Singular Learning Theory

Generative thermodynamic computers turn thermal noise into structured data through Langevin dynamics. We train these systems with a local update at each integration step. The reverse-path Onsager-Machlup objective yields a coupling gradient that is a symmetric sum of local residual-state correlations. We apply this gradient immediately rather than accumulating it over a full trajectory. In digital simulations using MNIST prototypes, online and trajectory-batch training reach similar validation losses on fixed noising paths. Models trained online release less heat on average in all five independently seeded pairs, with both models' parameters held fixed during sampling. Auxiliary classifier and nearest-prototype measures change modestly, while pairwise diversity decreases. The response to noise depends strongly on where the errors enter: independent zero-mean errors in the formed updates produce little heat change over a finite range of noise amplitudes, whereas residual offset and temporal correlation have much larger effects. Storing trained couplings requires substantially less precision than resolving deterministic updates during training. Together, these results establish a local online training method and show how update timing, noise structure, and precision affect generative thermodynamic computing.

Teacher-student systems, in which a teacher neural network generates training labels so that a student neural network can learn to implement the same function, are widely used as an abstract setting to study learning. However, the structure of the teachers is often overlooked by assuming randomly-generated, normally-distributed parameters. This hides substantial variation in how learnable different teachers are. We formalize learnability as the success rate of converging to the global minimum, as a function of overparameterization, learning algorithm, student initialization distribution, and teacher geometry. We both identify an easy distribution that maximizes node dissimilarity and a hard distribution that minimizes it, and show that these two distributions induce markedly different success rates across a large range of settings and for different activation functions. To explain the gap, we study the loss landscape of small neural networks that contain two distinct kinds of suboptimal local minima, out-of-bounds (OOB) minima at the edge of the data distribution and interior minima within. Assuming infinite data and a fast readout layer, we analytically reduce the loss landscape of small networks to two dimensions, showing that the region of attraction of interior minima changes as a function of teacher structure. In larger networks, maximally dissimilar teachers induce more interior minima, while minimally dissimilar teachers induce more OOB minima. Motivated by these analyses, we show that differentially increasing the learning rate of the readout layer and decreasing the learning rate of the inner biases increases success rates. These findings provide an important step in narrowing the gap between the study of teacher-student networks and more structured functions that arise in practice.

The dominance of Neural Networks (NNs) in RL is partially due to their incremental learning capability, which naturally suits the online, non-stationary nature of self-play training. However, gradient-boosted trees like LightGBM are widely recognised as the state of the art for tabular data in supervised learning, often outperforming NNs in accuracy and efficiency. Game states are inherently tabular---discrete actions, categorical card identities, structured board positions---which makes them an ideal candidate for tree-based methods. We introduce LUGL (Local Updates, Global Learning), a framework that decouples data collection from model fitting, enabling non-incremental learners such as GBTs to operate in RL settings where they would otherwise fail due to distributional shift. LUGL alternates between a local updates phase, where the agent plays self-play games and accumulates tabular updates (Q-values, V-values, policies, or regret values) in a finite table, and a global learning phase, where the table is used to train a function approximator that generalises to unseen states before the table is reset. We test our approach in four standard perfect-information games (Tic-tac-toe, Connect-4, Othello, and Hex) and five imperfect-information games (Kuhn's poker, Leduc Hold'em, Liar's Dice, Goofspiel, and Flop5 Hold'em), and show that our results are competitive with or superior to DQN and DeepCFR. Our experiments demonstrate that the community's strong bias towards NNs in game-playing may be unwarranted, since LightGBM-based agents achieve competitive or superior performance across all tested benchmarks.

Local learning methods offer an alternative to end-to-end backpropagation, but their unstructured local objectives can produce weak feature learning in deep networks. We study whether the order of hidden-state optimization can address this limitation. We propose a boundary-first inference schedule that partitions a model into chunks, first coordinates hidden states at chunk boundaries, and then refines representations within each chunk. We instantiate this schedule in predictive coding networks (PCNs), a local-learning framework in which hidden activities and prediction errors are explicitly exposed during inference. On CIFAR-10, the resulting boundary-first predictive-coding instantiation improves accuracy over standard predictive coding by $9.77\%$ under a standard parametrization and by $5.51\%$ under a $μ$-parametrization. Diagnostic analyses further show more non-trivial early-layer updates, lower initial-to-final CKA, and more diverse layerwise gradients, consistent with stronger feature learning. These results support boundary-first, chunk-based inference as a practical design principle for predictive-coding training and motivate its study in broader local-learning systems.

Probabilistic Circuits (PCs) are generative models that support exact inference and, unlike deep neural networks, admit an exact and tractable measure of loss-surface curvature: the trace of the Hessian of the log-likelihood. Recent work regularizes this trace globally to bias learning toward flatter, better generalizing optima. We show that treating sharpness as a global regularizer can be misspecified for PCs, whose curvature is inherently compositional. We prove that each sum node's contribution to the Hessian trace factorizes exactly into its circuit flow, which measures how heavily the node is used, and a local sharpness term determined by its output distribution. This decomposition provides insights into why global sharpness regularization is depth biased and can lead to underfitting. Building on it, we introduce an adaptive sharpness aware regularizer that penalizes nodes based on intrinsic local curvature and preserves closed form EM updates. We also show that empirically, this targeted regularization recovers the generalization that global regularization sacrifices while retaining the robustness and benefits of sharpness aware learning.

Invariant learning seeks representations that remain predictive across environments, yet the behavior of its objectives along the regularization path is often opaque. We address this objective-behavior gap by viewing representation learning as multimode magnetization and deriving, from concrete invariant-learning objectives, a Landau-type effective free energy whose low-order coefficients form objective signatures and induce distinct regularization phenotypes. Effective quadratic corrections move the phase boundary and enable finite-strength mode elimination; quartic corrections regulate post-onset amplitude and typically leave residual loading at finite strength; higher-order structure governs non-monotone tails, instability, and collapse at large regularization. In a canonical bilinear model, the theory yields closed-form phase boundaries and steady-state loadings, as well as distinct critical strengths for shortcut and stable modes that define a selective-retention window. Controlled experiments confirm the predicted phase boundaries, loadings, and regularization phenotypes. In one- and two-hidden-layer ReLU networks, the same signatures remain predictive of qualitative regularization-path behavior despite depth-dependent shifts in scale. A matrix extension generalizes the framework to coupled collective modes and yields a spectral phase-boundary criterion. Together, the framework turns low-order objective signatures into predictions of regularization phenotypes and, ultimately, of what models learn as regularization varies.

Hanneke, Moran, and Waknine \cite{HannekeMoranWaknine2024} asked how the agnostic PAC learning curve of the direct sum $C^r$ depends on the single-instance learning curve $\epsagn(n\mid C)$ and on $r$. We show that the single-instance learning rate does not determine the direct-sum rate. Let $\F$ be the class of the two constant binary functions and let $\G$ consist of the zero function and the identity function. Both classes have agnostic learning curve of order $n^{-1/2}$.

We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.

