatial consequence of prediction.
Ibne Farabi Shihab, Abu Sa-Adat Mohamed Moon-Im Al Ahsan, Md Najmus Swaqeebcs.LG
Average-reward reinforcement-learning regret is known up to logarithmic factors, but the numerical content of published guarantees is difficult to compare because probability mode, structural parameter, logarithmic normalization, prior information, and planning assumptions differ. We introduce a constant-aware comparison protocol and derive an explicit finite lower certificate for communicating MDPs. The construction is a binary tree of two-state blocks; its proof uses exact trajectory-level Bernoulli KL divergence and keeps action budget, diameter, occupancy, navigation cost, and terminal bias explicit. A common closed-form envelope improves the published coefficient $0.015$ across a finite frontier: $0.0200$ in a moderate regime and up to $0.0291$ under stronger action, diameter, and horizon conditions, a $94\%$ increase. The limiting coefficient is $\frac1{32}\sqrt{(A-3)/A}$. For upper bounds, we give an auditable composition rule for a span-constrained optimistic learner, but do not claim a coefficient while adaptive directional-variance and planning certificates remain open. We also formalize valid expectation conversion and constant comparability. Controlled diagnostics test diameter dependence, bonus-by-width interactions, span misspecification, and the finite lower certificate on its exact family.
Schrödinger bridge models restore a clean signal from a degraded observation by following the conditional bridges of a reference process, yet this reference is chosen heuristically, typically white noise with a hand-tuned schedule. We develop PRISM, a theory of bridge reference design. We characterize the time-varying Gaussian references that remain exactly tractable with per-mode schedules: precisely those whose instantaneous covariances commute. We then prove an invisibility principle: with the exact drift and unlimited solver steps, every admissible reference recovers the true posterior. The choice of reference therefore matters only under finite computational resources. For a fixed step budget, we derive the finite-step objective in closed form and prove that every optimal noise spectrum is proportional to Pk, the spectrum of information destroyed by the sensor, with a mode-independent constant x*(T) = (2 ln T)^-1/2 (1 + o(1)). The analysis shows that noise color and temporal scheduling are interchangeable, and regularization provably shifts the optimal reference toward white noise. Experiments in Gaussian settings confirm the predicted orderings and the closed-form loss floors. On FFHQ, the distortion-- perception trade-off and spectral localization transfer, but white noise outperforms the matched reference; a pre-registered study that changes the training regime refutes ridge whitening as the explanation. A 2x2 mechanism study then traces the inversion to the non-Gaussian per-mode statistics of real images. PRISM turns reference design from a hyperparameter sweep into a calculation in the Gaussian regime, and locates exactly where real images break it.
Black-box language-model reliability is commonly pursued by sampling, prompting, voting, verifying, or iteratively revising individual answers. We ask a prior question: \emph{what determines whether a collection of black-box responses is recoverable at all?} We represent responses to typed transformations of a query as a \emph{relational response field} (RRF). Edge transports encode how valid responses must change under paraphrase, scaling, decomposition, refactoring, or other task symmetries; anchors encode independently trusted evidence such as execution or a verifier. For relation operator $D$, anchor operator $A$, and at most $k$ corrupted response nodes, we identify $γ_k(D,A)$ as the intrinsic difficulty of black-box response recovery. It is positive exactly when every $k$-node corruption is identifiable; it gives a deterministic stability bound proportional to $1/γ_k$; and a matching two-point minimax lower bound shows that no estimator can improve this dependence. Thus consistency is not truth: relation-only methods are blind to null directions, including shared hallucinations. We derive sparse field-repair algorithms while separating information-theoretic identifiability from the stronger null-space conditions required by convex optimization. Controlled theorem tests and black-box mathematics/code experiments evaluate four theory-fixed consequences: consistency--truth separation, anchor phase transitions, redundancy saturation, and cross-model, cross-task prediction of repair difficulty. The results support $γ_k(D,A)$ as a measurable property of a response-recovery instance, rather than a score attached to one repair heuristic.
Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying \textit{c}-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative \textit{c}-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative \textit{c}-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).
