The canonical deep learning approach for learning requires computing a gradient term at each layer by back-propagating the error signal from the output towards each learnable parameter. Given the stacked structure of neural networks, where each layer builds on the representation of the layer below, this approach leads to hierarchical representations. More abstract features live on the top layers of the model, while features on lower layers are expected to be less abstract. In contrast to this, we introduce a new learning method named NoProp, which does not rely on either forward or backwards propagation. Instead, NoProp takes inspiration from diffusion and flow matching methods, where each layer independently learns to denoise a noisy target. We believe this work takes a first step towards introducing a new family of gradient-free learning methods, that does not learn hierarchical representations -- at least not in the usual sense. NoProp needs to fix the representation at each layer beforehand to a noised version of the target, learning a local denoising process that can then be exploited at inference. We demonstrate the effectiveness of our method on MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks. Our results show that NoProp is a viable learning algorithm which achieves superior accuracy, is easier to use and computationally more efficient compared to other existing back-propagation-free methods. By departing from the traditional gradient based learning paradigm, NoProp alters how credit assignment is done within the network, enabling more efficient distributed learning as well as potentially impacting other characteristics of the learning process.

We consider the task of privately obtaining prediction error guarantees in ordinary least-squares regression problems with Gaussian covariates (with unknown covariance structure). We provide the first sample-optimal polynomial time algorithm for this task under both pure and approximate differential privacy. We show that any improvement to the sample complexity of our algorithm would violate either statistical-query or information-theoretic lower bounds. Additionally, our algorithm is robust to a small fraction of arbitrary outliers and achieves optimal error rates as a function of the fraction of outliers. In contrast, all prior efficient algorithms either incurred sample complexities with sub-optimal dimension dependence, scaling with the condition number of the covariates, or obtained a polynomially worse dependence on the privacy parameters. Our technical contributions are two-fold: first, we leverage resilience guarantees of Gaussians within the sum-of-squares framework. As a consequence, we obtain efficient sum-of-squares algorithms for regression with optimal robustness rates and sample complexity. Second, we generalize the recent robustness-to-privacy framework [HKMN23, (arXiv:2212.05015)] to account for the geometry induced by the covariance of the input samples. This framework crucially relies on the robust estimators to be sum-of-squares algorithms, and combining the two steps yields a sample-optimal private regression algorithm. We believe our techniques are of independent interest, and we demonstrate this by obtaining an efficient algorithm for covariance-aware mean estimation, with an optimal dependence on the privacy parameters.

Best subset selection in linear regression is well known to be nonconvex and computationally challenging to solve, as the number of possible subsets grows rapidly with increasing dimensionality of the problem. As a result, finding the global optimal solution via an exact optimization method for a problem with dimensions of 1000s may take an impractical amount of CPU time. This suggests the importance of finding suboptimal procedures that can provide good approximate solutions using much less computational effort than exact methods. In this work, we introduce a new procedure and compare it with other popular suboptimal algorithms to solve the best subset selection problem. Extensive computational experiments using synthetic and real data have been performed. The results provide insights into the performance of these methods in different data settings. The new procedure is observed to be a competitive suboptimal algorithm for solving the best subset selection problem for high-dimensional data.

Predictive coding (PC) is an influential theory of information processing in the brain, providing a biologically plausible alternative to backpropagation. It is motivated in terms of Bayesian inference, as hidden states and parameters are optimised via gradient descent on variational free energy. However, implementations of PC rely on maximum \textit{a posteriori} (MAP) estimates of hidden states and maximum likelihood (ML) estimates of parameters, limiting their ability to quantify epistemic uncertainty. In this work, we investigate a Bayesian extension to PC that estimates a posterior distribution over network parameters. This approach, termed Bayesian Predictive coding (BPC), preserves the locality of PC and results in closed-form Hebbian weight updates. Compared to PC, our BPC algorithm converges in fewer epochs in the full-batch setting and remains competitive in the mini-batch setting. Additionally, we demonstrate that BPC offers uncertainty quantification comparable to existing methods in Bayesian deep learning, while also improving convergence properties. Together, these results suggest that BPC provides a biologically plausible method for Bayesian learning in the brain, as well as an attractive approach to uncertainty quantification in deep learning.

We investigate the problem of learning a Single Index Model (SIM)- a popular model for studying the ability of neural networks to learn features - from anisotropic Gaussian inputs by training a neuron using vanilla Stochastic Gradient Descent (SGD). While the isotropic case has been extensively studied, the anisotropic case has received less attention and the impact of the covariance matrix on the learning dynamics remains unclear. For instance, Mousavi-Hosseini et al. (2023b) proposed a spherical SGD that requires a separate estimation of the data covariance matrix, thereby oversimplifying the influence of covariance. In this study, we analyze the learning dynamics of vanilla SGD under the SIM with anisotropic input data, demonstrating that vanilla SGD automatically adapts to the data's covariance structure. Leveraging these results, we derive upper and lower bounds on the sample complexity using a notion of effective dimension that is determined by the structure of the covariance matrix instead of the input data dimension.

