Discrete-Valued Neural Communication

1 Introduction . In AI , there has long been a tension between subsymbolic and symbolic architectures . Subsymbolic architectures , like neural networks , utilize continuous representations and statistical computation . Symbolic architectures , like production systems ( Laird et al. , 1986 ) and traditional expert systems , use discrete , structured representations and logical computation . Each architecture has its strengths : subsymbolic computation is useful for perception and control , symbolic computation for higher level , abstract reasoning . A challenge in integrating these approaches is developing unified learning procedures . As a step toward bridging the gap , recent work in deep learning has focused on constructing structured architectures with multiple components that interact with one another . For instance , graph neural networks are composed of distinct nodes ( Kipf et al. , 2019 ; Scarselli et al. , 2008 ; Kipf et al. , 2018 ; ∗co-first author . Address correspondence to liudianbo @ gmail.com , alex6200 @ gmail.com , and kkawaguchi @ fas.harvard.edu . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Santoro et al. , 2017 ; Raposo et al. , 2017 ; Bronstein et al. , 2017 ; Gilmer et al. , 2017 ; Tacchetti et al. , 2018 ; Van Steenkiste et al. , 2018 ) , transformers are composed of positional elements ( Bahdanau et al. , 2014 ; Vaswani et al. , 2017 ) , and modular models are divided into slots or modules with bandwidth limited communication ( Jacobs et al. , 1991 ; Bottou and Gallinari , 1991 ; Goyal and Bengio , 2020 ; Ronco et al. , 1997 ; Reed and De Freitas , 2015 ; Lamb et al. , 2021 ; Andreas et al. , 2016 ; Rosenbaum et al. , 2017 ; Fernando et al. , 2017 ; Shazeer et al. , 2017 ; Rosenbaum et al. , 2019 ; Kucinski et al. , 2021 ) . Although these structured models exploit the discreteness in their architectural components , the present work extends these models to leverage discreteness of representations , which is an essential property of symbols . We propose to learn a common codebook that is shared by all components for inter-component communication . The codebook permits only a discrete set of communicable values . We hypothesize that communication based on the use and re-use of discrete symbols will provide two benefits : • Discrete symbols limit the bandwidth of representations whose interpretation needs to be learned and synchronized across modules . It may therefore serve as a common language for interaction and facilitate learning . • The use of shared discrete symbols will promote systematic generalization by allowing for the re-use of previously encountered symbols in new situations . This makes it easier to hot-swap one component for another when new out-of-distribution ( OOD ) settings arise that require combining existing components in novel ways . Our work is inspired by cognitive science , neuroscience , and mathematical considerations . From the cognitive science perspective , we can consider different components of structured neural architectures to be analogous to autonomous agents in a distributed system whose ability to communicate stems from sharing the same language . If each agent speaks a different language , learning to communicate would be slow and past experience would be of little use when the need arises to communicate with a new agent . If all agents learn the same language , each benefits from this arrangement . To encourage a common language , we limit the expressivity of the vocabulary to discrete symbols that can be combined combinatorially ( Tomasello , 2009 ) . From the neuroscience perspective , we note that various areas in the brain , including the hippocampus ( Sun et al. , 2020 ; Quiroga et al. , 2005 ; Wills et al. , 2005 ) , the prefrontal cortex ( Fujii and Graybiel , 2003 ) , and sensory cortical areas ( Tsao et al. , 2006 ) are tuned to discrete variables ( concepts , actions , and objects ) , suggesting the evolutionary advantage of such encoding , and its contribution to the capacity for generalization in the brain . From a theoretical perspective , we present analysis suggesting that multi-head discretization of inter-component communication increases model sensitivity