Nonlinear Inductive Matrix Completion based on One-layer Neural Networks

The goal of a recommendation system is to predict the interest of a user in a given item by exploiting the existing set of ratings as well as certain user/item features. A standard approach to modeling this problem is Inductive Matrix Completion where the predicted rating is modeled as an inner product of the user and the item features projected onto a latent space. In order to learn the parameters effectively from a small number of observed ratings, the latent space is constrained to be low-dimensional which implies that the parameter matrix is constrained to be low-rank. However, such bilinear modeling of the ratings can be limiting in practice and non-linear prediction functions can lead to significant improvements. A natural approach to introducing non-linearity in the prediction function is to apply a non-linear activation function on top of the projected user/item features. Imposition of non-linearities further complicates an already challenging problem that has two sources of non-convexity: a) low-rank structure of the parameter matrix, and b) non-linear activation function. We show that one can still solve the non-linear Inductive Matrix Completion problem using gradient descent type methods as long as the solution is initialized well. That is, close to the optima, the optimization function is strongly convex and hence admits standard optimization techniques, at least for certain activation functions, such as Sigmoid and tanh. We also highlight the importance of the activation function and show how ReLU can behave significantly differently than say a sigmoid function. Finally, we apply our proposed technique to recommendation systems and semi-supervised clustering, and show that our method can lead to much better performance than standard linear Inductive Matrix Completion methods.

1 Introduction

Matrix Completion (MC) or Collaborative filtering [CR09, GUH16] is by now a standard technique to model recommendation systems problems where a few user-item ratings are available and the goal is to predict ratings for any user-item pair. However, standard collaborative filtering suffers from two drawbacks: 1) Cold-start problem: MC can’t give prediction for new users or items, 2) Missing side-information: MC cannot leverage side-information that is typically present in recommendation systems such as features for users/items. Consequently, several methods [ABEV06, Ren10, XJZ13, JD13] have been proposed to leverage the side information together with the ratings. Inductive matrix completion (IMC) [ABEV06, JD13] is one of the most popular methods in this class.

IMC models the ratings as the inner product between certain linear mapping of the user/items’ features, i.e., , where is the predicted rating of user for item , are the feature vectors. Parameters () can typically be learned using a small number of observed ratings [JD13].

However, the bilinear structure of IMC is fairly simplistic and limiting in practice and might lead to fairly poor accuracy on real-world recommendation problems. For example, consider the Youtube recommendation system [CAS16] that requires predictions over videos. Naturally, a linear function over the pixels of videos will lead to fairly inaccurate predictions and hence one needs to model the videos using non-linear networks. The survey paper by [ZYS17] presents many more such examples, where we need to design a non-linear ratings prediction function for the input features, including [LLL16] for image recommendation, [WW14] for music recommendation and [ZYL16] for recommendation systems with multiple types of inputs.

We can introduce non-linearity in the prediction function using several standard techniques, however, if our parameterization admits too many free parameters then learning them might be challenging as the number of available user-item ratings tend to be fairly small. Instead, we use a simple non-linear extension of IMC that can control the number of parameters to be estimated. Note that IMC based prediction function can be viewed as an inner product between certain latent user-item features where the latent features are a linear map of the raw user-item features. To introduce non-linearity, we can use a non-linear mapping of the raw user-item features rather than the linear mapping used by IMC. This leads to the following general framework that we call non-linear inductive matrix completion (NIMC),

(1)

where are the feature vectors, is their rating and are non-linear mappings from the raw feature space to the latent space.

The above general framework reduces to standard inductive matrix completion when are linear mappings and further reduces to matrix completion when are unit vectors for -th item and -th user respectively. When is used as the feature vector and is restricted to be a two-block (one for and the other for ) diagonal matrix, then the above framework reduces to the dirtyIMC model [CHD15]. Similarly, can also be neural networks (NNs), such as feedforward NNs [SCH16, CAS16], convolutional NNs for images and recurrent NNs for speech/text.

In this paper, we focus on a simple nonlinear activation based mapping for the user-item features. That is, we set and where is a nonlinear activation function . Note that if is ReLU then the latent space is guaranteed to be in non-negative orthant which in itself can be a desirable property for certain recommendation problems.