Hayder Tirmazi, Sam Markelon, Allison Bishop +1cs.DS cs.AI
Large Language Models (LLMs) have a bounded context window. The context window is the maximum input size an LLM can consume for a single inference. AI agents rely on a process called context compaction to fit their state within the context window when calling an LLM. Despite its ubiquity, context compaction has received essentially no formal analysis. In this paper, we initiate a formal study of context compaction. We first introduce a framework consisting of two games that capture the two algorithmic strategies for context compaction used by contemporary AI agents in practice. The Context Selection Game models context compaction algorithms that select a subset of an agent's accumulated state to retain. The Context Generation Game models context compaction algorithms that summarize an agent's state by an arbitrary message of bounded length. We then prove an equivalence between the Context Generation Game and one-way communication complexity. The minimum context compaction budget for answering a set of queries within a target error is equal to the one-way communication complexity of the induced communication problem at the same error. Known bounds from communication complexity therefore transfer directly to context compaction. We also show that the Context Selection Game corresponds to a restricted class of one-way communication protocols. Any gap between selection and generation is therefore a gap between two classes of communication protocols. We prove that there exists a set of queries for which generation needs strictly less budget than selection. The equivalence between the Context Generation Game and one-way communication also lets us measure how well a deployed context compaction algorithm performs on a query relative to the optimal strategy. As an example, we present a case study that evaluates Anthropic's context compaction endpoint on set membership queries.
The temperature that maximizes pass@$k$ is often low for a small sampling budget and higher for a large budget. This pattern has been reported from Codex through recent multi-sample inference studies. It is not an algebraic property of pass@$k$: as Slocum et al. (ICLR 2025) observe, for one fixed task the maximizing temperature is independent of $k$. Building on that fixed-task observation and the hard/easy-task explanation, we give a formal population-level sufficient condition for the aggregate pattern. For task $X$, let $p_t(X)$ be one-sample success probability at temperature $t$, and define the conditional log-success response $m_t(u)=\mathbb{E}[\dot p_t(X)\mid p_t(X)=u]/u$. If $m_t(u)$ is nonincreasing in current success probability, then the normalized temperature derivative of aggregate pass@$k$ is nondecreasing in $k$. Consequently, derivative signs are nested across budgets; if each temperature-performance curve is strictly single-peaked, its unique maximizer is nondecreasing in $k$. The proof identifies the mechanism as a monotone-likelihood-ratio power tilt toward lower-success tasks. We derive a closed-form two-stratum phase diagram, including upward and downward regimes, and show that the marginal temperature derivative admits an exact $\mathrm{Beta}(2,k)$ kernel representation whose kernel concentrates at one-sample success of order $1/k$. Interpreting that scale as task-level localization additionally requires a regular, nonvanishing density-response factor near zero. A signed-moment representation yields diagnostic shape restrictions, while a short appendix records exact discrete refinements of the existing multi-configuration allocation formulation. No language model is trained, and no model query is used as an experimental measurement: the contribution is a conditional theory of an established empirical phenomenon, with assumptions that can be tested in future work.
Fairoz Nower Khan, Nabuat Zaman Nahim, Peizhong Jucs.LG
Flow matching assumes fully observed training data, which many real-world applications rarely provide. We propose Missing-Data Flow Matching, which treats the missing coordinates of training samples as latent variables and averages the flow matching loss over the values they could take. We first prove the correction is exact rather than approximate. Under missing completely at random with true completions, the incomplete-data objective equals the complete-data objective, so missingness changes nothing about what flow matching learns and the entire difficulty relocates to the completion model. Our finite-sample analysis then answers design questions that the algorithm leaves open, and the answers are not the ones intuition suggests. Missingness transfers estimator variance rather than adding it, one completion per example already matches complete-data variance exactly, and under a fixed evaluation budget one completion is optimal. A learned completion model contributes a single irreducible bias, which we bound by its expected conditional Wasserstein distance to the true completion law. Experiments numerically validate the theoretical predictions, show that deterministic rather than frozen imputation is what collapses the generated distribution, and place our method alongside strong classical and deep imputation baselines on real tabular data.