and reduces underlying dimensions ( Section 2 ) . These sources of inspiration lead us to the proposed method of Discrete-Valued Neural Communication ( DVNC ) . Architectures like graph neural networks ( GNNs ) , transformers , and slot-based or modular neural networks consist of articulated specialist components , for instance , nodes in GNNs , positions in transformers , and slots/modules for modular models . We evaluate the efficacy of DVNC in GNNs , transformers , and in a modular recurrent architecture called RIMs . For each of these structured architectures , we keep the original architecture and all of its specialist components the same . The only change is that we impose discretization in the communication between components ( Figure 1 ) . Our work is organized as follows . First , we introduce DVNC and present theoretical analysis showing that DVNC improves sensitivity and reduces metric entropy of models ( the logarithm of the covering number ) . Then we explain how DVNC can be incorporated into different model architectures . And finally we report experimental results showing improved OOD generalization with DVNC . 2 Discrete-Value Neural Communication and Theoretical Analysis . In this section , we begin with the introduction of Discrete-Value Neural Communication ( DVNC ) and proceed by conducting a theoretical analysis of DVNC affects the sensitivity and metric entropy of models . We then explain how DVNC can be used within several different architectures . Discrete-Value Neural Communication ( DVNC ) The process of converting data with continuous attributes into data with discrete attributes is called discretization ( Chmielewski and Grzymala-Busse , 1996 ) . In this study , we use discrete latent variables to quantize information communicated among different modules in a similar manner as in Vector Quantized Variational AutoEncoder ( VQ-VAE ) ( Oord et al. , 2017 ) . Similar to VQ-VAE , we introduce a discrete latent space vector e ∈ RL× ( m/G ) where L is the size of the discrete latent space ( i.e. , an L-way categorical variable ) , and m is the dimension of each latent embedding vector ej . Here , L and m are both hyperparameters . In addition , by dividing each target vector into G segments or discretization heads , we separately quantize each head and concatenate the results ( Figure 2 ) . More concretely , the discretization process for each vector h ∈ H ⊂ Rm is described as follows . First , we divide a vector h into G segments s1 , s2 , . . . , sG with h = CONCATENATE ( s1 , s2 , . . . , sG ) , where each segment si ∈ Rm/G with mG ∈ N + . Second , we discretize each segment si separately : eoi = DISCRETIZE ( si ) , where oi = argmin j∈ { 1 , ... , L } ||si − ej || . Finally , we concatenate the discretized results to obtain the final discretized vector Z as Z = CONCATENATE ( DISCRETIZE ( s1 ) , DISCRETIZE ( s2 ) , ... , DISCRETIZE ( sG ) ) . The multiple steps described above can be summarized by Z = q ( h , L , G ) , where q ( · ) is the whole discretization process with the codebook , L is the codebook size , and G is number of segments per vector . It is worth emphasizing that the codebook e is shared across all communication vectors and heads , and is trained together with other parts of the model . The overall loss for model training is : L = Ltask + 1 G ( G∑ i || sg ( si ) − eoi ||22 + β G∑ i ||si − sg ( eoi ) ||22 ) ( 1 ) where Ltask is the loss for specific task , e.g. , cross entropy loss for classification or mean square error loss for regression , sg refers to a stop-gradient operation that blocks gradients from flowing into its argument , and β is a hyperparameter which controls the reluctance to change the code . The second term ∑G i || sg ( si ) − eoi ||22 is the codebook loss , which only applies to the discrete latent vector and brings the selected eoi close to the output segment si . The third term ∑G i ||si − sg ( eoi ) ||22 is the commitment loss , which only applies to the target segment si and trains the module that outputs si to make si stay close to the chosen discrete latent vector eoi . We picked β = 0.25 as in the original VQ-VAE paper ( Oord et al. , 2017 ) . We initialized e using k-means clustering on vectors h with k = L and