Note that parameter estimation in both IMC and NIMC models is hard due to non-convexity of the corresponding optimization problem. However, for "nice" data, several strong results are known for the linear models, such as [CR09, JNS13, GJZ17] for MC and [JD13, XJZ13, CHD15] for IMC. However, non-linearity in NIMC models adds to the complexity of an already challenging problem and has not been studied extensively, despite its popularity in practice.

In this paper, we study a simple one-layer neural network style NIMC model mentioned above. In particular, we formulate a squared-loss based optimization problem for estimating parameters and . We show that under a realizable model and Gaussian input assumption, the objective function is locally strongly convex within a "reasonably large" neighborhood of the ground truth. Moreover, we show that the above strong convexity claim holds even if the number of observed ratings is nearly-linear in dimension and polynomial in the conditioning of the weight matrices. In particular, for well-conditioned matrices, we can recover the underlying parameters using only user-item ratings, which is critical for practical recommendation systems as they tend to have very few ratings available per user. Our analysis covers popular activation functions, e.g., sigmoid and ReLU, and discuss various subtleties that arise due to the activation function. Finally we discuss how we can leverage standard tensor decomposition techniques to initialize our parameters well. We would like to stress that practitioners typically use random initialization itself, and hence results studying random initialization for NIMC model would be of significant interest.

As mentioned above, due to non-linearity of activation function along with non-convexity of the parameter space, the existing proof techniques do not apply directly to the problem. Moreover, we have to carefully argue about both the optimization landscape as well as the sample complexity of the algorithm which is not carefully studied for neural networks. Our proof establishes some new techniques that might be of independent interest, e.g., how to handle the redundancy in the parameters for ReLU activation. To the best of our knowledge, this is one of the first theoretically rigorous study of neural-network based recommendation systems and will hopefully be a stepping stone for similar analysis for "deeper" neural networks based recommendation systems. We would also like to highlight that our model can be viewed as a strict generalization of a one-hidden layer neural network, hence our result represents one of the few rigorous guarantees for models that are more powerful than one-hidden layer neural networks [LY17, BGMSS18, ZSJ17].

Finally, we apply our model on synthetic datasets and verify our theoretical analysis. Further, we compare our NIMC model with standard linear IMC on several real-world recommendation-type problems, including user-movie rating prediction, gene-disease association prediction and semi-supervised clustering. NIMC demonstrates significantly superior performance over IMC.

1.1 Related work

Collaborative filtering: Our model is a non-linear version of the standard inductive matrix completion model [JD13]. Practically, IMC has been applied to gene-disease prediction [ND14], matrix sensing [ZJD15], multi-label classification[YJKD14], blog recommender system [SCLD15], link prediction [CHD15] and semi-supervised clustering [CHD15, SCH16]. However, IMC restricts the latent space of users/items to be a linear transformation of the user/item’s feature space. [SCH16] extended the model to a three-layer neural network and showed significantly better empirical performance for multi-label/multi-class classification problem and semi-supervised problems.

Although standard IMC has linear mappings, it is still a non-convex problem due to the bilinearity . To deal with this non-convex problem, [JD13, Har14] provided recovery guarantees using alternating minimization with sample complexity linear in dimension. [XJZ13] relaxed this problem to a nuclear-norm problem and also provided recovery guarantees. More general norms have been studied [RSW16, SWZ17a, SWZ17b, SWZ18], e.g. weighted Frobenius norm, entry-wise norm. More recently, [ZDG18] uses gradient-based non-convex optimization and proves a better sample complexity. [CHD15] studied dirtyIMC models and showed that the sample complexity can be improved if the features are informative when compared to matrix completion. Several low-rank matrix sensing problems [ZJD15, GJZ17] are also closely related to IMC models where the observations are sampled only from the diagonal elements of the rating matrix. [Ren10, LY16] introduced and studied an alternate framework for ratings prediction with side-information but the prediction function is linear in their case as well.

