Deep Learning Through A Telescoping Lens: A Simple Model Provides Empirical Insights On Grokking, Gradient Boosting & Beyond

Deep learning sometimes appears to work in unexpected ways. In pursuit of a deeper understanding of its surprising behaviors, we investigate the utility of a simple yet accurate model of a trained neural network consisting of a sequence of first-order approximations telescoping out into a single empirically operational tool for practical analysis. Across three case studies, we illustrate how it can be applied to derive new empirical insights on a diverse range of prominent phenomena in the literature -- including double descent, grokking, linear mode connectivity, and the challenges of applying deep learning on tabular data -- highlighting that this model allows us to construct and extract metrics that help predict and understand the a priori unexpected performance of neural networks. We also demonstrate that this model presents a pedagogical formalism allowing us to isolate components of the training process even in complex contemporary settings, providing a lens to reason about the effects of design choices such as architecture & optimization strategy, and reveals surprising parallels between neural network learning and gradient boosting.

Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise

Constructing valid prediction intervals rather than point estimates is a well-established approach for uncertainty quantification in the regression setting. Models equipped with this capacity output an interval of values in which the ground truth target will fall with some prespecified probability. This is an essential requirement in many real-world applications where simple point predictions' inability to convey the magnitude and frequency of errors renders them insufficient for high-stakes decisions. Quantile regression is a leading approach for obtaining such intervals via the empirical estimation of quantiles in the (non-parametric) distribution of outputs. This method is simple, computationally inexpensive, interpretable, assumption-free, and effective. However, it does require that the specific quantiles being learned are chosen a priori. This results in (a) intervals that are arbitrarily symmetric around the median which is sub-optimal for realistic skewed distributions, or (b) learning an excessive number of intervals. In this work, we propose Relaxed Quantile Regression (RQR), a direct alternative to quantile regression based interval construction that removes this arbitrary constraint whilst maintaining its strengths. We demonstrate that this added flexibility results in intervals with an improvement in desirable qualities (e.g. mean width) whilst retaining the essential coverage guarantees of quantile regression.

Why do Random Forests Work? Understanding Tree Ensembles as Self-Regularizing Adaptive Smoothers

Despite their remarkable effectiveness and broad application, the drivers of success underlying ensembles of trees are still not fully understood. In this paper, we highlight how interpreting tree ensembles as adaptive and self-regularizing smoothers can provide new intuition and deeper insight to this topic. We use this perspective to show that, when studied as smoothers, randomized tree ensembles not only make predictions that are quantifiably more smooth than the predictions of the individual trees they consist of, but also further regulate their smoothness at test-time based on the dissimilarity between testing and training inputs. First, we use this insight to revisit, refine and reconcile two recent explanations of forest success by providing a new way of quantifying the conjectured behaviors of tree ensembles objectively by measuring the effective degree of smoothing they imply. Then, we move beyond existing explanations for the mechanisms by which tree ensembles improve upon individual trees and challenge the popular wisdom that the superior performance of forests should be understood as a consequence of variance reduction alone. We argue that the current high-level dichotomy into bias- and variance-reduction prevalent in statistics is insufficient to understand tree ensembles -- because the prevailing definition of bias does not capture differences in the expressivity of the hypothesis classes formed by trees and forests. Instead, we show that forests can improve upon trees by three distinct mechanisms that are usually implicitly entangled. In particular, we demonstrate that the smoothing effect of ensembling can reduce variance in predictions due to noise in outcome generation, reduce variability in the quality of the learned function given fixed input data and reduce potential bias in learnable functions by enriching the available hypothesis space.

A U-turn on Double Descent: Rethinking Parameter Counting in Statistical Learning

Conventional statistical wisdom established a well-understood relationship between model complexity and prediction error, typically presented as a U-shaped curve reflecting a transition between under- and overfitting regimes. However, motivated by the success of overparametrized neural networks, recent influential work has suggested this theory to be generally incomplete, introducing an additional regime that exhibits a second descent in test error as the parameter count p grows past sample size n - a phenomenon dubbed double descent. While most attention has naturally been given to the deep-learning setting, double descent was shown to emerge more generally across non-neural models: known cases include linear regression, trees, and boosting. In this work, we take a closer look at evidence surrounding these more classical statistical machine learning methods and challenge the claim that observed cases of double descent truly extend the limits of a traditional U-shaped complexity-generalization curve therein. We show that once careful consideration is given to what is being plotted on the x-axes of their double descent plots, it becomes apparent that there are implicitly multiple complexity axes along which the parameter count grows. We demonstrate that the second descent appears exactly (and only) when and where the transition between these underlying axes occurs, and that its location is thus not inherently tied to the interpolation threshold p=n. We then gain further insight by adopting a classical nonparametric statistics perspective. We interpret the investigated methods as smoothers and propose a generalized measure for the effective number of parameters they use on unseen examples, using which we find that their apparent double descent curves indeed fold back into more traditional convex shapes - providing a resolution to tensions between double descent and statistical intuition.

