Statistical learning theory and Occam's razor: Regularization

The principle of Occam's razor, which instructs us to prefer simplicity in inductive inference, has attracted much scrutiny both in the philosophy of science and in machine learning. In either field, however, a justification for the principle has been elusive. In this paper, building on an earlier "core argument," I spell out a justification from statistical learning theory for the procedure of regularization: for trading off fit for simplicity. The means-ends argument is that in order to profit from theoretical reliability and "what-you-see-is-what-you-get" guarantees, one must implement a certain preference for simplicity over fit. This is a genuine methodological justification, which neither collapses to a purely pragmatic principle that we prefer simplicity for its own sake, nor to an ontological assumption that the truth is simple.

Toward a Unified Statistical Theory of Unsupervised Pretraining and Supervised Neural Knowledge Graph Learning

Knowledge graph learning provides a powerful framework for representing and inferring structured knowledge, with broad practical applications. However, the scarcity of relation-specific labeled triples per entity hinders the training of expressive models, and the ad hoc design of scoring functions limits generalizability and lacks theoretical grounding. We address both issues with a theoretically grounded, end-to-end training framework that extends and subsumes existing methods. Our framework is a two-stage procedure: unsupervised pretraining over heterogeneous corpora followed by supervised learning with multiple relation types. We establish a nonasymptotic risk bound that disentangles pretraining representation error from labeled-sample complexity, formally quantifying the benefit of large-scale unlabeled data for downstream knowledge prediction. Synthetic experiments validate each theoretical component, and real-world experiments confirm the effectiveness of our approach on large-scale knowledge graph benchmarks.

Learning Ergodic Dynamical Systems from a Finite Trajectory

We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.

Brownian Kernel Ladders

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.

Near-Optimal Pure Machine Unlearning for Smooth Strongly Convex Losses

Machine unlearning is motivated by legal and user-facing requirements to remove the influence of individuals' data from trained models, such as the right to be forgotten. Prior work has developed algorithms and error bounds for unlearning in smooth strongly convex stochastic optimization, but the fundamental statistical cost of unlearning has remained unclear. We nearly resolve this problem by proving upper and lower bounds on the excess population risk of approximate $\varepsilon$-unlearning; our bounds are tight up to a condition-number factor. For mean estimation over the unit ball, our upper and lower bounds match. The optimal rate is the usual statistical error plus an unlearning penalty that interpolates between the retraining-from-scratch rate and an exponentially smaller term as $\varepsilon/d$ grows, where $d$ is the dimension of the model. In particular, when $\varepsilon \gg d$, our $\varepsilon$-unlearning algorithm offers an exponential accuracy improvement over retraining the model from scratch and differentially private baselines. On the other hand, when $\varepsilon \le d$, retraining from scratch is optimal.

Statistical Consistency and Generalization of Contrastive Representation Learning

Contrastive representation learning (CRL) underpins many modern foundation models. Despite recent theoretical progress, existing analyses suffer from several key limitations: (i) the statistical consistency of CRL remains poorly understood; (ii) available generalization bounds deteriorate as the number of negative samples increases, contradicting the empirical benefits of large negative sets; and (iii) the retrieval performance of CRL has received limited theoretical attention. In this paper, we develop a unified statistical learning theory for CRL. For downstream tasks, we evaluate retrieval quality using an AUC-type population criterion and show that the contrastive loss is \emph{statistically consistent} with optimal ranking. We further establish a \emph{calibration-style inequality} that quantitatively relates excess contrastive risk to excess retrieval suboptimality. For upstream training, we study both supervised and self-supervised contrastive objectives and derive generalization bounds of order $O(1/m + 1/\sqrt{n})$ and $O(1/\sqrt{m} + 1/\sqrt{n})$, respectively, where $m$ denotes the number of negative samples and $n$ the number of anchor points. These bounds not only explain the empirical advantages of large negative sets but also reveal an explicit trade-off between $m$ and $n$. Extensive experiments on large-scale vision--language models corroborate our theoretical predictions.