M EASURING THE I NTRINSIC D IMENSION OF O BJECTIVE L ANDSCAPES
A BSTRACT
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, bu 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.
I NTRODUCTION 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.
Dauphin et al.
(2014) 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.
Goodfellow et al. (2015) 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.
D EFINING AND E STIMATING I NTRINSIC D IMENSION
We introduce the intrinsic dimension of an objective landscape with an illustrative toy problem.
Let θ ( D ) ∈
R D be a parameter vector in a parameter space of dimension D , let θ ( D )be a randomly chosen initial parameter vector, and let θ ( D ) ∗ be the ﬁnal parameter vector arrived at via optimization.
Consider a toy optimization problem where D = 1000 and where θ ( D ) optimized to minimize a squared error cost function that requires the ﬁrst 100 elements to sum to 1, the second 100 elements to sum to 2, and so on until the vector has been divided into 10 groups with their requisite 10 sums.
We may start from a θ ( D )that is drawn from a Gaussian distribution and optimize in R D to ﬁnd a θ ( D ) ∗ that solves the problem with cost arbitrarily close to zero.
Solutions to this problem are highly redundant.
With a little algebra, one can ﬁnd that the manifold of solutions is a 990 dimensional hyperplane: from any point that has zero cost, there areorthogonal directions one can move and remain at zero cost.
Denoting as s the dimensionality of the solution set, we deﬁne the intrinsic dimensionality d int of a solution as the codimension of the solution set inside of R D :
[Equation]
Here the intrinsic dimension d int is 10 (1000 = 10 + 990), with 10 corresponding intuitively to the number of constraints placed on the parameter vector.
Though the space is large ( D = 1000 ), the number of things one needs to get right is small ( d int = 10 ).
2. M EASURING I NTRINSIC D IMENSION VIA R ANDOM S UBSPACE T RAINING
The above example had a simple enough form that we obtained d int = 10 by calculation.
But in general we desire a method to measure or approximate d int for more complicated problems, including problems with data-dependent objective functions, e.g. neural network training.
Random subspace optimization provides such a method.
[Figure]
Standard optimization, which we will refer to hereafter as the direct method of training, entails evaluating the gradient of a loss with respect to θ ( D ) and taking steps directly in the space of θ ( D ) .
To train in a random subspace, we instead deﬁne θ ( D ) in the following way:
[Equation]
where P is a randomly generated D × d projection matrix 1 and θ ( d ) is a parameter vector in a generally smaller space R d .
θ ( D )and P are randomly generated and frozen (not trained), so the system has only d degrees of freedom.
We initialize θ ( d ) to a vector of all zeros, so initially θ ( D )
= θ ( D ) 0 .
This convention serves an important purpose for neural network training: it allows the network to beneﬁt from beginning in a region of parameter space designed by any number of good initialization schemes (Glorot & Bengio, 2010; He et al., 2015) to be well-conditioned, such that gradient descent via commonly used optimizers will tend to work well.
Training proceeds by computing gradients with respect to θ ( d ) and taking steps in that space.
Columns of P are normalized to unit length, so steps of unit length in θ ( d ) chart out unit length motions of θ ( D ) .
Columns of P may also be orthogonalized if desired, but in our experiments we relied simply on the approximate orthogonality of high dimensional random vectors.
By this construction P forms an approximately orthonormal basis for a randomly oriented d dimensional subspace of R D , with the origin of the new coordinate system at θ ( D ) 0 .
Fig.
1 (left and middle) shows an illustration of the related vectors.
Consider a few properties of this training approach.
If d = D and P is a large identity matrix, we recover exactly the direct optimization problem.
If d = D but P is instead a random orthonormal basis for all of R D (just a random rotation matrix), we recover a rotated version of the direct problem.
Note that for some “rotation-invariant” optimizers, such as SGD and SGD with momentum, rotating the basis will not change the steps taken nor the solution found, but for optimizers with axis-aligned assumptions, such as RMSProp (Tieleman & Hinton, 2012) and Adam (Kingma & Ba, 2014), the path taken through θ ( D ) space by an optimizer will depend on the rotation chosen.
Finally, in the general case where d < D and solutions exist in D , solutions will almost surely (with probability 1) not be found if d is less than the codimension of the solution.
On the other hand, when d ≥ D − s , if the solution set is a hyperplane, the solution will almost surely intersect the subspace, but for solution sets of arbitrary topology, intersection is not guaranteed.
Nonetheless, by iteratively increasing d , re-running optimization, and checking for solutions, we obtain one estimate of d int .
We try this sweep of d for our toy problem laid out in the beginning of this section, measuring (by convention as described in the next section) the positive performance (higher is better) instead of loss.
3 As expected, the solutions are ﬁrst found at d = 10 (see Fig.
1, right), conﬁrming our intuition that for this problem, d int = 10 .
2. D ETAILS AND CONVENTIONS
In the rest of this paper, we measure intrinsic dimensions for particular neural network problems and draw conclusions about the associated objective landscapes and solution sets.
Because modeling real data is more complex than the above toy example, and losses are generally never exactly zero, we ﬁrst choose a heuristic for classifying points on the objective landscape as solutions vs. non-solutions.
