A trained Large Language Model (LLM) contains much of human knowledge. Yet, it is difficult to gauge the extent or accuracy of that knowledge, as LLMs do not always ``know what they know'' and may even be actively misleading. In this work, we give a general method for detecting semantic concepts in the internal activations of LLMs. Furthermore, we show that our methodology can be easily adapted to steer LLMs toward desirable outputs. Our innovations are the following: (1) we use a nonlinear feature learning method to identify important linear directions for predicting concepts from each layer; (2) we aggregate features across layers to build powerful concept detectors and steering mechanisms. We showcase the power of our approach by attaining state-of-the-art results for detecting hallucinations, harmfulness, toxicity, and untruthful content on seven benchmarks. We highlight the generality of our approach by steering LLMs towards new concepts that, to the best of our knowledge, have not been previously considered in the literature, including: semantic disambiguation, human languages, programming languages, hallucinated responses, science subjects, poetic/Shakespearean English, and even multiple concepts simultaneously. Moreover, our method can steer concepts with numerical attributes such as product reviews. We provide our code (including a simple API for our methods) at https://github.com/dmbeaglehole/neural_controllers .

Deep Neural Collapse (DNC) refers to the surprisingly rigid structure of the data representations in the final layers of Deep Neural Networks (DNNs). Though the phenomenon has been measured in a wide variety of settings, its emergence is only partially understood. In this work, we provide substantial evidence that DNC formation occurs primarily through deep feature learning with the average gradient outer product (AGOP). This takes a step further compared to efforts that explain neural collapse via feature-agnostic approaches, such as the unconstrained features model. We proceed by providing evidence that the right singular vectors and values of the weights are responsible for the majority of within-class variability collapse in DNNs. As shown in recent work, this singular structure is highly correlated with that of the AGOP. We then establish experimentally and theoretically that AGOP induces neural collapse in a randomly initialized neural network. In particular, we demonstrate that Deep Recursive Feature Machines, a method originally introduced as an abstraction for AGOP feature learning in convolutional neural networks, exhibits DNC.

Understanding the mechanisms through which neural networks extract statistics from input-label pairs is one of the most important unsolved problems in supervised learning. Prior works have identified that the gram matrices of the weights in trained neural networks of general architectures are proportional to the average gradient outer product of the model, in a statement known as the Neural Feature Ansatz (NFA). However, the reason these quantities become correlated during training is poorly understood. In this work, we explain the emergence of this correlation. We identify that the NFA is equivalent to alignment between the left singular structure of the weight matrices and a significant component of the empirical neural tangent kernels associated with those weights. We establish that the NFA introduced in prior works is driven by a centered NFA that isolates this alignment. We show that the speed of NFA development can be predicted analytically at early training times in terms of simple statistics of the inputs and labels. Finally, we introduce a simple intervention to increase NFA correlation at any given layer, which dramatically improves the quality of features learned.

Understanding the mechanism of how convolutional neural networks learn features from image data is a fundamental problem in machine learning and computer vision. In this work, we identify such a mechanism. We posit the Convolutional Neural Feature Ansatz, which states that covariances of filters in any convolutional layer are proportional to the average gradient outer product (AGOP) taken with respect to patches of the input to that layer. We present extensive empirical evidence for our ansatz, including identifying high correlation between covariances of filters and patch-based AGOPs for convolutional layers in standard neural architectures, such as AlexNet, VGG, and ResNets pre-trained on ImageNet. We also provide supporting theoretical evidence. We then demonstrate the generality of our result by using the patch-based AGOP to enable deep feature learning in convolutional kernel machines. We refer to the resulting algorithm as (Deep) ConvRFM and show that our algorithm recovers similar features to deep convolutional networks including the notable emergence of edge detectors. Moreover, we find that Deep ConvRFM overcomes previously identified limitations of convolutional kernels, such as their inability to adapt to local signals in images and, as a result, leads to sizable performance improvement over fixed convolutional kernels.

In recent years neural networks have achieved impressive results on many technological and scientific tasks. Yet, the mechanism through which these models automatically select features, or patterns in data, for prediction remains unclear. Identifying such a mechanism is key to advancing performance and interpretability of neural networks and promoting reliable adoption of these models in scientific applications. In this paper, we identify and characterize the mechanism through which deep fully connected neural networks learn features. We posit the Deep Neural Feature Ansatz, which states that neural feature learning occurs by implementing the average gradient outer product to up-weight features strongly related to model output. Our ansatz sheds light on various deep learning phenomena including emergence of spurious features and simplicity biases and how pruning networks can increase performance, the "lottery ticket hypothesis." Moreover, the mechanism identified in our work leads to a backpropagation-free method for feature learning with any machine learning model. To demonstrate the effectiveness of this feature learning mechanism, we use it to enable feature learning in classical, non-feature learning models known as kernel machines and show that the resulting models, which we refer to as Recursive Feature Machines, achieve state-of-the-art performance on tabular data.

``Benign overfitting'', the ability of certain algorithms to interpolate noisy training data and yet perform well out-of-sample, has been a topic of considerable recent interest. We show, using a fixed design setup, that an important class of predictors, kernel machines with translation-invariant kernels, does not exhibit benign overfitting in fixed dimensions. In particular, the estimated predictor does not converge to the ground truth with increasing sample size, for any non-zero regression function and any (even adaptive) bandwidth selection. To prove these results, we give exact expressions for the generalization error, and its decomposition in terms of an approximation error and an estimation error that elicits a trade-off based on the selection of the kernel bandwidth. Our results apply to commonly used translation-invariant kernels such as Gaussian, Laplace, and Cauchy.