Stochastic learning objectives are typically written as expectations over abstract random variables. Actual training, however, uses concrete random inputs that enter both the realized loss and its gradient. Structure in these inputs that is accessible to the learning system can therefore be learned and exploited. Much prior structured-noise work asks how noise should be distributed or designed; we instead ask what structure in the concrete realized randomness becomes exploitable by the learner. We develop this general view and analyze its mechanism in diffusion models: in noise prediction, clean data and realized noise jointly form the noisy input, so the model can improve prediction by learning clean-data regularities or exploiting structure introduced through the noise, and the two routes can interact. Using pseudorandom streams as controlled, reproducible instances of structured noise, we provide mechanistic evidence on MNIST and CIFAR-10: random-role ablations localize the dominant effect to diffusion noise; in a diffusion probe, structured-noise training can reduce prediction loss below the IID reference, but replacing the test noise with IID reverses this advantage; and shuffling the same values largely removes the source-dependent loss reduction. This learned dependence can also affect generation. The same view offers a unified interpretation of data-dependent noise assignment, noise-based backdoors, and temporally correlated noise in video diffusion: although these methods introduce different structures, all alter what the model can exploit through noise and its interaction with clean-data learning. Our results indicate that diffusion noise is not merely a passive stochastic perturbation, but a learnable---and therefore potentially designable---input dimension.

In social learning, agents learn not only from direct evidence but also through interactions with their peers. We investigate the role of imprecision in such interactions and ask whether it can improve the effectiveness of the collective learning process. To that end we propose a model of social learning where beliefs are equivalent to formulas in a propositional language, and where agents learn from each other by combining their beliefs according to a fusion operator. The latter is parametrised so as to allow for different levels of imprecision, where a more imprecise fusion operator tends to generates a more imprecise fused belief when the two combined beliefs differ. In this context we describe both difference equation models and agent-based simulations of social learning under a variety of conditions and with different initial biases. The results presented suggest that for populations with a strong initial bias towards incorrect beliefs some level of imprecision in fusion can improve learning accuracy across a range of learning conditions. Furthermore, such benefits of imprecision are consistent with a stability analysis of the fixed points of the proposed difference equation models.

We argue that a single structural fact organizes a wide range of phenomena in contemporary AI safety: a semantic safety constraint (e.g., the agent does not escape its sandbox) is an off-support object. Formally, if q is the data distribution and $p(\cdot\mid w)$ the model, the safety predicate B is not measurable with respect to $σ(\text{model}, q)$, whereas the real log-canonical threshold (RLCT) of singular learning theory (SLT) is. From this non-invariance we derive, as corollaries rather than independent observations: (i) why reward hacking and sandbox escape arise under outcome-based optimization; (ii) why encoding such constraints through Bayesian prior design or soft penalty weighting has poor leverage in singular models; (iii) why hard invariants belong in the harness and soft dispositions in the model; (iv) why the same B is nonetheless soundly and locally certifiable by formal verification, exactly as the local learning coefficient (LLC) locally pins the same RLCT --- with two precise points of disanalogy; and (v) why the residual difficulty, identifying which off-support region matters, coincides with performative prediction and self-referential functional dynamics, where SLT's analytic machinery breaks down. We use the July 2026 OpenAI--Hugging Face evaluation incident as the motivating case. Numerical experiments code and related proofs in lean are available at https://github.com/xiangze/Preventing_Jailbreak_as_regularization

We introduce CMP (Cognitive Memory Primitive), a continual-learning architecture that repre?sents inputs as sparse relational codes, stores them in a two-tier competitive memory, and learns through local updates without end-to-end backpropagation through its feature-generating system. We investigate whether combining sparse representations, local learning, and persistent memory can reduce catastrophic forgetting relative to conventional backpropagation-based continual?learning approaches. On a controlled domain-incremental byte-level language modeling protocol, CMP demonstrates substantially lower backward transfer than a parameter-matched Trans?former trained with online Elastic Weight Consolidation (EWC). Across a three-seed replicated 15-domain experiment, CMP exhibits stable forgetting behavior, while separate head-to-head comparisons and domain-order analyses show consistently lower forgetting than the evaluated Transformer baseline under the reported experimental settings. We report these findings alongside a substantial single-domain accuracy gap relative to the Transformer, a null result on a vision benchmark, and a documented failure to combine CMP with an independent accuracy-improving mechanism, reflecting our commitment to reporting both positive and negative outcomes. These results suggest that the combination of sparse representations, local learning, and persistent memory is a promising direction for continual learning, while motivating further investigation into the respective roles of learning rules, representations, and architectural design in mitigating catastrophic forgetting.

Machine Unlearning aims to remove undesired information from trained models without full retraining from scratch. Despite recent progress, the loss landscape and optimization geometry of unlearning are poorly understood. In this paper, we study machine unlearning through the lens of mode connectivity--the phenomenon that independently trained models can often be connected by smooth low-loss paths in parameter space. We introduce {\em mode connectivity in unlearning} (MCU) and evaluate it across a range of settings, including curriculum learning, second-order optimization, and connectivity across different unlearning methods. We find that many unlearned models lie in connected basins with smooth retain/forget behavior, while changes in training dynamics can move solutions into different basins. MCU also reveals that models within the same basin can differ substantially on privacy metrics, and that unlearning progresses nonlinearly from the original model to the unlearned model. In addition, linear connectivity suggests that most approximate unlearning methods are mechanistically distinct from retraining. Finally, MCU-based ensembling can improve generalization and robustness to relearning attacks, and MCU smoothness correlates with unlearning difficulty. To our knowledge, this is the first study of machine unlearning through the lens of mode connectivity.

Local regularization assigns each hypothesis a test-point-dependent score and predicts with a minimum-score hypothesis consistent with the sample. Asilis et al. asked whether this principle characterizes multiclass PAC learnability. We give a negative answer. There is a countable class of Daniely--Shalev-Shwartz dimension at most two with realizable PAC sample complexity $$O\!\left(\frac{1}{\varepsilon}\log\frac{1}δ\right),$$ that no local regularizer learns. Hypotheses are edges of complete graphs and instances are tournaments. At a test tournament, the scores fix an edge ranking while the training sample independently removes competitors. Cyclic triangles force enough inversions that surviving competitors produce constant population error at arbitrarily large sample sizes.

Multiclass classification is a fundamental problem across a wide range of domains. It is still challenging due to possession of high inter-class similarity, class imbalance datasets, and variability in data distributions. Rule-based classifiers such as XGBoost often achieve stronger performance on structured features, but they are limited in capturing smooth functional relationships among variables. Similarly, neural network models can represent complex nonlinear interactions but frequently suffer from overfitting and generalization issues. To address these limitations, we propose LFS-FRAME, a Leakage-Free Stacked ensemble framework that integrates functional learning using Kolmogorov-Arnold Networks (KAN) and rule-based learning via XGBoost for robust multiclass classification. The proposed framework constructs unbiased meta-features by employing a strict out-of-fold stacking strategy to ensure complete isolation between training and validation data hence preventing performance leakage. By learning over probabilistic outputs from heterogeneous base learners, the meta-classifier effectively exploits both global functional patterns and sharp decision boundaries present in the complex data. Experimental evaluations on multi-class datasets demonstrate that LFS-FRAME improves performance metrics, and overall accuracy is 89.85% in identifying major families and 81.74% in identifying sub-families relative to strong single-model baselines. These results highlight the effectiveness of leakage-free functional and rule-based stacking for reliable and generalizable multiclass classification.

