Measuring the Intrinsic Dimension of Objective Landscapes

    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.

    Authors

    Chunyuan Li, Heerad Farkhoor, Rosanne Liu, Jason Yosinski

    Conference

    ICLR 2018

    Full Paper

    ‘Measuring the Intrinsic Dimension of Objective Landscapes’ (PDF)

    Uber AI

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    Rosanne Liu
    Rosanne is a senior research scientist and a founding member of Uber AI. She obtained her PhD in Computer Science at Northwestern University, where she used neural networks to help discover novel materials. She is currently working on the multiple fronts where machine learning and neural networks are mysterious. She attempts to write in her spare time.
    Jason Yosinski
    Jason Yosinski is a founding member of Uber AI Labs and there leads the Deep Collective research group. He is known for contributions to understanding neural network modeling, representations, and training. Prior to Uber, Jason worked on robotics at Caltech, co-founded two web companies, and started a robotics program in Los Angeles middle schools that now serves over 500 students. He completed his PhD working at the Cornell Creative Machines Lab, University of Montreal, JPL, and Google DeepMind. He is a recipient of the NASA Space Technology Research Fellowship, has co-authored over 50 papers and patents, and was VP of ML at Geometric Intelligence, which Uber acquired. His work has been profiled by NPR, the BBC, Wired, The Economist, Science, and the NY Times. In his free time, Jason enjoys cooking, reading, paragliding, and pretending he's an artist.