E
XPONENTIALLY
VANISHING SUB - OPTIMAL LOCAL MINIMA IN MULTILAYER NEURAL NETWORKS
Anonymous authors
Paper under double-blind review
A BSTRACT
Background: Statistical mechanics results (Dauphin et al.
; Choromanska et al.
2015)) suggest that local minima with high error are exponentially rare in high dimensions.
However, to prove low error guarantees for Multilayer Neural
Networks (MNNs), previous works so far required either a heavily modiﬁed MNN model or training method, strong assumptions on the labels ( e g. , “near” linear separability) or an unrealistically wide hidden layer with Ω ( N ) units.
Results: We examine a MNN with one hidden layer of piecewise linear units, a single output and a quadratic loss.
We prove that, with high probability in the limit of N → ∞ datapoints, the volume of differentiable regions of the empiric loss containing sub-optimal differentiable local minima is exponentially vanishing in comparison with the same volume of global minima, given standard normal input of dimension d 0 = ˜Ω (cid:16) √ N (cid:17) , and a more realistic number of d 1 = ˜Ω ( N/d 0 ) hidden units.
We demonstrate our results numerically: for example, 0% binary classiﬁcation training error on CIFAR with only N/d 0 ≈ 16 hidden neurons.
I NTRODUCTION
Motivation.
Multilayer Neural Networks (MNNs), trained with simple variants of stochastic gradient descent (SGD), have achieved state-of-the-art performances in many areas of machine learning (LeCun et al., 2015).
However, theoretical explanations seem to lag far behind this empirical success (though many hardness results exist , e.g. , (Síma, 2002; Shamir, 2016)).
For example, as a common rule-of-the-thumb, a MNN should have at least as many parameters as training samples.
However, it is unclear why such over-parameterized MNNs often exhibit remarkably small generalization error ( i.e. , difference between “training error” and “test error”), even without explicit regularization (Zhang et al., 2017a).
Moreover, it has long been a mystery why MNNs often achieve low training error (Dauphin et al., 2014).
SGD is only guaranteed to converge to critical points in which the gradient of the expected loss is zero (Bottou, 1998), and, speciﬁcally, to local minima (Pemantle, 1990) (this is true also for regular gradient descent (Lee et al., 2016)).
Since loss functions parameterized by MNN weights are non-convex, it is unclear why does SGD often work well – rather than converging to sub-optimal local minima with high training error, which are known to exist (Fukumizu & Amari, 2000; Swirszcz et al., 2016).
Understanding this behavior is especially relevant in important cases where SGD does get stuck (He et al., 2016) – where training error may be a bottleneck in further improving performance.
Ideally, we would like to quantify the probability to converge to a local minimum as a function of the error at this minimum, where the probability is taken with the respect to the randomness of the initialization of the weights, the data and SGD.
Speciﬁcally, we would like to know, under which conditions this probability is very small if the error is high, as was observed empirically ( e.g. , (Dauphin et al., 2014; Goodfellow et al., 2015)).
However, this seems to be a daunting task for realistic MNNs, since it requires a characterization of the sizes and distributions of the basins of attraction for all local minima.
Previous works (Dauphin et al., 2014; Choromanska et al., 2015), based on statistical physics analogies, suggested a simpler property of MNNs: that with high probability, local minima with high error diminish exponentially with the number of parameters.
Though proving such a geometric property with realistic assumptions would not guarantee convergence to global minima, it appears to be a necessary ﬁrst step in this direction (see discussion on section 6).
It was therefore pointed out as an open problem at the Conference of Learning Theory (COLT) 2015.
However, one has to be careful and use realistic MMN architectures, or this problem becomes “too easy”.
For example, one can easily achieve zero training error (Nilsson, 1965; Baum, 1988) – if the MNN’s last hidden layer has more neurons than training samples.
Such extremely wide MNNs are easy to optimize (Yu, 1992; Huang et al., 2006; Livni et al., 2014; Haeffele & Vidal, 2015; Shen, 2016;
Nguyen & Hein, 2017).
In this case, the hidden layer becomes linearly separable in classiﬁcation tasks, with high probability over the random initialization of the weights.
Thus, by training the last layer we get to a global minimum (zero training error).
However, such extremely wide layers are not very useful, since they result in a huge number of weights, and serious overﬁtting issues.
Also, training only the last layer seems to take little advantage of the inherently non-linear nature of MNNs.
Therefore, in this paper we are interested to understand the properties of local and global minima, but
at a more practical number of parameters – and when at least two weight layers are trained.
For example, Alexnet (Krizhevsky, 2014) is trained using about 1.2 million ImageNet examples, and has about 60 million parameters – 16 million of these in the two last weight layers.
Suppose we now train the last two weight layers in such an over-parameterized MNN.
When do the sub-optimal local minima become exponentially rare in comparison to the global minima?
Main contributions.
We focus on MNNs with a single hidden layer and piecewise linear units, optimized using the Mean Square Error (MSE) in a supervised binary classiﬁcation task (Section 2).
We deﬁne N as the number of training samples, d l as the width of the l -th activation layer, and g ( x ) ˙ <
h ( x ) as an asymptotic inequality in the leading order (formally: lim x →∞ log g ( x ) log h ( x ) < 1 ).
We examine Differentiable Local Minima (DLMs) of the MSE: sub-optimal DLMs where at least a fraction of (cid:15) > 0 of the training samples are classiﬁed incorrectly, and global minima where all samples are classiﬁed correctly.
Our main result, Theorem 10, states that, with high probability, the total volume of the differentiable regions of the MSE containing sub-optimal DLMs is exponentially vanishing in comparison to the same volume of global minima, given that:
Assumption 1.
The datapoints (MNN inputs) are sampled from a standard normal distribution.
Assumption 2.
N → ∞ , d 0 ( N ) and d 1 ( N ) increase with N , while (cid:15) ∈ (0 , 1) is a constant 1 .
Assumption 3.
The input dimension scales as √ N ˙ <d 0 ˙ ≤ N .
Assumption 4.
The hidden layer width scales as
[Equation]
Importantly, we use a standard, unmodiﬁed, MNN model, and make no assumptions on the target function.
Moreover, as the number of parameters in the MNN is approximately d 0 d 1 , we require only “asymptotically mild” over-parameterization: d 0 d 1 ˙ >N log 4 N from eq.
For example, if d 0 ∝ N , we only require d 1 ˙ > log 4 N neurons.
This improves over previously known results (Yu, 1992; Huang et al., 2006;
Livni et al., 2014; Shen, 2016; Nguyen & Hein, 2017) – which require an extremely wide hidden layer with d 1 ≥ N neurons (and thus Nd 0 parameters) to remove sub-optimal local minima with high probability.
In section 5 we validate our results numerically.
We show that indeed the training error becomes low when the number of parameters is close to N .
For example, with binary classiﬁcation on CIFAR and ImageNet, with only 16 and 105 hidden neurons (about N/d 0 ), respectively, we obtain less then 0 .
1% training error.
Additionally, we ﬁnd that convergence to non-differentiable critical points does not appear to be very common.
Lastly, in section 6 we discuss our results might be extended, such as how to apply them to “mildly” non-differentiable critical points.
Plausibility of assumptions.
Assumption 1 is common in this type of analysis (Andoni et al., 2014;
Choromanska et al., 2015; Xie et al., 2016; Tian, 2017; Brutzkus & Globerson, 2017).
At ﬁrst it may appear rather unrealistic, especially since the inputs are correlated in typical datasets.
However, this no-correlation part of the assumption may seem more justiﬁed if we recall that datasets are many times whitened before being used as inputs.
Alternatively, if, as in our motivating question, we consider the input to the our simple MNN to be the output of the previous layers of a deep MNN with ﬁxed random weights, this also tends to de-correlate inputs (Poole et al., 2016, Figure 3).
The remaining part of assumption 1, that the distribution is normal, is indeed strong, but might be relaxed in the future, e.g. using central limit theorem type arguments.
In assumption 2 we use this asymptotic limit to simplify our proofs and ﬁnal results.
Multiplicative constants and ﬁnite (yet large) N results can be found by inspection of the proofs.
We assume a constant error (cid:15) since typically the limit (cid:15) → 0 is avoided to prevent overﬁtting.
In assumption 3, for simplicity we have d 0 ˙ ≤ N , since in the case d 0 ≥ N the input is generically linearly separable, and sub-optimal local minima are not a problem (Gori & Tesi, 1992; Safran &
Shamir, 2016).
Additionally, we have √ N ˙ <d 0 , which seems very reasonable, since for example, d 0 /N
≈ 0 .
016 , 0 . 061 and 0 .
055 MNIST, CIFAR and ImageNet, respectively.
In assumption 4, for simplicity we have d 1 ˙ <N , since, as mentioned earlier, if d 1 ≥ N the hidden layer is linearly separable with high probability, which removes sub-optimal local minima.
The other bound N log 4 N ˙ <
d 0 d 1 is our main innovation – a large over-parameterization which is nevertheless asymptotically mild and improves previous results.
Previous work.
So far, general low (training or test) error guarantees for MNNs could not be found – unless the underlying model (MNN) or learning method (SGD or its variants) have been signiﬁcantly modiﬁed.
For example, (Dauphin et al., 2014) made an analogy with high-dimensional random Gaussian functions, local minima with high error are exponentially rare in high dimensions; (Choromanska et al., 2015; Kawaguchi, 2016) replaced the units (activation functions) with independent random variables; (Pennington & Bahri, 2017) replaces the weights and error residuals with independent random variables; (Baldi, 1989; Saxe et al., 2014; Hardt & Ma, 2017; Lu & Kawaguchi, 2017; Zhou & Feng, 2017) used linear units; (Zhang et al., 2017b) used unconventional units ( e.g. , polynomials) and very large hidden layers ( d 1 = poly ( d 0 ) , typically (cid:29) N ); (Brutzkus & Globerson, 2017; Du et al., 2017; Shalev-Shwartz et al., 2017) used a modiﬁed convnet model with less then dparameters (therefore, not a universal approximator (Cybenko, 1989; Hornik, 1991))
; (Tian, 2017;
Soltanolkotabi et al., 2017; Li & Yuan, 2017) assume the weights are initialized very close to those of the teacher generating the labels; and (Janzamin et al., 2015; Zhong et al., 2017) use a non-standard tensor method during training.
Such approaches fall short of explaining the widespread success of standard MNN models and training practices.
Other works placed strong assumptions on the target functions.
For example, to prove convergence of the training error near the global minimum, (Gori & Tesi, 1992) assumed linearly separable datasets, while (Safran & Shamir, 2016) assumed strong clustering of the targets (“near” linear-separability).
Also, (Andoni et al., 2014) showed a p -degree polynomial is learnable by a MNN, if the hidden layer is very large ( d 1 = Ω (cid:16) d 6 p 0 (cid:17) , typically (cid:29) N ) so learning the last weight layer is sufﬁcient.
However, these are not the typical regimes in which MNNs are required or used.
In contrast, we make no assumption on the target function.
Other closely related results (Soudry & Carmon, 2016; Xie et al., 2016) also used unrealistic assumptions, are discussed in section 6, in regards to the details of our main results.
Therefore, in contrast to previous works, the assumptions in this paper are applicable in some situations ( e.g. , Gaussian input) where a MNN trained using SGD might be used and be useful ( e.g. , have a lower test error then a linear classier).
P RELIMINARIES AND NOTATION
Model.
We examine a Multilayer Neural Network (MNN) with a single hidden layer and a scalar output.
