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Register a new userAnalyzing Monotonic Linear Interpolation in Neural Network Loss Landscapes
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James Lucas
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Linear interpolation between initial neural network parameters and converged parameters after training with stochastic gradient descent (SGD) typically leads to a monotonic decrease in the training objective. This Monotonic Linear Interpolation (MLI) property, first observed by Goodfellow et al. (2014) persists in spite of the non-convex objectives and highly non-linear training dynamics of neural networks. Extending this work, we evaluate several hypotheses for this property that, to our knowledge, have not yet been explored. Using tools from differential geometry, we draw connections between the interpolated paths in function space and the monotonicity of the network - providing sufficient conditions for the MLI property under mean squared error. While the MLI property holds under various settings (e.g. network architectures and learning problems), we show in practice that networks violating the MLI property can be produced systematically, by encouraging the weights to move far from initialization. The MLI property raises important questions about the loss landscape geometry of neural networks and highlights the need to further study their global properties.
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The success of deep learning is due, to a large extent, to the remarkable effectiveness of gradient-based optimization methods applied to large neural networks. The purpose of this work is to propose a modern view and a general mathematical framework for loss landscapes and efficient optimization in over-parameterized machine learning models and systems of non-linear equations, a setting that includes over-parameterized deep neural networks. Our starting observation is that optimization problems corresponding to such systems are generally not convex, even locally. We argue that instead they satisfy PL$^*$, a variant of the Polyak-Lojasiewicz condition on most (but not all) of the parameter space, which guarantees both the existence of solutions and efficient optimization by (stochastic) gradient descent (SGD/GD). The PL$^*$ condition of these systems is closely related to the condition number of the tangent kernel associated to a non-linear system showing how a PL$^*$-based non-linear theory parallels classical analyses of over-parameterized linear equations. We show that wide neural networks satisfy the PL$^*$ condition, which explains the (S)GD convergence to a global minimum. Finally we propose a relaxation of the PL$^*$ condition applicable to almost over-parameterized systems.
Viewing neural network models in terms of their loss landscapes has a long history in the statistical mechanics approach to learning, and in recent years it has received attention within machine learning proper. Among other things, local metrics (such as the smoothness of the loss landscape) have been shown to correlate with global properties of the model (such as good generalization). Here, we perform a detailed empirical analysis of the loss landscape structure of thousands of neural network models, systematically varying learning tasks, model architectures, and/or quantity/quality of data. By considering a range of metrics that attempt to capture different aspects of the loss landscape, we demonstrate that the best test accuracy is obtained when: the loss landscape is globally well-connected; ensembles of trained models are more similar to each other; and models converge to locally smooth regions. We also show that globally poorly-connected landscapes can arise when models are small or when they are trained to lower quality data; and that, if the loss landscape is globally poorly-connected, then training to zero loss can actually lead to worse test accuracy. Based on these results, we develop a simple one-dimensional model with load-like and temperature-like parameters, we introduce the notion of an emph{effective loss landscape} depending on these parameters, and we interpret our results in terms of a emph{rugged convexity} of the loss landscape. When viewed through this lens, our detailed empirical results shed light on phases of learning (and consequent double descent behavior), fundamental versus incidental determinants of good generalization, the role of load-like and temperature-like parameters in the learning process, different influences on the loss landscape from model and data, and the relationships between local and global metrics, all topics of recent interest.
Most existing interpretable methods explain a black-box model in a post-hoc manner, which uses simpler models or data analysis techniques to interpret the predictions after the model is learned. However, they (a) may derive contradictory explanations on the same predictions given different methods and data samples, and (b) focus on using simpler models to provide higher descriptive accuracy at the sacrifice of prediction accuracy. To address these issues, we propose a hybrid interpretable model that combines a piecewise linear component and a nonlinear component. The first component describes the explicit feature contributions by piecewise linear approximation to increase the expressiveness of the model. The other component uses a multi-layer perceptron to capture feature interactions and implicit nonlinearity, and increase the prediction performance. Different from the post-hoc approaches, the interpretability is obtained once the model is learned in the form of feature shapes. We also provide a variant to explore higher-order interactions among features to demonstrate that the proposed model is flexible for adaptation. Experiments demonstrate that the proposed model can achieve good interpretability by describing feature shapes while maintaining state-of-the-art accuracy.
Solving partial differential equations (PDEs) by parametrizing its solution by neural networks (NNs) has been popular in the past a few years. However, different types of loss functions can be proposed for the same PDE. For the Poisson equation, the loss function can be based on the weak formulation of energy variation or the least squares method, which leads to the deep Ritz model and deep Galerkin model, respectively. But loss landscapes from these different models give arise to different practical performance of training the NN parameters. To investigate and understand such practical differences, we propose to compare the loss landscapes of these models, which are both high dimensional and highly non-convex. In such settings, the roughness is more important than the traditional eigenvalue analysis to describe the non-convexity. We contribute to the landscape comparisons by proposing a roughness index to scientifically and quantitatively describe the heuristic concept of roughness of landscape around minimizers. This index is based on random projections and the variance of (normalized) total variation for one dimensional projected functions, and it is efficient to compute. A large roughness index hints an oscillatory landscape profile as a severe challenge for the first order optimization method. We apply this index to the two models for the Poisson equation and our empirical results reveal a consistent general observation that the landscapes from the deep Galerkin method around its local minimizers are less rough than the deep Ritz method, which supports the observed gain in accuracy of the deep Galerkin method.
The distributional reinforcement learning (RL) approach advocates for representing the complete probability distribution of the random return instead of only modelling its expectation. A distributional RL algorithm may be characterised by two main components, namely the representation and parameterisation of the distribution and the probability metric defining the loss. This research considers the unconstrained monotonic neural network (UMNN) architecture, a universal approximator of continuous monotonic functions which is particularly well suited for modelling different representations of a distribution (PDF, CDF, quantile function). This property enables the decoupling of the effect of the function approximator class from that of the probability metric. The paper firstly introduces a methodology for learning different representations of the random return distribution. Secondly, a novel distributional RL algorithm named unconstrained monotonic deep Q-network (UMDQN) is presented. Lastly, in light of this new algorithm, an empirical comparison is performed between three probability quasimetrics, namely the Kullback-Leibler divergence, Cramer distance and Wasserstein distance. The results call for a reconsideration of all probability metrics in distributional RL, which contrasts with the dominance of the Wasserstein distance in recent publications.
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