Few-shot class-incremental learning (FSCIL) aims to incrementally learn novel classes with only a few samples while avoiding forgetting base classes. However, current methods show a tendency to misclassify novel-class samples into base classes, which we find to be caused by the excessive focus on base-class-discriminative regions on novel-class samples. In this work, we aim to explore the underlying mechanism for an interpretation and solution. We first provide a compositional view to analyze the transferred and reused spatial patterns on novel-class samples. Then, through extensive experiments and theoretical analysis, we identify both empirically and theoretically that a shortcut exists in the model's base-class training, which naturally forms the excessive focus on only the most discriminative regions (primitives), which we term as the regional shortcut. Finally, based on this interpretation, to address this problem, we propose a compositional-learning-based method to learn two primitive sets (a common set and a discriminative set), which alleviates the regional shortcut by constraining the model to learn and utilize the common primitive set for base- and novel-class recognition. Extensive experiments on standard FSCIL benchmarks demonstrate the effectiveness of our approach, yielding consistent improvements over existing state-of-the-art methods in both accuracy and interpretability.

Causal inference has become a central issue across various fields, including computer science, statistics, economics, education, healthcare, and medicine. The broad applicability of this discipline has garnered increased research funding and attention. In recent years, the estimation of causal effects from observational data has gained traction due to the vast amounts of collected data and the lower costs compared to randomized controlled trials. Advances in causal effect estimation methods have enhanced service personalization tools. For instance, these tools can help identify the most effective type of treatment (considering both cost and success rate) for each patient among different medical service options. This paper proposes an innovative method for estimating the heterogeneity of treatment effects. The structure of the proposed model is based on a deep neural network and a pseudo-single learner. The proposed method has been compared with other state-of-the-art methods on the IHDP benchmark. Acceptable results have been obtained by using one estimator to estimate the potential outcomes of two treatment groups. Accordingly, this paves the way for further development and improvement of the proposed method.

Agent Skills package reusable procedural knowledge as external artifacts for frozen language-model agents, yet existing optimizers do not jointly resolve where a failure occurs in a workflow, which mechanism caused it, and how relevant knowledge from third-party Skills should be reused locally. We introduce Workflow-Localized Mechanism Learning (WML). Its Node--Mechanism Attribution identifies the failed workflow node, implicated mechanisms, and smallest valid edit target, routing single-mechanism defects to L3 resources and relational defects across mechanisms to L2 composition protocols. A six-module Workflow-Guided Skill Optimization (WGSO) loop then selects provenance- and scope-aware third-party knowledge, applies bounded patches, evaluates candidates, and stores verified outcomes in optimizer-side memory. On SpreadsheetBench, WML reaches 90.33 +/- 1.53 and 74.67 +/- 3.51 Hard Accuracy with DeepSeek and Qwen3.6-Flash, respectively; without additional optimization, the learned Skills transfer to WikiTableQuestions with 84.00 +/- 2.00 and 83.00 +/- 2.00 Denotation Accuracy. On Compiler-Supported50, WML attains both the highest hard-PASS rate and the lowest cost per successful task; compiled execution sharply reduces tokens and calls relative to a direct SkillAgent while retaining most of its successful tasks. Code and artifacts are available at https://github.com/xiaolin9595/workflow-localized-mechanism-learning.

We study Gaussian averaging as a smooth surrogate for quantized neural models. Under bounded local oscillation, we derive a local dimension-dependent bound on |f-g|, linking Gaussian smoothing to the stability analysis of discontinuous networks. We compute closed-form Gaussian averages of the rectified linear unit (ReLU) and sign activation functions, and illustrate the mechanism on a high-dimensional binary perceptron, where layer-preactivation aggregation under an explicit quantization-noise surrogate yields the Gaussian envelope used in inference-side smoothing and training-side smooth surrogate gradients.

Structural generalization has been measured repeatedly by several benchmarks, yet it has never been formally defined. We give a definition that translates the two premises (compositional structure and unbounded generalization) into mathematical language. The definition itself is neutral: a compiler that hard-codes the rules satisfies it just as well. But structural generalization becomes a scientific question only insofar as the capacity can autonomously emerge from finite data. This question pits the computational lower bound $\mathrm{NC}^1$ against the learnable ceiling $\mathrm{TC}^0$ of pure Transformers. Under a Montagovian instantiation, each compositional rule splits into two projections: a syntactic face ($F_γ$) and a semantic face ($G_γ$). Tree evaluation on the $G_γ$ side is an instantiation of BFVP, which is $\mathrm{NC}^1$-complete (Buss, 1987). A pure Transformer must learn both faces at once, but Kraus et al. (2026) prove that its learnable class $\subseteq \mathrm{TC}^0$. Under the standard assumption $\mathrm{TC}^0 \neq \mathrm{NC}^1$, a pure Transformer cannot learn structural generalization. Neuro-symbolic systems achieve the best benchmark scores precisely because they inject $G_γ$, sidestepping the genuinely hard half. Benchmark scores cannot distinguish "learned" from "given." This is what this paper sets out to make clear.

We introduce CMP (Cognitive Memory Primitive), an architecture that represents inputs as sparse relational codes, stores them in a two-tier competitive memory, and learns entirely through local, gradient-free updates, with no backpropagation anywhere in the network. We use this architecture to test a specific hypothesis: that catastrophic forgetting, usually treated as a training-time defect to be patched with replay or regularization, is instead a structural consequence of how backpropagation assigns credit and that a learning rule that is local and sparse by construction should resist it without a patch. On a controlled domain-incremental protocol across 15 text domains, three-seed replicated, CMP's backward transfer is 15-19x better than a matched-size Transformer trained with online EWC, and the result survives a domain-order control (reported as a range, +0.24 to +0.44, rather than a single figure). We report this alongside a real, substantial accuracy gap versus the Transformer baseline, a null result on a recognized vision benchmark, and a diagnosed, unresolved failure attempting to combine this architecture with a separate mechanism that improves raw accuracy, disclosed because an honest negative result is more useful than an omitted one. The central claim is narrow and falsifiable: local, sparse, non-backpropagation learning measurably resists catastrophic forgetting better than backpropagation with its standard fix, under conditions we state precisely.

This paper develops a categorical framework -- Learning in Infinitesimal Non-Compositional Sketches (LINCS) -- as the repair of non-compositionality: failures of diagrams to factor through quotient sketches lifted to the tangent category setting. Machine learning problems are specified as sketches: graphs with commutativity conditions $\mathcal D$, limit cones $\mathcal L$, and colimit cocones $\mathcal K$, generalizing the usual scalarization of loss functions or vector space assumptions. Non-compositionality is defined purely as failure of a universal factorization problem, not as arithmetic error between the desired and actual predictions. Given a learning sketch $\mathbb S=(S,\mathcal D,\mathcal L,\mathcal K)$, whose underlying graph is $S$, and a model $D:J \rightarrow C$, the base defect is the obstruction to factorization $\mbox{Obs}(\mbox{Fact}_{\mathbb S}(D))$. The tangent lift applies the tangent functor $T$ to obtain $TD:J \rightarrow C$, and LINCS is defined as the obstruction $\mbox{Obs}(\mbox{Fact}_{\mathbb S}(TD))$ -- asking whether infinitesimal perturbations preserve the compositionality constraints.The paper also introduces Tangent Learning Sketches, which are sketches equipped with Cockett-Cruttwell tangent structure. The paper defines the INC endofunctor, which iterates the tangent lift, producing a tower $D,TD,T^2D, \cdots$ of factorization problems. ML is thereby formulated as the search for a coalgebraic fixed point where successive tangent unfoldings stabilize ($νT_{\mbox{INC}}$). Using the Aczel--Mendler theorem, we prove existence of a final INC coalgebra whenever $T_{\mbox{INC}}$ admits a set-based class realization that creates its final carrier. A detailed experimental evaluation of LINCS is underway in a number of concrete ML settings, including deep learning, large language models, and reinforcement learning, and is described in companion papers.