The MNN is trained on a ﬁnite training set of N datapoints (features) X (cid:44) (cid:2)
x (1) , . . .
, x ( N ) (cid:3) ∈ R d 0 × N with their target labels y
(cid:44) (cid:2) y (1) , . . .
, y ( N ) (cid:3) (cid:62) ∈ { 0 , 1 } N – each datapoint-label pair (cid:0) x ( n ) , y ( n ) (cid:1) is independently sampled from some joint distribution P X,Y .
We deﬁne W =
[ w 1 , . .
. , w d 1 ] (cid:62) ∈ R
d 1 × d 0 and z ∈ R d 1 as the ﬁrst and second weight layers (bias terms are ignored for simplicity), respectively, and f ( · ) as the common leaky rectiﬁer linear unit (LReLU (Maas et al., 2013))
[Equation]
for some ρ (cid:54) = 1 (so the MNN is non-linear) , where both functions f and a operate component-wise (e.g. , for any matrix M : ( f ( M )) ij =
f ( M ij ) ).
Thus, the output of the MNN on the entire dataset can be written as
[Equation]
We use the mean square error (MSE) loss for optimization
[Equation]
where (cid:107)·(cid:107) is the standard euclidean norm.
Also, we measure the empiric performance as the fraction of samples that are classiﬁed correctly using a decision threshold at y = 0 .
5 , and denote this as the mean classiﬁcation error, or MCE 2 .
Note that the variables e , MSE , MCE and other related variables ( e.g. , their derivatives) all depend on W , z , X , y and ρ , but we keep this dependency implicit, to avoid cumbersome notation.
Additional Notation.
We deﬁne g ( x ) ˙
<
h
( x ) if and only if lim x →∞ log g ( x ) log h ( x ) < 1 (and similarly ˙ ≤ and ˙= ).
We denote “ M ∼ N ” when M is a matrix with entries drawn independently from a standard normal distribution ( i.e. , ∀ i, j : M ij ∼ N (0 , 1) ).
The Khatari-rao product ( cf.
(Allman et al., 2009)) of two matrices, A = (cid:2) a (1) , . .
, a ( N ) (cid:3) ∈
R
d 1 × N and X = (cid:2) x (1) , . . .
, x ( N ) (cid:3) ∈ R d 0 × N is deﬁned as A
[Equation]
X (cid:44) (cid:104)
[Equation]
[Equation]
where a ⊗ x = (cid:2) a 1 x (cid:62) , . .
. , a d 1 x (cid:62) (cid:3) (cid:62) is the Kronecker product.
B ASIC P ROPERTIES OF D IFFERENTIABLE L OCAL MINIMA
MNNs are typically trained by minimizing the loss over the training set, using Stochastic Gradient
Descent (SGD), or one of its variants ( e.g. , Adam (Kingma & Ba, 2014)).
Under rather mild conditions (Pemantle, 1990; Bottou, 1998), SGD asymptotically converges to local minima of the loss.
For simplicity, we focus on differentiable local minima (DLMs) of the MSE (eq.
In section 4 we will show that sub-optimal DLMs are exponentially rare in comparison to global minima.
Non-differentiable critical points, in which some neural input (pre-activation) is exactly zero, are shown to be numerically rare in section 5, and are left for future work, as discussed in section 6.
Before we can provide our results, in this section we formalize a few necessary notions.
For example, one has to deﬁne how to measure the amount of DLMs in the over-parameterized regime: there is an inﬁnite number of such points, but they typically occupy only a measure zero volume in the weight space.
Fortunately, using the differentiable regions of the MSE (deﬁnition 1), the DLMs can partitioned to a ﬁnite number of equivalence groups, so all DLMs in each region have the same error (Lemma 2).
Therefore, we use the volume of these regions (deﬁnition 3) as the relevant measure in our theorems.
Differentiable regions of the MSE.
The MSE is a piecewise differentiable function of W , with at most 2 d 1 N differentiable regions, deﬁned as follows.
Deﬁnition 1.
For any A ∈ { ρ, 1 } d 1 × N we deﬁne the corresponding differentiable region
[Equation]
Also, any DLM ( W , z ) , for which W ∈ D
A ( X ) is denoted as “in D A ( X ) ”.
Note that D A ( X ) is an open set, since a (0) is undeﬁned (from eq. 2.1).
Clearly, for all W ∈ D
A ( X ) the MSE is differentiable, so any local minimum can be non-differentiable only if it is not in any differentiable region.
Also, all DLMs in a differentiable region are equivalent, as we prove on appendix section 7:
Lemma 2.
At all DLMs in D A ( X ) the residual error e is identical, and furthermore
[Equation]
The proof is directly derived from the ﬁrst order necessary condition of DLMs ( ∇ MSE = 0 ) and their stability.
Note that Lemma 2 constrains the residual error e in the over-parameterized regime:
d 0 d 1 ≥ N .
In this case eq.
(3.2) implies e = 0 , if rank ( A ◦ X )
= N .
Therefore, we must have rank ( A ◦ X ) < N for sub-optimal DLMs to exist.
Later, we use similar rank-based constraints to bound the volume of differentiable regions which contain DLMs with high error.
Next, we deﬁne this volume formally.
Angular Volume.
From its deﬁnition (eq. (3.1)) each region D A ( X ) has an inﬁnite volume in R d 1 × d 0 : if we multiply a row of W by a positive scalar, we remain in the same region.
Only by rotating the rows of W can we move between regions.
We measure this “angular volume” of a region in a probabilistic way: we randomly sample the rows of W from an isotropic distribution, e.g. , standard Gaussian: W ∼ N , and measure the probability to fall in D A ( X ) , arriving to the following
Deﬁnition 3.
For any region R ⊂ R d 1 × d 0 .
The angular volume of R is
[Equation]
M AIN R ESULTS
Some of the DLMs are global minima, in which e =
and so,
MCE = MSE = 0 , while other DLMs are sub-optimal local minima in which MCE >(cid:15) > 0 .
We would like to compare the angular volume (deﬁnition 3) corresponding to both types of DLMs.
Thus, we make the following deﬁnitions.
Deﬁnition 4.
We deﬁne 3 L (cid:15) ⊂
R d 1 × d 0 as the union of differentiable regions containing sub-optimal
DLMs with MCE > (cid:15) , and G ⊂ R d 1 × d 0 as the union of differentiable regions containing global minima with MCE = 0 .
Deﬁnition 5.
We deﬁne the constant γ (cid:15) as γ (cid:15) (cid:44) 0 .
23 max [lim N →∞ ( d 0 ( N ) /N ) , (cid:15) ] 3 / 4 if ρ (cid:54) = { 0 , 1 } , and γ (cid:15) (cid:44) 0 .
23 (cid:15) 3 / 4 if ρ = 0 .
In this section, we use assumptions 1-4 (stated in section 1) to bound the angular volume of the region
L (cid:15) encapsulating all sub-optimal DLMs, the region G , encapsulating all global minima, and the ratio between the two.
Angular volume of sub-optimal DLMs.
First, in appendix section 8 we prove the following upper bound in expectation
Theorem 6.
Given assumptions 1-4, the expected angular volume of sub-optimal DLMs, with
MCE > (cid:15) > 0 , is exponentially vanishing in N as
[Equation]
and, using Markov inequality, its immediate probabilistic corollary
Corollary 7.
Given assumptions 1-4, for any δ > 0 (possibly a vanishing function of N ), we have, with probability 1 − δ , that the angular volume of sub-optimal DLMs, with MCE > (cid:15) > 0 , is exponentially vanishing in N
[Equation] [Equation]
[Equation] [Equation]
[Equation] [Equation]
Proof idea of Theorem 6: we ﬁrst show that in differentiable regions with MCE > (cid:15) > 0 , the condition in Lemma 2, ( A ◦ X ) e = 0 , implies that A = a ( WX ) must have a low rank.
Then, we show that, when X ∼ N and W ∼ N , the matrix A = a ( WX ) has a low rank with exponentially low probability.
Combining both facts, we obtain the bound.
Existence of global minima.
Next, to compare the volume of sub-optimal DLMs with that of global minima, in appendix section 9 we show ﬁrst that, generically, global minima do exist (using a variant of the proof of (Baum, 1988, Theorem 1)):
Theorem 8.
For any y ∈ { 0 , 1 } N and X ∈ R d 0 × N almost everywhere 4 we ﬁnd matrices W ∗ ∈
R d ∗ 1 × d 0 and z ∗ ∈ R d ∗ 1 , such that y =
f ( W ∗ X ) (cid:62) z ∗ , where d ∗ 1 (cid:44) 4 (cid:100) N/ (2 d 0 − 2) (cid:101) and ∀ i, n : w (cid:62) i x ( n ) (cid:54) = 0 .
Therefore, every MNN with d 1 ≥ d ∗ 1 has a DLM which achieves zero error e = 0 .
Recently (Zhang et al., 2017a, Theorem 1) similarly proved that a 2-layer MNN with approximately 2 N parameters can achieve zero error.
However, that proof required N neurons (similarly to (Nilsson, 1965; Baum, 1988; Yu, 1992; Huang et al., 2006; Livni et al., 2014; Shen, 2016)), while Theorem 8 here requires much less: approximately d ∗ 1 ≈ 2 N/d 0 .
Also, (Hardt & Ma, 2017, Theorem 3.2) showed a deep residual network with N log N parameters can achieve zero error.
In contrast, here we require just one hidden layer with 2 N parameters.
Note the construction in Theorem 8 here achieves zero training error by overﬁtting to the data realization, so it is not expected to be a “good” solution in terms of generalization.
To get good generalization, one needs to add additional assumptions on the data ( X and y ).
Such a possible (common yet insufﬁcient for MNNs) assumption is that the problem is “realizable”, i.e. , there exist a small “solution MNN”, which achieves low error.
For example, in the zero error case:
Assumption 5.
(Optional)
The labels are generated by some teacher y =
f ( W ∗ X )
(cid:62) z ∗ with weight matrices W ∗
∈
R
d ∗ 1 × d 0 and z ∗ ∈ R d ∗ 1 independent of X , for some d ∗ 1 ˙ <N/d 0 .
This assumption is not required for our main result (Theorem 10) – it is merely helpful in improving the following lower bound on V ( G ) .
Angular volume of global minima.
We prove in appendix section 10:
Theorem 9.
Given assumptions 1-3, we set δ ˙= (cid:113) 8 π d − 1 /
[Equation]
and d
[Equation]
=
[Equation]
N
[Equation]
d
[Equation]
, or if assumption 5 holds, we set d ∗ 1 as in this assumption.
Then, with probability 1 − δ , the angular volume of global minima is lower bounded as,
[Equation]
Proof idea: First, we lower bound V ( G ) with the angular volume of a single differentiable region of one global minimum ( W ∗ , z ∗ ) – either from Theorem 8, or from assumption 5.
Then we show that this angular volume is lower bounded when W ∼ N , given a certain angular margin between the datapoints in X and the rows of W ∗ .
We then calculate the probability of obtaining this margin when X ∼ N .
Combining both results, we obtain the ﬁnal bound.
Main result: angular volume ratio.
Finally, combining Theorems 6 and 9 it is straightforward to prove our main result in this paper, as we do in appendix section 11:
Theorem 10.
Given assumptions 1-3, we set δ .
= (cid:113)
π d − 1 /
[Equation]
0 /N .