Neural networks: Nonlinear activation functions play an important role in neural networks. Recently, several powerful results have been discovered for learning one-hidden-layer feedforward neural networks [Tia17, ZSJ17, JSA15, LY17, BGMSS18, GKKT17], convolutional neural networks [BG17, ZSD17, DLT18a, DLT18b, GKM18]. However, our problem is a strict generalization of the one-hidden layer neural network and is not covered by the above mentioned results.

Notations. For any function , we define to be . For two functions , we use the shorthand (resp. ) to indicate that (resp. ) for an absolute constant . We use to mean for constants . We use to denote .

Roadmap. We first present the formal model and the corresponding optimization problem in Section 2. We then present the local strong convexity and local linear convergence results in Section 3. Finally, we demonstrate the empirical superiority of NIMC when compared to linear IMC (Section 4).

2 Problem Formulation

Consider a user-item recommender system, where we have users with feature vectors , items with feature vectors and a collection of partially-observed user-item ratings, . That is is the rating that user gave for item . For simplicity, we assume ’s and ’s are sampled i.i.d. from distribution and , respectively. Each element of the index set is also sampled independently and uniformly with replacement from .

In this paper, our goal is to predict the rating for any user-item pair with feature vectors and , respectively. We model the user-item ratings as:

(2)

where , and is a non-linear activation function. Under this realizable model, our goal is to recover from a collection of observed entries, . Without loss of generality, we set . Also we treat as a constant throughout the paper. Our analysis requires to be full column rank, so we require . And w.l.o.g., we assume , i.e., the smallest singular value of both and is .

Note that this model is similar to one-hidden layer feed-forward network popular in standard classification/regression tasks. However, as there is an inner product between the output of two non-linear layers, and , it cannot be modeled by a single hidden layer neural network (with same number of nodes). Also, for linear activation function, the problem reduces to inductive matrix completion [ABEV06, JD13].

Now, to solve for , , we optimize a simple squared-loss based optimization problem, i.e.,

(3)

Naturally, the above problem is a challenging non-convex optimization problem that is strictly harder than two non-convex optimization problems which are challenging in their own right: a) the linear inductive matrix completion where non-convexity arises due to bilinearity of , and b) the standard one-hidden layer neural network (NN). In fact, recently a lot of research has focused on understanding various properties of both the linear inductive matrix completion problem [GJZ17, JD13] as well as one-hidden layer NN [GLM18, ZSJ17].

In this paper, we show that despite the non-convexity of Problem (3), it behaves as a convex optimization problem close to the optima if the data is sampled stochastically from a Gaussian distribution. This result combined with standard tensor decomposition based initialization [ZSJ17, KCL15, JSA15] leads to a polynomial time algorithm for solving (3) optimally if the data satisfies certain sampling assumptions in Theorem 2.1. Moreover, we also discuss the effect of various activation functions, especially the difference between a sigmoid activation function vs RELU activation (see Theorem 3.2 and Theorem 3.4).

Informally, our recovery guarantee can be stated as follows,

Theorem 2.1 (Informal Recovery Guarantee).

Consider a recommender system with a realizable model Eq. (2) with sigmoid activation, Assume the features and are sampled i.i.d. from the normal distribution and the observed pairs are i.i.d. sampled from uniformly at random. Then there exists an algorithm such that can be recovered to any precision with time complexity and sample complexity (refers to ) polynomial in the dimension and the condition number of , and logarithmic in .

3 Main Results

Our main result shows that when initialized properly, gradient-based algorithms will be guaranteed to converge to the ground truth. We first study the Hessian of empirical risk for different activation functions, then based on the positive-definiteness of the Hessian for smooth activations, we show local linear convergence of gradient descent. The proof sketch is provided in Appendix C.

The positive definiteness of the Hessian does not hold for several activation functions. Here we provide some examples. Counter Example 1) The Hessian at the ground truth for linear activation is not positive definite because for any full-rank matrix , is also a global optimal. Counter Example 2) The Hessian at the ground truth for ReLU activation is not positive definite because for any diagonal matrix with positive diagonal elements, is also a global optimal. These counter examples have a common property: there is redundancy in the parameters. Surprisingly, for sigmoid and tanh, the Hessian around the ground truth is positive definite. More surprisingly, we will later show that for ReLU, if the parameter space is constrained properly, its Hessian at a given point near the ground truth can also be proved to be positive definite with high probability.