The heuristic we choose is to threshold network performance at some level relative to a baseline model, where generally we take as baseline the best directly trained model.
In supervised classiﬁcation settings, validation accuracy is used as the measure of performance, and in reinforcement learning scenarios, the total reward (shifted up or down such that the minimum reward is 0) is used.
Accuracy and reward are preferred to loss to ensure results are grounded to real-world performance and to allow comparison across models with differing scales of loss and different amounts of regularization included in the loss.
We deﬁne d int100 as the intrinsic dimension of the “100%” solution: solutions whose performance is statistically indistinguishable from baseline solutions.
However, when attempting to measure d int100 , we observed it to vary widely, for a few confounding reasons: d int100 can be very high — nearly as high as D — when the task requires matching a very well-tuned baseline model, but can drop signiﬁcantly when the regularization effect of restricting parameters to a subspace boosts performance by tiny amounts.
While these are interesting effects, we primarily set out to measure the basic difﬁculty of problems and the degrees of freedom needed to solve (or approximately solve) them rather than these subtler effects.
Thus, we found it more practical and useful to deﬁne and measure d
int90 as the intrinsic dimension of the “90%” solution: solutions with performance at least 90% of the baseline.
We chose 90% after looking at a number of dimension vs. performance plots (e.g. Fig. 2) as a reasonable trade off between wanting to guarantee solutions are as good as possible, but also wanting measured d int values to be robust to small noise in measured performance.
If too high a threshold is used, then the dimension at which performance crosses the threshold changes a lot for only tiny changes in accuracy, and we always observe tiny changes in accuracy due to training noise.
If a somewhat different (higher or lower) threshold were chosen, we expect most of conclusions in the rest of the paper to remain qualitatively unchanged.
In the future, researchers may ﬁnd it useful to measure d int using higher or lower thresholds.
R ESULTS AND D ISCUSSION 3. MNIST
We begin by analyzing a fully connected (FC) classiﬁer trained on MNIST.
We choose a network with layer sizes 784–200–200–10, i.e. a network with two hidden layers of width 200; this results in a total number of parameters D = 199 , 210 .
A series of experiments with gradually increasing subspace dimension d produce monotonically increasing performances, as shown in Fig.
2 (left).
By checking the subspace dimension at which performance crosses the 90% mark, we measure this network’s intrinsic dimension d
int90 at about 750.
Some networks are very compressible.
A salient initial conclusion is that 750 is quite low.
At that subspace dimension, only 750 degrees of freedom (0.4%) are being used and 198,460 (99.6%) unused to obtain 90% of the performance of the direct baseline model.
A compelling corollary of this result is a simple, new way of creating and training compressed networks, particularly networks for applications in which the absolute best performance is not critical.
To store this network, one need only store a tuple of three items: ( i ) the random seed to generate the frozen θ ( D ) 0 , ( ii ) the
[Figure]
random seed to generate P and ( iii ) the 750 ﬂoating point numbers in θ ( d ) ∗ .
It leads to compression (assuming 32-bit ﬂoats) by a factor of 260 × from 793kB to only 3.2kB, or 0.4% of the full size.
Such compression could be very useful for scenarios where storage or bandwidth are limited, e.g. including neural networks in downloaded mobile apps or on web pages.
This compression approach differs from other neural network compression methods in the following aspects.
( i )
While it has previously been appreciated that large networks waste parameters (Dauphin & Bengio, 2013) and weights contain redundancy (Denil et al., 2013) that can be exploited for posthoc compression (Wen et al., 2016), this paper’s method constitutes a much simpler approach to compression, where training happens once, end-to-end, and where any parameterized model is an allowable base model.
( ii ) Unlike layerwise compression models (Denil et al., 2013; Wen et al., 2016), we operate in the entire parameter space, which could work better or worse, depending on the network.
( iii )
Compared to methods like that of Louizos et al.
(2017), who take a Bayesian perspective and consider redundancy on the level of groups of parameters (input weights to a single neuron) by using group-sparsity-inducing hierarchical priors on the weights, our approach is simpler but not likely to lead to compression as high as the levels they attain.
( iv ) Our approach only reduces the number of degrees of freedom, not the number of bits required to store each degree of freedom, e.g. as could be accomplished by quantizing weights (Han et al., 2016).
Both approaches could be combined.
( v ) There is a beautiful array of papers on compressing networks such that they also achieve computational savings during the forward pass (Wen et al., 2016; Han et al., 2016;
Yang et al., 2015); subspace training does not speed up execution time during inference.
( vi ) Finally, note the relationships between weight pruning, weight tying, and subspace training: weight pruning is equivalent to ﬁnding, post-hoc, a subspace that is orthogonal to certain axes of the full parameter space and that intersects those axes at the origin.
Weight tying, e.g. by random hashing of weights into buckets (Chen et al., 2015), is equivalent to subspace training where the subspace is restricted to lie along the equidistant “diagonals” between any axes that are tied together.
Robustness of intrinsic dimension.
Next, we investigate how intrinsic dimension varies across
FC networks with a varying number of layers and varying layer width.
4 We perform a grid sweep of networks with number of hidden layers L chosen from { 1, 2, 3, 4, 5 } and width W chosen from { 50, 100, 200, 400 } .