Then, with probability 1 − δ , the angular volume of sub-optimal DLMs, with MCE > (cid:15) > 0 , is exponentially vanishing in N, in comparison to the angular volume of global minima with MCE =
[Equation]
N UMERICAL E XPERIMENTS
Theorem 10 implies that, with “asymptotically mild” over-parameterization ( i.e. in which #parameters = ˜Ω ( N ) ), differentiable regions in weight space containing sub-optimal DLMs (with high MCE) are
[Table]
ImageNet (downsampled to 64 × 64 )
[Table]
exponentially small in comparison with the same regions for global minima.
Since these results are asymptotic in N → ∞ , in this section we examine it numerically for a ﬁnite number of samples and parameters.
We perform experiments on random data, MNIST, CIFAR10 and ImageNetILSVRC2012.
In each experiment, we used ReLU activations ( ρ = 0 ), a binary classiﬁcation target (we divided the original classes to two groups), MSE loss for optimization (eq.
(2.3)), and MCE to determine classiﬁcation error.
Additional implementation details are given in appendix part III.
First, on the small synthetic Gaussian random data (matching our assumptions) we perform a scan on various networks and dataset sizes.
With either one or two hidden layers (Figure 5.1) , the error goes to zero when the number of non-redundant parameters (approximately d 0 d 1 ) is greater than the number of samples, as suggested by our asymptotic results.
Second, on the non-syntehtic datasets, MNIST, CIFAR and ImageNet (In ImageNet we downsampled the images to size 64 × 64 , to allow input whitening)
we only perform a simulation with a single 1-hidden layer MNN for which #parameters ≈ N , and again ﬁnd (Table 1) that the ﬁnal error is zero (for MNIST and CIFAR) or very low (ImageNet).
Lastly, in Figure 5.2 we ﬁnd that, on the Gaussian dataset, the inputs to the hidden neurons converge to a distinctly non-zero value.
This indicates we converged to differentiable critical points – since nondifferentiable critical points must have zero neural inputs.
Note that occasionally, during optimization, we could ﬁnd some neural inputs with very low values near numerical precision level, so convergence to non-differentiable minima may be possible.
However, as explained in the next section, as long as the number of neural inputs equal to zero are not too large, our bounds also hold for these minima.
D ISCUSSION
In this paper we examine Differentiable Local Minima (DLMs) of the empiric loss of Multilayer
Neural Networks (MNNs) with one hidden layer, scalar output, and LReLU nonlinearities (section 2).
We prove (Theorem 10) that with high probability the angular volume (deﬁnition 3) of sub-optimal
DLMs is exponentially vanishing in comparison to the angular volume of global minima (deﬁnition 4), under assumptions 1-4.
This results from an upper bound on sub-optimal DLMs (Theorem 6) and a lower bound on global minima (Theorem 9).
Convergence of SGD to DLMs.
These results suggest a mechanism through which low training error is obtained in such MNNs.
However, they do not guarantee it.
One issue is that sub-optimal DLMs may have exponentially large basins of attraction.
We see two possible paths that might address this issue in future work, using additional assumptions on y .
One approach is to show that, with high probability, no sub optimal DLM falls within the vanishingly small differentiable regions we bounded in Theorem 6.
Another approach would be to bound the size of these basins of attraction, by showing that sufﬁciently large of number of differentiable regions near the DLM are also vanishingly small (other methods might also help here (Freeman & Bruna, 2016)).
Another issue is that
SGD might get stuck near differentiable saddle points, if their Hessian does not have strictly negative eigenvalues ( i.e. , the strict saddle property (Sun et al., 2015)).
It should be straightforward to show that such points also have exponentially vanishing angular volume, similar to sub-optimal DLMs.
Lastly, SGD might also converge to non-differentiable critical points, which we discuss next.
Non-differentiable critical points.
The proof of Theorem 6 stems from a ﬁrst order necessary condition (Lemma 2): ( A ◦ X ) e = 0 , which is true for any DLM.
However, non-differentiable critical points, in which some neural inputs are exactly zero, may also exist (though, numerically, they don’t seem very common – see Figure 5.2).
In this case, to derive a similar bound, we can replace the condition with P ( A ◦ X ) e = 0 , where P is a projection matrix to the subspace orthogonal to the non-differentiable directions.
As long as there are not too many zero neural inputs, we should be able to obtain similar results.
For example, if only a constant ratio r of the neural inputs are zero, we can simply choose P to remove all rows of ( A ◦ X ) corresponding to those neurons, and proceed with exactly the same proof as before, with d 1 replaced with (1 − r )
d 1 .
It remains a theoretical challenge to ﬁnd reasonable assumptions under which the number of non-differentiable directions ( i.e. , zero neural inputs) does not become too large.
Related results.
Two works have also derived related results using the ( A ◦ X ) e = 0 condition from Lemma 2.
In (Soudry & Carmon, 2016), it was noticed that an inﬁnitesimal perturbation of A makes the matrix A ◦ X full rank with probability 1 (Allman et al., 2009, Lemma 13) – which entails that e = 0 at all DLMs.
Though a simple and intuitive approach, such an inﬁnitesimal perturbation is problematic: from continuity, it cannot change the original MSE at sub-optimal DLMs – unless the weights go to inﬁnity, or the DLM becomes non-differentiable – which are both undesirable results.
An extension of this analysis was also done to constrain e using the singular values of A ◦ X (Xie et al., 2016), deriving bounds that are easier to combine with generalization bounds.
Though a promising approach, the size of the sub-optimal regions (where the error is high) does not vanish exponentially in the derived bounds.
More importantly, these bounds require assumptions on the activation kernel spectrum γ m , which do not appear to hold in practice ( e.g. , (Xie et al., 2016, Theorems 1,3) require mγ m (cid:29) 1 to hold with high probability, while mγ m < 10 − 2 in (Xie et al., 2016, Figure 1)).
Modiﬁcations and extensions.
There are many relatively simple extensions of these results: the
Gaussian assumption could be relaxed to other near-isotropic distributions ( e.g. , sparse-land model, (Elad, 2010, Section 9.2)) and other convex loss functions are possible instead of the quadratic loss.
More challenging directions are extending our results to MNNs with multi-output and multiple hidden layers, or combining our training error results with novel generalization bounds which might be better suited for MNNs ( e.g. , (Feng et al., 2016; Sokolic et al., 2016; Dziugaite & Roy, 2017)) than previous approaches (Zhang et al., 2017a).
Supplementary information - Appendix
The appendix is divided into three parts.
In part I we prove all the main theorems mentioned in the paper.
Some of these rely on other technical results, which we prove later in part II.
Lastly, in part III we give additional numerical details and results.
First, however, we deﬁne additional notation (some already deﬁned in the main paper) and mention some known results, which we will use in our proofs.
E XTENDED P RELIMINARIES
• The indicator function I ( A ) (cid:44) (cid:26) 1 , if A 0 , else , for any event
A .
• Kronecker’s delta δ ij (cid:44) I
( i = j ) .
• The Matrix I d as the identity matrix in R d × d , and I d × k is the relevant R d × k upper left sub-matrix of the identity matrix.
• [ L ] (cid:44) { 1 , 2 , . . .
, L } • The vector m n as the n ’th column of a matrix M , unless deﬁned otherwise (then m n will be a row of M ).
• M > 0 implies that ∀ i, j : M ij > 0 .
• M S is the matrix composed of the columns of M that are in the index set S .
• A property holds “ M -almost everywhere” (a.e. for short), if the set of entries of M for which the property does not hold has zero measure (Lebesgue).
• (cid:107) v (cid:107)
= (cid:80) di
=1 I ( v i > 0) is the L 0 “norm” that counts the number of non-zero values in v ∈ R d .
• If x ∼ N ( µ , Σ ) the x is random Gaussian vector.
• φ ( x ) (cid:44) 1 √ 2 π exp (cid:0) − 12 x 2 (cid:1) as the univariate Gaussian probability density function.
Φ ( x ) (cid:44) (cid:82) x
−∞ φ ( u ) du as the Gaussian cumulative distribution function.
• B ( x, y ) as the
beta function.
Lastly, we recall the well known Markov Inequality:
Fact 11.
(Markov Inequality)
For any random variable X ≥ 0 , we have ∀ η >
P ( X ≥ η
[Equation]
≤ E X η .
Part I
Proofs of the main results F IRST ORDER CONDITION : P ROOF OF L EMMA
Lemma 12.
(Lemma 2 restated)
At all DLMs in D A ( X ) the residual error e is identical, and furthermore ( A ◦ X ) e = 0 .
Proof.
Let W =
[ w 1 , . . . ,
w d 1 ] (cid:62)
∈
D A ( X ) , G (cid:44)
A ◦ X ∈ R d 0 d 1 × N , ˜ W = diag ( z ) W =
[ ˜ w 1 , . .
˜ w d 1 ] (cid:62) and ˜ w (cid:44) vec (cid:16) ˜ W (cid:62) (cid:17)
∈ R d 0 d 1 , where diag ( v ) is the diagonal matrix with v in its diagonal, and vec ( M ) is vector obtained by stacking the columns of the matrix M on top of one another.
Then, we can re-write the MSE (eq.
(2.3)) as
[Equation]
where G (cid:62) ˜ w is the output of the MNN.
Now, if ( W , z ) is a DLM of the MSE in eq.
(2.3), then there is no inﬁnitesimal perturbation of ( W , z ) which reduces this MSE.
Next, for each row i , we will show that ∂ MSE /∂
˜ w i = 0 , since otherwise we can ﬁnd an inﬁnitesimal perturbation of ( W , z ) which decreases the MSE, contradicting the assumption that ( W , z ) is a local minimum.
For each row i , we divide into two cases:
First, we consider the case z i (cid:54) = 0 .
In this case, any inﬁnitesimal perturbation q i in ˜w i can be produced by an inﬁnitesimal perturbation in w i : ˜ w i
+ q i
=
( w i
+ q i /z i )
z i .
Therefore, unless the gradient ∂ MSE /∂
˜ w i is equal to zero, we can choose an inﬁnitesimal perturbation q i in the opposite direction to this gradient, which will decrease the MSE.
Second, we consider the case z i = 0 .
In this case, the MSE is not affected by changes made exclusively to w i .
Therefore, all w i derivatives of the MSE are equal to zero ( ∂ k MSE /∂
k w i , to any order k ) .
Also, since we are at a differentiable local minimum, ∂ MSE /∂z i = 0 .
Thus, using a
Taylor expansion, if we perturb ( w i , z i ) by ( ˆw i , ˆ z i ) then the MSE is perturbed by
[Equation]
Therefore, unless ∂ 2 MSE / ( ∂ w i ∂z i ) =
we can choose ˆw i and a sufﬁciently small ˆ z i such that the
MSE is decreased.
Lastly, using the chain rule
[Equation]
Thus, ∂ MSE /∂
˜ w i = 0 .
This implies that ˜ w is also a DLM 5 of eq.
(7.2), which entails
[Equation]
Since G =
A ◦ X and e = y − G (cid:62)
˜
w this proves eq.
Now, for any two solutions ˜ w 1 and ˜ wof eq.
(7.3), we have
[Equation]
Multiplying by ( ˜ w 2 − ˜ w 1 ) (cid:62) from the left we obtain
[Equation]
⇒
[Equation]
Therefore, the MNN output and the residual error e are equal for all DLMs in D A ( X ) .
S UB - OPTIMAL DIFFERENTIABLE LOCAL MINIMA : P ROOF OF T HEOREMAND ITS COROLLARY
Theorem 13.
