Separable nonnegative matrix factorization (SNMF) has been widely used for low-rank representation and clustering of nonnegative data, owing to its ability to produce part-based and interpretable decompositions. In particular, SNMF is closely related to graph clustering and community detection. To enhance sparsity and identifiability of the learned factors, we propose an $\ell_1^p/\ell_2$-regularized SNMF model based on a powered ratio-of-norms regularizer. The resulting formulation is nonconvex and nonsmooth, which poses significant challenges for optimization. To address this, we develop efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM). The proposed methods decompose the original problem into tractable subproblems, leveraging closed-form proximal operators associated with the powered norm terms. We establish descent and limiting criticality properties for the DCA scheme and convergence under standard assumptions for the ADMM scheme. Extensive numerical experiments on synthetic datasets and hand gesture classification tasks demonstrate that the proposed approach achieves competitive or improved performance in anchor identification and classification accuracy compared with existing SNMF methods, while maintaining competitive computational efficiency.
Two of the most fundamental questions in statistical learning theory are the following: which prediction problems are learnable, and how should they be learned? For the former, elegant answers often take the form of combinatorial dimensions. The latter question, however, has proved considerably more elusive: all known general-purpose multiclass learners rely on intricate orientations of exponentially large one-inclusion structures, and familiar algorithmic principles such as proper learning and regularization remain poorly understood. Motivated by prior work, we ask whether learning reduces to proper learning---possibly over a larger hypothesis class---and whether proper or improper multiclass learning can ultimately be captured by suitable regularizers. Our primary results answer both questions negatively, resolving three open problems from prior work. First, we exhibit a learnable multiclass problem that cannot be embedded in any properly learnable class, meaning learning cannot be reduced to proper learning by enlarging the hypothesis class. Second, we demonstrate that proper learning can require training error and characterize this phenomenon precisely: every properly learnable class admits a proper learner making $o(m)$ errors on samples of size $m$, but every prescribed sublinear scale $a_m=o(m)$ is necessary for some properly learnable problem. Third, regularization is not a general learner: we exhibit a properly learnable class that cannot be learned by any Structural Risk Minimization (SRM) learner, and a learnable class that cannot be learned by any local regularizer. We complement these impossibility results with a positive theory that gives two sufficient conditions for SRM learnability and characterizes SRM representability through integrability of revealed preferences.
Random feature methods provide a scalable approximation to kernel ridge regression (KRR), but the regularization parameter that yields the oracle learning rate depends on unknown smoothness and capacity parameters. In this work, we propose a neighboring early-stopping rule for adaptive regularization in KRR with random features (KRR-RF). The method uses a grid that is uniform in inverse regularization and compares only adjacent estimators, reducing the number of discrepancy comparisons relative to standard all-pairs Lepskii-type procedures. Both the neighboring discrepancy and its empirical complexity term can be computed directly in the random feature space, without constructing the exact kernel Gram matrix. We establish a high-probability comparison bound for neighboring KRR-RF estimators and show that, under standard source and capacity conditions together with suitable grid and random feature budget conditions, the selected estimator attains the oracle polynomial learning rate up to logarithmic factors. The result allows the regularization parameter to be selected without prior knowledge of the source and capacity exponents and covers both well-specified and partially misspecified regimes. Our analysis is based on an empirical random feature effective dimension that connects the observable stopping threshold with the population complexity of the random feature model. Simulation and real-data experiments illustrate the prediction performance and computational behavior of the proposed method in comparison with standard tuning procedures.
We study a tree-structured regularizer over class-prototype layouts in a hierarchical-classification model and ask whether the choice of latent manifold for the prototypes (Euclidean R^d vs. the Poincare ball B^d_c) affects how well that regularizer can be satisfied without distorting the data likelihood. The two manifolds differ only in their volume growth: hyperbolic space grows exponentially with radius and embeds trees with provably lower distortion than R^d of matched dimension, so the structured regularizer should be cheaper to satisfy on B^d_c. Across 150 seed-replicated regularized maximum-likelihood fits spanning embedding dimension, curvature, and regularizer strength on WikiArt (27 styles, 81,446 paintings, frozen CLIP ViT-B/16 features), we find a single robust effect: Poincare prototypes preserve the topology of the nearest-neighbor graph in latent space substantially better than matched Euclidean prototypes (sibling recall@5 +8.7 pp, cousin recall +15.2 pp; paired-t p < 10^-4, sign agreement 0.94), and the gap holds across three reference-tree definitions (hand-built lineage, CLIP-derived, and DINOv2-derived). On classification, Euclidean prototypes are tied with logistic regression on raw encoder features, indicating no detectable contribution from the latent geometry; only the hyperbolic fit improves on a k-NN encoder baseline for local retrieval. Global tree-fidelity comparisons are unstable across reference trees and we do not claim a winner. The results give an empirical separation, on a real hierarchical-classification problem, between two natural latent geometries for a class-structured regularizer.
