Scalable optimization has been playing crucial roles in the success of deep learning, which has immense applications in artificial intelligence. Remarkably, optimization issues are often addressed through designing new models that make the resulting training objective functions easier to be optimized. For example, over-parameterization[LSSS14]IS15], and residual networks [HZRS16a, HZRS16b] are often considered as ways to improve the optimization landscape of the resulting objective functions.
How do we design models and objective functions that allow efficient optimization with guarantees? Towards understanding this question in a principled way, this paper studies learning neural networks with one hidden layer. Roughly speaking, we will show that when the input is from Gaussian distribution and under certain simplifying assumptions on the weights, we can design an objective function , such that
[a] all local minima of are global minima
[b] all the global minima are the desired solutions, namely, the ground-truth parameters (up to permutation and some fixed transformation).
We note that designing such objective functions is challenging because 1) the natural loss objective does have bad local minimum, and 2) due to the permutation invariance111Permuting the rows of and the coordinates of correspondingly preserves the functionality of the network., the objective function inherently has to contain an exponential number of isolated local minima.
1.1 Setup and known issues with proper learning
We aim to learn a neural network with a one-hidden-layer using a non-convex objective function. We assume input comes from Gaussian distribution and the label comes from the model
where are the ground-truth parameters, is a element-wise non-linear function, and is a noise vector with zero mean. Here we can without loss of generality assume comes from spherical Gaussian distribution . 222This is because if , then we can whiten the data by taking and define . We note that and therefore we main the functionality of the model.
For technical reasons, we will further assume and that has non-negative entries.
The most natural learning objective is perhaps the loss function, given the additive noise. Concretely, we can parameterize with training parameters of the same dimension as and correspondingly,
and then use stochastic gradient descent to optimize the loss function. When we have enough training examples, we are effectively minimizing the following population risk with stochastic updates,
However, empirically stochastic gradient descent cannot converge to the ground-truth parameters in the synthetic setting above when , even if we have access to an infinite number of samples, and
is a orthogonal matrix. Such empirical results have been reported in[LSSS14] previously, and we also provide our version in Figure 1 of Section 6. This is consistent with observations and theory that over-parameterization is crucial for training neural networks successfully [LSSS14, HMR16, SC16].
These empirical findings suggest that the population risk has spurious local minima with inferior error compared to that of the global minimum. This phenomenon occurs even if we assume we know or is merely just the all one’s vector. Empirically, such landscape issues seem to be alleviated by over-parameterization. By contrast, our method described in the next section does not require over-parameterization and might be suitable for applications that demand the recovery of the true parameters.
1.2 Our contributions
Towards learning with the same number of training parameters as the ground-truth model, we first study the landscape of the population risk and give an analytic formula for it — as an explicit function of the ground-truth parameter and training parameter with the randomness of the data being marginalized out. The formula in equation (2.3) shows that is implicitly attempting to solve simultaneously a finite number of low-rank tensor decomposition problems with commonly shared components.
Inspired by the formula, we design a new training model whose associated loss function — named and formally defined in equation (2.6) — corresponds to the loss function for decomposing a matrix (2-nd order tensor) and a 4-th order tensor (Theorem 2.2). Empirically, stochastic gradient descent on learns the network as shown in experiment section (Section 6).
Despite the empirical success of , we still lack a provable guarantee on the landscape of . The second contribution of the paper is to design a more sophisticated objective function whose landscape is provably nice — all the local minima of are proven to be global, and they correspond to the permutation of the true parameters. See Theorem 2.3.
Moreover, the value and the gradient of can be estimated using samples, and there are no constraints in the optimization. These allow us to use straightforward stochastic gradient descent (see guarantees in [GHJY15, JGN17]) to optimize and converge to a local minimum, which is also a global minimum (Corollary 2.4).
Finally, we also prove a finite-sample complexity result. We will show that with a polynomial number of samples, the empirical version of share almost the same landscape properties as itself (Theorem 2.7). Therefore, we can also use an empirical version of as a surrogate in the optimization.