( Theorem 6 restated )
Given assumptions 1-4, the expected angular volume of suboptimal DLMs, with MCE > (cid:15) > 0 , is exponentially vanishing in N as
[Equation]
where γ (cid:15) (cid:44) 0 .
23 max [lim N →∞ ( d 0 ( N ) /N ) , (cid:15) ] 3 / 4 if ρ (cid:54) = { 0 , 1 } , and γ (cid:15) (cid:44) 0 .
23 (cid:15) 3 / 4 if ρ = 0 .
To prove this theorem we upper bound the angular volume of L (cid:15) (deﬁnition 4), i.e. , differentiable regions in which there exist DLMs with MCE > (cid:15) > 0 .
Our proof uses the ﬁrst order necessary condition for DLMs from Lemma 2, ( A ◦ X ) e = 0 , to ﬁnd which conﬁgurations of A allow for a high residual error e with MCE > (cid:15) > 0 .
In these conﬁgurations A ◦ X cannot have full rank, and therefore, as we show (Lemma 14 below), A = a ( WX ) must have a low rank.
However,
A = a ( WX ) has a low rank with exponentially low probability when X ∼ N and W ∼ N (Lemmas 15 and 16 below).
Thus, we derive an upper bound on E X ∼N V ( L (cid:15) ( X , y )) .
Before we begin, let us recall some notation:
[ L ] (cid:44) { 1 , 2 , . . .
, L } , M > 0 implies that ∀ i, j : M ij > 0 , M S is the matrix composed of the columns of M that are in the index set S , (cid:107) v (cid:107) 0 as the L 0 “norm” that counts the number of non-zero values in v .
First we consider the case ρ (cid:54) = 0 .
Also, we denote
K r (cid:44) max [ N(cid:15), rd 0 ] .
First we consider the case ρ (cid:54) = 0 .
From deﬁnition 3 of the angular volume
E
[Equation]
0 (cid:17)
[Equation]
where
If we are at DLM a in D A ( X ) , then Lemma 2 implies ( A ◦ X ) e = 0 .
Also, if e ( n ) = 0 on some sample, we necessarily classify it correctly
and therefore MCE ≤ (cid:107) e (cid:107) 0 /N .
Since MCE > (cid:15) in L (cid:15) this implies that N(cid:15) < (cid:107) e (cid:107) 0 .
Thus, this inequality holds for v = e . 2.
We apply assumption 1, that X ∼ N . 3.
Assumption 4 implies d 0 d 1 ˙ >N log 4 N ≥ N .
Thus, we can apply the following Lemma
, proven in appendix section 12.1:
Lemma 14.
Let X ∈ R d 0 × N , A ∈ { ρ, 1 } d 1 × N , S ⊂ [ N ] and d 0 d 1 ≥ N .
Then, simultaneously for every possible A and S such that
[Equation]
we have that, X -a.e.
, (cid:64) v
∈
R N such that v n (cid:54) = 0 ∀ n ∈ S and ( A ◦ X ) v =
Recall that K r (cid:44) max [ N(cid:15), rd 0 ] .
We use the union bound over all possible ranks r ≥ 1 : we ignore the r = 0 case since for ρ (cid:54) =
(see eq.
there is zero probability that rank ( a ( WX S ))
=
for some non-empty S .
For each rank r ≥ 1 , it is required that | S | > K r = max [ N(cid:15), rd 0 ] , so | S | =
K r is a relaxation of the original condition, and thus its probability is not lower.
We again use the union bound over all possible subsets S of size K r .
Thus, from eq.
[Equation]
we have
[Equation]
Since we take the expectation over X , the location of S does not affect the probability.
Therefore, we can set without loss of generality S =
[ K r ] .
Note that r ≤ N/d 0 ˙ < min [ d 0 , d 1 ] from assumptions 3 and 4.
Thus, with k = K
r ≥ d 0 , we apply the following Lemma, proven in appendix section 12.2:
Lemma 15.
Let X ∈ R d 0 × k be a random matrix with independent and identically distributed columns, and
W ∈ R d 1 × d 0 an independent standard random Gaussian matrix.
Then, in the limit min [ k, d 0 , d 1 ] ˙ >r ,
[Equation]
Note that K r ≥ N(cid:15)
˙= N > 2 d 1 , and min [ K r , d 0 , d 1 ] ˙ >d 0 d 1 /K r ˙ > 1 from assumptionsand 4.
Thus, we apply the following Lemma (with C
= X (cid:62) , B = W (cid:62) , M = d 0 , L = dand N = K r / 2 ), proven in appendix section 12.3:
Lemma 16.
Let C ∈ R N × M and B ∈ R M ×
L be two independent standard random
Gaussian matrices.
Without loss of generality, assume N ≥ L , and denote α (cid:44) ML/N .
Then, in the regime M ≤ N and in the limit min [ N, M, L ] ˙ >α ˙ > 1 , we have
[Equation]
We
use rd 0 ≤ N , (cid:18) NK r
[Equation]
≤ 2 N , K r ≤ N , and d 1 ˙ <N (from assumption
and r
≤ N 2 /d 20 ˙ <
N (from
assumption
to
simplify
the
combintaorial expressions.
First, note
that
r = 1 is
the
maximal term in
the
sum, so we can neglect the other, exponentially smaller, terms.
Second, from assumption 3 we have d 0 ˙ ≤ N , so
[Equation]
N
[Equation]
Third, from assumption 4 we have N log 4 N ˙ <
d 0 d 1 , so the 2 N log N term is negligible.
Thus,
[Equation]
which proves the Theorem for the case ρ (cid:54)
= 0 .
Next, we consider the case ρ = 0 .
In this case, we need to change transition (4) in eq.
(8.1), so the sum starts from r = 0 , since now we can have rank ( a ( WX S ))
= 0 .
Following exactly the same logic (except the modiﬁcation to the sum), we only need to modify transition (5) in eq.
(8.2) – since now the maximal term in the sum is at r = 0 .
This entails γ (cid:15) = 0 .
23 (cid:15) 3 / 4 .
(cid:4)
Corollary 17.
(Corollary 7 restated)
Given assumptions 1-4, for any δ > 0 (possibly a vanishing function of N ), we have, with probability 1 − δ , that the angular volume of sub-optimal DLMs, with
MCE > (cid:15) > 0 , is exponentially vanishing in N as
[Equation]
Proof.
Since V ( L (cid:15) ( X , y )) ≥ 0 we can use Markov’s Theorem (Fact 11) ∀ η > 0 :
[Equation] [Equation]
denoting η = 1 δ E X
∼N V ( L (cid:15) ( X , y )) , and using Theorem (6) we prove the corollary.
[Equation]
[Equation]
where we note that replacing a regular inequality < with inequality in the leading order ˙ ≤ only removes constraints, and therefore increases the probability.
C ONSTRUCTION OF GLOBAL MINIMA : P ROOF OF T HEOREM 8:
Recall the LReLU non-linearity
[Equation]
in eq.
(2.1), where ρ (cid:54) = 1 .
Theorem 18.
(Theorem 8 restated) For any y ∈ { 0 , 1 } N and X ∈ R d 0 × N almost everywhere we ﬁnd matrices W ∗ ∈ R d ∗ 1 × d 0 and z
∗ ∈ R d ∗ 1 , such that y =
f ( W ∗ X ) (cid:62) z ∗ , where d ∗ 1 (cid:44) 4 (cid:100) N/ (2 d 0 − 2) (cid:101) and ∀ i, n : w (cid:62) i x ( n ) (cid:54) = 0 .
Therefore, every MNN with d 1 ≥ d ∗ 1 has a DLM which achieves zero error e = 0 .
We prove the existence of a solution ( W ∗ , z ∗ ), by explicitly constructing it.
This construction is a variant of (Baum, 1988, Theorem 1), except we use LReLU without bias and MSE – instead of threshold units with bias and MCE.
First, we note that for any (cid:15) 1 > (cid:15) 2 > 0 , the following trapezoid function can be written as a scaled sum of four LReLU:
[Equation]
Next, we examine the set of data points which are classiﬁed to 1 : S + (cid:44) (cid:8) n ∈ [ N ] | y ( n ) = 1 (cid:9) .
Without loss of generality, assume |S + | ≤ N 2 .
We partition S + to
[Equation]
subsets (cid:8) S
+ i (cid:9)
K
i =1 , each with no more than d 0 − 1 samples.
For almost any dataset we can ﬁnd K hyperplanes passing through the origin, with normals { ˜w i } Ki =1 such that each hyperplane contains all d 0 − 1 points in subset S + i , i.e. ,
[Equation]
but no other point, so ∀ n / ∈ S
+ i : ˜ w (cid:62) i x ( n ) (cid:54) = 0 ,
If (cid:15) 1 , (cid:15) 2 in eq.
(9.1) are sufﬁciently small ( ∀ n / ∈ S
+ i : (cid:12)(cid:12)
˜ w (cid:62) i x ( n ) (cid:12)(cid:12) > (cid:15) 1 ) then we have τ (cid:16) ˜ w (cid:62) i x ( n ) (cid:17) = (cid:26) 1 , if n ∈ S
+ i 0 , else .
Then we have
K (cid:88)
[Equation]
(cid:16)
[Equation]
(cid:17) = (cid:26)
[Equation] [Equation]
0 , else (9.3) which gives the correct classiﬁcation on all the data points.
Thus, from eq.
(9.1), we can construct a
MNN with d ∗ 1 = 4 K hidden neurons which achieves zero error.
This is straightforward to do if we have a bias in each neuron.
To construct this MNN even without bias, we ﬁrst ﬁnd a vector
ˆ w i such that
[Equation] [Equation]
Note that this is possible since (cid:104) X S
+ i , ˜
w i (cid:105) has full rank X -a.e.
(the matrix X S + i ∈
R
d 0 × d 0 − 1 has,
X -a.e.
, one zero left eigenvector, which is ˜ w i , according to eq.
Additionally, we can set (cid:107) ˜ w i (cid:107) =
(cid:107) ˆ w i (cid:107) , (9.5) since changing the scale of w i would not affect the validity of eq.
Then, we denote
[Equation]
Note, from eqs.
(9.2) and (9.4) that this choice satisﬁes
[Equation]
Also, to ensure that ∀ n / ∈ S + i the sign of w ( j ) (cid:62) i x ( n ) does not change for different j , for some β, γ <we deﬁne
[Equation] [Equation]
where with probability 1 , min n/ ∈S + i (cid:12) (cid:12) ˜ w (cid:62) i x ( n ) (cid:12) (cid:12) > 0 and max
n/ ∈S
+ i (cid:12) (cid:12) ˆ w (cid:62) i x ( n ) (cid:12) (cid:12) > 0 .
Deﬁning
[Equation]
and combining all the above facts
[Equation]
we have
[Equation]
1 , if n ∈ S
[Equation]
i 0 , else .
Thus, for
[Equation] [Equation]
we obtain a MNN that implements f (cid:16) W
[Equation]
x ( n ) (cid:17) (cid:62) z ∗ = (cid:26) 1 , if n ∈ S + 0 , else and thus achieves zero error.
Clearly, from this construction, if w i is a row of W ∗ , then ∀ n ∈ S + i , ∀ i : (cid:12)(cid:12) w (cid:62) i x ( n ) (cid:12)(cid:12) ≥ (cid:15) 2 , and with probability 1 ∀ n / ∈ S
+ i , ∀ i : (cid:12)(cid:12) w (cid:62) i x ( n ) (cid:12)(cid:12) > 0 , so this construction does not touch any non-differentiable region of the MSE.
