On Generalization Bounds of a Family of Recurrent Neural Networks

1 INTRODUCTION . Recurrent Neural Networks ( RNNs ) have successfully revolutionized sequential data analysis , and been widely applied to many real world problems , such as natural language processing ( Cho et al. , 2014 ; Bahdanau et al. , 2014 ; Sutskever et al. , 2014 ) , speech recognition ( Graves et al. , 2006 ; Mikolov et al. , 2010 ; Graves , 2012 ; Graves et al. , 2013 ) , computer vision ( Gregor et al. , 2015 ; Xu et al. , 2015 ; Donahue et al. , 2015 ; Karpathy and Fei-Fei , 2015 ) , healthcare ( Lipton et al. , 2015 ; Choi et al. , 2016a ; b ) , and robot control ( Ku and Lee , 1995 ; Lee and Teng , 2000 ; Yoo et al. , 2006 ) . Quite a few of these applications can be approached easily in our daily life , such as Google Translate , Google Now , Apple Siri , etc . The sequential modeling nature of RNNs is significantly different from feedforward neural networks , though they both have neurons as the basic components . RNNs exploit the internal state ( also known as hidden unit ) to process the sequence of inputs , which naturally captures the dependence of the sequence . RNNs can also be viewed as nonlinear dynamical systems , and reduced to linear dynamical systems given identity activation operators . Besides the vanilla version , RNNs have many other variants . A large class of variants incorporate the so-called “ gated ” units to trim RNNs for different tasks . Typical examples include Long Short-Term Memory ( LSTM , Hochreiter and Schmidhuber ( 1997 ) ) , Gated Recurrent Unit ( GRU , Jozefowicz et al . ( 2015 ) ) and Minimal Gated Unit ( MGU , Zhou et al . ( 2016 ) ) . The success of RNNs owes not only to their special network structures and the ability to fit training data , but also to their good generalization property : They can provide accurate predictions on unseen data . For example , Graves et al . ( 2013 ) report that after training with merely 462 speech samples , deep LSTM RNNs achieve a test set error of 17.7 % on TIMIT phoneme recognition benchmark , which is the best recorded score . Mikolov et al . ( 2010 ) also show that RNNs outperform significantly state-of-the-art backoff models for speech recognition . When using RNNs in Wall Street Journal task to predict the next word in textual data given the context , word error rate is reduced around 18 % compared to backoff models trained on the same amount of data , and 12 % when backoff model is trained on 5 times more data . Despite of the popularity of RNNs in applications , their theory is less studied than other feedforward neural networks ( Bartlett et al. , 2017 ; Neyshabur et al. , 2017 ; Golowich et al. , 2017 ; Li et al. , 2018 ) . There are still several long lasting fundamental questions regarding the approximation , trainability , and generalization of RNNs . In this paper , we propose to understand the generalization ability of RNNs and their variants . We aim to answer two questions from a theoretical perspective : Q.1 ) Do RNNs suffer from significant curse of dimensionality ? Q.2 ) What are the advantages of MGU and LSTM over vanilla RNNs ? The investigation of generalization properties of neural networks including RNNs has a long history . Most of these works adopt a layer wise analysis and establish their results by induction . For example , Haussler ( 1992 ) establishes a complexity bound for feedforward neural networks . The result assumes average Lipschitz constant of each layer to be greater than 1 and involves the product of these Lipschitz constants . Thus , the resulting bound is inevitably exponential in the depth of the network . Later , Dasgupta and Sontag ( 1996 ) and Koiran ( 1998 ) adopt a VC-dimension argument to show complexity bounds of RNNs that are polynomial in the size of the network . Both of the works , however , are based on oversimplified assumptions : Dasgupta and Sontag ( 1996 ) only consider linear RNNs for binary classification tasks ; Koiran ( 1998 ) assumes