Deep Learning is Singular, and That's Good

In singular models , the optimal set of parameters forms an analytic set with singularities and classical statistical inference can not be applied to such models . This is significant for deep learning as neural networks are singular and thus “ dividing ” by the determinant of the Hessian or employing the Laplace approximation are not appropriate . Despite its potential for addressing fundamental issues in deep learning , singular learning theory appears to have made little inroads into the developing canon of deep learning theory . Via a mix of theory and experiment , we present an invitation to singular learning theory as a vehicle for understanding deep learning and suggest important future work to make singular learning theory directly applicable to how deep learning is performed in practice . 1 INTRODUCTION . It has been understood for close to twenty years that neural networks are singular statistical models ( Amari et al. , 2003 ; Watanabe , 2007 ) . This means , in particular , that the set of network weights equivalent to the true model under the Kullback-Leibler divergence forms a real analytic variety which fails to be an analytic manifold due to the presence of singularities . It has been shown by Sumio Watanabe that the geometry of these singularities controls quantities of interest in statistical learning theory , e.g. , the generalisation error . Singular learning theory ( Watanabe , 2009 ) is the study of singular models and requires very different tools from the study of regular statistical models . The breadth of knowledge demanded by singular learning theory – Bayesian statistics , empirical processes and algebraic geometry – is rewarded with profound and surprising results which reveal that singular models are different from regular models in practically important ways . To illustrate the relevance of singular learning theory to deep learning , each section of this paper illustrates a key takeaway idea1 . The real log canonical threshold ( RLCT ) is the correct way to count the effective number of parameters in a deep neural network ( DNN ) ( Section 4 ) . To every ( model , truth , prior ) triplet is associated a birational invariant known as the real log canonical threshold . The RLCT can be understood in simple cases as half the number of normal directions to the set of true parameters . We will explain why this matters more than the curvature of those directions ( as measured for example by eigenvalues of the Hessian ) laying bare some of the confusion over “ flat ” minima . For singular models , the Bayes predictive distribution is superior to MAP and MLE ( Section 5 ) . In regular statistical models , the 1 ) Bayes predictive distribution , 2 ) maximum a posteriori ( MAP ) estimator , and 3 ) maximum likelihood estimator ( MLE ) have asymptotically equivalent generalisation error ( as measured by the Kullback-Leibler divergence ) . This is not so in singular models . We illustrate in our experiments that even “ being Bayesian ” in just the final layers improves generalisation over MAP . Our experiments further confirm that the Laplace approximation of the predictive distribution Smith & Le ( 2017 ) ; Zhang et al . ( 2018 ) is not only theoretically inappropriate but performs poorly . Simpler true distribution means lower RLCT ( Section 6 ) . In singular models the RLCT depends on the ( model , truth , prior ) triplet whereas in regular models it depends only on the ( model , prior ) pair . The RLCT increases as the complexity of the true distribution relative to the supposed model increases . We verify this experimentally with a simple family of ReLU and SiLU networks . 1The code to reproduce all experiments in the paper will be released on Github . For now , see the zip file . 2 RELATED WORK . In classical learning theory , generalisation is explained by measures of capacity such as the l2 norm , Radamacher complexity , and VC dimension ( Bousquet et al. , 2003 ) . It has become clear however that these measures can not capture the empirical success of DNNs ( Zhang et al. , 2017 ) . For instance , over-parameterised neural networks can easily fit random labels ( Zhang et al. , 2017 ; Du et al. , 2018 ; Allen-Zhu et al. , 2019b ) indicating that complexity measures such as Rademacher complexity are very large . There is also a slate of work on generalisation bounds in deep learning . Uniform convergence bounds ( Neyshabur et al. , 2015 ; Bartlett et al. , 2017 ; Neyshabur & Li , 2019 ; Arora et al. , 2018 ) usually can not provide non-vacuous bounds . Data-dependent bounds ( Brutzkus et al. , 2018 ; Li & Liang , 2018 ; Allen-Zhu et al. , 2019a ) consider the “ classifiability ” of the data distribution in generalisation analysis of neural networks . Algorithm-dependent bounds ( Daniely , 2017 ; Arora et al. , 2019 ; Yehudai & Shamir , 2019 ; Cao & Gu , 2019 ) consider the relation of Gaussian initialisation and the training dynamics of ( stochastic ) gradient descent to kernel methods ( Jacot et al. , 2018 ) . In contrast to many of the aforementioned works , we are interested in estimating the conditional distribution q ( y|x ) . Specifically , we measure the generalisation error of some estimate q̂n ( y|x ) in terms of the Kullback-Leibler divergence between q and q̂n , see ( 8 ) . The next section gives a crash course on singular learning theory . The rest of the paper illustrates the key ideas listed in the introduction . Since we cover much ground in this short note , we will review other relevant work along the way , in particular literature on “ flatness ” , the Laplace approximation in deep learning , etc . 