3.1 Local Geometry and Local Linear Convergence for Sigmoid and Tanh

We define two natural condition numbers for the problem that captures the "hardness" of the problem:

Definition 3.1.

Define and , where , , and denotes the -th singular value of with the ordering .

First we show the result for sigmoid and tanh activations.

Theorem 3.2 (Positive Definiteness of Hessian for Sigmoid and Tanh).

Let the activation function in the NIMC model (2) be sigmoid or tanh and let be as defined in Definition 3.1. Then for any and any given , if

then with probability at least , the smallest eigenvalue of the Hessian of Eq. (3) is lower bounded by:

Remark. Theorem 3.2 shows that, given sufficiently large number of user-items ratings and a sufficiently large number of users/items themselves, the Hessian at a point close enough to the true parameters , , is positive definite with high probability. The sample complexity, including and , have a near-linear dependency on the dimension, which matches the linear IMC analysis [JD13]. Strong convexity parameter as well as the sample complexity depend on the condition number of as defined in Definition 3.1. Although we don’t explicitly show the dependence on , both sample complexity and the minimal eigenvalue scale as a polynomial of . The proofs can be found in Appendix C.

As the above theorem shows the Hessian is positive definite w.h.p. for a given that is close to the optima. This result along with smoothness of the activation function implies linear convergence of gradient descent that samples a fresh batch of samples in each iteration as shown in the following, whose proof is postponed to Appendix E.1.

Theorem 3.3.

Let be the parameters in the -th iteration. Assuming , then given a fresh sample set, , that is independent of and satisfies the conditions in Theorem 3.2, the next iterate using one step of gradient descent, i.e., satisfies

with probability , where is the step size and is the lower bound on the eigenvalues of the Hessian and is the upper bound on the eigenvalues of the Hessian.

Remark. The linear convergence requires each iteration has a set of fresh samples. However, since it converges linearly to the ground-truth, we only need iterations, therefore the sample complexity is only logarithmic in . This dependency is better than directly using Tensor decomposition method [JSA15], which requires samples. Note that we only use Tensor decomposition to initialize the parameters. Therefore the sample complexity required in our tensor initialization does not depend on .

3.2 Empirical Hessian around the Ground Truth for ReLU

We now present our result for ReLU activation. As we see in Counter Example 2, without any further modification, the Hessian for ReLU is not locally strongly convex due to the redundancy in parameters. Therefore, we reduce the parameter space by fixing one parameter for each pair, . In particular, we fix when minimizing the objective function, Eq. (3), where is -th element in the first row of . Note that as long as , can be fixed to any other non-zero values. We set just for simplicity of the proof. The new objective function can be represented as

(4)

where is the first row of and .

Surprisingly, after fixing one parameter for each pair, the Hessian using ReLU is also positive definite w.h.p. for a given around the ground truth.

Theorem 3.4 (Positive Definiteness of Hessian for ReLU).

Define . For any and any given , if

then with probability , the minimal eigenvalue of the objective for ReLU activation function, Eq. (4), is lower bounded,

Remark. Similar to the sigmoid/tanh case, the sample complexity for ReLU case also has a linear dependency on the dimension. However, here we have a worse dependency on the condition number of the weight matrices. The scale of can also be important and in practice one needs to set it carefully. Note that although the activation function is not smooth, the Hessian at a given point can still exist with probability , since ReLU is smooth almost everywhere and there are only a finite number of samples. However, owing to the non-smoothness, a proof of convergence of gradient descent method for ReLU is still an open problem.

3.3 Initialization

To achieve the ground truth, our algorithm needs a good initialization method that can initialize the parameters to fall into the neighborhood of the ground truth. Here we show that this is possible by using tensor method under the Gaussian assumption.

In the following, we consider estimating . Estimating is similar.