We analyze a variant of stochastic gradient descent with initial regularization (SGDIR) and derive dimension-free upper bounds on its expected excess risk for the squared loss. In the noiseless case, we obtain new bounds for both averaged and non-averaged SGDIR under moment, source, and capacity assumptions. For a particular value of the source parameter, these bounds are of order $m^{-2}\log^{2}m$, where the number of training samples is of order $m$. For another value of the source parameter, we obtain, for any $ε>0$, bounds of order $m^{-3+ε}$, provided that the capacity parameter exceeds $ε^{-1}$. We also establish a lower bound that matches our upper bounds in certain regimes up to a polylogarithmic factor. In the noisy case, we provide an instance-based comparison between SGDIR and ridge regression. Under general assumptions and a mild lower bound on the regularization parameter, we show that the expected excess risk of SGDIR is no larger than that of ridge regression, up to a polylogarithmic factor. Numerical experiments on synthetic and real data are consistent with our theoretical findings.
Siyuan Tang, Gongjun Xu, Ji Zhustat.ME cs.LG stat.ML
Mixed-type data containing both numerical and categorical variables arise in many scientific and real-world applications. Existing representation learning and generative modeling approaches typically focus either on reconstruction accuracy or unconditional data generation, but often fail to recover the full conditional distribution of the data while preserving interpretable structural relationships between heterogeneous variable types. In this work, we introduce Conditional-Independence-Regularized Distributional Autoencoders, a framework for learning low-dimensional representations of mixed-type data through conditional distribution matching and structural regularization. Our method combines an energy-score-based objective for numerical variables, a likelihood-based objective for categorical variables, and an auxiliary conditional independence regularization term encouraging the learned representation to capture the dependence between numerical and categorical components. We provide theoretical analysis showing that the optimal representation balances unexplained numerical variability, conditional entropy of categorical variables, and residual conditional dependence. Empirically, the proposed method achieves strong performance on both synthetic and real-world datasets, substantially improving categorical distribution recovery, achieving competitive overall conditional distribution recovery, and preserving mixed-type dependence structure. The code has been made available at GitHub.
Brian B. Moser, Ahmed Anwar, Tobias Christian Nauen +5cs.LG cs.AI
Continual learning regularizers like EWC fight forgetting by penalizing changes from previous-task parameters with per-parameter importance, typically diagonal Fisher values. Per-parameter looks more flexible than per-layer, but each layer's diagonal Fisher is a weak summary of its actual curvature, missing the top-eigenvalue information that controls forgetting. Adversarial bit-flip attacks and Hessian-spectrum studies show that this missing per-layer sensitivity spans orders of magnitude in neural networks. Under a block-diagonal Hessian assumption, the layer-level analogue of EWC's existing diagonal assumption, we prove three things. Forgetting decomposes as a sum of per-layer terms weighted by each layer's top Hessian eigenvalue. Diagonal-Fisher weights cannot recover this eigenvalue. For instance, two layers with identical Fisher averages can have top eigenvalues differing by a factor as large as the layer width. For the same level of forgetting, uniform regularization loses new-task performance by an amount scaling with the layer condition number. Our theoretical analysis leads to a simple recipe: protect early layers strongly, let deeper layers move. We apply this recipe to EWC and SLCA and show clear improvements in average performance and forgetting metrics.
Catastrophic forgetting remains a fundamental obstacle to continual learning, where neural networks lose previously acquired knowledge while learning new tasks. Existing methods primarily mitigate forgetting through parameter regularization or experience replay, while the representation-space dynamics associated with forgetting remain less understood. We investigate latent representation evolution during sequential learning and introduce representation flux, a geometric measure of sample-level representation displacement across training. We show that representation flux is strongly associated with catastrophic forgetting across multiple benchmarks, with temporal analyses indicating that elevated flux can precede subsequent performance degradation. Representation displacement is also associated with confidence degradation, while complementary geometric properties provide additional information about sample-level forgetting. Motivated by these observations, we propose FlowLess-R, a representation-space regularization method that constrains replay representations relative to stored references while allowing continued learning. FlowLess-R is architecture-agnostic and integrates into replay-based methods through a representation-matching term. Experiments on SplitMNIST, SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet show improved final average accuracy and reduced forgetting with ER, DER++, and ER-ACE. Our results identify representation flux as an informative geometric marker of forgetting and show that stabilizing latent representations provides a simple strategy for mitigating catastrophic forgetting.
