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Title: Measuring the Intrinsic Dimension of Objective Landscapes

Paper content:
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.
Training a neural network to model a given dataset entails several steps.
First, the network designer chooses a loss function and a network architecture for a given dataset.
The architecture is then initialized by populating its weights with random values drawn from some distribution.
Finally, the network is trained by adjusting its weights to produce a loss as low as possible.
We can think of the training procedure as traversing some path along an objective landscape.
Note that as soon as a dataset and network architecture are specified, the landscape in its entirety is completely determined.
It is instantiated and frozen; all subsequent parameter initialization, forward and backward propagation, and gradient steps taken by an optimizer are just details of how the frozen space is explored.Consider a network parameterized by D weights.
We can picture its associated objective landscape as a set of "hills and valleys" in D dimensions, where each point in R D corresponds to a value of the loss, i.e., the elevation of the landscape.
If D = 2, the map from two coordinates to one scalar loss can be easily imagined and intuitively understood by those living in a three-dimensional world with similar hills.
However, in higher dimensions, our intuitions may not be so faithful, and generally we must be careful, as extrapolating low-dimensional intuitions to higher dimensions can lead to unreliable conclusions.
The difficulty of understanding high-dimensional landscapes notwithstanding, it is the lot of neural network researchers to spend their efforts leading (or following?) networks over these multi-dimensional surfaces.
Therefore, any interpreted geography of these landscapes is valuable.Several papers have shed valuable light on this landscape, particularly by pointing out flaws in common extrapolation from low-dimensional reasoning.
BID4 showed that, in contrast to conventional thinking about getting stuck in local optima (as one might be stuck in a valley in our familiar D = 2), local critical points in high dimension are almost never valleys but are instead saddlepoints: structures which are "valleys" along a multitude of dimensions with "exits" in a multitude of other dimensions.
The striking conclusion is that one has less to fear becoming hemmed in on all sides by higher loss but more to fear being waylaid nearly indefinitely by nearly flat regions.
BID9 showed another property: that paths directly from the initial point to the final point of optimization are often monotonically decreasing.
Though dimension is high, the space is in some sense simpler than we thought: rather than winding around hills and through long twisting corridors, the walk could just as well have taken a straight line without encountering any obstacles, if only the direction of the line could have been determined at the outset.In this paper we seek further understanding of the structure of the objective landscape by restricting training to random slices through it, allowing optimization to proceed in randomly generated subspaces of the full parameter space.
Whereas standard neural network training involves computing a gradient and taking a step in the full parameter space (R D above), we instead choose a random d-dimensional subspace of R D , where generally d < D, and optimize directly in this subspace.
By performing experiments with gradually larger values of d, we can find the subspace dimension at which solutions first appear, which we call the measured intrinsic dimension of a particular problem.
Examining intrinsic dimensions across a variety of problems leads to a few new intuitions about the optimization problems that arise from neural network models.We begin in Sec. 2 by defining more precisely the notion of intrinsic dimension as a measure of the difficulty of objective landscapes.
In Sec. 3 we measure intrinsic dimension over a variety of network types and datasets, including MNIST, CIFAR-10, ImageNet, and several RL tasks.
Based on these measurements, we draw a few insights on network behavior, and we conclude in Sec. 4.
In this paper, we have defined the intrinsic dimension of objective landscapes and shown a simple method -random subspace training -of approximating it for neural network modeling problems.
We use this approach to compare problem difficulty within and across domains.
We find in some cases the intrinsic dimension is much lower than the direct parameter dimension, and hence enable network compression, and in other cases the intrinsic dimension is similar to that of the best tuned models, and suggesting those models are better suited to the problem.Further work could also identify better ways of creating subspaces for reparameterization: here we chose random linear subspaces, but one might carefully construct other linear or non-linear subspaces to be even more likely to contain solutions.
Finally, as the field departs from single stackof-layers image classification models toward larger and more heterogeneous networks BID11 BID14 often composed of many modules and trained by many losses, methods like measuring intrinsic dimension that allow some automatic assessment of model components might provide much-needed greater understanding of individual black-box module properties.
In the main paper, we attempted to find d int90 across 20 FC networks with various depths and widths.
A grid sweep of number of hidden layers from {1,2,3,4,5} and width of each hidden layer from {50,100,200,400} is performed, and all 20 plots are shown in FIG7 .
For each d we take 3 runs and plot the mean and variance with blue dots and blue error bars.
d int90 is indicated in plots (darkened blue dots) by the dimension at which the median of the 3 runs passes 90% performance threshold.
The variance of d int90 is estimated using 50 bootstrap samples.
Note that the variance of both accuracy and measured d int90 for a given hyper-parameter setting are generally small, and the mean of performance monotonically increases (very similar to the single-run result) as d increases.
This illustrates that the difference between lucky vs. unlucky random projections have little impact on the quality of solutions, while the subspace dimensionality has a great impact.
We hypothesize that the variance due to different P matrices will be smaller than the variance due to different random initial parameter vectors θ 0 , and aspects of the network depending on smaller numbers of random samples will exhibit greater variance.
Hence, in some other experiments we rely on single runs to estimate the intrinsic dimension, though slightly more accurate estimates could be obtained via multiple runs.In similar manner to the above, in FIG8 we show the relationship between d int90 and D across 20 networks but using a per-model, directly trained baseline.
Most baselines are slightly below 100% accuracy.
This is in contrast to FIG3 , which used a simpler global baseline of 100% across all models.
Results are qualitatively similar but with slightly lower intrinsic dimension due to slightly lower thresholds.

assistant: text: We train in random subspaces of parameter space to measure how many dimensions are really needed to find a solution.