trained the codebook together with other parts of the model by gradient descent . When there were multiple h vectors to discretize in a model , the mean of the codebook and commitment loss across all h vectors was used . Unpacking this equation , it can be seen that we adapted the vanilla VQ-VAE loss to directly suit our discrete communication method ( Oord et al. , 2017 ) . In particular , the VQ-VAE loss was adapted to handle multi-headed discretization by summing over all the separate discretization heads . In the next subsection , we use the following additional notation . The function φ is arbitrary and thus can refer to the composition of an evaluation criterion and the rest of the network following discretization . Given any function φ : Rm → R and any family of sets S = { S1 , . . . , SK } with S1 , . . . , SK ⊆ H , we define the corresponding function φSk by φSk ( h ) = 1 { h ∈ Sk } φ ( h ) for all k ∈ [ K ] , where [ K ] = { 1 , 2 , . . . , K } . Let e ∈ E ⊂ RL×m be fixed and we denote by ( Qk ) k∈ [ LG ] all the possible values after the discretization process : i.e. , q ( h , L , G ) ∈ ∪k∈ [ LG ] { Qk } for all h ∈ H. Theoretical Analysis This subsection shows that adding the discretization process has two potential advantages : ( 1 ) it improves noise sensitivity or robustness and ( 2 ) it reduces metric entropy . These are proved in Theorems 1–2 , illustrated by examples ( Table 1 ) , and explored in analytical experiments using Gaussian-distributed vectors ( Figure 3 ) . To understand the advantage on noise sensitivity or robustness , we note that there is an additional error incurred by noise without discretization , i.e. , the second term ς̄kRH/ √ n ≥ 0 in the bound of Theorem 2 ( ς̄k and RH are defined in Theorem 2 ) . This term is defined to be the noise sensitivity ( or robustness ) term . This term due to noise disappears with discretization in the bound of Theorem 1 as the discretization process reduces the sensitivity to noise . This is because the discretization process lets the communication become invariant to noise within the same category ; e.g. , the communication is invariant to different notions of “ cats ” . To understand the advantage of discretization on dimensionality , we can see that it reduces the metric entropy of m ln ( 4 √ nm ) without discretization ( in Theorem 2 ) to that of G ln ( L ) with discretization ( in Theorem 1 ) . As a result , the size of the codebook L affects the underlying dimension in a weak ( logarithmic ) fashion , while the number of dimensions m and the number of discretization heads G scale the underlying dimension in a linear way . Thus , the discretization process successfully lowers the metric entropy for any n ≥ 1 as long as G ln ( L ) < m ln ( 4 √ m ) . This is nearly always the case as the number of discretization heads G is almost always much smaller than the number of units m. Intuitively , a discrete language has combinatorial expressiveness , making it able to model complex phenomena , but still lives in a much smaller space than the world of unbounded continuous-valued signals ( as G can be much smaller than m ) . Theorem 1 . ( with discretization ) Let Sk = { Qk } for all k ∈ [ LG ] . Then , for any δ > 0 , with probability at least 1 − δ over an iid draw of n examples ( hi ) ni=1 , the following holds for any φ : Rm → R and all k ∈ [ LG ] : if |φSk ( h ) | ≤ α for all h ∈ H , then∣∣∣∣∣Eh [ φSk ( q ( h , L , G ) ) ] − 1n n∑ i=1 φSk ( q ( hi , L , G ) ) ∣∣∣∣∣ = O ( α √ G ln ( L ) + ln ( 2/δ ) 2n ) , ( 2 ) where no constant is hidden in O. Theorem 2 . ( without discretization ) Assume that ‖h‖2 ≤ RH for all h ∈ H ⊂ Rm . Fix C ∈ argminC̄ { |C̄| : C̄ ⊆ Rm , H ⊆ ∪c∈C̄B [ c ] } where B [ c ] = { x ∈ Rm : ‖x − c‖2 ≤ RH/ ( 2 √ n ) } . Let Sk = B [ ck ] for all k ∈ [ |C| ] where ck ∈ C and ∪k { ck } = C. Then , for any δ > 0 , with probability at least 1− δ over an iid draw of n examples ( hi ) ni=1 , the following holds for any φ : Rm → R and all k ∈ [ |C| ] : if |φSk ( h ) | ≤ α for all h ∈ H and |φSk ( h ) −φSk ( h′ ) | ≤ ςk‖h−h′‖2 for all h , h′ ∈ Sk , then∣∣∣∣∣Eh [ φSk ( h ) ] − 1n n∑ i=1 φSk ( hi ) ∣∣∣∣∣ = O ( α √ m ln ( 4 √ nm ) + ln ( 2/δ ) 2n + ς̄kRH√ n ) , ( 3 ) where no constant is hidden in O and ς̄k = ςk ( 