Generative models, including diffusion models, are increasingly used as foundation models and adapted through sequential fine-tuning, making continual learning an essential problem setting. However, continual learning in such generative models remains poorly understood: after a task change, what aspects of the learned distribution are most easily lost, and what replay samples should be prioritized? We address these questions through the modern Hopfield energy. Recent links between modern Hopfield networks (MHNs) and diffusion models allow analyses in MHNs to be transferred to diffusion models. We introduce intrinsic forgetting as an increase in Hopfield energy after the task change. In tractable settings in an MHN, we prove that high-energy, outlier-like samples undergo a larger energy increase than cluster-like samples, implying that samples located in sharp, isolated basins are more forgettable. We further analyze memory replay and show that replay is particularly effective for high-energy samples, enabling an energy-based selection of replay samples. We validate these predictions in experiments on MHNs and two diffusion models under continual-learning settings: Stable Diffusion and a pixel-space DDPM. In these diffusion models, Hopfield energy tracks reconstruction-based forgetting, and replay experiments reveal energy-dependent mitigation of forgetting that is consistent with the MHN analysis.

Large language models (LLMs) are increasingly used as automated evaluators of AI systems, including in high-stakes applications. In this role, LLMs are used to generate judgments about the quality, appropriateness, or even safety of model outputs. This approach is motivated by practical constraints. Expert human ratings are costly and difficult to scale, whereas LLM ratings can be produced quickly at low cost. However, current approaches to deploying LLM evaluators are ad hoc, typically limited to reporting agreement metrics between human and LLM judges as a justification for substitution of human ratings, and lack a formal basis for study design. This paper (1) shifts the role of the LLM judge from substitutive to auxiliary, and (2) formulates the LLM-as-a-judge paradigm as one of augmenting human evaluation through a two-stage sampling design, where LLM evaluations are measured for all observations at the first stage and human ratings are partially observed for a subsample at the second stage. We propose to use a doubly robust estimator from the missing data literature, which takes advantage of the robustness property against the prediction model, since the missingness model is known by design. Using the asymptotic variance of this estimator, we propose how sample sizes of human and LLM ratings can be determined to achieve a targeted level of power. We also show that a study can be efficiently designed by allocating more human ratings for types of evaluations where the predictability of LLM ratings is not high. To the best of our knowledge, there is very little guidance on how much human oversight should be retained when validating benchmarks.

We revisit the problem of robust linear regression under Gaussian covariates with an unknown covariance matrix of condition number $κ$. For this fundamental problem, significant gaps remain in our understanding of the trade-offs among sample complexity, condition number, runtime, and prediction error for efficient algorithms. Our first result is a near-linear-time algorithm that uses $\widetilde{O}(d/ε^4)$ samples, where $d$ is the dimension and $ε$ is the corruption rate, and achieves prediction error $O(\sqrt{εκ})$ under the condition $εκ\lesssim 1$, improving over all prior works. We complement this result with a Statistical Query (SQ) lower bound showing that efficient SQ algorithms achieving error $o(\sqrt{εκ})$ when $εκ\lesssim 1$ require queries that take $Ω(d^2)$ samples to simulate. Finally, we prove a low-degree polynomial lower bound that gives fine-grained evidence that, without assumptions such as $εκ\lesssim 1$, efficient algorithms may require $\tildeΩ\left(\min\{dε^{2}κ^{2},\ ε^{2}d^{2}\}\right)$ samples to significantly outperform the trivial estimator that always guesses $0$.

We study a prototypical situation when a learned predictor can discover useful low-dimensional structure in data, while using fewer samples than are needed for accurate prediction. Specifically, we consider the problem of recovering a multi-index polynomial $f^*(x)=h(Ux)$, with $U\in\mathbb{R}^{r\times d}$ and $r\ll d$, from finitely many data/label pairs. Importantly, the target function depends on input $x$ only through the projection onto an unknown $r$-dimensional central subspace. The algorithm we analyze is appealingly simple: fit kernel ridge regression (KRR) to the data and compute the Average Gradient Outer Product (AGOP) from the fitted predictor. Our main results show that under reasonable assumptions the top $r$-dimensional eigenspace of AGOP provably recovers the central subspace, even in regimes when the prediction error remains large. Specifically, if the target function $f^*$ has degree $p^*$, it is known that $n\asymp d^{p^*}$ samples are necessary for KRR to achieve accurate prediction. In contrast, we show that if a low degree $p$ component of $f^*$ already carries all relevant directions for prediction, subspace recovery occurs in the much lower sample regime $n\asymp d^{p+δ}$ for any $δ\in(0,1)$. Our results thus demonstrate a separation between prediction and representation, and provide an explanation for why iterative kernel methods such as Recursive Feature Machines (RFM) can be sample-efficient in practice.