(cid:4)
G LOBAL MINIMA : P ROOF OF T HEOREM
Theorem 19.
(Theorem 9 restated).
Given assumptions 1-3, we set δ ˙= (cid:113) 8 π d − 1 / 2+ 2 d 1 / 2 0 √ log d 0 /N and d ∗ 1 = 2 N/d 0 , or if assumption 5 holds, we set d ∗ 1 as in this assumption.
Then, with probability 1 − δ , the angular volume of global minima is lower bounded as,
[Equation]
In this section we lower bound the angular volume of G (deﬁnition 4), i.e. , differentiable regions in which there exist DLMs with MCE = 0 .
We lower bound V ( G ) using the angular volume corresponding to the differentiable region containing a single global minimum.
From assumption 4, we have d 0 d 1 ˙ >N , so we can apply Theorem 8 and say that the labels are generated using a ( X , y )
-dependent
MNN:
y =
f ( W ∗ X )
(cid:62) z ∗ with target weights W ∗ =
(cid:104)
w ∗(cid:62) 1 , . .
w ∗(cid:62) d ∗ 1 (cid:105) (cid:62)
∈ R d ∗ 1 × d 0 and z ∗ ∈ R d 1 .
If, in addition, assumption 5 holds then we can assume W ∗ and z ∗ are independent from ( X , y ) .
In both cases, the following differentiable region
[Equation]
also contains a differentiable global minimum (just set w i =
w ∗ i , z i =
z ∗ i ∀ i ≤ d ∗ 1 , and z i =∀ i > d ∗ 1 ), and therefore ∀ X , y and their corresponding W ∗ , we have
[Equation]
Also, we will make use of the following deﬁnition.
Deﬁnition 20.
Let X have an angular margin α from W ∗ if all datapoints (columns in X ) are at an angle of at least α from all the weight hyperplanes (rows of W ∗ ) , i.e. , X is in the set
[Equation]
Using the deﬁnitions in eqs.
(10.3) and (10.1), we prove the Theorem using the following three
Lemmas.
First, In appendix section 13.2 we prove
Lemma 21.
For any α, if W ∗ is independent from W then, in the limit N → ∞ ,
∀ X ∈
M
α
( W ∗ ) with log sin α ˙ >
d − 1 0 log d
[Equation]
Second, in appendix section 13.3 we prove
Lemma 22.
Let W ∗ ∈
R d ∗ 1 × d 0 a ﬁxed matrix independent of X .
Then, in the limit N → ∞ with d ∗ 1 ˙ ≤ d 0 ˙ ≤ N , the probability of not having an angular margin sin α = 1 / ( d ∗ 1 d 0 N ) (eq. (10.3)) is upper bounded by
[Equation]
Lastly, in appendix section 13.4 we prove
Lemma 23.
Let X ∈ R d 0 ×
N be a standard random Gaussian matrix of datapoints.
Then we can ﬁnd, with probability 1, ( X , y ) -dependent matrices W ∗ and z ∗ as in Theorem 8 (where d ∗ 1 (cid:44) 4 (cid:100) N/ (2 d 0 − 2) (cid:101) ) .
Moreover, in the limit N → ∞ , where N/d 0 ˙ ≤ d 0 ˙ ≤ N , for any y , we can bound the probability of not having an angular margin (eq. (10.3)) with sin α = 1 / ( d ∗ 1 d 0 N ) by
[Equation]
Recall that ∀ X , y and their corresponding W ∗ , we have G ( X , y ) ⊂
˜ G
( X
[Equation]
W ∗ ) (eq.
Thus, combining Lemmas 21 with sin α = 1 / ( d ∗ 1 d 0 N ) together with either Lemma 22 or 23, we prove the ﬁrst (left) inequality of Theorem 9:
[Equation]
Next, if d ∗ 1 = 2 N/d 0 or d ∗ 1 ˙ <N/d 0 (is assumption 5 holds), we obtain the second (right) inequality
[Equation]
V OLUME
RATIO OF GLOBAL AND LOCAL MINIMA : P ROOF OF T HEOREM
Theorem 24.
(Theorem 10 restated)
Given assumptions 1-3, we set δ
[Equation]
+ 2 d 1 / 2 0 √ log d 0 /N .
Then, with probability 1 − δ , the angular volume of sub-optimal DLMs, with
MCE > (cid:15) > 0 , is exponentially vanishing in N, in comparison to the angular volume of global minima with MCE =
[Equation]
To prove this theorem we ﬁrst calculate the expectation of the angular volume ratio given the X -event that the bound in Theorem 9 holds (given assumptions 1-3),
i
[Equation]
e. , V ( G ( X , y )) ˙ ≥ exp ( − 2 N log N ) .
Denoting this event 6 as M
[Equation]
we ﬁnd:
[Equation]
where
We apply Theorem 9. 2.
We use the following fact
Fact 25.
For any variable X ≥ 0 and event A (where ¯ A is its complement)
[Equation]
We apply Theorem 6.
We apply Theorem 9.
We use assumption 4, which implies γ (cid:15) N 3 / 4 [ d 1 d 0 ] 1 / 4 ˙ > 2 N log N .
For simplicity, in the reminder of the proof we denote
[Equation]
X
[Equation]
From Markov inequality (Fact 11), since R ( X ) ≥ 0 , we have ∀ η ( N ) > 0 :
[Equation]
On the other hand, from fact 25, we have
[Equation] [Equation]
Combining Eqs.
(11.2)-(11.3) we obtain
[Equation]
and so
[Equation]
We choose
[Equation]
so that
[Equation]
Then, from Theorem 9 we have
[Equation]
so we obtain the ﬁrst (left) inequality in the Theorem (10)
[Equation]
[Equation]
Lastly, we note that assumption 4 implies γ (cid:15) N 3 /
[ d 1 d 0 ] 1 / 4 ˙ >N log N , which proves the second (right) inequality of the theorem.
[Equation]
Part II Proofs of technical results
In this part we prove the technical results used in part I.
U PPER BOUNDING THE ANGULAR VOLUME OF SUB - OPTIMAL DIFFERENTIABLE LOCAL MINIMA : P ROOFS OF L EMMAS USED IN S ECTION 12. P ROOF OF L EMMA
In this section we will prove Lemma 14 in subsection 12.3.3.
Recall the following deﬁnition
Deﬁnition 26.
Let
A =
[ a 1 , . .
. , a N ] ; X =
[ x 1 , . .
, x N ] , where X ∈ R d 0 × N and A ∈ R d 1 × N .
The Khatari-Rao product between the two matrices is deﬁned as
[Equation]
Lemma 27.
(Lemma 14 restated)
Let X ∈ R d 0 × N , A ∈ { ρ, 1 } d 1 × N , S ⊂ [ N ] and d 0 d 1 ≥ N .
Then, simultaneously for every possible A and S such that
[Equation]
we have that, X -a.e.
, (cid:64) v ∈ R N such that v n (cid:54) = 0 ∀ n ∈ S and ( A ◦ X ) v = 0 .
Proof.
We examine speciﬁc A ∈ { ρ, 1 } d 1 × N and S ⊂ [ N ] , and such that | S | ≤ d S d 0 , where we deﬁned d S (cid:44) rank ( A S ) .
We assume that d S ≥ 1 , since otherwise the proof is trivial.
Also, we assume by contradiction
that ∃ v ∈ R N such that v i (cid:54) = 0 ∀ i ∈ S and ( A ◦ X ) v = 0 .
Without loss of generality, assume that S = { 1 , 2 , ..., | S |} and that a 1 , a 2 , ..., a d S are linearly independent.
Then
[Equation] [Equation]
for every 1 ≤ k ≤ d 1 .
From the deﬁnition of S we must have v n (cid:54) = 0 for every 1 ≤ n ≤ | S | .
Since a 1 , a 2 , ..., a d S are linearly independent, the rows of A d S
=
[ a 1 , a 2 , ..., a d S ] span a d S -dimensional space.
Therefore, it is possible to ﬁnd a matrix R such that RA d S
=
[ I d S
[Equation]
d S , 0 d S ×
[Equation]
d 1 − d S ) ] (cid:62) , wherei × j is the all zeros matrix with i columns and j rows.
Consider now A S ◦ X S , i.e. , the matrix composed of the columns of A ◦ X in S .
Applying R (cid:48)
=
R ⊗ I d 0 to A S ◦ X S , turns (12.2) into d 0 d S equations in the variables v 1 , ..., v | S | , of the form
[Equation]
for every 1 ≤ k ≤ d S .
We prove by induction that for every 1 ≤ d ≤ d S , the ﬁrst d 0 d equations are linearly independent, except for a set of matrices X of measure 0.
This will immediately imply | S | > d S d 0 , or else eq.
12.2 cannot be true for v (cid:54) = 0 .
which will contradict our assumption, as required.
The induction can be viewed as carrying out Gaussian elimination of the system of equations described by (12.3), where in each elimination step we characterize the set of matrices X that for which that step is impossible, and show it has measure 0.
For d = 1 , the ﬁrst d 0 equations read v 1 x 1 + (cid:80) | S | n = d S +1 v n ˜ a 1 ,n x n = 0 , and since v 1 (cid:54) = 0 , we must have x 1 ∈ Span (cid:8) ˜ a 1 ,d S +1 x d S +1 , ..., ˜ a 1 , | S | x | S | (cid:9) .
However, except for a set of measurewith respect to x 1 (a linear subspace of R d 0 with dimension less than d 0 ), this can only happen if dim Span (cid:8) ˜ a 1 ,d S +1 x d S +1 , ..., ˜ a 1 , | S | x | S | (cid:9) =
d 0 , which implies | S | ≥ d S − 1 + d 0 > d 0 and also that the ﬁrst d 0 rows are linearly independent (since there are d 0 independent columns).
For a general d , we begin by performing Gaussian elimination on the ﬁrst ( d − 1) d 0 equations, resulting in a new set of r d equations, such that every new equation contains one variable that appears in no other new equation.
Let C be the set of the indices (equivalently, columns) of these variables r d variables.
From (12.3) it is clear none of the variables v d , v d
+1 , ..., v d S appear in the ﬁrst ( d − 1) dequations, and therefore C ⊆ S (cid:48)
=
S \ { d, d + 1 , ..., d S } .
By our induction assumptions, except for a set of measure 0, the ﬁrst ( d − 1) d 0 are independent, which means that | C | = r d = ( d − 1) d 0 .
We now extend the Gaussian elimination to the next d 0 equations, and eliminate all the variables in C from them.
The result of the elimination can be written down as,
[Equation]
(12.4) where Y is a square matrix of size d 0 whose coefﬁcients depend only on { ˜ a k,n } n ∈ C,d>k ≥ 1 and on { x n } n ∈ C , and in particular do not depend on x d and { x n } n ∈ S (cid:48) \ C .
Now set ˜ x n =
(˜ a d,n I d 0 − Y ) x n for n ∈ S (cid:48) \ C .
As in the case of d = 1 , since v d (cid:54) = 0 , x d ∈ Span { ˜ x n } n ∈ S (cid:48) \ C .
Therefore, for all values of x d ∈ R d 0 but a set of measure zero (linear subspace of with dimension less than d 0 ), we must have dim Span { ˜ x n } n ∈ S (cid:48) \ C =
d 0 .
From the independence of { ˜ x n } n ∈ S (cid:48) \ C on x d it follows that dim Span { ˜ x n } n ∈ S (cid:48) \ C
= d 0 holds a.e. with respect to the Lebesgue measure over x .
