Large language models (LLMs) are increasingly deployed in settings where the available context is incomplete or degraded. We argue that an LLM generating answers under incomplete context can be viewed as an implicit imputer, and evaluated against a criterion from the multiple imputation (MI) literature: uncertainty should scale with the amount of missing information. We assess this criterion on SQuAD, using a controlled framework in which context availability is varied across five levels. We evaluate two answer-level uncertainty measures that can be estimated from repeated sampling: sampling-based confidence (empirical mode frequency) and response entropy. Confidence fails to reflect increasing missingness: it remains high even as accuracy collapses. Entropy, by contrast, increases with context removal, consistent with the MI analogy, and explains substantially more variance in accuracy than confidence across all evidence levels (quadratic $R^2$ gap up to 0.057). We further introduce a black-box diagnostic $ρ_R(α)$ that estimates the proportion of baseline uncertainty resolved by context level $α$, requiring only repeated sampling with and without context. These results suggest that entropy is a more responsive black-box uncertainty measure than confidence under incomplete context.

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We introduce Metric-Aware Principal Component Analysis (MAPCA), a unified framework for scale-invariant representation learning based on the generalised eigenproblem max Tr(W^T Sigma W) subject to W^T M W = I, where M is a symmetric positive definite metric matrix. The choice of M determines the representation geometry. The canonical beta-family M(beta) = Sigma^beta, beta in [0,1], provides continuous spectral bias control between standard PCA (beta=0) and output whitening (beta=1), with condition number kappa(beta) = (lambda_1/lambda_p)^(1-beta) decreasing monotonically to isotropy. The diagonal metric M = D = diag(Sigma) recovers Invariant PCA (IPCA), a method rooted in Frisch (1928) diagonal regression, as a distinct member of the broader framework. We prove that scale invariance holds if and only if the metric transforms as M_tilde = CMC under rescaling C, a condition satisfied exactly by IPCA but not by the general beta-family at intermediate values. Beyond its classical interpretation, MAPCA provides a geometric language that unifies several self-supervised learning objectives. Barlow Twins and ZCA whitening correspond to beta=1 (output whitening); VICReg's variance term corresponds to the diagonal metric. A key finding is that W-MSE, despite being described as a whitening-based method, corresponds to M = Sigma^{-1} (beta = -1), outside the spectral compression range entirely and in the opposite spectral direction to Barlow Twins. This distinction between input and output whitening is invisible at the level of loss functions and becomes precise only within the MAPCA framework.

We derive the asymptotic risk function of regularized empirical risk minimization (ERM) estimators tuned by $n$-fold cross-validation (CV). The out-of-sample prediction loss of such estimators converges in distribution to the squared-error loss (risk function) of shrinkage estimators in the normal means model, tuned by Stein's unbiased risk estimate (SURE). This risk function provides a more fine-grained picture of predictive performance than uniform bounds on worst-case regret, which are common in learning theory: it quantifies how risk varies with the true parameter. As key intermediate steps, we show that (i) $n$-fold CV converges uniformly to SURE, and (ii) while SURE typically has multiple local minima, its global minimum is generically well separated. Well-separation ensures that uniform convergence of CV to SURE translates into convergence of the tuning parameter chosen by CV to that chosen by SURE.

The typical training of neural networks using large stepsize gradient descent (GD) under the logistic loss often involves two distinct phases, where the empirical risk oscillates in the first phase but decreases monotonically in the second phase. We investigate this phenomenon in two-layer networks that satisfy a near-homogeneity condition. We show that the second phase begins once the empirical risk falls below a certain threshold, dependent on the stepsize. Additionally, we show that the normalized margin grows nearly monotonically in the second phase, demonstrating an implicit bias of GD in training non-homogeneous predictors. If the dataset is linearly separable and the derivative of the activation function is bounded away from zero, we show that the average empirical risk decreases, implying that the first phase must stop in finite steps. Finally, we demonstrate that by choosing a suitably large stepsize, GD that undergoes this phase transition is more efficient than GD that monotonically decreases the risk. Our analysis applies to networks of any width, beyond the well-known neural tangent kernel and mean-field regimes.

In this paper we consider an alternative strategy.

Symmetries are prevalent in deep learning and can significantly influence the learning dynamics of neural networks. In this paper, we examine how exponential symmetries - a broad subclass of continuous symmetries present in the model architecture or loss function - interplay with stochastic gradient descent (SGD). We first prove that gradient noise creates a systematic motion (a "Noether flow") of the parameters θ along the degenerate direction to a unique initialization-independent fixed point θ

The following lemma is standard in the literature, see e.g.