Fig.
S6 in the Supplementary Information shows performance vs. subspace dimension plots in the style of Fig.
2 for all 20 networks, and Fig.
3 shows each network’s d
int plotted against its native dimension D .
As one can see, D changes by a factor of 24 . 1 between the smallest and largest networks, but d int90 changes over this range by a factor of only 1 . 33 , with much of this possibly due to noise.
Thus it turns out that the intrinsic dimension changes little even as models grown in width or depth!
The striking conclusion is that every extra parameter added to the network — every extra dimension added to D — just ends up adding one dimension to the redundancy of the solution, s .
red interval is the standard derivation of the measurement of d
Though the of native parameters D varies by a factor of d int90 varies by only 1 . with much of that possibly due to noise, showing that d int90 is a fairly robust measure across a model family and each extra ends up adding an extra dimension directly to the redundancy
Standard deviation was estimated via bootstrap
[Figure]
Often the most accurate directly trained models for a problem have far more parameters than needed (Zhang et al., 2017); this may be because they are just easier to train and our observation suggests a reason why: with larger models, solutions have greater redundancy and in a sense “cover” more of the space.
5 To our knowledge, this is the ﬁrst time this phenomenon has been directly measured.
We should also be careful not to claim that all FC nets on MNIST will have an intrinsic dimension of around 750; instead, we should just consider that we have found for this architecture/dataset combination a wide plateau of hyperparamter space over which intrinsic dimension is approximately constant.
Are random subspaces really more parameter-efﬁcient for FC nets?
One might wonder to what extent claiming 750 parameters is meaningful given that performance achieved (90%) is far worse than a state of the art network trained on MNIST.
With such a low bar for performance, could a directly trained network with a comparable number of trainable parameters be found that achieves the same performance?
We generated 1000 small networks (depth randomly chosen from { 1, 2, 3, 4, 5 } , layer width randomly from { 2, 3, 5, 8, 10, 15, 20, 25 } , seed set randomly) in an attempt to ﬁnd high-performing, small FC networks, but as Fig.
4 (left) shows, a gap still exists between the subspace dimension and the smallest direct FC network giving the same performance at most levels of performance.
Measuring d int90 on a convolutional network.
Next we measure d
int90 of a convolutional network, LeNet (D=44,426).
Fig.
2 (right) shows validation accuracy vs. subspace dimension d , and we ﬁnd d int90 = 290 , or a compression rate of about 150 × for this network.
As with the FC case above, we also do a sweep of random networks, but notice that the performance gap of convnets between direct and subspace training methods becomes closer for ﬁxed budgets, i.e., the number of trainable parameters.
Further, the performance of direct training varies signiﬁcantly, depending on the extrinsic design of convet architectures.
We interpret these results in terms of the Minimum Description Length below.
Relationship between Intrinsic Dimension and Minimum Description Length (MDL).
As discussed earlier, the random subspace training method leads naturally to a compressed representation of a network, where only d ﬂoating point numbers need to be stored.
We can consider this d as an upper bound on the MDL of the problem solution.
6 We cannot yet conclude the extent to which this bound is loose or tight, and tightness may vary by problem.
However, to the extent that it is tighter than previous bounds ( e.g., just the number of parameters D ) and to the extent that it is correlated with the actual MDL, we can use this interpretation to judge which solutions are more well-suited to the problem in a principled way.
As developed by Rissanen (1978) and further by Hinton &
Van Camp (1993), holding accuracy constant, the best model is the one with the shortest MDL.
Thus, there is some rigor behind our intuitive assumption that LeNet is a better model than an FC network for MNIST image classiﬁcation, because its intrinsic dimension is lower ( d int90 of 290 vs.
In this particular case we are lead to a predictable conclusion, but as models become larger, more complex, and more heterogeneous, conclusions of this type will often not be obvious.
Having a simple method of approximating MDL may prove extremely useful for guiding model exploration, for example, for the countless datasets less well-studied than MNIST and for models consisting of separate sub-models that may be individually designed and evaluated (Ren et al., 2015; Kaiser et al., 2017).
In this latter case, considering the MDL for a sub-model could provide a more detailed view of that sub-model’s properties than would be available by just analyzing the system’s overall validation performance.
Finally, note that although our approach is related to a rich body of work on estimating the “intrinsic dimension of a dataset” (Camastra & Vinciarelli, 2002; K ´ egl, 2003; Fukunaga & Olsen, 1971; Levina & Bickel, 2005; Tenenbaum et al., 2000), it differs in a few respects.
Here we do not measure the number of degrees of freedom necessary to represent a dataset (which requires representation of a global p ( X ) and per-example properties and thus grows with the size of the dataset), but those required to represent a model for part of the dataset (here p ( y | X ) , which intuitively might saturate at some complexity even as a dataset grows very large).
That said, in the following section we do show measurements for a corner case where the model must memorize per-example properties.
Are convnets always better on MNIST?
Measuring d int90 on shufﬂed data.
Zhang et al.
(2017) provocatively showed that large networks normally thought to generalize well can nearly as easily be trained to memorize entire training sets with randomly assigned labels or with input pixels provided in random order.
Consider two identically sized networks: one trained on a real, non-shufﬂed dataset and another trained with shufﬂed pixels or labels.