Submodular Information Measures (SIMs) have recently emerged as a powerful framework for representation learning and multimodal learning. In particular, the SCORE framework~\cite{majee2024score} demonstrated that SIMs can serve as effective objectives for supervised contrastive learning. Despite their empirical success, however, the geometric and statistical properties induced by different submodular information measures remain poorly understood. In this work, we develop a unified theoretical framework connecting SIMs to classical concepts in representation learning and statistical pattern recognition. We show that Total Information (TI) objectives characterize intra-class structure: Graph Cut TI recovers within-class variance, LogDet TI recovers generalized variance and covariance volume, and Facility Location TI induces imbalance-aware separation that emphasizes rare and confusable classes. We further show that Mutual Information (MI) objectives capture complementary notions of inter-class structure: Graph Cut MI is closely related to centroid separation and Fisher-style discrimination, LogDet MI captures covariance-aware separation through Mahalanobis distance, and Facility Location MI measures nearest-mode representational overlap. We validate these theoretical characterizations using controlled synthetic experiments that independently vary variance, covariance, class imbalance, class separation, and multimodal overlap. Across all settings, the empirical behavior closely matches the proposed theory. Our results provide the first unified geometric and statistical understanding of submodular information measures and offer principled guidance for selecting and designing SIM-based objectives for representation learning.
Transformer language models are usually analyzed through vector geometry, yet ordered context and rotary position encoding introduce explicit phase structure into query-key interactions. This paper develops a bounded spectral framework for examining rotary phase alignment, hidden-state continuity, and semantic drift without treating language models as literal physical wave systems. It first identifies ordered hidden-state sequences, rather than vocabulary indices, as valid domains for spectral decomposition. It then derives the Rotary Position Embedding (RoPE) attention score as a sum of magnitude-weighted cosine terms and proves a local stability lemma: uniformly bounded phase displacement limits degradation of the corresponding pre-softmax score. To extend phase analysis beyond native RoPE coordinates, the paper defines complex modal coordinates over fixed orthonormal direction pairs and introduces a weighted coherence functional for hidden-state trajectories. These constructions support a strict distinction between representational continuity and execution-boundary admissibility. Internal coherence may describe preservation of task-relevant relations, but it cannot authorize a consequential transition. Positioned against existing geometric, spectral, phase-modulation, representation-analysis, and mechanistic-interpretability accounts, the framework contributes a theoretical and methodological program for determining when spectral structure explains continuity and when governance must remain an external predicate over execution.
Yue Yao, Caleb N. Ellington, Jingyun Jia +9stat.ML cs.LG stat.ME
Modern predictive systems are expected to adapt their behavior to the specific situation they are facing. A clinical model should not treat every patient the same; a retrieval-augmented model should change its answer when given different evidence; a mixture-of-experts model should route different inputs to different experts. We call this capability context-adaptive inference: before predicting, the system uses information about the current context to specialize its parameters or computation for that instance. This article provides a unified view of context-adaptive inference across three traditions that are usually treated separately: (i) explicit adaptation in statistics (e.g. varying-coefficient models, local regression, hierarchical sharing), (ii) rapid task-specific adaptation in meta-learning and transfer, and (iii) implicit adaptation in large foundation models via prompting, retrieval, and expert routing. We formalize these approaches under a common objective: to map context $c$ to adapted parameters $θ(c)$, then to predict via $f(x; θ(c))$. Under squared loss, linear prediction heads, and fixed features, we prove that explicit parameter adaptation and implicit routing are mathematically equivalent to kernel ridge regression on joint features of inputs and context. Building on this bridge, we propose practical design principles and evaluation metrics including adaptation-efficiency, routing stability, and context-specific robustness to guide when to specialize, how to constrain that specialization, and how to audit context-adaptive models in deployment. Finally, we identify open problems in identifiability, robustness under distribution shift, and efficient large-scale adaptation, outlining design principles for methods that are scalable, reliable, and transparent in real-world settings.
Recent work has shown that large language models (LLMs) can iteratively improve their outputs by incorporating generated samples and their corresponding evaluation scores as in-context examples. Despite these empirical findings, the theoretical foundations underlying this phenomenon remain poorly understood. In this paper, we show that score-conditioned In-Context Learning (ICL) admits a structural correspondence to policy gradient optimization. We first provide a constructive proof that self-attention mechanisms can implement reward-weighted aggregation analogous to the REINFORCE algorithm under specific weight matrix configurations, and discuss the relationship between this construction and the behavior of pretrained transformers. The correspondence is directional in hidden-state space and holds exactly only under the stated simplifying conditions; we quantify its strength empirically. Within our simplified hidden-state model, we furthermore derive an exact upper bound on the distribution shift induced by a bounded attention update, yielding a trust-region-like analogy to KL-constrained policy optimization. We validate our theory through extensive experiments across multiple LLMs, demonstrating that LLMs effectively utilize score information to shift output distributions toward high-scoring exemplars, and that attention weights exhibit a strong correlation with example scores.