Plasticity -- a neural network's ability to adapt to new tasks -- is critical for continual and transfer learning. Existing measures, such as effective rank, dead neuron fraction, and weight norm, lack theoretical grounding and correlate poorly with performance on new tasks. We introduce local redundancy, an information-theoretic measure derived from universal compression theory. We define local redundancy as the worst-case redundancy of a local model family -- parameters in an infinitesimal neighborhood along gradient directions -- and show this is a principled measure of plasticity. Although local redundancy is intractable to compute exactly, we prove that the expected squared gradient norm on a synthetic memorization task provides an efficiently computable lower bound. Experiments on continual image classification and time series transfer learning demonstrate that local redundancy predicts downstream performance better than existing measures and enables pretraining checkpoint selection where validation loss plateaus.

The Forward-Forward (FF) algorithm trains each layer locally, so that a scalar goodness - the sum of squared activations - is high on real inputs and low on contrastive ones, with activations normalized between layers. Both choices are usually treated as heuristics. Under an explicit generative model they are not: the squared goodness is the sufficient statistic of a likelihood-ratio test between two zero-mean populations differing in scale, and the FF threshold is its boundary. It generalizes: anisotropic populations yield a Mahalanobis goodness, the plain square being its isotropic case; heavy-tailed populations yield a saturating statistic whose slope is a posterior precision - divisive normalization - with bounded evidence and an advantage only under aggregation. The same lens characterizes the inter-layer normalization: it must remove the length while preserving per-coordinate energy, explaining a depth collapse we observe under unit-norm normalization; and the pairwise objective admits a scale-inflation shortcut that a whitened goodness removes.

Continual learning commonly relies on post-hoc mechanisms such as replay, elastic regularization, or distillation. This work argues that forgetting should instead be modeled directly as interference between tasks. In the frozen-feature regime, forgetting from learning a new task is exactly the interference energy induced on the old task. In deep networks, the same quantity is recovered through path-averaged curvature with minimal additional forward passes. When task supports are disjoint, forgetting can be eliminated structurally and when task supports overlap in conflicting directions, a non-zero distortion floor is unavoidable. The same geometry optimally merges models through task-aware orthogonalization. From this analysis we derive Interference-Gated Functional Allocation (IGFA), a replay-free, Fisher-free method that shares directions when tasks align and protects them when they conflict. Across benchmarks, IGFA achieves lossless retention when tasks are structurally separable and moves unavoidable cost from irreversible forgetting into deferred but recoverable plasticity when they are not. It matches the strongest replay-free structural baselines on dissimilar-task streams and improves on unconditional projection when similarity makes transfer worth preserving.

The problem of learning constant-depth circuits holds profound implications for computational learning theory. In a seminal result, by introducing the low-degree algorithm, Linial, Mansour, and Nisan (J. ACM 1993) presented a quasipolynomial-time learner for $\mathsf{AC}^0$ under the uniform distribution. However, obtaining comparable learning guarantees for broader classes of correlated distributions has remained a longstanding challenge. Recently, Chandrasekaran, Gaitonde, Moitra, and Vasilyan (arXiv 2026) extended these guarantees to Gibbs distributions on bounded-degree graphical models with both strong spatial mixing and polynomial growth. In this paper, we give a quasipolynomial-time learner for $\mathsf{AC}^0$ under graphical models that admit efficient local samplers, circumventing the polynomial-growth requirement in prior work. The key ingredient is a new low-degree approximation for Gibbs distributions, established by simulating and suitably truncating the classical Glauber dynamics. As applications, this framework yields learners for two-spin systems, including the hard-core model and Ising model, on arbitrary bounded-degree graphs, in regimes approaching their respective sampling thresholds.

Infinite-width limits are a standard way to reason about neural networks, but it is not automatic that the limiting learner has the same complexity-theoretic inductive bias as large finite networks. We study this question for Bayesian neural networks at the mean-field, or critical feature-learning, scaling. The central quantity is the \emph{reduced entropy} $$s_\infty(y,\varepsilon)=\limsup_N -\frac{1}{N}\log π_N^0(L\le \varepsilon),$$ the intensive prior cost of representing a target function $y$ to population mean-squared error $\varepsilon$. Our main result is a width-robust learnability theorem. At fixed depth, a family of Boolean-cube targets is learnable from polynomially many samples at infinite width if and only if it is learnable at polynomial width, if and only if its reduced entropy is polynomially bounded. Equivalently, up to polynomial slack in accuracy, the Bayesian mean-field learner generalizes exactly on the targets that can be represented by polynomial-size networks. The forward direction is proved by a form of subsampling: from the infinitely many hidden neurons in the mean-field solution, one can select polynomially many representatives and still preserve the learned function on every input simultaneously. At the critical scaling this subsampling has both an active'' component, which keeps the data-dependent low-dimensional statistics, and a lazy'' component, which resamples the entropy-dominated directions from the prior. Thus the infinite-width mean-field limit gives a clean analytic description of learning without introducing spurious width-dependent generalization power.

We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function trees built from a finite vocabulary of smooth operators (e.g., $\{+,\times,\sin,\exp\}$ and affine maps). We prove that the relevant generalization quantity, Rademacher complexity, hence the excess risk, does not necessarily blow up exponentially with the number of distinct symbolic structures, but is controlled by (i) the depth $d$ and (ii) the Lipschitz constants of the base operators along the composed computation graph. Concretely, under mild Lipschitz conditions on operators and bounded affine leaves, a finite-union bound over a vocabulary of size $K=|\mathcal{H}_{\mathrm{base}}|$ together with Maurer-type vector contraction yields $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{d}) \leq (Kb\sqrt{2}L)^{d-1}\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})$ with arity bound $b$; corresponding high-probability risk bounds scale as $\mathcal{O}(L^{d}/\sqrt{n})$ when $K,b=O(1)$ and $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})=O(n^{-1/2})$. We complement the theory with a modular codebase that trains differentiable operator trees (not MLPs) on synthetic "physics-like" targets of controlled depth and shows that the empirical generalization gap correlates positively with the predicted complexity term $(\widehat{L}^{d})/\sqrt{n}$.

In the optimization of neural networks, gradient dynamics are influenced by critical points that arise from the model's architecture. These critical points occur where the Jacobian of the model's parametrization is rank-deficient, and are the most pronounced singularities studied in Singular Learning Theory. We investigate such points in deep fully-connected networks with monomial activations via tools from polynomial algebra such as Mason's Theorem. We show that, for sufficiently large activation degree, criticality occurs precisely at subnetworks, i.e., at parameter configurations where some neurons are inactive or redundant. This offers a mathematical perspective on the implicit bias in deep neural networks, explaining the tendency of these models to converge toward simpler functions.