Define , where . Define . Then where and . When , we can approximately recover and from the empirical version of using non-orthogonal tensor decomposition [KCL15]. When is sigmoid, . Given , we can estimate , since is a monotonic function w.r.t. . Applying Lemma B.7 in [ZSJ17], we can bound the approximation error of empirical and population using polynomial number of samples. By [KCL15], we can bound the estimation error of and . Finally combining Theorem 3.2, we are able to show the recovery guarantee for sigmoid activation, i.e., Theorem 2.1.

Although tensor initialization has nice theoretical guarantees and sample complexity, it heavily depends on Gaussian assumption and realizable model assumption. In contrast, practitioners typically use random initialization.

4 Experiments

Dataset NIMC IMC NIMC- RFF IMC- RFF mushroom 8124 112 2 5 0 0.0049 0 0 20 0 0.0010 0 0 segment 2310 19 7 5 0.0543 0.0694 0.0197 0.0257 20 0.0655 0.0768 0.0092 0.0183 covtype 1708 54 7 5 0.1671 0.1733 0.1548 0.1529 20 0.1555 0.1600 0.1200 0.1307 letter 15000 16 26 5 0.0590 0.0704 0.0422 0.0430 20 0.0664 0.0760 0.0321 0.0356 yalefaces 2452 100 38 5 0.0315 0.0329 0.0266 0.0273 20 0.0212 0.0277 0.0064 0.0142 usps 7291 256 10 5 0.0211 0.0361 0.0301 0.0185 20 0.0184 0.0320 0.0199 0.0152
Figure 1: The left two figures (top: sigmoid, bottom: ReLU) plot rate of success of GD over synthetic data. White blocks denote success rate. The right table presents error in semi-supervised clustering using NIMC and IMC.

In this section, we show experimental results on both synthetic data and real-world data. Our experiments on synthetic data are intended to verify our theoretical analysis, while the real-world data shows the superior performance of NIMC over IMC. We apply gradient descent with random initialization to both NIMC and IMC.

4.1 Synthetic Data

We first generate some synthetic datasets to verify the sample complexity and the convergence of gradient descent using random initialization. We fix . For sigmoid, set the number of samples and the number of observations . For ReLU, set and . The sampling rule follows our previous assumptions. For each pair, we make 5 trials and take the average of the successful recovery times. We say a solution () successfully recovers the ground truth parameters when the solution achieves 0.001 relative testing error, i.e., where is a newly sampled testing dataset. For both ReLU and sigmoid, we minimize the original objective function (3). We illustrate the recovery rate in left figures in Figure 1. As we can see, ReLU requires more samples/observations than that for sigmoid for exact recovery (note the scales of and are different in the two figures). This is consistent with our theoretical results. Comparing Theorem 3.2 and Theorem 3.4, we can see the sample complexity for ReLU has a worse dependency on the conditioning of than sigmoid. We can also see that when is sufficiently large, the number of observed ratings required remains the same for both methods. This is also consistent with the theorems, where is near-linear in and is independent of .

4.2 Semi-supervised Clustering

We apply NIMC to semi-supervised clustering and follow the experimental setting in GIMC [SCH16]. In this problem, we are given a set of items with their features, , where is the number of items and is the feature dimension, and an incomplete similarity matrix , where if -th item and -th item are similar and if -th item and -th item are dissimilar. The goal is to do clustering using both existing features and the partially observed similarity matrix. We build the dataset from a classification dataset where the label of each item is known and will be used as the ground truth cluster. We first compute the similarity matrix from the labels and sample entries uniformly as the observed entries. Since there is only one features we set in the objective function Eq. (3).