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.
Ordinal regression, also called ordinal classification, is classification of ordinal data, in which the underlying target variable is categorical and considered to have a natural ordinal relation. Previous works have indicated that, in many real-world ordinal data, the conditional probability distribution (CPD) of the target variable given a value of the explanatory variable would be unimodal in a large domain of the explanatory variable and close to be unimodal even in a remaining domain. Therefore, unimodality-promoting regularized learning (UPRL), which promotes a predicted CPD closer to be unimodal with the aim of decreasing a prediction variance without inducing much bias for ordinal data of the unimodality, is promising to improve the prediction performance especially with small-size training data. In this study, we show that previous UPRL methods promote a predicted CPD to not only become closer to be unimodal but also have a larger scale (in other words, be smoother or less-confident). Therefore, we develop a novel method that more strictly reflects the idea of UPRL and evades a scale-related bias, and verify through experimental comparison that the unimodality-promotion indeed contributes to improve the prediction performance. Additionally, while our proposed UPRL method could perform better for smaller-scale data or with larger-size training data compared to a previous UPRL method, our analysis explains this experimental observation in terms of the presence or absence of an unexpected scale-related bias.
Multivariate time series forecasting presents unique challenges because future variables often co-evolve under shared system dynamics. While existing studies mainly focus on cross-variable dependencies in historical observations, dependencies among future values are much less explored. Specifically, modern forecasting models largely follow the Direct Forecasting (DF) paradigm, generating multi-step forecasts with point-wise objectives that do not explicitly constrain cross-variable structure. In this work, we show that the DF objective is mismatched in the presence of cross-variable and lagged dependencies, revealing an objective gap. To address this issue, we propose \textbf{C}ross-\textbf{V}ariable \textbf{Loss} (CvLoss), a plug-in structural regularizer that constrains forecast residuals on a cross-variable graph. CvLoss penalizes inconsistent edge-wise residual differences over forecast patches, encouraging consistency across both synchronous and asynchronous interactions. Our experiments show that CvLoss consistently improves competitive forecasting models, outperforms representative learning objectives, and is compatible with a variety of forecasting backbones.
Andrei A. Klishin, J. Nathan Kutz, Krithika Manoharstat.ML cs.LG math.DS physics.data-an
Latent low-dimensional structure in datasets of natural and engineered systems enables their sparse sensing, or full-state reconstruction from historical data and very few carefully chosen localized measurements. Depending on the reconstruction algorithm, sensor locations, and measurement noise, the reconstruction risk curves demonstrate a diversity of patterns including a dramatic peak in error known as double descent in Machine Learning literature. Here we explore those scenarios under a unified Data-Noise Averaging theory. Qualitatively, we formulate sufficient criteria for double descent to emerge through a catastrophic amplification of a pathological signal in reconstruction. Quantitatively, we predict the detailed risk curves at a fraction of computational cost, trace reconstruction instability to individual sensors and their combinations, and provide regularization mechanisms to mitigate the instability. We demonstrate results for both static reconstruction of Sea Surface Temperature patterns and time integration of a reduced order model of a PDE.
Neural networks increasingly combine data across populations, time periods, and operating conditions to improve generalization. This raises a reliability question: whether a model refitted on pooled data preserves an action ordering supported by both sources. Case-Based Decision Theory (CBDT) formalizes this requirement through its composition axiom, which requires source-supported preferences to survive their union. We study when this property holds for fixed-representation neural networks with ordinary least squares (OLS) output heads. First, we show that pooled refitting recomputes the inverse-Gram geometry used to weight source evidence, which can reverse shared preferences, and derive exact and approximate preservation conditions. Next, we introduce a scale-invariant Gram mismatch measure for prioritizing candidate pools and geometry-oriented regularization for shaping source geometry during training. Finally, we develop a three-stage audit that traces strict pairwise reversals through decision changes to task-defined utility loss. Experiments spanning a load-based bidding proxy and medical and financial decision proxies reveal stable and reversal-prone pooling regimes: the load audit identifies a measurable nonzero class of source-consensus-relative harmful decisions under the proxy utility, while cross-domain audits show that comparable mismatch can correspond to sharply different preservation rates. Geometry-oriented objectives occupy distinct descriptive accuracy-consistency-geometry-harm operating points. Together, the framework makes compositional reliability measurable and operational through screening, analytic certification, geometry-oriented training, and decision-consequence auditing.