Whenever dim Span { ˜ x n } n ∈ S (cid:48) \ C = d 0 we must have | S (cid:48) \ C | ≥ d 0 and therefore
[Equation]
Moreover, dim Span { ˜ x n } n ∈ S (cid:48) \ C =
d 0 implies that the d 0 equations v d x d + (cid:80) n ∈ S (cid:48) \ C v n ˜ x n =are independent.
Thus, we may perform another step of Gaussian elimination on these d 0 equations, forming d 0 new equations each with a variable unique to it.
Denoting by C (cid:48) the set of these dvariables, it is seen from (12.4) that C (cid:48) ⊆
( S (cid:48) ∪ { d } ) \ C and in particular C (cid:48) is disjoint from C .
Thus, considering the ﬁrst ( d − 1) d 0 equations together with the new d 0 equations, we see that there is a set C ∪ C (cid:48) of d 0 d variables, such that each variable in C ∪ C (cid:48) appears only in one of the d 0 d equations, and each of the d 0 d contains only a single variable in C ∪ C (cid:48) .
This means that the ﬁrst d 0 d must be linearly independent for all values of X except for a set of Lebesgue measure zero, completing the induction.
Thus, we have proven, that for some A ∈ { ρ, 1 } d 1 × N and S ⊂ [ N ] such that | S | ≤ rank ( A S ) dthe event
[Equation]
has zero measure.
The event discussed in the theorem is a union of these events:
[Equation]
and it also has zero measure, since it is a ﬁnite union of zero measure events.
For completeness we note the following corollary, which is not necessary for a our main results.
Corollary 28.
If N ≤ d 1 d 0 , then rank ( A ◦ X ) = N , X -a.e.
, if and only if,
[Equation]
Proof.
We deﬁne d S (cid:44) rank ( A S ) and A ◦ X .
The necessity of the condition | S | ≤ d 0 d S holds for every X , as can be seen from the following counting argument.
Since the matrix A S has rank d S , there exists an invertible row transformation matrix R , such that RA S has only d S non-zero rows.
Consider now G S =
A S ◦ X S , i.e. , the matrix composed of the columns of G in S .
We have
[Equation]
where R (cid:48)
=
R ⊗ I d 0 is also an invertible row transformation matrix, which applies R separately on the d 0 sub-matrices of G S that are constructed by taking one every d 0 rows.
Since G (cid:48) S has at most d 0 d S non-zero rows, the rank of G S cannot exceed d 0 d S .
Therefore, if | S | > d 0 d S , G S will not have full column rank, and hence neither will G .
To demonstrate sufﬁciency a.e., suppose G does not have full column rank.
Let S be the minimum set of columns of G which are linearly dependent.
Since the columns of G S are assumed linearly dependent there exists v ∈ R | S | such (cid:107) v (cid:107)
= | S | and
G S v = 0 .
Using Lemma 28 we complete the proof.
12. P ROOF OF L EMMA
In this section we will prove Lemma 15 in subsection 12.3.3.
This proof relies on two rather basic results, which we ﬁrst prove in subsections 12.2.1 and 12.2.2.
12.2. N UMBER OF DICHOTOMIES INDUCED BY A HYPERPLANE
Fact 29.
A hyperplane w ∈ d 0 can separate a given set of points X = (cid:2) x (1) , . . .
, x ( N ) (cid:3) ∈ R d 0 × N into several different dichotomies, i.e., different results for sign (cid:0) w (cid:62) X (cid:1) .
The number of dichotomies is upper bounded as follows:
[Equation]
h
[Equation]
−
[Equation]
Proof.
See (Cover, 1965, Theorem 1) for a proof of the left inequality as equality (the Schläﬂi
Theorem) in the case that the columns of X are in “general position” (which holds X -a.e, see deﬁnition in (Cover, 1965)) .
If X is not in general position then this result becomes an upper bound, since some dichotomies might not be possible.
Next, we prove the right inequality.
For N = 1 and N = 2 the inequality trivially holds.
For N ≥ 3 , we have
[Equation]
where in (1) we used the bound (cid:18)
Nk
[Equation]
≤ N k , in (2) we used the sum of a geometric series.
12.2. A BASIC PROBABILISTIC BOUND
Lemma 30.
Let H = (cid:2)
h (cid:62) 1 , . . .
, h (cid:62) d 1 (cid:3) (cid:62) ∈ {− 1 , 1 } d 1 × k be a deterministic binary matrix, W = (cid:2) w (cid:62) 1 , . . .
, w (cid:62) d 1 (cid:3) (cid:62) ∈
R
d 1 × d 0 be an independent standard random Gaussian matrix, and X ∈ R d 0 × k be a random matrix with independent and identically distributed columns.
[Equation]
Proof.
By direct calculation
[Equation]
where
We
used the independence of the w i .
We deﬁne ˆ S ± ( h ) (cid:44) (cid:8) S ⊂ [ k ] : ± h (cid:62) S > 0 (cid:9) as the sets in which h is always positive/negative, and ˆ S ( h ) as the maximal set between these two.
Note that w i has a stan-
dard
normal distribution which is symmetric to sign ﬂips, so ∀ S : P (cid:0) w (cid:62) i X S >
| X (cid:1) =
P (cid:0) w (cid:62) i X S < 0 | X (cid:1) .
Note that (cid:12)(cid:12)(cid:12) ˆ S ( h ) (cid:12)(cid:12)(cid:12) ≥ (cid:98) k/ 2 (cid:99) .
Therefore, we deﬁne S ∗ =
[Equation]
We used the independence of the w i .
The maximum is a single term in the following sum of non-negative terms.
Taking the expectation over X , since the columns of X are independent and identically distributed, the location of S does not affect the probability.
Therefore, we can set without loss of generality S =
[ (cid:98) k/ 2 (cid:99) ] .
12.2. M AIN PROOF : B OUND ON THE NUMBER OF CONFIGURATIONS FOR A BINARY MATRIX WITH CERTAIN RANK
Recall the function a ( · ) from eq.
[Equation]
where ρ (cid:54) = 1 .
Lemma 31.
(Lemma 15 restated).
Let X ∈ R d 0 × k be a random matrix with independent and identically distributed columns, and
W ∈ R d 1 × d 0 an independent standard random Gaussian matrix.
Then, in the limit min [ k, d 0 , d 1 ] ˙ >r ,
[Equation]
Proof.
We denote A = a ( WX ) ∈ { ρ, 1 } d 1 × k .
For any such A for which rank ( A ) =
r , we have a collection of r rows that span the remaining rows.
There are (cid:18) dr (cid:19) possible locations for these r spanning rows.
In these rows there exist a collection of r columns that span the remaining columns.
There are (cid:18) k r (cid:19) possible locations for these r spanning columns.
At the intersection of the spanning rows and columns, there exist a full rank sub-matrix D .
We denote ˜
A as the matrix A which rows and columns are permuted so that D is the lower right block
[Equation]
where D is an invertible r × r matrix, and we divided X and W to the corresponding block matrices
[Equation]
with W 2 ∈ R r × d 0 rows and X 2 ∈ R d 0 × r .
Since rank (cid:16) ˜
A (cid:17) =
r , the ﬁrst d 1 − r rows are contained in the span of the last r rows.
Therefore, there exists a matrix Q such that QC =
Z and QD
=
B .
Since D is invertible, this implies that
Q = B D − 1 and therefore
[Equation]
i.e. , B , C and D uniquely determine Z .
Using the union bound over all possible permutations from A to ˜ A , and eq.
(12.9), we have
P (rank ( A ) = r )
[Equation]
[Equation] [Equation]
Using Lemma
[Equation]
we have
[Equation]
an upper bound which does not depend on H .
So all that remains is to compute the sum:
[Equation]
where
Given ( W 1 , X 1 ) , and eq.
(12.8), the indicator function in eq.
(12.12) is equal to zero
only if P (cid:16) a ( W 1 X 2 ) [ a ( W 2 X 2 )] − 1 a ( W 2 X 1 ) =
A | W 1 , X 1 (cid:17) = 0 , and one otherwise.
This sum counts the number of values of H consistent with W 1 and X 1 .
Conditioned on ( W 1 , X 1 ) , D =
[ a ( W 2 X 2 )] − 1 , B = a ( W 1 X 2 ) and C = a ( W 2 X 1 ) can have multiple values, depending on W 2 and X 2 .
Also, any single value for ( D , B , C ) results in a single value of H .
Therefore, the number of possible values of H in eq.
(12.12) is upper bounded by the product of the number of possible values of D , B and C , which is product in the following equation.
The function (cid:80)
h ∈
−
1 , 1 } ( k − r
I (cid:0) ∃ w : sign (cid:0) w (cid:62) X 1 (cid:1) =
h (cid:62) (cid:1) counts the number of dichotomies that can be induced by the linear classiﬁer w on X 1 .
Using eq.
(12.7) we can bound this number by 2 ( k − r ) d 0 .
Similarly, the other sum can be bounded by 2 ( d 1 − r ) r .
Combining eqs.
(12.10), (12.11) and (12.13) we obtain
P (rank ( A ) = r ) ≤
[Equation]
Next, we take the log.
To upper bound (cid:18) Nk (cid:19) , for small k we use (cid:18) Nk (cid:19) ≤ N k , while for k = N/ 2 , we use (cid:18) NN /
[Equation]
≤ 2 N .
Thus, we obtain log
[Equation]
Recalling that W 1 ∈ R ( d 1 − r ) × d 0 while W ∈ R
d 1 × d 0 , we obtain from Jensen’s inequality
[Equation]
Taking the limit min [ k, d 0 , d 1 ] ˙ >r on eqs.
(12.14) and (12.15) we obtain
P (rank
[Equation]
A
[Equation]
12. P ROOF OF L EMMA
In this section we will prove Lemma 16 in subsection 12.3.3.
This proof relies on more elementary results, which we ﬁrst prove in subsections 12.3.1 and 12.3.2.
12.3. O RTHANT PROBABILITY OF A RANDOM G AUSSIAN VECTOR
Recall that φ ( x ) and Φ ( x ) are, respectively, the probability density function and cumulative distribution function for a scalar standard normal random variable.
Deﬁnition 32.
We deﬁne the following functions ∀
[Equation] [Equation]
[Equation] [Equation]
where the inverse function g − 1 ( x ) :
[0 , ∞ )
→ [0 , ∞ ) is well deﬁned since g ( x ) monotonically increase from 0 to ∞ , for x ≥ 0 .
Lemma 33.
Let z ∼ N (0 , Σ ) be a random Gaussian vector in R K , with a covariance matrix Σ ij = (cid:0) 1 − θK − 1 (cid:1) δ mn + θK − 1 where K (cid:29) θ > 0 .
Then, recalling ψ ( θ ) in eq.
(12.17), we have log P
( ∀ i : z i > 0) ≤ − Kψ
( θ )
+ O (log K ) .
Proof.
Note that we can write z = u + η , where u ∼ N (cid:0) 0 , (cid:0) 1 − θK − 1 (cid:1)
I K (cid:1) , and η ∼ N (cid:0) 0 , θK − 1 (cid:1) .
Using this notation, we have
[Equation]
where in (1) we changed the variable of integration to ξ = (cid:112) θ/ ( K − θ ) η .
We denote, for a ﬁxed θ ,
[Equation]
and ξ 0 as its global maximum.