As noted by Zhang et al.
(2017), externally the networks are very similar, and the training loss may even be identical at the ﬁnal epoch.
However, the intrinsic dimension of each may be measured to expose the differences in problem difﬁculty.
When training on a dataset with shufﬂed pixels — pixels for each example in the dataset subject to a random permutation, chosen once for the entire dataset — the intrinsic dimension of an FC network remains the same at 750, because FC networks are invariant to input permutation.
But the intrinsic dimension of a convnet increases from 290 to 1400, even higher than an FC network.
Thus while convnets are better suited to classifying digits given images with local structure, when this structure is removed, violating convolutional assumptions, our measure can clearly reveal that many
[Table]
more degrees of freedom are now required to model the underlying distribution.
When training on
MNIST with shufﬂed labels — the label for each example is randomly chosen — we redeﬁne our measure of d int90 relative to training accuracy (validation accuracy is always at chance).
We ﬁnd that memorizing random labels on the 50,000 example MNIST training set requires a very high dimension, d int90 = 190 , 000 , or 3.8 ﬂoats per memorized label.
Sec.
S5.2 gives a few further results, in particular that the more labels are memorized, the more efﬁcient memorization is in terms of ﬂoats per label.
Thus, while the network obviously does not generalize to an unseen validation set
[Table]
it would seem “generalization” within a training set may be occurring as the network builds a shared infrastructure that makes it possible to more efﬁciently memorize labels
[Table]
3. CIFAR-10 AND I MAGE N ET
We scale to larger supervised classiﬁcation problems by considering CIFAR-10 (Krizhevsky & Hinton, 2009) and ImageNet (Russakovsky et al., 2015).
When scaling beyond MNIST-sized networks with D on the order of 200k and d on the order of 1k, we ﬁnd it necessary to use more efﬁcient methods of generating and projecting from random subspaces.
This is particularly true in the case of ImageNet, where the direct network can easily require millions of parameters.
In Sec.
S7, we describe and characterize scaling properties of three methods of projection: dense matrix projection, sparse matrix projection (Li et al., 2006), and the remarkable Fastfood transform (Le et al., 2013).
We generally use the sparse projection method to train networks on CIFAR-10 and the Fastfood transform for ImageNet.
Measured d int90 values for CIFAR-10 and are ImageNet given in Table 1, next to all previous
MNIST results and RL results to come.
For CIFAR-10 we ﬁnd qualitatively similar results to
MNIST, but with generally higher dimension (9k vs. 750 for FC and 2.9k vs. 290 for LeNet).
It is also interesting to observe the difference of d int90 across network architectures.
For example, to achieve a global > 50% validation accuracy on CIFAR-10, FC, LeNet and ResNet approximately requires d int90 = 9 k, 2 . 9 k and 1 k, respectively, showing that ResNets are more efﬁcient.
Full results and experiment details are given in Sec. S8 and Sec. S9.
Due to limited time and memory issues, training on ImageNet has not yet given a reliable estimate for d int90 except that it is over 500k.
3. R EINFORCEMENT L EARNING ENVIRONMENTS
Measuring intrinsic dimension allows us to perform some comparison across the divide between supervised learning and reinforcement learning.
In this section we measure the intrinsic dimension of three control tasks of varying difﬁculties using both value-based and policy-based algorithms.
The value-based algorithm we evaluate is the Deep Q-Network (DQN) (Mnih et al., 2013), and the policy-based algorithm is Evolutionary Strategies (ES) (Salimans et al., 2017).
Training details are given in Sec.
S6.2.
For all tasks, performance is deﬁned as the maximum-attained (over training iterations) mean evaluation reward (averaged over 30 evaluations for a given parameter setting).
In
Fig.
5, we show results of ES on three tasks: median of observed performance values across many runs for given d , and the uncertainty bar shows the maximum and minimum observed performance values.
The line corresponds to the usual 90% baseline derived from the best directly-trained network (the
[Figure]
horizontal line).
A dot is darkened signifying the ﬁrst d that allows a satisfactory performance.
We ﬁnd that the inverted pendulum task is surprisingly easy, with d int100 = d
int90 = 4 , meaning that only four parameters are needed to perfectly solve the problem (see Stanley & Miikkulainen (2002) for a similarly small solution found via evolution).
The walking humanoid task is more difﬁcult: solutions are found reliably by dimension 700, a similar complexity to that required to model MNIST with an FC network, and far less than modeling CIFAR-10 with a convnet.
Finally, to play Pong on Atari (directly from pixels) requires a network trained in a 6k dimensional subspace, making it on the same order of modeling CIFAR-10.
For an easy side-by-side comparison we list all intrinsic dimension values found for all problems in Table 1.
For more complete ES results see Sec.
S6.2, and Sec.
S6.1 for DQN results.
C ONCLUSIONS AND F UTURE D IRECTIONS
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 (Ren et al., 2015; Kaiser et al., 2017) 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.
A CKNOWLEDGMENTS
The authors gratefully acknowledge Zoubin Ghahramani, Peter Dayan, Sam Greydanus, Jeff Clune, and Ken Stanley for insightful discussions, Joel Lehman for initial idea validation, Felipe Such,
Edoardo Conti and Xingwen Zhang for helping scale the ES experiments to the cluster, Vashisht
Madhavan for insights on training Pong, Shrivastava Anshumali for conversations about random projections, and Ozan Sener for discussion of second order methods.