Behraj Khan, Behroz Mirza, Syed Ahmad Chan Bukhari +1cs.LG
Covariate shift across training-data partitions biases model selection and parameter estimation in cross-validation, lifelong learning, and federated learning. We propose \textit{Partition-Induced Covariate-shift Correction} (\texttt{PIcsC}), a Fisher information-based regularization framework that mitigates distribution mismatch between data partitions and a reference distribution. \texttt{PIcsC} approximates partition divergence using the Fisher Information Matrix (FIM) and incorporates the resulting statistic as a regularizer during optimization. The same formulation applies to both centrally partitioned datasets (batches or cross-validation folds) and inherently distributed data (federated clients or decentralized nodes), requiring only partition-local gradient statistics rather than raw data. We further introduce a conditional adaptation mechanism that combines FIM shift with KL divergence to detect significant distribution shifts and activates regularization only when necessary. Experiments on more than 40 datasets demonstrate consistent improvements under both natural and synthetic covariate shift. On fragmented batch and fold settings, \texttt{PIcsC} reduces fragmentation-induced performance degradation by more than 20\% and 25\%, respectively. On seven federated learning benchmarks, it consistently outperforms FedAvg, FedProx, and SCAFFOLD by 3 -5 percentage points without requiring client-specific personalization. These results demonstrate that Fisher information provides an effective and unified mechanism for mitigating partition-induced covariate shift across both centralized and distributed learning.
Additive models buy interpretability by forbidding feature interactions, a constraint that neural instantiations enforce architecturally. We introduce the quadrilateral loss, a differentiable penalty that treats additivity as a measurable behavior instead: a second-order mixed difference on pairs of training points swapping one coordinate, which vanishes if and only if the coordinate carries no interaction, remains informative for piecewise-linear networks, and equals in expectation the per-coordinate interaction mass of the interventional Shapley-GAM. The loss turns additivity into a dial - most learned interactions prove removable almost for free, and on small datasets a moderate penalty improves accuracy and additivity simultaneously - and into an online observable: its per-feature surrender curves show, across seeds and datasets, that pre-regularization interaction magnitude barely predicts what a regularized model retains, undermining post-hoc interaction rankings. Against this instrument we compare routes to exact additivity, spanning structural masks, behavioral penalties (optionally crystallized into exact structure), weight decay, backfitting, the shared-section model, and bagged boosted stumps: constraining behavior before structure dominates weight-space constraints, rankings reverse between data regimes, and converging routes agree on the shape functions themselves. Three silent failure modes we document share one anatomy: guarantees imported into settings that quietly void their preconditions.
Predictive models deployed at scale influence future data, a phenomenon called performativity. And there is always one way to cope: Train the model on new data, deploy it again, and repeat. This process, called retraining or repeated risk minimization, creates a feedback loop between model and data that real-world learning systems can't avoid. Results on performative prediction shed light on this dynamic: If the model's influence on the data is small, retraining reaches a fixed point. What remains open is why fixed points should naturally exist, and what governs retraining when the model's influence is strong. In this work we develop a new perspective on retraining -- the stable signal principle -- that addresses these questions. We start from the assumption that the prediction target has at least some small model-independent component, a stable signal, such as the intrinsic quality of an item. We prove that when a nonzero stable signal exists, repeated risk minimization, suitably regularized, converges geometrically to the direction of this stable signal. This is true even if the model's influence on the target is arbitrarily large relative to the stable signal. Regularization emerges naturally as a force to control performativity, rather than to promote generalization, revealing a new facet of an old concept. We extend the analysis to a broad family of affine retraining operators under arbitrary model-induced feature changes, heterogeneous time-varying effects, and nonlinear responses. The stable signal perspective also applies to data feedback loops in language modeling, providing new explanations for the stability of language model training from model-generated data.