Since q is twice differentiable, we can use Laplace’s method ( e.g. , (Butler, 2007)) to simplify eq.
[Equation]
To ﬁnd ξ 0 , we differentiate q ( ξ ) and equate to zero to obtain
[Equation]
12.22) which implies (recall eq.
[Equation]
This is a monotonically increasing function from 0 to ∞ in the range ξ ≥ 0 .
Its inverse function can also be deﬁned in that range g − 1 ( θ ) :
[0 , ∞ ] → [0 , ∞ ] .
This implies that this equation has only one solution, ξ 0 = g − 1 ( θ ) .
Since lim ξ →∞ q ( ξ ) =
−∞ , this ξ 0 is indeed the global maximum of q ( ξ ) .
Substituting this solution into q ( ξ ) , we get (recall eq.
[Equation]
Using eq.
(12.18), (12.21) and (12.24) we obtain:
[Equation]
Next, we generalize the previous Lemma to a general covariance matrix.
Corollary 34.
Let u ∼ N (0 , Σ ) be a random Gaussian vector in R K for which ∀ n : Σ nn = 1 , and θ ≥ K max n,m : n (cid:54)
=
m Σ nm > 0 .
Then, again, for large K log P ( ∀ i : u i > 0) ≤ − Kψ ( θ ) + O (log K ) .
Proof.
We deﬁne ˜
u ∼ N (cid:16) 0 , ˜ Σ (cid:17) , with ˜Σ mn = (cid:0) 1 − θK − 1 (cid:1)
δ mn + θK − 1 .
Note that ∀ n : Σ nn = ˜Σ nn = 1 and ∀ m (cid:54) =
n : Σ mn ≤ ˜Σ mn .
Therefore, from Slepian’s Lemma (Slepian, 1962, Lemma 1),
[Equation]
Using Lemma 33 on ˜ u completes the proof.
12.3. M UTUAL COHERENCE BOUNDS
Deﬁnition 35.
We deﬁne the mutual coherence of the columns of a matrix A
=
[ a 1 , · · · , a N ] ∈ R M × N as the maximal angle between different columns
[Equation]
Note that γ ( A ) ≤ 1 and from (Welch, 1974), for N ≥ M , γ ( A ) ≥ (cid:113) N − M M ( N − 1) .
Lemma 36.
Let A =
[ a 1 , · · · , a N ] ∈ R M × N be a standard random Gaussian matrix, and γ ( A ) is the mutual coherence of it columns (see deﬁnition 35).
Then
[Equation]
Proof.
In this case
[Equation]
we have from (Chen & Peng, 2016, Appendix 1):
[Equation]
for any a ∈ (0 , 1) .
Setting a = 1 − (cid:15)/
[Equation]
where in (1) we can assume that (cid:15) ≤ 1 , since for (cid:15) ≥
[Equation]
we have P ( γ ( A ) > (cid:15) )
=
(recall γ ( A ) ≤ 1 ).
Lemma 37.
Let B =
[ b 1 , ·
· · , b L ] ∈ R M
×
L be a standard random Gaussian matrix and mutual coherence γ as in deﬁnition 35.
Then, ∀ (cid:15) > 0 and ∀ K ∈
[ L ] :
[Equation]
[Table]
Proof.
We upper bound this probability by partitioning the set of column vectors into (cid:98) L/K (cid:99) subsets
S i of size | S i | =
K and require that in each subset the mutual coherence is lower bounded by (cid:15) .
Since the columns are independent, we have
[Equation]
where in (1) we used the bound from Lemma 36.
12.3. M AIN PROOF : O RTHANT PROBABILITY OF A PRODUCT G AUSSIAN MATRICES
Lemma 38.
(Lemma 16 restated).
Let C
=
[ c 1 , · · · , c N ] (cid:62) ∈ R N × M and B ∈ R M
×
L be two independent random Gaussian matrices.
Without loss of generality, assume N ≥ L , and denote α (cid:44) ML/N .
Then, in the regime M ≤ N and in the limit min [ N, M, L ] ˙ >α ˙ > 1 , we have
[Equation]
Proof.
For some θ > 0 , and subset S such that | S | = K < L , we have
[Equation]
where in the last equality we used the fact that the rows of C are independent and identically distributed.
We choose a speciﬁc subset
[Equation]
to minimize the second term and then upper bound it using Lemma 37 with θ = K(cid:15) ; additionally, we apply Corollary 34 on the ﬁrst term with the components of the vector u being
[Equation]
which is a Gaussian random vector with mean zero and covariance Σ for which ∀
i : Σ ii = 1 and ∀ i (cid:54) =
j : Σ ij ≤ (cid:15) =
θK − 1 .
Thus, we obtain
[Equation]
(12.25) where we recall ψ ( θ ) is deﬁned in eq.
Next, we wish to select good values for θ and K , which minimize this bound for large ( M, N, L, K ) .
Thus, keeping only the ﬁrst order terms in each exponent (assuming L (cid:29) K (cid:29) 1) , we aim to minimize the function as much as possible
[Equation]
Note that the ﬁrst term is decreasing in K , while the second term increases.
Therefore, for any θ the minimum of this function in K would be approximately achieved when both terms are equal, i.e. ,
[Equation]
so we choose
[Equation]
Substituting K ( θ ) into f ( K, θ ) yields
[Equation]
To minimize this function in θ , we need to maximize the function ψ 3 ( θ ) θ 2 (which has a single maximum).
Doing this numerically gives us
[Equation]
Substituting eqs.
(12.27) and (12.28) into eq.
(12.25), we obtain
[Equation]
where in the last line we used N ≥ L , N ≥ M and min [ N, M, L ] ˙ >α ˙ > 1 .
Taking the log, and denoting α (cid:44) ML/N , we thus obtain log P ( CB > 0) ≤
[Equation] [Equation]
Nα 1 / 4 + O
( N log α ) ,
Therefore, in the limit that N → ∞ and α ( N ) → ∞ , with α ( N ) ˙ <N , we have
[Equation]
L OWER BOUNDING THE ANGULAR VOLUME OF GLOBAL MINIMA : P ROOF OF L EMMAS USED IN SECTION13. A NGLES BETWEEN RANDOM G AUSSIAN VECTORS
To prove the results in the next appendix sections
[Equation]
we will rely on the following basic Lemma.
Lemma 39.
For any vector y and x ∼ N (0 , I d 0 )
[Equation]
we have
[Equation]
(cid:18)(cid:12)(cid:12)(cid:12)(cid:12)
[Equation]
where we recall that B ( x, y ) is the beta function.
Proof.
Since N (
[Equation]
I d 0 ) is spherically symmetric
[Equation]
we can set y =
, 0] (cid:62) , without loss of generality.
Therefore,
[Equation]
the Beta distribution, since x 21 ∼ χ 2 (1) and (cid:80) d 0 i =2 x 2 i ∼ χ 2 ( d 0 − 1) are independent chi-square random variables.
Suppose Z ∼ B ( α, β ) , α ∈ (0 , 1) , and β > 1 .
P ( Z > u ) =
[Equation]
Therefore, for (cid:15) > 0 ,
[Equation]
which proves eq.
Similarly, for α ∈ (0 , 1) and β >
[Equation]
Therefore, for (cid:15) > 0 ,
P
[Equation]
(cid:107) x (cid:107) (cid:107)
[Equation]
(cid:107) (cid:12)(cid:12)(cid:12)(cid:12) <
u
[Equation]
(cid:33) ≤ 2 u
B (cid:0)
[Equation]
d
[Equation]
1 2 (cid:1) , which proves eq.
13. P ROOF OF L EMMA 21:
Given three matrices: datapoints, X =
[Equation]
∈
R d
[Equation]
N , weights W = (cid:2) w (cid:62) 1 , . .
, w (cid:62) d 1 (cid:3) (cid:62) ∈
R
d 1 × d 0 , and target weights W ∗ = (cid:104) w ∗(cid:62) 1 , . . . , w ∗(cid:62) d ∗ 1 (cid:105) (cid:62) ∈
R d ∗ 1 × d 0 , with d ∗ 1 ≤ d 1 ,we recall the following deﬁnitions:
[Equation]
) and
[Equation]
Using these deﬁnitions, in this section we prove the following Lemma.
Lemma 40.
(Lemma 21 restated).
For any α, if W ∗ is independent from W then, in the limit
N
→ ∞ , ∀ X ∈ M α
( W ∗ ) with log sin α ˙ >
d − 1 0 log d
P W ∼N (cid:16)
[Equation]
G
[Equation]
X
[Equation]
˙
[Equation]
sin
[Equation]
Proof.
To lower bound P W ∼N (cid:16) W ∈ ˜ G
( X , W ∗ ) (cid:17)
∀ X ∈ M α
( W ∗ ) , we deﬁne the event that all weight hyperplanes (with normals w i ) have an angle of at least α from the corresponding target hyperplanes (with normals w ∗ i ).
[Equation]
In order that sign (cid:0) w (cid:62) i x
[Equation]
n )
[Equation]
= sign (cid:0) w ∗(cid:62) 1 x ( n ) (cid:1) , w i must be rotated in respect to w ∗ i by an angle greater then the angular margin α , which is the minimal the angle between x ( n ) and the solution hyperplanes (with normals w ∗ i ).
Therefore, we have that, given X ∈ M α ( W ∗ ) ,
[Equation]
[Equation]
And so, ∀ X ∈ M α ( W ∗ ) :
[Equation]
where in (1) we used eq.
(13.5), in (2) we used the independence of { w i } d ∗ 1 i =1 and in (3) we used eq.
(13.1) from Lemma 39.
Lastly, to simplify this equation we use the asymptotic expansion of the beta function B (cid:0) 12 , x (cid:1) =
(cid:112) π/x
+ O (cid:0) x − 3 / 2 (cid:1) for large x :
[Equation]
We obtain the Lemma in the limit N → ∞ when log sin α ˙ >
d − 1 0 log d 0 .
P ROOF OF L EMMA 22:
Lemma 41.
(Lemma 22 restated).
Let W ∗ =
(cid:104) w (cid:62) 1 , . .
w
[Equation]
(cid:62) ∈ R d ∗ 1 × d 0 a ﬁxed matrix independent of X .
Then, in the limit N → ∞ with d ∗ 1 ˙ ≤ d 0 ˙ ≤ N , the probability of not having an angular margin sin α = 1 / ( d ∗ 1 d 0 N ) (eq. (13.3)) is upper bounded by
[Equation]
Proof.
We deﬁne
[Equation]
and M αn ( W ∗ ) (cid:44) (cid:84) d ∗ 1 i =1 M αn,i ( W ∗ ) .
Since M ( W ∗ ) = (cid:84) Nn =1 M αn ( W ∗ ) , we have
[Equation]
where in (1) we used the independence of (cid:8) x ( n ) (cid:9) N n =1 , in (2) we use the union bound, and in (3) we use eq.
(13.2) from Lemma 39.
Taking the log and we using the asymptotic expansion of the beta function B (cid:0) 12 , x (cid:1)
= (cid:112) π/x + O (cid:0) x − 3 / 2 (cid:1) for large x , we get
[Equation]
W ∗
[Equation]
log
[Equation]
where in the last line we recalled sin α = 1 /N .
Recalling that d ∗ 1 ˙ ≤ d 0 ˙ ≤ N , we ﬁnd
[Equation]
13. P ROOF OF L EMMA 23:
Lemma 42.
(Lemma 23 restated).
Let X ∈ R d 0 ×
N be a standard random Gaussian matrix of datapoints.