We are also grateful to Paul
Mikesell, Leon Rosenshein, Alex Sergeev and the entire OpusStack Team inside Uber for providing our computing platform and for technical support.
S A DDITIONAL MNIST RESULTS AND INSIGHTS S5. S WEEPING DEPTHS AND WIDTHS ; MULTIPLE RUNS TO ESTIMATE VARIANCE
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 Fig.
S6.
For each d we takeruns 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 ( D )because there are dD i.i.d.
samples used to create P (at least in the dense case) but only D samples used to create θ ( D ) 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 Fig.
S7 we show the relationship between d int90 and D acrossnetworks but using a per-model, directly trained baseline.
Most baselines are slightly below 100% accuracy.
This is in contrast to Fig.
3, 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.
S5. A DDITIONAL DETAILS ON SHUFFLED MNIST DATASETS
Two kinds of shufﬂed MNIST datasets are considered:
• The shufﬂed pixel dataset : the label for each example remains the same as the normal dataset, but a random permutation of pixels is chosen once and then applied to all images in the training and test sets.
FC networks solve the shufﬂed pixel datasets exactly as easily as the base dataset, because there is no privileged ordering of input dimension in FC networks; all orderings are equivalent.
• The shufﬂed label dataset : the images remain the same as the normal dataset, but labels are randomly shufﬂed for the entire training set.
Here, as in (Zhang et al., 2017), we only evaluate training accuracy, as test set accuracy remains forever at chance level (the training set X and y convey no information about test set p ( y | X ) , because the shufﬂed relationship in test is independent of that of training).
On the full shufﬂed label MNIST dataset (50k images), we trained an FC network ( L = 5 , W = 400 , which had d int90 = 750 on standard MNIST), it yields d int90 = 190 k. We can interpret this as requiring 3.8 ﬂoats to memorize each random label (at 90% accuracy).
Wondering how this scales with dataset size, we estimated d
int90 on shufﬂed label versions of MNIST at different scales and found curious results, shown in Table S2 and Fig. S8.
As the dataset memorized becomes smaller, the number of ﬂoats required to memorize each label becomes larger.
Put another way, as dataset size
[Figure]
V
[Figure]
li d a t i o n a cc u r a c y
[Figure]
[Figure]
number of parameters D for 20 models of varying width (from 50 to 400) and depth (number of hidden layers from 1 to 5) trained on MNIST.
The vertical red interval is the standard derivation of measured d int90 .
As opposed to Fig.
3, which used a global, shared baseline across all models, here a per-model baseline is used.
The number of native parameters varies by a factor of 24.1, but d int90 varies by only 1.42.
The per-model baseline results in higher measured d int90 for larger models because they have a higher baseline performance than the shallower models.
[Table]
increases, the intrinsic dimension also increases, but not as fast as linearly.
The best interpretation is not yet clear, but one possible interpretation is that networks required to memorize large training sets make use of shared machinery for memorization.
In other words, though performance does not generalize to a validation set, generalization within a training set is non-negligible even though labels are random.
S5. T RAINING STABILITY
An interesting tangential observation is that random subspace training can in some cases make optimization more stable.
First, it helps in the case of deeper networks.
Fig.
S9 shows training results for FC networks with up to 10 layers.
SGD with step 0.1, and ReLUs with He initialization is used.
Multiple networks failed at depths 4, and all failed at depths higher than 4, despite the activation function and initialization designed to make learning stable (He et al., 2015).
Second, for MNIST with shufﬂed labels, we noticed that it is difﬁcult to reach high training accuracy using the direct training method with SGD, though both subspace training with SGD and either type of training with
Adam reliably reach 100% memorization as d increases (see Fig. S8).
Because each random basis vector projects across all D direct parameters, the optimization problem may be far better conditioned in the subspace case than in the direct case.
A related potential downside is that projecting across D parameters which may have widely varying scale could result in ignoring parameter dimensions with tiny gradients.
This situation is similar to that faced by methods like SGD, but ameliorated by RMSProp, Adam, and other methods that rescale per-dimension step sizes to account for individual parameter scales.
Though convergence of the subspace approach seems robust, further work may be needed to improve network amenability to subspace training: for example by ensuring direct parameters are similarly scaled by clever initialization or by inserting a pre-scaling layer between the projected subspace and the direct parameters themselves.
S5. T HE ROLE OF OPTIMIZERS
Another ﬁnding through our experiments with MNIST FC networks has to do with the role of optimizers.
The same set of experiments are run with both SGD (learning rate 0.1) and ADAM (learning
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rate 0.001), allowing us to investigate the impact of stochastic optimizers on the intrinsic dimension achieved.
The intrinsic dimension d int90 are reported in Fig.
S10 (a)(b).
In addition to two optimizers we also use two baselines:
Global baseline that is set up as 90% of best performance achieved across all models, and individual baseline that is with regards to the performance of the same model in direct training.