Matt L. Wiemann, Peter Melchior, Andrew K. Saydjarics.LG
Noise injection is a well-known technique in stochastic optimization. We report its surprising effectiveness with an interleaved (on-off-on-off...) rather than the usual monotonic decay schedule. We present a theoretical analysis of noise injection, which confirms that corruption by impulse noise approximates a Jacobian regularization, whereas Gaussian noise acts as a curvature penalty. This regularization behavior has been invoked to explain why noise injection increases model robustness. But the interleaved nature of our proposed schedule produces superior results even for the optimization objective: mixing phases of noisy data permits the optimizer to escape local minima and increase exploration without the risk of catastrophically forgetting the important features from the clean data. To stabilize this training scheme against the rapid changes of the loss when switching between clean and noisy data, we introduce a gradient-norm stabilization technique that scales noisy updates based on clean gradient magnitudes. We compare this method with other common augmentation methods and find substantial improvements in corruption tolerance and robustness to real-world distribution shifts on CIFAR-100-C, ImageNet-C, and ImageNet-R for ResNet and ViT architectures, with the best results being achieved by stacking our method on top of other augmentations. Through saliency and attention maps we show that the effect of interleaved noise injection stems from penalizing the failure modes encouraged by the inductive bias of the models: impulse noise works against the locality bias of convolutional (ResNet) architectures, and Gaussian noise reduces the tendency of attention-based models to pick up large-scale spurious features. Interleaved noise injection is therefore an effective tool to improve the test performance on clean, noisy, and out-of-distribution data at essentially zero computational cost.
Retail demand forecasts are reused across replenishment, capacity, labor, and transportation planning cycles. Point-error objectives do not constrain abrupt movement between adjacent forecasts, while post-hoc smoothing acts only after model fitting. We ask whether a training-time penalty on consecutive within-series movement can improve horizontal forecast-path stability without materially changing point accuracy. The penalty is evaluated in a temporal-structured pipeline combining recent-demand embeddings with calendar, price, hierarchy, item, and store features. On selected M5 demand series at 1000, 3000, and 4000-series scales, the stability-aware hybrid model improves Forecast Stability Score over XGBoost by 6.91%, 6.66%, and 7.68%, respectively, while RMSE changes remain within 0.72% across three random seeds. Post-hoc exponential smoothing attains lower raw movement but incurs a larger RMSE cost; training-time regularization preserves more point accuracy and performs favorably under normalized stability. These findings extend forecast evaluation from point-error minimization toward an accuracy-stability trade-off perspective for operational retail forecasting.
Grokking is a phenomenon in which neural networks initially memorize training data and only later exhibit strong generalization after prolonged optimization. Despite extensive recent study, the factors influencing the emergence and timing of grokking remain incompletely understood. We investigate the relationship between representation geometry and delayed generalization. We find that dimensionality collapse consistently precedes the onset of grokking in all evaluated settings. Motivated by these observations, we introduce Geometric Dimensionality Regularization (GeomDR), a simple spectral regularizer that modifies the effective dimensionality of hidden representations during training. Across modular addition, modular division, and permutation composition tasks, GeomDR consistently alters grokking dynamics and can substantially accelerate the onset of generalization depending on the intervention schedule and target dimensionality. In several settings, grokking is accelerated by up to 52 times relative to standard AdamW training. Similar qualitative effects are observed in both multilayer perceptrons and transformers. Together, these results suggest that representation geometry can serve as an effective control signal for grokking and provide evidence that geometric interventions offer a practical approach for studying and influencing delayed generalization in neural networks.
Marlon Becker, Jonas Konrad, Luis Garcia Rodriguez +1cs.LG
We introduce a straightforward yet effective method to empirically study memorization in deep neural networks for classification tasks. Our approach augments each training sample with auxiliary random labels, which are then predicted by a random label prediction head (RLP-head). RLP-heads can be attached at arbitrary depths of a network, predicting random labels from the corresponding intermediate representation and thereby enabling analysis of how memorization capacity evolves across layers. By interpreting the RLP-head performance as an empirical estimate of Rademacher complexity, we obtain a direct measure of both sample-level memorization and model capacity. We leverage this random label accuracy metric to analyze generalization and overfitting in different models and datasets. Building on this approach, we further propose a novel regularization technique based on the output of the RLP-head, which demonstrably reduces memorization. Interestingly, our experiments reveal that reducing memorization can either improve or impair generalization, depending on the dataset and training setup. These findings challenge the traditional assumption that overfitting is equivalent to memorization and suggest new hypotheses to reconcile these seemingly contradictory results. The source code is available at https://github.com/MarlonBecker/RandomLabelHeads
We study ridge-regularized log-density-ratio estimation in the Gaussian location model with a common covariance matrix. By affine invariance, the model is written as q $\sim$ N(0, I), p $\sim$ N($Δ$, I), with linear features, where $Δ$ is a mean vector. The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient, and the spectral estimator recently introduced in [1] replaces a single variational problem by a continuum of ridge-regularized least-squares problems. We derive high-dimensional deterministic asymptotic equivalents when the numbers of observations and dimension tend to infinity with fixed ratios. The regularized variational limit is characterized by a scalar entropy minimization problem derived from the convex-Gaussian-min-max theorem (CGMT), while the regularized spectral limit follows from deterministic equivalents for resolvents of weighted sums of two independent Gaussian sample covariance matrices. We use these formulas to compare population risks, with experiments focused on fixed-signal aspect-ratio sweeps and optimized regularization. Our conclusion is that with many observations, under the criteria and asymptotic regimes analyzed here, the well-specified variational estimator has the smaller risk, while with fewer observations, the spectral estimator is favored because its covariance-based construction has lower variance. We also study how a nuclear penalty can be used and partially analyzed to perform feature learning.
