Learning curves show the expected performance with respect to training set size. This is often used to evaluate and compare models, tune hyper-parameters and determine how much data is needed for a specific performance. However, the distributional properties of performance are frequently overlooked on learning curves. Generally, only an average with standard error or standard deviation is used. In this paper, we analyze the distributions of generalization performance on the learning curves. We compile a high-fidelity learning curve database, both with respect to training set size and repetitions of the sampling for a fixed training set size. Our investigation reveals that generalization performance rarely follows a Gaussian distribution for classical classifiers, regardless of dataset balance, loss function, sampling method, or hyper-parameter tuning along learning curves. Furthermore, we show that the choice of statistical summary, mean versus measures like quantiles affect the top model rankings. Our findings highlight the importance of considering different statistical measures and use of non-parametric approaches when evaluating and selecting machine learning models with learning curves.

Learning curves describe how the performance of a model evolves with increasing training data. Although more data is generally expected to improve model performance, in practice models can exhibit non-monotonic behavior where additional data leads to performance degradation. Sample-wise double descent is one particular example. We address the question of how a learner can have a provably monotone learning curve. For isotropic Gaussian covariates under a Gaussian noise model and a linear predictor, we prove that a single step of steepest descent guarantees sample-wise monotonicity in the learning curve, if the step size does not exceed an upper bound. Furthermore, we present a practical procedure that ensures monotonicity without explicit regularization or cross-validation, using initialization from the previous training set size. Experiments on real-world datasets demonstrate that this method achieves monotone behavior and improved sample efficiency compared to ordinary least squares and optimally regularized ridge regression. We also explore extensions to binary classification, where monotonicity depends on the chosen performance metric. While our guarantees are derived under simplifying assumptions, they provide both theoretical and practical insights for constructing monotone learners and for understanding and mitigating sample-wise double descent behavior.

Learning curves depict how a model’s expected performance changes with varying training set sizes, unlike training curves, showing a gradient-based model’s performance with respect to training epochs. Extrapolating learning curves can be useful for determining the performance gain with additional data. Parametric functions, that assume monotone behaviour of the curves, are a prevalent methodology to model and extrapolate learning curves. However, learning curves do not necessarily follow a specific parametric shape: they can have peaks, dips, and zigzag patterns. These unconventional shapes can hinder the extrapolation performance of commonly used parametric curve-fitting models. In addition, the objective functions for fitting such parametric models are non-convex, making them initialization-dependent and brittle. In response to these challenges, we propose a convex, data-driven approach that extracts information from available learning curves to guide the extrapolation of another targeted learning curve. Our method achieves this through using a learning curve database. Using the initial segment of the observed curve, we determine a group of similar curves from the database and reduce the dimensionality via Functional Principle Component Analysis FPCA. These principal components are used in a semi-parametric kernel ridge regression (SPKR) model to extrapolate targeted curves. The solution of the SPKR can be obtained analytically and does not suffer from initialization issues. To evaluate our method, we create a new database of diverse learning curves that do not always adhere to typical parametric shapes. Our method performs better than parametric non-parametric learning curve-fitting methods on this database for the learning curve extrapolation task.

Data-driven modeling in mechanics is evolving rapidly based on recent machine learning advances, especially on artificial neural networks. As the field matures, new data and models created by different groups become available, opening possibilities for cooperative modeling. However, artificial neural networks suffer from catastrophic forgetting, i.e. they forget how to perform an old task when trained on a new one. This hinders cooperation because adapting an existing model for a new task affects the performance on a previous task trained by someone else. The authors developed a continual learning method that addresses this issue, applying it here for the first time to solid mechanics. In particular, the method is applied to recurrent neural networks to predict history-dependent plasticity behavior, although it can be used on any other architecture (feedforward, convolutional, etc.) and to predict other phenomena. This work intends to spawn future developments on continual learning that will foster cooperative strategies among the mechanics community to solve increasingly challenging problems. We show that the chosen continual learning strategy can sequentially learn several constitutive laws without forgetting them, using less data to achieve the same error as standard (non-cooperative) training of one law per model.