A fundamental task in AI is providing performance guarantees for predictions made in unseen domains. In practice, there can be substantial uncertainty about the distribution of new data, and corresponding variability in the performance of existing predictors. Building on the theory of partial identification and transportability, this paper introduces new results for bounding the value of a functional of the target distribution, such as the generalization error of a classifier, given data from source domains and assumptions about the data generating mechanisms, encoded in causal diagrams. Our contribution is to provide the first general estimation technique for transportability problems, adapting existing parameterization schemes such Neural Causal Models to encode the structural constraints necessary for cross-population inference. We demonstrate the expressiveness and consistency of this procedure and further propose a gradient-based optimization scheme for making scalable inferences in practice. Our results are corroborated with experiments.

Domain generalization (DG) aims to learn models that can generalize well to unseen domains by training only on a set of source domains. Sharpness-Aware Minimization (SAM) has been a popular approach for this, aiming to find flat minima in the total loss landscape. However, we show that minimizing the total loss sharpness does not guarantee sharpness across individual domains. In particular, SAM can converge to fake flat minima, where the total loss may exhibit flat minima, but sharp minima are present in individual domains. Moreover, the current perturbation update in gradient ascent steps is ineffective in directly updating the sharpness of individual domains. Motivated by these findings, we introduce a novel DG algorithm, Decreased-overhead Gradual Sharpness-Aware Minimization (DGSAM), that applies gradual domain-wise perturbation to reduce sharpness consistently across domains while maintaining computational efficiency. Our experiments demonstrate that DGSAM outperforms state-of-the-art DG methods, achieving improved robustness to domain shifts and better performance across various benchmarks, while reducing computational overhead compared to SAM.

We develop new uncertainty propagation methods for feed-forward neural network architectures with leaky ReLU activation functions subject to random perturbations in the input vectors. In particular, we derive analytical expressions for the probability density function (PDF) of the neural network output and its statistical moments as a function of the input uncertainty and the parameters of the network, i.e., weights and biases. A key finding is that an appropriate linearization of the leaky ReLU activation function yields accurate statistical results even for large perturbations in the input vectors. This can be attributed to the way information propagates through the network. We also propose new analytically tractable Gaussian copula surrogate models to approximate the full joint PDF of the neural network output. To validate our theoretical results, we conduct Monte Carlo simulations and a thorough error analysis on a multi-layer neural network representing a nonlinear integro-differential operator between two polynomial function spaces. Our findings demonstrate excellent agreement between the theoretical predictions and Monte Carlo simulations.

--This paper presents a comprehensive comparative analysis of prominent clustering algorithms--K-means, DB-SCAN, and Spectral Clustering--on high-dimensional datasets. We introduce a novel evaluation framework that assesses clustering performance across multiple dimensionality reduction techniques (PCA, t-SNE, and UMAP) using diverse quantitative metrics. Experiments conducted on MNIST, Fashion-MNIST, and UCI HAR datasets reveal that preprocessing with UMAP consistently improves clustering quality across all algorithms, with Spectral Clustering demonstrating superior performance on complex manifold structures. Our findings show that algorithm selection should be guided by data characteristics, with K-means excelling in computational efficiency, DBSCAN in handling irregular clusters, and Spectral Clustering in capturing complex relationships. This research contributes a systematic approach for evaluating and selecting clustering techniques for high-dimensional data applications.

SHAP is one of the most popular local feature-attribution methods. Given a function f and an input x, it quantifies each feature's contribution to f(x). Recently, SHAP has been increasingly used for global insights: practitioners average the absolute SHAP values over many data points to compute global feature importance scores, which are then used to discard unimportant features. In this work, we investigate the soundness of this practice by asking whether small aggregate SHAP values necessarily imply that the corresponding feature does not affect the function. Unfortunately, the answer is no: even if the i-th SHAP value is 0 on the entire data support, there exist functions that clearly depend on Feature i. The issue is that computing SHAP values involves evaluating f on points outside of the data support, where f can be strategically designed to mask its dependence on Feature i. To address this, we propose to aggregate SHAP values over the extended support, which is the product of the marginals of the underlying distribution. With this modification, we show that a small aggregate SHAP value implies that we can safely discard the corresponding feature. We then extend our results to KernelSHAP, the most popular method to approximate SHAP values in practice. We show that if KernelSHAP is computed over the extended distribution, a small aggregate value justifies feature removal. This result holds independently of whether KernelSHAP accurately approximates true SHAP values, making it one of the first theoretical results to characterize the KernelSHAP algorithm itself. Our findings have both theoretical and practical implications. We introduce the Shapley Lie algebra, which offers algebraic insights that may enable a deeper investigation of SHAP and we show that randomly permuting each column of the data matrix enables safely discarding features based on aggregate SHAP and KernelSHAP values.