Backpropagation-trained dense neural networks are powerful function approximators, but they couple learning across many parameters and can overwrite previous associations when tasks conflict. This paper describes Local Pheromone Network, a small research prototype for sparse, local, manually updated neural networks. In Local Pheromone Network, each output unit reads only a fixed local neighborhood of input units subject to geometric distance and molecular-tag compatibility. Each synapse stores a weight, a short-term pheromone trace, a long-term pheromone trace, and an optional consolidation state. Training does not call automatic differentiation. Instead, every layer performs a pheromone-weighted Hebbian-style update on a budgeted subset of local synapses selected from local error and co-activity. The update budget adapts online: it shrinks when loss improves and expands toward recently active neighborhoods when loss worsens. Optional mechanisms add structural plasticity, local replay, output masks for partitioned learning, and a target-free local contrastive step. We present the implementation, learning rule, and preliminary experiments on synthetic regression, partitioned memory, conflicting memory, consolidated conflict, structural plasticity, replay, and a synthetic long-context hybrid memory task. The prototype learns local linear rules, preserves partitioned memories through tags and masks, reduces forgetting under consolidation, and uses replay under conflict.

Singular Learning Theory leverages the Local Learning Coefficient (LLC) to quantify the geometry of neural network loss landscapes. However, mean-energy LLC estimators depend explicitly on an additive loss baseline, typically an estimate of the local minimum. During transient, off-equilibrium training phases, this minimum is unknown; substituting it with the lowest noisy mini-batch loss induces a systematic minimization bias that distorts the geometric measurement. In this paper, we propose the Shift-Invariant Variance Estimator (SIVE), a variance-based local LLC probe that structurally eliminates the unknown additive baseline through the variance operator. Combining this shift-invariant observable with an explicit correction derived from the Law of Total Variance, SIVE separates geometric loss fluctuations from mini-batch evaluation noise. Controlled experiments on analytically tractable toy models show that SIVE recovers the expected finite-temperature geometric signal in regimes where anchored mean estimators fail. Applied to deep neural networks, SIVE provides a robust, localized online diagnostic for tracking structural phase transitions throughout training.

We study internalization processes, by which neural-network-based systems absorb an explicit computational procedure into their own weights, and how they facilitate learning. We investigate how transformers internalize the simulation of semiautomata by internalizing chain-of-thought (CoT) tokens, which classes of semiautomata are harder to internalize, and expose the flip side of internalization, that is, a progressive degradation of out-of-distribution performance. We then provide the first provable analysis of successful internalization: for the task of learning parities, we show that a simplified one-layer transformer provably first learns the target with explicit CoT supervision and then internalizes the autoregressive generation as CoT tokens are progressively removed, learning to directly compute the parity. This task is computationally hard to learn from data without CoT supervision. Finally, we discuss how learning through internalization relates to the \textit{Positive Distribution Shift} phenomenon recently introduced by~\citet{Med+26}.

Forward-Forward (FF) learning [Hinton, 2022] replaces backpropagation with strictly layer-local goodness updates. Recent FF-CNN work has narrowed the gap to BP on 32x32 benchmarks, raising the question of whether layer-local training is becoming a viable alternative at realistic scale. To probe this rigorously, we develop DTG-FF -- dynamic temperature goodness, decoupled normalization, and multi-layer fusion -- as an instrument that sets FF-family state of the art across nine real-data benchmarks (91.8% CIFAR-10 and the first FF baseline at ImageNet-100 224x224), and use it to audit how far layer-local training actually scales. (1) Real-data scaling. Under identical recipe and backbone, an architecture-matched BP-DeepSup baseline beats DTG-FF by 2.40/5.93 pp on CIFAR-10/CIFAR-100, and the gap widens with class count. At 224x224 the same instrument reaches only 49.4% -- the first FF baseline at this scale, versus typical BP above 75% [Tian et al., 2020] -- exposing a real-data ceiling invisible at 32x32. (2) Synthetic vs. real K-conflict. DTG-FF increasingly outperforms BP as class count K grows on synthetic teacher-student tasks, yet on real images the FF-BP gap reverses sign and widens with K. A within-dataset CIFAR-100 coarse vs. fine probe isolates label-hierarchy from image distribution: synthetic K-sweeps confound output dimensionality with fine-grained discrimination difficulty and thereby overstate FF transferability. (3) Systems audit. FF can be implemented without storing depth-wide activations, but on commodity 8 GB hardware standard BP+gradient-accumulation reaches 4.18 GB / 157 imgs/s versus DTG-FF's 7.90 GB / 138 imgs/s, so a memory-based justification for FF at this scale is not supported under fair baselines.

Machine unlearning aims to remove the influence of selected training examples without full retraining. Standard evaluations often summarize unlearning quality with aggregate metrics, such as accuracy- and forgetting-based scores, which can hide localized failures. We study this failure mode at the example level by comparing the predictions of an unlearned model to those of the model retrained after deletion. We show that this pointwise discrepancy can be highly non-uniform: for gradient-ascent and random-labeling methods, with and without retain-set fine-tuning, it grows with geometric proximity to the forget set. We call this phenomenon localized collateral forgetting. Our analysis identifies a mechanism behind the effect: surrogate targets used during unlearning can be inconsistent with the local prediction structure induced by retraining, and this inconsistency propagates through shared representations to nearby examples. Motivated by this mechanism, we propose Local Teacher Distillation, a simple mitigation strategy that replaces random targets with soft labels from a small teacher trained only on retained neighbors of the forget set. On CIFAR-100 partial-class deletion, this local teacher brings the unlearned model substantially closer to retraining, especially near the forget set, while maintaining competitive aggregate unlearning metrics.

Symbolic regression (SR) seeks closed-form mathematical expressions that fit observed data. Neural SR methods amortize the search by training an encoder to map observations directly to expressions in a single pass, but this amortized inference leaves a residual amortization gap between its one-shot prediction and the true posterior. We propose Latent Equation Embedding (LEE), a framework that closes this gap through iterative amortized inference in a functionally grounded latent space. LEE learns a shared latent space Z equipped with three components: an encoder f_theta that jointly embeds symbolic tokens and numerical observations into a single latent vector z; an expression decoder g_expr that reconstructs formulas from z; and an evaluation decoder g_eval that predicts function values from z, explicitly grounding the latent space in functional behavior. At inference, LEE performs iterative refinement by re-encoding decoded expressions jointly with observations, progressively improving the latent estimate. LEE uses the encoder itself as a learned inference optimizer: each re-encoding step implicitly computes the mismatch between the candidate and the data. Because g_eval is differentiable in z, we additionally interleave continuous gradient descent with discrete re-encoding, yielding a hybrid iterative and gradient refinement procedure. On SRBench across three noise levels, against 19 baselines spanning genetic programming, symbolic-neural hybrids, and pre-trained Transformers, LEE produces expressions 2--10x simpler than the strongest accuracy-oriented baselines, including Operon, GP-GOMEA, TPSR, RAG-SR, and GenSR, with complexity 8--11 versus 20--90. These results advance the low-complexity region of the accuracy-complexity Pareto frontier and show graceful degradation as noise increases.