S A DDITIONAL R EINFORCEMENT L EARNING RESULTS AND DETAILS S6. DQN EXPERIMENTS
DQN on Cartpole We start with a simple classic control game CartPole − v0 in OpenAI
Gym (Brockman et al., 2016).
A pendulum starts upright, and the goal is to prevent it from falling over.
The system is controlled by applying a force of LEFT or RIGHT to the cart.
The full game ends when one of two failure conditions is satisﬁed: the cart moves more than 2.4 units from the center (where it started), or the pole is more than 15 degrees from vertical (where it started).
A reward of +1 is provided for every time step as long as the game is going.
We further created two easier environments Pole and Cart , each conﬁned by one of the failure modes only.
A DQN is used, where the value network is parameterized by an FC ( L = 2 , W = 400 ).
For each subspace d at least 5 runs are conducted, the mean of which is used to computed d int90 , and the baseline is set as 195.0 7 .
The results are shown in Fig.
S11.
The solid line connects mean rewards within a run over the last 100 episodes, across different d s. Due to the noise-sensitiveness of RL games the course is not monotonic any more.
The intrinsic dimension for CartPole , Pole and Cart is d int90 = 25 , 23 and 7 , respectively.
This reveals that the difﬁculty of optimization landscape of these games is remarkably low, as well as interesting insights such as driving a cart is much easier than keeping a pole straight, the latter being the major cause of difﬁculty when trying to do both.
S6. E VOLUTIONARY S TRATEGIES (ES) COMPLETE RESULTS
We carry out with ES 3 RL tasks: InvertedPendulum − v1 , Humanoid − v1 , Pong − v0 .
The hyperparameter settings for training are in Table S3.
Inverted
pendulum The InvertedPendulum − v1 environment uses the MuJoCo physics simulator (Todorov et al., 2012) to instantiate the same problem as CartPole − v0 in a realistic setting.
We expect that even with richer environment dynamics, as well as a different RL algorithm – ES – the intrinsic dimensionality should be similar.
As seen in Fig.
5, the measured intrinsic dimensionality d int90 = 4 is of the same order of magnitude, but smaller.
Interestingly, although the environment
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dynamics are more complex than in CartPole − v0 , using ES rather than DQN seems to induce a simpler objective landscape.
Learning to walk A more challenging problem is Humanoid − v1 in MuJoCo simulator.
Intuitively, one might believe that learning to walk is a more complex task than classifying images.
Our results show the contrary – that the learned intrinsic dimensionality of d int90 = 700 is similar to that of MNIST on a fully-connected network ( d int90 = 650 ) but signiﬁcantly less than that of even a convnet trained on CIFAR-10 ( d int90 = 2 , 500 ).
Fig.
5 shows the full results.
Interestingly, we
[Table]
Atari Pong Finally, using a base convnet of approximately D = 1 M in the Pong − v0 pixels-toactions environment (using 4-frame stacking).
The agent receives an image frame (size of 210 × 160 × 3 ) and the action is to move the paddle UP or DOWN.
We were able to determine d int90 = 6 , 000 .
S T HREE M ETHODS OF RANDOM PROJECTION
Scaling the random subspace training procedure to large problems requires an efﬁcient way to map from R d into a random d -dimensional subspace of R D that does not necessarily include the origin.
Algebraically, we need to left-multiply a vector of parameters v ∈ R d by a random matrix M ∈ R D × d , whose columns are orthonormal, then add an offset vector θ 0 ∈ R D .
If the low-dimensional parameter vector in R d is initialized to zero, then specifying an offset vector is equivalent to choosing an initialization point in the original model parameter space R D .
A na ¨ ıve approach to generating the random matrix M is to use a dense D ×
d matrix of independent standard normal entries, then scale each column to be of length 1.
The columns will be approximately orthogonal if D is large because of the independence of the entries.
Although this approach is sufﬁcient for low-rank training of models with few parameters, we quickly run into scaling limits because both matrix-vector multiply time and storage of the matrix scale according to O ( Dd ) .
We were able to successfully determine the intrinsic dimensionality of MNIST ( d =225) using a LeNet ( D =44,426), but were unable to increase d beyond 1,000 when applying a LeNet ( D =62,006)
to
CIFAR-10, which did not meet the performance criterion to be considered the problems intrinsic dimensionality.
Random matrices need not be dense for their columns to be approximately orthonormal.
In fact, a method exists for “very sparse” random projections (Li et al., 2006), which achieves a density of 1 √ D .
To construct the D × d matrix, each entry is chosen to be nonzero with probability 1 √ D .
If chosen, then with equal probability, the entry is either positive or negative with the same magnitude in either case.
The density of 1 √ D implies √ Dd nonzero entries, or O ( √ Dd ) time and space complexity.
Implementing this procedure allowed us to ﬁnd the intrinsic dimension of d =2,500 for CIFARusing a LeNet mentioned above.
Unfortunately, when using Tensorﬂow’s SparseTensor implementation we did not achieve the theoretical √ D -factor improvement in time complexity (closer to a constant 10x).
Nonzero elements also have a signiﬁcant memory footprint of 24 bytes, so we could not scale to larger problems with millions of model parameters and large intrinsic dimensionalities.
We need not explicitly form and store the transformation matrix.