Floor van Maarschalkerwaart, Subhadip Mukherjee, Christoph Brune +1math.OC cs.LG
Learned reconstruction operators for inverse problems are typically trained under a fixed noise model, and generalize poorly when the distribution during testing differs from the one assumed during training. Distributionally robust optimization (DRO) addresses this by optimizing against the worst-case distribution within a prescribed ambiguity set, but standard Wasserstein DRO perturbs the full joint distribution uniformly, which can be overly conservative and ignores the physics of the measurement process. We develop a structured DRO framework in which the ambiguity set is restricted to structured perturbations aligned with the data-acquisition process. This allows us to learn data-driven reconstruction operators that remain robust to distributional shifts. By constraining perturbations to subsets such as $P(Y|X)$, our framework models uncertainty in the forward operator and noise model more faithfully, accommodating any noise model expressible as a stochastic forward operator. We establish strong duality for this general formulation and derive explicit finite-dimensional dual representations for perturbations in the joint, marginal, and conditional distributions. A central result is an explicit worst-case risk bound that induces Tikhonov regularization on the Lipschitz constant of the reconstruction operator, and is less conservative relative to standard DRO for well-posed problems. Numerical experiments on deblurring and sinogram-to-CT reconstruction demonstrate improved robustness, stability, and interpretability over standard DRO and MSE baselines. In the linear setting, the learned operator becomes effectively low-rank, truncating at the intrinsic dimension of the data and recovering a data-driven analogue of truncated-SVD regularization.
Hugo L. Hammer, Vajira Thambawita, Kristoffer Herland Hellton +1cs.LG
Interpretable Mesomorphic Neural Networks (IMNs) offer a promising framework that combines the predictive power of deep neural networks with the interpretability of linear models. However, the original formulation lacks safeguards to ensure that the learned interpretations are in fact reliable. In particular, the network is free to concentrate all explanatory variance into a single weight of the linear output layer, achieving strong predictive performance while producing interpretations that are largely meaningless. Paradoxically, the L1 penalty proposed to encourage sparse solutions exacerbates this problem by further incentivizing such degenerate configurations. To address this vulnerability, we introduce Local Fidelity Regularization (LFR), a novel penalty term that prevents degenerate weight collapse by aligning the linear output weights with local data variations. This structural constraint guarantees faithful explanations and substantially improves the reliability of model interpretations. Furthermore, empirical evaluations across the OpenML benchmark suite demonstrate that LFR does not compromise accuracy for explainability; rather, it achieved improved AUROC over the unregularized IMN. By yielding results highly competitive with state-of-the-art black-box models, LFR provides the dual benefit of reliable interpretability and superior predictive performance. Source code and usage instructions are available at https://github.com/hugohammer/LFR-IMN.git.
Modern neural networks can fit corrupted training labels, making noisy-label learning a useful setting for studying memorization-driven overfitting. Most regularization methods modify the objective, architecture, or data distribution; here we instead study a geometric intervention on the optimizer update itself. We evaluate OrthoGrad, which removes the component of each weight gradient parallel to the current weight vector, in noisy-label image classification. On MNIST with small-data regimes, OrthoGrad improves test accuracy most clearly for CNNs while reducing corrupted-label fitting. Mechanism diagnostics based on weight norms and gradient-weight cosine similarity suggest that the projection has the strongest effect when the raw gradient contains a nontrivial radial component, and becomes weaker in larger-data regimes where gradients are already nearly orthogonal to weights. Additional CIFAR-10 ResNet-18 experiments show that the method can alter memorization trajectories but does not prevent eventual noisy-label memorization. These results support orthogonal update constraints as a useful diagnostic for studying learning dynamics, while showing that OrthoGrad is regime-dependent rather than universally regularizing.