In the context of artificial neural networks, subliminal learning refers to the transfer of task-relevant knowledge or unintended biases from teacher to student models through distillation on task-unrelated input$\unicode{x2013}$output pairs. Prior explanations tie this effect to shared or closely matched teacher$\unicode{x2013}$student initialization. We show that a closely matched initialization is not necessary. Instead, subliminal learning is governed by compatible output heads. Using a controlled MNIST setting, we split outputs into an auxiliary head (for auxiliary, task-unrelated noise signals) and a class head (for classification) to demonstrate subliminal learning occurs$\unicode{x2014}$even when we randomly initialize hidden layers and remove layers, add new layers, or change the architecture (MLP-to-CNN). Compatible auxiliary heads enable transfer of a recoverable teacher signal, bringing the student's representations closer to the teacher's. When the class heads remain compatible as well, students trained only on task-unrelated noise can approach, and in favorable regimes match, teacher-level task performance. Our setting enables us to develop a theory that explains the mechanism of subliminal learning and to derive upper bounds on when subliminal learning fails. Together, our results turn subliminal learning from a surprising transfer effect into a theoretically grounded mechanism with predictable limits.

Humans, animals, and modern machine learning models exhibit impressive abilities to learn complex behaviors and generalize these behaviors to unseen situations. This ability requires us to learn rules and regularities that allow for such generalizations. At the same time, in most complex environments, any rule will have its exceptions. How do learning systems balance between learning general regularities and memorizing exceptions? We argue that a lack of task paradigms has hindered the study of this essential ability. To address this gap, we introduce a novel task, transitive inference with exceptions, that tests for relational generalization and memorization of an exception to the relational rule. We then analytically characterize the behavior of a simple, theoretically tractable model of neural network learning (kernel ridge regression) across a broad family of representations and task parameters. We find that these models can balance between relational generalization and memorization, but unlike for transitive inference without an exception, successful generalization is sensitive to the specific representational geometry. We explain why this task is more challenging mechanistically by drawing on our analytical theory. Finally, we validate our theoretical insights in pretrained language models that are finetuned on ordered relations, finding that these models successfully generalize according to the transitive rule, but also make the kinds of systematic mistakes predicted by our theory. Overall, our theory shows how learning systems can balance between relational generalization and memorization, explains how this can go wrong, and emphasizes the need for new task paradigms designed to probe this ability.

The brain learns abstract representations of high-dimensional sensory input, but the plasticity rules that enable such learning are unknown. We study biologically plausible algorithms on the Random Hierarchy Model (RHM), an artificial dataset designed to investigate how deep neural networks learn the intrinsic hierarchical structure of high-dimensional data. We focus on two types of local learning rules that avoid both a long convergence time and the use of a symmetric error network. The first type uses direct feedback signals to approximate error propagation from the output layer. The second type uses layerwise self-supervised contrastive or non-contrastive loss functions that do not explicitly approximate errors at the output layer. We show that all rules of the first type fail to solve the tasks of the RHM and trace this failure back to input-specific nonlinearities (`masking') that are implemented in full backpropagation and are essential for learning complex tasks. However, algorithms of the second type are able to learn the hierarchical hidden structure of the RHM tasks and are as data-efficient as supervised backpropagation training, while being compatible with known rules of synaptic plasticity in cortex.

In-context learning (ICL) excels at new tasks from minimal examples, yet we still lack a mechanistic explanation of how few-shot prompts shape a model's function vector (FV)--a causal activation direction that drives task behavior on the ICL query. Across tasks and models, an $n$-shot FV is well-approximated by a linear combination of example-level sub-FVs, suggesting additive and composable contributions from individual demonstrations. Beyond additivity, we show that models contextualize individual examples' representations based on prior examples to adaptively reweight which demonstrations dominate the FV: attention shifts toward examples that are more informative and less ambiguous under the context. Finally, a causal decomposition separates Query-Key routing from Value updates, finding that contextualization's most consistent contributions to FV quality arise from Query-Key alignment--particularly in ambiguous settings--while Value-mediated effects are more heterogeneous. Together, these results unify additive superposition with context-dependent attention reweighting into a mechanistic, testable account of how few-shot prompts implement tasks.

Continual learning enables neural networks to learn tasks sequentially without forgetting previously acquired knowledge. However, neural networks suffer from catastrophic forgetting, where learning new tasks degrades performance on earlier ones. We address this problem with Shapley Neuron Valuation (SNV), a principled framework that quantifies Neuron importance in continual learning, grounded in cooperative game theory. SNV selectively freezes important Neurons while keeping others plastic, enabling buffer-free continual learning without expanding architecture. Experiments on ImageNet-1k show that SNV consistently outperforms existing buffer-free methods. In particular, SNV improves accuracy by +2.88% in the class incremental learning and +6.46% in the task incremental learning scenarios compared to the second baseline.

The rapid expansion of spiking neural networks (SNNs) has led to a proliferation of training algorithms that differ widely in biological inspiration, computational structure, and hardware suitability. Despite this progress, the field lacks a unified, fine-grained taxonomy that systematically organizes these approaches and clarifies their conceptual relationships. This survey provides a comprehensive taxonomy of SNN training algorithms, spanning surrogate-gradient backpropagation, local and three-factor learning rules, biologically inspired plasticity mechanisms, ANN-to-SNN conversion pipelines, and non-standard optimization strategies. We analyze each class in terms of its computational principles, learning signals, and locality properties. To support reproducible research, we release NeuroTrain, an open-source snnTorch-based framework that implements a representative set of these algorithms within a unified, modular, and extendable framework, enabling consistent benchmarking across datasets, architectures, and training regimes. By consolidating fragmented literature and providing a reusable benchmarking framework, this survey identifies common patterns, highlights open challenges, and outlines promising directions for future work on scalable, efficient SNN training.

Neural networks suffer from catastrophic forgetting in class-incremental learning (CIL) settings. Rehearsal$\unicode{x2013}$replaying a subset of past samples$\unicode{x2013}$is a well-established mitigation strategy. However, recent results suggest that, despite balanced rehearsal allocation, some classes are forgotten substantially more than others. Despite its relevance, this imbalanced forgetting phenomenon remains underexplored. This work shows that imbalanced forgetting arises systematically and severely in rehearsal-based CIL and investigates it extensively. Specifically, we construct, from a principled analysis, three last-layer coefficients that capture different gradient-level sources of interference affecting each past class during an incremental step. We then demonstrate that, together, they reliably predict how past classes will rank in terms of forgetting at the end of that step. While predictive performance alone does not establish causality, these results support the interpretation of the coefficients as a plausible mechanistic account linking last-layer gradient-level interactions during training to class-level forgetting outcomes. Notably, one coefficient$\unicode{x2013}$capturing self-induced interference$\unicode{x2013}$emerges as the strongest predictor, with controlled experiments providing evidence consistent with this coefficient being influenced by the new-class interference coefficient. Overall, our findings provide valuable insights and suggest promising directions for mitigating imbalanced forgetting by reducing class-wise disparities in the identified sources of interference.

Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: Which classes are learnable in it? For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

Strategic classification studies learning settings in which individuals can modify their features, at a cost, in order to influence the classifier's decision. A central question is how the sample complexity of the induced (strategic) hypothesis class depends on the complexities of the underlying hypothesis class and the cost structure governing feasible manipulations. Prior work has shown that in several natural settings, such as linear classifiers with norm costs, the induced complexity can be controlled. We begin by showing that such guarantees fail in general - even in simple cases: there exist hypothesis classes of VC dimension $1$ on the real line such that, even under the simplest interval neighborhoods, the induced class has infinite VC dimension. Thus, strategic behavior can turn an easy learning problem into a non-learnable one. To overcome this, we introduce structure via a geometric definability assumption: both the hypothesis class and the cost-induced neighborhood relation can be defined by first-order formulas over $\mathbb{R}_{\mathtt{exp}}$. Intuitively, this means that hypotheses and costs can be described using arithmetic operations, exponentiation, logarithms, and comparisons. This captures a broad range of natural classes and cost functions, including $\ell_p$ distances, Wasserstein distance, and information-theoretic divergences. Under this assumption, we prove that learnability is preserved, with sample complexity controlled by the complexity of the defining formulas.

We propose Adaptive Multi-Scale Goodness Aggregation (AMSGA), a novel extension of the Forward-Forward (FF) algorithm designed to improve stability, robustness, and generalization in local-learning neural networks. AMSGA addresses several limitations of the original FF framework by introducing multi-scale goodness aggregation across local, intermediate, and global representations; adaptive curriculum-guided hard negative mining; layer-dependent adaptive thresholds; and a warm-up cosine annealing learning-rate schedule for improved optimization stability. Together, these modifications strengthen the FF paradigm while preserving its biologically plausible and memory-efficient properties. Experiments on MNIST and Fashion-MNIST demonstrate consistent performance improvements over the baseline FF algorithm, achieving up to +1.45% improvement on MNIST and +1.50% improvement on Fashion-MNIST without significant computational overhead. Our results suggest that local learning methods can become substantially more competitive when goodness estimation and training dynamics are carefully designed.

Learning parity functions, more general modular addition, is a challenging machine learning task due to its input sensitivity. A recent study substantially scaled modular addition learning in both the number of summands and the modulus. Its key idea is to increase zeros in training sequences, reducing the effective number of summands and thus controlling training difficulty; however, this induces covariate shift between training and test input distributions. This study theoretically and empirically analyzes this side effect and proposes a covariate-shift-free method for modular addition. Specifically, we introduce an auxiliary modulus $Kq$ during training, which reduces wrap-around frequency and problem difficulty while preserving the same input distribution across training and testing. Experiments show strong scalability and sample efficiency: even for large input length $N$, large modulus $q$, and small datasets -- where the sparse method fails to learn -- our method achieves equal or better match accuracy and relaxed $τ$-accuracy. For example, at $N=64$ and $q=974269$, our method trained on 100K samples achieves $97.0\%$ $τ$-accuracy at $τ=0.05$, while the sparse method achieves only $9.5\%$ with the same data size and $93.9\%$ even when extended to 1M samples.

We introduce GC-ART (Global Curve Adaptive Rational Tone-mapping), a lightweight differentiable pre-processing module for robust image classification. GC-ART predicts an endpoint-pinned rational tone curve from per-channel soft histograms using a 643-parameter MLP, then applies the curve pointwise before the classifier. The module is trained end-to-end with cross-entropy and a soft monotonicity penalty. On CIFAR-10 with a CIFAR-style ResNet-18, GC-ART matches clean accuracy with the unenhanced baseline and other learned enhancers, improves over the baseline on multiplicative darkening, and achieves the best learned-method result on contrast corruption (48.45% vs. 46.27% for the baseline and 47.13% for Zero-DCE++). These results suggest that histogram-conditioned rational curves can learn useful global tone corrections, including contrast-expanding behavior, while preserving edge locations by construction through pointwise mapping. GC-ART also uses substantially fewer FLOPs than convolutional learned enhancers at 32 x 32. The current hyperparameters are untuned, leaving room for systematic improvement.

This demo paper describes the development of the AI Teaching & Learning Assistant, a modular Moodle plugin that leverages Retrieval-Augmented Generation (RAG) to deliver high-quality, hallucination-free education. The system employs a dual-centric design, providing students with interactive, Socratic-based tutoring and educators with a "human-in-the-loop" workspace for supervised content generation. By grounding Large Language Model (LLM) responses in teacher-provided materials, the assistant addresses the risks of misinformation while encouraging deep conceptual mastery. Evaluation via the Ragas (LLM-as-a-Judge) framework and a preliminary user study confirms its effectiveness, achieving faithfulness scores up to 0.97 and a 4.00/5.00 recommendation rate.

We study the complexity of smoothed agnostic learning of halfspaces on $\{\pm 1\}^n$ under uniform marginals in the model of~\cite{KM25}, where each input coordinate is independently flipped with probability $σ\in (0, {1}/{2})$. We show that $L^1$ polynomial regression achieves runtime and sample complexity $\tilde{O}(n^{O(\log(1/\varepsilon)/σ)})$, and prove a nearly matching Statistical Query complexity lower bound of $n^{Ω(\log(1+σ/\varepsilon^2)/σ)}$. This complements the recent work of~\cite{DK26}, which established analogous bounds in the continuous setting under Gaussian marginals.

Recent work revisiting measurability in the fundamental theorem of statistical learning imposes Borel measurability of ghost-gap suprema. We show that, at the one-sided ghost-gap interface actually used by the standard symmetrization proof, this requirement is stronger than necessary. For any Borel-parameterized concept class on a Polish domain, the bad event "there exists a hypothesis whose ghost empirical error exceeds its training empirical error by at least ε/2" is analytic. By Choquet capacitability, it is therefore measurable in the completion of every finite Borel measure. We then construct a concept class whose bad event is null-measurable but not Borel, giving a strict separation from the Borel supremum condition. Finally, we prove closure under patching, fixed and countable interpolation, and fiber-product amalgamation, showing that the weaker regularity level is stable under natural concept-class constructors. In the realizable setting, where targets belong to the class and are measurable, these results weaken the measurability hypothesis needed by the symmetrization route from finite VC dimension to PAC learnability. The main results and the descriptive-set-theoretic infrastructure used by them are formalized in Lean 4.

The Super Learner (SL) is a widely used ensemble method that combines point predictions from a library of learners based on their predictive performance. Interval predictions are of considerable practical interest because they allow uncertainty in predictions produced by an individual learner or an ensemble to be quantified. Several methods have been proposed for constructing interval predictions based on the SL, however, these approaches are typically justified using asymptotic arguments or rely on computationally intensive procedures such as the bootstrap. Conformal prediction (CP) is a machine learning framework for constructing prediction intervals with finite-sample and asymptotic coverage guarantees under mild conditions. We propose coupling CP with the SL through a natural construction that mirrors the original SL framework, using individual learner weights and combining learner-specific conformity scores via a weighted majority vote. We characterize the properties of the resulting SL-based prediction intervals for continuous outcomes. We cover settings under exchangeability, potential violations of exchangeability, and data-generating mechanisms exhibiting heteroscedasticity, sparsity, and other forms of distributional heterogeneity. A comprehensive simulation study shows that the conformalized SL achieves valid finite-sample coverage with competitive performance relative to the true data-generating mechanism. A central contribution of this work is an application to predicting creatinine levels using socio-demographic, biometric, and laboratory measurements. This example demonstrates the benefits of an ensemble with carefully selected learners designed to capture key aspects of complex regression functions, including non-linear effects, interactions, sparsity, heteroscedasticity, and robustness to outliers.

