Measuring the Intrinsic Dimension of Objective Landscapes (1804.08838v1)

Abstract: Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

Analyzing Intrinsic Dimension in Neural Network Objective Landscapes

The paper "Measuring the Intrinsic Dimension of Objective Landscapes" offers an empirically driven exploration into the geometric complexity of neural network solutions. By defining the intrinsic dimension of an optimization problem as the smallest dimensionality of a parameter subspace in which a model needs to train to achieve a certain performance level, the authors explore how this concept provides insights into the real degrees of freedom necessary for model learning.

Methodology

The core methodology involves projecting the high-dimensional parameter space of neural networks into random lower-dimensional subspaces and investigating the dimensions at which acceptable solutions first emerge. This intrinsic dimension is juxtaposed against the direct parameter count, offering a metric more aligned with computational tractability and meaningful information retention. The paper leverages straightforward optimization in these subspaces by adapting conventional training techniques and evaluates performance relative to predefined baselines in classification and reinforcement learning tasks.

Key Findings

1. Intrinsic Dimensionality and Parameter Efficiency: One principal insight is that many neural network models operate effectively in a parameter space significantly smaller than what is traditionally utilized. In some instances, it is found that less than 1% of the original parameter count suffices to attain 90% of baseline accuracy. This revelation underlines the substantial redundancy in most practical neural networks and points to the possibility of vast reductions in parameter space without markedly compromising on performance.
2. Comparison Across Model Variants: Through a systematic paper of fully connected networks (FCNs) of varying architectures on the MNIST dataset, the authors show that intrinsic dimensionality remains relatively stable across models despite varying model sizes. This consistency indicates that additional complexity in neural architectures primarily contributes to redundancy rather than enhanced essential problem-solving capability.
3. Application to Various Problem Domains: The method was applied across several tasks in computer vision (MNIST, CIFAR-10, ImageNet) and reinforcement learning (Atari, Mujoco), corroborating the pervasiveness of redundancy in neural network models. In reinforcement learning contexts, the varying intrinsic dimensions highlight task-dependent complexities, suggesting potential in leveraging this metric for cross-domain complexity assessments.
4. Architectural Insights Through Intrinsic Dimensions: By comparing architectures such as LeNet and ResNet across datasets, the research attributes a lower intrinsic dimension to convolutional networks versus fully connected counterparts, thus validating convolutional patterns as inherently more parameter-efficient for certain pattern recognition tasks.

Implications and Future Directions

This empirical approach to understanding neural network landscapes challenges the convention of heavy parameter reliance and provides a foundation for novel compression techniques based on intrinsic dimensionality estimates. It opens pathways for designing more efficient models that leverage the minimal essential dimensional subspaces, potentially reducing both memory requirements and training times significantly.

Theoretically, assessing intrinsic dimension could interlink with concepts like minimum description length (MDL) and provide a tangible measure of model efficiency, aiding in architecture selection and hyperparameter tuning. Practically, this understanding could influence the design of more resource-sensitive applications, especially in edge computing and environments with restricted processing capabilities.

Going forward, further research could explore non-linear transformations for even more precise subspace assessments, drawing connections between learned models and data manifold structures. This paper lays foundational work for automatic intrinsic dimension estimation, prompting more discussions in academic and applied machine learning sectors on genuinely necessary model complexity and the potential for substantial paradigm shifts in neural network design philosophies.