We present a large-scale empirical study isolating the contributions of the Derivative Regularization penalty (DREG). Across a fully-crossed factorial sweep of 960 experiments spanning 4 activations, 6 regularizers, 8 datasets, and 5 random seeds, we ask: when, where, and why does DREG work? Our results establish three principal findings. First, DREG achieves the highest overall and clean-regime accuracy among all regularizers evaluated (significantly so against the unregularized baseline, Weight Decay, and IGPen; Wilcoxon $p \leq 0.031$). It ranks second in noise robustness behind Spectral Normalization (SN) - the only two layer-wise regularizers in the study. Second, DREG is globally the best-performing regularizer under GELU, the default activation in modern transformer architectures, particularly on both messy vision and messy NLP benchmarks, suggesting direct applicability to frontier deep learning settings. Third, DREG's advantage over competing regularizers is most pronounced under data scarcity, consistent with its role as a geometric inductive bias that substitutes for the regularizing effect of data volume. Throughout, DREG is applied with a single fixed hyperparameter $λ= 10^{-2.5}$ and no per-dataset tuning, supporting its characterization as a plug-and-play regularizer for neural networks with nontrivial Jacobian structure. These findings are consistent with DREG's design: concentrating regularization pressure on layers where the activation derivative is largest, rather than constraining the network uniformly.
Kernel methods requiring matrix inversion -- particularly Least-Squares Twin Support Vector Machines (LSTSVM) -- suffer from exponential eigenvalue decay in their system matrices, producing severely ill-conditioned problems where standard Tikhonov regularization applies uniform damping regardless of eigenvector reliability. We propose Differential Spectral Damping (DSD), a regularization formula that adapts its penalty to localized eigengap structure: preserving eigenvectors with large spectral gaps (reliable per Davis-Kahan perturbation theory) while aggressively suppressing those with small gaps (directionally corrupted beyond recovery). We motivate DSD through a principled design procedure grounded in the Davis-Kahan $\sin(Θ)$ theorem, systematically deriving the requirements for a reliability-aware damping function and selecting the exponential form for its smoothness, differentiability, and natural saturation properties. Through rigorous paired testing with fairly optimized baselines (including gradient-optimized Tikhonov receiving equal optimization opportunity), we demonstrate that DSD improves LSTSVM classification accuracy by +4.8 percentage points on real-world GINA ($d=970$, Cohen's $d = 4.49$, $p < 0.0001$), +10.4 percentage points at $d=200$, and +2.6 percentage points on Madelon ($d=500$) -- all using only principled spectral initialization while Tikhonov receives grid search. For pre-image reconstruction on manifold data, DSD ties Tikhonov at high perturbation noise ($p=0.99$) but slightly underperforms at lower noise levels; both reduce naive inversion error by $66\times$. We characterize the precise operating regime ($d \geq 100$, condition number $> 10^3$) and document where simpler methods suffice, providing practitioners with clear deployment guidance.
Alexandre Lemire Paquin, Brahim Chaib-Draa, Philippe Giguèrecs.LG stat.ML
We study PAC-Bayes derandomization for smooth loss functions. Our goal is to obtain generalization bounds that hold with high probability for deterministic predictors by exploiting smoothness properties of both the loss and the predictor class. We show that passing from the Gibbs predictor to the deterministic predictor at the posterior mean has a precise cost, given by the generalization gap of the Jensen gap class. We control this class through its Rademacher complexity, leading to bounds for deterministic predictors that involve flatness quantities expressed in terms of parameter Jacobians and Hessians of the score map. The framework applies to both bounded and unbounded smooth loss functions, and we specialize the results to linear predictors and smooth neural networks. Finally, the Jacobian and Hessian quantities appearing in the theory motivate a practical regularizer. For BatchNorm networks, we compute this regularizer with respect to effective BatchNorm weights obtained by folding the BatchNorm transformation into the adjacent affine weights. Experiments on CIFAR-10 illustrate the behavior of this regularizer under different batch sizes.