We initialize and to be the same Gaussian random matrix, then apply gradient descent. This guarantees and to keep identical during the optimization process. Once converges, we take the top left singular vectors of to do k-means clustering. The clustering error is defined as in [SCH16]. Like [SCH16], we define the clustering error as follows,

where is the ground-truth clustering and is the predicted clustering. We compare NIMC of a ReLU activation function with IMC on six datasets using raw features and random Fourier features (RFF). The random Fourier feature is and each entry of is i.i.d. sampled from . We use Random Fourier features in order to see how increasing the depth of the neural network changes the performance. However, our analysis only works for one-layer neural networks, therefore, we use Random Fourier features, which can be viewed as using two-layer neural networks but with the first-layer parameters fixed.

is chosen such that a linear classifier using these random features achieves the best classification accuracy. is set as for all datasets. Datasets mushroom, segment, letter,usps,covtype are downloaded from libsvm website. We subsample covtype dataset to balance the samples from different classes. We preprocess yalefaces dataset as described in [KTWA14]. As shown in the right table in Figure 1, when using raw features, NIMC achieves better clustering results than IMC for all the cases. This is also true for most cases when using Random Fourier features.

4.3 Recommendation Systems

Recommender systems are used in many real situations. Here we consider two tasks.

Movie recommendation for users. We use Movielens[Res97] dataset, which has not only the ratings users give movies but also the users’ demographic information and movies’ genre information. Our goal is to predict ratings that new users will give the existing movies. We randomly split the users into existing users (training data) and new users (testing data) with ratio 4:1. The user features include 21 types of occupations, 7 different age ranges and one gender information; the movie features include 18-19 (18 for ml-1m and 19 for ml-100k) genre features and 20 features from the top 20 right singular values of the training rating matrix (which has size #training users -by- #movies). In our experiments, we set to be 50. Here are our results on datasets ml-1m and ml-100k. For NIMC, we use ReLU activation. As shown in Table 1, NIMC achieves much smaller RMSE than IMC for both ml-100k and ml-1m datasets.

Dataset #movies #users # ratings # movie feat. # user feat.
RMSE
NIMC
RMSE
IMC
ml-100k 1682 943 100,000 39 29 1.034 1.321
ml-1m 3883 6040 1,000,000 38 29 1.021 1.320
Table 1: Test RMSE for recommending new users with movies on Movielens dataset.
(a) (b) (c) (d)
Figure 2: NIMC v.s. IMC on gene-disease association prediction task.

Gene-Disease association prediction. We use the dataset collected by [ND14], which has 300 gene features and 200 disease features. Our goal is to predict associated genes for a new disease given its features. Since the dataset only contains positive labels, this is a problem called positive-unlabeled learning [HND15] or one-class matrix factorization [YHDL17]. We adapt our objective function to the following objective,

(5)

where is the association matrix, is the set of indices for observed associations, is the complementary set of and is the penalty weight for unobserved associations. There are totally 12331 genes and 3209 diseases in the dataset. We randomly split the diseases into training diseases and testing diseases with ratio 4:1. The results are presented in Fig 2. We follow [ND14] and use the cumulative distribution of the ranks as a measure for comparing the performances of different methods, i.e., the probability that any ground-truth associated gene of a disease appears in the retrieved top- genes for this disease.

In Fig 2(a), we show how changes the performance of NIMC and IMC. In general, the higher , the better the performance. The performance of IMC becomes stable when is larger than 100, while the performance of NIMC is still increasing. Although IMC performs better than NIMC when is small, the performance of NIMC increases much faster than IMC when increases. is fixed as and in the experiment for Fig 2(a). In Fig. 2(b), we present how in Eq. (5) affects the performance. We tried over to check how the value of changes the performance. As we can see, and give the best results. Fig. 2(c) shows the probability that any ground-truth associated gene of a disease appears in the retrieved top- genes for this disease w.r.t. different ’s. Here we fix , and . Fig. 2(d) shows the precision-recall curves for different methods when , and .

5 Conclusion

In this paper, we studied a nonlinear IMC model that represents one of the simplest inductive model for neural-network-based recommender systems. We study local geometry of the empirical risk function and show that, close to the optima, the function is strongly convex for both ReLU and sigmoid activations. Therefore, using a smooth activation function like sigmoid activation along with standard tensor initialization, gradient descent recovers the underlying model with polynomial sample complexity and time complexity. Thus we provide the first theoretically rigorous result for the non-linear recommendation system problem, which we hope will spur further progress in the area of deep-learning based recommendation systems. Our experimental results on synthetic data matches our analysis and the results on real-world benchmarks for semi-supervised clustering and recommendation systems show a superior performance over linear IMC.