RNNs take the first coordinate of their hidden states as outputs . More recently , Bartlett et al . ( 2017 ) propose a new technique for developing generalization bounds for feedforward neural networks based on empirical Rademacher complexity under the PAC-Learning framework . Neyshabur et al . ( 2017 ) further adapt the technique to establish their generalization bound using the PAC-Bayes approach . Then the follow-up work Zhang et al . ( 2018 ) use the PAC-Bayes approach to establish a generalization bound for vanilla RNNs . Our theory is partially motivated by Bartlett et al . ( 2017 ) , but is quite different from Bartlett et al . ( 2017 ) and Zhang et al . ( 2018 ) . In particular , our analysis exploits the compositional nature of RNNs , and decouples the spectral norms of weight matrices and the number of weight parameters . This makes our analysis conceptually much simpler ( e.g . avoid layer wise analysis ) , and also yields better generalization bound than Zhang et al . 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< /latexit > Taking a sequence to sequence multiclass classification problem as an example , we observe m sequences of data points ( xi , t , zi , t ) T t=1 , where xi , t ∈ Rdx and the class label zi , t ∈ { 1 , . . . , K } for all t = 1 , ... , T and i = 1 , ... , m. Each sequence is drawn independently from some underlying distribution over Rdx×T × { 1 , . . . , K } . The vanilla RNNs compute hi , t and yi , t iteratively as follows , hi , t = σh ( Uhi , t−1 +Wxi , t ) , and yi , t = σy ( V hi , t ) , where σy and σh are activation operators , hi , t ∈ Rdh is the hidden state with hi,0 = 0 , and U ∈ Rdh×dh , V ∈ Rdy×dh , and W ∈ Rdh×dx are weight matrices . The activation operators σh and σy are entrywise , i.e. , σh ( [ v1 , . . . , vd ] > ) = [ σh ( v1 ) , . . . , σh ( vd ) ] > , and Lipschitz with parameters ρh and ρy respectively . For simplicity , we assume σh ( · ) = tanh ( · ) , σy ( 0 ) = 0 , and ρy = 1 . Extensions to general activation operators are given in Section 2 . For a new testing sequence ( xt , zt ) T t=1 , we predict the label sequence using z̃t = argmaxj [ yt ] j , for all t = 1 , . . . , T. To establish the generalization bound , we need to define the “ model complexity ” of vanilla RNNs . In this paper , we adopt the empirical Rademacher complexity ( ERC , see more details in Section 2 ) , which has been widely used in the existing literature on PAC-Learning . For many nonparametric function classes , we often need complicated argument to upper bound their ERC . Our analysis , however , shows that we can upper bound the ERC of vanilla RNNs in a very simple manner by exploiting their Lipschitz continuity with respect to ( w.r.t ) the model parameters , since they are essentially in parametric forms . More specifically , denote Ft = { ft : { x1 , ... , xt } 7→ yt } as the class of mappings from the first t inputs to the t-th output computed by vanilla RNNs . For a matrix A , ‖A‖2 denotes the spectral norm , and for a vector v , ‖v‖2 denotes the Euclidean norm . Define xt−1 x−1 = t for x = 1 . Then , informally speaking , the “ model complexity ” of vanilla RNNs satisfies Complexity = O ( dmin { √ d , ‖W‖2 ‖U‖t2 − 1 ‖U‖2 − 1 } ‖V ‖2 √ log ( t √ d ‖U‖t2 − 1 ‖U‖2 − 1 ) ) , where d = √ dxdh + d2h + dhdy . We then give the generalization bound in the following statement . Theorem 1 ( informal ) . Given a collection of samples S = { ( xi , t , zi , t ) T t=1 , i = 1 , ... , m } with ‖xi , t‖2 ≤ 1 and a new testing sequence ( xt , zt ) Tt=1 , with probability at least 1 − δ over S , for every margin value γ > 0 and ft ∈ F for integer t ≤ T , we have , P ( z̃t 6= zt ) ≤ R̂γ , t +O Complexity√ mγ + √ log 1δ m , ( 1 ) where R̂γ , t = 1m ∑m i=1 1 ( [ yi , t ] zi , t ≤ maxj 6=zi , t [ yi , t ] j + γ ) . Please refer to Section 2 for a complete statement . The generalization bound in Theorem 1 can be interpreted under three different scenarios1 : ( I ) When ‖U‖2 < 1 , the generalization bound is Õ ( d√ mγ ) , which