Interpreting machine-learning models has attracted increasing attention, particularly in the physical sciences, where one often seeks to understand the underlying mechanisms rather than merely make predictions. Multiple linear regression is often regarded as an interpretable alternative to more complex models, such as deep neural networks, because its predictions are expressed as explicit weighted sums of input features. However, when input features are strongly correlated, namely in the presence of multicollinearity, the learned weights can exhibit large dataset-to-dataset fluctuations and oscillatory behavior across physically similar features, making their interpretation difficult or even impossible. Although the instability of the weights under multicollinearity is well known in statistics, its consequences for physical interpretation, in particular its connection to oscillatory weights across physically similar features, have not been systematically clarified. Here, we theoretically discuss the mechanism behind this loss of interpretability by analyzing the eigenmodes of the feature correlation matrix. We show that small-eigenvalue modes associated with multicollinearity amplify fluctuations in the weights and generate oscillatory patterns that do not necessarily reflect meaningful contributions. We test this theoretical picture numerically on physics datasets and show that Ridge regularization suppresses these unstable modes, although the resulting weights must still be interpreted with caution. We further confirm the generality of our findings beyond physics by analyzing a diverse collection of publicly available datasets. Our results clarify why, in the presence of multicollinearity, physical interpretation can remain difficult even for linear regression models.
Mahdi Mohammadigohari, Giuseppe Di Fatta, Giuseppe Nicosia +1cs.LG
We introduce Brownian kernel ladders (BKLs), a recursive hierarchy of integral reproducing kernel Hilbert spaces built from linear functionals by repeatedly integrating Brownian pullback kernels indexed by functions from the preceding layer. The nonnegative 1-homogeneity of the Brownian kernel yields a kernel-preserving canonical spherical normalization and propagates square-root regularity through the hierarchy. Allowing all canonical ladder measures to vary produces a full adaptive BKL envelope with an infimal complexity. For this envelope, we prove depth-dependent Hölder and pointwise estimates, quasi-Banach structure, nestedness, and, under a geometric trace condition, strict growth with ballwise separation. We also establish existence of regularized empirical-risk minimizers for continuous losses uniformly bounded below, with almost-everywhere uniqueness of population predictions under strict convexity and pointwise uniqueness under full support. For statistical estimation, we study one realized ladder and finite dictionaries fixed independently of the estimation sample. For a dictionary of $M$ ladders, the Gaussian complexity of the union of radius-$r$ top-layer RKHS balls has $n^{-1/2}$ dependence, no explicit ambient-dimension factor, and model-selection factor $1+\sqrt{2\ln M}$. Corresponding high-probability oracle and excess-risk bounds follow; polynomial-size dictionaries retain a near-parametric rate. The theory separates adaptive representational richness from the statistical cost of ladder selection.
Yuan-chin Ivan Changcs.LG stat.AP stat.CO stat.ME stat.ML
Recurrent neural networks maintain a hidden state $h_t$, but its probabilistic meaning is often unclear. We study hidden-state stability through \emph{backward coherence}: the extent to which $h_t$ can be reconstructed from $h_{t+1}$ by a learned backward projector $g_φ$. Under contraction and summable backward drift, the hidden-state sequence forms a quasi-reverse-martingale. This yields almost-sure convergence, rates under mixing, an interpretable limiting representation, finite pathwise stopping times, and a theoretical framework for time-uniform confidence sequences. Simulations support the theory. Backward-coherence regularisation reduces the empirical quasi-martingale total $\hat Q$ by $43$--$58%$, reaches stability $28$--$44%$ earlier than an unregularised RNN, and gives tracking-error recovery consistent with geometric bounds. Additional tests confirm echo-state forgetting rates bounded by $ρ$ and verify the increment-sum tube $R_t$ with $100%$ simultaneous coverage, although $R_t$ is conservative; in practice, the defect-tail proxy $\hat Q_t$ is the more useful monitor. The backward-coherence loss is also equivalent to minimising a Kullback--Leibler divergence in a Gaussian backward model, linking the method to variational inference. Extensions cover $φ$-mixing inputs, change-point tracking, and finite-sample concentration. Three real-data studies further validate the approach. On PhysioNet 2012 ICU data, the Reverse Martingale RNN (RMRNN) matches RNN mortality-prediction AUC while reaching stable representations 13 hours earlier. On FRED-MD, it reduces one-month-ahead forecast error by about fourfold under concept drift. On UCI Human Activity Recognition, it maintains lower post-transition tracking error with geometric decay. The guarantees apply under the stated assumptions; universality is not claimed.