1 n ∑n i=1 1 { hi ∈ B [ ck ] } ) . The two proofs of these theorems use the same steps and are equally tight as shown in Appendix D. Equation ( 3 ) is also as tight as that of related work as discussed in Appendix C.2 . The set S is chosen to cover the original continuous spaceH in Theorem 2 ( via the -covering C ofH ) , and its discretized space in Theorem 1 . Equations ( 2 ) – ( 3 ) hold for all functions φ : Rm → R , including the maps that depend on the samples ( hi ) ni=1 via any learning processes . For example , we can set φ to be an evaluation criterion of the latent space h or the composition of an evaluation criterion and any neural network layers that are learned with the samples ( hi ) ni=1 . In Appendix A , we present additional theorems , Theorems 3–4 , where we analyze the effects of learning the map x 7→ h and the codebook e via input-target pair samples ( ( xi , yi ) ) ni=1 . Intuitively , the proof shows that we achieve the improvement in sample efficiency when G and L are small , with the dependency on G being significantly stronger ( details in Appendix ) . Moreover , the dependency of the bound on the Lipschitz constant ςk is eliminated by using discretization . Our theorems 1–2 are applicable to all of our models for recurrent neural networks , transformers , and graph neural networks ( since the function φ is arbitrary ) in the following subsections . Communication along edges of Graph Neural Network One model where relational information is naturally used is in a graph neural network for modelling object interactions and predicting the next time frame . We denote node representations by ζti , edge representations by t i , j , and actions applied to each node by ati . Without DVNC , the changes of each nodes after each time step is computed by ζt+1i = ζ t i + ∆ζ t i , where ∆ζ t i = fnode ( ζ t i , a i t , ∑ j 6=i i , j t ) and t i , j = fedge ( ζ t i , z t j ) . In this present work , we discretize the sum of all edges connected to each node with DVNC , as so : ∆ζti = fnode ( ζ t i , a i t , q ( ∑ j 6=i i , j t , L , G ) ) . Communication Between Positions in Transformers In a transformer model without DVNC , at each layer , the scaled dot product multi-head soft attention is applied to allow the model to jointly attend to information from different representation subspaces at different positions ( Vaswani et al. , 2017 ) as : Output = residual + MULTIHEADATTENTION ( B , K , V ) , where MULTIHEADATTENTION ( B , K , V ) = CONCATENATE ( head1 , head2 , ..... headn ) WO and headi = SOFTATTENTION ( BWBi , KW K i , V W V i ) . Here , W O , WB , WK , and WV are projection matrices , and B , K , and V are queries , keys , and values respectively . In this present work , we applied the DVNC process to the results of the attention in the last two layers of transformer model , as so : Output = residual + q ( MULTIHEADATTENTION ( B , K , V ) , L , G ) . Communication with Modular Recurrent Neural Networks There have been many efforts to introduce modularity into RNN . Recurrent independent mechanisms ( RIMs ) activated different modules at different time step based on inputs ( Goyal et al. , 2019 ) . In RIMs , outputs from different modules are communicated to each other via soft attention mechanism . In the original RIMs method , we have ẑt+1i = RNN ( z t i , x t ) for active modules , and ẑt+1i′ = z t i′ for inactive modules , where t is the time step , i is index of the module , and xt is the input at time step t. Then , the dot product query-key soft attention is used to communication output from all modules i ∈ { 1 , . . . , M } such that ht+1i = SOFTATTENTION ( ẑ t+1 1 , ẑ t+1 2 , .... ẑ t+1 M ) . In this present work , we applied the DVNC process to the output of the soft attention , like so : zt+1i = ẑ t+1 i + q ( h t+1 i , L , G ) . Appendix E presents the pseudocode for RIMs with discretization .

Summary: The submission discusses discrete communication between neural network components and proposes 'discrete-valued neural communication (DVNC)', i.e. discretization of continuous representations in neural networks. The key contribution is a segmented discretization of vectors (see Eq. 1), in contrast to a joint discretization in a prior work (Oord et al., 2017).