only has a logarithmic dependence on t ; ( II ) When ‖U‖2 = 1 , the generaliza- tion bound is Õ ( dt√ mγ ) , which has a linear dependence on d and t ; ( III ) When ‖U‖2 > 1 , the generalization bound is Õ ( √ d3t√ mγ ) , which has a polynomial dependence on d and t. We theoretically justify that vanilla RNNs do not suffer from significant curse of dimensionality . Because they compute outputs yt recursively using the same weight matrices , and their hidden states ht are entrywise bounded . Compared with the generalization bound in Zhang et al . ( 2018 ) , which is of the order Õ ( dt2‖W‖2‖V ‖2 max { 1 , ‖U‖t2 } √ mγ ) , our bound is tighter by a factor of t2 for case ( I ) , a factor of t for case ( II ) . Additionally , Zhang et al . ( 2018 ) fail to incorporate the boundedness condition of hidden state into their analysis , thus the generalization bound is exponential in t for case ( III ) . Our generalization bound , however , is still polynomial in d and t for case ( III ) . Moreover , ( II ) is closely related to a few recent results on imposing orthogonal constraints on weight matrices to stabilize the training of RNNs ( Saxe et al. , 2013 ; Le et al. , 2015 ; Arjovsky et al. , 2016 ; Vorontsov et al. , 2017 ; Zhang et al. , 2018 ) . We remark that from a learning theory perspective , ( II ) also implies that the orthogonal constraints can potentially help generalization . We also present refined generalization bounds with additional matrix norm assumptions . These assumptions allow us to derive norm-based generalization bounds . We draw a comparison among these bounds and highlight their advantage under different scenarios . Our theory can be further extended to several variants , including MGU and LSTM RNNs . Specifically , we show that the gated units in MGU and LSTM RNNs can introduce extra decaying factors to further reduce the dependence on d and t in generalization . Such an advantage in generalization make these RNNs do not suffer from significant curse of dimensionality either . To the best of our knowledge , these are the first results on generalization guarantees for these neural networks . The rest of the paper is organized as follows : Section 2 presents the generalization bound of vanilla RNNs ; Section 3 presents the proof outline of the generalization bound ; Section 4 presents refined generalization bounds and their comparison ; Section 5 presents the generalization bound of MGU and LSTM RNNs ; Section 7 discusses related works and collects open problems . Notations : Given a vector v ∈ Rd , we denote the vector Euclidean norm by ‖v‖22 = ∑d i=1 |vi|2 , and the infinity norm by ‖v‖∞ = maxj |vj | . Given a matrix M ∈ Rm×n , we denote the spectral norm by ‖M‖2 as the largest singular value ofM , the Frobenius norm by ‖M‖2F = trace ( MM > ) , and the ( 2 , 1 ) norm by ‖M‖2,1 = ∑n i=1 ‖M : ,i‖2 . Given a function f , we denote the function infinity norm by ‖f‖∞ = sup |f | . We adopt the standard O ( · ) notation , which is defined as f ( x ) = O ( g ( x ) ) for x → ∞ if and only if there exists M > 0 and x0 , such that |f ( x ) | ≤ Mg ( x ) for x ≥ x0 . We use Õ ( · ) to denote O ( · ) with hidden log factors . 1To ease the discussion , we assume ‖U‖2 does not scale with t. Therefore , ‖U‖2 < 1 is equivalent to ‖U‖2 ≤ 1−∆ for a constant ∆ > 0 . A precise statement can be found following Theorem 2 .

Summary: The paper focuses on the generalization performance of RNNs and its variant in a theoretical perspective. Compared to the previous result (Zhang et al., 2018) for RNNs, this paper refines the generalization bounds for vanilla RNNs in all cases and fills the blank for RNN variants like MGU and LSTM. Specifically, in the work of Zhang et al. (2018), the complexity term quadratically depends on the layer (or say, current sequence length, denoted by t in original paper), making it less instructive. This paper improves it to at most linear dependence and can achieve at logarithmic dependence in some cases, which should be accredited.