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Appendix

Appendix A Notation

For any positive integer , we use to denote the set . For random variable , let denote the expectation of (if this quantity exists).

For any vector , we use to denote its norm.

We provide several definitions related to matrix . Let denote the determinant of a square matrix . Let denote the transpose of . Let denote the Moore-Penrose pseudoinverse of . Let denote the inverse of a full rank square matrix. Let denote the Frobenius norm of matrix . Let denote the spectral norm of matrix . Let to denote the -th largest singular value of .

We use to denote the indicator function, which is if holds and otherwise. Let denote the identity matrix. We use to denote an activation function. We use to denote a Gaussian distribution . For integer , we use to denote .

For any function , we define to be . In addition to notation, for two functions , we use the shorthand (resp. ) to indicate that (resp. ) for an absolute constant . We use to mean for constants .

Appendix B Preliminaries

We state some useful facts in this section.

Fact B.1.

Let . Let denote the vector where the -th entry is , . Let denote the vector that the -th entry is , . We have the following properties,

Proof.

Using the definition, it is easy to see that (\@slowromancapi@), (\@slowromancapii@) and (\@slowromancapiii@) are holding.

Proof of (\@slowromancapiv@), we have

where the last step follows by (\@slowromancapii@) and (\@slowromancapiii@). ∎

Fact B.2.

Let . Let denote the vector where the -th entry is , . Let denote the vector that the -th entry is , . We have the following properties,

Proof.

Proof of (\@slowromancapi@). We have

Proof of (\@slowromancapii@). It is similar to (\@slowromancapi@).

Proof of (\@slowromancapiii@). We have

Proof of (\@slowromancapiv@). We have

where the second step follows by , the third step follows by . ∎

Appendix C Proof Sketch

At high level the proofs for Theorem 3.2 and Theorem 3.4 include the following steps. 1) Show that the population Hessian at the ground truth is positive definite. 2) Show that population Hessians near the ground truth are also positive definite. 3) Employ matrix Bernstein inequality to bound the population Hessian and the empirical Hessian.

We now formulate the Hessian. The Hessian of Eq. (3), , can be decomposed into two types of blocks, (),

where (, resp.) is the -th column of (-th column of , resp.). Note that each of the above second-order derivatives is a matrix.

The first type of blocks are given by:

where , , and

For sigmoid/tanh activation function, the second type of blocks are given by:

(6)

For ReLU/leaky ReLU activation function, the second type of blocks are given by:

Note that the second term of Eq. (6) is missing here as are fixed, the number of samples is finite and almost everywhere.

In this section, we will discuss important lemmas/theorems for Step 1 in Appendix C.1 and Step 2,3 in Appendix C.3.

c.1 Positive definiteness of the population hessian

The corresponding population risk for Eq. (3) is given by:

(7)

where . For simplicity, we also assume and are normal distributions.

Now we study the Hessian of the population risk at the ground truth. Let the Hessian of at the ground-truth be , which can be decomposed into the following two types of blocks (),

To study the positive definiteness of , we characterize the minimal eigenvalue of by a constrained optimization problem,

(8)

where denotes that . Obviously, due to the squared loss and the realizable assumption. However, this is not sufficient for the local convexity around the ground truth, which requires the positive (semi-)definiteness for the neighborhood around the ground truth. In other words, we need to show that is strictly greater than , so that we can characterize an area in which the Hessian still preserves positive definiteness (PD) despite the deviation from the ground truth.

Challenges. As we mentioned previously there are activation functions that lead to redundancy in parameters. Hence one challenge is to distill properties of the activation functions that preserve the PD. Another challenge is the correlation introduced by when it is non-orthogonal. So we first study the minimal eigenvalue for orthogonal and orthogonal and then link the non-orthogonal case to the orthogonal case.

c.2 Warm up: orthogonal case

In this section, we consider the case when are unitary matrices, i.e., . (). This case is easier to analyze because the dependency between different elements of or can be disentangled. And we are able to provide lower bound for the Hessian. Before we introduce the lower bound, let’s first define the following quantities for an activation function .

(9)