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

We present a generic algorithm for learning and approximate inference with an intuitive epistemic interpretation: iteratively focus on a subset of the model and resolve inconsistencies using the parameters under control. This framework, which we call Local Inconsistency Resolution (LIR) is built upon Probabilistic Dependency Graphs (PDGs), which provide a flexible representational foundation capable of capturing inconsistent beliefs. We show how LIR unifies and generalizes a wide variety of important algorithms in the literature, including the Expectation-Maximization (EM) algorithm, belief propagation, adversarial training, GANs, and GFlowNets. In the last case, LIR actually suggests a more natural loss, which we demonstrate improves GFlowNet convergence. Each method can be recovered as a specific instance of LIR by choosing a procedure to direct focus (attention and control). We implement this algorithm for discrete PDGs and study its properties on synthetically generated PDGs, comparing its behavior to the global optimization semantics of the full PDG.

Parallel reasoning enhances Large Reasoning Models (LRMs) but incurs prohibitive costs due to futile paths caused by early errors. To mitigate this, path pruning at the prefix level is essential, yet existing research remains fragmented without a standardized framework. In this work, we propose the first systematic taxonomy of path pruning, categorizing methods by their signal source (internal vs. external) and learnability (learnable vs. non-learnable). This classification reveals the unexplored potential of learnable internal methods, motivating our proposal of STOP (Super TOken for Pruning). Extensive evaluations across LRMs ranging from 1.5B to 20B parameters demonstrate that STOP achieves superior effectiveness and efficiency compared to existing baselines. Furthermore, we rigorously validate the scalability of STOP under varying compute budgets - for instance, boosting GPT-OSS-20B accuracy on AIME25 from 84% to nearly 90% under fixed compute budgets. Finally, we distill our findings into formalized empirical guidelines to facilitate optimal real-world deployment. Code, data and models are available at https://bijiaxihh.github.io/STOP

Catastrophic forgetting remains a fundamental challenge in continual learning, in which models often forget previous knowledge when fine-tuned on a new task. This issue is especially pronounced in class incremental learning (CIL), which is the most challenging setting in continual learning. Existing methods to address catastrophic forgetting often sacrifice either model interpretability or accuracy. To address this challenge, we introduce ClassIncremental Concept Bottleneck Model (CI-CBM), which leverage effective techniques, including concept regularization and pseudo-concept generation to maintain interpretable decision processes throughout incremental learning phases. Through extensive evaluation on seven datasets, CI-CBM achieves comparable performance to black-box models and outperforms previous interpretable approaches in CIL, with an average 36% accuracy gain. CICBM provides interpretable decisions on individual inputs and understandable global decision rules, as shown in our experiments, thereby demonstrating that human understandable concepts can be maintained during incremental learning without compromising model performance. Our approach is effective in both pretrained and non-pretrained scenarios; in the latter, the backbone is trained from scratch during the first learning phase. Code is publicly available at github.com/importAmir/CI-CBM.

Synaptic plasticity is metabolically expensive, yet animals continuously update their internal models without exhausting energy reserves. However, when artificial neural networks are trained, the network parameters are typically updated on every sample that is presented, even if the sample was classified correctly. Inspired by the human negativity bias and error-related negativity, we propose 'memorized mistake-gated learning' -- a biologically plausible plasticity rule where synaptic updates are strictly gated by current and past classification errors. This reduces the number of updates the network needs to make by $50\%\sim80\%$. Mistake gating is particularly well suited in two cases: 1) For incremental learning where new knowledge is acquired on a background of pre-existing knowledge, 2) For online learning scenarios when data needs to be stored for later replay, as mistake-gating reduces storage buffer requirements. The algorithm can be implemented in a few lines of code, adds no hyper-parameters, and comes at negligible computational overhead. Learning on mistakes is an energy efficient and biologically relevant modification to commonly used learning rules that is well suited for continual learning.

A key capability of modern neural networks is their capacity to simultaneously learn underlying rules and memorize specific facts or exceptions. Yet, theoretical understanding of this dual capability remains limited. We introduce the Rules-and-Facts (RAF) model, a minimal solvable setting that enables precise characterization of this phenomenon by bridging two classical lines of work in the statistical physics of learning: the teacher-student framework for generalization and Gardner-style capacity analysis for memorization. In the RAF model, a fraction $1 - \varepsilon$ of training labels is generated by a structured teacher rule, while a fraction $\varepsilon$ consists of unstructured facts with random labels. We characterize when the learner can simultaneously recover the underlying rule - allowing generalization to new data - and memorize the unstructured examples. Our results quantify how overparameterization enables the simultaneous realization of these two objectives: sufficient excess capacity supports memorization, while regularization and the choice of kernel or nonlinearity control the allocation of capacity between rule learning and memorization. Experiments on a CIFAR10-based RAF task show that the qualitative trade-off persists for fixed kernels, whereas representations learned by deeper neural networks substantially mitigate it, preserving factual recall while improving generalization. The RAF model provides a theoretical foundation for understanding how modern neural networks can infer structure while storing rare or non-compressible information.

Interpretable machine learning is essential in high-stakes domains where decision-making requires accountability, transparency, and trust. While rule-based models offer global and exact interpretability, learning rule sets that simultaneously achieve high predictive performance and low, human-understandable complexity remains challenging. To address this, we introduce TT-Sparse, a flexible neural building block that leverages differentiable truth tables as nodes to learn sparse, effective connections. A key contribution of our approach is a new soft TopK operator with straight-through estimation for learning discrete, cardinality-constrained feature selection in an end-to-end differentiable manner. Crucially, the forward pass remains sparse, enabling efficient computation and exact symbolic rule extraction. As a result, each node (and the entire model) can be transformed exactly into compact, globally interpretable DNF/CNF Boolean formulas via Quine-McCluskey minimization. Extensive empirical results across 28 datasets spanning binary, multiclass, and regression tasks show that the learned sparse rules exhibit superior predictive performance with lower complexity compared to existing state-of-the-art methods.

Understanding how the structure of language can be learned from sentences alone is a central question in both cognitive science and machine learning. Studies of the internal representations of Large Language Models (LLMs) support their ability to parse text when predicting the next word, while representing semantic notions independently of surface form. Yet, which data statistics make these feats possible, and how much data is required, remain largely unknown. Probabilistic context-free grammars (PCFGs) provide a tractable testbed for studying these questions. However, prior work has focused either on the post-hoc characterization of the parsing-like algorithms used by trained networks; or on the learnability of PCFGs with fixed syntax, where parsing is unnecessary. Here, we (i) introduce a tunable class of PCFGs in which both the degree of ambiguity and the correlation structure across scales can be controlled; (ii) provide a learning mechanism -- an inference algorithm inspired by the structure of deep convolutional networks -- that links learnability and sample complexity to specific language statistics; and (iii) validate our predictions empirically across deep convolutional and transformer-based architectures. Overall, we propose a unifying framework where correlations at different scales lift local ambiguities, enabling the emergence of hierarchical representations of the data.