Causal queries are often only partially identifiable from observational data, and experiments that could tighten the resulting bounds are typically costly. We study the problem of selecting, prior to observing experimental outcomes, a cost-constrained subset of experiments that maximally tightens bounds on a target query. We formalize this as the max-potency problem, where epistemic potency measures the worst-case reduction in bound width guaranteed by an experiment, and show that this problem is NP-hard via a reduction from 0-1 knapsack. Building on the polynomial-programming framework of Duarte et al. (2023), we give a general procedure for evaluating epistemic potency in discrete settings. To control the super-exponential search space, we introduce two graphical pruning criteria that depend only on the causal graph and the query: a novel path-interception rule that exploits district structure to certify zero potency in linear time, and an identifiability check based on the ID algorithm. On Erdos-Renyi random graphs and 11 bnlearn benchmark networks, the two criteria together prune 50-88% of candidate experiments on average without solving a single polynomial program. For the general subset search, we show that ID-pruned experiments are combinatorially inert, yielding a super-exponential reduction in the number of subsets evaluated. We close with an end-to-end demonstration on observational NHANES data, selecting optimal experiments for estimating the effect of physical activity on diabetes.

We show that if the conditional distribution p(C | T) factors through a sufficient statistic ϕ(T), then the Information Bottleneck (IB) problem for (T, C) is exactly equivalent to the IB problem for (ϕ(T), C). The reduction is loss-free: it preserves the full IB curve, the Lagrangian optimum at every trade-off parameter \b{eta}, and the optimal representations up to pullback through ϕ. As a result, the computational complexity of solving the IB problem is governed by the dimension of the sufficient statistic rather than the ambient dimension of the source. This identifies an exact structural condition under which the generic IB problem becomes tractable, and gives a formal bridge between the discrete and linear-Gaussian regimes. We then show that the classical Gaussian IB solution of Chechik, Globerson, Tishby and Weiss is an immediate corollary of this reduction, and we state a nonlinear-Gaussian generalisation. A small numerical example illustrates the practical consequence: when a low-dimensional sufficient statistic is available, the exact IB curve can be computed on the reduced problem at a cost determined by the statistic rather than by the ambient source dimension.

Our results settle the complexity status regarding these parameters number of dimensions and number of ReLUs if the network is assumed to compute the ReLU case, we show fixed-parameter tractability for the combined parameter four ReLUs (or two linear threshold neurons) with zero training error. Finally, in We also answer a question by Froese et al. [2022, JAIR] proving W[1]-hardness for dimensions, which excludes any polynomial-time algorithm for constant dimension. Khalife and Basu [2022, IPCO] showing that both problems are NP-hard for two eral questions are still open. We answer questions by Arora et al. [2018, ICLR] and complexity of these problems has been studied numerous times in recent years, sevsidering ReLU and linear threshold activation functions.

Convolutional neural networks (CNNs) have become increasingly difficult to deploy in resource-constrained environments due to their large memory and computational requirements. Although low-rank compression methods can reduce this burden, most existing approaches compress spatial and channel redundancy independently and therefore do not fully exploit the localised structure within convolutional feature maps. This paper proposes a hierarchical spatio-channel low-rank compression framework for CNNs that exploits redundancy across spatial regions and channel activations. Unlike conventional methods, which apply a uniform decomposition across an entire layer, the proposed approach first partitions feature maps into spatial regions, then groups channels according to their co-activation patterns within each region, and finally applies rank-adaptive SVD to each resulting spatio-channel cluster. The method is evaluated on an AlexNet-based brain tumour MRI classification model and compared with Global SVD and Tucker decomposition under \(3\times\) and \(6\times\) compression budgets. Our method outperforms both baselines, reducing FLOPs from \(8.21\,\mathrm{G}\) to \(1.55\,\mathrm{G}\) (\(81.1\%\) reduction), achieving a \(1.38\times\) inference speed-up, and increasing classification accuracy from \(87.76\%\) to \(89.80\%\). The method also improves the macro \(F_1\)-score and performance on challenging classes such as meningioma. A hyper-parameter trade-off analysis demonstrates that the framework provides Pareto-optimal configurations, enabling control over the balance between compression and predictive performance. Moderate clustering with adaptive rank selection yields strong results. Bootstrap standard errors are reported for all classification metrics.