1.3 Related work
The work of Arora et al. [ABGM14] is one of the early results on provable algorithms for learning deep neural networks, where the authors give an algorithm for learning deep generative models with sparse weights. Livni et al. [LSSS14], Zhang et al. [ZLJ16, ZLWJ17], and Daniely et al. [DFS16] study the learnability of special cases of neural networks using ideas from kernel methods. Janzamin et al. [JSA15]
give a polynomial-time algorithm for learning one-hidden-layer neural networks with twice-differential activation function and known input distributions, using the ideas from tensor decompositions.
A series of recent papers study the theoretical properties of non-convex optimization algorithms for one-hidden-layer neural networks. Brutzkus and Globerson [BG17] and Tian [Tia17] analyze the landscape of the population risk for one-hidden-layer neural networks with Gaussian inputs under the assumption that the weights vector associated to each hidden variable (that is, the filters) have disjoint supports. Li and Yuan [LY17]
prove that stochastic gradient descent recovers the ground-truth parameters when the parameters are known to be close to the identity matrix. Zhang et al.[ZPS17] studies the optimization landscape of learning one-hidden-layer neural networks with a specific activation function, and they design a specific objective function that can recover a single column of the weight matrix. Zhong et al. [ZSJ17]
studies the convergence of non-convex optimization from a good initializer that is produced by tensor methods. Our algorithm works for a large family of activation functions (including ReLU) and any full-rank weight matrix. To our best knowledge, we give the first global convergence result for gradient-based methods for our general setting.333The work of [JSA15, ZSJ17] are closely related, but they require tensor decomposition as the algorithm/initialization.
The optimization landscape properties have also been investigated on simplified neural networks models. Kawaguchi [Kaw16] shows that the landscape of deep neural nets does not have bad local minima but has degenerate saddle points. Hardt and Ma [HM17] show that re-parametrization using identity connection as in residual networks [HZRS16a] can remove the degenerate saddle points in the optimization landscape of deep linear residual networks. Soudry and Carmon [SC16] showed that an over-parameterized neural network does not have bad differentiable local minimum. Hardt et al. [HMR16] analyze the power of over-parameterization in a linear recurrent network (which is equivalent to a linear dynamical system.)
The optimization landscape has also been analyzed for other machine learning problems, including SVD/PCA phase retrieval/synchronization, orthogonal tensor decomposition, dictionary learning, matrix completion, matrix sensing[BH89, SJ13, GHJY15, SQW15, BBV16, GLM16, BNS16, GJZ17]. Our analysis techniques build upon that for tensor decomposition in [GHJY15] — we add two additional regularization terms to deal with spurious local minimum caused by the weights and to remove the constraints.
We use to denote the set of natural numbers and real numbers respectively. We use to denote the Euclidean norm of a vector and spectral norm of a matrix. We use to denote the Frobenius/Euclidean norm of a matrix or high-order tensor. For a vector , let denotes its infinity norm and for a matrix , let be a shorthand for where is the vectorization of . For a vector , let denotes the second largest absolute values of the entries for . We note that is not a norm.
We use to denote the Kronecker product of and , and is a shorthand for where appears times. For vectors and denote the tensor product. We use
to denote the largest and smallest eigenvalues of a square matrix. Similarly,and
are used to denote the largest and smallest singular values. We denote the identity matrix in dimensionby , or Id when the dimension is clear from the context.
In the analysis, we rely on many properties of Hermite polynomials. We use to denote the -th normalized Hermite polynomial. These polynomials form an orthonormal basis. See Section 4.1 for an introduction of Hermite polynomials.
We will define other notations when we first use them.
2 Main Results
2.1 Connecting Population Risk with Tensor Decomposition
We first show that a natural loss for the one-hidden-layer neural network can be interpreted as simultaneously decomposing tensors of different orders.
A straightforward approach of learning the model (1.1) is to parameterize the prediction by
where are the training parameters. Naturally, we can use as the empirical loss, which means the population risk is
Throughout the paper, we use to denote the row vectors of and similarly for . That is, we have and . Let and ’s be the coordinates of and respectively.
We give the following analytic formula for the population risk defined above.
Connection to tensor decomposition: We see from equation (2.3) that the population risk of is essentially an average of infinite number of loss functions for tensor decomposition. For a fixed , we have that the -th summand in equation (2.3) is equal to (up to the scaling factor )
where is a -th order tensor in . We note that the objective naturally attempts to decompose the -order rank- tensor into rank-1 components .