The Fastfood transform (Le et al., 2013) was initially developed as an efﬁcient way to compute a nonlinear, high-dimensional feature map φ ( x ) for a vector x .
A portion of the procedure involves implicitly generating a D × d matrix with approximately uncorrelated standard normal entries, using only O ( D ) space, which can be multiplied by v in O
( D log d ) time using a specialized method.
The method relies on the fact that
Hadamard matrices multiplied by Gaussian vectors behave like dense Gaussian matrices.
In detail, to implicitly multiply v by a random square Gaussian matrix M with side-lengths equal to a power of two, the matrix is factorized into multiple simple matrices: M = HG Π HB , where B is a random diagonal matrix with entries +-1 with equal probability, H is a Hadamard matrix, Π is a random permutation matrix, and G is a random diagonal matrix with independent standard normal entries.
Multiplication by a Hadamard matrix can be done via the Fast Walsh-Hadamard Transform in O ( d log d ) time and takes no additional space.
The other matrices have linear time and space complexities.
When D > d , multiple independent samples of M can be stacked to increase the output dimensionality.
When d is not a power of two, we can zero-pad v appropriately.
Stacking
Dd samples of M results in an overall time complexity of O ( Dd d log d )
=
O ( D log d ) , and a space complexity of O
( Dd d ) =
O ( D ) .
In practice, the reduction in space footprint allowed us to scale to much larger problems, including the Pong RL task using a 1M parameter convolutional network for the policy function.
Table S4 summarizes the performance of each of the three methods theoretically and empirically.
Time
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Space complexity
D = 100k
D = 1M
D =
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Dense
O ( Dd ) O ( Dd )
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s
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Sparse
O ( √ Dd ) O ( √ Dd )
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0002 s
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Fastfood
O ( D log d )
O (
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s
The Fastfood timings are based on a Tensorﬂow implementation of the Fast Walsh-Hadamard Transform, and could be drastically reduced with an efﬁcient CUDA implementation.
Asterisks mean that we encountered an out-of-memory error, and the values are extrapolated from the largest successful run (a few powers of two smaller).
For example, we expect sparse to outperform Fastfood if it didn’t run into memory issues.
Figure S4 compares the computational time for direct and subspace training (various projections) methods for each update.
Our subspace training is more computational expensive, because the subspace training method has to propagate the signals through two modules: the layers of neural networks, and the projection between two spaces.
The direct training only propagates signals in the layers of neural networks.
We have made efforts to reduce the extra computational cost.
For example, the sparse projection less than doubles the time cost for a large range of subspace dimensions.
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S A DDITIONAL CIFAR-10 R ESULTS
FC networks We consider the CIFAR-10 dataset and test the same set of FC and LeNet architectures as on MNIST.
For FC networks, d int90 values for all 20 networks are shown in S13 (a) plotted against the native dimension D of each network; D changes by a factor of 12.16 between the smallest and largest networks, but d int90 changes over this range by a factor of 5.0.
However, much of this change is due to change of baseline performance.
In S13 (b), we instead compute the intrinsic dimension with respect to a global baseline: 50% validation accuracy.
d int90 changes over this range by a factor of 1.52.
This indicates that various FC networks share similar intrinsic dimension ( d int90 = 5000 ∼ 8000 ) to achieve the same level of task performance.
For LeNet ( D = 62 , 006 ), the validation accuracy vs. subspace dimension d is shown in Fig.
S14 (a), the corresponding d int90 = 2900 .
It yields a compression rate of 5%, which is 10 times larger than LeNet on MNIST.
It shows that CIFAR-10 images are signiﬁcantly more difﬁcult to be correctly classiﬁed
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than MNIST.
In another word, CIFAR-10 is a harder problem than MNIST, especially given the fact that the notion of “problem solved” (baseline performance) is deﬁned as 99% accuracy on MNIST and 58% accuracy on CIFAR-10.
On the CIFAR-10 dataset, as d increases, subspace training tends to overﬁtting; we study the role of subspace training as a regularizer below.
ResNet vs. LeNet We test ResNets, compare to LeNet, and ﬁnd they make efﬁcient use of parameters.
We adopt the smallest 20-layer structure of ResNet with 280k parameters, and ﬁnd out in Fig.
S14 (b) that it reaches LeNet baseline with d int90 = 1000 ∼ 2000 (lower than the d int90 of
LeNet), while takes a larger d
int90 ( 20 , 000 ∼ 50 , 000 ) to reach reach its own, much higher baseline.
The role of regularizers Our subspace training can be considered as a regularization scheme, as it restricts the solution set.
We study and compare its effects with two traditional regularizers with an
FC network ( L =2 , W =200 ) on CIFAR
-10 dataset, including (cid:96) 2 penalty on the weights ( i.e., weight decay) and Dropout.
• (cid:96) 2 penalty Various amount of (cid:96) 2 penalty from { 10 − 2 , 10 − 3 , 5 × 10 − 4 , 10 − 4 , 10 − 5 , 0 } are considered.
The accuracy and negative log-likelihood (NLL) are reported in Figure S (a) (b), respectively.
As expected, larger amount of weight decay reduces the gap between training and testing performance for both direct and subspace training methods, and eventually closes the gap ( i.e., (cid:96) 2 penalty = 0.01).