3 SINGULAR LEARNING THEORY . To understand why classical measures of capacity fail to say anything meaningful about DNNs , it is important to distinguish between two different types of statistical models . Recall we are interested in estimating the true ( and unknown ) conditional distribution q ( y|x ) with a class of models { p ( y|x , w ) : w ∈ W } where W ⊂ Rd is the parameter space . We say the model is identifiable if the mapping w 7→ p ( y|x , w ) is one-to-one . Let q ( x ) be the distribution of x . The Fisher information matrix associated with the model { p ( y|x , w ) : w ∈W } is the matrix-valued function on W defined by I ( w ) ij = ∫ ∫ ∂ ∂wi [ log p ( y|x , w ) ] ∂ ∂wj [ log p ( y|x , w ) ] q ( y|x ) q ( x ) dxdy , if this integral is finite . Following the conventions in Watanabe ( 2009 ) , we have the following bifurcation of statistical models . A statistical model p ( y|x , w ) is called regular if it is 1 ) identifiable and 2 ) has positive-definite Fisher information matrix . A statistical model is called strictly singular if it is not regular . Let ϕ ( w ) be a prior on the model parameters w. To every ( model , truth , prior ) triplet , we can associate the zeta function , ζ ( z ) = ∫ K ( w ) zϕ ( w ) dw , z ∈ C , where K ( w ) is the Kullback-Leibler ( KL ) divergence between the model p ( y|x , w ) and the true distribution q ( y|x ) : K ( w ) : = ∫ ∫ q ( y|x ) log q ( y|x ) p ( y|x , w ) q ( x ) dx dy . ( 1 ) For a ( model , truth , prior ) triplet ( p ( y|x , w ) , q ( y|x ) , ϕ ) , let −λ be the maximum pole of the corresponding zeta function . We call λ the real log canonical threshold ( RLCT ) ( Watanabe , 2009 ) of the ( model , truth , prior ) triplet . The RLCT is the central quantity of singular learning theory . By Watanabe ( 2009 , Theorem 6.4 ) the RLCT is equal to d/2 in regular statistical models and bounded above by d/2 in strictly singular models if realisability holds : let W0 = { w ∈W : p ( y|x , w ) = q ( y|x ) } be the set of true parameters , we say q ( y|x ) is realisable by the model class if W0 is non-empty . The condition of realisability is critical to standard results in singular learning theory . Modifications to the theory are needed in the case that q ( y|x ) is not realisable , see the condition called relatively finite variance in Watanabe ( 2018 ) . Neural networks in singular learning theory . Let W ⊆ Rd be the space of weights of a neural network of some fixed architecture , and let f ( x , w ) : RN ×W −→ RM be the associated function . We shall focus on the regression task and study the model p ( y|x , w ) = 1 ( 2π ) M/2 exp ( − 12‖y − f ( x , w ) ‖ 2 ) ( 2 ) but singular learning theory can also apply to classification , for instance . It is routine to check ( see Appendix A.1 ) that for feedforward ReLU networks not only is the model strictly singular but the matrix I ( w ) is degenerate for all nontrivial weight vectors and the Hessian of K ( w ) is degenerate at every point of W0 . RLCT plays an important role in model selection . One of the most accessible results in singular learning theory is the work related to the widely-applicable Bayesian information criterion ( WBIC ) Watanabe ( 2013 ) , which we briefly review here for completeness . Let Dn = { ( xi , yi ) } ni=1 be a dataset of input-output pairs . Let Ln ( w ) be the negative log likelihood Ln ( w ) = − 1 n n∑ i=1 log p ( yi|xi , w ) ( 3 ) and p ( Dn|w ) = exp ( −nLn ( w ) ) . The marginal likelihood of a model { p ( y|x , w ) : w ∈ W } is given by p ( Dn ) = ∫ W p ( Dn|w ) ϕ ( w ) dw and can be loosely interpreted as the evidence for the model . Between two models , we should prefer the one with higher model evidence . However , since the marginal likelihood is an intractable integral over the parameter space of the model , one needs to consider some approximation . The well-known Bayesian Information Criterion ( BIC ) derives from an asymptotic approximation of − log p ( Dn ) using the Laplace approximation , leading to BIC = nLn ( wMLE ) + d2 log n. Since we want the marginal likelihood of the data for some given model to be high one should almost never adopt a DNN according to the BIC , since in such models d may be very large . However , this argument contains a serious mathematical error : the Laplace approximation used to derive BIC only applies to regular statistical models , and DNNs are not regular . The correct criterion for both regular and strictly singular models was shown in Watanabe ( 2013 ) to be nLn ( w0 ) +λ log nwherew0 ∈W0 and λ is the RLCT . Since DNNs are highly singular λ may be much smaller than d/2 ( Section 6 ) it is possible for DNNs to have high marginal likelihood – consistent with their empirical success .

Summary: This paper is more like a review of singular learning theory and its implication on deep learning. The authors point out that deep neural networks are singular models and ways to characterize generalization error for regular models cannot produce satisfactory results in this setting. Then the authors introduce the singular learning theory, which has been developed for decades. Then, a series of topics for deep learning, such as flatness and generalization, are studied within the framework singular learning theory, with a combination of theoretical analysis and numerical experiments. The paper is clearly written and well organized.