Issues with optimizing :
It turns out that optimizing the population risk using stochastic gradient descent is empirically difficult. Figure 1 shows that in a synthetic setting where the noise is zero, the test error empirically doesn’t converge to zero for sufficiently long time with various learning rate schemes, even if we are using fresh samples in iteration. This suggests that the landscape of the population risk has some spurious local minimum that is not a global minimum. See Section 6 for more details on the experiment setup.
An empirical fix:
Inspired by the connection to tensor decomposition objective described earlier in the subsection, we can design a new objective function that takes exactly the same form as the tensor decomposition objective function . Concretely, let’s define
where and and are the 2nd and 4th normalized probabilists’ Hermite polynomials [Wik17b]. We abuse the notation slightly by using the same notation to denote the its element-wise application on a vector. Now for each example we use as loss function. The corresponding population risk is
Now by an extension of Theorem 2.1, we have that the new population risk is equal to the .
It turns out stochastic gradient descent on the objective (with projection to the set of matrices with row norm 1) converges empirically to the ground truth or one of its equivalent permutations. (See Figure 2.) However, we don’t know of any existing work for analyzing the landscape of the objective (or for any ). We conjecture that the landscape of doesn’t have any spurious local minimum under certain mild assumptions on . Despite recent attempts on other loss functions for tensor decomposition [GM17], we believe that analyzing is technically challenging and its resolution will be potentially enlightening for the understanding landscape of loss function with permutation invariance. See Section 6 for more experimental results.
2.2 Landscape design for orthogonal
The population risk defined in equation (2.6) — though works empirically for randomly generated ground-truth — doesn’t have any theoretical guarantees. It’s also possible that when are chosen adversarially or from a different distribution, SGD no longer converges to the ground-truth.
To solve this problem, we design another objective function , such that the optimizer of still corresponds to the ground-truth, and has provably nice landscape — all local minima of are global minima.
In this subsection, for simplicity, we work with the case when is an orthogonal matrix and state our main result. The discussion of the general case is deferred to the end of this Section and Section A.
We define our objective function as
where is defined as
and is defined as
The rationale behind of the choices of and will only be clearer and relevant in later sections. For now, the only relevant property of them is that both are smooth functions whose derivatives are easily computable.
We remark that we can sample using the samples straightforwardly — it’s defined as an average of functions of examples and the parameters. We also note that only parameter appears in the loss function. We will infer the value of
using straightforward linear regression after we get the (approximately) accurate value of.
Due to technical reasons, our method only works for the case when for every . We will assume this throughout the rest of the paper. The general case is left for future work. Let , , and . Our result will depend on the value of Essentially we treat as an absolute constant that doesn’t scale in dimension. The following theorem characterizes the properties of the landscape of .
Let be a sufficiently small universal constant (e.g. suffices) and suppose the activation function satisfies . Assume , , and is an orthogonal matrix. The function defined as in equation (2.8) satisfies that
A matrix is a local minimum of if and only if can be written as where is a permutation matrix and is a diagonal matrix with .555More precisely, Furthermore, this means that all local minima of are also global.
Any saddle point has a strictly negative curvature in the sense that where
Suppose is an approximate local minimum in the sense that satisfies
Then can be written as where is a permutation matrix, is a diagonal matrix satisfying the same bound as in bullet 1, and .
As a direct consequence, is -close to a global minimum in Euclidean distance, where hides polynomial dependency on and other parameters.
The theorem above implies that we can learn (up to permutation of rows and sign-flip) if we take to be sufficiently large and optimize using stochastic gradient descent. In this case, the diagonal matrix in bullet 1 is sufficiently close to identity (up to sign flip) and therefore a local minimum is close to up to permutation of rows and sign flip. The sign of each can be recovered easily after we recover (see Lemma 2.5 below.)
Stochastic gradient descent converges to a local minimum [GHJY15] (under the additional property as established in bullet 2 above), which is also a global minimum for the function . We will prove the theorem in Section 5 as a direct corollary of Theorem 5.1. The technical bullet 2 and 3 of the theorem is to ensure that we can use stochastic gradient descent to converge to a local minimum as stated below.666In the most general setting, converging to a local minimum of a non-convex function is NP-hard.
In the setting of Theorem 2.3, we can use stochastic gradient descent to optimize function (with fresh samples at each iteration) and converge to an approximate global minimum that is -close to a global minimum in time .