Subspace training itself exhibits strong regularization ability, especially when d is small, at which the performance gap between training and testing is smaller.
• Dropout Various dropout rates from { 0 . 5 , 0 . 4 , 0 . 3 , 0 . 2 , 0 . 1 , 0 } are considered.
The accuracy and NLL are reported in Figure S16.
Larger dropout rates reduce the gap between training and testing performance for both direct and subspace training methods.
When observing testing NLL, subspace training tends to overﬁt the training dataset less.
• Subspace training as implicit regularization Subspace training method performs implicit regularization, as it restricts the solution set.
We visualized the testing NLL in Figure S17.
Subspace training method outperforms direct method when d is properly chosen (when (cid:96)penalty < 5 × 10 − 4 , or dropout rate < 0 . 1 ), suggesting the potential of this method as a better alternative to traditional regularizers.
When d is large, the method also overﬁts the training
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dataset.
Note that the these methods perform regularization in different ways: weight decay enforces the learned weights concentrating around zeros, while subspace training directly reduces the number of dimensions of the solution space.
S I MAGE N ET
To investigate even larger problems
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we attempted to measure d int90 for an ImageNet classiﬁcation network.
We use a relatively smaller network, SqueezeNet by Iandola et al.
(2016), with 1.24M parameters.
Larger networks suffered from memory issues.
A direct training produces Top-1 accuracy of 55.5%.
We vary intrinsic dimension from 50k, 100k, 200k, 500k, 800k, and record the validation accuracies as shown in Fig.
S18.
The training of each intrinsic dimension takes about 6 to 7 days, distributed across 4 GPUs.
Due to limited time, training on ImageNet has not yet produced a reliable estimate for d int90 except that it is over 500k.
S I NVESTIGATION OF C ONVOLUTIONAL N ETWORKS
Since the learned d int90 can be used as a robust measure to study the ﬁtness of neural network architectures for speciﬁc tasks, we further apply it to understand the contribution of each component in convolutional networks for image classiﬁcation task.
The convolutional network
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2 penalty and subspace training.
Weight decay interacts with d
int90 since it changes the objective landscapes through various loss functions
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V a li d a t i o n a cc u r a c y 0.
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is a special case of FC network in two aspects: local receptive ﬁelds and weight-tying.
Local receptive ﬁelds force each ﬁlter to “look” only at a small, localized region of the image or layer below.
Weight-tying enforces that each ﬁlter shares the same weights, which reduces the number of learnable parameters.
We performed control experiments to investigate the degree to which each component contributes.
Four variants of LeNet are considered:
• Standard LeNet 6 kernels ( 5 × 5 ) – max-pooling ( 2 × 2 ) – 16 kernels ( 5 × 5 ) – max-pooling ( 2 × 2 ) – 120 FC – 84 FC – 10 FC • Untied-LeNet
The same architecture with the standard LeNet is employed, except that weights are unshared, i.e., a different set of ﬁlters is applied at each different patch of the input.
For example in Keras, the LocallyConnected2D layer is used to replace the Conv2D layer.
• FCTied-LeNet
The same set of ﬁlters is applied at each different patch of the input.
we break local connections by applying ﬁlters to global patches of the input.
Assume the image size is H × H , the architecture is 6 kernels ( (2 H − 1) ×
(2 H − 1) ) – max-pooling ( 2 × 2 ) – 16 kernels ( ( H − 1) ×
( H − 1) ) – max-pooling ( 2 × 2 ) – 120 FC – 84 FC – 10 FC.
The padding type is same .
• FC-LeNet
Neither local connections or
tied weights is employed, we mimic LeNet with its FC implementation.
The same number of hidden units as the standard LeNet are used at each layer.
The results are shown in Fig.
S19 (a)(b).
We set a crossing-line accuracy ( i.e., threshold) for each task, and investigate d int90 needed to achieve it.
For MNIST and CIFAR-10, the threshold is 90% and 45%, respectively.
For the above LeNet variants, d int90 = 290 , 600 , 425 , 2000 on MNIST, and d int90 = 1000 , 2750 , 2500 , 35000 on CIFAR-10.
Experiments show both tied-weights and local connections are important to the model.
That tied-weights should matter seems sensible.
However, models with maximal convolutions (convolutions covering the whole image) may have had the same intrinsic dimension as smaller convolutions, but this turns out not to be the case.
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S S UMMARIZATION OF d int
We summarize d int90 of the objective landscape on all different problems and neural network architectures in Table S5 and Fig.
S20.
“SP” indicates shufﬂed pixel, and “PL” for shufﬂed label, and “FC-5” for a 5-layer FC.
d int90 indicates the minimum number of dimensions of trainable parameters required to properly solve the problem, and thus reﬂects the difﬁculty level of problems.
Further work could also identify better ways of creating subspaces for reparameterization: here we chose random linear subspaces, but one might carefully construct linear or non-linear subspaces to be even more efﬁcient, or pre-scale or otherwise transform the direct parameters before projecting to or from the subspace to make projections more likely to intersect solutions.
Finally, as the ﬁeld departs from single stack-of-layers image classiﬁcation models toward larger and more heterogeneous networks (Ren et al., 2015; Kaiser et al., 2017) 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.
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