Neural scaling laws describe how loss decreases as models, data, and compute grow, but they do not answer a prior question: for a fixed task, what is the minimum model capacity required to solve it? We study this through the Entropic Bound, a spectral notion of task-intrinsic capacity for Transformers. We first prove that, in a linear attention surrogate, the intrinsic rank $r^*$ of the token-mixing operator is a tight lower bound: any rank-deficient model incurs unavoidable excess risk, and the bound is achievable at $r^*$. We further show that gradient descent recovers this rank under standard low-rank implicit-bias assumptions, confirm all three properties empirically, and show $r^*$ is recoverable from data before training. We then ask whether this transfers to real attention. A naive transfer fails, and a controlled interpolation ladder localizes the cause precisely: it is not softmax and not a rank constraint, but the input-conditioned nature of attention's mixing operator, which a static weight kernel cannot summarize. Motivated by this, we introduce an attention-native intrinsic rank -- the minimum query-key kernel rank realizing the task within the attention class -- and show that under this definition the full Entropic Bound structure (deficiency, achievability, recovery) is restored for both linear and softmax attention, with the energy effective rank as the estimator robust to softmax distortion. Finally, we map the boundary of data-only predictability: $r^*$ is exactly recoverable for linear QK attention, even without the value map at scale, while softmax attention admits only partial pre-training recovery due to nonlinear inversion and kernel-value identifiability effects. Our results reframe the Entropic Bound from a post-hoc descriptor into an attention-native capacity measure with a precisely characterized predictability frontier.
Surprisal theory holds that the human processing difficulty of a linguistic unit in context is an affine function of its surprisal under some language model. I argue this claim is a tautology without further constraint: for any non-negative difficulty measure over units in context, there exists a language model whose surprisal is an affine function of it under mild technical conditions. Therefore, because any pattern of difficulty is consistent with some language model, without an additional constraint on the language model, surprisal theory makes no falsifiable predictions. The tautology was long obscured by an assumption implicit in two decades of psycholinguistic work---that the relevant language model is the distribution that generated the training corpus, so that improving corpus fit improves predictions of human behavior. Recent empirical work has undermined this assumption, demonstrating that better corpus models can be worse predictors of processing difficulty. I conclude that breaking the tautology requires a rationalist intervention, i.e., the relevant language model must be derived from a non-empirically motivated model of the comprehender, which could be based on, for instance, memory constraints or processing goals, and that, thus, does not depend on the behavioral data surprisal theory is meant to explain.
Abigail Woodring, Adrian Chan, Rana Muhammad Shahroz Khan +3cs.LG cs.CL
Fine-tuning has been widely used to adapt large language models (LLMs) for domain-specific tasks. Parameter efficient fine-tuning (PEFT) methods such as low-rank adaptation (LoRA) are frequently used to reduce computational costs. PortLLM is a training-free and data-free scheme used to adapt LLMs after continual pretraining. Although the initial PortLLM results show that LoRA patches exhibit short-term temporal portability, the long-term performance of PortLLM across several updates of continual pretraining remains underexplored. Furthermore, the intriguing effectiveness of PortLLM is not well understood from a theoretical standpoint. We address these two open questions by (1) performing an extensive empirical study of the long-term temporal portability of PortLLM patches across 10 continual pretraining steps using base models Mistral, Gemma, and Qwen; and (2) offering two theoretical analyses to explain our observation that the simple PortLLM method achieves competitive performance. We find empirically that the portability persists across longer time duration, indicating that repeated fine-tuning is not required when the base model is periodically updated. We find theoretically that near-orthogonality of high-dimensional vectors is a key justification for temporal portability. Our analyses also demonstrate a geometric perspective of the loss landscape in facilitating the theoretical comparison of different adaptation options.