After approximately recovering the matrix , we can also recover the coefficient easily. Note that fixing , we can fit using simply linear regression. For the ease of analysis, we analyze a slightly different algorithm. The lemma below is proved in Section B.
Given a matrix whose rows have unit norm, and are -close to in Euclidean distance up to permutation and sign flip with . Then, we can give estimates (using e.g., Algorithm 1) such that there exists a permutation where and is row-wise -close to .
The key step towards analyzing objective function is the following theorem that gives an analytic formula for .
The function satisfies
Finite sample complexity bounds
: Extending Theorem 2.3, we can characterize the landscape of the empirical risk , which implies that stochastic gradient on also converges approximately to the ground-truth parameters with polynomial number of samples.
In the setting of Theorem 2.3, suppose we use empirical samples to approximate and obtain empirical risk . There exists a fixed polynomial such that if , then with high probability the landscape of
, then with high probability the landscape ofvery similar properties to that of .
Precisely, if is an approximate local minimum in the sense that and , then can be written as where is a permutation matrix, is a diagonal matrix and .
All of the results above assume that
is orthogonal. Since the local minimum are preserved by linear transformation of the input space, these results can be extended to the general case when
is not orthogonal but full rank (with some additional technicality) or the case when the dimension is larger than the number of neurons (). See Section A for details.
3 Overview: Landscape Design and Analysis
, we envision a family of possible objective functions for which we have unbiased estimators via samples. In Section3.2, we pick a specific function that feeds our needs: a) it has no spurious local minimum; b) the global minimum corresponds to the ground-truth parameters.
3.1 Which objective can be estimated by samples?
Recall that in equation (2.2) of Theorem 2.1 we give an analytic formula for the straightforward population risk . Although the population risk doesn’t perform well empirically, the lesson that we learn from it help us design better objective functions. One of the key fact that leads to the proof of Theorem 2.1 is that for any continuous and bounded function , we have that
Here and are the -th Hermite coefficient of the function and . That is, letting the -th normalized probabilists’ Hermite polynomials [Wik17b] and be the standard inner product between functions, we have .
Note that can be chosen arbitrarily to extract different terms. For example, by choosing , we obtain that
That is, we can always access functions forms that involves weighted sum of the powers of , as in RHS of equation (3.1).
Using a bit more technical tools in Fourier analysis (see details in Section 4), we claim that most of the symmetric polynomials over variables can be estimated by samples:
Claim 3.1 (informal).
For an arbitrary polynomial over a single variable, there exits a corresponding function such that
Moreover, for an any polynomial over two variables, there exists corresponding such that
3.2 Which objective has no spurious local minima?
As discussed briefly in the introduction, one of the technical difficulties to design and analyze objective functions for neural networks comes from the permutation invariance — if a matrix is a good solution, then any permutation of the rows of still gives an equally good solution (if we also permute the coefficients in accordingly). We only know of a very limited number of objective functions that guarantee to enjoy permutation invariance and have no spurious local minima [GHJY15].
We start by considering the objective function used in [GHJY15],
Note that here we overload the notation by using ’s to denote a set of fixed vectors that we wanted to recover and using ’s to denote the variables. Careful readers may notice that doesn’t fall into the family of functions that we described in the previous section (that is, RHS equation of (3.2) and (3.3)), because it lacks the weighting ’s. We will fix this issue later in the subsection. Before that we first summarize the nice properties of the landscape of .
For the simplicity of the discussion, let’s assume forms an orthonormal matrix in the rest of the subsection. Then, any permutation and sign-flip of the rows of leads to a global minimum of — when with a permutation matrix and a sign matrix (diagonal with ), we have that because one of and has to be zero for all 777Note that is orthogonal, and ).
It turns out that these permutations/sign-flips of are also the only local minima888We note that since there are constraints here, by local minimum we mean the local minimum on the manifold defined by the constraints. of function . To see this, notice that is a degree-2 polynomial of . Thus if we pick an index and fix every row except for , then is a quadratic function over unit vector
– reduces to an smallest eigenvector problem. Eigenvector problems are known to have no spurious local minimum. Thus the corresponding function (w.r.t) has no spurious local minimum. It turns out the same property still holds when we treat all the rows as variables and add the row-wise norm constraints (see proof in [GHJY15]).