Then we can ﬁnd, with probability 1, ( X , y ) -dependent matrices W ∗ and z ∗ as in
Theorem 8 (where d ∗ 1 (cid:44) 4 (cid:100) N/ (2 d 0 − 2) (cid:101) ) .
Moreover, in the limit N → ∞ , where N/d 0 ˙ ≤ d 0 ˙ ≤ N , for any y , we can bound the probability of not having an angular margin (eq.
(13.3)) with sin
α = 1 / ( d ∗ 1 d 0 N ) by
[Equation]
N
Proof.
In this proof we heavily rely on the notation and results from the proof of in appendix section 9.
Without loss of generality we assume S +
=
[ d 0 − 1] .
Unfortunately, we can’t use Lemma 41 – this proof is signiﬁcantly more complicated since the constructed solution W ∗ depends on X (we keep this dependence implicit, for brevity).
Similarly to the proof of Lemma 41, we deﬁne,
[Equation]
and M αi ( W
[Equation]
) (cid:44)
[Equation]
Nn
[Equation]
M
[Equation] [Equation]
so
[Equation]
( W
[Equation]
d
[Equation]
1 i =
M αi ( W ∗ ) .
We have
P
[Equation]
where in (1) we used the union bound
[Equation]
and in (2) we used the fact that, from symmetry, ∀ i : P ( X / ∈ M αi ( W ∗ ))
=
P ( X / ∈ M α 1 ( W ∗ )) .
Next, we examine the minimal angular margin in M α 1 ,n : separately for ∀ n < d 0 and ∀ n ≥ d 0 .
Recalling the construction of W in appendix section 9, we have, for ∀ n < d 0 :
[Equation]
n )
[Equation] [Equation]
, (13.8) where in (1) we used ∀ n < d 0 : x ( n ) (cid:62) ˆ w 1 = 1 and x ( n ) (cid:62) ˜w 1 = 0 , from the construction of ˜w 1 and ˆ w 1 (eqs.
(9.2), (9.5), and (9.4)), and in (2) we used the fact that ˆ w (cid:62) 1 ˜w 1 = 0 from eq.
(9.4) together with (cid:107) ˜w 1 (cid:107) =
(cid:107) ˆ w 1 (cid:107) from eq.
(9.5), and (cid:15) 2 = γ(cid:15) 1 from eq.
For ∀ n ≥ d 0 : min i,n ≥ d(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) x ( n ) (cid:62) w ∗ i (cid:13)(cid:13) x ( n ) (cid:13)(cid:13) (cid:107) w ∗ i (cid:107) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) =
min n ≥ d 0 , ± (cid:12)(cid:12) (cid:12) ( ˜w 1 ± (cid:15) 1 ˆ w 1 ) (cid:62) x ( n ) (cid:12)(cid:12) (cid:12) (cid:107) ˜w 1 ± (cid:15) 1 ˆ w 1 (cid:107) (cid:13)(cid:13) x ( n ) (cid:13)(cid:13) ≥ (1 − γβ ) (cid:15)γβ (cid:112) 1 + (cid:15) 21 min n ≥ d(cid:12) (cid:12) ˆw (cid:62) 1 x ( n ) (cid:12) (cid:12) (cid:107) ˆw 1 (cid:107) (cid:13)(cid:13) x ( n ) (cid:13)(cid:13) , (13.9) where we used the fact that ∀ n ≥ d 0 : (cid:15) 2 (cid:12)(cid:12)
ˆ w (cid:62) 1 x ( n ) (cid:12)(cid:12) ≤ γβ (cid:12)(cid:12)
˜w (cid:62) 1 x ( n ) (cid:12)(cid:12) , from eq.
(9.7), and also that ˆ w (cid:62) 1 ˜w 1 = 0 from eq.
We substitute eqs.
(13.8) and (13.9) into P ( X ∈ M α 1 ( W ∗ )) :
P ( X ∈ M α 1 ( W ∗ )) ≥ P
[Equation] [Equation]
>
[Equation] [Equation]
min
[Equation] [Equation]
(1) ≥ P
[Equation]
γκ
[Equation]
max
[Equation]
> sin α, (1 − γβ ) γβ κ min n ≥ dx ( n )(cid:13)(cid:13) x ( n ) (cid:13)(cid:13) > sin α, (cid:15) 1 (cid:112) 1 + (cid:15) 21 > κ (cid:33) (13.10) (2) ≥ P (cid:18) γκ η sin α > (cid:107) ˆw 1 (cid:107) , η > max n<d 0 (cid:13)(cid:13)(cid:13) x ( n ) (cid:13)(cid:13)(cid:13)(cid:19) P (cid:32) (1 − γβ ) γβ κ min n ≥ dx ( n )(cid:13)(cid:13) x ( n ) (cid:13)(cid:13) > sin α, (cid:15) 1 (cid:112) 1 + (cid:15) 2 1 > κ (cid:33) , where in (1) we rotate the axes so that ˆ w 1 ∝ [1 , 0 , 0 . . .
, 0] axes ˜ w 1 ∝ [0 , 1 , 0 , 0 . . .
, 0] – this is possible due to the spherical symmetry of x ( n ) , and the fact that ˆ w 1 and ˜ w 1 are functions of x ( n ) for n < d 0 (from eqs.
(9.4) and (9.2)), and as such, they are independent from x ( n ) for n ≥ d 0 , in (2) we use that fact that (cid:107) ˆw 1 (cid:107) and max n<d 0 (cid:13)(cid:13) x ( n ) (cid:13)(cid:13) are functions of x ( n ) for n < d 0 , and as such, they are independent from x ( n ) for n ≥ d 0 .
Thus,
[Equation]
where in (1) we use the union bound on both probability terms
[Equation]
All that remains is to calculate each remaining probability term in eq.
First, we have
[Equation] [Equation]
where in (1) we used eq.
(9.7), in (2) we recall that in eq.
(13.10) we rotated the axes so that ˆ w 1 ∝ [1 , 0 , 0 . . .
, 0] axes ˜ w 1 ∝ [0 , 1 , 0 , 0 . . .
, 0] , in (3) we used the independence of different x ( n ) , and in (4) we used the fact that the ratio of two independent Gaussian variables is distributed according to the symmetric Cauchy distribution, which has the cumulative distribution function P ( X > x ) = 12 − 1 π arctan ( x ) , and therefore P ( | X | > x ) = 1 − 2 π arctan ( x ) .
Second, we use eq.
[Equation]
min
[Equation] [Equation]
n
[Equation]
n
[Equation]
> γβ sin α (
[Equation]
>
[Equation] [Equation]
(cid:35)
[Equation]
Third, (cid:13)(cid:13) x ( n ) (cid:13)(cid:13) 2 is distributed according to the chi-square distribution of order d 0 , so for η 2 > d 0 ,
P (cid:18)
[Equation]
x ( n
[Equation]
2 ≥ η
[Equation]
η
[Equation]
η
[Equation]
d 0 (cid:1)
[Equation]
d 0 (cid:1) d 0 / 2 .
Therefore,
[Equation]
max
[Equation]
Lastly, we bound (cid:107) ˜ w 1 (cid:107) =
(cid:107) ˆ w 1 (cid:107) (from eq.
From eq.
(9.4), we have
[Equation]
where X [ d 0 − 1] has a singular value decomposition
[Equation]
with σ i being the singular values, and u i and v i being the singular vectors.
The singular values are ordered from smallest to largest, and σ 1 = 0 with u 1 = ˜ w 1 , from eq.
With probability 1, the other d 0 − 1 singular value are non-zero: they are the square roots of the eigenvalues of the random matrix X (cid:62)
[ d 0 − 1]
X [ d 0 − 1] ∈ R d 0 − 1 × d 0 − 1 .
Taking the squared norm of eq.
(13.15), we have
[Equation]
where the last inequality stems from the fact that u (cid:62) 1 ˆ w 1 =
˜w
[Equation]
1 ˆ w 1 = 0 (from eq.
(9.4)), so the minimal possible value is attained when u (cid:62) 2 ˆ w 1 = (cid:107) ˆ w 1 (cid:107) .
The minimal nonzero singular value, σ 2 , can be bounded using the following result from (Rudelson & Vershynin, 2010, eq. (3.2))
[Equation]
Since
[Equation]
we have,
[Equation]
Combining this with eq.
(13.16) we get
[Equation]
Lastly, combining eqs.
(13.12), (13.13), (13.14) and (13.17) into eqs.
(13.7) and (13.11), we get, for η 2 > d 0 ,
P ( X ∈ M α
[Equation]
W ∗
[Equation] [Equation]
where in the last line we take β = γ = κ = 1 / √ 2 , η = d 1 / 2 0 √ log d 0 , sin α = 1 / ( d ∗ 1 d 0 N ) .
Using the asymptotic expansion of the beta function B (cid:0) 1 2 , x (cid:1) =
(cid:112) π/x
+ O (cid:0) x − 3 / 2 (cid:1) for large x , we obtain, for sin
α = 1 / ( d ∗ 1 d 0 N )
[Equation]
d
[Equation]
(cid:18)
[Equation]
(cid:18)
[Equation]
(cid:18)
[Equation]
d(cid:19)(cid:19)(cid:19)(cid:33) ·
[Equation]
O
[Equation]
(cid:19)(cid:19)(cid:33)(cid:33)(cid:33) =
[Equation]
(cid:18)
[Equation]
(cid:19)(cid:19)(cid:33) .
Thus, taking the log, and using log (1 − x ) = − x + O (cid:0) x 2 (cid:1) , we obtain, for sin α
= 1 / ( d ∗ 1 d 0 N ) log P ( X ∈ M α ( W ∗ )) ≥ log
[Equation]
−
[Equation]
(cid:19)(cid:19)(cid:33)(cid:33) = −
[Equation]
(cid:19)(cid:19)(cid:33) .
Recall that d ∗ 1 (cid:44) 4 (cid:100) N/ (2 d 0 − 2) (cid:101) ˙= N/d 0 .
Taking the limit N → ∞ , d 0 → ∞ with d ∗ 1 ˙ ≤ d 0 ˙ ≤ N , we have P
[Equation]
[Table]
N
Part III Numerical Experiments - implementation details
Code and trained models for CIFAR and ImageNet results is available here https://github.
com/MNNsMinima/Paper .
In MNIST, CIFAR and ImageNet we performed binary classiﬁcation on between the original odd and even class numbers.
In we performed this binary classiﬁcation between digits 0 − 4 and 5 − 9 .
Weights were initialized to be uniform with mean zero and variance 2 /d , where d is fan-in (here the width of the previous neuron layer), as suggested in (He et al., 2015).
In each epoch we randomly permuted the dataset and used the Adam (Kingma & Ba, 2014) optimization method (a variant of SGD) with β 1 = 0 .
9 , β 2 = 0 .
99 , ε = 10 − 8 .
Different learning rates and mini-batch sizes were selected for each dataset and architecture.
In CIFAR10 and ImageNet we used a learning-rate of α = 10 − 3 and a mini-batch size of 1024 ; also, ZCA whitening of the training samples was done to remove correlations between the input dimensions, allowing faster convergence.
We deﬁne L as the number of weight layers.
For the random dataset we use a mini-batch size of (cid:98) min ( N/ 2 , d/ 2) (cid:99) with learning rate α = 0 .
1 and 0 .
05 , for L = 2 and 3 , respectively.
In the random data parameter scans the training was done for no more than 4000 epochs – we stopped if MCE = 0 was reached.