Riccardo Poiani, Martino Bernasconi, Andrea Cellics.LG
Reinforcement Learning is a cornerstone technique for modern large reasoning models. Usually, for difficult tasks such as code generation and theorem proving, the agent is evaluated by generating $K$ responses rather than sampling a single response, and performance is then measured using a retry-aware metric such as $\max$@$k$. Despite their practical importance, the theoretical foundations of learning under such criteria remain limited. In this work, we provide a theoretical study of the $\max$@$k$ learning problem in finite-horizon reinforcement learning. We show that optimizing the $\max$@$k$ objectives is fundamentally different from standard expected-return maximization. In particular, we prove that Markovian policies are in general insufficient, identify a compact state augmentation that restores optimality, and explicitly characterize the performance gap that can arise between history-dependent and non-history-dependent policies. Moreover, we show that learning $\max$@$k$-optimal policies is statistically harder than standard reinforcement learning and provide an efficient algorithm that achieves the optimal sample complexity rate.
We present a continuous geometric framework that models the discrete algebraic operations of the Transformer architecture as an integro-differential equation (IDE) on a semantic fiber bundle $\calE = \calM \times \R^d$. Beginning from a single geometric axiom -- that the token sequence forms a discrete $1$-manifold equipped with a canonical measure lattice -- we translate every core component of the modern Transformer (RMSNorm, RoPE, Softmax Attention, FFN, Residual Stream, SGD, Weight Decay) into a cohesive vocabulary of differential geometry, measure theory, and stochastic calculus. The resulting framework yields quantitative predictions spanning entropic optimal transport (Attention as a Schrödinger bridge) and non-equilibrium thermodynamics (SGD as Itô diffusion violating detailed balance). We conduct a six-part experimental campaign across five architectures (Qwen3, LLaMA\nobreakdash-3.1, Gemma\nobreakdash-3, GPT-2, Mistral) spanning $124$M to $8$B parameters. The empirical observables are quantitatively consistent with the geometric predictions: the $ε^{-1/2}$ Lipschitz scaling calibration at machine precision ($R^2 = 1.000$), the Lie--Trotter operator-splitting torsion, the symmetric ablation instability confirming the Dual-Law of Topological Stability, the $\calO(1/\sqrt{k})$ thermodynamic suppression of Poincaré recurrence on the RoPE torus, the thermodynamic context-limit phase transition, and the Non-Equilibrium Steady State parameter vortex -- verified across two optimizers (AdamW and Pure SGD) to exclude momentum artifacts. The results demonstrate that analyzing Transformers through the lens of continuous stochastic differential geometry provides a predictive descriptive vocabulary for the stability limits, context bounds, and optimization dynamics of Large Language Models.
Sachin Dev Duggal, Pradyumna Swarnalatha Ramanna, Alexandros Vassiliadescs.AI
This paper proposes a theoretical framework for understanding intelligence as a process of atomic compression and compositional reuse. We argue that cognitive, biological, computational, and organizational systems achieve scalable intelligence by decomposing complex phenomena into reusable atomic units that can be recombined into higher-order structures. Drawing on evidence from cognitive science, information theory, evolutionary biology, software engineering, medicine, legal reasoning, education, music, and artificial intelligence, the paper develops the concept of atomic units as fundamental compression layers that support efficiency, transfer, interpretability, and evolvability. The central contribution is the Compression Calculus, a formal framework for comparing surface-level representations with atomic representations and for describing how compression gains compound across abstraction layers. We introduce the Compounding Cascade thesis, according to which each additional layer of abstraction multiplicatively increases representational efficiency rather than merely adding incremental savings. The paper further argues that contemporary AI systems often operate at suboptimal levels of representation, relying on token-level processing or document-level retrieval rather than stable, concept-level atomic structures. In this view, large language models are best understood not as complete knowledge architectures, but as dynamic fusion engines capable of navigating, sequencing, and recombining atomic units. The framework provides a foundation for designing self-evolving knowledge systems that can discover, refine, and compose new primitives over time. By reframing intelligence as compression through compositional abstraction, the paper offers a unifying perspective on expertise, knowledge representation, explainable AI, and the future architecture of adaptive intelligent systems.