However, there are two issues with using objective function . The obvious one is that it doesn’t involve the coefficients ’s and thus doesn’t fall into the forms of equation (3.3). Optimistically, we would hope that for nonnegative ’s the weighted version of below would also enjoy the similar landscape property
When ’s are positive, indeed the global minimum of are still just all the permutations of the .999This is the main reason why we require . However, when , we found that starts to have spurious local minima . It seems that spurious local minimum often occurs when a row of is a linear combination of a smaller number of rows of . See Section D for a concrete example.
To remove such spurious local minima, we add a regularization term below that pushes each row of to be close to one of the rows of ,
We see that for each fixed , the part in that involves has the form
This is commonly used objective function for decomposing tensor . It’s known that for orthogonal ’s, the only local minima are [GHJY15]. Therefore, intuitively pushes each of the ’s towards one of the ’s. 101010However, note that by itself doesn’t work because it does not prevent the solutions where all the ’s are equal to the same . Choosing to be small enough, it turns out that doesn’t have any spurious local minimum as we will show in Section 5.
Another issue with the choice of is that we are still having a constraint minimization problem. Such row-wise norm constraints only make sense when the ground-truth is orthogonal and thus has unit row norm. A straightforward generalization of to non-orthogonal case requires some special constraints that also depend on the covariance matrix , which in turn requires a specialized procedure to estimate. Instead, we move the constraints into the objective function by considering adding another regularization term that approximately enforces the constraints.
It turns out the following regularizer suffices for the orthogonal case,
Moreover, we can extend this easily to the non-orthogonal case (see Section A) without estimating any statistics of in advance. We note that is not the Lagrangian multiplier and it does change the global minima slightly. We will take to be large enough so that has to be close to 1. As a summary, we finally use the unconstrained objective
Since and are degree-4 polynomials of , the analysis of is much more delicate, and we cannot use much linear algebra as we could for . See Section 5 for details.
Finally we note that a feature of this objective is that it only takes as variables. We will estimate the value of after we recover the value of . (see Section B). ·
4 Analytic Formula for Population Risks
4.1 Basics on Hermite Polynomials
In this section, we briefly review Hermite polynomials and Fourier analysis on Gaussian space. Let be the probabilists’ Hermite polynomial [Wik17b], and let be the normalized Hermite polynomials. The normalized Hermite polynomial forms a complete orthonormal basis in the function space in the following sense111111We denote by the weighted space, namely, . For two functions that map to , define the inner product with respect to the Gaussian measure as
The polynomials are orthogonal to each other under this inner product:
Here if and otherwise . Given a function , let the -th Hermite coefficient of be defined as
Since forms a complete orthonormal basis, we have the expansion that
We will leverage several other nice properties of the Hermite polynomials in our proofs. The following claim connects the Hermite polynomial to the coefficients of Taylor expansion of a certain exponential function. It can also serve as a definition of Hermite polynomials.
Claim 4.1 ([O’d14, Equation 11.8]).
We have that for ,
The following Claims shows that the expectation can be computed easily when
are (correlated) Gaussian random variables.
Claim 4.2 ([O’d14, Section 11.2]).
Let be -correlated standard normal variables (that is, both , have marginal distribution and ). Then,
As a direct corollary, we can compute by expanding in the Hermite basis and applying the Claim above.
Let be two functions from to such that . Then, for any unit vectors , we have that
4.2 Analytic Formula for population risk and
Let , and with parameter and . Define the population risk as
Suppose and and ’s and ’s have unit norm. Then,
where are the -th Hermite coefficients of the function and respectively.
We can see that Theorem 2.1 follows from choosing and Theorem 2.2 follows from choosing . The key intuition here is that we can decompose into a weighted combination of Hermite polynomials, and each Hermite polynomial influence the population risk more or less independently (because they are orthogonal polynomials with respect to the Gaussian measure).
4.3 Analytic Formula for population risk
In this section we show that the population risk (defined as in equation (2.8)) has the following analytical formula:
Let be defined as in equation (2.10), we have that
Let be defined as in equation (2.9), then we have that
We start with a simple but fundamental lemma. Essentially all the result in this section follows from expanding the two sides of equation (4.1) below.
Let be two fixed vectors and . Then, for any ,
Using the fact that , we have that,