TANGOS: Regularizing Tabular Neural Networks through Gradient Orthogonalization and Specialization

Despite their success with unstructured data, deep neural networks are not yet a panacea for structured tabular data. In the tabular domain, their efficiency crucially relies on various forms of regularization to prevent overfitting and provide strong generalization performance. Existing regularization techniques include broad modelling decisions such as choice of architecture, loss functions, and optimization methods. In this work, we introduce Tabular Neural Gradient Orthogonalization and Specialization (TANGOS), a novel framework for regularization in the tabular setting built on latent unit attributions. The gradient attribution of an activation with respect to a given input feature suggests how the neuron attends to that feature, and is often employed to interpret the predictions of deep networks. In TANGOS, we take a different approach and incorporate neuron attributions directly into training to encourage orthogonalization and specialization of latent attributions in a fully-connected network. Our regularizer encourages neurons to focus on sparse, non-overlapping input features and results in a set of diverse and specialized latent units. In the tabular domain, we demonstrate that our approach can lead to improved out-of-sample generalization performance, outperforming other popular regularization methods. We provide insight into why our regularizer is effective and demonstrate that TANGOS can be applied jointly with existing methods to achieve even greater generalization performance.

Improving Adaptive Conformal Prediction Using Self-Supervised Learning

Conformal prediction is a powerful distribution-free tool for uncertainty quantification, establishing valid prediction intervals with finite-sample guarantees. To produce valid intervals which are also adaptive to the difficulty of each instance, a common approach is to compute normalized nonconformity scores on a separate calibration set. Self-supervised learning has been effectively utilized in many domains to learn general representations for downstream predictors. However, the use of self-supervision beyond model pretraining and representation learning has been largely unexplored. In this work, we investigate how self-supervised pretext tasks can improve the quality of the conformal regressors, specifically by improving the adaptability of conformal intervals. We train an auxiliary model with a self-supervised pretext task on top of an existing predictive model and use the self-supervised error as an additional feature to estimate nonconformity scores. We empirically demonstrate the benefit of the additional information using both synthetic and real data on the efficiency (width), deficit, and excess of conformal prediction intervals.

Joint Training of Deep Ensembles Fails Due to Learner Collusion

Ensembles of machine learning models have been well established as a powerful method of improving performance over a single model. Traditionally, ensembling algorithms train their base learners independently or sequentially with the goal of optimizing their joint performance. In the case of deep ensembles of neural networks, we are provided with the opportunity to directly optimize the true objective: the joint performance of the ensemble as a whole. Surprisingly, however, directly minimizing the loss of the ensemble appears to rarely be applied in practice. Instead, most previous research trains individual models independently with ensembling performed post hoc. In this work, we show that this is for good reason - joint optimization of ensemble loss results in degenerate behavior. We approach this problem by decomposing the ensemble objective into the strength of the base learners and the diversity between them. We discover that joint optimization results in a phenomenon in which base learners collude to artificially inflate their apparent diversity. This pseudo-diversity fails to generalize beyond the training data, causing a larger generalization gap. We proceed to demonstrate the practical implications of this effect finding that, in some cases, a balance between independent training and joint optimization can improve performance over the former while avoiding the degeneracies of the latter.

Spike-inspired Rank Coding for Fast and Accurate Recurrent Neural Networks

Biological spiking neural networks (SNNs) can temporally encode information in their outputs, e.g. in the rank order in which neurons fire, whereas artificial neural networks (ANNs) conventionally do not. As a result, models of SNNs for neuromorphic computing are regarded as potentially more rapid and efficient than ANNs when dealing with temporal input. On the other hand, ANNs are simpler to train, and usually achieve superior performance. Here we show that temporal coding such as rank coding (RC) inspired by SNNs can also be applied to conventional ANNs such as LSTMs, and leads to computational savings and speedups. In our RC for ANNs, we apply backpropagation through time using the standard real-valued activations, but only from a strategically early time step of each sequential input example, decided by a threshold-crossing event. Learning then incorporates naturally also _when_ to produce an output, without other changes to the model or the algorithm. Both the forward and the backward training pass can be significantly shortened by skipping the remaining input sequence after that first event. RC-training also significantly reduces time-to-insight during inference, with a minimal decrease in accuracy. The desired speed-accuracy trade-off is tunable by varying the threshold or a regularization parameter that rewards output entropy. We demonstrate these in two toy problems of sequence classification, and in a temporally-encoded MNIST dataset where our RC model achieves 99.19% accuracy after the first input time-step, outperforming the state of the art in temporal coding with SNNs, as well as in spoken-word classification of Google Speech Commands, outperforming non-RC-trained early inference with LSTMs.