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Title: Over-parameterization Improves Generalization in the XOR Detection Problem

Paper content:
Empirical evidence suggests that neural networks with ReLU activations generalize better with over-parameterization.
However, there is currently no theoretical analysis that explains this observation.
In this work, we study a simplified learning task with over-parameterized convolutional networks that empirically exhibits the same qualitative phenomenon.
For this setting, we provide a theoretical analysis of the optimization and generalization performance of gradient descent.
Specifically, we prove data-dependent sample complexity bounds which show that over-parameterization improves the generalization performance of gradient descent.
Most successful deep learning models use a number of parameters that is larger than the number of parameters that are needed to get zero-training error.
This is typically referred to as overparameterization.
Indeed, it can be argued that over-parameterization is one of the key techniques that has led to the remarkable success of neural networks.
However, there is still no theoretical account for its effectiveness.One very intriguing observation in this context is that over-parameterized networks with ReLU activations, which are trained with gradient based methods, often exhibit better generalization error than smaller networks BID11 Novak et al., 2018) .
This somewhat counterintuitive observation suggests that first-order methods which are trained on over-parameterized networks have an inductive bias towards solutions with better generalization performance.
Understanding this inductive bias is a necessary step towards a full understanding of neural networks in practice.Providing theoretical guarantees for this phenomenon is extremely challenging due to two main reasons.
First, to show a generalization gap, one needs to prove that large networks have better sample complexity than smaller ones.
However, current generalization bounds that are based on complexity measures do not offer such guarantees.
Second, analyzing the dynamics of first-order methods on networks with ReLU activations is a major challenge.
Indeed, there do not exist optimization guarantees even for simple learning tasks such as the classic XOR problem in two dimensions.
1 To advance this issue, we focus on a particular learning setting that captures key properties of the over-parameterization phenomenon.
We consider a high-dimensional extension of the XOR problem, which we refer to as the "XOR Detection problem (XORD)".
The XORD is a pattern recognition task where the goal is to learn a function which classifies binary vectors according to whether they contain a two-dimensional binary XOR pattern (i.e., (1, 1) or (−1, −1)).
This problem contains the classic XOR problem as a special case when the vectors are two dimensional.
We consider learning this function with gradient descent trained on an over-parameterized convolutional neural network (i.e., with multiple channels) with ReLU activations and three layers: convolutional, max pooling and fully connected.
As can be seen in FIG0 , over-parameterization improves generalization in this problem as well.
Therefore it serves as a good test-bed for understanding the role of over-parameterization.
1 We are referring to the problem of learning the XOR function given four two-dimensional points with binary entries, using a moderate size one-hidden layer neural network (e.g., with 50 hidden neurons).
Note that there are no optimization guarantees for this setting.
Variants of XOR have been studied in BID10 ; Sprinkhuizen-Kuyper & Boers (1998) but these works only analyzed the optimization landscape and did not provide guarantees for optimization methods.
We provide guarantees for this problem in Sec. 9.
3).
The figure shows the test error obtained for different number of channels k.
The blue curve shows test error when restricting to cases where training error was zero.
It can be seen that increasing the number of channels improves the generalization performance.
Experimental details are provided in Section 8.2.1..
In this work we provide an analysis of optimization and generalization of gradient descent for XORD.
We show that for various input distributions, ranges of accuracy and confidence parameters, sufficiently over-parameterized networks have better sample complexity than a small network which can realize the ground truth classifier.
To the best of our knowledge, this is the first example which shows that over-paramaterization can provably improve generalization for a neural network with ReLU activations.Our analysis provides a clear distinction between the inductive bias of gradient descent for overparameterized and small networks.
It reveals that over-parameterized networks are biased towards global minima that detect more patterns in the data than global minima found by small networks.
2 Thus, even though both networks succeed in optimization, the larger one has better generalization performance.
We provide experiments which show that the same phenomenon occurs in a more general setting with more patterns in the data and non-binary input.
We further show that our analysis can predict the behavior of over-parameterized networks trained on MNIST and guide a compression scheme for over-parameterized networks with a mild loss in accuracy (Sec. 6).
In this paper we consider a simplified learning task on binary vectors and show that overparameterization can provably improve generalization performance of a 3-layer convolutional network trained with gradient descent.
Our analysis reveals that in the XORD problem overparameterized networks are biased towards global minima which detect more relevant patterns in the data.
While we prove this only for the XORD problem and under the assumption that the training set contains diverse points, our experiments clearly show that a similar phenomenon occurs in other settings as well.
We show that this is the case for XORD with non-diverse points FIG0 ) and in the more general OBD problem which contains 60 patterns in the data and is not restricted to binary inputs FIG1 .
Furthermore, our experiments on MNIST hint that this is the case in MNIST as well FIG5 .By
clustering the detected patterns of the large network we could achieve better accuracy with a small network. This
suggests that the larger network detects more patterns with gradient descent even though its effective size is close to that of a small network.We believe that these insights and our detailed analysis can guide future work for showing similar results in more complex tasks and provide better understanding of this phenomenon. It would
also be interesting to further study the implications of such results on model compression and on improving training algorithms. Behnam Neyshabur
, Zhiyuan Li, Srinadh Bhojanapalli, Yann LeCun, and Nathan Srebro. We tested the generalization
performance in the setup of Section3. We considered networks with
number of channels 4,6,8,20,50,100 and 200 . The distribution in this setting
has p + = 0.5 and p − = 0.9 and the training sets are of size 12 (6 positive, 6 negative). Note that in this case the training
set contains non-diverse points with high probability. The ground truth network can be realized
by a network with 4 channels. For each number of channels we trained a
convolutional network 100 times and averaged the results. In each run we sampled a new training set
and new initialization of the weights according to a gaussian distribution with mean 0 and standard deviation 0.00001. For each number of channels c, we ran gradient
descent with learning rate 0.04 c and stopped it if it did not improve the cost for 20 consecutive iterations or if it reached 30000 iterations. The last iteration was taken for the calculations
. We plot both average test error over all 100 runs
and average test error only over the runs that ended at 0% train error. In this case, for each number of channels 4, 6, 8
, 20, 50, 100 ,200 the number of runs in which gradient descent converged to a 0% train error solution is 62, 79, 94, 100, 100, 100, 100, respectively. Figure 5 shows that setting γ = 5 gives better performance
than setting γ = 1 in the XORD problem. The setting is similar to the setting of Section 8.2.1. Each
point is an average test error of 100 runs. . Because the
result is a lower bound, it is desirable to understand
the behaviour of gradient descent for values outside these ranges. In Figure 6 we empirically show that for values outside these ranges, there is a generalization gap between gradient descent for k = 2 and gradient descent for larger k.

assistant: text: We show in a simplified learning task that over-parameterization improves generalization of a convnet that is trained with gradient descent.